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{'hep-ph-0405286-1-0-0': 'In the context of the brane-world scenarios with compactified extra dimensions, we study the production of brane fluctuations (branons) in hadron colliders ([MATH], [MATH] and [MATH]) in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-1-0-1': 'From the absence of monojets events at HERA and Tevatron (run I), we set bounds on these parameters and we also study how such bounds could be improved at Tevatron (run II) and the future LHC.', 'hep-ph-0405286-1-0-2': 'The single photon channel is also analyzed for the two last colliders.', 'hep-ph-0405286-1-1-0': 'PACS: 11.25Mj, 11.10Lm, 11.15Ex', 'hep-ph-0405286-1-2-0': 'plain', 'hep-ph-0405286-1-3-0': '20true cm', 'hep-ph-0405286-1-4-0': '# Introduction', 'hep-ph-0405286-1-5-0': 'Since rigid objects do not exist in relativistic theories, it is clear that brane fluctuations must play a role in the so called brane world, proposed some years ago by Arkani-Hamed, Dimopoulos and Dvali (ADD scenarios [CITATION]), where the Standard Model (SM) particles are confined to live in the world brane and only gravitons are free to move along the [MATH] dimensional bulk space (see [CITATION] for recent reviews).', 'hep-ph-0405286-1-5-1': 'This fact turns out to be particularly true when the brane tension scale [MATH] being the brane tension) is much smaller than the [MATH] dimensional or fundamental gravitational scale [MATH], i.e., [MATH].', 'hep-ph-0405286-1-5-2': 'In this case the only relevant low-energy modes of the ADD scenarios are the SM particles and branons which are the quantized brane oscillations.', 'hep-ph-0405286-1-5-3': 'Branons can be understood as the (pseudo)Goldstone bosons corresponding to the spontaneous breaking of translational invariance in the bulk space produced by the presence of the brane.', 'hep-ph-0405286-1-5-4': 'It has been pointed out that branons could solve some of the problems of the brane-world scenarios such as the divergent virtual contributions from the Kaluza-Klein tower at the tree level or non-unitarity of the graviton production cross-sections [CITATION].', 'hep-ph-0405286-1-5-5': 'As Goldstone bosons, branons are in principle massless, but in the cases where the metric of the extra dimensions cannot be factorized, they can become massive [CITATION].', 'hep-ph-0405286-1-5-6': 'This is similar to the case of pions which, being the Goldstone bosons of the spontaneous breaking of chiral symmetry, acquire some mass due to the explicit breaking of the symmetry induced by the quark masses.', 'hep-ph-0405286-1-6-0': 'In previous works the different effective actions have been obtained, namely: the effective action for the SM fields on the brane, that for the branon self-interactions and finally that corresponding to the interaction between SM fields and branons [CITATION].', 'hep-ph-0405286-1-6-1': 'In general, this branon effective action can be parameterized by the number of branons [MATH], the tension scale [MATH] and the branon masses (for an explicit construction see [CITATION]).', 'hep-ph-0405286-1-6-2': 'Using the effective action it is possible to obtain the different Feynman rules, the amplitudes and finally the cross-sections for branon production from SM particles.', 'hep-ph-0405286-1-6-3': 'In [CITATION] the case of electron-positron colliders has been considered.', 'hep-ph-0405286-1-6-4': 'By using the Large Electron-Positron Collider (LEP) data it is possible to set important bounds on the tension scale and on the branon mass for a given branon number.', 'hep-ph-0405286-1-6-5': 'Other restrictions have also been set from astrophysical and cosmological considerations due to the fact that branon dark matter can present relevant abundances [CITATION].', 'hep-ph-0405286-1-7-0': 'In this work we study branon production in hadron colliders and also in electron-proton colliders such as HERA.', 'hep-ph-0405286-1-7-1': 'Most of these cross sections have been studied by Creminelli and Strumia for the massless branon case [CITATION].', 'hep-ph-0405286-1-7-2': 'We reproduce their results and extend the analysis for an arbitrary branon mass.', 'hep-ph-0405286-1-7-3': 'The paper is organized as follows: In Sec.II we shortly review the branon effective action.', 'hep-ph-0405286-1-7-4': 'In Sec.III we consider the case of proton-(anti)proton colliders like Tevatron or the future Large Hadron Collider (LHC).', 'hep-ph-0405286-1-7-5': 'In Sec.IV electron(positron)-proton colliders like HERA are studied.', 'hep-ph-0405286-1-7-6': 'In Sec.V we show the main results for the relevant examples and in Sec.VI we set the conclusions.', 'hep-ph-0405286-1-8-0': '# Effective action', 'hep-ph-0405286-1-9-0': 'The relevant effective action describing the low-energy interactions of SM particles and branons was derived in [CITATION], where the necessary vertices are detailed.', 'hep-ph-0405286-1-9-1': 'The branon effective action can be expanded according to the number of branon fields appearing in each term: [EQUATION] where the zeroth-order term is just a constant and the second-order is just the free action: [EQUATION] with [MATH] the branon fields where [MATH] and [MATH] is the squared mass matrix which, without loss of generality, can be assumed to be diagonal.', 'hep-ph-0405286-1-9-2': 'The effective action for the SM particles and their interactions with branons is given by [EQUATION] where [MATH] is the SM Lagrangian and [MATH] is the SM energy-momentum tensor defined as: [EQUATION] where [MATH] is some arbitrary metric on the world brane and [MATH] is the Minkowski metric.', 'hep-ph-0405286-1-10-0': 'In this work we are interested in the interactions between quarks and gluons or photons.', 'hep-ph-0405286-1-10-1': 'Thus, for Dirac fermions with masses [MATH] belonging to some representation of a gauge group, such as [MATH] or [MATH], with generators [MATH], the Lagrangian is [EQUATION] where the covariant derivative is defined as [MATH], [MATH] being the appropriate gauge coupling.', 'hep-ph-0405286-1-10-2': 'Thus the energy-momentum tensor is given by [EQUATION] from where it is possible to find vertices such as [MATH] and [MATH].', 'hep-ph-0405286-1-10-3': 'For gauge fields [MATH] the appropriate Lagrangian for perturbation theory is: [EQUATION] where as usual [MATH] and [MATH] is the Fadeev-Popov Lagrangian including the gauge fixing and the ghost terms.', 'hep-ph-0405286-1-10-4': 'The energy-momentum tensor is: [EQUATION] from where we can obtain the [MATH], [MATH] and [MATH] vertices.', 'hep-ph-0405286-1-11-0': 'Therefore, by using these energy-momentum tensors and the effective action above, it is possible to obtain the different Feynman rules involving branons.', 'hep-ph-0405286-1-11-1': 'One important observation is that in all the vertices obtained above, branons appear always by pairs.', 'hep-ph-0405286-1-11-2': 'In fact they interact in a way similar to gravitons since they couple to the energy momentum tensor.', 'hep-ph-0405286-1-11-3': 'This can be seen by making the formal identification of the graviton field [MATH] which appear in linearized gravity with [EQUATION] where [MATH] and [MATH] the Planck mass.', 'hep-ph-0405286-1-11-4': 'Of course the physical meaning is completely different for branons and gravitons.', 'hep-ph-0405286-1-11-5': 'In any case branons are expected to be weakly interacting and then they will scape to detection.', 'hep-ph-0405286-1-11-6': 'Hence their typical signature will be missing energy and momentum with a continuum spectrum.', 'hep-ph-0405286-1-11-7': 'In the following sections we will study the production mechanisms relevant for the different hadronic colliders.', 'hep-ph-0405286-1-12-0': '# Proton-(anti)proton colliders', 'hep-ph-0405286-1-13-0': 'For the case of proton-antiproton colliders like Tevatron, the most important processes for branon production are quark-antiquark annihilation or gluon fusion giving a gluon and a branon pair; and (anti)quark-gluon interaction giving an (anti)quark and a branon pair.', 'hep-ph-0405286-1-13-1': 'Therefore the expected experimental signal will be in both cases one monojet [MATH] and missing energy and momentum.', 'hep-ph-0405286-1-13-2': 'This is a very clear signature that in principle can be easily identified.', 'hep-ph-0405286-1-13-3': 'Another potentially interesting process is the quark-antiquark annihilation giving a photon and a branon pair.', 'hep-ph-0405286-1-13-4': 'In this case the signature is one single photon and missing energy and momentum.', 'hep-ph-0405286-1-14-0': 'The Feynman diagrams contributing to the main subprocesses [MATH], [MATH], [MATH] and [MATH] are shown in Fig. [REF], Fig. [REF] and Fig. [REF].', 'hep-ph-0405286-1-15-0': 'From these diagrams and the Feynman rules coming from the effective action of the previous section, it is possible to obtain the differential cross section: [EQUATION] where [MATH], [MATH], [MATH] and [MATH], [MATH] and [MATH] being the anti-quark and quark four-momenta respectively, [MATH] the gluon four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-1-15-1': 'We have assumed for the sake of simplicity that all the branons are degenerated with a common mass [MATH] and that all the quarks are massless.', 'hep-ph-0405286-1-15-2': 'We have also neglected the effects of the top quark.', 'hep-ph-0405286-1-15-3': 'In addition we have the well-known relation [MATH].', 'hep-ph-0405286-1-15-4': 'The contribution to the total cross section of the process [MATH] coming from this subprocess is given by [EQUATION] where [MATH] and [MATH] are the distribution functions of the anti-quark inside the antiproton and of the quark of flavor [MATH] inside the proton respectively, and [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and antiquark.', 'hep-ph-0405286-1-15-5': 'The different limits of the integrals can be written in terms of the cuts used to define the total cross-section.', 'hep-ph-0405286-1-15-6': 'For example, in order to be able to detect clearly the monojet one must impose a minimal value for its transverse energy [MATH] and a pseudorapidity range given by [MATH] and [MATH].', 'hep-ph-0405286-1-15-7': 'Then we have the limits [MATH], [MATH] and [MATH].', 'hep-ph-0405286-1-15-8': 'On the other hand [MATH] and [MATH] where [MATH] is the total center of mass energy squared of the process and [EQUATION]', 'hep-ph-0405286-1-15-9': 'In addition the dots in ([REF]) represent the contribution of the case in which the quark comes from the antiproton and the anti-quark comes from the proton.', 'hep-ph-0405286-1-16-0': 'The cross-section of the subprocess [MATH] is given by [EQUATION] where the Mandelstan variables are defined as in the previous case, with [MATH] and [MATH] being the initial gluon four-momenta, [MATH] the final gluon four-momentum and [MATH] the total branon four-momentum.', 'hep-ph-0405286-1-16-1': 'Then the contribution to the total cross section from the [MATH] reaction is [EQUATION]', 'hep-ph-0405286-1-16-2': 'Here [MATH] is the gluon distribution function of the (anti)proton, [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the initial gluons and the integration limits remain the same.', 'hep-ph-0405286-1-16-3': 'From the above equations, it is possible to compute the total cross-section [MATH] in terms of the cut in the gluon (monojet) transverse energy [MATH].', 'hep-ph-0405286-1-17-0': 'For the [MATH] process the cross-section is given by [EQUATION] with [MATH] and [MATH] being the quark and the gluon four-momenta respectively, [MATH] the final state quark four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-1-17-1': 'The Mandelstam variables are defined as in previous cases.', 'hep-ph-0405286-1-17-2': 'The cross-section for the conjugate process [MATH] is exactly the same.', 'hep-ph-0405286-1-17-3': 'Then the total cross section for the the reaction [MATH] is [EQUATION]', 'hep-ph-0405286-1-17-4': 'In this equation [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and gluon.', 'hep-ph-0405286-1-17-5': 'The different integration limits are defined as in the previous case in terms of the minimal transverse energy of the quark (monojet) [MATH] and the dots refer to the case where the initial gluon is coming from the proton and the quark is coming from the antiproton.', 'hep-ph-0405286-1-17-6': 'In addition we have the contribution from the conjugate case where we take an anti-quark from the proton and a gluon from the antiproton and conversely.', 'hep-ph-0405286-1-17-7': 'This amount just to a factor of two.', 'hep-ph-0405286-1-18-0': 'From all the above equations it is possible to compute the total cross-section [MATH] in terms of the cut in the jet transverse energy [MATH].', 'hep-ph-0405286-1-19-0': 'For the subprocess [MATH] the cross-section is given by [EQUATION]', 'hep-ph-0405286-1-19-1': 'Here the notation is similar to the [MATH] case with the obvious differences in couplings, color and charge factors.', 'hep-ph-0405286-1-19-2': 'Thus [EQUATION]', 'hep-ph-0405286-1-19-3': 'All the previous discussion about branon production in [MATH] reactions can be easily translated to the [MATH] case.', 'hep-ph-0405286-1-19-4': 'The only point is to change the antiproton distribution functions of the different partons by the corresponding proton ones.', 'hep-ph-0405286-1-20-0': '# Electron(positron)-proton colliders', 'hep-ph-0405286-1-21-0': 'For electron(positron)-proton colliders like HERA, the most interesting branon creating process is branon photoproduction, where a photon emitted by the electron(positron) interacts with a quark(antiquark) from the proton giving a quark (antiquark) and a branon pair.', 'hep-ph-0405286-1-21-1': 'Thus the experimental signature is again one monojet [MATH] plus missing energy and momentum.', 'hep-ph-0405286-1-21-2': 'The relevant Feynman diagrams are shown in Fig. [REF] and the corresponding differential cross-section for the subprocess [MATH] is [EQUATION] where [MATH] and [MATH], [MATH] being the photon, [MATH] the proton quark, [MATH] the final quark and [MATH] the total branon momenta respectively.', 'hep-ph-0405286-1-21-3': 'The total cross-section for the process [MATH] is given by [EQUATION] [MATH] and [MATH] are defined in this case as [MATH] and [MATH] with [MATH] and [MATH] being the proton and electron(positron) momenta respectively.', 'hep-ph-0405286-1-21-4': 'Thus at high energies compared with the proton mass [MATH] where [MATH].', 'hep-ph-0405286-1-21-5': 'The integral limits [MATH], [MATH], [MATH] and [MATH] are defined like in the proton-(anti)proton collider case.', 'hep-ph-0405286-1-22-0': 'The photon spectrum [MATH] can be obtained from the well-known Weizs[MATH]cker-Williams approximation [CITATION]: [EQUATION] with [MATH] and [MATH] being the electron mass.', 'hep-ph-0405286-1-23-0': 'The cross-section [MATH] can be obtained in a similar way.', 'hep-ph-0405286-1-23-1': 'Then the total contribution to monojet plus missing energy and momentum production for large enough [MATH] coming from branons can be written as the sum of [MATH] and [MATH].', 'hep-ph-0405286-1-24-0': '# Results', 'hep-ph-0405286-1-25-0': 'By using the cross-sections shown in the previous sections it is possible to compute the expected number of branon pairs produced in the different hadron colliders in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-1-25-1': 'To this end we have used the distribution functions which can be found in [CITATION].', 'hep-ph-0405286-1-25-2': 'The values of the electromagnetic and strong couplings have been taken at the electroweak boson masses, namely [MATH] and [MATH].', 'hep-ph-0405286-1-25-3': 'However our final results do not depend too much on the precise value of these couplings.', 'hep-ph-0405286-1-25-4': 'In fact our main source of error is the use of an effective action to describe the SM particles and branon interactions since, in principle, this is only guaranteed for energies well below [MATH].', 'hep-ph-0405286-1-26-0': 'As discussed in the introduction, our main goal in this work is to study the bounds that can be set on the [MATH], [MATH] and [MATH] parameters coming from hadron colliders.', 'hep-ph-0405286-1-26-1': 'We will present all our limits at the 95% confidence level.', 'hep-ph-0405286-1-26-2': 'In particular, for the electron(positron)-proton case, HERA is the most relevant experiment.', 'hep-ph-0405286-1-26-3': 'In fact, the ZEUS collaboration has studied the jet production in charged current deep inelastic [MATH] scattering.', 'hep-ph-0405286-1-26-4': 'Its results are perfectly compatible with the SM background and therefore, we can set some bounds on the branon production and hence on the [MATH], [MATH] and [MATH] parameters.', 'hep-ph-0405286-1-26-5': 'These data were taken from 1995 to 2000 at a maximum CM energy of [MATH] GeV.', 'hep-ph-0405286-1-26-6': 'The total integrated luminosity was [MATH] pb[MATH] and the cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for more details).', 'hep-ph-0405286-1-26-7': 'By using the same cuts with our cross-sections for monojet plus a branon pair production, we find the bound [MATH] GeV for massless branons.', 'hep-ph-0405286-1-26-8': 'For a branon mass larger than [MATH] GeV there is no restrictions on the [MATH] value because of kinematical reasons.', 'hep-ph-0405286-1-26-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] where we have assumed [MATH].', 'hep-ph-0405286-1-26-10': 'For other [MATH] values one just has to take into account that the bound scales like [MATH] since all the cross-sections are proportional to [MATH].', 'hep-ph-0405286-1-27-0': 'In the [MATH] case the most relevant experimental information so far is the one obtained at the Tevatron (Run I).', 'hep-ph-0405286-1-27-1': 'The [MATH] detector has studied the monojet channel and CDF the single photon one.', 'hep-ph-0405286-1-27-2': 'As far as the number of events found in both cases is compatible with the SM background, we can set new bounds on the branon theory parameters.', 'hep-ph-0405286-1-27-3': 'For light branons the most important bound comes from the [MATH] data.', 'hep-ph-0405286-1-27-4': 'These data were taken from 1994 to 1996 at a CM energy of [MATH] TeV and correspond to an integrated luminosity [MATH] pb[MATH].', 'hep-ph-0405286-1-27-5': 'The cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for the details of the analysis).', 'hep-ph-0405286-1-27-6': 'The total number of monojets observed was [MATH] and the expected number from the SM plus cosmic rays events was [MATH].', 'hep-ph-0405286-1-27-7': 'By using our cross sections for monojet plus a branon pair production with these cuts we get the bound [MATH] GeV for light branons.', 'hep-ph-0405286-1-27-8': 'The restrictions for [MATH] improve up to a branon mass of [MATH] GeV.', 'hep-ph-0405286-1-27-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] for [MATH].', 'hep-ph-0405286-1-28-0': 'In a similar way we can use the CDF data on single photon production.', 'hep-ph-0405286-1-28-1': 'In this case the total luminosity collected was [MATH] pb[MATH] and the pseudorapidity cut was [MATH].', 'hep-ph-0405286-1-28-2': 'For the transverse photon energy several cuts were considered (for example 55 GeV at the [MATH] efficiency).', 'hep-ph-0405286-1-28-3': 'The total expected background for this process was [MATH], without taking into account the QCD contribution (see [CITATION] for the details of the analysis), and the number of events found was [MATH].', 'hep-ph-0405286-1-28-4': 'Comparing this result with our computations for photon plus one branon pair production, we find the bound [MATH] GeV for massless branons and no bound for [MATH] larger than [MATH] GeV.', 'hep-ph-0405286-1-28-5': 'The bound obtained for the rest of the cases is shown also in Fig. [REF].', 'hep-ph-0405286-1-29-0': 'In addition to this analysis corresponding to the Tevatron data (Run I), it is also interesting to make some estimation about the bounds that could be set from future experiments such as Tevatron (Run II) and the LHC.', 'hep-ph-0405286-1-29-1': 'In the case of the Tevatron (Run II), which is already in progress, the main novelties are a CM energy which equals [MATH] TeV and an expected integrated luminosity [MATH] at the end of the run of about [MATH] pb[MATH].', 'hep-ph-0405286-1-29-2': 'The detectors are also improved so that the pseudorapidity cuts can be taken as [MATH] for [MATH] and [MATH] for CDF.', 'hep-ph-0405286-1-29-3': 'This would result in a factor of [MATH] on the statistical significance when compared to the Run I, with integrated luminosity [MATH], provided that the CM energy and the cuts were the same.', 'hep-ph-0405286-1-29-4': 'For massless branons, the bound on [MATH] scales as the CM energy [MATH].', 'hep-ph-0405286-1-29-5': 'Even more important is the possibility of exploring higher branon masses, since the kinematical limit is given by [MATH].', 'hep-ph-0405286-1-29-6': 'In Fig. [REF] we show the expected bounds from the Run II in the [MATH] plane, again for [MATH].', 'hep-ph-0405286-1-30-0': 'The LHC will produce [MATH] collisions at a CM energy of [MATH] TeV and the integrated luminosity will be something about [MATH] pb[MATH].', 'hep-ph-0405286-1-30-1': 'In order to estimate the bounds on the [MATH], [MATH] and [MATH] parameters that will be possible to obtain at the LHC, we have proceeded in a similar way as in the Tevatron case, with the obvious changes in the distribution functions due to the fact that now we are dealing with [MATH] instead of [MATH] collisions.', 'hep-ph-0405286-1-30-2': 'We have kept the same cuts except for the transverse energy which has been corrected in order to maintain the same proportion relative to the CM energy.', 'hep-ph-0405286-1-30-3': 'Again the best bounds for [MATH] come from monojet production, which for [MATH] turns out to be [MATH] GeV.', 'hep-ph-0405286-1-30-4': 'For low [MATH] the best bound for [MATH] is given by the single photon channel ([MATH]).', 'hep-ph-0405286-1-30-5': 'The LHC sensitivity for other values in the [MATH] plane can be found in Fig. [REF] for [MATH].', 'hep-ph-0405286-1-31-0': '# Conclusions', 'hep-ph-0405286-1-32-0': 'In this work we have studied the flexible brane-world scenario, where the brane tension scale [MATH] is much smaller than the fundamental [MATH]-dimensional gravitational scale [MATH].', 'hep-ph-0405286-1-32-1': 'In this case, the relevant low-energy degrees of freedom are the SM particles and the brane fluctuations or branons.', 'hep-ph-0405286-1-32-2': 'From the corresponding effective action, we have calculated the relevant cross-sections for different branon searches in hadronic colliders.', 'hep-ph-0405286-1-32-3': 'We have used the information coming from HERA and the first Tevatron Run in order to get different exclusion plots on the branon mass [MATH] and the tension scale [MATH] plane for a given branon number [MATH].', 'hep-ph-0405286-1-32-4': 'Monojet production turns out to be the most efficient process for light branons, whereas the single photon channel is the most important one for heavy branons.', 'hep-ph-0405286-1-33-0': 'We have also extended the analysis to future hadronic colliders.', 'hep-ph-0405286-1-33-1': 'The corresponding sensitivity regions for the second Tevatron run and the LHC have also been plotted (see Table [REF] for a summary of the analysis).', 'hep-ph-0405286-1-34-0': 'These analysis improve those already done for electron-positron colliders for heavy branons, whereas for light branons, the results are similar [CITATION].', 'hep-ph-0405286-1-34-1': 'The Tevatron (run I) limit [MATH] GeV can be compared to the analogous limit from LEP II [MATH] GeV [CITATION].', 'hep-ph-0405286-1-34-2': 'On the other hand, LHC could detect branons up to a mass of several TeV ([MATH] GeV) improving even the CLIC prospects ([MATH] GeV) [CITATION].', 'hep-ph-0405286-1-35-0': 'The study of branons in colliders can be complemented with other bounds coming from astrophysics and cosmology.', 'hep-ph-0405286-1-35-1': 'In fact, as shown in [CITATION], the branon relic abundance can have cosmological consequences.', 'hep-ph-0405286-1-35-2': 'Other issues related to branon phenomenology, such as their radiative corrections to the SM processes, or their distinctive signatures at colliders with respect to the KK gravitons will be analyzed elsewhere.'}

{'hep-ph-0405286-2-0-0': 'In the context of the brane-world scenarios with compactified extra dimensions, we study the production of brane fluctuations (branons) in hadron colliders ([MATH], [MATH] and [MATH]) in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-2-0-1': 'From the absence of monojets events at HERA and Tevatron (run I), we set bounds on these parameters and we also study how such bounds could be improved at Tevatron (run II) and the future LHC.', 'hep-ph-0405286-2-0-2': 'The single photon channel is also analyzed for the two last colliders.', 'hep-ph-0405286-2-1-0': 'PACS: 11.25Mj, 11.10Lm, 11.15Ex', 'hep-ph-0405286-2-2-0': 'plain', 'hep-ph-0405286-2-3-0': '20true cm', 'hep-ph-0405286-2-4-0': '# Introduction', 'hep-ph-0405286-2-5-0': 'Since rigid objects do not exist in relativistic theories, it is clear that brane fluctuations must play a role in the so called brane world, proposed some years ago by Arkani-Hamed, Dimopoulos and Dvali (ADD scenarios [CITATION]), where the Standard Model (SM) particles are confined to live in the world brane and only gravitons are free to move along the [MATH] dimensional bulk space (see [CITATION] for recent reviews).', 'hep-ph-0405286-2-5-1': 'This fact turns out to be particularly true when the brane tension scale [MATH] being the brane tension) is much smaller than the [MATH] dimensional or fundamental gravitational scale [MATH], i.e., [MATH].', 'hep-ph-0405286-2-5-2': 'In this case the only relevant low-energy modes of the ADD scenarios are the SM particles and branons which are the quantized brane oscillations.', 'hep-ph-0405286-2-5-3': 'Branons can be understood as the (pseudo)Goldstone bosons corresponding to the spontaneous breaking of translational invariance in the bulk space produced by the presence of the brane.', 'hep-ph-0405286-2-5-4': 'It has been pointed out that branons could solve some of the problems of the brane-world scenarios such as the divergent virtual contributions from the Kaluza-Klein tower at the tree level or non-unitarity of the graviton production cross-sections [CITATION].', 'hep-ph-0405286-2-5-5': 'As Goldstone bosons, branons are in principle massless, but in the cases where the metric of the extra dimensions cannot be factorized, they can become massive [CITATION].', 'hep-ph-0405286-2-5-6': 'This is similar to the case of pions which, being the Goldstone bosons of the spontaneous breaking of chiral symmetry, acquire some mass due to the explicit breaking of the symmetry induced by the quark masses.', 'hep-ph-0405286-2-6-0': 'In previous works the different effective actions have been obtained, namely: the effective action for the SM fields on the brane, that for the branon self-interactions and finally that corresponding to the interaction between SM fields and branons [CITATION].', 'hep-ph-0405286-2-6-1': 'In general, this branon effective action can be parameterized by the number of branons [MATH], the tension scale [MATH] and the branon masses (for an explicit construction see [CITATION]).', 'hep-ph-0405286-2-6-2': 'Using the effective action it is possible to obtain the different Feynman rules, the amplitudes and finally the cross-sections for branon production from SM particles.', 'hep-ph-0405286-2-6-3': 'In [CITATION] the case of electron-positron colliders has been considered.', 'hep-ph-0405286-2-6-4': 'By using the Large Electron-Positron Collider (LEP) data it is possible to set important bounds on the tension scale and on the branon mass for a given branon number.', 'hep-ph-0405286-2-6-5': 'Other restrictions have also been set from astrophysical and cosmological considerations due to the fact that branon dark matter can present relevant abundances [CITATION].', 'hep-ph-0405286-2-7-0': 'In this work we study branon production in hadron colliders and also in electron-proton colliders such as HERA.', 'hep-ph-0405286-2-7-1': 'Most of these cross sections have been studied by Creminelli and Strumia for the massless branon case [CITATION].', 'hep-ph-0405286-2-7-2': 'We reproduce their results and extend the analysis for an arbitrary branon mass.', 'hep-ph-0405286-2-7-3': 'The paper is organized as follows: In Sec.II we shortly review the branon effective action.', 'hep-ph-0405286-2-7-4': 'In Sec.III we consider the case of proton-(anti)proton colliders like Tevatron or the future Large Hadron Collider (LHC).', 'hep-ph-0405286-2-7-5': 'In Sec.IV electron(positron)-proton colliders like HERA are studied.', 'hep-ph-0405286-2-7-6': 'In Sec.V we show the main results for the relevant examples and in Sec.VI we set the conclusions.', 'hep-ph-0405286-2-8-0': '# Effective action', 'hep-ph-0405286-2-9-0': 'The relevant effective action describing the low-energy interactions of SM particles and branons was derived in [CITATION], where the necessary vertices are detailed.', 'hep-ph-0405286-2-9-1': 'The branon effective action can be expanded according to the number of branon fields appearing in each term: [EQUATION] where the zeroth-order term is just a constant and the second-order is just the free action: [EQUATION] with [MATH] the branon fields where [MATH] and [MATH] is the squared mass matrix which, without loss of generality, can be assumed to be diagonal.', 'hep-ph-0405286-2-9-2': 'The effective action for the SM particles and their interactions with branons is given by [EQUATION] where [MATH] is the SM Lagrangian and [MATH] is the SM energy-momentum tensor defined as: [EQUATION] where [MATH] is some arbitrary metric on the world brane and [MATH] is the Minkowski metric.', 'hep-ph-0405286-2-10-0': 'In this work we are interested in the interactions between quarks and gluons or photons.', 'hep-ph-0405286-2-10-1': 'Thus, for Dirac fermions with masses [MATH] belonging to some representation of a gauge group, such as [MATH] or [MATH], with generators [MATH], the Lagrangian is [EQUATION] where the covariant derivative is defined as [MATH], [MATH] being the appropriate gauge coupling.', 'hep-ph-0405286-2-10-2': 'Thus the energy-momentum tensor is given by [EQUATION] from where it is possible to find vertices such as [MATH] and [MATH].', 'hep-ph-0405286-2-10-3': 'For gauge fields [MATH] the appropriate Lagrangian for perturbation theory is: [EQUATION] where as usual [MATH] and [MATH] is the Fadeev-Popov Lagrangian including the gauge fixing and the ghost terms.', 'hep-ph-0405286-2-10-4': 'The energy-momentum tensor is: [EQUATION] from where we can obtain the [MATH], [MATH] and [MATH] vertices.', 'hep-ph-0405286-2-11-0': 'Therefore, by using these energy-momentum tensors and the effective action above, it is possible to obtain the different Feynman rules involving branons.', 'hep-ph-0405286-2-11-1': 'One important observation is that in all the vertices obtained above, branons appear always by pairs.', 'hep-ph-0405286-2-11-2': 'In fact they interact in a way similar to gravitons since they couple to the energy momentum tensor.', 'hep-ph-0405286-2-11-3': 'This can be seen by making the formal identification of the graviton field [MATH] which appear in linearized gravity with [EQUATION] where [MATH] and [MATH] the Planck mass.', 'hep-ph-0405286-2-11-4': 'Of course the physical meaning is completely different for branons and gravitons.', 'hep-ph-0405286-2-11-5': 'In any case branons are expected to be weakly interacting and then they will scape to detection.', 'hep-ph-0405286-2-11-6': 'Hence their typical signature will be missing energy and momentum.', 'hep-ph-0405286-2-11-7': 'Since branons are produced by pairs, the energy spectrum of any other particle present in the final state will be continuous.', 'hep-ph-0405286-2-11-8': 'In the following sections we will study the production mechanisms relevant for the different hadronic colliders.', 'hep-ph-0405286-2-12-0': '# Proton-(anti)proton colliders', 'hep-ph-0405286-2-13-0': 'For the case of proton-antiproton colliders like Tevatron, the most important processes for branon production are quark-antiquark annihilation or gluon fusion giving a gluon and a branon pair; and (anti)quark-gluon interaction giving an (anti)quark and a branon pair.', 'hep-ph-0405286-2-13-1': 'Therefore the expected experimental signal will be in both cases one monojet [MATH] and missing energy and momentum.', 'hep-ph-0405286-2-13-2': 'This is a very clear signature that in principle can be easily identified.', 'hep-ph-0405286-2-13-3': 'Another potentially interesting process is the quark-antiquark annihilation giving a photon and a branon pair.', 'hep-ph-0405286-2-13-4': 'In this case the signature is one single photon and missing energy and momentum.', 'hep-ph-0405286-2-14-0': 'The Feynman diagrams contributing to the main subprocesses [MATH], [MATH], [MATH] and [MATH] are shown in Fig. [REF], Fig. [REF] and Fig. [REF].', 'hep-ph-0405286-2-15-0': 'From these diagrams and the Feynman rules coming from the effective action of the previous section, it is possible to obtain the differential cross section: [EQUATION] where [MATH], [MATH], [MATH] and [MATH], [MATH] and [MATH] being the anti-quark and quark four-momenta respectively, [MATH] the gluon four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-2-15-1': 'We have assumed for the sake of simplicity that all the branons are degenerated with a common mass [MATH] and that all the quarks are massless.', 'hep-ph-0405286-2-15-2': 'We have also neglected the effects of the top quark.', 'hep-ph-0405286-2-15-3': 'In addition we have the well-known relation [MATH].', 'hep-ph-0405286-2-15-4': 'The contribution to the total cross section of the process [MATH] coming from this subprocess is given by [EQUATION] where [MATH] and [MATH] are the distribution functions of the anti-quark inside the antiproton and of the quark of flavor [MATH] inside the proton respectively, and [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and antiquark.', 'hep-ph-0405286-2-15-5': 'The different limits of the integrals can be written in terms of the cuts used to define the total cross-section.', 'hep-ph-0405286-2-15-6': 'For example, in order to be able to detect clearly the monojet one must impose a minimal value for its transverse energy [MATH] and a pseudorapidity range given by [MATH] and [MATH].', 'hep-ph-0405286-2-15-7': 'Then we have the limits [MATH], [MATH] and [MATH].', 'hep-ph-0405286-2-15-8': 'On the other hand [MATH] and [MATH] where [MATH] is the total center of mass energy squared of the process and [EQUATION]', 'hep-ph-0405286-2-15-9': 'In addition the dots in ([REF]) represent the contribution of the case in which the quark comes from the antiproton and the anti-quark comes from the proton.', 'hep-ph-0405286-2-16-0': 'The cross-section of the subprocess [MATH] is given by [EQUATION] where the Mandelstan variables are defined as in the previous case, with [MATH] and [MATH] being the initial gluon four-momenta, [MATH] the final gluon four-momentum and [MATH] the total branon four-momentum.', 'hep-ph-0405286-2-16-1': 'Then the contribution to the total cross section from the [MATH] reaction is [EQUATION]', 'hep-ph-0405286-2-16-2': 'Here [MATH] is the gluon distribution function of the (anti)proton, [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the initial gluons and the integration limits remain the same.', 'hep-ph-0405286-2-16-3': 'From the above equations, it is possible to compute the total cross-section [MATH] in terms of the cut in the gluon (monojet) transverse energy [MATH].', 'hep-ph-0405286-2-17-0': 'For the [MATH] process the cross-section is given by [EQUATION] with [MATH] and [MATH] being the quark and the gluon four-momenta respectively, [MATH] the final state quark four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-2-17-1': 'The Mandelstam variables are defined as in previous cases.', 'hep-ph-0405286-2-17-2': 'The cross-section for the conjugate process [MATH] is exactly the same.', 'hep-ph-0405286-2-17-3': 'Then the total cross section for the the reaction [MATH] is [EQUATION]', 'hep-ph-0405286-2-17-4': 'In this equation [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and gluon.', 'hep-ph-0405286-2-17-5': 'The different integration limits are defined as in the previous case in terms of the minimal transverse energy of the quark (monojet) [MATH] and the dots refer to the case where the initial gluon is coming from the proton and the quark is coming from the antiproton.', 'hep-ph-0405286-2-17-6': 'In addition we have the contribution from the conjugate case where we take an anti-quark from the proton and a gluon from the antiproton and conversely.', 'hep-ph-0405286-2-17-7': 'This amount just to a factor of two.', 'hep-ph-0405286-2-18-0': 'From all the above equations it is possible to compute the total cross-section [MATH] in terms of the cut in the jet transverse energy [MATH].', 'hep-ph-0405286-2-19-0': 'For the subprocess [MATH] the cross-section is given by [EQUATION]', 'hep-ph-0405286-2-19-1': 'Here the notation is similar to the [MATH] case with the obvious differences in couplings, color and charge factors.', 'hep-ph-0405286-2-19-2': 'Thus [EQUATION]', 'hep-ph-0405286-2-19-3': 'All the previous discussion about branon production in [MATH] reactions can be easily translated to the [MATH] case.', 'hep-ph-0405286-2-19-4': 'The only point is to change the antiproton distribution functions of the different partons by the corresponding proton ones.', 'hep-ph-0405286-2-20-0': '# Electron(positron)-proton colliders', 'hep-ph-0405286-2-21-0': 'For electron(positron)-proton colliders like HERA, the most interesting branon creating process is branon photoproduction, where a photon emitted by the electron(positron) interacts with a quark(antiquark) from the proton giving a quark (antiquark) and a branon pair.', 'hep-ph-0405286-2-21-1': 'Thus the experimental signature is again one monojet [MATH] plus missing energy and momentum.', 'hep-ph-0405286-2-21-2': 'The relevant Feynman diagrams are shown in Fig. [REF] and the corresponding differential cross-section for the subprocess [MATH] is [EQUATION] where [MATH] and [MATH], [MATH] being the photon, [MATH] the proton quark, [MATH] the final quark and [MATH] the total branon momenta respectively.', 'hep-ph-0405286-2-21-3': 'The total cross-section for the process [MATH] is given by [EQUATION] [MATH] and [MATH] are defined in this case as [MATH] and [MATH] with [MATH] and [MATH] being the proton and electron(positron) momenta respectively.', 'hep-ph-0405286-2-21-4': 'Thus at high energies compared with the proton mass [MATH] where [MATH].', 'hep-ph-0405286-2-21-5': 'The integral limits [MATH], [MATH], [MATH] and [MATH] are defined like in the proton-(anti)proton collider case.', 'hep-ph-0405286-2-22-0': 'The photon spectrum [MATH] can be obtained from the well-known Weizs[MATH]cker-Williams approximation [CITATION]: [EQUATION] with [MATH] and [MATH] being the electron mass.', 'hep-ph-0405286-2-23-0': 'The cross-section [MATH] can be obtained in a similar way.', 'hep-ph-0405286-2-23-1': 'Then the total contribution to monojet plus missing energy and momentum production for large enough [MATH] coming from branons can be written as the sum of [MATH] and [MATH].', 'hep-ph-0405286-2-24-0': '# Results', 'hep-ph-0405286-2-25-0': 'By using the cross-sections shown in the previous sections it is possible to compute the expected number of branon pairs produced in the different hadron colliders in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-2-25-1': 'To this end we have used the distribution functions which can be found in [CITATION].', 'hep-ph-0405286-2-25-2': 'The values of the electromagnetic and strong couplings have been taken at the electroweak boson masses, namely [MATH] and [MATH].', 'hep-ph-0405286-2-25-3': 'However our final results do not depend too much on the precise value of these couplings.', 'hep-ph-0405286-2-25-4': 'In fact our main source of error is the use of an effective action to describe the SM particles and branon interactions since, in principle, this is only guaranteed for energies well below [MATH].', 'hep-ph-0405286-2-26-0': 'As discussed in the introduction, our main goal in this work is to study the bounds that can be set on the [MATH], [MATH] and [MATH] parameters coming from hadron colliders.', 'hep-ph-0405286-2-26-1': 'We will present all our limits at the 95% confidence level.', 'hep-ph-0405286-2-26-2': 'In particular, for the electron(positron)-proton case, HERA is the most relevant experiment.', 'hep-ph-0405286-2-26-3': 'In fact, the ZEUS collaboration has studied the jet production in charged current deep inelastic [MATH] scattering.', 'hep-ph-0405286-2-26-4': 'Its results are perfectly compatible with the SM background and therefore, we can set some bounds on the branon production and hence on the [MATH], [MATH] and [MATH] parameters.', 'hep-ph-0405286-2-26-5': 'These data were taken from 1995 to 2000 at a maximum CM energy of [MATH] GeV.', 'hep-ph-0405286-2-26-6': 'The total integrated luminosity was [MATH] pb[MATH] and the cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for more details).', 'hep-ph-0405286-2-26-7': 'By using the same cuts with our cross-sections for monojet plus a branon pair production, we find the bound [MATH] GeV for massless branons.', 'hep-ph-0405286-2-26-8': 'For a branon mass larger than [MATH] GeV there is no restrictions on the [MATH] value because of kinematical reasons.', 'hep-ph-0405286-2-26-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] where we have assumed [MATH].', 'hep-ph-0405286-2-26-10': 'For other [MATH] values one just has to take into account that the bound scales like [MATH] since all the cross-sections are proportional to [MATH].', 'hep-ph-0405286-2-27-0': 'In the [MATH] case the most relevant experimental information so far is the one obtained at the Tevatron (Run I).', 'hep-ph-0405286-2-27-1': 'The [MATH] detector has studied the monojet channel and CDF the single photon one.', 'hep-ph-0405286-2-27-2': 'As far as the number of events found in both cases is compatible with the SM background, we can set new bounds on the branon theory parameters.', 'hep-ph-0405286-2-27-3': 'For light branons the most important bound comes from the [MATH] data.', 'hep-ph-0405286-2-27-4': 'These data were taken from 1994 to 1996 at a CM energy of [MATH] TeV and correspond to an integrated luminosity [MATH] pb[MATH].', 'hep-ph-0405286-2-27-5': 'The cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for the details of the analysis).', 'hep-ph-0405286-2-27-6': 'The total number of monojets observed was [MATH] and the expected number from the SM plus cosmic rays events was [MATH].', 'hep-ph-0405286-2-27-7': 'By using our cross sections for monojet plus a branon pair production with these cuts we get the bound [MATH] GeV for light branons.', 'hep-ph-0405286-2-27-8': 'The restrictions for [MATH] improve up to a branon mass of [MATH] GeV.', 'hep-ph-0405286-2-27-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] for [MATH].', 'hep-ph-0405286-2-28-0': 'In a similar way we can use the CDF data on single photon production.', 'hep-ph-0405286-2-28-1': 'In this case the total luminosity collected was [MATH] pb[MATH] and the pseudorapidity cut was [MATH].', 'hep-ph-0405286-2-28-2': 'For the transverse photon energy several cuts were considered (for example 55 GeV at the [MATH] efficiency).', 'hep-ph-0405286-2-28-3': 'The total expected background for this process was [MATH], without taking into account the QCD contribution (see [CITATION] for the details of the analysis), and the number of events found was [MATH].', 'hep-ph-0405286-2-28-4': 'Comparing this result with our computations for photon plus one branon pair production, we find the bound [MATH] GeV for massless branons and no bound for [MATH] larger than [MATH] GeV.', 'hep-ph-0405286-2-28-5': 'The bound obtained for the rest of the cases is shown also in Fig. [REF].', 'hep-ph-0405286-2-29-0': 'In addition to this analysis corresponding to the Tevatron data (Run I), it is also interesting to make some estimation about the bounds that could be set from future experiments such as Tevatron (Run II) and the LHC.', 'hep-ph-0405286-2-29-1': 'In the case of the Tevatron (Run II), which is already in progress, the main novelties are a CM energy which equals [MATH] TeV and an expected integrated luminosity [MATH] at the end of the run of about [MATH] pb[MATH].', 'hep-ph-0405286-2-29-2': 'The detectors are also improved so that the pseudorapidity cuts can be taken as [MATH] for [MATH] and [MATH] for CDF.', 'hep-ph-0405286-2-29-3': 'This would result in a factor of [MATH] on the statistical significance when compared to the Run I, with integrated luminosity [MATH], provided that the CM energy and the cuts were the same.', 'hep-ph-0405286-2-29-4': 'For massless branons, the bound on [MATH] scales as the CM energy [MATH].', 'hep-ph-0405286-2-29-5': 'Even more important is the possibility of exploring higher branon masses, since the kinematical limit is given by [MATH].', 'hep-ph-0405286-2-29-6': 'In Fig. [REF] we show the expected bounds from the Run II in the [MATH] plane, again for [MATH].', 'hep-ph-0405286-2-30-0': 'The LHC will produce [MATH] collisions at a CM energy of [MATH] TeV and the integrated luminosity will be something about [MATH] pb[MATH].', 'hep-ph-0405286-2-30-1': 'In order to estimate the bounds on the [MATH], [MATH] and [MATH] parameters that will be possible to obtain at the LHC, we have proceeded in a similar way as in the Tevatron case, with the obvious changes in the distribution functions due to the fact that now we are dealing with [MATH] instead of [MATH] collisions.', 'hep-ph-0405286-2-30-2': 'We have kept the same cuts except for the transverse energy which has been corrected in order to maintain the same proportion relative to the CM energy.', 'hep-ph-0405286-2-30-3': 'Again the best bounds for [MATH] come from monojet production, which for [MATH] turns out to be [MATH] GeV.', 'hep-ph-0405286-2-30-4': 'For low [MATH] the best bound for [MATH] is given by the single photon channel ([MATH]).', 'hep-ph-0405286-2-30-5': 'The LHC sensitivity for other values in the [MATH] plane can be found in Fig. [REF] for [MATH].', 'hep-ph-0405286-2-31-0': '# Conclusions', 'hep-ph-0405286-2-32-0': 'In this work we have studied the flexible brane-world scenario, where the brane tension scale [MATH] is much smaller than the fundamental [MATH]-dimensional gravitational scale [MATH].', 'hep-ph-0405286-2-32-1': 'In this case, the relevant low-energy degrees of freedom are the SM particles and the brane fluctuations or branons.', 'hep-ph-0405286-2-32-2': 'From the corresponding effective action, we have calculated the relevant cross-sections for different branon searches in hadronic colliders.', 'hep-ph-0405286-2-32-3': 'We have used the information coming from HERA and the first Tevatron Run in order to get different exclusion plots on the branon mass [MATH] and the tension scale [MATH] plane for a given branon number [MATH].', 'hep-ph-0405286-2-32-4': 'Monojet production turns out to be the most efficient process for light branons, whereas the single photon channel is the most important one for heavy branons.', 'hep-ph-0405286-2-33-0': 'We have also extended the analysis to future hadronic colliders.', 'hep-ph-0405286-2-33-1': 'The corresponding sensitivity regions for the second Tevatron run and the LHC have also been plotted (see Table [REF] for a summary of the analysis).', 'hep-ph-0405286-2-34-0': 'These analysis improve those already done for electron-positron colliders for heavy branons, whereas for light branons, the results are similar [CITATION].', 'hep-ph-0405286-2-34-1': 'The Tevatron (run I) limit [MATH] GeV can be compared to the analogous limit from LEP II [MATH] GeV [CITATION].', 'hep-ph-0405286-2-34-2': 'According to the previous estimations, the Tevatron run II could also improve the bound [MATH] GeV obtained by LEP-II.', 'hep-ph-0405286-2-34-3': 'On the other hand, LHC could detect branons up to a mass of several TeV ([MATH] GeV) improving even the CLIC prospects ([MATH] GeV) [CITATION].', 'hep-ph-0405286-2-35-0': 'The study of branons in colliders can be complemented with other bounds coming from astrophysics and cosmology (see Fig. 10).', 'hep-ph-0405286-2-35-1': 'In fact, as shown in [CITATION], the branon relic abundance can have cosmological consequences.', 'hep-ph-0405286-2-35-2': 'Other issues related to branon phenomenology, such as their radiative corrections to the SM processes, or their distinctive signatures at colliders with respect to the KK gravitons will be analyzed elsewhere.', 'hep-ph-0405286-2-36-0': 'Note added: After this paper was completed, we were informed by M. Spiropulu that CDF collaboration had performed a monojet study [CITATION] which could improve the bounds in a more detailed analysis.'}

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'hep-ph-0405286-2-15-7'], ['hep-ph-0405286-1-15-8', 'hep-ph-0405286-2-15-8'], ['hep-ph-0405286-1-15-9', 'hep-ph-0405286-2-15-9']]

[['hep-ph-0405286-1-35-0', 'hep-ph-0405286-2-35-0']]

[]

[['hep-ph-0405286-1-11-6', 'hep-ph-0405286-2-11-6']]

[]

['hep-ph-0405286-1-1-0', 'hep-ph-0405286-1-2-0', 'hep-ph-0405286-1-3-0', 'hep-ph-0405286-1-14-0', 'hep-ph-0405286-1-19-2', 'hep-ph-0405286-2-1-0', 'hep-ph-0405286-2-2-0', 'hep-ph-0405286-2-3-0', 'hep-ph-0405286-2-14-0', 'hep-ph-0405286-2-19-2']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0405286

cond-mat-0305190

{'cond-mat-0305190-1-0-0': 'We report on the effect of the back-action of a Single Cooper Pair Transistor electrometer (SCPT1) in measurements of charge [MATH] on the island of a second SCPT2.', 'cond-mat-0305190-1-0-1': 'The measurement of [MATH] is [MATH]-periodic in its gate charge [MATH] when the electrometer is biased at [MATH] or [MATH].', 'cond-mat-0305190-1-0-2': 'We show that this is due to quasiparticle poisoning of SCPT2 at a rate proportional to the number of quasiparticle tunneling events in SCPT1.', 'cond-mat-0305190-1-0-3': 'We are able to eliminate this back-action and recover a 2e-periodic [MATH] dependence using a new measurement method based on switching current modulation of SCPT1.', 'cond-mat-0305190-1-1-0': 'The superconducting box, e.g. a small island of Al film very weakly coupled to the outside circuitry by Josephson junctions, has shown considerable promise as a qubit for quantum information processing where the two states can be represented by superpositions of 0 or 1 excess Cooper pairs in the box [CITATION].', 'cond-mat-0305190-1-1-1': 'Measurement of the quantum state of this so-called charge qubit without inducing unwanted decoherence is a significant problem as is quasiparticle poisoning, i.e. the introduction of unpaired electrons (quasiparticles) into the box.', 'cond-mat-0305190-1-1-2': 'At temperatures of 10 mK or so, where experiments are commonly done, the number of quasiparticle should, in principle, be negligible.', 'cond-mat-0305190-1-1-3': 'However, such quasiparticle poisoning, due perhaps to the measurement process itself, is commonly observed.', 'cond-mat-0305190-1-1-4': 'A manifestation of this is seen in the so called Coulomb staircase.', 'cond-mat-0305190-1-1-5': 'When a charge, [MATH], is capacitively induced on the junctions of the box, one expects Cooper pairs to tunnel resonantly into or out of the box at [MATH], where [MATH] is an odd integer, to maintain the lowest energy charge state of the box.', 'cond-mat-0305190-1-1-6': 'This results in the Coulomb staircase of the charge in the box [MATH] with period 2[MATH] in [MATH].', 'cond-mat-0305190-1-1-7': 'On the other hand, if there are quasiparticles in the system, then maintaining the lowest charging energy state also leads to quasiparticle tunneling.', 'cond-mat-0305190-1-1-8': 'This gives rise to splitting of the steps in the Coulomb staircase, which shifts toward e-periodicity as the number of quasiparticles increases [CITATION].', 'cond-mat-0305190-1-1-9': 'As a result, the lowest energy state of the box at [MATH], no longer corresponds to a resonant state of the Cooper pair tunneling.', 'cond-mat-0305190-1-1-10': 'For the box qubit this means that relaxation does not bring the system back to its computational ground state at its operating point.', 'cond-mat-0305190-1-1-11': 'Since the ability to prepare the initial state of the qubit is an absolutely necessary condition for quantum computing, solving the problem of quasiparticle poisoning in charge qubits is essential.', 'cond-mat-0305190-1-2-0': '(a) Schematic of the measurement showing SCPT electrometer (E) coupled to the second SCPT (B).', 'cond-mat-0305190-1-2-1': 'Tunnel junctions are represented by double box symbols.', 'cond-mat-0305190-1-2-2': '(b) Average charge of B as a function of its gate voltage when electrometer operates in VM mode.', 'cond-mat-0305190-1-2-3': '(c) Switching current of B as a function of its gate voltage.', 'cond-mat-0305190-1-2-4': 'There is an undetermined shift in gate voltage between measurements in (b) and in (c) because of drift of background charge.', 'cond-mat-0305190-1-3-0': 'Quasiparticle poisoning has been a serious road block for groups working to build a charge qubit [CITATION].', 'cond-mat-0305190-1-3-1': 'The purpose of this paper is to investigate the effects of the measurement process on quasiparticle poisoning and to develop approaches to minimize these effects.', 'cond-mat-0305190-1-3-2': 'The Single Cooper Pair Transistor (SCPT) electrometer can be operated in several different modes when measuring the charge of the box [CITATION].', 'cond-mat-0305190-1-3-3': 'Commonly the charge measurements of the box are done by operating the SCPT electrometer in the voltage modulation mode (VM).', 'cond-mat-0305190-1-3-4': 'In this mode the SCPT is biased at a sufficiently high voltage that quasiparticles are generated, so it effectively functions as a SET where the source drain voltage is modulated by [MATH] with a period of [MATH].', 'cond-mat-0305190-1-3-5': 'However it is known that the switching current of SCPTs, i.e. the current at which it switches hystereticaly from the low voltage or phase-diffusion branch to [MATH] is also charge sensitive [CITATION] and can be used for charge measurement.', 'cond-mat-0305190-1-3-6': 'We refer to this as the switching current mode (SW).', 'cond-mat-0305190-1-3-7': 'The SW mode of operation has been analyzed [CITATION], but until now no measurements of the charge on a superconducting island or box using a SW mode electrometer have been reported.', 'cond-mat-0305190-1-3-8': 'This paper presents the results of the measurements of the island charge of a SCPT used in a box configuration (B) by a SCPT electrometer (E) operated in either the VM or SW mode.', 'cond-mat-0305190-1-3-9': 'The results demonstrate that measurement-induced poisoning, which leads to an e-periodic Coulomb staircase when using the electrometer in the voltage modulation mode, can be eliminated in the switching current mode.', 'cond-mat-0305190-1-4-0': 'The sample, shown in Fig. 1(a), consists of symmetric arraignment of the two nominally identical SCPTs, E and B, which are capacitively coupled.', 'cond-mat-0305190-1-4-1': 'Due to the symmetry of the circuit the roles of E and B are interchangeable.', 'cond-mat-0305190-1-4-2': 'The parameters of the circuit are as follows: E has a normal state resistance [MATH] k[MATH], charging energy [MATH]eV and Josephson energy [MATH]eV.', 'cond-mat-0305190-1-4-3': 'For B these parameters are [MATH] k[MATH], [MATH]eV and [MATH]eV.', 'cond-mat-0305190-1-4-4': '[MATH] is the average of the Josephson energies of two junctions as determined from the values of [MATH] and superconducting gap [MATH]eV by using the Ambakoekar-Baratoff formula [CITATION].', 'cond-mat-0305190-1-4-5': '[MATH] is determined from the amplitude of maximum voltage modulation of the devices.', 'cond-mat-0305190-1-4-6': 'The coupling capacitance between E and B is determined to be 80 aF from their measured coupling of 4.8%.', 'cond-mat-0305190-1-4-7': 'All devices are made using standard two angle shadow evaporation without having normal metal quasiparticle traps close to junctions.', 'cond-mat-0305190-1-4-8': 'The sample is placed in microwave tight copper can located on a temperature regulated stage of a dilution refrigerator having a base temperature of 6 mK.', 'cond-mat-0305190-1-4-9': 'All the measurement leads are filtered by low temperature microwave filters [CITATION] that are thermally anchored to the mixing chamber.', 'cond-mat-0305190-1-5-0': 'Figure 1(b) shows the charge on the island of B, [MATH], measured with E in the VM mode.', 'cond-mat-0305190-1-5-1': 'During this measurement the source and drain leads of B are at a common potential with respect to its gate.', 'cond-mat-0305190-1-5-2': 'Figure 1(c) presents the switching current modulation of B measured with the bias current through E, [MATH], set equal to zero.', 'cond-mat-0305190-1-5-3': 'As can be seen, the switching current modulation of B is 2e-periodic as expected at low temperature, but the charge of B, measured by the electrometer, is e-periodic.', 'cond-mat-0305190-1-5-4': 'Similar dependences of [MATH] on [MATH] were measured with E biased in either of its voltage sensitive regions, i.e. near the gap where [MATH] or near the Josephson-quasiparticle peak where [MATH].', 'cond-mat-0305190-1-5-5': 'To determine if the e-periodicity of [MATH] is due to the back-action of E on B, the quasiparticle poisoning rate of B, [MATH], was measured for a range of bias conditions of E.', 'cond-mat-0305190-1-5-6': 'In addition to this, we studied how the biases of two other SCPTs located on the same chip but coupled more weakly to B effected.', 'cond-mat-0305190-1-6-0': 'Switching current histograms of B measured for different bias conditions of E. Corresponding time delay from the beginning of the current ramp is shown in the upper horizontal axes.', 'cond-mat-0305190-1-6-1': 'The histograms are shifted on vertical axis for clarity.', 'cond-mat-0305190-1-6-2': '"odd" ("even") peaks in the histograms correspond to switching from the odd (even) parity states of B. Inset: IV-characteristic of E. Arrows indicate bias conditions at which switching current histograms of the main Figure were measured.', 'cond-mat-0305190-1-6-3': 'Diamonds mark positions where [MATH] has been measured for data in Fig. 3.', 'cond-mat-0305190-1-7-0': 'The details of the technique for determining [MATH] from switching current distributions have been reported previously [CITATION].', 'cond-mat-0305190-1-7-1': 'Briefly, as [MATH] is linearly ramped in time, the switching current histogram of B with its gate biased near [MATH] exhibits two peaks if the number of quasiparticles on the island changes during the measurement of histogram.', 'cond-mat-0305190-1-7-2': 'One peak, [MATH], which is close to the maximum of the [MATH] characteristic occurs when the island has an even number of electrons (even state) and the other -much lower current - peak, [MATH], is near the predicted switching current minimum at [MATH] when one quasiparticle occupies the island (odd state) [Fig. 2].', 'cond-mat-0305190-1-7-3': 'If [MATH] and B has not switched, it must be in the even state.', 'cond-mat-0305190-1-7-4': 'The entry of a quasiparticle onto the island effectively changes [MATH] to [MATH] which for [MATH] will cause B to switch rapidly to the running state.', 'cond-mat-0305190-1-7-5': 'This can be used to obtain the poisoning rate [MATH].', 'cond-mat-0305190-1-7-6': 'Previous studies have shown that [MATH] is independent of [MATH] in the region between the peaks giving an exponential decay of the probability of the even state.', 'cond-mat-0305190-1-7-7': 'Several of these histograms with increasing [MATH] as [MATH] and [MATH] increase are shown in Fig. 2.', 'cond-mat-0305190-1-7-8': 'The relatively low bandwidth of our filters limits these measurement to [MATH] ms[MATH].', 'cond-mat-0305190-1-8-0': 'First we determine when all the electrodes of electrometer are disconnected from the measurement circuitry and grounded.', 'cond-mat-0305190-1-8-1': 'In this case we still observe a small residual rate [MATH] ms[MATH].', 'cond-mat-0305190-1-8-2': 'This non-zero rate can be caused, e.g. , by the presence of impurity levels in the superconducting gap of Al [CITATION].', 'cond-mat-0305190-1-8-3': 'For the present discussion it is clear this small residual rate is not related to the back-action of E. [MATH] is unchanged if E is biased on its supercurrent branch or when E is biased on its return current branch at low voltage [MATH] as shown in Fig. 3.', 'cond-mat-0305190-1-8-4': 'For [MATH]V, [MATH] decreases rapidly, becoming too short to measure for slightly higher voltages.', 'cond-mat-0305190-1-8-5': 'The modulation characteristics of the current [MATH] with gate voltage also change at this point from being 2e-periodic for [MATH]V to e-periodic for [MATH]V.', 'cond-mat-0305190-1-8-6': 'A similar cross-over from e to 2e-periodicity at a voltage [MATH] has been seen in a similar system recently [CITATION].', 'cond-mat-0305190-1-9-0': 'To understand how the E generates quasiparticles in B when it is biased at the usual operating voltages of VM mode, i.e. [MATH] and [MATH], we study the rate as a function of bias current and voltage of two other SCPT devices fabricated on the same chip.', 'cond-mat-0305190-1-9-1': 'These two SCPT, which we call E1 and E2, are much more weakly coupled to B and therefore allow us to measure for bias currents, [MATH], that are several orders of magnitude higher than is possible using E.', 'cond-mat-0305190-1-9-2': 'The island of SCPT E1 is located 96 [MATH]m from the island of B and has [MATH] k[MATH], while the corresponding parameters for E2 are 143 [MATH]m and [MATH] k[MATH].', 'cond-mat-0305190-1-9-3': 'For these devices, we can measure the rate up to voltages [MATH].', 'cond-mat-0305190-1-9-4': 'Again, we see a small initial increase of [MATH] at [MATH]V and then sharp increases at voltages [MATH] and [MATH].', 'cond-mat-0305190-1-9-5': 'These voltages correspond approximately to the Josephson-quasiparticle tunneling and sequential quasiparticle tunneling thresholds in an SCPT and are accompanied by sharp increases in [MATH].', 'cond-mat-0305190-1-9-6': 'The inset of Fig. 3 shows the rate as a function of the currents in E1 and E2 above these two thresholds.', 'cond-mat-0305190-1-9-7': 'To test the hypothesis that [MATH] is proportional to the total number of quasiparticle tunneling events, the currents for electrometer voltages above [MATH] are multiplied by two.', 'cond-mat-0305190-1-9-8': 'This is because there are twice as many quasiparticle tunneling events for voltages [MATH] through the junctions of E than for [MATH] for a given current.', 'cond-mat-0305190-1-9-9': 'As one can see, this scaling collapses all of the data from each electrometer to a common line.', 'cond-mat-0305190-1-10-0': 'From these data one can conclude that the quasiparticle current of electrometer is the source of back-action noise leading to quasiparticle poisoning and e-periodic Coulomb staircase of B when measured by E. Further, the quasiparticle generation in B is proportional to the total number of quasiparticle tunneling events per second through E.', 'cond-mat-0305190-1-10-1': 'This relationship could indicate that the back-action of E results from the shot noise of tunneling quasiparticles.', 'cond-mat-0305190-1-10-2': 'On the other hand, this back-action could also be the result of the recombination of quasiparticles in E into pairs.', 'cond-mat-0305190-1-10-3': 'This quasiparticle recombination produces phonons and in smaller extent photons of energy [MATH].', 'cond-mat-0305190-1-10-4': 'These phonons/photons, which propagate from E to B without energy relaxation could generate quasiparticles in B. Determining the details of the interaction between E and B will require further work.', 'cond-mat-0305190-1-10-5': 'However, it is interesting to note that this sort of recombination noise would likely be suppressed by having normal metal leads close to the junctions.', 'cond-mat-0305190-1-10-6': 'This may provide an explanation of previous results [CITATION] in which it was possible to observe a 2e-periodic Coulomb staircase using a VM electrometer measuring a superconducting box, which had normal metal "quasiparticle traps" close to the junctions.', 'cond-mat-0305190-1-11-0': 'Average charge of B as a function of its gate charge when the electrometer operates in SW mode: solid line, quasiparticle flushing (see text) before each charge measurement; dashed line, no flushing.', 'cond-mat-0305190-1-11-1': 'Vertical dotted lines mark the positions at which the maxima of [MATH] appear.', 'cond-mat-0305190-1-11-2': 'Inset shows grey scale image of density of switching events of E as a function of gate charge and charge on the island of B.', 'cond-mat-0305190-1-11-3': 'The bright line shows the predicted average charge in B as a function of the gate charge for a coherent superposition of 0 and 1 Cooper pair states with [MATH].', 'cond-mat-0305190-1-11-4': 'Residual quasiparticle poisoning appears in the grey area pointed by the arrow.', 'cond-mat-0305190-1-12-0': 'In an attempt to eliminate the poisoning due the measurement electrometer, we have investigated using it in the switching current mode.', 'cond-mat-0305190-1-12-1': 'For this type of measurement, the electrometer is operated on its switching current (phase-diffusion) branch until after the measurement is made.', 'cond-mat-0305190-1-12-2': 'So, its voltage is well below that for which a significant increase in [MATH] is observed.', 'cond-mat-0305190-1-12-3': 'For these measurements, [MATH] is set near [MATH], where the transfer function of E, [MATH], is maximum, ([MATH]3.6 nA/e).', 'cond-mat-0305190-1-12-4': 'For each charge measurement [MATH] at fixed [MATH] several hundred switching events is measured at a ramp rate of [MATH]130 nA/s.', 'cond-mat-0305190-1-12-5': 'The switching current, which is related the charge on the island of B by non-linear transfer function [MATH], is then inverted to obtain [MATH].', 'cond-mat-0305190-1-12-6': 'The result of this measurement after correction for non-linearity of the electrometer and averaging over measurements at fixed [MATH] is shown with dashed line in Fig. 4.', 'cond-mat-0305190-1-12-7': 'The dotted vertical lines in Fig. 4 show the locations of the measured peaks in the switching current of [MATH].', 'cond-mat-0305190-1-12-8': 'These peaks are located at [MATH] and so serve to calibrate the scale and origin (modulo [MATH]) the abscissa.', 'cond-mat-0305190-1-13-0': 'The Coulomb staircase in Fig. 4 (dashed line) still shows the split steps at [MATH] characteristic of quasiparticles.', 'cond-mat-0305190-1-13-1': 'This is to be expected based on the measurements of [MATH] even if there is no quasiparticle generation by E before a measurement.', 'cond-mat-0305190-1-13-2': 'The residual rate, [MATH] ms[MATH], which is not related to the back-action of electrometer, leads to trapping of quasiparticles by the electrostatic potential of island for [MATH] near [MATH] with a probability approaching 100% for measurement times much greater than [MATH].', 'cond-mat-0305190-1-13-3': 'The observed difference in the length of the two steps implies that the escape rate for a quasiparticle near [MATH] is greater than [MATH].', 'cond-mat-0305190-1-13-4': 'One would expect this to happen if the density of quasiparticles in the leads of B is sufficiently low.', 'cond-mat-0305190-1-14-0': 'The probability of an even parity state of B for [MATH] can be greatly increased by flushing the quasiparticle from B before each measurement.', 'cond-mat-0305190-1-14-1': 'To prepare the even parity state, a voltage pulse [MATH] is applied across B just prior to each measurement of its charge by E.', 'cond-mat-0305190-1-14-2': 'The amplitude of [MATH] is chosen such that [MATH], in order to release quasiparticles trapped by the electrostatic potential of the island without generating new quasiparticles, which would increase [MATH].', 'cond-mat-0305190-1-14-3': '(Note that a similar flushing action occurs automatically on the return branch at the end of each switching current measurement in B or E).', 'cond-mat-0305190-1-14-4': 'Switching histograms show that this procedure prepares the even state with a probability of about 85%.', 'cond-mat-0305190-1-14-5': 'Immediately after B is reset to its even state, the measurement ramp of [MATH] begins.', 'cond-mat-0305190-1-14-6': 'The result of this procedure is shown in the top curve in Fig. 4.', 'cond-mat-0305190-1-14-7': 'As one can see, the splitting of the step at [MATH] has vanished.', 'cond-mat-0305190-1-14-8': 'The imperfect preparation of the initial state and the residual rate of [MATH] ms[MATH] along with the ramping time of the current [MATH] still lead to switching events of E ,which correspond to an unpaired electron on the island of B [inset of Fig.4].', 'cond-mat-0305190-1-14-9': 'At [MATH] , where the switching events of E corresponding to even and odd parity states of B can be well distinguished, the odd parity state of B accounts for 30 of the total switching events.', 'cond-mat-0305190-1-14-10': 'This is in good agreement with the estimate (28) based on [MATH], the ramping time of the current [MATH]3 ms) and an initial probability of the even state (85).', 'cond-mat-0305190-1-14-11': 'Neglecting the poisoned events curve [MATH] vs. [MATH] results from the mixing of 0 and 1 Cooper pair states and is determined by the ratio [MATH] [CITATION].', 'cond-mat-0305190-1-14-12': 'The solid curve in the inset of Fig. 4 is calculated using the independently measured value of [MATH] 0.44 and is in reasonable agreement with the data.', 'cond-mat-0305190-1-15-0': 'In conclusion, our measurements clearly show that operation of SCPT electrometer at voltages [MATH]V leads to substantial generation of quasiparticles on the island of second SCPT which the electrometer measures.', 'cond-mat-0305190-1-15-1': 'The rate of quasiparticle generation depends linearly on the total number of quasiparticle tunneling events per second through the junctions of the electrometer.', 'cond-mat-0305190-1-15-2': 'To overcome this back-action from the electrometer, we operate it in the mode which uses switching current modulation for charge detection.', 'cond-mat-0305190-1-15-3': 'Using this mode of operation, we are able to recover the 2e-periodic Cooper staircase of the SCPT, which is expected both theoretically and from the 2e-periodicity of its switching current.', 'cond-mat-0305190-1-16-0': 'The authors thank D. V. Averin, J. R. Friedman and K. K. Likharev for useful discussions and W. Chen and V. V. Kuznetsov for technical assistance.', 'cond-mat-0305190-1-16-1': 'Work is supported in part by AFOSR grant No. F49620010001.'}

{'cond-mat-0305190-2-0-0': 'We report on the effect of the back-action of a Single Cooper Pair Transistor electrometer (E) on the measurement of charge on the island of a superconducting box (B).', 'cond-mat-0305190-2-0-1': 'The charge is e-periodic in the gate bias of B when E is operated near voltages [MATH] or [MATH].', 'cond-mat-0305190-2-0-2': 'We show that this is due to quasiparticle poisoning of B at a rate proportional to the number of quasiparticle tunneling events in E per second.', 'cond-mat-0305190-2-0-3': 'We are able to eliminate this back action and recover 2e charge periodicity using a new measurement method based on switching current modulation of E.', 'cond-mat-0305190-2-1-0': 'The superconducting box, e.g. a small island of Al film very weakly coupled to the outside circuitry by Josephson junctions, has shown considerable promise as a qubit for quantum information processing where the two states can be represented by superpositions of 0 or 1 excess Cooper pairs in the box [CITATION].', 'cond-mat-0305190-2-1-1': 'Measurement of the quantum state of this so-called charge qubit without inducing unwanted decoherence is a significant problem as is quasiparticle poisoning, i.e. the introduction of an unpaired electron (quasiparticle) into the box.', 'cond-mat-0305190-2-1-2': 'At temperatures of 10 mK or so, where experiments are commonly done, the number of quasiparticles should, in principle, be negligible.', 'cond-mat-0305190-2-1-3': 'However, such quasiparticle poisoning, due perhaps to the measurement process itself, is commonly observed.', 'cond-mat-0305190-2-1-4': 'A manifestation of this is seen in the so called Coulomb staircase.', 'cond-mat-0305190-2-1-5': 'When a charge, [MATH], is capacitively induced on the box, one expects Cooper pairs to tunnel resonantly into or out of the box at [MATH], where [MATH] is an odd integer, to maintain the lowest energy charge state of the box.', 'cond-mat-0305190-2-1-6': 'This results in the Coulomb staircase of the charge in the box [MATH] with period [MATH] in [MATH].', 'cond-mat-0305190-2-1-7': 'On the other hand, if there are quasiparticles in the system, then maintaining the lowest charging energy state also leads to quasiparticle tunneling.', 'cond-mat-0305190-2-1-8': 'This gives rise to splitting of the steps in the Coulomb staircase, which shifts toward e-periodicity as the number of quasiparticles increases [CITATION].', 'cond-mat-0305190-2-1-9': 'As a result, the lowest energy state of the box at [MATH] no longer corresponds to a resonant state of the Cooper pair tunneling.', 'cond-mat-0305190-2-1-10': 'For the box qubit, this means that relaxation does not bring the system back to its computational ground state at its operating point.', 'cond-mat-0305190-2-1-11': 'Since the ability to prepare the initial state of the qubit is an absolutely necessary condition for quantum computing, quasiparticle poisoning has been a serious road block for groups working to build a charge qubit [CITATION].', 'cond-mat-0305190-2-1-12': 'Solving this problem in charge qubits is essential.', 'cond-mat-0305190-2-2-0': 'The purpose of this paper is to investigate the effects of measurement (i.e. back-action) on the measured charge in the box and to develop approaches to minimize these effects.', 'cond-mat-0305190-2-2-1': 'For this study, we use two capacitively coupled Single Cooper Pair Transistors (SCPT)s, one of which acts as an electrometer (E) and the other as a superconducting box (B).', 'cond-mat-0305190-2-2-2': 'The latter gives a good representation of the box and at the same time allows us to study quasiparticle poisoning effects without needing to operate E.', 'cond-mat-0305190-2-2-3': 'The SCPT electrometer can be operated in several different modes when measuring the charge of B [CITATION].', 'cond-mat-0305190-2-2-4': 'Commonly, the charge measurements of B are done by operating E in the voltage modulation mode (VM).', 'cond-mat-0305190-2-2-5': 'In this mode, E is biased at a sufficiently high voltage that quasiparticles are generated, so it effectively functions as a SET where the source drain voltage is modulated by [MATH] with a period of [MATH].', 'cond-mat-0305190-2-2-6': 'However, it is known that the switching current of a SCPT, i.e. the current at which it switches hystereticaly from the low voltage or phase-diffusion branch to [MATH], is also charge sensitive [CITATION] and can be used for charge measurement.', 'cond-mat-0305190-2-2-7': 'We refer to this as the switching current mode (SW).', 'cond-mat-0305190-2-2-8': 'The SW mode of operation has been analyzed [CITATION], but until now no measurements of the charge on the island of a box using a SW mode of the electrometer have been reported.', 'cond-mat-0305190-2-2-9': 'We present the results of the measurements of the island charge in B by E operated in either the VM or SW mode.', 'cond-mat-0305190-2-2-10': 'The results demonstrate that measurement-induced poisoning, which leads to an e-periodic Coulomb staircase when using E in the VM mode, can be eliminated in the SW mode.', 'cond-mat-0305190-2-3-0': 'The parameters of the sample (Fig. 1.', 'cond-mat-0305190-2-3-1': 'a) are as follows: E has a normal state resistance [MATH] k[MATH], charging energy [MATH]eV and Josephson energy [MATH]eV.', 'cond-mat-0305190-2-3-2': 'For B these parameters are [MATH] k[MATH], [MATH]eV and [MATH]eV.', 'cond-mat-0305190-2-3-3': '[MATH] is the average of the Josephson energies of two junctions as determined from the values of [MATH] and the superconducting gap [MATH]eV by using the Ambegaokar-Baratoff formula.', 'cond-mat-0305190-2-3-4': '[MATH] is determined from the amplitude of maximum voltage modulation of the devices.', 'cond-mat-0305190-2-3-5': 'The coupling capacitance between E and B is determined to be 80 aF from their measured coupling of 4.8%.', 'cond-mat-0305190-2-3-6': 'All devices are made using standard two angle shadow evaporations without having normal metal quasiparticle traps close to junctions.', 'cond-mat-0305190-2-3-7': 'The sample is placed in microwave tight copper can located on a temperature regulated stage of a dilution refrigerator having a base temperature of 6 mK.', 'cond-mat-0305190-2-3-8': 'All the measurement leads are filtered by low temperature microwave filters [CITATION] that are thermally anchored to the mixing chamber.', 'cond-mat-0305190-2-4-0': 'Figure 1.', 'cond-mat-0305190-2-4-1': 'b shows the charge on the island of B measured with E in the VM mode.', 'cond-mat-0305190-2-4-2': 'During this measurement, the source and drain leads of B are at a common potential with respect to its gate.', 'cond-mat-0305190-2-4-3': 'Figure 1.c presents the switching current modulation of B measured with the bias current through E, [MATH], set equal to zero.', 'cond-mat-0305190-2-4-4': 'In this and following measurements, the bias current of B, [MATH], is ramped at a rate of [MATH] nA/s.', 'cond-mat-0305190-2-4-5': 'As can be seen, the switching current modulation of B is 2e-periodic as expected at low temperature, but the charge of B, measured by the electrometer, is e-periodic.', 'cond-mat-0305190-2-4-6': 'Similar dependences of [MATH] on [MATH] were measured with E biased in either of its voltage sensitive regions, i.e. near the gap where [MATH] or near the Josephson-quasiparticle peak where [MATH].', 'cond-mat-0305190-2-4-7': 'To determine if the e-periodicity of [MATH] is due to the back-action of E on B, the quasiparticle poisoning rate of B, [MATH], was measured for a range of bias conditions of E.', 'cond-mat-0305190-2-4-8': 'In addition to this, we studied how the biases of two other SCPTs located on the same chip, but coupled more weakly to B, affected [MATH].', 'cond-mat-0305190-2-4-9': 'Switching current histograms of B measured for different bias conditions of E. Corresponding time delay from the beginning of the current ramp is shown in the upper horizontal axis.', 'cond-mat-0305190-2-4-10': 'The histograms are shifted on vertical axis for clarity.', 'cond-mat-0305190-2-4-11': 'The "odd" ("even") peaks in the histograms correspond to switching from the odd (even) parity states of B. Inset: IV-characteristic of E. Arrows indicate bias conditions at which switching current histograms of the main figure were measured.', 'cond-mat-0305190-2-4-12': 'Diamonds mark positions where [MATH] has been measured for data in Fig. 3.', 'cond-mat-0305190-2-5-0': 'The details of the technique for determining [MATH] from switching current distributions have been reported previously [CITATION].', 'cond-mat-0305190-2-5-1': 'Briefly, as [MATH] is linearly ramped in time, the switching current histogram of B, with its gate biased near [MATH], exhibits two peaks if the number of quasiparticles on the island changes during the measurement of histogram.', 'cond-mat-0305190-2-5-2': 'One peak, [MATH], which is close to the maximum of the switching-current characteristic [MATH], occurs when the island has an even number of electrons (even state) and the other -much lower current- peak, [MATH], is near the predicted switching current minimum at [MATH] when one quasiparticle occupies the island (odd state) (Fig. 2).', 'cond-mat-0305190-2-5-3': 'If [MATH] and B has not switched, it must be in the even state.', 'cond-mat-0305190-2-5-4': 'The entry of a quasiparticle onto the island effectively changes [MATH] to [MATH] which for [MATH] will cause B to switch rapidly to the running state, giving the time of the poisoning and thus the quasiparticle poisoning rate [MATH].', 'cond-mat-0305190-2-5-5': 'Previous studies have shown that [MATH] is independent of [MATH] in the region between the peaks, giving an exponential decay of the even state.', 'cond-mat-0305190-2-5-6': 'Several of these histograms with increasing [MATH] as [MATH] and [MATH] increase are shown in Fig. 2.', 'cond-mat-0305190-2-5-7': 'The relatively low bandwidth of our filters limits these measurement to [MATH] ms[MATH].', 'cond-mat-0305190-2-6-0': 'First we determine [MATH] when all the electrodes of the electrometer are disconnected from the measurement circuitry and grounded.', 'cond-mat-0305190-2-6-1': 'In this case we still observe a small residual rate [MATH] ms[MATH] for [MATH].', 'cond-mat-0305190-2-6-2': 'This non-zero rate can be caused, e.g., by the presence of impurity levels in the superconducting gap of Al [CITATION].', 'cond-mat-0305190-2-6-3': 'For the present discussion it is clear that this small residual rate is not related to the back-action of E. [MATH] is unchanged if E is biased on its supercurrent branch or when E is biased on its return current branch at low voltage [MATH]V [MATH] as shown in Fig. 3.', 'cond-mat-0305190-2-6-4': 'For [MATH]V, [MATH] decreases rapidly, becoming too short to measure for slightly higher voltages.', 'cond-mat-0305190-2-6-5': 'The modulation characteristics of the current [MATH] with gate voltage also changes at this point from being 2e-periodic for [MATH]V to e-periodic for [MATH]V. Similar cross-overs from e to 2e periodicity at a voltage [MATH] have previously been seen in related systems (see e.g. [CITATION]).', 'cond-mat-0305190-2-6-6': 'Quasiparticle poisoning rate [MATH] of B as a function of the voltage across E. Inset: [MATH] as a function of the total quasiparticle tunneling current in E1 (diamonds) and of E2 (triangles).', 'cond-mat-0305190-2-6-7': 'The open and filled symbols correspond to bias conditions [MATH] or [MATH], respectively.', 'cond-mat-0305190-2-6-8': '[MATH] or [MATH] for the filled or open symbols, respectively.', 'cond-mat-0305190-2-7-0': 'In order to study how [MATH] depends on the voltage and current of E through its entire operating range, and in particular near [MATH] and [MATH], we use two other SCPTs fabricated on the same chip but much more weakly coupled to B.', 'cond-mat-0305190-2-7-1': 'Since these two SCPTs, which we call E1 and E2, are much more weakly coupled to B, they allow us to measure [MATH] for bias currents, [MATH], that are several orders of magnitude higher than is possible using E.', 'cond-mat-0305190-2-7-2': 'The island of SCPT E1 is located 96 [MATH]m from the island of B and has [MATH] k[MATH], while the corresponding parameters for E2 are 143 [MATH]m and [MATH] k[MATH].', 'cond-mat-0305190-2-7-3': 'For these devices, we can measure the rate [MATH] up to voltages [MATH].', 'cond-mat-0305190-2-7-4': 'Again, we see a small initial increase of [MATH] at [MATH]V and then sharp increases at voltages [MATH] and [MATH].', 'cond-mat-0305190-2-7-5': 'These voltages correspond approximately to the Josephson-quasiparticle tunneling and sequential quasiparticle tunneling thresholds in a SCPT and are accompanied by sharp increases in [MATH].', 'cond-mat-0305190-2-7-6': 'The inset of Fig. 3 shows the rate [MATH] as a function of the currents in E1 and E2 above these two thresholds.', 'cond-mat-0305190-2-7-7': 'To test the hypothesis that [MATH] is proportional to the total number of quasiparticle tunneling events, the currents for the electrometer voltages [MATH] are multiplied by two.', 'cond-mat-0305190-2-7-8': 'This is done since, for a given current at voltages [MATH], there are twice as many quasiparticle tunneling events through the junctions of E as for [MATH].', 'cond-mat-0305190-2-7-9': 'As one can see, this scaling collapses all of the data from each electrometer to a common line.', 'cond-mat-0305190-2-8-0': 'From these data one can conclude that the quasiparticle current of electrometer is the source of back-action noise leading to quasiparticle poisoning and an e-periodic Coulomb staircase of B when measured by E. Further, the quasiparticle generation in B is proportional to the total number of quasiparticle tunneling events per second through E.', 'cond-mat-0305190-2-8-1': 'This relationship could indicate that the back-action of E results from the shot noise of tunneling quasiparticles.', 'cond-mat-0305190-2-8-2': 'On the other hand, this back-action could also be the result of the recombination of quasiparticles in E into pairs.', 'cond-mat-0305190-2-8-3': 'This quasiparticle recombination produces phonons and to a smaller extent photons of energy [MATH].', 'cond-mat-0305190-2-8-4': 'These phonons/photons, which propagate from E to B without energy relaxation could generate quasiparticles in B. Determining the details of the interaction between E and B will require further work.', 'cond-mat-0305190-2-8-5': 'However, it is interesting to note that this sort of recombination noise would likely be suppressed by having normal metal leads close to the junctions.', 'cond-mat-0305190-2-8-6': 'This may provide an explanation of previous results [CITATION] in which it was possible to observe a 2e-periodic Coulomb staircase using a VM electrometer measuring a superconducting box, which had normal metal "quasiparticle traps" close to the junctions.', 'cond-mat-0305190-2-8-7': 'Average charge of B as a function of its gate charge when the electrometer operates in SW mode: bottom curve, quasiparticle flushing (see text) before each charge measurement; top curve, no flushing.', 'cond-mat-0305190-2-8-8': 'The top curve is shifted in vertical direction for clarity.', 'cond-mat-0305190-2-8-9': 'The flushing pulse has amplitude 150 [MATH]V and duration 25 ms. The dotted lines mark the positions at which the maxima of [MATH] appear.', 'cond-mat-0305190-2-8-10': 'Inset shows the average switching current of E as a function of its gate charge.', 'cond-mat-0305190-2-8-11': 'The arrows marked by "0","1" and "P" correspond to switching currents of E when the island of B has 0 or 1 excess Cooper pairs or when it is in the poisoned state, respectively.', 'cond-mat-0305190-2-9-0': 'Quasiparticle poisoning of B due to the measurement of its charge can, in principle, be eliminated by the operation of E in the SW mode, where its voltage remains well below [MATH] until the measurement is made.', 'cond-mat-0305190-2-9-1': 'The SW mode of operation of the electrometer is illustrated in the inset of Fig. 4, which shows its switching current vs. gate charge transfer-function.', 'cond-mat-0305190-2-9-2': 'In general, [MATH] contains a component proportional to [MATH], in addition to the externally applied bias.', 'cond-mat-0305190-2-9-3': 'So, to measure changes in [MATH] as a function of [MATH], we bias E near point "0", where the switching current is very sensitive to variations of external charge ( inset of Fig. 4), and record several hundred switching events of E for each value of [MATH].', 'cond-mat-0305190-2-9-4': 'These switching data are then corrected for the measured nonlinearity in the transfer-function of E and averaged.', 'cond-mat-0305190-2-9-5': 'The top curve in Fig. 4.', 'cond-mat-0305190-2-9-6': 'shows the result obtained using this procedure.', 'cond-mat-0305190-2-10-0': 'While these data are no longer strictly e-periodic, as in Fig. 1b, the Coulomb staircase in Fig. 4 (top line) still shows the split steps around [MATH] characteristic of quasiparticle poisoning.', 'cond-mat-0305190-2-10-1': 'We can see that this is consistent with the measured residual rate, [MATH], as follows.', 'cond-mat-0305190-2-10-2': 'The edges of these steps should occur for an energy difference between the even and odd states, [MATH], such that the rates for even to odd ([MATH]) and odd to even ([MATH]) transitions of the island are equal.', 'cond-mat-0305190-2-10-3': 'This ratio is given by [MATH] [CITATION], where [MATH] is the sum of the quasiparticle densities on the leads and the island in the even state.', 'cond-mat-0305190-2-10-4': 'For the odd state, this sum is [MATH], where [MATH] is the volume of the island.', 'cond-mat-0305190-2-10-5': 'The second term in [MATH] accounts for the extra quasiparticle occupying the island in the odd state, which increases the density by [MATH]m[MATH].', 'cond-mat-0305190-2-10-6': '[MATH] can be estimated from the measured residual poisoning rate [MATH], the normal density of states of the Al film and the junction resistances, giving [MATH]m[MATH].', 'cond-mat-0305190-2-10-7': 'Taking the electron temperature of B to be 15 mK, which is reasonable since B is completely passive in these measurements, gives a length for the short step of [MATH] in agreement with the data shown in Fig. 4.', 'cond-mat-0305190-2-11-0': 'The effects which result from the residual rate [MATH] can be greatly reduced by flushing the quasiparticle from the island of B before each measurement.', 'cond-mat-0305190-2-11-1': 'As one possible approach to prepare the even parity state, we apply a voltage pulse [MATH] across B just prior to each measurement.', 'cond-mat-0305190-2-11-2': 'The amplitude of [MATH] is chosen such that [MATH], in order to release the quasiparticle from the electrostatic potential of the island but yet not to generate any new quasiparticles by the pulse [CITATION].', 'cond-mat-0305190-2-11-3': 'Switching histograms of B show that this procedure prepares the even state with a probability of about 85%.', 'cond-mat-0305190-2-11-4': 'Immediately after B is flushed, the measurement ramp of [MATH] begins.', 'cond-mat-0305190-2-11-5': 'The result of this procedure is shown in the bottom curve in Fig. 4.', 'cond-mat-0305190-2-11-6': 'As one can see, the quasiparticle-induced splitting of the step at [MATH] is no longer apparent.', 'cond-mat-0305190-2-11-7': 'However, individual histograms still show about 30% of the switching events in E near [MATH] are from the poisoned state of B.', 'cond-mat-0305190-2-11-8': 'This is consistent with the imperfect preparation of the initial state and additional poisoning with rate [MATH] in the finite time between flushing and the switching event.', 'cond-mat-0305190-2-11-9': 'Thus we see that, with the effects of residual poisoning greatly reduced by the flushing, the measurement of [MATH] by E in the SW mode gives results consistent with the 2e-periodic switching-current distribution of B.', 'cond-mat-0305190-2-12-0': 'In conclusion, our measurements clearly show that operation of a SCPT electrometer at voltages [MATH]V causes a substantial generation of quasiparticles in the circuit of the superconducting box leading to an e-periodic Coulomb staircase.', 'cond-mat-0305190-2-12-1': 'The rate of quasiparticle poisoning in the box depends linearly on the total number of quasiparticle tunneling events per second through the junctions of the electrometer.', 'cond-mat-0305190-2-12-2': 'To overcome this back-action from the electrometer, we operate it in a mode which uses switching-current modulation for charge detection.', 'cond-mat-0305190-2-12-3': 'Using this mode of operation, we are able to recover the 2e-periodic Coulomb staircase of the SCPT in the box configuration, which is expected both theoretically and from the 2e-periodicity of its switching current.', 'cond-mat-0305190-2-13-0': 'The authors thank D. V. Averin, J. R. Friedman and K. K. Likharev for useful discussions and W. Chen and V. V. Kuznetsov for technical assistance.', 'cond-mat-0305190-2-13-1': 'Work is supported in part by AFOSR grant No. F49620010001.'}

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0305190

1804.01505

{'1804.01505-1-0-0': 'The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly-interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei.', '1804.01505-1-0-1': 'We present a general algorithm for quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.', '1804.01505-1-1-0': 'Quantum computers should enable dramatic new capabilities in simulating quantum many-body systems, particularly their dynamic properties [CITATION].', '1804.01505-1-1-1': 'Quantum dynamics is in general extremely difficult to treat on a classical computer except for a few special cases such as very low-energy scattering where spectral decomposition in finite volumes enable direct connections between spectra and phase shifts in the scattering of 2- or 3-clusters [CITATION] or very high-energy scattering that can be treated as nearly non-interacting final states, including y-scaling in neutron or electron scattering [CITATION] or inclusive deep inelastic scattering in QCD [CITATION].', '1804.01505-1-1-2': 'The general problem is essentially intractable because of quantum interference, the rapidly oscillating phases that arise in the relevant path integrals.', '1804.01505-1-2-0': 'Perhaps the simplest quantum dynamics problem is the dynamic linear response, framed as the response of a quantum system to a small perturbation.', '1804.01505-1-2-1': 'Examples are ubiquitous, including for example neutron scattering on materials, photon scattering in atomic systems, and electron and neutrino scattering from atomic nuclei.', '1804.01505-1-2-2': 'The response of the system can in principle tell us much about the structure of the system being probed as well as important properties of the dynamics.', '1804.01505-1-2-3': 'In the case of neutrinos scattered by nuclei it is also used to infer properties of the neutrino itself including masses, mixing angles, the mass hierarchy and CP violation in the neutrino sector (eg. [CITATION]).', '1804.01505-1-3-0': 'The ability to accurately calculate the dynamic response over a wide range of energy and momentum transfers, augmented by the possibility of determining specific features of the final states, would revolutionize our ability to extract information from many kinds of scattering experiments.', '1804.01505-1-3-1': 'Some information on quantum dynamics can be obtained using classical computers even for relatively large systems, typically by computing imaginary-time correlation functions [CITATION].', '1804.01505-1-3-2': 'Even for systems where the ground state or thermal ensembles can be simulated free of any sign problem, it is extremely difficult to invert these correlation functions to obtain the exact dynamic response.', '1804.01505-1-3-3': 'In this paper we discuss methods to determine the dynamic response on a quantum computer, as well as to detect important features of explicit final states that can be directly compared to experimental data.', '1804.01505-1-3-4': 'Our approach is particularly well suited for problems defined on a lattice, but these lattice methods can of course also be used to simulate systems in the continuum over a wide but finite range of energy and momenta, for example in lattice studies of cold atoms [CITATION] and nuclear systems [CITATION].', '1804.01505-1-3-5': 'Also we restrict ourselves to the response from the quantum ground state (T=0), generalizations to finite temperature are possible by preparing states in thermal equilibrium rather than the ground state [CITATION].', '1804.01505-1-4-0': 'We note that the similar problem of evaluating chemical reaction rates [CITATION] and time-dependent correlation functions [CITATION] have been already investigated in the quantum-chemistry literature.', '1804.01505-1-4-1': 'Our proposed algorithm improves on these earlier techniques in that our strategy is completely general, does not depend on simplifying assumptions on the excitation operator (for example, being able to diagonalize it as in [CITATION]) and requires only a polynomial number of measurements (instead of exponential like in [CITATION]).', '1804.01505-1-4-2': 'Also, working directly in frequency space allows us a direct access to the final states of a reaction which can be further analyzed.', '1804.01505-1-4-3': 'This is particularly important for neutrinos where the momentum and energy transfer are a priori unknown.', '1804.01505-1-5-0': 'Furthermore we are able to provide rigorous cost and error estimates of the computed dynamical properties.', '1804.01505-1-5-1': 'Available algorithms for evaluating energy spectra [CITATION] can in principle be adapted to compute response functions but they require resolution of individual excited states which grows exponentially in number for large systems.', '1804.01505-1-6-0': 'The paper is organized as follows, in Sec. [REF] we provide detailed definition of the Dynamical Response Function and describe the implementation of our method.', '1804.01505-1-6-1': 'In Sec. [REF] we provide an example of final state characterization by discussing the estimation of the one- and two-body momentum distribution and conclude in Sec. [REF].', '1804.01505-1-7-0': '# Method', '1804.01505-1-8-0': 'In the linear regime the response of a system of interacting particles due to a perturbative probe characterized by the excitation operator [MATH] can be fully characterized using the Dynamical Response Function, which can be expressed as [EQUATION] where [MATH] is the ground-state of the system with energy [MATH], [MATH] are the final states of the reaction with energies [MATH] and [MATH] is the energy transfer.', '1804.01505-1-8-1': "It is convenient to rescale the response function so that it's zero moment (the integral over frequencies) is [MATH]; this can be achieved by defining [EQUATION]", '1804.01505-1-8-2': 'The final normalization can be restored by either using the knowledge of one of the sum rules or by direct evaluation of the ground state expectation value [MATH].', '1804.01505-1-8-3': 'Understanding this, in the following we will drop the superscript [MATH].', '1804.01505-1-9-0': 'Our goal is to estimate the dynamical response function [MATH] with energy resolution [MATH] and a precision [MATH] with probability [MATH].', '1804.01505-1-9-1': 'We will indicate the difference between the largest eigenvalue of [MATH] and the ground state energy by: [MATH].', '1804.01505-1-9-2': 'Note that this quantity grows only polynomially with system size for most Hamiltonians of interest (see discussion below).', '1804.01505-1-10-0': 'In the following we will assume to have access to three black-box quantum procedures (oracles):', '1804.01505-1-11-0': 'Even though the oracle [MATH] may be impractical to implement for a general Hamiltonian, for most systems of interest many different algorithms are available in the literature ([CITATION]) and some have already be tested on simple nuclear systems [CITATION].', '1804.01505-1-11-1': 'Also, close to optimal strategies to implement the time-evolution operator for sparse Hamiltonians are known [CITATION] and for Hubbard-type Hamiltonians (like those derived within lattice-EFT [CITATION]) efficient implementations of Trotter steps with sub-linear circuit depth are available [CITATION].', '1804.01505-1-11-2': 'For the common case where [MATH] is a one-body operator the latter strategies can be used to implement [MATH] efficiently.', '1804.01505-1-12-0': 'Our scheme is composed of two quantum circuits', '1804.01505-1-13-0': 'For typical situations where the implementation of [MATH] requires considerable effort the success probability of the first routine can be amplified to [MATH] with additional [MATH] calls to the oracle [MATH].', '1804.01505-1-13-1': 'An alternative algorithm which removes the dependence of [MATH] on [MATH] but is more difficult to make deterministic is also presented in Sec. [REF].', '1804.01505-1-14-0': 'This whole circuit needs to be run a number of times given approximately by [EQUATION] independent of the target resolution [MATH].', '1804.01505-1-15-0': 'In summary, for a given choice of the excitation operator [MATH] our algorithm can be described by the following steps:', '1804.01505-1-16-0': 'iteration number less than [MATH] prepare the ground state using [MATH] run the first quantum algorithm (Sec. [REF]) algorithm succeeds we have prepared [MATH] run the second quantum algorithm (Sec. [REF]) store result for classical post-processing final state information needed measure final state (eg.', '1804.01505-1-16-1': 'Sec [REF])', '1804.01505-1-17-0': 'In the next sections we describe in detail the implementation of the two quantum routines introduced above.', '1804.01505-1-17-1': 'We also present examples obtained by classical simulation of a simple 2D fermionic system described by the Hubbard hamiltonian [EQUATION] where [MATH] indicates the nearest-neighbor lattice sites and [MATH] denotes the number operator.', '1804.01505-1-17-2': 'The results shown here were obtained for [MATH] "nucleons", [MATH] lattice sites and [MATH].', '1804.01505-1-17-3': 'These parameters are chosen to give a bound state considerably smaller than the lattice.', '1804.01505-1-18-0': '## State preparation algorithm', '1804.01505-1-19-0': 'The first problem we have to solve is the preparation of the state [MATH] given a quantum register initialized in the ground-state [MATH].', '1804.01505-1-19-1': "Let's start by adding an ancilla qubit and defining the unitary operator [EQUATION] where the Pauli [MATH] operator acts on the ancilla and the final matrix representation is on the basis spanned by the states [MATH] of the ancilla.", '1804.01505-1-19-2': 'Note that this unitary can be implemented efficiently with just [MATH] calls to a controlled version of the oracle [MATH] and additional [MATH] one-qubit gates.', '1804.01505-1-20-0': 'By initializing the ancilla register to [MATH], applying [MATH] and measuring the state [MATH] we have effectively produced [EQUATION] which differs from the wanted state by corrections of order [MATH].', '1804.01505-1-20-1': 'The error in the implementation of the unitary [MATH] needs to be at least of the same order, which means a simple single Trotter step will suffice.', '1804.01505-1-20-2': 'The state preparation has a success probability of [EQUATION]', '1804.01505-1-20-3': 'This approach for the application of a non-unitary transformation is similar in spirit to earlier work (see eg. [CITATION]) and it suffers from a possibly very low efficiency since we may need [MATH] trials to succeed.', '1804.01505-1-20-4': 'One option is to perform the algorithm at a few relatively large values of [MATH] and fit a quadratic function to extrapolate out the error from the final response function.', '1804.01505-1-20-5': 'This approach is however complicated if one is interested also in properties of the final states.', '1804.01505-1-20-6': 'A second approach, already proposed in [CITATION], is to repeat the application of the unitary [MATH] until success.', '1804.01505-1-20-7': 'This works because [MATH] is approximately the identity.', '1804.01505-1-20-8': 'In order to obtain a success probability [MATH] we will need [MATH] repetitions.', '1804.01505-1-20-9': "In addition, if the inverse [MATH] of the ground-state preparation circuit is available then it's possible to use Amplitude Amplification [CITATION] to gain a quadratic speedup over this .", '1804.01505-1-21-0': 'Note that by using the normalized state [MATH] we will compute the normalized response function Eq. [REF].', '1804.01505-1-21-1': 'If no sum-rules are known one can estimate the normalization constant by estimating the success probability Eq. [REF] at different values of [MATH] and extrapolating.', '1804.01505-1-22-0': 'Since the state preparation through the unitary [MATH] is only approximate, the parameter [MATH] would need to depend on the final target accuracy.', '1804.01505-1-22-1': 'As mentioned in the introduction an alternative scheme that avoids this problem by removing the error in Eq. [REF] can be obtained by representing the excitation operator [MATH] as a linear combination of [MATH] unitary matrices [EQUATION] which can be efficiently implemented employing additional [MATH] ancilla qubits using known techniques [CITATION].', '1804.01505-1-22-2': 'The success probability in this case is given by [EQUATION] which depending on the particular case may be larger than Eq. [REF].', '1804.01505-1-22-3': 'The main drawback of this approach is that Amplitude Amplification is the only process that can make the algorithm deterministic since upon failure the output state can in general be very different from the starting point.', '1804.01505-1-23-0': '## Response Function estimation', '1804.01505-1-24-0': 'We now present our strategy to obtain the response function trough the standard Phase Estimation Algorithm (PEA) [CITATION].', '1804.01505-1-24-1': 'It is convenient to shift and scale the original Hamiltonian: [EQUATION] so that we map the energy spectrum to [MATH].', '1804.01505-1-25-0': 'By direct calculation we see that the response function [MATH] obtained from [MATH] is related to the original one by [EQUATION] for a scaled frequency [MATH].', '1804.01505-1-26-0': 'Our goal is to estimate [MATH] efficiently.', '1804.01505-1-26-1': 'We do this by using PEA on an auxiliary register of [MATH] qubits with the evolution operators [EQUATION] for [MATH].', '1804.01505-1-26-2': 'The resulting circuit will have depth [MATH], where the first term comes from the inverse Quantum Fourier Transform [CITATION] and [MATH] is the gate count needed for a time evolution of [MATH] using the oracle [MATH].', '1804.01505-1-26-3': 'The resulting probability of measuring the [MATH] ancilla qubits in the binary representation of the integer [MATH] is (see eg. [CITATION] for more details) [EQUATION] where [MATH] is the well-known Fejer kernel from Fourier analysis (see eg. [CITATION]).', '1804.01505-1-26-4': 'The probability distribution [MATH] is a good approximation of [MATH] since this kernel can be seen as a representation of the delta function with width [MATH].', '1804.01505-1-26-5': 'Therefore if we require a frequency resolution [MATH] we will need [MATH] auxiliary qubits and a polynomial number of applications of the time evolution operator to obtain a sample from [MATH].', '1804.01505-1-27-0': 'As mentioned above, for most Hamiltonians of interest the energy gap [MATH] scales only polynomially with the size of the system.', '1804.01505-1-28-0': 'We now need to estimate [MATH] from [MATH] samples drawn from it.', '1804.01505-1-28-1': 'Since [MATH] is a discrete variable an efficient way of reconstructing the probability distribution is by producing an histogram [MATH] from the samples.', '1804.01505-1-28-2': "Using Hoeffding's inequality [CITATION] we find that [EQUATION] which implies in order to obtain a precision [MATH] with probability [MATH] we need approximately [EQUATION] independent samples.", '1804.01505-1-29-0': 'In Fig. [REF] we plot the approximate response [MATH] for the model Hamiltonian Eq. [REF] at three different values of [MATH] (6,8,12).', '1804.01505-1-29-1': 'By comparing with the exact result shown as black dots, we see that for [MATH] the effect of energy resolution is negligible but already with [MATH] we obtain a rather accurate estimate for [MATH].', '1804.01505-1-29-2': 'Even [MATH] reproduces important features of the response, which in experiments is convoluted with the detector resolution.', '1804.01505-1-29-3': 'The inset shows the convergence of the maximum error [EQUATION] as a function of the sample size [MATH].', '1804.01505-1-29-4': 'Response functions relevant for [MATH] and [MATH] scattering are typically smooth at high energy and hence require small [MATH] and short propagation times.', '1804.01505-1-30-0': 'Finally, in order to obtain a negligible bias from the state preparation we need the parameter [MATH] to scale as [EQUATION] for some constant [MATH].', '1804.01505-1-30-1': 'Note that the Hamiltonian evolution implemented in [MATH] has to have an error [MATH] to be negligible (luckily algorithms with only logarithmic dependence on [MATH] are known [CITATION]).', '1804.01505-1-31-0': 'This concludes the proof of the scalings [REF] and [REF].', '1804.01505-1-32-0': '# Final state measurements', '1804.01505-1-33-0': 'In electron- or neutrino-nuclear scattering experiments [CITATION] one would like to infer the probability [MATH] that the probe transferred energy-momentum [MATH] to the nucleus and simultaneously that the final state includes a nucleon (or neutron or proton) of momentum [MATH].', '1804.01505-1-33-1': 'More concretely this amounts to an inference procedure of the form [EQUATION] where [MATH] results from the experimental measure, [MATH] is the momentum distribution of the final states for a process with given [MATH] and [MATH].', '1804.01505-1-33-2': 'The prior probability [MATH] depends on the static response of the nucleus and the characteristic of the probe beam and can be updated given the other ones by a Bayesian procedure.', '1804.01505-1-33-3': 'The above section explains how to obtain [MATH] with a given accuracy and in the following we will show how to evaluate few-body momentum distributions given by the final state of the algorithm above.', '1804.01505-1-33-4': 'Note that after measuring the [MATH] ancilla qubits of Sec. [REF] the main register will be left in a state [MATH] composed by a linear superposition of final states corresponding to energy transfer [MATH].', '1804.01505-1-33-5': 'Imagine we want now to compute exclusive 1 and 2-body momentum distributions [EQUATION] where [MATH] is the number operator for a state with momentum [MATH], spin [MATH] and isospin [MATH].', '1804.01505-1-33-6': 'We can define a unitary operator [MATH] (which is efficiently implementable) and run the following circuit with an ancilla qubit [EQUATION]', '1804.01505-1-33-7': 'By using the idempotence of [MATH] we find [EQUATION] and we can then extract the expectation value by estimating these probabilities.', '1804.01505-1-33-8': 'Note that we may use the same procedure with [MATH] to estimate [MATH] (and possibly higher body momentum distributions).', '1804.01505-1-33-9': 'We can get a better strategy by reusing the final state of circuit [REF] upon measuring the ancilla in [MATH] and running it again with [MATH] since the probabilities now will be [EQUATION]', '1804.01505-1-33-10': 'Note that [MATH] will in general be contaminated by final state interactions but we can access a better approximation to an asymptotic state by evolving it in time using [MATH].', '1804.01505-1-34-0': 'This measurement procedure will need to then be repeated a polynomial number of times for all the observables of interest.', '1804.01505-1-34-1': 'Given the expensive procedure needed to generate the final states a better strategy to estimate multiple observables per iteration may be needed for greater efficiency.', '1804.01505-1-34-2': "One option is using state reconstruction techniques developed in quantum tomography [CITATION] or devising strategies tailored to the particular system studied and it's encoding on the quantum computer.", '1804.01505-1-35-0': '# Conclusions', '1804.01505-1-36-0': 'We presented a complete quantum algorithm for calculating the linear response of a quantum system to external perturbations with controllable accuracy.', '1804.01505-1-36-1': 'This is achieved by probabilistically preparing the perturbed state (even though a deterministic preparation with polynomial cost is in general available) and then analyzing it by using the standard Phase Estimation Algorithm [CITATION].', '1804.01505-1-36-2': 'Our approach is efficient (scaling is polynomial in system size and required accuracy) and provides direct access to the final states resulting from the perturbation, a property that potentially makes it extremely valuable to the interpretation of ongoing and planned scattering experiments.'}

{'1804.01505-2-0-0': 'The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly-interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei.', '1804.01505-2-0-1': 'We present a general algorithm for quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.', '1804.01505-2-1-0': 'Quantum computers should enable dramatic new capabilities in simulating quantum many-body systems, particularly their dynamic properties [CITATION].', '1804.01505-2-1-1': 'Quantum dynamics is in general extremely difficult to treat on a classical computer except for a few special cases such as very low-energy scattering where spectral decomposition in finite volumes enable direct connections between spectra and phase shifts in the scattering of 2- or 3-clusters [CITATION] or very high-energy scattering that can be treated as nearly non-interacting final states, including y-scaling in neutron or electron scattering [CITATION] or inclusive deep inelastic scattering in QCD [CITATION].', '1804.01505-2-1-2': 'The general problem is essentially intractable because of quantum interference, the rapidly oscillating phases that arise in the relevant path integrals.', '1804.01505-2-2-0': 'Perhaps the simplest quantum dynamics problem is the dynamic linear response, framed as the response of a quantum system to a small perturbation.', '1804.01505-2-2-1': 'Examples are ubiquitous, including for example neutron scattering on materials, photon scattering in atomic systems, and electron and neutrino scattering from atomic nuclei.', '1804.01505-2-2-2': 'The response of the system can in principle tell us much about the structure of the system being probed as well as important properties of the dynamics.', '1804.01505-2-2-3': 'In the case of neutrinos scattered by nuclei it is also used to infer properties of the neutrino itself including masses, mixing angles, the mass hierarchy and CP violation in the neutrino sector (eg. [CITATION]).', '1804.01505-2-3-0': 'The ability to accurately calculate the dynamic response over a wide range of energy and momentum transfers, augmented by the possibility of determining specific features of the final states, would revolutionize our ability to extract information from many kinds of scattering experiments.', '1804.01505-2-3-1': 'Some information on quantum dynamics can be obtained using classical computers even for relatively large systems, typically by computing imaginary-time correlation functions [CITATION].', '1804.01505-2-3-2': 'Even for systems where the ground state or thermal ensembles can be simulated free of any sign problem, it is extremely difficult to invert these correlation functions to obtain the exact dynamic response.', '1804.01505-2-3-3': 'In this paper we discuss methods to determine the dynamic response on a quantum computer, as well as to detect important features of explicit final states that can be directly compared to experimental data.', '1804.01505-2-3-4': 'Our approach is particularly well suited for problems defined on a lattice, but these lattice methods can of course also be used to simulate systems in the continuum over a wide but finite range of energy and momenta, for example in lattice studies of cold atoms [CITATION] and nuclear systems [CITATION].', '1804.01505-2-3-5': 'Also we restrict ourselves to the response from the quantum ground state (T=0), generalizations to finite temperature are possible by preparing states in thermal equilibrium rather than the ground state [CITATION].', '1804.01505-2-4-0': 'We note that the similar problem of evaluating chemical reaction rates [CITATION] and time-dependent correlation functions [CITATION] have been already investigated in the quantum-chemistry literature.', '1804.01505-2-4-1': 'Our proposed algorithm improves on these earlier techniques in that our strategy is completely general, does not depend on simplifying assumptions on the excitation operator (for example, being able to diagonalize it as in [CITATION]) and requires only a polynomial number of measurements (instead of exponential like in [CITATION]).', '1804.01505-2-4-2': 'Also, working directly in frequency space allows us a direct access to the final states of a reaction which can be further analyzed.', '1804.01505-2-4-3': 'This is particularly important for neutrinos where the momentum and energy transfer are a priori unknown.', '1804.01505-2-5-0': 'Furthermore we are able to provide rigorous cost and error estimates of the computed dynamical properties.', '1804.01505-2-5-1': 'Available algorithms for evaluating energy spectra [CITATION] can in principle be adapted to compute response functions but they require resolution of individual excited states which grows exponentially in number for large systems.', '1804.01505-2-6-0': 'The paper is organized as follows, in Sec. [REF] we provide detailed definition of the Dynamical Response Function and describe the implementation of our method.', '1804.01505-2-6-1': 'In Sec. [REF] we provide an example of final state characterization by discussing the estimation of the one- and two-body momentum distribution and conclude in Sec. [REF].', '1804.01505-2-7-0': '# Method', '1804.01505-2-8-0': 'In the linear regime the response of a system of interacting particles due to a perturbative probe characterized by the excitation operator [MATH] can be fully characterized using the Dynamical Response Function, which can be expressed as [EQUATION] where [MATH] is the ground-state of the system with energy [MATH], [MATH] are the final states of the reaction with energies [MATH] and [MATH] is the energy transfer.', '1804.01505-2-8-1': "It is convenient to rescale the response function so that it's zero moment (the integral over frequencies) is [MATH]; this can be achieved by defining [EQUATION]", '1804.01505-2-8-2': 'The final normalization can be restored by either using the knowledge of one of the sum rules or by direct evaluation of the ground state expectation value [MATH].', '1804.01505-2-8-3': 'Understanding this, in the following we will drop the superscript [MATH].', '1804.01505-2-9-0': 'Our goal is to estimate the dynamical response function [MATH] with energy resolution [MATH] and a precision [MATH] with probability [MATH].', '1804.01505-2-9-1': 'We will indicate the difference between the largest eigenvalue of [MATH] and the ground state energy by: [MATH].', '1804.01505-2-9-2': 'Note that this quantity grows only polynomially with system size for most Hamiltonians of interest (see discussion below).', '1804.01505-2-10-0': 'In the following we will assume to have access to three black-box quantum procedures (oracles):', '1804.01505-2-11-0': 'Even though the oracle [MATH] may be impractical to implement for a general Hamiltonian, for most systems of interest many different algorithms are available in the literature ([CITATION]) and some have already be tested on simple nuclear systems [CITATION].', '1804.01505-2-11-1': 'Also, close to optimal strategies to implement the time-evolution operator for sparse Hamiltonians are known [CITATION] and for Hubbard-type Hamiltonians (like those derived within lattice-EFT [CITATION]) efficient implementations of Trotter steps with sub-linear circuit depth are available [CITATION].', '1804.01505-2-11-2': 'For the common case where [MATH] is a one-body operator the latter strategies can be used to implement [MATH] efficiently.', '1804.01505-2-12-0': 'Our scheme is composed of two quantum circuits', '1804.01505-2-13-0': 'For typical situations where the implementation of [MATH] requires considerable effort the success probability of the first routine can be amplified to [MATH] with additional [MATH] calls to the oracle [MATH].', '1804.01505-2-13-1': 'An alternative algorithm which removes the dependence of [MATH] on [MATH] but is more difficult to make deterministic is also presented in Sec. [REF].', '1804.01505-2-14-0': 'This whole circuit needs to be run a number of times given approximately by [EQUATION] independent of the target resolution [MATH].', '1804.01505-2-15-0': 'In summary, for a given choice of the excitation operator [MATH] our algorithm can be described by the following steps:', '1804.01505-2-16-0': 'iteration number less than [MATH] prepare the ground state using [MATH] run the first quantum algorithm (Sec. [REF]) algorithm succeeds we have prepared [MATH] run the second quantum algorithm (Sec. [REF]) store result for classical post-processing final state information needed measure final state (eg.', '1804.01505-2-16-1': 'Sec [REF])', '1804.01505-2-17-0': 'In the next sections we describe in detail the implementation of the two quantum routines introduced above.', '1804.01505-2-17-1': 'We also present examples obtained by classical simulation of a simple 2D fermionic system described by the Hubbard hamiltonian [EQUATION] where [MATH] indicates the nearest-neighbor lattice sites and [MATH] denotes the number operator.', '1804.01505-2-17-2': 'The results shown here were obtained for [MATH] "nucleons", [MATH] lattice sites and [MATH].', '1804.01505-2-17-3': 'These parameters are chosen to give a bound state considerably smaller than the lattice.', '1804.01505-2-18-0': '## State preparation algorithm', '1804.01505-2-19-0': 'The first problem we have to solve is the preparation of the state [MATH] given a quantum register initialized in the ground-state [MATH].', '1804.01505-2-19-1': "Let's start by adding an ancilla qubit and defining the unitary operator [EQUATION] where the Pauli [MATH] operator acts on the ancilla and the final matrix representation is on the basis spanned by the states [MATH] of the ancilla.", '1804.01505-2-19-2': 'Note that this unitary can be implemented efficiently with just [MATH] calls to a controlled version of the oracle [MATH] and additional [MATH] one-qubit gates.', '1804.01505-2-20-0': 'By initializing the ancilla register to [MATH], applying [MATH] and measuring the state [MATH] we have effectively produced [EQUATION] which differs from the wanted state by corrections of order [MATH].', '1804.01505-2-20-1': 'The error in the implementation of the unitary [MATH] needs to be at least of the same order, which means a simple single Trotter step will suffice.', '1804.01505-2-20-2': 'The state preparation has a success probability of [EQUATION]', '1804.01505-2-20-3': 'This approach for the application of a non-unitary transformation is similar in spirit to earlier work (see eg. [CITATION]) and it suffers from a possibly very low efficiency since we may need [MATH] trials to succeed.', '1804.01505-2-20-4': 'One option is to perform the algorithm at a few relatively large values of [MATH] and fit a quadratic function to extrapolate out the error from the final response function.', '1804.01505-2-20-5': 'This approach is however complicated if one is interested also in properties of the final states.', '1804.01505-2-20-6': 'A second approach, already proposed in [CITATION], is to repeat the application of the unitary [MATH] until success.', '1804.01505-2-20-7': 'This works because [MATH] is approximately the identity.', '1804.01505-2-20-8': 'In order to obtain a success probability [MATH] we will need [MATH] repetitions.', '1804.01505-2-20-9': "In addition, if the inverse [MATH] of the ground-state preparation circuit is available then it's possible to use Amplitude Amplification [CITATION] to gain a quadratic speedup over this .", '1804.01505-2-21-0': 'Note that by using the normalized state [MATH] we will compute the normalized response function Eq. [REF].', '1804.01505-2-21-1': 'If no sum-rules are known one can estimate the normalization constant by estimating the success probability Eq. [REF] at different values of [MATH] and extrapolating.', '1804.01505-2-22-0': 'Since the state preparation through the unitary [MATH] is only approximate, the parameter [MATH] would need to depend on the final target accuracy.', '1804.01505-2-22-1': 'As mentioned in the introduction an alternative scheme that avoids this problem by removing the error in Eq. [REF] can be obtained by representing the excitation operator [MATH] as a linear combination of [MATH] unitary matrices [EQUATION] which can be efficiently implemented employing additional [MATH] ancilla qubits using known techniques [CITATION].', '1804.01505-2-22-2': 'The success probability in this case is given by [EQUATION] which depending on the particular case may be larger than Eq. [REF].', '1804.01505-2-22-3': 'The main drawback of this approach is that Amplitude Amplification is the only process that can make the algorithm deterministic since upon failure the output state can in general be very different from the starting point.', '1804.01505-2-23-0': '## Response Function estimation', '1804.01505-2-24-0': 'We now present our strategy to obtain the response function trough the standard Phase Estimation Algorithm (PEA) [CITATION].', '1804.01505-2-24-1': 'It is convenient to shift and scale the original Hamiltonian: [EQUATION] so that we map the energy spectrum to [MATH].', '1804.01505-2-25-0': 'By direct calculation we see that the response function [MATH] obtained from [MATH] is related to the original one by [EQUATION] for a scaled frequency [MATH].', '1804.01505-2-26-0': 'Our goal is to estimate [MATH] efficiently.', '1804.01505-2-26-1': 'We do this by using PEA on an auxiliary register of [MATH] qubits with the evolution operators [EQUATION] for [MATH].', '1804.01505-2-26-2': 'The resulting circuit will have depth [MATH], where the first term comes from the inverse Quantum Fourier Transform [CITATION] and [MATH] is the gate count needed for a time evolution of [MATH] using the oracle [MATH].', '1804.01505-2-26-3': 'The resulting probability of measuring the [MATH] ancilla qubits in the binary representation of the integer [MATH] is (see eg. [CITATION] for more details) [EQUATION] where [MATH] is the well-known Fejer kernel from Fourier analysis (see eg. [CITATION]).', '1804.01505-2-26-4': 'The probability distribution [MATH] is a good approximation of [MATH] since this kernel can be seen as a representation of the delta function with width [MATH].', '1804.01505-2-26-5': 'Therefore if we require a frequency resolution [MATH] we will need [MATH] auxiliary qubits and a polynomial number of applications of the time evolution operator to obtain a sample from [MATH].', '1804.01505-2-27-0': 'As mentioned above, for most Hamiltonians of interest the energy gap [MATH] scales only polynomially with the size of the system.', '1804.01505-2-28-0': 'We now need to estimate [MATH] from [MATH] samples drawn from it.', '1804.01505-2-28-1': 'Since [MATH] is a discrete variable an efficient way of reconstructing the probability distribution is by producing an histogram [MATH] from the samples.', '1804.01505-2-28-2': "Using Hoeffding's inequality [CITATION] we find that [EQUATION] which implies in order to obtain a precision [MATH] with probability [MATH] we need approximately [EQUATION] independent samples.", '1804.01505-2-29-0': 'In Fig. [REF] we plot the approximate response [MATH] for the model Hamiltonian Eq. [REF] at three different values of [MATH] (6,8,12).', '1804.01505-2-29-1': 'By comparing with the exact result shown as black dots, we see that for [MATH] the effect of energy resolution is negligible but already with [MATH] we obtain a rather accurate estimate for [MATH].', '1804.01505-2-29-2': 'Even [MATH] reproduces important features of the response, which in experiments is convoluted with the detector resolution.', '1804.01505-2-29-3': 'The inset shows the convergence of the maximum error [EQUATION] as a function of the sample size [MATH].', '1804.01505-2-29-4': 'Response functions relevant for [MATH] and [MATH] scattering are typically smooth at high energy and hence require small [MATH] and short propagation times.', '1804.01505-2-30-0': 'Finally, in order to obtain a negligible bias from the state preparation we need the parameter [MATH] to scale as [EQUATION] for some constant [MATH].', '1804.01505-2-30-1': 'Note that the Hamiltonian evolution implemented in [MATH] has to have an error [MATH] to be negligible (luckily algorithms with only logarithmic dependence on [MATH] are known [CITATION]).', '1804.01505-2-31-0': 'This concludes the proof of the scalings [REF] and [REF].', '1804.01505-2-32-0': '# Final state measurements', '1804.01505-2-33-0': 'In electron- or neutrino-nuclear scattering experiments [CITATION] one would like to infer the probability [MATH] that the probe transferred energy-momentum [MATH] to the nucleus and simultaneously that the final state includes a nucleon (or neutron or proton) of momentum [MATH].', '1804.01505-2-33-1': 'More concretely this amounts to an inference procedure of the form [EQUATION] where [MATH] results from the experimental measure, [MATH] is the momentum distribution of the final states for a process with given [MATH] and [MATH].', '1804.01505-2-33-2': 'The prior probability [MATH] depends on the static response of the nucleus and the characteristic of the probe beam and can be updated given the other ones by a Bayesian procedure.', '1804.01505-2-33-3': 'The above section explains how to obtain [MATH] with a given accuracy and in the following we will show how to evaluate few-body momentum distributions given by the final state of the algorithm above.', '1804.01505-2-33-4': 'Note that after measuring the [MATH] ancilla qubits of Sec. [REF] the main register will be left in a state [MATH] composed by a linear superposition of final states corresponding to energy transfer [MATH].', '1804.01505-2-33-5': 'Imagine we want now to compute exclusive 1 and 2-body momentum distributions [EQUATION] where [MATH] is the number operator for a state with momentum [MATH], spin [MATH] and isospin [MATH].', '1804.01505-2-33-6': 'We can define a unitary operator [MATH] (which is efficiently implementable) and run the following circuit with an ancilla qubit [EQUATION]', '1804.01505-2-33-7': 'By using the idempotence of [MATH] we find [EQUATION] and we can then extract the expectation value by estimating these probabilities.', '1804.01505-2-33-8': 'Note that we may use the same procedure with [MATH] to estimate [MATH] (and possibly higher body momentum distributions).', '1804.01505-2-33-9': 'We can get a better strategy by reusing the final state of circuit [REF] upon measuring the ancilla in [MATH] and running it again with [MATH] since the probabilities now will be [EQUATION]', '1804.01505-2-33-10': 'Note that [MATH] will in general be contaminated by final state interactions but we can access a better approximation to an asymptotic state by evolving it in time using [MATH].', '1804.01505-2-34-0': 'This measurement procedure will need to then be repeated a polynomial number of times for all the observables of interest.', '1804.01505-2-34-1': 'Given the expensive procedure needed to generate the final states a better strategy to estimate multiple observables per iteration may be needed for greater efficiency.', '1804.01505-2-34-2': "One option is using state reconstruction techniques developed in quantum tomography [CITATION] or devising strategies tailored to the particular system studied and it's encoding on the quantum computer.", '1804.01505-2-35-0': '# Conclusions', '1804.01505-2-36-0': 'We presented a complete quantum algorithm for calculating the linear response of a quantum system to external perturbations with controllable accuracy.', '1804.01505-2-36-1': 'This is achieved by probabilistically preparing the perturbed state (even though a deterministic preparation with polynomial cost is in general available) and then analyzing it by using the standard Phase Estimation Algorithm [CITATION].', '1804.01505-2-36-2': 'Our approach is efficient (scaling is polynomial in system size and required accuracy) and provides direct access to the final states resulting from the perturbation, a property that potentially makes it extremely valuable to the interpretation of ongoing and planned scattering experiments.'}

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[]

[]

[]

[]

['1804.01505-1-10-0', '1804.01505-1-12-0', '1804.01505-1-15-0', '1804.01505-1-16-1', '1804.01505-1-31-0', '1804.01505-2-10-0', '1804.01505-2-12-0', '1804.01505-2-15-0', '1804.01505-2-16-1', '1804.01505-2-31-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1804.01505

1605.00633

{'1605.00633-1-0-0': 'The interaction of charged particles and photons with intense electromagnetic fields gives rise to multi-photon Compton and Breit-Wheeler processes.', '1605.00633-1-0-1': 'These are usually described in the framework of the external field approximation, where the electromagnetic field is assumed to have infinite energy.', '1605.00633-1-0-2': 'However, the multi-photon nature of these processes implies the absorption of a significant number of photons, which scales as the external field amplitude cubed.', '1605.00633-1-0-3': 'As a result, the interaction of a highly charged electron bunch with an intense laser pulse can lead to significant depletion of the laser pulse energy, thus rendering the external field approximation invalid.', '1605.00633-1-0-4': 'We provide relevant estimates for this depletion and find it to become important in the interaction between fields of amplitude [MATH] and electron bunches with charges of the order of nC.', '1605.00633-1-1-0': 'Introduction.', '1605.00633-1-1-1': 'The interaction of charged particles with ultra-intense electromagnetic (EM) pulses is the cornerstone of a newly emerging area of research, high intensity particle physics.', '1605.00633-1-1-2': 'It is located at the intersection of quantum electrodynamics (QED) and the theory of strong EM background fields.', '1605.00633-1-1-3': 'The latter significantly alter the physics of typical QED processes, leading to effects not encountered in perturbative quantum field theory [CITATION].', '1605.00633-1-1-4': 'Recently, there has been a surge of interest in these processes due to the rapid planning and realization of new laser facilities, which will be able to deliver EM pulses of unprecedented intensities to test the predictions of high intensity particle physics [CITATION].', '1605.00633-1-1-5': 'Moreover, the development of compact laser accelerators [CITATION] able to generate multi-GeV focused electron beams [CITATION] adds another component necessary to carry out these studies.', '1605.00633-1-2-0': 'Here, we will hence assume that the strong EM field is provided by an ultra-intense laser (pulse) with wave vector [MATH], central frequency [MATH] in the optical regime and electric field magnitude [MATH].', '1605.00633-1-2-1': 'The (nonlinear) interactions of this strong field with photons and charged particles are parametrized in terms of the following parameters : (i) the dimensionless amplitude of the EM vector potential, [MATH], a Lorentz and gauge invariant [CITATION] (ii) the QED critical field, [MATH] [CITATION], (iii) the strong field invariants [MATH] and [MATH] [CITATION].', '1605.00633-1-2-2': 'Here, [MATH] and [MATH] are electron charge and mass, [MATH] is the EM field tensor, while [MATH] and [MATH] denote the 4-momenta of electron and photon probing the laser.', '1605.00633-1-2-3': 'The parameter [MATH] is usually referred to as the classical nonlinearity parameter, since its physical meaning is the energy gain of an electron (in units of its rest energy) traversing a reduced wavelength, [MATH], of the field.', '1605.00633-1-2-4': 'For [MATH] the electron/positron motion in such a field becomes relativistic.', '1605.00633-1-2-5': 'The quantum parameter [MATH] characterizes a distinct feature of QED, the ability to produce new particles.', '1605.00633-1-2-6': 'This happens when an energy of [MATH] is delivered across an electron Compton wavelength, [MATH], which is precisely achieved by [MATH] [CITATION].', '1605.00633-1-2-7': 'The parameters [MATH] and [MATH] have been introduced to take into account the interaction of charged particles and photons with the strong EM field.', '1605.00633-1-2-8': 'For example, [MATH] is just the EM field strength in the electron rest frame in units of [MATH].', '1605.00633-1-2-9': 'Quantum effects become of crucial importance when [MATH] or [MATH].', '1605.00633-1-3-0': 'For large field amplitudes, [MATH], the interaction of electrons/positrons and photons with strong EM fields involves the absorption of a large number of photons from the field.', '1605.00633-1-3-1': 'Clearly, these correspond to an energy loss of the laser background field, which may or may not be negligible.', '1605.00633-1-3-2': 'Revisiting the seminal results of [CITATION] on multi-photon Compton and Breit-Wheeler processes [CITATION], we have been led to the conclusion that there is a parameter range, for which effects of multi-photon absorption can no longer be neglected.', '1605.00633-1-3-3': 'These processes have been receiving a lot of interest recently [CITATION], with a focus on the final particle states (a frequency shifted photon or electron positron pairs).', '1605.00633-1-4-0': 'In this letter we want to change perspective and study in detail the dependence of nonlinear Compton scattering on the initial multi-photon states, that is on the number of laser photons absorbed.', '1605.00633-1-4-1': 'This will allow us to establish a threshold for the validity of the external field approximation and discuss some immediate consequences.', '1605.00633-1-4-2': 'These findings should have a direct impact on the analysis of QED backreaction on the classical EM field [CITATION].', '1605.00633-1-4-3': 'It should also be of great importance for the study of EM avalanches [CITATION], since background depletion will significantly alter the energy partitioning of the processes.', '1605.00633-1-4-4': 'Recall that an EM avalanche is formed when Compton and Breit-Wheeler processes occur subsequently in an EM field of two colliding laser pulses with typical intensities of about [MATH] W/cm[MATH].', '1605.00633-1-4-5': 'This gives rise to an exponential growth of the number of emitted electrons, positrons, and photons due to the fact that the charged particles are being constantly accelerated by the EM field.', '1605.00633-1-5-0': 'Classical Depletion.', '1605.00633-1-5-1': 'The external field approximation should be valid as long as the number of photons absorbed from the laser, [MATH], is small compared to the total number [MATH] of photons in the pulse, which we take to be strongly focussed to volume [MATH].', '1605.00633-1-5-2': 'A natural criterion for depletion is then provided by the equality [MATH].', '1605.00633-1-5-3': 'The number of laser photons is proportional to intensity or field strength squared, [MATH].', '1605.00633-1-5-4': 'Here [MATH] denotes the fine structure constant.', '1605.00633-1-5-5': 'The number of absorbed photons is [MATH], where [MATH] is the number of electrons in the bunch and [MATH] is the energy loss upon radiating power [MATH] per laser period [MATH].', '1605.00633-1-5-6': 'This power, a Lorentz invariant, can be estimated classically by making an analogy with synchrotron radiation [CITATION].', '1605.00633-1-5-7': 'To this end we go to a boosted frame, where the electron is on average at rest.', '1605.00633-1-5-8': "If the laser is circularly polarized, the electron moves on a circle like in a synchrotron, and we can use Larmor's formula, [EQUATION]", '1605.00633-1-5-9': "Here [MATH] is the electron 4-velocity, and we have introduced [MATH], the laser frequency 'seen' in the average rest frame (ARF) by an electron, which, in the lab, has relativistic parameters [MATH] and [MATH].", '1605.00633-1-5-10': 'The number of absorbed photons per laser period [MATH] is then [EQUATION]', '1605.00633-1-5-11': 'Taking into account the fact that the characteristic frequency of the radiation emitted by an electron in this frame is [MATH] [CITATION], we obtain for the number of photons emitted per laser period [MATH].', '1605.00633-1-5-12': 'Thus, in order for the electron to emit one high frequency ([MATH]) photon it needs to absorb [MATH] photons from the EM field.', '1605.00633-1-5-13': 'Hence, we obtain the important result that the number [MATH] of absorbed photons scales as [MATH] for large [MATH].', '1605.00633-1-5-14': 'Equating [MATH], we see that depletion requires [EQUATION]', '1605.00633-1-5-15': 'For an electron bunch containing a charge of 1 nC, a laser with [MATH] is needed.', '1605.00633-1-5-16': 'Even classically, one expects that [MATH] values of this magnitude imply significant radiation reaction with ensuing changes of the particle trajectories [CITATION].', '1605.00633-1-5-17': 'Rather than including this classically, we utilize the more general quantum theory straight away.', '1605.00633-1-6-0': 'Quantum Depletion.', '1605.00633-1-6-1': 'Let us next take quantum effects into account.', '1605.00633-1-6-2': 'According to [CITATION] the number of absorbed photons scales as in the classical case [MATH] for [MATH].', '1605.00633-1-6-3': 'Then the number of absorbed photons per period is [EQUATION] where [MATH] is the radiation length of the electron in a strong EM field [CITATION].', '1605.00633-1-6-4': 'Thus, on average, there is one photon emission when the electron travels a distance [MATH].', '1605.00633-1-6-5': 'Note that the classical behaviour ([REF]) is recovered in the limit [MATH] using [MATH].', '1605.00633-1-6-6': 'In the deep quantum regime, [MATH], we only need the asymptotic expression [MATH] [CITATION] and obtain the following quantum formula for the threshold of depletion: [EQUATION] which replaces ([REF]).', '1605.00633-1-6-7': 'Again, for an electron beam of 1 nC a laser with intensity [MATH] is required.', '1605.00633-1-6-8': 'Intensities of this magnitude should become reality in the near future [CITATION].', '1605.00633-1-6-9': 'The critical value of [MATH] depends weakly on the initial electron energy ([MATH]) as shown in Fig. 1, where [MATH] is shown for different values of [MATH].', '1605.00633-1-6-10': 'Thus, taking quantum effects into account increases the critical value of [MATH], needed to deplete the laser pulse for a given value of initial electron momentum.', '1605.00633-1-6-11': 'This is due to the fact that the classical approach overestimates the amount of radiation emitted by an electron.', '1605.00633-1-6-12': 'In any case, both classical and quantum estimates, ([REF]) and ([REF]) respectively, show that, when a sufficiently charged electron bunch collides with an intense laser pulse, depletion of the laser pulse can become significant, with the originally strong EM field turning weak.', '1605.00633-1-6-13': 'The required number of electrons is quite typical for an EM avalanche [CITATION], where an intense laser produces a copious amount of high energy photons and subsequently electron-positron pairs.', '1605.00633-1-7-0': 'This provides sufficient motivation to investigate depletion in some quantitative detail.', '1605.00633-1-7-1': 'To this end, we introduce particle emission probabilities [MATH], which are differential in the number of photons [MATH] absorbed from the laser field.', '1605.00633-1-7-2': 'The average amount of energy [MATH] drawn from the laser field in a single photon emission or pair production is then given in terms of expectation values, [MATH], with the average number of absorbed laser photons given by [EQUATION] with normalization integral [MATH].', '1605.00633-1-8-0': 'Compton Process.', '1605.00633-1-8-1': 'In a monochromatic plane wave laser field, taken to be circularly polarized for simplicity, the variable [MATH] is discrete and describes the emission of higher harmonics due to absorption of [MATH] laser photons, [MATH].', '1605.00633-1-8-2': 'Energy momentum conservation implies [MATH] with [MATH] and [MATH].', '1605.00633-1-8-3': 'The partial probabilities or rates (i.e. events per time) are [CITATION]', '1605.00633-1-9-0': '[EQUATION] where the various arguments are [MATH], [MATH], [MATH], [MATH].', '1605.00633-1-9-1': 'The total probability for Compton photon emission is the sum over all harmonics: [MATH].', '1605.00633-1-9-2': 'For large values of [MATH] the number of harmonics that contribute to the sum grows as [MATH] and the photon number [MATH] can be considered as quasi-continuous so that the sum turns into an integral over [MATH].', '1605.00633-1-9-3': 'In this regime the formation length of the Compton process scales like [MATH] [CITATION], and the laser field can be approximated as a locally constant crossed field.', '1605.00633-1-9-4': 'Formally, this is achieved by employing the Watson representation of the Bessel functions: [MATH], [MATH].', '1605.00633-1-9-5': 'Asymptotically, when [MATH], the probability to absorb [MATH] photons from the laser while emitting a single high-frequency photon of momentum [MATH] is found to be [EQUATION] where [MATH].', '1605.00633-1-9-6': 'The dependence of the distribution [MATH] on [MATH] is shown in Fig. 2a.', '1605.00633-1-9-7': 'One can see a maximum corresponding to the most probable number of photons being absorbed.', '1605.00633-1-9-8': 'This maximum shifts towards lower values of [MATH] as the initial electron energy is increased.', '1605.00633-1-9-9': 'Moreover, in Fig. 2b we display the dependence of [MATH] on the parameter [MATH], which indeed demonstrates an increase of the number of absorbed photons with the EM field strength, but indicates a dependence different from [MATH] - the most probable number of absorbed photons also depends on the parameter [MATH].', '1605.00633-1-9-10': 'A fit to our numerical results yields the parametrization [MATH], which is represented by the black dashed curves in Fig. 2b.', '1605.00633-1-9-11': 'The number of electrons needed to deplete the laser should equal [MATH].', '1605.00633-1-9-12': 'This is 10-100 times larger than the simple estimate ([REF]).', '1605.00633-1-9-13': 'Thus, using the results from Fig. 2b for [MATH], the number of electrons needed to deplete the laser is [MATH], which is still within reach of EM avalanches [CITATION].', '1605.00633-1-10-0': 'The probabilities [MATH] determine the number distribution of absorbed photons.', '1605.00633-1-10-1': 'However, we are also interested in the spectral distribution [MATH] of the scattered photon longitudinal momentum (recall [MATH]).', '1605.00633-1-10-2': 'The two distributions are obviously related via the chain rule, [EQUATION] but we do not know the functional relation [MATH].', '1605.00633-1-10-3': 'An approximate way to determine the latter is as follows.', '1605.00633-1-10-4': 'From the [MATH]-integral in [REF] we see that the integrand is sharply peaked at [MATH].', '1605.00633-1-10-5': 'Using energy momentum conservation we can solve [MATH] for [MATH] with the result [EQUATION] valid for [MATH].', '1605.00633-1-10-6': 'Thus, when a Compton photon with a certain value of [MATH] is emitted, the number of laser photons drawn from the laser field approximately equals [MATH] as given in ([REF]).', '1605.00633-1-11-0': 'Let us use this result to examine how the most probable emission angle depends on the longitudinal photon momentum in the case of electrons colliding head-on with a laser pulse.', '1605.00633-1-11-1': 'To this end, we follow [CITATION] and assume quasi momentum conservation, [MATH] with quasi momentum [MATH] and analogously for [MATH], such that [MATH], the effective mass squared [CITATION].', '1605.00633-1-11-2': 'Let us write the scattered photon momentum as [MATH] where [MATH].', '1605.00633-1-11-3': 'We can then find [MATH] from quasi-momentum conservation.', '1605.00633-1-11-4': 'Assuming a head-on collision of electrons and laser ([MATH]) the following answer is obtained: [EQUATION]', '1605.00633-1-11-5': 'This identity is manifestly invariant with respect to boosts collinear with the laser direction [MATH].', '1605.00633-1-11-6': 'It defines an ellipse in the [MATH] plane for given values of [MATH], [MATH] and [MATH] (see Fig. 3).', '1605.00633-1-11-7': 'However, the differential probability [MATH] has a sharp maximum at [MATH] defined by ([REF]).', '1605.00633-1-11-8': 'Thus, only one point from the ellipse ([REF]) effectively contributes to the probability.', '1605.00633-1-11-9': 'It can be obtained by rewriting ([REF]) as [MATH].', '1605.00633-1-11-10': 'Plugging this into ([REF]) yields the surprisingly simple result [EQUATION] when [MATH], which implies that the transverse scattered momentum, [MATH], grows linearly with the light-front component [MATH].', '1605.00633-1-11-11': 'Recall that for a head-on collision, [MATH], where [MATH] is the rapidity, so that ([REF]) reduces to [EQUATION]', '1605.00633-1-11-12': 'Interestingly, this result is independent of the laser frequency, [MATH].', '1605.00633-1-11-13': 'Note also that quasi-momentum conservation implies that [MATH], the change in the electron light-front momentum component.', '1605.00633-1-11-14': 'This momentum transfer is a purely quantum effect as [MATH] upon reinstating [MATH], an observation consistent with the fact that [MATH] is conserved classically.', '1605.00633-1-12-0': 'Let us define the scattering or emission angle in the usual manner, [MATH].', '1605.00633-1-12-1': 'Denoting by [MATH] the emission angle obeying [MATH] [CITATION], ([REF]) can be recast in the following form, [EQUATION] which is indeed consistent with the findings of [CITATION]: As long as [MATH], the photons are predominantly emitted in the forward direction, with [MATH].', '1605.00633-1-12-2': 'However, as [MATH] increases, significant photon emission takes place in the perpendicular direction.', '1605.00633-1-12-3': 'This can be understood classically, in particular in the ARF where [MATH] so that [MATH] as it must be for circular (synchrotron) motion in the transverse plane.', '1605.00633-1-13-0': 'In order to visualize the dependence of the emission probability on the photon momentum and the number of absorbed photons we present Fig. 3.', '1605.00633-1-13-1': 'It shows the [MATH] plane together with two series of curves determined by momentum conservation ([REF]) and parameter choices [MATH] and [MATH] for fixed [MATH] and different values of [MATH].', '1605.00633-1-13-2': 'On top we have added the distribution of [MATH], which has a narrow peak corresponding to the most probable value of [MATH] and the emission angle [MATH] from ([REF]).', '1605.00633-1-13-3': 'One can show that this emission angle can be obtained classically through [MATH], the appropriate ratio of the rms values of the classical electron momentum components in the laser field [MATH], [MATH].', '1605.00633-1-13-4': 'As these are an important input for PIC codes, one could use this agreement as an independent check of their validity, in particular of the semi-classical approximation employed.', '1605.00633-1-14-0': 'Breit-Wheeler Process.', '1605.00633-1-14-1': 'For the Breit-Wheeler process, [MATH], the differential probabilities corresponding to [REF] can be found from [CITATION].', '1605.00633-1-14-2': 'For reasons of brevity we refrain from displaying them here and just recall that there is now a threshold in photon number [MATH].', '1605.00633-1-14-3': 'As before, we can derive a relation between the laser photon number [MATH] and the normalized longitudinal momentum of the produced positron, [EQUATION]', '1605.00633-1-14-4': 'Using momentum conservation and ([REF]), an analysis completely analogous to that of nonlinear Compton scattering can be performed for the Breit-Wheeler process as well.', '1605.00633-1-15-0': 'Conclusion.', '1605.00633-1-15-1': 'In this letter we have reconsidered the multi-photon Compton and Breit-Wheeler processes in strong EM fields, focussing on the energy loss of the laser due to absorption, which transforms the initially strong fields into weak ones.', '1605.00633-1-15-2': 'We found this to happen if the laser pulses are sufficiently intense ([MATH]) and interact with electron beams containing a charge of the order of tens of nC.', '1605.00633-1-15-3': 'In order to have actual depletion, a lower bound for the number of electrons in a volume of a laser wavelength cubed needs to be exceeded, namely [MATH].', '1605.00633-1-15-4': 'It is expected that this threshold will be overcome in the case of EM avalanches.', '1605.00633-1-15-5': 'As a consequence, laser depletion will not just be due to the creation of [MATH] pairs as considered previously, but must also be taken into account in laser photon absorption.', '1605.00633-1-16-0': 'We have further analyzed the photon emission rates differential in multi-photon number [MATH] and discovered that they strongly peak near the value [MATH].', '1605.00633-1-16-1': 'This value of [MATH] determines the direction of the photon emission relative to the initial electron momentum direction in terms of an emission or scattering angle, [MATH], which ranges between [MATH] (forward scattering, [MATH]) and [MATH] (back scattering, [MATH]).', '1605.00633-1-16-2': 'The latter will dominate in the EM avalanche regime, i.e. in colliding laser pulses or during interactions of laser pulses with solid density foils or plasmas of near-critical density.', '1605.00633-1-16-3': 'The classical interpretation of the emission angle [MATH] in terms of averages over trajectories should yield a new test of the PIC codes currently in use.', '1605.00633-1-17-0': 'We acknowledge support from the Office of Science of the US DOE under Contract No. DE-AC02-05CH11231.', '1605.00633-1-17-1': 'MM was supported by the Swedish Research Council grants 2012-5644 and 2013-4248.', '1605.00633-1-17-2': 'MM would like to thank Anton Ilderton for fruitful discussions.', '1605.00633-1-17-3': 'The authors acknowledge the hospitality of the Kavli Institute for Theoretical Physics (KITP), where this research was initiated during the Frontiers of Intense Laser Physics Program and so was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.'}

{'1605.00633-2-0-0': 'The interaction of charged particles and photons with intense electromagnetic fields gives rise to multi-photon Compton and Breit-Wheeler processes.', '1605.00633-2-0-1': 'These are usually described in the framework of the external field approximation, where the electromagnetic field is assumed to have infinite energy.', '1605.00633-2-0-2': 'However, the multi-photon nature of these processes implies the absorption of a significant number of photons, which scales as the external field amplitude cubed.', '1605.00633-2-0-3': 'As a result, the interaction of a highly charged electron bunch with an intense laser pulse can lead to significant depletion of the laser pulse energy, thus rendering the external field approximation invalid.', '1605.00633-2-0-4': 'We provide relevant estimates for this depletion and find it to become important in the interaction between fields of amplitude [MATH] and electron bunches with charges of the order of 10 nC.', '1605.00633-2-1-0': 'The interaction of charged particles with ultra-intense electromagnetic (EM) pulses is the cornerstone of a newly emerging area of research, high intensity particle physics, located at the intersection of quantum electrodynamics (QED) and the theory of strong EM background fields.', '1605.00633-2-1-1': 'The latter significantly alter the physics of typical QED processes, leading to effects not encountered in perturbative quantum field theory [CITATION].', '1605.00633-2-1-2': 'Recently, there has been a surge of interest in these processes due to the planning and realization of new laser facilities, which will be able to deliver EM pulses of unprecedented intensities to test the predictions of high intensity particle physics [CITATION].', '1605.00633-2-1-3': 'Moreover, the development of compact multi-GeV laser electron accelerators [CITATION] adds another component necessary to carry out these studies.', '1605.00633-2-2-0': 'Here, we will assume that the strong EM field is provided by an ultra-intense laser (pulse) with wave vector [MATH], central frequency [MATH] in the optical regime and electric field magnitude [MATH].', '1605.00633-2-2-1': 'The interactions of this strong field with photons and charged particles are parametrized in terms of the following parameters: (i) the (Lorentz and gauge invariant [CITATION]) dimensionless amplitude of the EM vector potential, [MATH], (ii) the QED critical field, [MATH] [CITATION], (iii) the strong field invariants [MATH] and [MATH] [CITATION].', '1605.00633-2-2-2': 'Here, [MATH] and [MATH] are electron charge and mass, [MATH] is the EM field tensor, while [MATH] and [MATH] denote the 4-momenta of electron and photon probing the laser.', '1605.00633-2-2-3': 'The parameter [MATH] is usually referred to as the classical nonlinearity parameter, since its physical meaning is the energy gain of an electron (in units [MATH]) traversing a reduced wavelength, [MATH], of the field.', '1605.00633-2-2-4': 'For [MATH] the electron/positron motion in such a field becomes relativistic.', '1605.00633-2-2-5': 'The parameter [MATH] characterizes a distinct feature of QED, the ability to produce new particles from vacuum.', '1605.00633-2-2-6': 'This happens when an energy of [MATH] is delivered across an electron Compton wavelength, [MATH], which is precisely achieved by [MATH] [CITATION].', '1605.00633-2-2-7': 'The parameters [MATH] and [MATH] characterizes the interaction of charged particles and photons with the strong EM field.', '1605.00633-2-2-8': 'For example, [MATH] is the EM field strength in the electron rest frame in units of [MATH].', '1605.00633-2-2-9': 'Quantum effects become of crucial importance when [MATH] or [MATH].', '1605.00633-2-3-0': 'For large field amplitudes, [MATH], the interaction of electrons/positrons and photons with strong EM fields involves the absorption of a large number of photons from the field.', '1605.00633-2-3-1': 'Clearly, these correspond to an energy loss of the laser background field, which may or may not be negligible.', '1605.00633-2-3-2': 'Revisiting the results on multi-photon Compton and Breit-Wheeler processes [CITATION], we find that there is indeed a parameter range, for which depletion of the laser becomes substantial.', '1605.00633-2-3-3': 'The processes in question have recently received a lot of interest [CITATION], albeit with a focus on the final states (a frequency shifted photon or electron positron pairs).', '1605.00633-2-4-0': 'In this letter we want to change perspective and study in detail the dependence of nonlinear Compton scattering on the initial multi-photon states, that is on the number of laser photons absorbed.', '1605.00633-2-4-1': 'This will allow us to establish a threshold for the validity of the external field approximation and discuss some immediate consequences.', '1605.00633-2-4-2': 'These findings should have a direct impact on the analysis of QED backreaction on the classical EM field [CITATION].', '1605.00633-2-4-3': 'It should also be of great importance for the study of EM avalanches [CITATION], since background depletion will significantly alter the energy partitioning of the processes.', '1605.00633-2-4-4': 'An avalanche is formed when Compton and Breit-Wheeler processes occur subsequently in an EM field of sufficiently high intensity, resulting in an exponential growth of the number of emitted particles.', '1605.00633-2-5-0': 'The external field approximation is valid when the number of photons absorbed from the laser, [MATH], is small compared to the total number [MATH] of photons in the pulse, which we take to be focussed to volume [MATH].', '1605.00633-2-5-1': 'A natural criterion for depletion is then provided by the equality [MATH].', '1605.00633-2-5-2': 'The number of laser photons is proportional to intensity or field strength squared, [MATH].', '1605.00633-2-5-3': 'Here [MATH] denotes the fine structure constant.', '1605.00633-2-5-4': 'The number of absorbed photons is [MATH], where [MATH] is the number of electrons in the bunch and [MATH] is the energy loss upon radiating power [MATH] per laser period [MATH].', '1605.00633-2-5-5': 'This power, a Lorentz invariant, can be estimated classically by making an analogy with synchrotron radiation [CITATION].', '1605.00633-2-5-6': 'In consequence, we will be able to estimate the number of photons absorbed from the field, the characteristic energy of an emitted photon and the angle of emission, implying a rather complete characterization of the processes.', '1605.00633-2-5-7': 'To this end we go to a boosted frame, where the electron is on average at rest.', '1605.00633-2-5-8': 'If the laser is circularly polarized, the electron moves on a circle like in a synchrotron with 4-velocity [MATH] where [MATH] characterizes the average rest frame (ARF).', '1605.00633-2-5-9': "Using Larmor's formula, the radiated power becomes [MATH].", '1605.00633-2-5-10': 'The boost to the ARF may be realized by choosing the initial electron momentum, [MATH], such that its light-front component equals [MATH], with [MATH] denoting the intensity-dependent effective mass [CITATION].', '1605.00633-2-5-11': 'At high energy, [MATH], the radiation is emitted in the plane of electron motion, which in the ARF is perpendicular to the laser axis.', '1605.00633-2-5-12': 'In the lab frame this transforms into an emission angle [EQUATION] determined by the ratio of longitudinal and transverse momenta in the ARF.', '1605.00633-2-5-13': 'For [MATH] the emission angle is [MATH], hence small, while for [MATH] there is significant emission in the transverse direction.', '1605.00633-2-5-14': 'In the ARF, [MATH].', '1605.00633-2-5-15': 'The number of absorbed photons per laser period [MATH] is then [EQUATION]', '1605.00633-2-5-16': 'So for [MATH], the radiated power, hence the number of absorbed photons per laser cycle, increases like [MATH].', '1605.00633-2-5-17': 'From synchrotron radiation it is known that the power radiated into the [MATH]-th harmonic asymptotically scales like [MATH] [CITATION], so that the total power is [MATH].', '1605.00633-2-5-18': 'We thus obtain the important result that the typical number [MATH] of laser photons, absorbed to yield emission of a single high-energy photon, scales like [MATH].', '1605.00633-2-6-0': 'Turning back to the question of beam depletion, we equate [MATH] to see that depletion requires [EQUATION]', '1605.00633-2-6-1': 'For an electron bunch containing a charge of 1 nC, a laser with [MATH] is needed.', '1605.00633-2-6-2': 'For such values of [MATH] the energy [MATH] of the emitted photons is of the order of the electron energy gain per laser period, and the emission angle significantly deviates from [MATH].', '1605.00633-2-6-3': 'Thus, in this case, one expects not just significant radiation reaction with ensuing changes of the particle trajectories [CITATION] but also strong recoil of the electron momentum.', '1605.00633-2-6-4': 'These features are best described in quantum theory to which we now turn.', '1605.00633-2-7-0': 'First, we estimate the depletion threshold by taking into account the discrete nature of photon emission.', '1605.00633-2-7-1': 'While the average number of absorbed photons, [MATH], still follows the classical scaling law [MATH] for [MATH] [CITATION], the classical formula ([REF]) is replaced by [EQUATION]', '1605.00633-2-7-2': 'Here, [MATH] is the radiation length of the electron in a strong EM field [CITATION] so that, on average, there is one photon emission per distance [MATH].', '1605.00633-2-7-3': 'The classical behavior ([REF]) is recovered in the limit [MATH] where [MATH].', '1605.00633-2-7-4': 'In the deep quantum regime, [MATH], we employ the asymptotic expression [MATH] [CITATION] to obtain a quantum formula for the threshold of depletion, [EQUATION] which supersedes ([REF]).', '1605.00633-2-7-5': 'Again, for an electron beam of 1 nC and [MATH], a laser with intensity [MATH] is required.', '1605.00633-2-7-6': 'Intensities of this magnitude should become reality in the near future [CITATION].', '1605.00633-2-7-7': 'The critical value of [MATH] depends weakly on the initial electron energy ([MATH]) as shown in Fig. 1, where [MATH] is shown for different values of [MATH].', '1605.00633-2-7-8': 'Thus, taking quantum effects into account increases the critical value of [MATH] needed to deplete the laser pulse for a given value of initial electron momentum.', '1605.00633-2-7-9': 'We note that, for [MATH], the depletion of the electron beam energy is quite strong [CITATION].', '1605.00633-2-7-10': 'This corresponds to the threshold for depletion of the laser beam going down from ([REF]) to ([REF]), as illustrated in Fig. 1.', '1605.00633-2-7-11': 'Both ([REF]) and ([REF]) show that, when a sufficiently charged electron bunch collides with an intense laser pulse, depletion of the laser pulse can become significant, with the originally strong EM field turning weak.', '1605.00633-2-7-12': 'The required number of electrons is quite typical for an EM avalanche [CITATION], where an intense laser produces a copious amount of high energy photons and subsequently electron-positron pairs.', '1605.00633-2-7-13': 'As the required electron densities are quite substantial, we briefly address the issue of coherence effects.', '1605.00633-2-7-14': 'To this end we note that the inter-electronic distance [MATH] remains much larger than the emitted photon wavelength [MATH].', '1605.00633-2-7-15': 'As a result, [MATH] (see Fig. 1), a well established criterion for hard photon emission to be incoherent [CITATION].', '1605.00633-2-8-0': 'Second, we refine the depletion threshold estimate by calculating the quantum corrections to the average number of absorbed photons taking into account the probabilistic nature of photon emission.', '1605.00633-2-8-1': 'In line with the current understanding of high-intensity laser matter interactions in the quantum regime, we model photon emission as succession of incoherent one-photon events [CITATION].', '1605.00633-2-8-2': 'There will be regimes where this assumption becomes challenged, for instance when extreme field strengths are reached such that [MATH].', '1605.00633-2-8-3': 'In this case, higher-order diagrams such as self-energy corrections [CITATION] and coherent multi-photon emission [CITATION] can no longer be neglected.', '1605.00633-2-8-4': 'A detailed investigation of higher-order effects is clearly beyond the scope of the present study, but we can at least state that [MATH] for our parameter range.', '1605.00633-2-8-5': 'It is thus sufficient to introduce one-particle emission probabilities [MATH], which are differential in the number of photons [MATH] absorbed from the laser field.', '1605.00633-2-8-6': 'The average amount of energy [MATH] drawn from the laser field in a single photon emission or pair production is then [MATH], with the average number of absorbed laser photons given by the expectation value [MATH] with with normalization integral [MATH].', '1605.00633-2-9-0': 'In a monochromatic plane wave laser field, taken to be circularly polarized for simplicity, the variable [MATH] is discrete and describes the emission of higher harmonics due to absorption of [MATH] laser photons.', '1605.00633-2-9-1': 'Introducing the usual quasi momentum [MATH] (and analogously for [MATH], whence [MATH]), kinematics become encoded in quasi momentum conservation, [MATH].', '1605.00633-2-9-2': 'The partial probabilities (per unit time), [MATH], were calculated long ago [CITATION] and give the total probability for Compton photon emission when summed over all harmonics: [MATH].', '1605.00633-2-9-3': 'For large values of [MATH], the number [MATH] of harmonics contributing grows like [MATH], hence can be assumed quasi-continuous.', '1605.00633-2-9-4': 'The sum may thus be replaced by an integral over [MATH] with integrand [CITATION]', '1605.00633-2-10-0': '[EQUATION]', '1605.00633-2-11-0': '[MATH] and [MATH] denote the Airy function and its derivative, their argument being [MATH], with [MATH], [MATH] and [MATH].', '1605.00633-2-11-1': 'The dependence of [MATH] on [MATH] is shown in Fig. 2b.', '1605.00633-2-11-2': 'The maximum corresponding to the most probable number of absorbed photons shifts towards lower values of [MATH] with increase of initial electron energy.', '1605.00633-2-11-3': 'In Fig. 2a the dependence of [MATH] on the parameter [MATH] shows an increase of the number of absorbed photons with the EM field strength, but indicates a dependence different from the classical behavior, [MATH]: the most probable number of absorbed photons also depends on the parameter [MATH] as given by the fit [MATH].', '1605.00633-2-11-4': 'Using [MATH], the threshold for depletion becomes [MATH].', '1605.00633-2-11-5': 'For instance, when [MATH], we find a value of [MATH], cf. Fig. 1 (right), which is larger than [MATH] predicted by the simple estimate ([REF]), but still within reach of EM avalanches [CITATION].', '1605.00633-2-12-0': 'We note that for classical synchrotron emission it is straightforward to relate radiated to absorbed power, because of the continuity of emission.', '1605.00633-2-12-1': 'In the quantum case a typical interaction of an electron beam with an intense laser pulse proceeds via multiple emissions, each of them potentially resulting in a significant change of the electron momentum.', '1605.00633-2-12-2': 'To characterize such interactions one uses simulation codes with QED modules, which take into account multi-photon Compton and Breit-Wheeler processes.', '1605.00633-2-12-3': 'For these codes to be used for depletion calculations, each Compton process needs to be characterized by photon energy, angle of emission and the number of absorbed photons.', '1605.00633-2-12-4': 'Furthermore, in numerical (QED-PIC) simulations of multi-stage emission processes, which lead to the formation of avalanches/cascades, photon and electron emission angles strongly determine the probability of the subsequent pair production or photon emission process, respectively [CITATION].', '1605.00633-2-12-5': 'We hence proceed by calculating these quantities.', '1605.00633-2-13-0': 'The probabilities [MATH] determine the number distribution of photons absorbed from the laser field in a single high-frequency photon emission.', '1605.00633-2-13-1': 'In what follows, we relate [MATH] to the distribution [MATH] of the scattered photon longitudinal momentum ([MATH]), which determines the intensity of the emitted high-energy photon radiation, via the chain rule: [EQUATION] where the functional relation [MATH] is unknown.', '1605.00633-2-13-2': 'From the [MATH]-integral in [REF] we see that the integrand is sharply peaked at [MATH].', '1605.00633-2-13-3': 'Using energy momentum conservation we can solve [MATH] for [MATH] with the result [EQUATION]', '1605.00633-2-13-4': 'This is valid for [MATH] and reproduces the leading order of the related result (18) in [CITATION].', '1605.00633-2-13-5': 'A direct numerical determination of [MATH] from ([REF]) shows excellent agreement with [REF] for the most important range of [MATH] (but deviates for [MATH]).', '1605.00633-2-13-6': 'Thus, when a Compton photon with a given value of [MATH] is emitted, the number of laser photons drawn from the laser field can safely be estimated using [REF] within the model of one-photon incoherent emission.', '1605.00633-2-14-0': 'To further illustrate the power of the result [REF], we employ it to determine the most probable emission angle without referring to an angular probability distribution.', '1605.00633-2-14-1': 'Let us write the scattered photon momentum as [MATH] where [MATH].', '1605.00633-2-14-2': 'We can then find [MATH] from quasi-momentum conservation.', '1605.00633-2-14-3': 'Assuming a head-on collision of electrons and laser ([MATH]) the following answer is obtained: [EQUATION]', '1605.00633-2-14-4': 'This identity is manifestly invariant with respect to boosts collinear with the laser direction [MATH].', '1605.00633-2-14-5': 'It defines an ellipse in the [MATH] plane for given values of [MATH], [MATH] and [MATH], see Fig. 3.', '1605.00633-2-14-6': 'Plugging ([REF]) into ([REF]) yields the tangent of the most probable photon emission angle, [EQUATION] where [MATH].', '1605.00633-2-14-7': 'This coincides with the classical emission angle ([REF]) and is indeed consistent with the findings of [CITATION]: As long as [MATH], the photons are predominantly emitted in the forward direction, with [MATH].', '1605.00633-2-14-8': 'However, as [MATH] increases, significant photon emission takes place in the perpendicular direction.', '1605.00633-2-14-9': 'This can be understood classically, in particular in the ARF where [MATH]), so that [MATH] as required for circular (synchrotron) motion in the transverse plane as well as by [REF].', '1605.00633-2-14-10': 'Equivalently, this follows from the classical equation of motion by calculating [MATH], the ratio of the rms values of the classical electron momentum components in the laser field [MATH], [MATH].', '1605.00633-2-15-0': 'Going back to Fig. 3 we see that the distribution of emitted photons is essentially supported on a straight line, [MATH] (with an angular spread of the order [MATH]), which intersects the ellipse ([REF]) in a single point.', '1605.00633-2-15-1': 'To relate back to the topic of depletion we recall Fig. 1 (right), which tells us that we have to stay away from the axes and the origin in the [MATH]-[MATH] plane according to our assumption of incoherent emission.', '1605.00633-2-15-2': "The 'safe' regime is thus [MATH], so that in terms of the emission angle we need to stay away from collinear emission, [MATH] or [MATH].", '1605.00633-2-15-3': 'Thus, in the generic regime of interest, [MATH], there is substantial transverse emission, cf. Fig. 3, right, for which the emission angle is about 50[MATH], with a depletion threshold of [MATH] according to Fig. 1.', '1605.00633-2-16-0': 'In this letter we have reconsidered the multi-photon Compton process in strong EM fields, focussing on the energy loss of the laser due to absorption, which transforms the initially strong fields into weak ones.', '1605.00633-2-16-1': 'We found that this phenomenon has an intensity threshold of [MATH], and requires [MATH] electrons per laser wavelength cubed, according to the numerical fit in Fig. 2.', '1605.00633-2-16-2': 'We have neglected coherent photon emission, which is valid when [MATH].', '1605.00633-2-16-3': 'It is expected that the depletion threshold will be overcome in the case of EM avalanches.', '1605.00633-2-16-4': 'Thus, laser depletion will not just be due to pair creation as considered previously, but must also be taken into account in laser photon absorption.', '1605.00633-2-17-0': 'We have further analyzed the photon emission rates differential in multi-photon number [MATH] and discovered that they strongly peak at a value [MATH], recall [REF], which determines the direction of the photon emission relative to the initial electron momentum direction in terms of an emission angle, [MATH], via [REF].', '1605.00633-2-17-1': 'For generic depletion parameters, [MATH], one finds substantial emission in transverse direction.', '1605.00633-2-17-2': 'In the collinear regime, [MATH] (forward scattering, [MATH]) and [MATH] (back scattering, [MATH]), coherent emission can no longer be neglected.', '1605.00633-2-17-3': 'Back scattering should dominate in the EM avalanche regime, i.e. in colliding laser pulses or during interactions of laser pulses with solid density foils or plasmas of near-critical density.', '1605.00633-2-17-4': 'The classical interpretation of the emission angle [MATH] in terms of averages over trajectories should yield a new test of the PIC codes currently in use.', '1605.00633-2-18-0': 'In future work, we want to understand the effect of depletion on the emission probabilities.', '1605.00633-2-18-1': 'This will require estimating the effect of a decreasing [MATH] on e.g. ([REF]), building on previous work such as [CITATION].', '1605.00633-2-19-0': 'We acknowledge support from the Office of Science of the US DOE under Contract No. DE-AC02-05CH11231.', '1605.00633-2-19-1': 'MM was supported by the Swedish Research Council grants 2012-5644 and 2013-4248.', '1605.00633-2-19-2': 'We would like to thank Anton Ilderton for fruitful discussions.', '1605.00633-2-19-3': 'The authors acknowledge the hospitality of the Kavli Institute for Theoretical Physics (KITP), where this research was initiated during the Frontiers of Intense Laser Physics Program and so was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1605.00633

cond-mat-0411186

{'cond-mat-0411186-1-0-0': 'The Mott-Hubbard metal-insulator transition is investigated in a two-band Hubbard model within dynamical mean-field theory.', 'cond-mat-0411186-1-0-1': "To this end, we use a suitable extension of Wilson's numerical renormalization group for the solution of the effective two-band single-impurity Anderson model.", 'cond-mat-0411186-1-0-2': "This method is non-perturbative and, in particular, allows to take into account the full exchange part of the Hund's rule coupling between the two orbitals.", 'cond-mat-0411186-1-0-3': 'We discuss in detail the influence of the various Coulomb interactions on thermodynamic and dynamic properties, for both the impurity and the lattice model.', 'cond-mat-0411186-1-0-4': "The exchange part of the Hund's rule coupling turns out to play an important role for the physics of the two-band Hubbard model and for the nature of the Mott-transition.", 'cond-mat-0411186-1-1-0': '# Introduction', 'cond-mat-0411186-1-2-0': 'Many materials with open [MATH]- or [MATH]-shells show metal-insulator transitions which are commonly classified to be of Mott-Hubbard type due to strong electron-electron correlations [CITATION].', 'cond-mat-0411186-1-2-1': 'The conventional model to study this type of transition is the one-band Hubbard model [CITATION], which in standard notation reads [EQUATION]', 'cond-mat-0411186-1-2-2': 'Major progress in understanding the physics of the Mott-Hubbard metal-insulator transition (MHMIT) of the model ([REF]) has been achieved in the last decade through the development of the dynamical mean-field theory (DMFT) [CITATION].', 'cond-mat-0411186-1-2-3': 'At [MATH] the MHMIT occurs at a critical value of the Coulomb parameter [MATH] [CITATION], where [MATH] denotes the bandwidth of the density of states at [MATH].', 'cond-mat-0411186-1-2-4': 'Interestingly, the transition is of first order [CITATION] for [MATH] with a second order end point at a [MATH] and [MATH].', 'cond-mat-0411186-1-3-0': 'Such a second order end point is also seen in the phase diagram of typical Mott-Hubbard systems like V[MATH]O[MATH] [CITATION].', 'cond-mat-0411186-1-3-1': 'Therefore, the one-band Hubbard model has been frequently used as a microscopic model for these materials (see Refs. rmp,roz95 and, in particular, Ref. Limelette which focusses on the critical regime close to the second order end point).', 'cond-mat-0411186-1-3-2': 'On the other hand, the justification to base the microscopic description on a one-band model (see Ref. castellani) has been questioned recently[CITATION].', 'cond-mat-0411186-1-4-0': 'For a proper description of materials such as transition metal oxides, the orbital structure of the relevant electronic degrees of freedom has to be taken into account.', 'cond-mat-0411186-1-4-1': 'This can lead to a fairly complicated form of the underlying tight-binding bandstructure (the kinetic energy term acquires a matrix structure).', 'cond-mat-0411186-1-4-2': 'Furthermore, additional local Coulomb matrix elements arise which describe the interactions between electrons in different orbitals.', 'cond-mat-0411186-1-5-0': 'The simplest possible extension of the model ([REF]) to the case of orbital degeneracy is the two-band Hubbard model.', 'cond-mat-0411186-1-5-1': 'It is a relevant model whenever the electronic degrees of freedom close to the Fermi level are two-fold degenerate, as for the [MATH] states in materials like LaMnO[MATH] or KCuF[MATH] [CITATION].', 'cond-mat-0411186-1-5-2': 'Here we investigate the two-band Hubbard model in the following form: [EQUATION] with the orbital index [MATH].', 'cond-mat-0411186-1-5-3': "The Coulomb parameters [MATH] and [MATH] describe the inter-orbital Coulomb interaction and Hund's exchange coupling, respectively.", 'cond-mat-0411186-1-5-4': 'The last term in ([REF]) is necessary to ensure rotational invariance of the interaction.', 'cond-mat-0411186-1-5-5': 'By virtue of this rotational invariance the Coulomb parameters are related by [MATH].', 'cond-mat-0411186-1-5-6': 'Generally, the hierarchy of interactions is [MATH].', 'cond-mat-0411186-1-6-0': 'A multi-band Hubbard model as in eq. ([REF]) displays a Mott transition at all integer fillings (not only at half filling as in the single-band case).', 'cond-mat-0411186-1-6-1': 'The additional Coulomb interactions also modify the value of [MATH] and, possibly, the character of the transition, as has already been investigated within the DMFT framework [CITATION].', 'cond-mat-0411186-1-7-0': 'Various theoretical and numerical techniques have been employed to investigate multi-band Hubbard models within DMFT.', 'cond-mat-0411186-1-7-1': 'The Quantum Monte Carlo method [CITATION], which is very successful in treating the multi-band Hubbard models in present applications of the LDA+DMFT approach [CITATION], cannot, however, handle the rotationally invariant form of the interaction in ([REF]) due to the sign problem.', 'cond-mat-0411186-1-7-2': 'On the other hand, exact diagonalization [CITATION], linearized DMFT [CITATION] and exact treatments in the limit of infinite orbital degeneracy [CITATION] are not able to reliably calculate dynamical properties.', 'cond-mat-0411186-1-8-0': "In this paper, we use Wilson's numerical renormalization group (NRG) [CITATION] to solve the effective two-orbital quantum impurity problem which appears in the DMFT for the two-band Hubbard model.", 'cond-mat-0411186-1-8-1': 'In Sec. II we start with some technical issues related to the NRG in the two-band case.', 'cond-mat-0411186-1-8-2': "Section III shows thermodynamic and dynamic quantities for the two-orbital single-impurity Anderson model, with the focus on the role of the Hund's coupling.", 'cond-mat-0411186-1-8-3': 'In Sec. IV we discuss the two-band Hubbard model; here we concentrate on the simplest case, i.e. degenerate orbitals and intra-orbital hopping only: [MATH].', 'cond-mat-0411186-1-8-4': 'The model eq. ([REF]) is studied on a Bethe lattice (mainly to compare our results with those from other approaches); the generalization to other lattices (other densities of states) is straightforward.', 'cond-mat-0411186-1-8-5': 'The paper is summarized in Sec. V.', 'cond-mat-0411186-1-9-0': '# NRG for multi-orbital models', 'cond-mat-0411186-1-10-0': 'The use of the NRG to solve the effective quantum impurity model appearing in the DMFT self-consistency has been extensively discussed in the literature [CITATION].', 'cond-mat-0411186-1-10-1': 'Here we focus on the additional problems arising in the two-orbital impurity Anderson model, which constitutes the effective local model arising in the DMFT for the two-orbital Hubbard model.', 'cond-mat-0411186-1-11-0': 'The two-orbital Anderson impurity Hamiltonian (in standard notation) is given by [EQUATION]', 'cond-mat-0411186-1-11-1': 'Within the NRG approach, the quantum impurity problem is mapped onto a semi-infinite chain form [CITATION] with the impurity at the first site of the chain and the conduction band written in a one-dimensional tight-binding form.', 'cond-mat-0411186-1-11-2': 'The model in the semi-infinite chain form is then solved by iterative diagonalization.', 'cond-mat-0411186-1-11-3': 'Therefore, the major obstacle in applying the NRG to multi-band models is the dramatic increase of the Hilbert space with each NRG step (in each step, one additional site of the semi-infinite chain is included).', 'cond-mat-0411186-1-12-0': 'A possible solution to this problem is to use very large values of the discretization parameter [MATH] so that the number of states can be reduced significantly.', 'cond-mat-0411186-1-12-1': 'Averaging over many discretizations (the so-called "Z-trick") has been shown to give reliable results for thermodynamic quantities (see Ref. [CITATION]), even for large values of [MATH].', 'cond-mat-0411186-1-12-2': 'However, this approach becomes at least cumbersome for the calculation of dynamic quantities which are required for the DMFT self-consistency.', 'cond-mat-0411186-1-13-0': 'Here we adopt a different strategy which allows to use a small value of the NRG discretization parameter [MATH] and to keep enough states in each step so that dynamic quantities can be calculated reliably.', 'cond-mat-0411186-1-13-1': 'This is achieved by introducing an asymmetric truncation scheme: instead of adding both orbital degrees of freedom simultaneously, the Hilbert space is truncated after adding each orbital individually.', 'cond-mat-0411186-1-13-2': 'In this way, the Hilbert space in the iterative diagonalization is reduced to a managable size; we can incorporate up to [MATH] states in each NRG step so that the required memory space is limited to sizes available on modern high performance computers.', 'cond-mat-0411186-1-13-3': 'In addition, the code is parallelized and shows almost linear speedup for up to 10 processors.', 'cond-mat-0411186-1-14-0': 'The asymmetric truncation scheme does not guarantee that orbital symmetry is preserved during the NRG iterations.', 'cond-mat-0411186-1-14-1': 'In fact, a slight violation of orbital symmetry is observed for very small energies, typically much lower than the Kondo temperature.', 'cond-mat-0411186-1-14-2': 'Figure [REF] shows a comparison of the local single-particle density of states (DOS) for the impurity Anderson model eq. ([REF]) with twofold degeneracy, particle-hole symmetry, and calculated with both schemes (symmetric and asymmetric truncation) in the Kondo limit.', 'cond-mat-0411186-1-14-3': 'Here, the NRG discretization parameter is [MATH] and [MATH] states were kept in each NRG step.', 'cond-mat-0411186-1-14-4': 'The model parameters are [MATH], [MATH] and [MATH].', 'cond-mat-0411186-1-14-5': 'As usual, [MATH] denotes the bare hybridization width ([MATH] is the conduction electron DOS at the Fermi energy).', 'cond-mat-0411186-1-14-6': 'Obviously, the asymmetric truncation (dashed line) leads to accurate results for both the value of the low-energy scale and the form of the Kondo resonance.', 'cond-mat-0411186-1-14-7': 'In addition, it does not violate particle-hole symmetry and orbital symmetry, at least on the scale shown in Fig. [REF].', 'cond-mat-0411186-1-15-0': 'Of course the above scheme is very susceptible to degeneracies of the system and likely to fail if the low-lying states are strongly degenerate.', 'cond-mat-0411186-1-15-1': 'Obviously, one has to check thoroughly whether degeneracies are respected in the course of a calculation and reject results where an apparent violation occurs.', 'cond-mat-0411186-1-15-2': 'However, at least for the application in mind, i.e. the two.orbital Hubbard model at half filling, the method appears to provide stable and reliable results.', 'cond-mat-0411186-1-16-0': "# Hund's rule coupling in the single-impurity model", 'cond-mat-0411186-1-17-0': "Before we turn to the application of the NRG to the two-orbital Hubbard model let us first discuss the effects of Hund's coupling for the single impurity model ([REF]).", 'cond-mat-0411186-1-17-1': 'For simplicity we consider a conduction band with constant DOS, [MATH], in the interval [MATH] and choose [MATH] as unit of energy.', 'cond-mat-0411186-1-17-2': 'The local energy [MATH] is chosen such that the model is particle-hole symmetric, i.e. [MATH].', 'cond-mat-0411186-1-17-3': 'As we will see, there is a profound difference between the cases with rotationally invariant and Ising-like exchange.', 'cond-mat-0411186-1-17-4': 'The two-orbital impurity Anderson model has already been investigated with NRG by several groups [CITATION].', 'cond-mat-0411186-1-18-0': "Here we want to concentrate on the influence of Hund's coupling on low energy scales and the possibility of a quantum phase transition.", 'cond-mat-0411186-1-18-1': 'To this end we present results for thermodynamic and dynamic properties.', 'cond-mat-0411186-1-18-2': 'To improve the accuracy of thermodynamic properties we employed Oliveira\'s "Z-trick" [CITATION], which allows to use a larger discretization [MATH] (we used [MATH]) and reduce the number of states kept ([MATH] after truncation here).', 'cond-mat-0411186-1-19-0': 'Let us begin with thermodynamic properties of the particle-hole symmetric two-orbital single impurity model in the Kondo limit, that is for a hybridization width [MATH] much smaller than the other bare energy scales [CITATION].', 'cond-mat-0411186-1-19-1': 'The temperature evolution of the effective squared moment and entropy for [MATH] and [MATH] is shown in Fig. [REF].', 'cond-mat-0411186-1-19-2': 'The calculation was done with the rotationally invariant exchange coupling.', 'cond-mat-0411186-1-19-3': 'In the atomic limit, any finite [MATH] leads to a spin triplet [MATH] as ground state with moment [MATH] and entropy [MATH].', 'cond-mat-0411186-1-19-4': 'Apparently, this situation is realized for intermediate temperatures for both [MATH] and [MATH].', 'cond-mat-0411186-1-19-5': 'At low temperatures, this local triplet is quenched by the conduction electrons like for an ordinary [MATH] Kondo effect.', 'cond-mat-0411186-1-19-6': 'For comparison we also include results for a single-orbital SIAM with the same values of [MATH] and [MATH].', 'cond-mat-0411186-1-19-7': 'Obviously, the two-orbital system has a considerably reduced low-energy scale [MATH], which in addition decreasing strongly with increasing [MATH].', 'cond-mat-0411186-1-19-8': 'A direct consequence of this substantial reduction of the low-energy scale for the application to the Hubbard model is that critical interactions for an MIT are also strongly reduced for finite [MATH] (see Sec. [REF] below).', 'cond-mat-0411186-1-20-0': 'At very low [MATH], additional shoulders appear in the thermodynamic properties.', 'cond-mat-0411186-1-20-1': 'We want to stress that this is not an artifact of the asymmetric truncation scheme but is also observed for a full calculation.', 'cond-mat-0411186-1-20-2': "The origin of these shoulders is at present not fully clear, but seems to be related to the [MATH] values used in the inplementation of Oliveira's [MATH]-trick.", 'cond-mat-0411186-1-21-0': 'The local DOS at [MATH] for [MATH] and [MATH] is shown in Fig. [REF].', 'cond-mat-0411186-1-21-1': 'The calculations were done with a discretization parameter [MATH] and [MATH] states kept after truncation.', 'cond-mat-0411186-1-21-2': 'On the scale used in the main panel of Fig. [REF] the Kondo resonance for [MATH] appears to be a vertical line, pointing to a strongly reduced Kondo temperature, too.', 'cond-mat-0411186-1-21-3': 'In addition, new structures on the scale of [MATH] appear as shoulders in the DOS.', 'cond-mat-0411186-1-22-0': 'If one zooms into the region [MATH] around the Fermi energy (see inset to Fig. [REF]), only the resonance for [MATH] remains visible with an energy scale well below [MATH].', 'cond-mat-0411186-1-22-1': 'This again confirms the result from the thermodynamic quantities, viz that with increasing [MATH] an exponential reduction of the Kondo temperature occurs.', 'cond-mat-0411186-1-22-2': 'It is quite obvious, that such a reduction in [MATH] will later manifest itself in a corresponding reduction of the critical [MATH] for the Mott-Hubbard metal- insulator transition within the DMFT.', 'cond-mat-0411186-1-23-0': "Let us point out that Friedel's sum rule implies as usual the constraint [MATH].", 'cond-mat-0411186-1-23-1': 'This constraint is fulfilled with high precision due to the calculation of the DOS via the self-energy according to Ref. [CITATION].', 'cond-mat-0411186-1-24-0': 'A completely different picture is obtained for an Ising-like exchange interaction in model ([REF]), which is realized by replacing [EQUATION] and neglecting the term [EQUATION].', 'cond-mat-0411186-1-24-1': 'In this case, the atomic ground state is doubly degenerate and consists of', 'cond-mat-0411186-1-25-0': 'the two states where two electrons with the same spin occupy different orbitals.', 'cond-mat-0411186-1-25-1': 'In contrast to the full exchange we find that the properties change quite dramatically with the ratio [MATH], where [MATH] denotes the hybridization width.', 'cond-mat-0411186-1-25-2': 'In Fig. [REF] we compare calculations for [MATH] and two different values of [MATH].', 'cond-mat-0411186-1-25-3': 'For large [MATH] we find the expected screening and corresponding formation of a Fermi liquid at low temperatures.', 'cond-mat-0411186-1-25-4': 'However, for small [MATH], this behaviour is replaced by the formation of a state reminiscent of a local moment with entropy [MATH] and finite effective local moment.', 'cond-mat-0411186-1-25-5': 'Obviously, the neglect of the spin-flip terms in Hund\'s exchange leads to a "critical" ratio [MATH] separating strong-coupling from local moment behavior.', 'cond-mat-0411186-1-25-6': 'We interpret this feature in the following way.', 'cond-mat-0411186-1-25-7': 'For the full exchange interaction, spin-flip scattering as in the conventional [MATH] case is presumably leading to a scenario similar to the standard Kondo effect.', 'cond-mat-0411186-1-25-8': 'For Ising-like Hund coupling [MATH], on the other hand, the atomic ground state consists, as already mentioned, of the two states where two electrons with the same spin occupy different orbitals.', 'cond-mat-0411186-1-25-9': 'Quite apparently, these two states cannot be connected by low-energy processes like spin-flips, i.e. the mechanism leading to the Kondo effect is not present here.', 'cond-mat-0411186-1-25-10': 'However, if the coupling to the band states is large enough such that the [MATH] Kondo temperature is larger than [MATH], the system can screen the spins for each orbital individually before the coupling [MATH] locks the system into the states with [MATH], leading to the observed strong-coupling behavior at large [MATH].', 'cond-mat-0411186-1-26-0': 'Presently, it is not clear whether the change in the impurity properties is connected to some quantum critical behavior like in the pseudo-gap model [CITATION].', 'cond-mat-0411186-1-26-1': 'The clarification of this question is of course of some interest in its own right and will be discussed in detail in a forthcoming publication.', 'cond-mat-0411186-1-27-0': 'The impurity DOS corresponding to the two different regimes are shown in Fig. [REF].', 'cond-mat-0411186-1-27-1': 'For [MATH] (full line) the typical structure is obtained.', 'cond-mat-0411186-1-27-2': 'Decreasing [MATH] below [MATH] completely changes the structure of the DOS.', 'cond-mat-0411186-1-27-3': 'Instead of a Kondo peak, we now find a structure with a pseudo-gap at the Fermi energy.', 'cond-mat-0411186-1-27-4': 'It is quite evident, that this "criticality" in the impurity model has profound effects on the MIT in the DMFT calculations for the Hubbard model.', 'cond-mat-0411186-1-28-0': '# MIT in the two-orbital Hubbard model', 'cond-mat-0411186-1-29-0': 'Theoretical investigations of multiorbital Hubbard models within DMFT have already led to a better understanding of various issues such as the nature of the Mott-transition as a function of orbital degeneracy [CITATION] and the structure of the spectral function in realistic treatments within the LDA+DMFT approach [CITATION].', 'cond-mat-0411186-1-29-1': 'Detailed results have been obtained for the dependence of the critical interaction strength [MATH] on the number of orbitals [MATH] and different integer fillings [MATH].', 'cond-mat-0411186-1-29-2': 'Numerical DMFT-QMC calculations have been performed for [MATH] [CITATION].', 'cond-mat-0411186-1-29-3': 'Remarkably, in the limit of large orbital degeneracy [MATH] an analytical treatment of the DMFT becomes possible for the MIT [CITATION].', 'cond-mat-0411186-1-29-4': 'A scaling [MATH] for the actual transition is found while [MATH] is obtained for the critical interaction where the insulating solution breaks down [CITATION].', 'cond-mat-0411186-1-29-5': 'This is consistent with the linear dependence for large [MATH] found in Refs. [CITATION] and with the square-root dependence reported in Ref. [CITATION].', 'cond-mat-0411186-1-29-6': "The inclusion of the Hund's rule exchange coupling [MATH] has been shown to significantly reduce the value of [MATH] [CITATION].", 'cond-mat-0411186-1-29-7': 'In particular, a qualitative change from continuous for [MATH] to discontinuous for any finite [MATH] has been observed in Ref. ohno (see also the Gutzwiller results in Ref. Bue97).', 'cond-mat-0411186-1-29-8': "A significant quantitative change of [MATH] when excluding the spin-dependent part from the exchange coupling in eq. [REF] has already been mentioned in Ref. ohno, but detailed results haven't been published yet.", 'cond-mat-0411186-1-29-9': 'Recently, the issue of possible orbital-selective Mott transitions has been investigated in Refs. koga04,liebsch03', 'cond-mat-0411186-1-30-0': 'Let us now discuss the results from the NRG for the particle-hole symmetric case.', 'cond-mat-0411186-1-30-1': 'To allow a direct comparison with earlier results, we use as non-interacting DOS the semielliptic form [MATH] (Bethe lattice) with the same bandwidth for both orbitals.', 'cond-mat-0411186-1-30-2': 'We begin by comparing results for the one-band Hubbard model ([REF]) (Fig. [REF]a) and a two-orbital Hubbard model with [MATH] and [MATH] (Fig. [REF]b).', 'cond-mat-0411186-1-31-0': 'As is well-known from earlier DMFT calculations [CITATION], the critical Coulomb parameter, [MATH], for the model with [MATH] increases strongly with the orbital degeneracy [MATH], [MATH] [CITATION].', 'cond-mat-0411186-1-31-1': 'This is also apparent from the results in Fig. [REF]b.', 'cond-mat-0411186-1-31-2': 'The system stays metallic up to the largest [MATH] shown.', 'cond-mat-0411186-1-31-3': 'The actual MIT occurs for a value [MATH] [CITATION].', 'cond-mat-0411186-1-31-4': 'From the discussion in section II it is clear that one has to be especially careful regarding the results for [MATH] in the asymmetric truncation scheme.', 'cond-mat-0411186-1-31-5': 'However, a close inspection of the level structure reveals that for [MATH]-values up to the actual MIT, where well-developed local moments exist, the formation of the local Fermi liquid is more efficient than the violation of local degeneracies.', 'cond-mat-0411186-1-31-6': 'This is further confirmed by the squared local moment shown in Fig. [REF].', 'cond-mat-0411186-1-31-7': 'We thus expect that the results up to [MATH] are reliable; the actual value of [MATH], however, will probably be underestimated.', 'cond-mat-0411186-1-32-0': "The influence of Hund's coupling on the development of the spectra and the occurence of the MIT can be seen in Fig. [REF]a, where results for different values of [MATH] and a full Hund's exchange [MATH] are presented[CITATION].", 'cond-mat-0411186-1-32-1': 'As has been noted before [CITATION], finite [MATH] substantially reduces', 'cond-mat-0411186-1-33-0': '[MATH].', 'cond-mat-0411186-1-33-1': 'Such a behavior is also seen in Fig. [REF]a.', 'cond-mat-0411186-1-33-2': 'The critical Coulomb parameter is reduced from [MATH] for [MATH] to [MATH].', 'cond-mat-0411186-1-34-0': 'Presently, the standard technique to solve quantum impurities with orbital degeneracy is Quantum Monte-Carlo (QMC).', 'cond-mat-0411186-1-34-1': "However, due to the minus sign problem, one has to restrict the Coulomb interaction to density-density type only, i.e. an Ising-like Hund's exchange.", 'cond-mat-0411186-1-34-2': 'The last term in ([REF]) has to be neglected completely.', 'cond-mat-0411186-1-34-3': 'This raises the following questions: What are the consequences of this approximation for the dynamics and in particular the MIT?', 'cond-mat-0411186-1-35-0': 'To answer this question (at least partially) we performed calculations with Ising-like exchange interaction as defined in the previous section (see Ref. [CITATION]) The results are shown in Fig. [REF]b.', 'cond-mat-0411186-1-35-1': 'At a first glance, the results are not very different, except that the critical [MATH] is further reduced to [MATH].', 'cond-mat-0411186-1-35-2': "On the other hand, the results for the impurity calculation in Figs. [REF] and [REF] already indicated that the replacement of Hund's exchange by an Ising-like term has more severe consequences than a mere quantitative change of energy scales.", 'cond-mat-0411186-1-35-3': 'In the following we show that this approximation indeed leads to a qualitative change in the physical properties of the Mott-Hubbard MIT.', 'cond-mat-0411186-1-36-0': 'Let us now turn to the nature of the Mott transition.', 'cond-mat-0411186-1-36-1': 'For a one-band model, it is now commonly accepted that the transition is of second order at [MATH] with a quasiparticle weight that vanishes smoothly as one approaches [MATH].', 'cond-mat-0411186-1-36-2': 'There is, however, a substantial region below [MATH], where the insulator is metastable [CITATION].', 'cond-mat-0411186-1-36-3': 'Previous work using the so-called linearized DMFT (L-DMFT) suggests that for orbitally degenerate systems with finite [MATH] this may be different [CITATION].', 'cond-mat-0411186-1-36-4': 'The authors of Ref. [CITATION] found a first order transition for small to intermediate [MATH] signalled by a jump in the quasiparticle weight at [MATH].', 'cond-mat-0411186-1-37-0': 'The NRG results for the inverse effective mass ([MATH] quasiparticle weight) for the case [MATH] are shown in Fig. [REF].', 'cond-mat-0411186-1-37-1': 'The circles were obtained from calculations using the full interaction, while the squares represent calculations with Ising-like interactions.', 'cond-mat-0411186-1-37-2': 'Apparently, the latter signal a strong first order transition at [MATH], while the former lead to a continuously vanishing quasiparticle weight.', 'cond-mat-0411186-1-37-3': 'For a fixed [MATH] (triangles in Fig. [REF]) the quasi-particle weight near [MATH] also shows a jump at [MATH] as predicted by L-DMFT [CITATION].', 'cond-mat-0411186-1-37-4': 'However, the magnitude of this jump comes out much smaller in our calculations.', 'cond-mat-0411186-1-38-0': 'The differences between the different calculations ([MATH], [MATH] and [MATH]) become more apparent when one looks at the local squared moment [MATH].', 'cond-mat-0411186-1-38-1': "For [MATH] this quantity has the value [MATH] (for [MATH] it is actually slightly larger), while deep in the Mott insulator it acquires the atomic value enforced by Hund's coupling, i.e. [MATH] for [MATH], [MATH] for finite full [MATH] and [MATH] for Ising-like [MATH].", 'cond-mat-0411186-1-38-2': 'This behavior is readily found in the calculated values of [MATH] in Fig. [REF].', 'cond-mat-0411186-1-38-3': 'In accordance with the results presented in Ref. [CITATION], the limiting value for [MATH] is approached smoothly for [MATH].', 'cond-mat-0411186-1-38-4': 'The same holds for [MATH], consistent with the results for the quasi-particle weight in Fig. [REF].', 'cond-mat-0411186-1-38-5': 'The slope, however, strongly increases when one approaches [MATH] from below.', 'cond-mat-0411186-1-38-6': 'For constant [MATH], the numerical results are not decisive, and could be interpreted as both a small discontinuity at [MATH] and a continuous approach with diverging slope.', 'cond-mat-0411186-1-38-7': "On the other hand, a quite strong discontinuity at [MATH] appears for an Ising-like Hund's coupling [MATH], signalling a rather strong first order transition in this case.", 'cond-mat-0411186-1-38-8': 'The above results are in rough agreement with the L-DMFT predictions [CITATION], although there one observes a first-order transition also for smaller values of [MATH].', 'cond-mat-0411186-1-39-0': 'The appearance of an unambiguous and rather strong first order transition for an Ising-like exchange coupling shows that in this case the physics underlying the Mott-Hubbard transition is very different from the one for the rotationally invariant exchange interaction.', 'cond-mat-0411186-1-39-1': 'As for the single impurity model, we believe that a transition between individually screened orbitals on the metallic side to a local moment regime enforced by the Ising coupling on the insulating side occurs as soon as [MATH] becomes of the order of the Fermi liquid scale.', 'cond-mat-0411186-1-39-2': 'Depending on the details of the non-interacting DOS (its value at the Fermi level and the band width) and the values of [MATH] and [MATH], this can lead in the worst case to a serious underestimation of [MATH] and possibly an incorrect description of the behavior of physical quantities close to the transition.', 'cond-mat-0411186-1-40-0': '# Summary and conclusions', 'cond-mat-0411186-1-41-0': "In this paper we presented first studies of the Mott-Hubbard transition in a two-orbital Hubbard model within the DMFT at [MATH] using Wislon's NRG.", 'cond-mat-0411186-1-41-1': 'We proposed an approximation to the NRG algorithm, namely an asymmetric truncation scheme, that enables us to do calculations at small enough NRG discretization parameter without running into compuational problems with too large Hilbert spaces.', 'cond-mat-0411186-1-41-2': 'We showed that this scheme works rather well in generic situations of the two-orbital single impurity Anderson model.', 'cond-mat-0411186-1-41-3': 'Of course, the additional approximation to the NRG makes it necesary to thoroughly check each individual calculation to ensure that no fundamental symmetries etc. are being violated by the asymmetric truncation.', 'cond-mat-0411186-1-42-0': 'As a first interesting result, we observed that for the particle-hole symmetric case a tremendous reduction in the low-energy scale [MATH] can be observed.', 'cond-mat-0411186-1-42-1': "Furthermore, [MATH] strongly decreases with increasing Hund's coupling [MATH].", 'cond-mat-0411186-1-42-2': 'Note that this is rather different from the case [MATH] in the multi-orbital case, where an increase of [MATH] orbital degeneracy, [MATH] single-orbital Kondo temperature) is known to occur.', 'cond-mat-0411186-1-42-3': '[CITATION]', 'cond-mat-0411186-1-43-0': 'A completely different behavior occurs if one replaces the rotationally invariant Hund exchange by an Ising-like one.', 'cond-mat-0411186-1-43-1': 'In this case, the levels of the atomic doublet with [MATH] enforced by the Ising coupling cannot be connected by Schrieffer-Wolff type spin-flip processes.', 'cond-mat-0411186-1-43-2': 'Thus, the Kondo effect can only occur for [MATH], while for larger [MATH] the systems is locked into a local moment enforced by the exchange coupling.', 'cond-mat-0411186-1-43-3': 'Note that this interpretation also implies that in the strong-coupling phase the spins on each individual orbital will be screened separately, while for the rotationally invariant case a full [MATH] system must be screened.', 'cond-mat-0411186-1-43-4': 'The details of the quantum phase transition between strong-coupling and local moment phases have not yet been analyzed.', 'cond-mat-0411186-1-43-5': 'However, in view of a possible relevance of an Ising-anisotropy in the presence of crystal fields, a further investigation of this problem is certainly interesting.', 'cond-mat-0411186-1-44-0': 'We also applied the NRG to the two-orbital Hubbard model in the framework of DMFT to investigate the Mott-Hubbard metal-insulator transition at [MATH] for the half-filled case.', 'cond-mat-0411186-1-44-1': "The major goal was here to eludicate the influence of Hund's coupling on the MIT and to investigate how the restriction to an Ising-like exchange changes the nature of the MIT.", 'cond-mat-0411186-1-44-2': 'Our results are in general agreement with previous ones.[', 'cond-mat-0411186-1-44-3': '[CITATION] In particular, for finite [MATH] quantities like the effective mass or [MATH] show diverging slopes as [MATH], possibly even discontinuities as proposed by the L-DMFT.[', 'cond-mat-0411186-1-45-0': "For an Ising-like Hund's exchange coupling the situation becomes qualitatively different.", 'cond-mat-0411186-1-45-1': 'As can be anticipated from the behavior of the impurity model, the MIT is strongly first order with clear jumps in the effective mass and [MATH].', 'cond-mat-0411186-1-45-2': 'Note that the former also implies a discontinuous vanishing of the quasi-particle peak at [MATH] as one reaches [MATH].', 'cond-mat-0411186-1-45-3': 'Also the physics underlying this transition is quite different compared to the rotationally invariant case, reflecting the lack of spin-flip scattering processes connecting the two states [MATH].', 'cond-mat-0411186-1-45-4': 'Thus, the metallic phase with Ising-like interaction will be characterized by individual screening of the spins on the two orbitals rather than a Kondo screening of a total spin [MATH].', 'cond-mat-0411186-1-45-5': 'Note that this subtlety will most likely influence low-energy properties on the metallic side close to [MATH], but be less important for "high-energy" properties like magnetic or orbital ordering.', 'cond-mat-0411186-1-46-0': "It is clear, that the investigations presented here are merely a starting point to systematically study properties of multi-orbital impurity models or correlated lattice models within the DMFT at [MATH] using Wilson's NRG.", 'cond-mat-0411186-1-46-1': 'The major advantage of this method is obviously its unmatched ability to handle exponentially small energy scales and neverthless provide reliable information on dynamics and thermodynamics even on high-energy scales.', 'cond-mat-0411186-1-46-2': 'Thus, at least for two-orbital models we are now in a position to systematically study their physical properties and address questions that are of fundamental interest for a realistic description of e.g. transition metal oxides but require local degrees of freedom beyond a simple one-band Hubbard model.', 'cond-mat-0411186-1-47-0': 'We acknowledge useful conversations with M. Vojta, A. Lichtenstein, and D. Vollhardt.', 'cond-mat-0411186-1-47-1': 'This work was supported by the DFG through the collaborative research center SFB 484, the Leibniz Computer center, the Computer center of the Max-Planck-Gesellschaft in Garching and the Norddeutsche Verbund fur Hoch- und Hochstleistungsrechnen.'}

{'cond-mat-0411186-2-0-0': 'The Mott-Hubbard metal-insulator transition is investigated in a two-band Hubbard model within dynamical mean-field theory.', 'cond-mat-0411186-2-0-1': "To this end, we use a suitable extension of Wilson's numerical renormalization group for the solution of the effective two-band single-impurity Anderson model.", 'cond-mat-0411186-2-0-2': "This method is non-perturbative and, in particular, allows to take into account the full exchange part of the Hund's rule coupling between the two orbitals.", 'cond-mat-0411186-2-0-3': 'We discuss in detail the influence of the various Coulomb interactions on thermodynamic and dynamic properties, for both the impurity and the lattice model.', 'cond-mat-0411186-2-0-4': "The exchange part of the Hund's rule coupling turns out to play an important role for the physics of the two-band Hubbard model and for the nature of the Mott-transition.", 'cond-mat-0411186-2-1-0': '# Introduction', 'cond-mat-0411186-2-2-0': 'Many materials with open [MATH]- or [MATH]-shells show metal-insulator transitions which are commonly classified to be of Mott-Hubbard type due to strong electron-electron correlations [CITATION].', 'cond-mat-0411186-2-2-1': 'The conventional model to study this type of transition is the one-band Hubbard model [CITATION], which in standard notation reads [EQUATION]', 'cond-mat-0411186-2-2-2': 'Major progress in understanding the physics of the Mott-Hubbard metal-insulator transition (MHMIT) of the model ([REF]) has been achieved in the last decade through the development of the dynamical mean-field theory (DMFT) [CITATION].', 'cond-mat-0411186-2-2-3': 'At [MATH] the MHMIT occurs at a critical value of the Coulomb parameter [MATH] [CITATION], where [MATH] denotes the bandwidth of the density of states at [MATH].', 'cond-mat-0411186-2-2-4': 'Interestingly, the transition is of first order [CITATION] for [MATH] with a second order end point at a [MATH] and [MATH].', 'cond-mat-0411186-2-3-0': 'Such a second order end point is also seen in the phase diagram of typical Mott-Hubbard systems like V[MATH]O[MATH] [CITATION].', 'cond-mat-0411186-2-3-1': 'Therefore, the one-band Hubbard model has been frequently used as a microscopic model for these materials (see Refs. rmp,roz95 and, in particular, Ref. Limelette which focusses on the critical regime close to the second order end point).', 'cond-mat-0411186-2-3-2': 'On the other hand, the justification to base the microscopic description on a one-band model (see Ref. castellani) has been questioned recently[CITATION].', 'cond-mat-0411186-2-4-0': 'For a proper description of materials such as transition metal oxides, the orbital structure of the relevant electronic degrees of freedom has to be taken into account.', 'cond-mat-0411186-2-4-1': 'This can lead to a fairly complicated form of the underlying tight-binding bandstructure (the kinetic energy term acquires a matrix structure).', 'cond-mat-0411186-2-4-2': 'Furthermore, additional local Coulomb matrix elements arise which describe the interactions between electrons in different orbitals.', 'cond-mat-0411186-2-5-0': 'The simplest possible extension of the model ([REF]) to the case of orbital degeneracy is the two-band Hubbard model.', 'cond-mat-0411186-2-5-1': 'It is a relevant model whenever the electronic degrees of freedom close to the Fermi level are two-fold degenerate, as for the [MATH] states in materials like LaMnO[MATH] or KCuF[MATH] [CITATION].', 'cond-mat-0411186-2-5-2': 'Here we investigate the two-band Hubbard model in the following form: [EQUATION] with the orbital index [MATH].', 'cond-mat-0411186-2-5-3': "The Coulomb parameters [MATH] and [MATH] describe the inter-orbital Coulomb interaction and Hund's exchange coupling, respectively.", 'cond-mat-0411186-2-5-4': 'The last term in ([REF]) is necessary to ensure rotational invariance of the interaction.', 'cond-mat-0411186-2-5-5': 'By virtue of this rotational invariance the Coulomb parameters are related by [MATH].', 'cond-mat-0411186-2-5-6': 'Generally, the hierarchy of interactions is [MATH].', 'cond-mat-0411186-2-6-0': 'A multi-band Hubbard model as in eq. ([REF]) displays a Mott transition at all integer fillings (not only at half filling as in the single-band case).', 'cond-mat-0411186-2-6-1': 'The additional Coulomb interactions also modify the value of [MATH] and, possibly, the character of the transition, as has already been investigated within the DMFT framework [CITATION].', 'cond-mat-0411186-2-7-0': 'Various theoretical and numerical techniques have been employed to investigate multi-band Hubbard models within DMFT.', 'cond-mat-0411186-2-7-1': 'The Quantum Monte Carlo method [CITATION], which is very successful in treating the multi-band Hubbard models in present applications of the LDA+DMFT approach [CITATION], cannot, however, handle the rotationally invariant form of the interaction in ([REF]) due to the sign problem (for recent progress in reducing this sign problem, see Ref. [CITATION]).', 'cond-mat-0411186-2-7-2': 'On the other hand, exact diagonalization [CITATION], linearized DMFT [CITATION] and exact treatments in the limit of infinite orbital degeneracy [CITATION] are not able to reliably calculate dynamical properties.', 'cond-mat-0411186-2-8-0': "In this paper, we use Wilson's numerical renormalization group (NRG) [CITATION] to solve the effective two-orbital quantum impurity problem which appears in the DMFT for the two-band Hubbard model.", 'cond-mat-0411186-2-8-1': 'In Sec. II we start with some technical issues related to the NRG in the two-band case.', 'cond-mat-0411186-2-8-2': "Section III shows thermodynamic and dynamic quantities for the two-orbital single-impurity Anderson model, with the focus on the role of the Hund's coupling.", 'cond-mat-0411186-2-8-3': 'In Sec. IV we discuss the two-band Hubbard model; here we concentrate on the simplest case, i.e. degenerate orbitals and intra-orbital hopping only: [MATH].', 'cond-mat-0411186-2-8-4': 'The model eq. ([REF]) is studied on a Bethe lattice (mainly to compare our results with those from other approaches); the generalization to other lattices (other densities of states) is straightforward.', 'cond-mat-0411186-2-8-5': 'The paper is summarized in Sec. V.', 'cond-mat-0411186-2-9-0': '# NRG for multi-orbital models', 'cond-mat-0411186-2-10-0': 'The use of the NRG to solve the effective quantum impurity model appearing in the DMFT self-consistency has been extensively discussed in the literature [CITATION].', 'cond-mat-0411186-2-10-1': 'Here we focus on the additional problems arising in the two-orbital impurity Anderson model, which constitutes the effective local model arising in the DMFT for the two-orbital Hubbard model.', 'cond-mat-0411186-2-11-0': 'The two-orbital Anderson impurity Hamiltonian (in standard notation) is given by [EQUATION]', 'cond-mat-0411186-2-11-1': 'Within the NRG approach, the quantum impurity problem is mapped onto a semi-infinite chain form [CITATION] with the impurity at the first site of the chain and the conduction band written in a one-dimensional tight-binding form.', 'cond-mat-0411186-2-11-2': 'The model in the semi-infinite chain form is then solved by iterative diagonalization.', 'cond-mat-0411186-2-11-3': 'Therefore, the major obstacle in applying the NRG to multi-band models is the dramatic increase of the Hilbert space with each NRG step (in each step, one additional site of the semi-infinite chain is included).', 'cond-mat-0411186-2-12-0': 'A possible solution to this problem is to use very large values of the discretization parameter [MATH] so that the number of states can be reduced significantly.', 'cond-mat-0411186-2-12-1': 'Averaging over many discretizations (the so-called "Z-trick") has been shown to give reliable results for thermodynamic quantities (see Ref. [CITATION]), even for large values of [MATH].', 'cond-mat-0411186-2-12-2': 'However, this approach becomes at least cumbersome for the calculation of dynamic quantities which are required for the DMFT self-consistency.', 'cond-mat-0411186-2-13-0': 'Here we adopt two different strategies which allow to use a small value of the NRG discretization parameter [MATH] and to keep enough states in each step so that dynamic quantities can be calculated reliably.', 'cond-mat-0411186-2-13-1': 'The first one is to explicitely include the orbital quantum number in the iterative construction of the basis states.', 'cond-mat-0411186-2-13-2': 'This additional quantum number significantly reduces the typical matrix size, so that O[MATH] states can be kept in each NRG iteration with reasonable computation time and memory consumption on modern SMP high-performance computers like e.g. the IBM Regatta.', 'cond-mat-0411186-2-13-3': 'The price to pay is that one has to omit the last term in the Coulomb interaction, i.e. [EQUATION] because it explicitely breaks the orbital symmetry.', 'cond-mat-0411186-2-13-4': 'However, it turns out that this term does not influence the thermodynamics, and dynamic quantities are only slightly affected via the multiplett structure of the Hubbard bands.', 'cond-mat-0411186-2-14-0': 'The second strategy is an asymmetric truncation scheme: instead of adding both orbital degrees of freedom simultaneously, the Hilbert space is truncated after adding each orbital individually.', 'cond-mat-0411186-2-14-1': 'This also leads to a significant reduction of the Hilbert space in the iterative diagonalization.', 'cond-mat-0411186-2-15-0': 'However, the asymmetric truncation scheme does not guarantee that orbital symmetry is preserved during the NRG iterations.', 'cond-mat-0411186-2-15-1': 'In fact, a slight violation of orbital symmetry is observed for very small energies, typically much lower than the Kondo temperature.', 'cond-mat-0411186-2-15-2': 'It turns out that for the DMFT-calculations presented in Sec. [REF], both methods give almost identical results, at least on the scale shown in the figures.', 'cond-mat-0411186-2-16-0': 'Figure [REF] shows a comparison of the local single-particle density of states (DOS) for the impurity Anderson model eq. ([REF]) with twofold degeneracy, particle-hole symmetry, and calculated with both schemes (symmetric truncation with orbital quantum number and asymmetric truncation) in the Kondo limit.', 'cond-mat-0411186-2-16-1': 'Here, the NRG discretization parameter is [MATH] and [MATH] states were kept in each NRG step.', 'cond-mat-0411186-2-16-2': 'The model parameters are [MATH], [MATH] and [MATH].', 'cond-mat-0411186-2-16-3': 'As usual, [MATH] denotes the bare hybridization width ([MATH] is the conduction electron DOS at the Fermi energy).', 'cond-mat-0411186-2-16-4': 'Obviously, the asymmetric truncation (dashed line) leads to accurate results for both the value of the low-energy scale and the form of the Kondo resonance.', 'cond-mat-0411186-2-16-5': 'In addition, it does not violate particle-hole symmetry and orbital symmetry, at least on the scale shown in Fig. [REF].', 'cond-mat-0411186-2-17-0': 'The aymmetric truncation scheme introduced here might be of advantage in cases where the orbital symmetry is violated from the outset so that the orbital quantum number cannot be used in the calculation to reduce the matrix size.', 'cond-mat-0411186-2-18-0': "# Hund's rule coupling in the single-impurity model", 'cond-mat-0411186-2-19-0': "Before we turn to the application of the NRG to the two-orbital Hubbard model let us first discuss the effects of Hund's coupling for the single impurity model ([REF]).", 'cond-mat-0411186-2-19-1': 'For simplicity we consider a conduction band with constant DOS, [MATH], in the interval [MATH] and choose [MATH] as unit of energy.', 'cond-mat-0411186-2-19-2': 'The local energy [MATH] is chosen such that the model is particle-hole symmetric, i.e. [MATH].', 'cond-mat-0411186-2-19-3': 'As we will see, there is a profound difference between the cases with rotationally invariant and Ising-like exchange.', 'cond-mat-0411186-2-19-4': 'The two-orbital impurity Anderson model has already been investigated with NRG by several groups [CITATION].', 'cond-mat-0411186-2-20-0': "Here we want to concentrate on the influence of Hund's coupling on low energy scales and the possibility of a quantum phase transition.", 'cond-mat-0411186-2-20-1': 'To this end we present results for thermodynamic and dynamic properties.', 'cond-mat-0411186-2-20-2': 'To improve the accuracy of thermodynamic properties we employed Oliveira\'s "Z-trick" [CITATION], which allows to use a larger discretization [MATH] (we used [MATH]) and reduce the number of states kept ([MATH] after truncation here).', 'cond-mat-0411186-2-21-0': 'Let us begin with thermodynamic properties of the particle-hole symmetric two-orbital single impurity model in the Kondo limit, that is for a hybridization width [MATH] much smaller than the other bare energy scales [CITATION].', 'cond-mat-0411186-2-21-1': 'The temperature evolution of the effective squared moment [MATH] and entropy [MATH] for [MATH], [MATH] and [MATH] is shown in Fig. [REF].', 'cond-mat-0411186-2-21-2': 'The calculations were done with the rotationally invariant exchange coupling and [MATH], but neglecting the term breaking orbital symmetry.', 'cond-mat-0411186-2-21-3': 'For comparison we also include results for a single-orbital SIAM - marked [MATH] in Fig. [REF] - with the same values of [MATH] and [MATH].', 'cond-mat-0411186-2-22-0': 'For [MATH], i.e. [MATH], we observe an intermediate "local-moment regime" with entropy [MATH] and effective moment [MATH] corresponding to the six degenerate states, two of them magnetic, in the atomic limit.', 'cond-mat-0411186-2-22-1': 'As expected from general [MATH] arguments [CITATION] this local moment is eventually Kondo screened with an enhanced Kondo scale [MATH].', 'cond-mat-0411186-2-22-2': 'As a technical sidemark let us point out that for [MATH], unlike finite [MATH], the asymmetric truncation does not work properly, leading to wrong results for [MATH] and [MATH] in the local moment regime.', 'cond-mat-0411186-2-23-0': 'In the atomic limit, any finite [MATH] leads to a spin triplet [MATH] as ground state with moment [MATH] and entropy [MATH].', 'cond-mat-0411186-2-23-1': 'Apparently, this situation is realized for intermediate temperatures for both [MATH] and [MATH].', 'cond-mat-0411186-2-23-2': 'At low temperatures, this local triplet is again quenched by the conduction electrons like for an ordinary [MATH] Kondo effect.', 'cond-mat-0411186-2-23-3': 'Obviously, the two-orbital system has a considerably reduced low-energy scale [MATH], which in addition decreases strongly with increasing [MATH].', 'cond-mat-0411186-2-23-4': 'At present we do not have a satisfactory explanation for this observation, but believe that it is related to the problem how spin-[MATH] electrons screen a true [MATH] object.', 'cond-mat-0411186-2-24-0': 'A direct consequence of this substantial reduction of the low-energy scale for the application to the Hubbard model is that critical interactions for an MIT are also strongly reduced for finite [MATH] (see Sec. [REF] below).', 'cond-mat-0411186-2-25-0': 'The local DOS at [MATH] for [MATH] and [MATH] is shown in Fig. [REF].', 'cond-mat-0411186-2-25-1': 'The calculations were done with a discretization parameter [MATH] and [MATH] states kept after truncation.', 'cond-mat-0411186-2-25-2': 'On the scale used in the main panel of Fig. [REF] the Kondo resonance for [MATH] appears to be a vertical line, pointing to a strongly reduced Kondo temperature, too.', 'cond-mat-0411186-2-25-3': 'In addition, new structures on the scale of [MATH] appear as shoulders in the DOS.', 'cond-mat-0411186-2-26-0': 'If one zooms into the region [MATH] around the Fermi energy (see inset to Fig. [REF]), only the resonance for [MATH] remains visible with an energy scale well below [MATH].', 'cond-mat-0411186-2-26-1': 'This again confirms the result from the thermodynamic quantities, viz that with increasing [MATH] an exponential reduction of the Kondo temperature occurs.', 'cond-mat-0411186-2-26-2': 'It is quite obvious, that such a reduction in [MATH] will later manifest itself in a corresponding reduction of the critical [MATH] for the Mott-Hubbard metal-insulator transition within the DMFT.', 'cond-mat-0411186-2-27-0': "Let us point out that Friedel's sum rule implies as usual the constraint [MATH].", 'cond-mat-0411186-2-27-1': 'This constraint is fulfilled with high precision due to the calculation of the DOS via the self-energy according to Ref. [CITATION].', 'cond-mat-0411186-2-28-0': 'A completely different picture is obtained for an Ising-like exchange interaction in model ([REF]), which is realized by replacing [EQUATION] and neglecting the term [EQUATION].', 'cond-mat-0411186-2-28-1': 'In this case, the atomic ground state is doubly degenerate and consists of', 'cond-mat-0411186-2-29-0': 'the two states where two electrons with the same spin occupy different orbitals.', 'cond-mat-0411186-2-29-1': 'In contrast to the full exchange we find that the properties change quite dramatically with the ratio [MATH], where [MATH] denotes the hybridization width.', 'cond-mat-0411186-2-29-2': 'In Fig. [REF] we compare calculations for [MATH] and two different values of [MATH].', 'cond-mat-0411186-2-29-3': 'For large [MATH] we find the expected screening and corresponding formation of a Fermi liquid at low temperatures.', 'cond-mat-0411186-2-29-4': 'However, for small [MATH], this behaviour is replaced by the formation of a state reminiscent of a local moment with entropy [MATH] and effective local moment [MATH].', 'cond-mat-0411186-2-29-5': 'Obviously, the neglect of the spin-flip terms in Hund\'s exchange leads to a "critical" ratio [MATH] separating strong-coupling from local moment behavior.', 'cond-mat-0411186-2-29-6': 'We interpret this feature in the following way.', 'cond-mat-0411186-2-29-7': 'For the full exchange interaction, spin-flip scattering as in the conventional [MATH] case is presumably leading to a scenario similar to the standard Kondo effect.', 'cond-mat-0411186-2-29-8': 'For Ising-like Hund coupling [MATH], on the other hand, the atomic ground state consists, as already mentioned, of the two states where two electrons with the same spin occupy different orbitals.', 'cond-mat-0411186-2-29-9': 'Quite apparently, these two states cannot be connected by low-energy processes like spin-flips, i.e. the mechanism leading to the Kondo effect is not present here.', 'cond-mat-0411186-2-29-10': 'However, if the coupling to the band states is large enough such that the [MATH] Kondo temperature is larger than [MATH], the system can screen the spins for each orbital individually before the coupling [MATH] locks the system into the states with [MATH], leading to the observed strong-coupling behavior at large [MATH].', 'cond-mat-0411186-2-30-0': 'Presently, it is not clear whether the change in the impurity properties is connected to some quantum critical behavior like in the pseudo-gap model [CITATION].', 'cond-mat-0411186-2-30-1': 'The clarification of this question is of course of some interest in its own right and will be discussed in detail in a forthcoming publication.', 'cond-mat-0411186-2-31-0': 'The impurity DOS corresponding to the two different regimes is shown in Fig. [REF].', 'cond-mat-0411186-2-31-1': 'For [MATH] (full line) the typical structure is obtained.', 'cond-mat-0411186-2-31-2': 'Decreasing [MATH] below [MATH] completely changes the structure of the DOS.', 'cond-mat-0411186-2-31-3': 'Instead of a Kondo peak, we now find a structure with a pseudo-gap at the Fermi energy.', 'cond-mat-0411186-2-31-4': 'It is quite evident, that this "criticality" in the impurity model has profound effects on the MIT in the DMFT calculations for the Hubbard model.', 'cond-mat-0411186-2-32-0': '# MIT in the two-orbital Hubbard model', 'cond-mat-0411186-2-33-0': 'Theoretical investigations of multi-orbital Hubbard models within DMFT have already led to a better understanding of various issues such as the nature of the Mott-transition as a function of orbital degeneracy [CITATION] and the structure of the spectral function in realistic treatments within the LDA+DMFT approach [CITATION].', 'cond-mat-0411186-2-33-1': 'Detailed results have been obtained for the dependence of the critical interaction strength [MATH] on the number of orbitals [MATH] and different integer fillings [MATH].', 'cond-mat-0411186-2-33-2': 'Numerical DMFT-QMC calculations have been performed for [MATH] [CITATION].', 'cond-mat-0411186-2-33-3': 'Remarkably, in the limit of large orbital degeneracy [MATH] an analytical treatment of the DMFT becomes possible for the MIT [CITATION].', 'cond-mat-0411186-2-33-4': 'For [MATH], a scaling [MATH] for the actual transition is found while [MATH] is obtained for the critical interaction where the insulating solution breaks down [CITATION].', 'cond-mat-0411186-2-33-5': 'This is consistent with the linear dependence for large [MATH] found in Refs. [CITATION] and with the square-root dependence reported in Ref. [CITATION].', 'cond-mat-0411186-2-33-6': "The inclusion of the Hund's rule exchange coupling [MATH] has been shown to significantly reduce the value of [MATH] [CITATION].", 'cond-mat-0411186-2-33-7': 'In particular, a qualitative change from continuous for [MATH] to discontinuous for any finite [MATH] has been observed in Ref. ohno (see also the Gutzwiller results in Ref. Bue97).', 'cond-mat-0411186-2-33-8': 'A significant quantitative change of [MATH] when excluding the spin-dependent part from the exchange coupling in eq. [REF] has already been mentioned in Ref. ohno, but detailed results have not been published yet.', 'cond-mat-0411186-2-33-9': 'Recently, the issue of possible orbital-selective Mott transitions has been investigated in Refs. koga04,liebsch03.', 'cond-mat-0411186-2-34-0': 'Let us now discuss the results from the NRG for the particle-hole symmetric case.', 'cond-mat-0411186-2-34-1': 'To allow a direct comparison with earlier results, we use as non-interacting DOS the semielliptic form [MATH] (Bethe lattice) with the same bandwidth for both orbitals.', 'cond-mat-0411186-2-34-2': 'As NRG discretization paramter we choose [MATH] and keep [MATH] states after truncation.', 'cond-mat-0411186-2-35-0': 'Except for [MATH], all the following results were obtained with the asymmetric truncation scheme introduced in Sec. II including all Coulomb interactions from the model ([REF]).', 'cond-mat-0411186-2-35-1': 'We have observed that the symmetric truncation scheme (taking into account the orbital quantum number and negelcting the last term in the interaction) leads to identical results for spectral functions, apart from some weak redistribution of spectral weight in the Hubbard bands due to a different atomic multiplett structure.', 'cond-mat-0411186-2-35-2': 'However, the latter effect is only barely noticable due to the broadening introduced in calculating continous spectra from the NRG.', 'cond-mat-0411186-2-35-3': 'More important, the critical values [MATH] for all two-band calculations within DMFT are not affected; possible problems due to the breaking of orbital symmetry by the asymmetric truncation scheme do not play a role here, except again for [MATH].', 'cond-mat-0411186-2-36-0': 'We begin by comparing results for the one-band Hubbard model ([REF]) (Fig. [REF]a) and a two-orbital Hubbard model with [MATH] and [MATH] (Fig. [REF]b).', 'cond-mat-0411186-2-37-0': 'As is well-known from earlier DMFT calculations [CITATION], the critical Coulomb parameter, [MATH], for the model with [MATH] increases strongly with the orbital degeneracy [MATH], [MATH] [CITATION].', 'cond-mat-0411186-2-37-1': 'This is also apparent from the results in Fig. [REF]b.', 'cond-mat-0411186-2-37-2': 'The system stays metallic up to the largest [MATH] shown.', 'cond-mat-0411186-2-37-3': 'The actual MIT occurs for a value [MATH].', 'cond-mat-0411186-2-38-0': 'The results in Figs. [REF] and [REF] are calculated with a fairly large value of [MATH] and broadening parameter [MATH] (see eq. (8) in Ref. [CITATION]) for both the single-band case in Fig. [REF]a and the two band case in Figs. [REF]b and [REF].', 'cond-mat-0411186-2-38-1': 'We therefore expect that our critical values [MATH] for the Mott transition differ from more precise calculations, and indeed we find them to be somewhat overestimated.', 'cond-mat-0411186-2-38-2': 'In the single-band case, for example, our result for [MATH] is slightly (10%) larger than the well established value [MATH] [CITATION].', 'cond-mat-0411186-2-38-3': 'A similar overestimation is also present for [MATH], where [MATH] is reported in literature [CITATION].', 'cond-mat-0411186-2-38-4': 'However, the qualitative features of the transition are not altered by this overestimation.', 'cond-mat-0411186-2-39-0': "The influence of Hund's coupling on the development of the spectra and the occurence of the MIT can be seen in Fig. [REF]a, where results for different values of [MATH] and a full Hund's exchange [MATH] are presented[CITATION].", 'cond-mat-0411186-2-39-1': 'As has been noted before [CITATION], finite [MATH] substantially reduces', 'cond-mat-0411186-2-40-0': '[MATH].', 'cond-mat-0411186-2-40-1': 'Such a behavior is also seen in Fig. [REF]a.', 'cond-mat-0411186-2-40-2': 'The critical Coulomb parameter is reduced from [MATH] for [MATH] to [MATH].', 'cond-mat-0411186-2-41-0': 'Presently, the standard technique to solve quantum impurities with orbital degeneracy is Quantum Monte-Carlo (QMC).', 'cond-mat-0411186-2-41-1': "However, due to the minus sign problem, one has to restrict the Coulomb interaction to density-density type only, i.e. an Ising-like Hund's exchange.", 'cond-mat-0411186-2-41-2': 'The last term in ([REF]) has to be neglected completely.', 'cond-mat-0411186-2-41-3': 'This raises the following questions: What are the consequences of this approximation for the dynamics and in particular the MIT?', 'cond-mat-0411186-2-42-0': 'To answer this question (at least partially) we performed calculations with Ising-like exchange interaction as defined in the previous section (see Ref. [CITATION]) The results are shown in Fig. [REF]b.', 'cond-mat-0411186-2-42-1': 'At a first glance, the results are not very different, except that the critical [MATH] is further reduced to [MATH].', 'cond-mat-0411186-2-42-2': "On the other hand, the results for the impurity calculation in Figs. [REF] and [REF] already indicate that the replacement of Hund's exchange by an Ising-like term has more severe consequences than a mere quantitative change of energy scales.", 'cond-mat-0411186-2-42-3': 'In the following we show that this approximation indeed leads to a qualitative change in the physical properties of the Mott-Hubbard MIT.', 'cond-mat-0411186-2-43-0': 'Let us now turn to the nature of the Mott transition.', 'cond-mat-0411186-2-43-1': 'For a one-band model, it is now commonly accepted that the transition is of second order at [MATH] with a quasiparticle weight that vanishes smoothly as one approaches [MATH].', 'cond-mat-0411186-2-43-2': 'There is, however, a substantial region below [MATH], where the insulator is metastable [CITATION].', 'cond-mat-0411186-2-43-3': 'Previous work using the so-called linearized DMFT (L-DMFT) suggests that for orbitally degenerate systems with finite [MATH] this may be different [CITATION].', 'cond-mat-0411186-2-43-4': 'The authors of Ref. [CITATION] found a first order transition for small to intermediate [MATH] signalled by a jump in the quasiparticle weight at [MATH].', 'cond-mat-0411186-2-44-0': 'The NRG results for the inverse effective mass ([MATH] quasiparticle weight) for the case [MATH] are shown in Fig. [REF].', 'cond-mat-0411186-2-44-1': 'The circles were obtained from calculations using the full interaction, while the squares represent calculations with Ising-like interactions.', 'cond-mat-0411186-2-44-2': 'Apparently, the latter signal a strong first order transition at [MATH], while the former lead to a continuously vanishing quasiparticle weight.', 'cond-mat-0411186-2-44-3': 'For a fixed [MATH] (triangles in Fig. [REF]) the quasi-particle weight near [MATH] also shows a jump at [MATH] as predicted by L-DMFT [CITATION].', 'cond-mat-0411186-2-44-4': 'However, the magnitude of this jump comes out much smaller in our calculations.', 'cond-mat-0411186-2-45-0': 'The differences between the different calculations ([MATH], [MATH] and [MATH]) become more apparent when one looks at the local squared moment [MATH].', 'cond-mat-0411186-2-45-1': "For [MATH] this quantity has the value [MATH] (for [MATH] it is actually slightly larger), while deep in the Mott insulator it acquires the atomic value enforced by Hund's coupling, i.e. [MATH] for [MATH], [MATH] for finite full [MATH] and [MATH] for Ising-like [MATH].", 'cond-mat-0411186-2-45-2': 'This behavior is readily found in the calculated values of [MATH] in Fig. [REF].', 'cond-mat-0411186-2-45-3': 'In accordance with the results presented in Ref. [CITATION], the limiting value for [MATH] is approached smoothly for [MATH].', 'cond-mat-0411186-2-45-4': 'The same holds for [MATH], consistent with the results for the quasi-particle weight in Fig. [REF].', 'cond-mat-0411186-2-45-5': 'The slope, however, strongly increases when one approaches [MATH] from below.', 'cond-mat-0411186-2-45-6': 'For constant [MATH], the numerical results are not decisive, and could be interpreted as both a small discontinuity at [MATH] and a continuous approach with diverging slope.', 'cond-mat-0411186-2-45-7': "On the other hand, a quite strong discontinuity at [MATH] appears for an Ising-like Hund's coupling [MATH], signalling a rather strong first order transition in this case.", 'cond-mat-0411186-2-45-8': 'The above results are in rough agreement with the L-DMFT predictions [CITATION], although there one observes a first-order transition also for smaller values of [MATH].', 'cond-mat-0411186-2-46-0': 'The appearance of an unambiguous and rather strong first order transition for an Ising-like exchange coupling shows that in this case the physics underlying the Mott-Hubbard transition is very different from the one for the rotationally invariant exchange interaction.', 'cond-mat-0411186-2-46-1': 'As for the single impurity model, we believe that a transition between individually screened orbitals on the metallic side to a local moment regime enforced by the Ising coupling on the insulating side occurs as soon as [MATH] becomes of the order of the Fermi liquid scale.', 'cond-mat-0411186-2-46-2': 'Depending on the details of the non-interacting DOS (its value at the Fermi level and the band width) and the values of [MATH] and [MATH], this can lead in the worst case to a serious underestimation of [MATH] and possibly an incorrect description of the behavior of physical quantities close to the transition.', 'cond-mat-0411186-2-47-0': '# Summary and conclusions', 'cond-mat-0411186-2-48-0': "In this paper we presented first studies of the Mott-Hubbard transition in a two-orbital Hubbard model within the DMFT at [MATH] using Wilson's NRG.", 'cond-mat-0411186-2-48-1': 'In addition to a standard NRG implementation using the orbital quantum number, we proposed an asymmetric truncation scheme which turns out to work rather well in both the two-orbital single impurity Anderson model and the two-band Hubbard model.', 'cond-mat-0411186-2-49-0': "As a first interesting result, we observed that for the particle-hole symmetric case a finite Hund's exchange [MATH] leads to a tremendous reduction in the low-energy scale [MATH].", 'cond-mat-0411186-2-49-1': 'This is in striking contrast to the result for [MATH], i.e. [MATH], where the behavior conventionally expected for an [MATH] Kondo model, viz [MATH], is found[CITATION].', 'cond-mat-0411186-2-49-2': 'At present, the precise theoretical reason for this rather unexpected strong influence of [MATH] on [MATH] is not clear.', 'cond-mat-0411186-2-49-3': 'Interestingly, it is also rather different from the case [MATH] where we find a mild increase of [MATH] with increasing [MATH].', 'cond-mat-0411186-2-49-4': 'Obviously, a detailed study of the physics of multi-orbital quantum impurity models has to be an important future aim.', 'cond-mat-0411186-2-50-0': 'A completely different behavior occurs if one replaces the rotationally invariant Hund exchange by an Ising-like one.', 'cond-mat-0411186-2-50-1': 'In this case, the levels of the atomic doublet with [MATH] enforced by the Ising coupling cannot be connected by Schrieffer-Wolff type spin-flip processes.', 'cond-mat-0411186-2-50-2': 'Thus, the Kondo effect can only occur for [MATH], while for larger [MATH] the systems is locked into a local moment enforced by the exchange coupling.', 'cond-mat-0411186-2-50-3': 'Note that this interpretation also implies that in the strong-coupling phase the spins on each individual orbital will be screened separately, while for the rotationally invariant case a full [MATH] system must be screened.', 'cond-mat-0411186-2-50-4': 'The details of the quantum phase transition between strong-coupling and local moment phases have not yet been analyzed.', 'cond-mat-0411186-2-50-5': 'However, in view of a possible relevance of an Ising-anisotropy in the presence of crystal fields, a further investigation of this problem is certainly interesting.', 'cond-mat-0411186-2-51-0': 'We also applied the NRG to the two-orbital Hubbard model in the framework of DMFT to investigate the Mott-Hubbard metal-insulator transition at [MATH] for the half-filled case.', 'cond-mat-0411186-2-51-1': "The major goal was here to eludicate the influence of Hund's coupling on the MIT and to investigate how the restriction to an Ising-like exchange changes the nature of the MIT.", 'cond-mat-0411186-2-51-2': 'Our results are in general agreement with previous ones.[', 'cond-mat-0411186-2-51-3': '[CITATION] In particular, for finite [MATH] quantities like the effective mass or [MATH] show diverging slopes as [MATH], possibly even discontinuities as proposed by the L-DMFT.[', 'cond-mat-0411186-2-52-0': "For an Ising-like Hund's exchange coupling the situation becomes qualitatively different.", 'cond-mat-0411186-2-52-1': 'As can be anticipated from the behavior of the impurity model, the MIT is strongly first order with clear jumps in the effective mass and [MATH].', 'cond-mat-0411186-2-52-2': 'Note that the former also implies a discontinuous vanishing of the quasi-particle peak at [MATH] as one reaches [MATH].', 'cond-mat-0411186-2-52-3': 'Also the physics underlying this transition is quite different compared to the rotationally invariant case, reflecting the lack of spin-flip scattering processes connecting the two states [MATH].', 'cond-mat-0411186-2-52-4': 'Thus, the metallic phase with Ising-like interaction will be characterized by individual screening of the spins on the two orbitals rather than a Kondo screening of a total spin [MATH].', 'cond-mat-0411186-2-52-5': 'Note that this subtlety will most likely influence low-energy properties on the metallic side close to [MATH], but be less important for "high-energy" properties like magnetic or orbital ordering.', 'cond-mat-0411186-2-53-0': "It is clear, that the investigations presented here are merely a starting point to systematically study properties of multi-orbital impurity models or correlated lattice models within the DMFT at [MATH] using Wilson's NRG.", 'cond-mat-0411186-2-53-1': 'The major advantage of this method is obviously its unmatched ability to handle exponentially small energy scales and nevertheless provide reliable information on dynamics and thermodynamics even on high-energy scales.', 'cond-mat-0411186-2-53-2': 'Thus, at least for two-orbital models we are now in a position to systematically study their physical properties and address questions that are of fundamental interest for a realistic description of, for example, transition metal oxides but require local degrees of freedom beyond a simple one-band Hubbard model.', 'cond-mat-0411186-2-54-0': 'We acknowledge useful conversations with F. Anders, A. Lichtenstein, M. Vojta, and D. Vollhardt.', 'cond-mat-0411186-2-54-1': 'This work was supported by the DFG through the collaborative research center SFB 484, the Leibniz Computer center, the Computer center of the Max-Planck-Gesellschaft in Garching and the Norddeutsche Verbund fur Hoch- und Hochstleistungsrechnen.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0411186

1807.05692

{'1807.05692-1-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-1-1-0': "Using similar assumptions as in Revuz and Yor's book [CITATION] we prove the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths.", '1807.05692-1-1-1': 'The main tool in our reasonings is a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous price paths.', '1807.05692-1-2-0': 'MSC: 60H20, 91G99', '1807.05692-1-3-0': '# Introduction', '1807.05692-1-4-0': 'The main purpose of this paper is to prove the existence and uniqueness of solutions of differential equations driven by model-free price paths.', '1807.05692-1-4-1': 'From pioneering works by Vovk [CITATION], [CITATION], [CITATION] [CITATION] it is well known that (typical) model-free price paths reveal many properties of local martingales.', '1807.05692-1-4-2': 'For example, it is possible to define quadratic variation as well as model-free version of stochastic integral with respect to cadlag model-free price paths (whose downward jumps satisfy some mild growth condition) [CITATION], [CITATION].', '1807.05692-1-4-3': 'The case of continuous price paths is understood much better than the case of cadlag paths.', '1807.05692-1-4-4': 'For example, for continuous paths there exists model-free versions of the Dambis, Dubins-Schwarz theorem [CITATION], they also possess local times [CITATION].', '1807.05692-1-5-0': 'But even in the space of continuous price paths there are still many topics which need to be understood better.', '1807.05692-1-5-1': 'One of such topics is the existence and uniqueness of solutions of differential equations driven by continuous model-free price paths.', '1807.05692-1-5-2': 'The first results in this direction are proven in [CITATION] even for Hilbert space-valued processes but under the assumption that one can also trade the difference [MATH], where [MATH] denotes the quadratic variation process of the coordinate process [MATH], and the measure [MATH] is majorized by the Lebesgue measure [MATH] multiplied by some constant.', '1807.05692-1-5-3': 'Our approach is different.', '1807.05692-1-5-4': 'We introduce outer expectation [MATH] of a process [MATH], where [MATH] is the space of all continuous functions [MATH] (representing possible evolutions of prices of some financial asset), which may be interpreted as the superhedging cost of not only the terminal value of [MATH], i. e. [MATH], but of any value [MATH], where [MATH] is a stopping time such that [MATH].', '1807.05692-1-5-5': 'Such modification allowed us to obtain a model-free version of the Burkholder-Davis-Gundy inequality (BDG inequality in short) for integrals driven by model-free, continuous price paths in a very direct way, from the pathwise version of the BDG inequality proven in [CITATION].', '1807.05692-1-6-0': 'In this paper we will consider the following SDE (or rather integral equation) driven by model-free continuous price paths [EQUATION] where [MATH] is the coordinate process, [MATH], [MATH] is the natural filtration of [MATH], [MATH] is [MATH] measurable, [MATH] are non-anticipating (the definition of non-anticipating functionals is given in Sect. [REF]) and [MATH] are Lipschitz in the sense that there exists [MATH] such that for all [MATH], [MATH] and [MATH] [EQUATION]', '1807.05692-1-6-1': 'This paper is organized as follows.', '1807.05692-1-6-2': 'In the next section we introduce necessary definitions and notation.', '1807.05692-1-6-3': "In the third section we present and prove a model-free version of the BDG inequality and in the last section we apply this inequality and Picard's iterations to prove the existence and uniqueness of the solution of ([REF]).", '1807.05692-1-7-0': 'We formulate and prove our results for continuous price paths attaining their values in [MATH] but it is possible to prove analogous results for continuous price paths attaining their values in [MATH], [MATH].', '1807.05692-1-7-1': 'Using our methods it is also possible (after small modification of stopping times used in the proof of Theorem [REF]) to prove the existence and uniquenes of the solution of ([REF]) in the case when the first integral in ([REF]) is replaced by the Lebesgue-Stieltjes integral with respect to [MATH] where [MATH] is some adapted, continuous, non-decreasing process such that [MATH].', '1807.05692-1-8-0': '# Definitions and notation', '1807.05692-1-9-0': 'Let [MATH] and [MATH] be the space of continuous functions [MATH] and let [MATH] denotes the the coordinate process.', '1807.05692-1-9-1': '[MATH] is the natural filtration of [MATH].', '1807.05692-1-9-2': 'In the sequel, saying that a process [MATH] is adapted we will mean that it is adapted to [MATH].', '1807.05692-1-10-0': 'Stopping times [MATH] with respect to [MATH] and the corresponding [MATH]-algebras [MATH] are defined as usual.', '1807.05692-1-11-0': 'A process [MATH] is a simple process (simple strategy) if there exist stopping times [MATH] and [MATH]-measurable bounded functions [MATH] such that for every [MATH] from some [MATH] on, and such that [EQUATION]', '1807.05692-1-11-1': 'For such [MATH] we define the corresponding integral process [MATH] [EQUATION] which is well-defined for all [MATH] and all [MATH]; here, for [MATH] we denote [MATH] .', '1807.05692-1-12-0': 'The family of simple strategies will be denoted by [MATH].', '1807.05692-1-12-1': 'For [MATH] a simple strategy [MATH] will be called (strongly) [MATH]-admissible if [MATH] for all [MATH] and all [MATH].', '1807.05692-1-12-2': 'The set of strongly [MATH]-admissible simple strategies will be denoted by [MATH].', '1807.05692-1-13-0': "Vovk's outer measure [MATH] of a set [MATH] is defined as the minimal superhedging price for [MATH], that is [EQUATION].", '1807.05692-1-13-1': 'A set [MATH] is called a null set if it has outer measure zero.', '1807.05692-1-13-2': 'A property (P) holds for typical price paths if the set [MATH] where (P) is violated is a null set.', '1807.05692-1-14-0': 'Now, let [MATH] be the [MATH]th Lebesgue partition of [MATH] which is defined as: [MATH] and for [MATH] [EQUATION]', '1807.05692-1-14-1': 'By convention [MATH].', '1807.05692-1-14-2': 'It is well known (see [CITATION]) that for [MATH] and typical price path [MATH] there exists the continuous limit [EQUATION] and this convergence is uniform in [MATH].', '1807.05692-1-15-0': 'The quadratic variation process of the integral process [MATH] is defined as [EQUATION]', '1807.05692-1-15-1': 'A useful tool which we aim to establish is a model-free version of the BDG inequality.', '1807.05692-1-15-2': 'It will be formulated for the outer expectation which is defined as follows.', '1807.05692-1-15-3': 'Let [MATH] be the family of stopping times [MATH] such that [MATH].', '1807.05692-1-15-4': 'For any process [MATH] we define [EQUATION]', '1807.05692-1-15-5': 'It is straightforward to prove that the expectation [MATH] is countably subadditive, monotone and positively homogeneous.', '1807.05692-1-16-0': 'By [MATH] we denote the family of processes [MATH] such that [EQUATION]', '1807.05692-1-17-0': '# Model-free version of the BDG inequality', '1807.05692-1-18-0': 'For any process [MATH] let us define [MATH].', '1807.05692-1-18-1': 'In this section we establish the following model-free version of the BDG inequality:', '1807.05692-1-19-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-1-20-0': 'PROOF.', '1807.05692-1-20-1': 'Let us recall the pathwise BDG inequalities of Beigblck and Siorpaes ([CITATION]): if for real numbers [MATH] and [MATH] we define [EQUATION] then for any [MATH] there exist positive constant [MATH] and numbers [MATH] such that [MATH] depends only on [MATH] [EQUATION] and such that for any [MATH] one has [EQUATION]', '1807.05692-1-20-2': 'Moreover, for [MATH] one has [MATH], [MATH] and the following estimate also holds [EQUATION]', '1807.05692-1-20-3': 'Let now [MATH] and ([REF]) be its representation.', '1807.05692-1-20-4': 'Let [MATH] be a non-decreasing rearrangement of [MATH], where [MATH] is the [MATH]th Lebesgue partition ([MATH]).', '1807.05692-1-20-5': 'For [MATH] and [MATH] we define [MATH] and for [MATH] [EQUATION].', '1807.05692-1-20-6': 'For [MATH] by [MATH] we denote the unique [MATH] such that [EQUATION].', '1807.05692-1-20-7': 'Let us notice that for typical for typical [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-1-20-8': 'Moreover, by the definition of [MATH] [EQUATION].', '1807.05692-1-20-9': 'We easily identify [EQUATION]', '1807.05692-1-20-10': 'Let us also define the simple strategy [MATH] which just after time [MATH] attains the position [EQUATION] i.e. [EQUATION].', '1807.05692-1-20-11': 'We estimate [EQUATION]', '1807.05692-1-20-12': 'Now, using ([REF]) and ([REF])-([REF]) we see that for [MATH] [EQUATION]', '1807.05692-1-20-13': 'Let now [MATH], and [MATH] be be finite reals such that [EQUATION] and let [MATH] be a sequence of [MATH]-admissible strategies such that [EQUATION]', '1807.05692-1-20-14': 'Now let us define [EQUATION] and let us consider the strategy [EQUATION].', '1807.05692-1-20-15': 'Directly from the definition it follows that [MATH] is [MATH]-admissible.', '1807.05692-1-20-16': 'Moreover, for each [MATH] and [MATH] by ([REF]), ([REF]), the definition of [MATH] and by ([REF]) we have that for sufficiently large [MATH] [EQUATION]', '1807.05692-1-20-17': 'Moreover, since for typical price paths we have that [EQUATION] uniformly in [MATH] then for typical price path [MATH] and [MATH] as [MATH].', '1807.05692-1-20-18': 'As a result we get that [EQUATION] and (since [MATH] may be as close to [MATH] as we please) we obtain ([REF]).', '1807.05692-1-20-19': 'QED', '1807.05692-1-21-0': 'Using ([REF]) and proceeding in a similar way as in the proof of ([REF]) we also get (for [MATH]) the estimate: [EQUATION]', '1807.05692-1-21-1': 'Now, we introduce the space [MATH] of (equivalence classess of) adapted processes [MATH] is equivalent with [MATH] if [MATH]) such that [MATH].', '1807.05692-1-21-2': 'Using standard arguments (see for example [CITATION]) we see that [MATH] equipped with the metric [EQUATION] is a complete metric space and the family of simple processes from [MATH] is dense in [MATH].', '1807.05692-1-22-0': 'Let us notice that for a simple process [MATH], the process [MATH] defined as [EQUATION] is also a simple process and if [MATH] then [EQUATION]', '1807.05692-1-22-1': 'Using this, ([REF]) and completeness of the space [MATH] we see that for any [MATH], any sequence of simple processes [MATH] such that [MATH] in [MATH] (i.e. [MATH]) and any [MATH], the sequence of integrals [MATH] converges in [MATH] to the process which is the integral [MATH].', '1807.05692-1-22-2': 'Moreover, the following analog of ([REF]) holds: [EQUATION] where [MATH], [MATH].', '1807.05692-1-23-0': 'Finally, let us introduce the space [MATH] of processes [MATH] such that for any [MATH], [MATH].', '1807.05692-1-24-0': '# Theorem on existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-1-25-0': 'In this section we prove the existence and uniqueness of the solution of SDE ([REF]).', '1807.05692-1-25-1': 'We will assume the following', '1807.05692-1-26-0': 'Instead of [MATH], [MATH] we will often write [MATH] and [MATH] resp.', '1807.05692-1-27-0': "Now we will use the just obtained model-free version of the BDG inequality and Picard's iterations to prove the following theorem.", '1807.05692-1-28-0': 'Under the assumptions stated above, integral equation ([REF]) has unique solution in the space [MATH].', '1807.05692-1-29-0': '## Existence', '1807.05692-1-30-0': 'Let us set [MATH], [MATH], [EQUATION] and for [MATH] such that [MATH] define [MATH], [EQUATION] (By the assumption on [MATH], ([REF]), ([REF]) and the calculation below (i.e. ([REF])) this definition is correct.)', '1807.05692-1-30-1': 'Now, by the BDG inequality ([REF]) we estimate [EQUATION]', '1807.05692-1-30-2': 'Thus [MATH] is a contraction on [MATH] and it has unique fixed point [MATH].', '1807.05692-1-30-3': 'Next, we define [EQUATION] (we apply the convention that [MATH] if [MATH]), and introduce the following operator [MATH], [EQUATION]', '1807.05692-1-30-4': 'Similarly as before, we prove that [MATH] is a contraction and has a fixed point [MATH].', '1807.05692-1-30-5': 'Moreover, [MATH] and [MATH] agree on the interval [EQUATION].', '1807.05692-1-30-6': 'Similarly, having defined [MATH], [MATH], [MATH], and its fixed point [MATH], [MATH], by induction, we define [EQUATION] and introduce the following operator [MATH], [EQUATION] and its fixed point [MATH] which agrees with [MATH] on the interval [EQUATION].', '1807.05692-1-31-0': 'Finally, setting [EQUATION] we get that [MATH] satisfies [EQUATION]', '1807.05692-1-31-1': 'This follows from the following lemma.', '1807.05692-1-32-0': 'Let [MATH] .', '1807.05692-1-32-1': 'Assume that [MATH] and the following inequalities hold: [EQUATION] for some non-negative integers [MATH] and [MATH].', '1807.05692-1-32-2': 'Then [EQUATION]', '1807.05692-1-32-3': 'PROOF.', '1807.05692-1-32-4': 'The proof follows by easy induction.', '1807.05692-1-32-5': 'For [MATH] from [MATH] and [MATH] it follows that [MATH] and [MATH] thus inequality ([REF]) holds.', '1807.05692-1-32-6': 'Assume that the thesis holds for some [MATH].', '1807.05692-1-32-7': 'Similarly as for [MATH], from the inequality [MATH] and [MATH] it follows that [MATH] and [MATH].', '1807.05692-1-32-8': 'Thus, from [EQUATION] it follows that [EQUATION] or [EQUATION].', '1807.05692-1-32-9': 'In both cases, from the induction hypothesis, [EQUATION] thus [EQUATION].', '1807.05692-1-32-10': 'QED', '1807.05692-1-33-0': '## Uniqueness', '1807.05692-1-34-0': 'In general, we can not guarantee that [MATH] but we will prove that [MATH] and it is the uniqe solution of ([REF]) in [MATH].', '1807.05692-1-35-0': 'First, we will prove that [MATH].', '1807.05692-1-35-1': 'We know that [MATH] for [MATH].', '1807.05692-1-35-2': 'Now, from Lemma [REF] it follows that for any fixed [MATH] and [MATH] we have that either [MATH] or [MATH] (otherwise we would have [EQUATION] and [EQUATION] which by Lemma [REF] would yield [EQUATION] ).', '1807.05692-1-35-3': 'Thus for [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-1-35-4': 'This proves that [MATH].', '1807.05692-1-36-0': 'To prove the uniqueness notice that if [MATH] was not unique in [MATH] then there would exist two processes [MATH] and [MATH] satisfying ([REF]) and such that [EQUATION] for some [MATH].', '1807.05692-1-36-1': 'However, using the same reasoning as in ([REF]) and the fact that [MATH] and [MATH] solve ([REF]) we can prove that [EQUATION] thus [EQUATION].', '1807.05692-1-36-2': 'Similarly, by induction (and subadditivity of [MATH]), we prove that for [MATH] [EQUATION].', '1807.05692-1-36-3': 'Now, for any fixed [MATH] and [MATH] for [MATH] we have [EQUATION] and [EQUATION] thus [EQUATION] which contradicts ([REF]).'}

{'1807.05692-2-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-2-1-0': "Using similar assumptions as in Revuz and Yor's book [CITATION] we prove the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths.", '1807.05692-2-1-1': 'The main tool in our reasonings is a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous price paths.', '1807.05692-2-2-0': 'MSC: 60H20, 91G99', '1807.05692-2-3-0': '# Introduction', '1807.05692-2-4-0': 'The main purpose of this paper is to prove the existence and uniqueness of solutions of differential equations driven by model-free price paths.', '1807.05692-2-4-1': 'From pioneering works by Vovk [CITATION], [CITATION], [CITATION], [CITATION] it is well known that (typical) model-free price paths reveal many properties of local martingales.', '1807.05692-2-4-2': 'For example, it is possible to define quadratic variation as well as model-free version of stochastic integral with respect to cadlag model-free price paths (whose downward jumps satisfy some mild growth condition) [CITATION], [CITATION].', '1807.05692-2-4-3': 'The case of continuous price paths is understood much better than the case of cadlag paths.', '1807.05692-2-4-4': 'For example, for continuous paths there exists a model-free version of the Dambis, Dubins-Schwarz theorem [CITATION], they also possess local times [CITATION].', '1807.05692-2-5-0': 'However, even in the space of continuous price paths there are still many topics which need to be understood better.', '1807.05692-2-5-1': 'One of such topics is the existence and uniqueness of solutions of differential equations driven by continuous model-free price paths.', '1807.05692-2-5-2': 'The first results in this direction are proven in [CITATION] even for Hilbert space-valued processes but under the assumption that one can also trade the difference [MATH], where [MATH] denotes the quadratic variation process of the coordinate process [MATH], and the measure [MATH] is majorized by the Lebesgue measure [MATH] multiplied by some constant.', '1807.05692-2-5-3': 'Our approach is different.', '1807.05692-2-5-4': 'We introduce outer expectation [MATH] of a process [MATH], where [MATH] is the space of all continuous functions [MATH] (representing possible evolutions of prices of some financial asset), which may be interpreted as the superhedging cost of not only the terminal value of [MATH], i. e. [MATH], but of any value [MATH], where [MATH] is a stopping time such that [MATH].', '1807.05692-2-5-5': 'Such modification allowed us to obtain a model-free version of the Burkholder-Davis-Gundy inequality (BDG inequality in short) for integrals driven by model-free, continuous price paths in a very direct way, from the pathwise version of the BDG inequality proven in [CITATION].', '1807.05692-2-6-0': 'In this paper we will consider the following differential equation (or rather integral equation) driven by model-free continuous price paths [EQUATION] where [MATH] is the coordinate process, [MATH], [MATH] is the natural filtration of [MATH], [MATH] is [MATH] measurable, [MATH] are non-anticipating (the definition of non-anticipating functionals is given in Sect. [REF]) and [MATH] are Lipschitz in the sense that there exists [MATH] such that for all [MATH], [MATH] and [MATH] [EQUATION]', '1807.05692-2-6-1': 'This paper is organized as follows.', '1807.05692-2-6-2': 'In the next section we introduce necessary definitions and notation.', '1807.05692-2-6-3': "In the third section we present and prove a model-free version of the BDG inequality and in the last section we apply this inequality and Picard's iterations to prove the existence and uniqueness of the solution of ([REF]).", '1807.05692-2-7-0': 'We formulate and prove our results for continuous price paths attaining their values in [MATH] but it is possible to prove analogous results for continuous price paths attaining their values in [MATH], [MATH].', '1807.05692-2-7-1': 'Using our methods it is also possible (after small modification of stopping times used in the proof of Theorem [REF]) to prove the existence and uniquenes of the solution of ([REF]) in the case when the first integral in ([REF]) is replaced by the Lebesgue-Stieltjes integral with respect to [MATH] where [MATH] is some adapted, continuous, non-decreasing process such that [MATH].', '1807.05692-2-8-0': '# Definitions and notation', '1807.05692-2-9-0': 'Let [MATH] and [MATH] be the space of continuous functions [MATH] and let [MATH] denotes the the coordinate process.', '1807.05692-2-9-1': '[MATH] is the natural filtration of [MATH].', '1807.05692-2-9-2': 'In the sequel, by saying that a process [MATH] is adapted we will mean that it is adapted to [MATH].', '1807.05692-2-10-0': 'Stopping times [MATH] with respect to [MATH] and the corresponding [MATH]-algebras [MATH] are defined as usual.', '1807.05692-2-11-0': 'A process [MATH] is a simple process (simple strategy) if there exist stopping times [MATH] and [MATH]-measurable, bounded functions [MATH] such that for every [MATH] from some [MATH] on, and such that [EQUATION]', '1807.05692-2-11-1': 'For such [MATH] we define the corresponding integral process [MATH] [EQUATION] which is well-defined for all [MATH] and all [MATH]; here, for [MATH] we denote [MATH] .', '1807.05692-2-12-0': 'The family of simple strategies will be denoted by [MATH].', '1807.05692-2-12-1': 'For [MATH] a simple strategy [MATH] will be called (strongly) [MATH]-admissible if [MATH] for all [MATH] and all [MATH].', '1807.05692-2-12-2': 'The set of strongly [MATH]-admissible simple strategies will be denoted by [MATH].', '1807.05692-2-13-0': "Vovk's outer measure [MATH] of a set [MATH] is defined as the minimal superhedging price for [MATH], that is [EQUATION].", '1807.05692-2-13-1': 'A set [MATH] is called a null set if it has outer measure zero.', '1807.05692-2-13-2': 'A property (P) holds for typical price paths if the set [MATH] where (P) is violated is a null set.', '1807.05692-2-14-0': 'Now, let [MATH] be the [MATH]th Lebesgue partition of [MATH] which is defined as: [MATH] and for [MATH] [EQUATION]', '1807.05692-2-14-1': 'By convention [MATH].', '1807.05692-2-14-2': 'It is well known (see [CITATION]) that for [MATH] and a typical price path [MATH] there exists the continuous limit [EQUATION] and this convergence is uniform in [MATH].', '1807.05692-2-15-0': 'The quadratic variation process of the integral process [MATH] is defined as [EQUATION]', '1807.05692-2-15-1': 'A useful tool which we aim to establish is a model-free version of the BDG inequality.', '1807.05692-2-15-2': 'It will be formulated for the outer expectation [MATH] which is defined as follows.', '1807.05692-2-15-3': 'Let [MATH] be the family of stopping times [MATH] such that [MATH].', '1807.05692-2-15-4': 'For any process [MATH] we define [EQUATION]', '1807.05692-2-15-5': 'It is straightforward to prove that the expectation [MATH] is countably subadditive, monotone and positively homogeneous.', '1807.05692-2-16-0': 'By [MATH] we denote the family of processes [MATH] such that [EQUATION]', '1807.05692-2-17-0': '# Model-free version of the BDG inequality', '1807.05692-2-18-0': 'For any process [MATH] let us define [MATH].', '1807.05692-2-18-1': 'In this section we establish the following model-free version of the BDG inequality:', '1807.05692-2-19-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-2-20-0': 'PROOF.', '1807.05692-2-20-1': 'Let us recall the pathwise BDG inequalities of Beigblck and Siorpaes ([CITATION]): if for real numbers [MATH] and [MATH] we define [EQUATION] then for any [MATH] there exist positive constant [MATH] and numbers [MATH] such that [MATH] depends only on [MATH] [EQUATION] and such that for any [MATH] one has [EQUATION]', '1807.05692-2-20-2': 'Moreover, for [MATH] one has [MATH], [MATH] and the following estimate also holds [EQUATION]', '1807.05692-2-20-3': 'Let now [MATH] and ([REF]) be its representation.', '1807.05692-2-20-4': 'Let [MATH] be a non-decreasing rearrangement of [MATH], where [MATH] is the [MATH]th Lebesgue partition ([MATH]).', '1807.05692-2-20-5': 'For [MATH] and [MATH] we define [MATH] and for [MATH] [EQUATION].', '1807.05692-2-20-6': 'For [MATH] by [MATH] we denote the unique [MATH] such that [EQUATION].', '1807.05692-2-20-7': 'Let us notice that for typical for typical [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-2-20-8': 'Moreover, by the definition of [MATH] [EQUATION].', '1807.05692-2-20-9': 'Let us also define the simple strategy [MATH] which just after time [MATH] attains the position [EQUATION] i.e. [EQUATION].', '1807.05692-2-20-10': 'We estimate [EQUATION]', '1807.05692-2-20-11': 'Let now [MATH], [MATH] and [MATH] be finite reals such that [EQUATION] and let [MATH] be a sequence of [MATH]-admissible strategies such that [EQUATION]', '1807.05692-2-20-12': 'Now let us define [EQUATION] and let us consider the strategy [EQUATION].', '1807.05692-2-20-13': 'Directly from the definition it follows that [MATH] is [MATH]-admissible.', '1807.05692-2-20-14': 'Moreover, since for typical price paths we have that [EQUATION] uniformly in [MATH] then for typical price path [MATH] as [MATH].', '1807.05692-2-20-15': 'Thus, for each [MATH] and [MATH] by ([REF]), by the estimate [MATH], ([REF]), the definition of [MATH] and by ([REF]) for sufficiently large [MATH] we have [EQUATION]', '1807.05692-2-20-16': 'As a result we get that [MATH] as [MATH] and [EQUATION].', '1807.05692-2-20-17': 'Since [MATH] may be as close to [MATH] as we please we obtain ([REF]).', '1807.05692-2-20-18': 'QED', '1807.05692-2-21-0': 'Using ([REF]) and proceeding in a similar way as in the proof of ([REF]) we also get (for [MATH]) the estimate: [EQUATION]', '1807.05692-2-21-1': 'Now, we introduce the space [MATH] of (equivalence classess of) adapted processes [MATH] is equivalent with [MATH] if [MATH]) such that [MATH].', '1807.05692-2-21-2': 'Using standard arguments (see for example [CITATION]) we see that [MATH] equipped with the metric [EQUATION] is a complete metric space and the family of simple processes from [MATH] is dense in [MATH].', '1807.05692-2-22-0': 'Let us notice that for a simple process [MATH], the process [MATH] defined as [EQUATION] is also a simple process and if [MATH] then [EQUATION]', '1807.05692-2-22-1': 'Using this, ([REF]) and completeness of the space [MATH] we see that for any [MATH], any sequence of simple processes [MATH] such that [MATH] in [MATH] (i.e. [MATH]) and any [MATH], the sequence of integrals [MATH] converges in [MATH] to the process which is the integral [MATH].', '1807.05692-2-22-2': 'Moreover, the following analog of ([REF]) holds: [EQUATION] where [MATH], [MATH].', '1807.05692-2-23-0': 'Finally, let us introduce the space [MATH] of processes [MATH] such that for any [MATH], [MATH].', '1807.05692-2-24-0': '# Theorem on existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-2-25-0': 'In this section we prove the existence and uniqueness of the solution of SDE ([REF]).', '1807.05692-2-25-1': 'We will assume the following', '1807.05692-2-26-0': 'Instead of [MATH], [MATH] we will often write [MATH] and [MATH] resp.', '1807.05692-2-27-0': "Now we will use the just obtained model-free version of the BDG inequality and Picard's iterations to prove the following theorem.", '1807.05692-2-28-0': 'Under the assumptions stated above, integral equation ([REF]) has unique solution in the space [MATH].', '1807.05692-2-29-0': '## Existence', '1807.05692-2-30-0': 'Let us set [MATH], [MATH], [EQUATION] and for [MATH] such that [MATH] define [MATH], [EQUATION] (By the assumption on [MATH], ([REF]), ([REF]) and the calculation below (i.e. ([REF])) this definition is correct.)', '1807.05692-2-30-1': 'Now, by the BDG inequality ([REF]) we estimate [EQUATION]', '1807.05692-2-30-2': 'Thus [MATH] is a contraction on [MATH] and it has unique fixed point [MATH].', '1807.05692-2-30-3': 'Next, we define [EQUATION] (we apply the convention that [MATH] if [MATH]), and introduce the following operator [MATH], [EQUATION]', '1807.05692-2-30-4': 'Similarly as before, we prove that [MATH] is a contraction and has a fixed point [MATH].', '1807.05692-2-30-5': 'Moreover, [MATH] and [MATH] agree on the interval [EQUATION].', '1807.05692-2-30-6': 'Similarly, having defined [MATH], [MATH], [MATH], and its fixed point [MATH], [MATH], by induction, we define [EQUATION] and introduce the following operator [MATH], [EQUATION] and its fixed point [MATH] which agrees with [MATH] on the interval [EQUATION].', '1807.05692-2-31-0': 'Finally, setting [EQUATION] we get that [MATH] satisfies [EQUATION]', '1807.05692-2-31-1': 'This follows from the following lemma.', '1807.05692-2-32-0': 'Let [MATH] .', '1807.05692-2-32-1': 'Assume that [MATH] and the following inequalities hold: [EQUATION] for some non-negative integers [MATH] and [MATH].', '1807.05692-2-32-2': 'Then [EQUATION]', '1807.05692-2-32-3': 'PROOF.', '1807.05692-2-32-4': 'The proof follows by easy induction.', '1807.05692-2-32-5': 'For [MATH] from [MATH] and [MATH] it follows that [MATH] and [MATH] thus inequality ([REF]) holds.', '1807.05692-2-32-6': 'Assume that the thesis holds for some [MATH].', '1807.05692-2-32-7': 'Similarly as for [MATH], from the inequality [MATH] and [MATH] it follows that [MATH] and [MATH].', '1807.05692-2-32-8': 'Thus, from [EQUATION] it follows that [EQUATION] or [EQUATION].', '1807.05692-2-32-9': 'In both cases, from the induction hypothesis, [EQUATION] thus [EQUATION].', '1807.05692-2-32-10': 'QED', '1807.05692-2-33-0': '## Uniqueness', '1807.05692-2-34-0': 'In general, we can not guarantee that [MATH] but we will prove that [MATH] and it is the uniqe solution of ([REF]) in [MATH].', '1807.05692-2-35-0': 'First, we will prove that [MATH].', '1807.05692-2-35-1': 'We know that [MATH] for [MATH].', '1807.05692-2-35-2': 'Now, from Lemma [REF] it follows that for any fixed [MATH] and [MATH] we have that either [MATH] or [MATH] (otherwise we would have [EQUATION] and [EQUATION] which by Lemma [REF] would yield [EQUATION] ).', '1807.05692-2-35-3': 'Thus for [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-2-35-4': 'This proves that [MATH].', '1807.05692-2-36-0': 'To prove the uniqueness notice that if [MATH] was not unique in [MATH] then there would exist two processes [MATH] and [MATH] satisfying ([REF]) and such that [EQUATION] for some [MATH].', '1807.05692-2-36-1': 'However, using the same reasoning as in ([REF]) and the fact that [MATH] and [MATH] solve ([REF]) we can prove that [EQUATION] thus [EQUATION].', '1807.05692-2-36-2': 'Similarly, by induction (and subadditivity of [MATH]), we prove that for [MATH] [EQUATION].', '1807.05692-2-36-3': 'Now, for any fixed [MATH] and [MATH] for [MATH] we have [EQUATION] and [EQUATION] thus [EQUATION] which contradicts ([REF]).'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '5': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1807.05692

{'1807.05692-3-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-3-1-0': "Using similar assumptions as in Revuz and Yor's book [CITATION] we prove the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths.", '1807.05692-3-1-1': 'The main tool in our reasonings is a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous price paths.', '1807.05692-3-2-0': 'MSC: 60H20, 91G99', '1807.05692-3-3-0': '# Introduction', '1807.05692-3-4-0': 'The main purpose of this paper is to prove the existence and uniqueness of solutions of differential equations driven by model-free price paths.', '1807.05692-3-4-1': 'From pioneering works by Vovk [CITATION], [CITATION], [CITATION], [CITATION] it is well known that (typical) model-free price paths reveal many properties of local martingales.', '1807.05692-3-4-2': 'For example, it is possible to define quadratic variation as well as model-free version of stochastic integral with respect to cadlag model-free price paths (whose downward jumps satisfy some mild growth condition) [CITATION], [CITATION].', '1807.05692-3-4-3': 'The case of continuous price paths is understood much better than the case of cadlag paths.', '1807.05692-3-4-4': 'For example, for continuous paths there exists a model-free version of the Dambis, Dubins-Schwarz theorem [CITATION], they also possess local times [CITATION].', '1807.05692-3-5-0': 'However, even in the space of continuous price paths there are still many topics which need to be understood better.', '1807.05692-3-5-1': 'One of such topics is the existence and uniqueness of solutions of differential equations driven by continuous model-free price paths.', '1807.05692-3-5-2': 'The first results in this direction are proven in [CITATION] even for Hilbert space-valued processes but under the assumption that one can also trade the difference [MATH], where [MATH] denotes the quadratic variation process of the coordinate process [MATH], and the measure [MATH] is majorized by the Lebesgue measure [MATH] multiplied by some constant.', '1807.05692-3-5-3': 'Our approach is different.', '1807.05692-3-5-4': 'We introduce outer expectation [MATH] of a process [MATH], where [MATH] is the space of all continuous functions [MATH] (representing possible evolutions of prices of some financial asset), which may be interpreted as the superhedging cost of not only the terminal value of [MATH], i. e. [MATH], but of any value [MATH], where [MATH] is a stopping time such that [MATH].', '1807.05692-3-5-5': 'Such modification allowed us to obtain a model-free version of the Burkholder-Davis-Gundy inequality (BDG inequality in short) for integrals driven by model-free, continuous price paths in a very direct way, from the pathwise version of the BDG inequality proven in [CITATION].', '1807.05692-3-6-0': 'In this paper we will consider the following differential equation (or rather integral equation) driven by model-free continuous price paths [EQUATION] where [MATH] is the coordinate process, [MATH], [MATH] is the natural filtration of [MATH], [MATH] is [MATH] measurable, [MATH] are non-anticipating (the definition of non-anticipating functionals is given in Sect. [REF]) and [MATH] are Lipschitz in the sense that there exists [MATH] such that for all [MATH], [MATH] and [MATH] [EQUATION]', '1807.05692-3-6-1': 'This paper is organized as follows.', '1807.05692-3-6-2': 'In the next section we introduce necessary definitions and notation.', '1807.05692-3-6-3': "In the third section we present and prove a model-free version of the BDG inequality and in the last section we apply this inequality and Picard's iterations to prove the existence and uniqueness of the solution of ([REF]).", '1807.05692-3-7-0': 'We formulate and prove our results for continuous price paths attaining their values in [MATH] but it is possible to prove analogous results for continuous price paths attaining their values in [MATH], [MATH].', '1807.05692-3-7-1': 'Using our methods it is also possible (after small modification of stopping times used in the proof of Theorem [REF]) to prove the existence and uniquenes of the solution of ([REF]) in the case when the first integral in ([REF]) is replaced by the Lebesgue-Stieltjes integral with respect to [MATH] where [MATH] is some adapted, continuous, non-decreasing process such that [MATH].', '1807.05692-3-8-0': '# Definitions and notation', '1807.05692-3-9-0': 'Let [MATH] and [MATH] be the space of continuous functions [MATH] and let [MATH] denotes the the coordinate process.', '1807.05692-3-9-1': '[MATH] is the natural filtration of [MATH].', '1807.05692-3-9-2': 'In the sequel, by saying that a process [MATH] is adapted we will mean that it is adapted to [MATH].', '1807.05692-3-10-0': 'Stopping times [MATH] with respect to [MATH] and the corresponding [MATH]-algebras [MATH] are defined as usual.', '1807.05692-3-11-0': 'A process [MATH] is a simple process (simple strategy) if there exist stopping times [MATH] and [MATH]-measurable, bounded functions [MATH] such that for every [MATH] from some [MATH] on, and such that [EQUATION]', '1807.05692-3-11-1': 'For such [MATH] we define the corresponding integral process [MATH] [EQUATION] which is well-defined for all [MATH] and all [MATH]; here, for [MATH] we denote [MATH] .', '1807.05692-3-12-0': 'The family of simple strategies will be denoted by [MATH].', '1807.05692-3-12-1': 'For [MATH] a simple strategy [MATH] will be called (strongly) [MATH]-admissible if [MATH] for all [MATH] and all [MATH].', '1807.05692-3-12-2': 'The set of strongly [MATH]-admissible simple strategies will be denoted by [MATH].', '1807.05692-3-13-0': "Vovk's outer measure [MATH] of a set [MATH] is defined as the minimal superhedging price for [MATH], that is [EQUATION].", '1807.05692-3-13-1': 'A set [MATH] is called a null set if it has outer measure zero.', '1807.05692-3-13-2': 'A property (P) holds for typical price paths if the set [MATH] where (P) is violated is a null set.', '1807.05692-3-14-0': 'Now, let [MATH] be the [MATH]th Lebesgue partition of [MATH] which is defined as: [MATH] and for [MATH] [EQUATION]', '1807.05692-3-14-1': 'By convention [MATH].', '1807.05692-3-14-2': 'It is well known (see [CITATION]) that for [MATH] and a typical price path [MATH] there exists the continuous limit [EQUATION] and this convergence is uniform in [MATH].', '1807.05692-3-15-0': 'The quadratic variation process of the integral process [MATH] is defined as [EQUATION]', '1807.05692-3-15-1': 'A useful tool which we aim to establish is a model-free version of the BDG inequality.', '1807.05692-3-15-2': 'It will be formulated for the outer expectation [MATH] which is defined as follows.', '1807.05692-3-15-3': 'Let [MATH] be the family of stopping times [MATH] such that [MATH].', '1807.05692-3-15-4': 'For any process [MATH] we define [EQUATION] where the first infimum is over all [MATH] such that [MATH].', '1807.05692-3-15-5': 'It is straightforward to prove that the expectation [MATH] is countably subadditive, monotone and positively homogeneous.', '1807.05692-3-16-0': 'By [MATH] we denote the family of processes [MATH] such that [EQUATION]', '1807.05692-3-17-0': '# Model-free version of the BDG inequality', '1807.05692-3-18-0': 'For any process [MATH] let us define [MATH].', '1807.05692-3-18-1': 'In this section we establish the following model-free version of the BDG inequality:', '1807.05692-3-19-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-3-20-0': 'PROOF.', '1807.05692-3-20-1': 'Let us recall the pathwise BDG inequalities of Beigblck and Siorpaes ([CITATION]): if for real numbers [MATH] and [MATH] we define [EQUATION] then for any [MATH] there exist positive constant [MATH] and numbers [MATH] such that [MATH] depends only on [MATH] [EQUATION] and such that for any [MATH] one has [EQUATION]', '1807.05692-3-20-2': 'Moreover, for [MATH] one has [MATH], [MATH] and the following estimate also holds [EQUATION]', '1807.05692-3-20-3': 'Let now [MATH] and ([REF]) be its representation.', '1807.05692-3-20-4': 'Let [MATH] be a non-decreasing rearrangement of [MATH], where [MATH] is the [MATH]th Lebesgue partition ([MATH]).', '1807.05692-3-20-5': 'For [MATH] and [MATH] we define [MATH] and for [MATH] [EQUATION].', '1807.05692-3-20-6': 'For [MATH] by [MATH] we denote the unique [MATH] such that [EQUATION].', '1807.05692-3-20-7': 'Let us notice that for typical for typical [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-3-20-8': 'Moreover, by the definition of [MATH] [EQUATION].', '1807.05692-3-20-9': 'Let us also define the simple strategy [MATH] which just after time [MATH] attains the position [EQUATION] i.e. [EQUATION].', '1807.05692-3-20-10': 'We estimate [EQUATION]', '1807.05692-3-20-11': 'Let now [MATH], [MATH] and [MATH] be finite reals such that [EQUATION] and let [MATH] be a sequence of [MATH]-admissible strategies such that [EQUATION]', '1807.05692-3-20-12': 'Now let us define [EQUATION] and let us consider the strategy [EQUATION].', '1807.05692-3-20-13': 'Directly from the definition it follows that [MATH] is [MATH]-admissible.', '1807.05692-3-20-14': 'Moreover, since for typical price paths we have that [EQUATION] uniformly in [MATH] then for typical price path [MATH] as [MATH].', '1807.05692-3-20-15': 'Thus, for each [MATH] and [MATH] by ([REF]), by the estimate [MATH], ([REF]), the definition of [MATH] and by ([REF]) for sufficiently large [MATH] we have [EQUATION]', '1807.05692-3-20-16': 'As a result we get that [MATH] as [MATH] and [EQUATION].', '1807.05692-3-20-17': 'Since [MATH] may be as close to [MATH] as we please we obtain ([REF]).', '1807.05692-3-20-18': 'QED', '1807.05692-3-21-0': 'Using ([REF]) and proceeding in a similar way as in the proof of ([REF]) we also get (for [MATH]) the estimate: [EQUATION]', '1807.05692-3-21-1': 'Now, we introduce the space [MATH] of (equivalence classess of) adapted processes [MATH] is equivalent with [MATH] if [MATH]) such that [MATH].', '1807.05692-3-21-2': 'Using standard arguments (see for example [CITATION]) we see that [MATH] equipped with the metric [EQUATION] is a complete metric space and the family of simple processes from [MATH] is dense in [MATH].', '1807.05692-3-22-0': 'Let us notice that for a simple process [MATH], the process [MATH] defined as [EQUATION] is also a simple process and if [MATH] then [EQUATION]', '1807.05692-3-22-1': 'Using this, ([REF]) and completeness of the space [MATH] we see that for any [MATH], any sequence of simple processes [MATH] such that [MATH] in [MATH] (i.e. [MATH]) and any [MATH], the sequence of integrals [MATH] converges in [MATH] to the process which is the integral [MATH].', '1807.05692-3-22-2': 'Moreover, the following analog of ([REF]) holds: [EQUATION] where [MATH], [MATH].', '1807.05692-3-23-0': 'Finally, let us introduce the space [MATH] of processes [MATH] such that for any [MATH], [MATH].', '1807.05692-3-24-0': '# Theorem on existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-3-25-0': 'In this section we prove the existence and uniqueness of the solution of SDE ([REF]).', '1807.05692-3-25-1': 'We will assume the following', '1807.05692-3-26-0': 'Instead of [MATH], [MATH] we will often write [MATH] and [MATH] resp.', '1807.05692-3-27-0': "Now we will use the just obtained model-free version of the BDG inequality and Picard's iterations to prove the following theorem.", '1807.05692-3-28-0': 'Under the assumptions stated above, integral equation ([REF]) has unique solution in the space [MATH].', '1807.05692-3-29-0': '## Existence', '1807.05692-3-30-0': 'Let us set [MATH], [MATH], [EQUATION] and for [MATH] such that [MATH] define [MATH], [EQUATION] (By the assumption on [MATH], ([REF]), ([REF]) and the calculation below (i.e. ([REF])) this definition is correct.)', '1807.05692-3-30-1': 'Now, by the BDG inequality ([REF]) we estimate [EQUATION]', '1807.05692-3-30-2': 'Thus [MATH] is a contraction on [MATH] and it has unique fixed point [MATH].', '1807.05692-3-30-3': 'Next, we define [EQUATION] (we apply the convention that [MATH] if [MATH]), and introduce the following operator [MATH], [EQUATION]', '1807.05692-3-30-4': 'Similarly as before, we prove that [MATH] is a contraction and has a fixed point [MATH].', '1807.05692-3-30-5': 'Moreover, [MATH] and [MATH] agree on the interval [EQUATION].', '1807.05692-3-30-6': 'Similarly, having defined [MATH], [MATH], [MATH], and its fixed point [MATH], [MATH], by induction, we define [EQUATION] and introduce the following operator [MATH], [EQUATION] and its fixed point [MATH] which agrees with [MATH] on the interval [EQUATION].', '1807.05692-3-31-0': 'Finally, setting [EQUATION] we get that [MATH] satisfies [EQUATION]', '1807.05692-3-31-1': 'This follows from the following lemma.', '1807.05692-3-32-0': 'Let [MATH] .', '1807.05692-3-32-1': 'Assume that [MATH] and the following inequalities hold: [EQUATION] for some non-negative integers [MATH] and [MATH].', '1807.05692-3-32-2': 'Then [EQUATION]', '1807.05692-3-32-3': 'PROOF.', '1807.05692-3-32-4': 'The proof follows by easy induction.', '1807.05692-3-32-5': 'For [MATH] from [MATH] and [MATH] it follows that [MATH] and [MATH] thus inequality ([REF]) holds.', '1807.05692-3-32-6': 'Assume that the thesis holds for some [MATH].', '1807.05692-3-32-7': 'Similarly as for [MATH], from the inequality [MATH] and [MATH] it follows that [MATH] and [MATH].', '1807.05692-3-32-8': 'Thus, from [EQUATION] it follows that [EQUATION] or [EQUATION].', '1807.05692-3-32-9': 'In both cases, from the induction hypothesis, [EQUATION] thus [EQUATION].', '1807.05692-3-32-10': 'QED', '1807.05692-3-33-0': '## Uniqueness', '1807.05692-3-34-0': 'In general, we can not guarantee that [MATH] but we will prove that [MATH] and it is the uniqe solution of ([REF]) in [MATH].', '1807.05692-3-35-0': 'First, we will prove that [MATH].', '1807.05692-3-35-1': 'We know that [MATH] for [MATH].', '1807.05692-3-35-2': 'Now, from Lemma [REF] it follows that for any fixed [MATH] and [MATH] we have that either [MATH] or [MATH] (otherwise we would have [EQUATION] and [EQUATION] which by Lemma [REF] would yield [EQUATION] ).', '1807.05692-3-35-3': 'Thus for [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-3-35-4': 'This proves that [MATH].', '1807.05692-3-36-0': 'To prove the uniqueness notice that if [MATH] was not unique in [MATH] then there would exist two processes [MATH] and [MATH] satisfying ([REF]) and such that [EQUATION] for some [MATH].', '1807.05692-3-36-1': 'However, using the same reasoning as in ([REF]) and the fact that [MATH] and [MATH] solve ([REF]) we can prove that [EQUATION] thus [EQUATION].', '1807.05692-3-36-2': 'Similarly, by induction (and subadditivity of [MATH]), we prove that for [MATH] [EQUATION].', '1807.05692-3-36-3': 'Now, for any fixed [MATH] and [MATH] for [MATH] we have [EQUATION] and [EQUATION] thus [EQUATION] which contradicts ([REF]).'}

{'1807.05692-4-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-4-1-0': "Using similar assumptions as in Revuz and Yor's book [CITATION] we prove the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths.", '1807.05692-4-1-1': 'The main tool in our reasonings is a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous price paths.', '1807.05692-4-2-0': 'MSC: 60H20, 91G99', '1807.05692-4-3-0': '# Introduction', '1807.05692-4-4-0': 'The main purpose of this paper is to prove the existence and uniqueness of solutions of differential equations driven by model-free price paths.', '1807.05692-4-4-1': 'From pioneering works by Vovk [CITATION], [CITATION], [CITATION], [CITATION] it is well known that (typical) model-free price paths reveal many properties of local martingales.', '1807.05692-4-4-2': 'For example, it is possible to define quadratic variation as well as model-free version of stochastic integral with respect to cadlag model-free price paths (whose downward jumps satisfy some mild growth condition) [CITATION], [CITATION].', '1807.05692-4-4-3': 'The case of continuous price paths is understood much better than the case of cadlag paths.', '1807.05692-4-4-4': 'For example, for continuous paths there exists a model-free version of the Dambis, Dubins-Schwarz theorem [CITATION], they also possess local times [CITATION].', '1807.05692-4-5-0': 'However, even in the space of continuous price paths there are still many topics which need to be understood better.', '1807.05692-4-5-1': 'One of such topics is the existence and uniqueness of solutions of differential equations driven by continuous model-free price paths.', '1807.05692-4-5-2': 'The first results in this direction are proven in [CITATION] even for Hilbert space-valued processes but under the assumption that one can also trade the difference [MATH], where [MATH] denotes the norm in the Hilbert space and [MATH] denotes the quadratic variation process of the coordinate process [MATH] (but defined in a different way than the usual tensor quadratic variation of a Hilbert space-valued semimartingale, see [CITATION]), and the measure [MATH] is majorized by the Lebesgue measure [MATH] multiplied by some constant.', '1807.05692-4-5-3': 'Our approach is different.', '1807.05692-4-5-4': 'We introduce outer expectation [MATH] of a process [MATH], where [MATH] is the space of all continuous functions [MATH] (representing possible evolutions of prices of [MATH] financial assets), which may be interpreted as the superhedging cost of not only the terminal value of [MATH], i. e. [MATH], but of any value [MATH], where [MATH] is a stopping time such that [MATH].', '1807.05692-4-5-5': 'Such modification allowed us to obtain a model-free version of the Burkholder-Davis-Gundy inequality (BDG inequality in short) for integrals driven by model-free, continuous price paths in a very direct way, from the pathwise version of the BDG inequality proven in [CITATION].', '1807.05692-4-6-0': 'In this paper we will consider the following differential equation (or rather integral equation) driven by model-free continuous price paths [EQUATION] where [MATH] is a stochastic process starting from [MATH] (i.e. [MATH]), [MATH] and [MATH], [MATH] are continuous, non-decreasing processes, starting from [MATH] such for all [MATH], [MATH], where [MATH] is a deterministic constant, [MATH] is the coordinate process, [MATH], [MATH] is the natural filtration of [MATH], [MATH] is [MATH] measurable, [MATH] and [MATH] are non-anticipating (the definition of non-anticipating functionals is given in Sect. [REF]) and [MATH] are Lipschitz in the sense that there exists [MATH] such that for all [MATH], [MATH] and [MATH] [EQUATION] (where [MATH] denotes the Euclidean norm on [MATH]).', '1807.05692-4-7-0': 'This paper is organized as follows.', '1807.05692-4-7-1': 'In the next section we introduce necessary definitions and notation.', '1807.05692-4-7-2': "In the third section we present and prove a model-free version of the BDG inequality and in the last section we apply this inequality and Picard's iterations to prove the existence and uniqueness of the solution of ([REF]).", '1807.05692-4-8-0': '# Definitions and notation', '1807.05692-4-9-0': 'Let [MATH], [MATH] and [MATH] be the space of continuous functions [MATH] and let [MATH] denotes the the coordinate process.', '1807.05692-4-9-1': '[MATH] is the natural filtration of [MATH].', '1807.05692-4-9-2': 'In the sequel, by saying that a process [MATH] is adapted we will mean that it is adapted to [MATH].', '1807.05692-4-9-3': 'We will assume that the processes [MATH] and [MATH] introduced in the Introduction are adapted.', '1807.05692-4-10-0': 'Stopping times [MATH] with respect to [MATH] and the corresponding [MATH]-algebras [MATH] are defined as usual.', '1807.05692-4-11-0': 'A process [MATH] is a simple process (simple strategy) if there exist stopping times [MATH] and [MATH]-measurable, bounded functions [MATH] such that for every [MATH] from some [MATH] on, and such that [EQUATION]', '1807.05692-4-11-1': 'For such [MATH] we define the corresponding integral process [MATH] [EQUATION] which is well-defined for all [MATH] and all [MATH]; here, for [MATH] we denote [MATH] and [MATH] denotes the inner product on [MATH].', '1807.05692-4-12-0': 'The family of simple strategies will be denoted by [MATH].', '1807.05692-4-12-1': 'For [MATH] a simple strategy [MATH] will be called (strongly) [MATH]-admissible if [MATH] for all [MATH] and all [MATH].', '1807.05692-4-12-2': 'The set of strongly [MATH]-admissible simple strategies will be denoted by [MATH].', '1807.05692-4-13-0': "Vovk's outer measure [MATH] of a set [MATH] is defined as the minimal superhedging price for [MATH], that is [EQUATION].", '1807.05692-4-13-1': 'A set [MATH] is called a null set if it has outer measure zero.', '1807.05692-4-13-2': 'A property (P) holds for typical price paths if the set [MATH] where (P) is violated is a null set.', '1807.05692-4-14-0': 'Next, we define a special sequence of partitions called the Lebesgue partitions generated by [MATH].', '1807.05692-4-14-1': 'The [MATH]th ([MATH]) Lebesgue partition [MATH] of [MATH] generated by [MATH] is defined for each [MATH] as: [MATH] and for [MATH] [EQUATION]', '1807.05692-4-14-2': 'By convention [MATH].', '1807.05692-4-14-3': 'Similarly, replacing [MATH] by [MATH] we define the Lebesgue partitions [MATH] generated by [MATH] for each [MATH].', '1807.05692-4-15-0': 'Next, we define the sequence of Lebesgue partitions generated by [MATH] as: [MATH] and for [MATH] [EQUATION].', '1807.05692-4-16-0': 'It is well known (see [CITATION]) that for [MATH] and a typical price path [MATH] there exists the continuous limit [EQUATION] and this convergence is uniform in [MATH].', '1807.05692-4-17-0': 'We will use the following notation: [MATH] and [EQUATION]', '1807.05692-4-17-1': 'The quadratic variation process of the real integral process [MATH] is defined as [EQUATION]', '1807.05692-4-17-2': 'For any process [MATH]) let us define [EQUATION] (and [MATH] is the Euclidean norm on [MATH]).', '1807.05692-4-17-3': 'We have the following estimate.', '1807.05692-4-18-0': 'Let [MATH], [MATH] and [MATH] be defined as [EQUATION].', '1807.05692-4-18-1': 'Then for any [MATH] [EQUATION].', '1807.05692-4-19-0': 'Using the inequality [MATH] (which follows from [MATH], [MATH]) and [MATH], we obtain [EQUATION]', '1807.05692-4-19-1': 'A useful tool which we aim to establish is a model-free version of the BDG inequality.', '1807.05692-4-19-2': 'It will be formulated for the outer expectation [MATH] which is defined as follows.', '1807.05692-4-19-3': 'Let [MATH] be the family of stopping times [MATH] such that [MATH].', '1807.05692-4-19-4': 'For any process [MATH] we define [EQUATION] where the first infimum is over all subsets [MATH] of typical paths, that is all [MATH] such that [MATH].', '1807.05692-4-19-5': 'It is straightforward to prove that the expectation [MATH] is countably subadditive, monotone and positively homogeneous.', '1807.05692-4-19-6': 'By [MATH] we denote the family of processes [MATH] such that [EQUATION].', '1807.05692-4-19-7': 'The outer expectation of non-negative processes which we have just introduced is not the same as the outer expectation of variables introduced by Vovk in [CITATION], where one looks only at the value of [MATH] at the terminal time.', '1807.05692-4-19-8': 'However, similar definitions to our already exist in literature in the context of pricing American options, see for example [CITATION].', '1807.05692-4-20-0': '# Model-free version of the BDG inequality', '1807.05692-4-21-0': 'In this section we establish the following model-free version of the BDG inequality:', '1807.05692-4-22-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-4-23-0': 'Let us recall the pathwise BDG inequalities of Beiglblock and Siorpaes ([CITATION]): if for real numbers [MATH] and [MATH] we define [EQUATION] then for any [MATH] there exist positive constant [MATH] and numbers [MATH] such that [MATH] depends only on [MATH] [EQUATION] and such that for any [MATH] one has [EQUATION]', '1807.05692-4-23-1': 'Moreover, for [MATH] one has [MATH], [MATH] and the following estimate also holds [EQUATION]', '1807.05692-4-23-2': 'Let now [MATH] and ([REF]) be its representation.', '1807.05692-4-23-3': 'Let [MATH] be a non-decreasing rearrangement of [MATH], where [MATH] is the [MATH]th Lebesgue partition ([MATH]).', '1807.05692-4-23-4': 'For [MATH] and [MATH] we define [MATH] and for [MATH] [EQUATION].', '1807.05692-4-23-5': 'For [MATH] by [MATH] we denote the unique [MATH] such that [EQUATION].', '1807.05692-4-23-6': 'Let us notice that for typical [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-4-23-7': 'Moreover, by the definition of [MATH] [EQUATION].', '1807.05692-4-23-8': 'Let us also define the simple strategy [MATH] which just after time [MATH] attains the position [EQUATION] i.e. [EQUATION].', '1807.05692-4-23-9': 'We estimate [EQUATION]', '1807.05692-4-23-10': 'Let now [MATH], [MATH] and [MATH] be finite reals such that [EQUATION] and let [MATH] be a sequence of [MATH]-admissible strategies such that [EQUATION] where [MATH] is a set of typical paths.', '1807.05692-4-23-11': 'Now let us define [EQUATION] and let us consider the strategy [EQUATION].', '1807.05692-4-23-12': 'Directly from the definition it follows that [MATH] is [MATH]-admissible.', '1807.05692-4-23-13': 'Moreover, for typical price paths we have that [EQUATION] uniformly in [MATH].', '1807.05692-4-23-14': 'Let [MATH] denotes the set of such paths.', '1807.05692-4-23-15': 'We have [MATH] and for each [MATH], [MATH] as [MATH].', '1807.05692-4-23-16': 'Thus for each [MATH] and [MATH] by ([REF]), by the estimate [MATH], ([REF]), the definition of [MATH] and by ([REF]), for sufficiently large [MATH] we have [EQUATION]', '1807.05692-4-23-17': 'As a result we get that for [MATH], [MATH] as [MATH] and [EQUATION].', '1807.05692-4-23-18': 'Since [MATH] may be as close to [MATH] as we please, we obtain ([REF]).', '1807.05692-4-24-0': 'Using ([REF]) and proceeding in a similar way as in the proof of ([REF]) we also get (for [MATH]) the estimate: [EQUATION]', '1807.05692-4-25-0': '## Multidimensional version of the model-free BDG inequality', '1807.05692-4-26-0': 'In this section we will prove the model-free BDG inequality in the case when [MATH] is a matrix-valued, simple process, i.e. [MATH] and [MATH] where [MATH], [MATH].', '1807.05692-4-26-1': 'The family of such processes will be denoted by [MATH].', '1807.05692-4-26-2': 'For [MATH] we define the integral [MATH] as the vector of integrals [EQUATION].', '1807.05692-4-26-3': 'Also, similarly to ([REF]) we define [EQUATION].', '1807.05692-4-26-4': 'Now we have the following generalisation of ([REF]).', '1807.05692-4-27-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-4-28-0': 'Using the inequality [MATH] valid for any real [MATH] we estimate for [MATH] [EQUATION].', '1807.05692-4-28-1': 'Next, using subadditivity of [MATH], ([REF]) and the monotonicity of [MATH] we obtain [EQUATION]', '1807.05692-4-29-0': '## Spaces [MATH], [MATH], [MATH] and [MATH]', '1807.05692-4-30-0': 'Now, we introduce the space [MATH] (resp. [MATH]) of (equivalence classes of) adapted processes [MATH] (resp. [MATH] is equivalent with [MATH] if [MATH]) such that [MATH].', '1807.05692-4-30-1': 'Using standard arguments (see for example [CITATION]) we see that [MATH] (resp. [MATH]) equipped with the metric [EQUATION] is a complete metric space and the family of simple processes from [MATH] (resp. [MATH]) is dense in [MATH] (resp. [MATH]).', '1807.05692-4-31-0': 'Let us notice that for a simple process [MATH] and [MATH] the process [MATH] defined as [EQUATION] is also a simple process and if [MATH] then using Lemma [REF] we get [EQUATION]', '1807.05692-4-31-1': 'Similarly, if [MATH] is a simple process we get [EQUATION]', '1807.05692-4-31-2': 'Using this, Proposition [REF] and completeness of the space [MATH] (resp. [MATH]) we see that for any [MATH] (resp. [MATH]), any sequence of simple processes [MATH] (resp. [MATH]) such that [MATH] in [MATH] (resp. [MATH]) (i.e. [MATH]) and any [MATH], the sequence of integrals [MATH] converges in [MATH] (resp. in [MATH]) to the process which is the integral [MATH].', '1807.05692-4-31-3': 'Moreover, the following analog of Proposition [REF] holds: [EQUATION]', '1807.05692-4-31-4': 'Finally, let us introduce the space [MATH] (resp. [MATH]) of processes [MATH] such that for any [MATH], [MATH] (resp. [MATH]).', '1807.05692-4-32-0': '# Theorem on existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-4-33-0': 'In this section we prove the existence and uniqueness of the solution of SDE ([REF]).', '1807.05692-4-33-1': 'Together with the assumptions stated in the previous sections we will assume the following:', '1807.05692-4-34-0': 'Instead of [MATH], [MATH] we will often write [MATH] and [MATH] resp.', '1807.05692-4-35-0': "Now we will use the just obtained model-free version of the BDG inequality and Picard's iterations to prove the following theorem.", '1807.05692-4-36-0': 'Under the assumptions stated above, integral equation ([REF]) has unique solution in the space [MATH].', '1807.05692-4-37-0': '## Existence', '1807.05692-4-38-0': 'Let us set [MATH], [MATH], [EQUATION] and for [MATH] such that [MATH] define [MATH], [EQUATION] (By the assumption on [MATH], ([REF]), ([REF]) and the calculation below (i.e. ([REF])) this definition is correct.)', '1807.05692-4-38-1': 'Now, by the Lipschitz property, the BDG inequality ([REF]) and Lemma [REF] we estimate [EQUATION]', '1807.05692-4-38-2': 'Thus [MATH] is a contraction on [MATH] and it has unique fixed point [MATH].', '1807.05692-4-38-3': 'Next, we define [EQUATION] (we apply the convention that [MATH] if [MATH]), and introduce the following operator [MATH], [EQUATION]', '1807.05692-4-38-4': 'Similarly as before, we prove that [MATH] is a contraction and has a fixed point [MATH].', '1807.05692-4-38-5': 'Moreover, [MATH] and [MATH] agree on the interval [EQUATION].', '1807.05692-4-38-6': 'Similarly, having defined [MATH], [MATH], [MATH], and its fixed point [MATH], [MATH], by induction, we define [EQUATION] and introduce the following operator [MATH], [EQUATION] and its fixed point [MATH] which agrees with [MATH] on the interval [EQUATION].', '1807.05692-4-39-0': 'Finally, setting [EQUATION] we get that [MATH] satisfies [EQUATION]', '1807.05692-4-39-1': 'This follows from the following lemma.', '1807.05692-4-40-0': 'Let [MATH] .', '1807.05692-4-40-1': 'Assume that [MATH] and the following inequalities hold: [EQUATION] for some non-negative integers [MATH] and [MATH].', '1807.05692-4-40-2': 'Then [EQUATION]', '1807.05692-4-40-3': 'The proof follows by easy induction.', '1807.05692-4-40-4': 'For [MATH] from [MATH] and [MATH] it follows that [MATH] and [MATH] thus inequality ([REF]) holds.', '1807.05692-4-40-5': 'Assume that the thesis holds for some [MATH].', '1807.05692-4-40-6': 'Similarly as for [MATH], from the inequality [MATH] and [MATH] it follows that [MATH] and [MATH].', '1807.05692-4-40-7': 'Thus, from [EQUATION] it follows that [EQUATION] or [EQUATION]', '1807.05692-4-40-8': 'In both cases, from the induction hypothesis, [EQUATION] thus [EQUATION]', '1807.05692-4-41-0': '## Uniqueness', '1807.05692-4-42-0': 'In general, we can not guarantee that [MATH] but we will prove that [MATH] and it is the unique solution of ([REF]) in [MATH].', '1807.05692-4-43-0': 'First, we will prove that [MATH].', '1807.05692-4-43-1': 'We know that [MATH] for [MATH].', '1807.05692-4-43-2': 'Now, from Lemma [REF] it follows that for any fixed [MATH] and [MATH] we have that either [MATH] or [MATH] (otherwise we would have [EQUATION] and [EQUATION] which by Lemma [REF] would yield [EQUATION] which is a contradiction).', '1807.05692-4-43-3': 'Thus for [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-4-43-4': 'This proves that [MATH].', '1807.05692-4-44-0': 'To prove the uniqueness notice that if [MATH] was not unique in [MATH] then there would exist two processes [MATH] and [MATH] satisfying ([REF]) and such that [EQUATION] for some [MATH].', '1807.05692-4-44-1': 'However, using the same reasoning as in ([REF]) and the fact that [MATH] and [MATH] solve ([REF]) we can prove that [EQUATION] thus [EQUATION].', '1807.05692-4-44-2': 'Similarly, by induction (and subadditivity of [MATH]), we prove that for [MATH] [EQUATION].', '1807.05692-4-44-3': 'Now, for any fixed [MATH] and [MATH] for [MATH] we have [EQUATION] and [EQUATION] thus [EQUATION] which contradicts ([REF]).'}

{'1807.05692-5-0-0': "Using similar assumptions as in Revuz and Yor's book [CITATION] we prove the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths.", '1807.05692-5-0-1': 'The main tool in our reasonings is a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous price paths.', '1807.05692-5-1-0': 'MSC: 60H20, 91G99', '1807.05692-5-2-0': '# Introduction', '1807.05692-5-3-0': 'The main purpose of this paper is to prove the existence and uniqueness of solutions of differential equations driven by typical, model-free price paths.', '1807.05692-5-3-1': 'Roughly speaking, there are such (continuous or cadlag) paths, representing possible evolution of prices of several financial assets, that do not allow to obtain infinite wealth by risking small amount and trading these assets.', '1807.05692-5-3-2': 'From pioneering works of Vovk [CITATION], [CITATION], [CITATION], [CITATION] it is well known that (typical) model-free price paths reveal many properties of local martingales.', '1807.05692-5-3-3': 'For example, it is possible to define quadratic variation as well as model-free version of stochastic integral with respect to cadlag model-free price paths (whose downward jumps satisfy some mild growth condition) [CITATION], [CITATION].', '1807.05692-5-3-4': 'The case of continuous price paths is understood much better than the case of cadlag paths.', '1807.05692-5-3-5': 'For example, for continuous paths there exists a model-free version of the Dambis, Dubins-Schwarz theorem [CITATION], [CITATION], they also possess local times [CITATION].', '1807.05692-5-4-0': 'However, even in the space of continuous price paths there are still many topics which need to be understood better.', '1807.05692-5-4-1': 'One of such topics is the existence and uniqueness of solutions of differential equations driven by continuous, model-free price paths.', '1807.05692-5-4-2': 'The first results in this direction are proven in [CITATION] even for Hilbert space-valued processes but under the assumption that one can also trade the difference [MATH] and the measure [MATH] is majorized by the Lebesgue measure [MATH] multiplied by some constant.', '1807.05692-5-4-3': '[MATH] denotes here the norm in the Hilbert space and [MATH] denotes the quadratic variation process of the coordinate process [MATH] but defined in a different way than the usual tensor quadratic variation of a Hilbert space-valued semimartingale, see [CITATION],', '1807.05692-5-5-0': 'Our approach is different.', '1807.05692-5-5-1': 'We introduce outer expectation [MATH] of a process [MATH], where [MATH] is the space of all continuous functions [MATH] (representing possible evolutions of prices of [MATH] financial assets), which may be interpreted as the superhedging cost of not only the terminal value of [MATH], i.e. [MATH], but of any value [MATH], where [MATH] is a stopping time such that [MATH].', '1807.05692-5-5-2': 'Unfortunately, with our definition we obtain higher superhedging prices than those in [CITATION], or even in the "standard" approach, where one looks only at the terminal values, however, the definition we use allow us to obtain a model-free version of the Burkholder-Davis-Gundy inequality (BDG inequality in short) for integrals driven by model-free, continuous price paths in a very direct way, from the pathwise version of the BDG inequality proven in [CITATION].', '1807.05692-5-6-0': 'In this paper we will consider the following differential (or rather integral) equation driven by continuous price paths [MATH]: [EQUATION] where [MATH] is a continuous, finite-variation process, [MATH] is the coordinate process, [MATH], [MATH] is the natural filtration of [MATH], [MATH] is [MATH]-measurable, [MATH] and [MATH] are non-anticipating (the definition of non-anticipating functionals and formal statement of all assumptions is given in Sect. [REF]), and Lipschitz in the sense that there exists [MATH] such that for all [MATH], [MATH] and [MATH] [EQUATION] where [MATH] denotes the Euclidean norm on [MATH].', '1807.05692-5-7-0': 'The first integral in ([REF]) is understood as the standard Lebesgue-Stieltjes integral, while the second as the model-free Ito integral, see [CITATION], [CITATION].', '1807.05692-5-8-0': 'Condition ([REF]) is sufficient for our purposes.', '1807.05692-5-8-1': 'The same condition is used in [CITATION] but it differs from those used in [CITATION].', '1807.05692-5-9-0': 'This paper is organized as follows.', '1807.05692-5-9-1': 'In the next section we introduce necessary definitions and notation.', '1807.05692-5-9-2': "In the third section we present and prove a model-free version of the BDG inequality and in the last section we apply this inequality and Picard's iterations to prove the existence and uniqueness of the solution of ([REF]).", '1807.05692-5-10-0': '# Definitions and notation', '1807.05692-5-11-0': 'Let [MATH], [MATH] and [MATH] be the space of continuous functions [MATH] and let [EQUATION] denote the the coordinate process.', '1807.05692-5-11-1': '[MATH] is the natural filtration of [MATH].', '1807.05692-5-11-2': 'In the sequel, by saying that a process [MATH] is adapted we will mean that it is adapted to [MATH].', '1807.05692-5-12-0': 'Stopping times [MATH] with respect to [MATH] and the corresponding [MATH]-algebras [MATH] are defined as usual.', '1807.05692-5-13-0': 'A process [MATH] is a simple process (simple strategy) if there exist stopping times [MATH] and [MATH]-measurable, bounded functions [MATH] such that for every [MATH] from some [MATH] and [EQUATION]', '1807.05692-5-13-1': 'For such [MATH] we define the corresponding integral process [MATH], [EQUATION] which is well-defined for all [MATH] and all [MATH]; here, for [MATH] we denote [MATH] and [MATH] denotes the inner product on [MATH].', '1807.05692-5-14-0': 'The family of simple strategies will be denoted by [MATH].', '1807.05692-5-14-1': 'For [MATH] a simple strategy [MATH] will be called (strongly) [MATH]-admissible if [MATH] for all [MATH] and all [MATH].', '1807.05692-5-14-2': 'The set of strongly [MATH]-admissible simple strategies will be denoted by [MATH].', '1807.05692-5-15-0': "Vovk's outer measure [MATH] of a set [MATH] is defined as the minimal superhedging price for [MATH], that is [EQUATION].", '1807.05692-5-15-1': 'A set [MATH] is called a null set if it has outer measure zero.', '1807.05692-5-15-2': 'A property [MATH] holds for typical price paths if the set [MATH] where [MATH] is violated is a null set.', '1807.05692-5-16-0': 'In the recent book [CITATION] Vladimir Vovk and Glenn Shafer consider also properties, which may hold up to some time [MATH] and as soon as they cease to hold, a trader may become infinitely rich.', '1807.05692-5-16-1': 'Such property [MATH] holds with instant enforcement and its complement [MATH] is called instantly blockable.', '1807.05692-5-16-2': 'For precise definitions we refer to [CITATION].', '1807.05692-5-16-3': 'Any instantly blockable property of [MATH] and [MATH] is evanescent in the sense that its projection onto [MATH] is a null set.', '1807.05692-5-17-0': 'Next, we define a special sequence of partitions called the Lebesgue partitions generated by [MATH].', '1807.05692-5-17-1': 'The [MATH]th ([MATH]) Lebesgue partition [MATH] of [MATH] generated by [MATH] is defined for each [MATH] as: [MATH] and for [MATH] [EQUATION]', '1807.05692-5-17-2': 'By convention [MATH].', '1807.05692-5-17-3': 'Similarly, replacing [MATH] by [MATH] we define the Lebesgue partitions [MATH] generated by [MATH] for each [MATH].', '1807.05692-5-18-0': 'Next, we define the sequence of Lebesgue partitions generated by [MATH] as: [MATH] and for [MATH] [EQUATION].', '1807.05692-5-19-0': 'It is well known (see [CITATION]) that for [MATH] and a typical price path [MATH] there exists the continuous limit [EQUATION] and this convergence is uniform in [MATH].', '1807.05692-5-20-0': 'We will use the following notation: [MATH] and [EQUATION]', '1807.05692-5-20-1': 'Let [MATH] be given by ([REF]).', '1807.05692-5-20-2': 'The quadratic variation process of the real integral process [MATH] is defined as [EQUATION]', '1807.05692-5-20-3': 'For any process [MATH]) let us define [EQUATION] (where [MATH] is the Euclidean norm on [MATH]).', '1807.05692-5-20-4': 'We have the following estimate.', '1807.05692-5-21-0': 'Let [MATH], [MATH] and [MATH] be defined as [EQUATION].', '1807.05692-5-21-1': 'Then [MATH] and for any [MATH] [EQUATION].', '1807.05692-5-22-0': 'Using the inequality [MATH] (which follows from [MATH], [MATH]) and [MATH], we obtain [EQUATION]', '1807.05692-5-22-1': 'A useful tool which we aim to establish is a model-free version of the BDG inequality.', '1807.05692-5-22-2': 'It will be formulated for the outer expectation [MATH] which is defined as follows.', '1807.05692-5-22-3': 'Let [MATH] be the family of stopping times [MATH] such that [MATH].', '1807.05692-5-22-4': 'For any process [MATH] we define [EQUATION] where the first infimum is over all subsets [MATH] of typical price paths, that is all [MATH] such that [MATH].', '1807.05692-5-23-0': 'It is straightforward to verify that the introduced outer expectation [MATH] is countably subadditive, monotone and positively homogeneous.', '1807.05692-5-24-0': 'By [MATH] we denote the family of processes [MATH] such that [EQUATION].', '1807.05692-5-25-0': 'The outer expectation of non-negative processes which we have just introduced is not the same as the outer expectation of variables introduced by Vovk in [CITATION], where one looks only at the value of [MATH] at the terminal time.', '1807.05692-5-25-1': 'However, similar definitions to ours already exist in literature in the context of pricing American options, see for example [CITATION].', '1807.05692-5-26-0': 'Vladimir Vovk (personal communication) noticed that the first infimum in ([REF]) would not be needed if it was over sets [MATH] which are complements of evanescent sets (see Remark [REF]) since in this case for any [MATH] and any [MATH] we have a sequence of simple strategies [MATH] such that for any [MATH], [MATH].', '1807.05692-5-26-1': 'Using these strategies (and [MATH]) it is easy for any [MATH] to construct strategies [MATH] for which [MATH] holds for all [MATH] .', '1807.05692-5-27-0': 'In fact, in the sequel we will use the set [MATH] where the convergence in ([REF]) holds uniformly, which is a complement of an evanescent set, see [CITATION].', '1807.05692-5-27-1': 'It is even possible to prove that the convergence in ([REF]) holds uniformly with instant enforcement, see [CITATION].', '1807.05692-5-28-0': '# Model-free version of the BDG inequality', '1807.05692-5-29-0': 'In this section we establish the following model-free version of the BDG inequality:', '1807.05692-5-30-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-5-31-0': 'Let us recall the pathwise BDG inequalities of Beiglblock and Siorpaes ([CITATION]): if for real numbers [MATH] and [MATH] we define [EQUATION] then for any [MATH] there exist positive constant [MATH] and numbers [MATH] such that [MATH] depends only on [MATH] [EQUATION] and such that for any [MATH] one has [EQUATION]', '1807.05692-5-31-1': 'Moreover, for [MATH] one has [MATH], [MATH] and the following estimate also holds [EQUATION]', '1807.05692-5-31-2': 'Let now [MATH] and ([REF]) be its representation.', '1807.05692-5-31-3': 'Let [MATH] be a non-decreasing rearrangement of [MATH], where [MATH] is the [MATH]th Lebesgue partition ([MATH]).', '1807.05692-5-31-4': 'For [MATH] and [MATH] we define [MATH] and for [MATH] [EQUATION] (to ease notation we write [MATH] instead of [MATH]).', '1807.05692-5-31-5': 'Let us notice that for [MATH] we have [EQUATION] and for [MATH] [EQUATION] where [MATH] is the set of typical paths, for which the quadratic variation along the sequence of Lebesgue partitions exist and the convergence in ([REF]) is uniform.', '1807.05692-5-32-0': 'Moreover, by the definition of [MATH] [EQUATION].', '1807.05692-5-32-1': 'Let us now define the simple strategy [MATH] which just after time [MATH] attains the position [EQUATION] i.e. [EQUATION].', '1807.05692-5-32-2': 'For [MATH] by [MATH] we denote the unique [MATH] such that [EQUATION].', '1807.05692-5-32-3': 'We estimate [EQUATION] where we used the fact that [MATH].', '1807.05692-5-33-0': 'Let now [MATH], [MATH] and [MATH] be finite reals such that [EQUATION] and let [MATH] be a sequence of [MATH]-admissible strategies such that [EQUATION]', '1807.05692-5-33-1': 'Now let us define [EQUATION] and let us consider the strategy [EQUATION].', '1807.05692-5-33-2': 'Directly from the definition it follows that [MATH] is [MATH]-admissible.', '1807.05692-5-33-3': 'Moreover, for [MATH] the convergence in ([REF]) is uniform in [MATH].', '1807.05692-5-33-4': 'We have [MATH] and for each [MATH], [MATH] as [MATH].', '1807.05692-5-33-5': 'Thus for each [MATH] and [MATH] by ([REF]), by the estimate [MATH], ([REF]), the definition of [MATH] and by ([REF]), for sufficiently large [MATH] we have [EQUATION]', '1807.05692-5-33-6': 'As a result we get that for [MATH], [MATH] as [MATH] and [EQUATION].', '1807.05692-5-33-7': 'Since [MATH] may be as close to [MATH] as we please, we obtain ([REF]).', '1807.05692-5-34-0': 'The proof of Theorem [REF] relies on the fact that having sequence of strategies [MATH] which dominate (in the sense of ([REF])) [MATH] we are able to construct strategies [MATH] which dominate [MATH].', '1807.05692-5-35-0': 'The constant [MATH] is not optimal.', '1807.05692-5-35-1': 'polish Adam Os"ekowski noticed (personal communication) that using so called Burkholder\'s method and functions constructed in [CITATION] it is possible to construct strategies which give [MATH], however, it is not clear if this is the best constant possible.', '1807.05692-5-36-0': 'Using ([REF]) and proceeding in a similar way as in the proof of Theorem [REF] we also get (for [MATH]) the estimate: [EQUATION]', '1807.05692-5-37-0': '## Multidimensional version of the model-free BDG inequality', '1807.05692-5-38-0': 'In this section we will prove the model-free BDG inequality in the case when [MATH] is a matrix-valued, simple process, i.e. [MATH] and [MATH] where [MATH], [MATH].', '1807.05692-5-38-1': 'The family of such processes will be denoted by [MATH].', '1807.05692-5-38-2': 'For [MATH] we define the integral [MATH] as the vector of integrals [EQUATION].', '1807.05692-5-38-3': 'Also, similarly to ([REF]) we define [EQUATION]', '1807.05692-5-38-4': 'Now we have the following generalisation of ([REF]).', '1807.05692-5-39-0': 'For any [MATH] [EQUATION] where [MATH].', '1807.05692-5-40-0': 'Using the inequality [MATH] valid for any real [MATH] we estimate for [MATH] [EQUATION].', '1807.05692-5-40-1': 'Next, using subadditivity of [MATH], ([REF]) and the monotonicity of [MATH] we obtain [EQUATION]', '1807.05692-5-41-0': '## Spaces [MATH], [MATH], [MATH] and [MATH]', '1807.05692-5-42-0': 'Now, we introduce the spaces of (equivalence classes of) adapted processes [MATH] (resp. [MATH]) such that [MATH] is equivalent with [MATH] if [MATH]).', '1807.05692-5-42-1': 'Using standard arguments (see for example [CITATION]) we see that these spaces equipped with the metric [EQUATION] are complete metric spaces.', '1807.05692-5-43-0': 'Let us notice that the fact [MATH] implies that for a typical price path [MATH] [EQUATION].', '1807.05692-5-44-0': 'Let [MATH] (resp. [MATH]) denote the closures of the sets of simple processes [MATH] (resp. [MATH]) in the defined spaces.', '1807.05692-5-44-1': 'Thus the simple processes are dense in [MATH] (resp. [MATH]).', '1807.05692-5-45-0': 'Let us notice that for a simple process [MATH] and [MATH] the process [MATH] defined as [EQUATION] is also a simple process and if [MATH] then using Lemma [REF] we get [EQUATION]', '1807.05692-5-45-1': 'Similarly, if [MATH] is a simple process we get [EQUATION]', '1807.05692-5-45-2': 'Using this, Theorem [REF] or Proposition [REF] and completeness of the space [MATH] (resp. [MATH]) we see that for any [MATH] (resp. [MATH]), any sequence of simple processes [MATH] (resp. [MATH]) such that [MATH] in [MATH] (resp. [MATH]) (i.e. [MATH]) and any [MATH], the sequence of integrals [MATH] converges in [MATH] (resp. in [MATH]) to the integral process [MATH].', '1807.05692-5-46-0': 'In analogy to ([REF]) for [MATH] we define [EQUATION] and in analogy to ([REF]) for [MATH] we define [EQUATION].', '1807.05692-5-47-0': 'Finally, let us introduce the space [MATH] (resp. [MATH]) of (adapted) processes [MATH] such that for any [MATH], [MATH] (resp. [MATH]).', '1807.05692-5-48-0': 'The following analog of Proposition [REF] holds: If [MATH] and [MATH] then [EQUATION] where [MATH].', '1807.05692-5-49-0': '# Theorem on existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free price paths', '1807.05692-5-50-0': 'In this section we prove the existence and uniqueness of the solution of SDE ([REF]).', '1807.05692-5-50-1': 'We will assume the following:', '1807.05692-5-51-0': '[MATH] is such that the process [MATH] defined by [MATH], [MATH], satisfies [MATH]; [MATH] and [MATH] are continuous, non-decreasing, adapted processes, starting from [MATH] such that for all [MATH], [MATH], where [MATH] is a deterministic constant; [MATH] and [MATH] are non-anticipating, by which we mean that for any adapted processes [MATH], [MATH] and [MATH] whenever [MATH] for all [MATH], and the processes [MATH], [MATH] are adapted (see also [CITATION]); [EQUATION] [MATH] and [MATH] satisfy condition ([REF]).', '1807.05692-5-52-0': 'Instead of [MATH], [MATH] we will often write [MATH] and [MATH] resp.', '1807.05692-5-53-0': "Now we will use the just obtained model-free version of the BDG inequality and Picard's iterations to prove the following theorem.", '1807.05692-5-54-0': 'Under the assumptions 1.-5.', '1807.05692-5-54-1': 'stated above, integral equation ([REF]) has unique solution in the space [MATH].', '1807.05692-5-55-0': 'The assumption that [MATH], where [MATH] is a deterministic constant seems to be important in the sense that when we allow [MATH] to be random then we can not prove that [MATH].', '1807.05692-5-56-0': 'Theorem [REF] implies the existence of the solution of ([REF]) in the space [MATH].', '1807.05692-5-56-1': 'More precisely, it implies the existence of a process [MATH] which is a uniform limit of simple processes (till the volatility measured by [MATH] is not too high) and such that for typical [MATH], [EQUATION]', '1807.05692-5-56-2': 'Naturally, for many equations, like for example the one-dimensional Black-Scholes equation [EQUATION] ([MATH] - deterministic) we can write the solution explicitly [EQUATION] and verify that it satisfies ([REF]) using the (model-free) Ito formula (see [CITATION]).', '1807.05692-5-56-3': 'However, for more general equations we often have no explicit solutions and the existence of a solution is not obvious.', '1807.05692-5-57-0': '## Proof of Theorem [REF]', '1807.05692-5-58-0': '### Existence', '1807.05692-5-59-0': 'Let us set [MATH], [MATH], [EQUATION] and define [MATH] such that for [MATH] , [EQUATION] (By the assumption on [MATH], ([REF]), ([REF]) and the calculation below (i.e. ([REF])) this definition is correct.)', '1807.05692-5-59-1': 'Now, by the Lipschitz property, the BDG inequality ([REF]) and similar reasoning as in the proof of Lemma [REF] we estimate [EQUATION]', '1807.05692-5-59-2': 'Thus [MATH] is a contraction on [MATH] and it has unique fixed point [MATH].', '1807.05692-5-59-3': 'Next, we define [EQUATION] (we apply the convention that [MATH] if [MATH]), and introduce the following operator [MATH], [EQUATION]', '1807.05692-5-59-4': 'Similarly as before, we prove that [MATH] is a contraction and has a fixed point [MATH].', '1807.05692-5-59-5': 'Moreover, [MATH] and [MATH] agree on the interval [EQUATION].', '1807.05692-5-59-6': 'Similarly, having defined [MATH], [MATH], [MATH], and its fixed point [MATH], [MATH], by induction, we define [EQUATION] and introduce the following operator [MATH], [EQUATION] and its fixed point [MATH] which agrees with [MATH] on the interval [EQUATION].', '1807.05692-5-60-0': 'Finally, setting [EQUATION] we get that [MATH] satisfies [EQUATION]', '1807.05692-5-60-1': 'This will follow from the following lemma.', '1807.05692-5-60-2': 'Let [MATH] .', '1807.05692-5-60-3': 'Assume that [MATH] and the following inequalities hold: [EQUATION] for some non-negative integers [MATH] and [MATH].', '1807.05692-5-60-4': 'Then [EQUATION]', '1807.05692-5-60-5': 'The proof follows by easy induction.', '1807.05692-5-60-6': 'For [MATH] from [MATH] and [MATH] it follows that [MATH] and [MATH] thus inequality ([REF]) holds.', '1807.05692-5-60-7': 'Assume that the thesis holds for some [MATH].', '1807.05692-5-60-8': 'Similarly as for [MATH], from the inequality [MATH] and [MATH] it follows that [MATH] and [MATH].', '1807.05692-5-60-9': 'Thus, from [EQUATION] it follows that [EQUATION] or [EQUATION]', '1807.05692-5-60-10': 'In both cases, from the induction hypothesis, [EQUATION] thus [EQUATION].', '1807.05692-5-61-0': 'For any [MATH] and any [MATH], [MATH] satisfies ([REF]).', '1807.05692-5-61-1': 'Thus, if for some even [MATH], [MATH] and [MATH] then [MATH] (in fact [MATH]) and [MATH] satisfies ([REF]) for all [MATH].', '1807.05692-5-61-2': 'Otherwise, if [MATH], by Lemma [REF] we would have [EQUATION] which would yield [EQUATION]', '1807.05692-5-62-0': '### Uniqueness', '1807.05692-5-63-0': 'In general, we can not guarantee that [MATH] but we will prove that [MATH].', '1807.05692-5-63-1': 'Moreover is the unique solution of ([REF]) in [MATH].', '1807.05692-5-64-0': 'First, we will prove that [MATH].', '1807.05692-5-64-1': 'We know that [MATH] for [MATH].', '1807.05692-5-64-2': 'Now, from Lemma [REF] it follows that for any fixed [MATH] and [MATH] we have that either [MATH] (which implies [MATH]) or [MATH] (otherwise we would have [EQUATION] and [EQUATION] which by Lemma [REF] would yield [EQUATION] which is a contradiction).', '1807.05692-5-64-3': 'Thus for [MATH] we have [EQUATION] and [EQUATION].', '1807.05692-5-64-4': 'This proves that [MATH].', '1807.05692-5-65-0': 'To prove the uniqueness notice that if [MATH] was not unique in [MATH] then there would exist two processes [MATH] and [MATH] satisfying ([REF]) and such that [EQUATION] for some [MATH].', '1807.05692-5-65-1': 'However, using the same reasoning as in ([REF]) and the fact that [MATH] and [MATH] solve ([REF]) we can prove that [EQUATION] thus [EQUATION].', '1807.05692-5-65-2': 'Similarly, by induction (and subadditivity of [MATH]), we prove that for [MATH] [EQUATION].', '1807.05692-5-65-3': 'Now, for any fixed [MATH] and [MATH] for [MATH] we have [EQUATION] and [EQUATION] thus [EQUATION] which contradicts ([REF]).'}

1904.11783

{'1904.11783-1-0-0': 'Word embeddings have recently been shown to reflect many of the pronounced societal biases (e.g., gender bias or racial bias).', '1904.11783-1-0-1': 'Existing studies are, however, limited in scope and do not investigate the consistency of biases across relevant dimensions like embedding models, types of texts, and different languages.', '1904.11783-1-0-2': 'In this work, we present a systematic study of biases encoded in distributional word vector spaces: we analyze how consistent the bias effects are across languages, corpora, and embedding models.', '1904.11783-1-0-3': 'Furthermore, we analyze the cross-lingual biases encoded in bilingual embedding spaces, indicative of the effects of bias transfer encompassed in cross-lingual transfer of NLP models.', '1904.11783-1-0-4': 'Our study yields some unexpected findings, e.g., that biases can be emphasized or downplayed by different embedding models or that user-generated content may be less biased than encyclopedic text.', '1904.11783-1-0-5': 'We hope our work catalyzes bias research in NLP and informs the development of bias reduction techniques.', '1904.11783-1-1-0': '# Introduction', '1904.11783-1-2-0': '# Data for Measuring Biases', '1904.11783-1-3-0': 'We first introduce the WEAT dataset [CITATION] and then describe XWEAT, our multilingual and cross-lingual extension of WEAT designed for comparative bias analyses across languages and in cross-lingual embedding spaces.', '1904.11783-1-4-0': '## WEAT', '1904.11783-1-5-0': 'The Word Embedding Association Test (WEAT) [CITATION] is an adaptation of the Implicit Association Test (IAT) [CITATION].', '1904.11783-1-5-1': 'Whereas IAT measures biases based on response times of human subjects to provided stimuli, WEAT quantifies the biases using semantic similarities between word embeddings of the same stimuli.', '1904.11783-1-5-2': 'For each bias test, WEAT specifies four stimuli sets: two sets of target words and two sets of attribute words.', '1904.11783-1-5-3': 'The sets of target words represent stimuli between which we want to measure the bias (e.g., for gender biases, one target set could contain male names and the other females names).', '1904.11783-1-5-4': 'The attribute words, on the other hand, represent stimuli towards which the bias should be measured (e.g., one list could contain pleasant stimuli like health and love and the other negative war and death).', '1904.11783-1-5-5': 'The WEAT dataset defines ten bias tests, each containing two target and two attribute sets.', '1904.11783-1-5-6': 'Table [REF] enumerates the WEAT tests and provides examples of the respective target and attribute words.', '1904.11783-1-6-0': '## Multilingual and Cross-Lingual WEAT', '1904.11783-1-7-0': 'We port the WEAT tests to the multilingual and cross-lingual settings by translating the test vocabularies consisting of attribute and target terms from English to six other languages: German (de), Spanish (es), Italian (it), Russian (ru), Croatian (hr), and Turkish (tr).', '1904.11783-1-7-1': 'We first automatically translate the vocabularies and then let native speakers of the respective languages (also fluent in English) fix the incorrect automatic translations (or introduce better fitting ones).', '1904.11783-1-7-2': 'Our aim was to translate WEAT vocabularies to languages from diverse language families for which we also had access to native speakers.', '1904.11783-1-7-3': 'Whenever the translation of an English term indicated the gender in a target language (e.g., Freund vs. Freundin in de), we asked the translator to provide both male and female forms and included both forms in the respective test vocabularies.', '1904.11783-1-7-4': 'This helps avoiding artificially amplifying the gender bias stemming from the grammatically masculine or feminine word forms.', '1904.11783-1-8-0': 'The monolingual tests in other languages are created by simply using the corresponding translations of target and attribute sets in those languages.', '1904.11783-1-8-1': 'For every two languages, L1 and L2 (e.g., de and it), we create two cross-lingual bias tests: we pair (1) target translations in L1 with L2 translations of attributes (e.g., for T2 we combine de target sets Klavier, Cello, Gitarre, and Gewehr, Schwert, Schleuder, with it attribute sets salute, amore, pace, and abuso, omicidio, tragedia, ), and vice versa, (2) target translations in L2 with attribute translations in L1 (e.g., for T2, it target sets pianoforte, violoncello, chitarra, and fucile, spada, fionda, with de attribute sets Gesundheit, Liebe, Frieden, and Missbrauch, Mord, Tragödie, ).', '1904.11783-1-8-2': 'We did not translate or modify proper names from WEAT sets 3-6.', '1904.11783-1-8-3': 'In our multilingual and cross-lingual experiments we, however, discard the (translations of) WEAT tests for which we cannot find more than 20% of words from some target or attribute set in the embedding vocabulary of the respective language.', '1904.11783-1-8-4': 'This strategy eliminates tests 3-5 and 10 which include proper American names, majority of which can not be found in distributional vocabularies of other languages.', '1904.11783-1-8-5': 'The exception to this is test 6, containing frequent English first names (e.g., Paul, Lisa), which we do find in distributional vocabularies of other languages as well.', '1904.11783-1-8-6': 'In summary, for languages other than en and for cross-lingual settings, we execute six bias tests (T1, T2, T6-T9).', '1904.11783-1-9-0': '# Methodology', '1904.11783-1-10-0': 'We adopt the general bias-testing framework from Caliskan183, but we span our study over multiple dimensions: (1) corpora - we analyze the consistency of biases across distributional vectors induced from different types of text; (2) embedding models - we compare biases across distributional vectors induced by different embedding models (on the same corpora); and (3) languages - we measure biases for word embeddings of different languages, trained from comparable corpora.', '1904.11783-1-10-1': 'Furthermore, unlike Caliskan183, we test whether biases depend on the selection of the similarity metric.', '1904.11783-1-10-2': 'Finally, given the ubiquitous adoption of cross-lingual embeddings [CITATION], we investigate biases in a variety of bilingual embedding spaces.', '1904.11783-1-11-0': 'Bias-Testing Framework.', '1904.11783-1-11-1': 'We first describe the WEAT framework [CITATION].', '1904.11783-1-11-2': 'Let [MATH] and [MATH] be two sets of targets, and [MATH] and [MATH] two sets of attributes (see [REF]).', '1904.11783-1-11-3': 'The tested statistic is the difference between [MATH] and [MATH] in average similarity of their terms with terms from [MATH] and [MATH]: -1em', '1904.11783-1-12-0': '[EQUATION]', '1904.11783-1-13-0': '-1em', '1904.11783-1-14-0': 'with association difference for term [MATH] computed as: -2.5em', '1904.11783-1-15-0': '[EQUATION]', '1904.11783-1-16-0': '-1em', '1904.11783-1-17-0': 'where [MATH] is the distributional vector of term [MATH] and [MATH] is a similarity or distance metric, fixed to cosine similarity in the original work [CITATION].', '1904.11783-1-17-1': 'The significance of the statistic is validated by comparing the score [MATH] with the scores [MATH] obtained for different equally sized partitions [MATH] of the set [MATH].', '1904.11783-1-17-2': 'The [MATH]-value of this permutation test is then measured as the probability of [MATH] computed over all permutations [MATH].', '1904.11783-1-17-3': 'The effect size, that is, the "amount of bias", is computed as the normalized measure of separation between association distributions: -1em', '1904.11783-1-18-0': '[EQUATION]', '1904.11783-1-19-0': '-1em where [MATH] denotes the mean and [MATH] standard deviation.', '1904.11783-1-20-0': 'Dimensions of Bias Analysis.', '1904.11783-1-20-1': 'We analyze the bias effects across multiple dimensions.', '1904.11783-1-20-2': 'First, we analyze the effect that different embedding models have: we compare biases of distributional spaces induced from English Wikipedia, using CBOW [CITATION], GloVe [CITATION], fastText [CITATION], and Dict2Vec algorithms [CITATION].', '1904.11783-1-20-3': 'Secondly, we investigate the effects of biases in different corpora: we compare biases between embeddings trained on the Common Crawl, Wikipedia, and a corpus of tweets.', '1904.11783-1-20-4': 'Finally, and (arguably) most interestingly, we test the consistency of biases across seven languages (see [REF]).', '1904.11783-1-20-5': 'To this end, we test for biases in seven monolingual fastText spaces trained on Wikipedia dumps of the respective languages.', '1904.11783-1-21-0': 'Biases in Cross-Lingual Embeddings.', '1904.11783-1-21-1': 'Cross-lingual embeddings (CLEs) are widely used in multilingual NLP and cross-lingual transfer of NLP models.', '1904.11783-1-21-2': 'Despite the ubiquitous usage of CLEs, the biases they potentially encode have not been analyzed so far.', '1904.11783-1-21-3': 'We analyze projection-based CLEs [CITATION], induced through post-hoc linear projections between monolingual embedding spaces [CITATION].', '1904.11783-1-21-4': 'The projection is commonly learned through supervision with few thousand word translation pairs.', '1904.11783-1-21-5': 'Most recently, however, a number of models have been proposed that learn the projection without any bilingual signal [CITATION]]artetxe2018robust,conneau2018word,hoshen2018nonadversarial,alvarez2018gromov.', '1904.11783-1-21-6': 'Let [MATH] and [MATH] be, respectively, the distributional spaces of the source (S) and target (T) language and let [MATH] be the word translation dictionary.', '1904.11783-1-21-7': 'Let [MATH] be the aligned subsets of monolingual embeddings, corresponding to word-aligned pairs from [MATH].', '1904.11783-1-21-8': 'We then compute the orthogonal matrix [MATH] that minimizes the Euclidean distance between [MATH] and [MATH] [CITATION]: [MATH], where [MATH].', '1904.11783-1-21-9': 'We create comparable bilingual dictionaries [MATH] by translating 5K most frequent en words to other six languages and induce a bilingual space for all [MATH] language pairs.', '1904.11783-1-22-0': '# Findings', '1904.11783-1-23-0': 'Here, we report and discuss the results of our multi-dimensional analysis.', '1904.11783-1-23-1': 'Table [REF] shows the effect sizes for WEAT T1-T10 based on Euclidean or cosine similarity between word vector representations trained on the en Wikipedia using fastText.', '1904.11783-1-24-0': 'We observe the highest bias effects for T6 (Male/Female - Career/Family), T9 (Physical/Mental deseases - Long-term/Short-term), and T1 (Insects/Flowera - Positive/Negative).', '1904.11783-1-24-1': 'Importantly, the results show that biases do not depend on the similarity metric.', '1904.11783-1-24-2': 'We observe nearly identical effects for cosine similarity and Euclidean distance for all WEAT tests.', '1904.11783-1-24-3': 'In the following experiments we thus analyze biases only for cosine similarity.', '1904.11783-1-25-0': 'Word Embedding Models.', '1904.11783-1-25-1': 'Table [REF] compares biases in embedding spaces induced with different models: CBOW, GloVe, fastText, and Dict2Vec.', '1904.11783-1-25-2': 'While the first three embedding methods are trained on Wikipedia only, Dict2Vec employs definitions from dictionaries (e.g., Oxford dictionary) as additional resources for identifying strongly related terms.', '1904.11783-1-25-3': "We only report WEAT test results T1, T2, and T7-T9 for Dict2Vec, as the Dict2Vec's vocabulary does not cover most of the proper names from the remaining tests.", '1904.11783-1-26-0': 'Somewhat surprisingly, the bias effects seem to vary greatly across embedding models.', '1904.11783-1-26-1': 'While GloVe embeddings are biased according to all tests, fastText and especially CBOW exhibit significant biases only for a subset of tests.', '1904.11783-1-26-2': 'We hypothesize that the bias effects reflected in the distributional space depend on the preprocessing steps of the embedding model.', '1904.11783-1-26-3': 'fastText, for instance, relies on embedding subword information, in order to avoid issues with representations of out-of-vocabulary and underrepresented terms: additional reliance on morpho-syntactic signal may make fastText more resilient to biases stemming from distributional signal (i.e., word co-occurrences).', '1904.11783-1-26-4': 'The fact that the embedding space induced with Dict2Vec exhibits larger bias effects may seem counterintuitive at first, since the dictionaries used for vector training should be more objective and therefore less biased than encyclopedic text.', '1904.11783-1-26-5': 'We believe, however, that the additional dictionary-based training objective only propagates the distributional biases across definitionally related words.', '1904.11783-1-26-6': 'Generally, we find these results to be important as they indicate that embedding models may accentuate or diminish biases expressed in text.', '1904.11783-1-27-0': 'Corpora.', '1904.11783-1-27-1': 'In Table [REF] we compare the biases of embeddings trained with the same model (GloVe) but on different corpora: Common Crawl (i.e., noisy web content), Wikipedia (i.e., encyclopedic text) and a corpus of tweets (i.e., user-generated content).', '1904.11783-1-28-0': 'Expectedly, the biases are slightly more pronounced for embeddings trained on the Common Crawl than for those obtained on encyclopedic texts (Wikipedia).', '1904.11783-1-28-1': 'Countering our intuition, the corpus of tweets seems to be consistently less biased (across all tests) than Wikipedia.', '1904.11783-1-28-2': 'In fact, the biases covered by tests T7-T10 are not even significantly present in the vectors trained on tweets.', '1904.11783-1-28-3': 'This finding is indeed surprising and the phenomenon warrants further investigation.', '1904.11783-1-29-0': 'Multilingual Comparison.', '1904.11783-1-29-1': 'Table [REF] compares the bias effects across the seven different languages.', '1904.11783-1-29-2': 'Whereas many of the biases are significant in all languages, de, hr, and tr consistently display smaller effect sizes.', '1904.11783-1-29-3': 'Intuitively, the amount of bias should be proportional to the size of the corpus.', '1904.11783-1-29-4': 'Wikipedias in tr and hr are the two smallest ones - thus they are expected to contain least biased statements.', '1904.11783-1-29-5': 'de Wikipedia, on the other hand, is the second largest and low bias effects here suggest that German texts are indeed less biased than texts in other languages.', '1904.11783-1-29-6': 'Additionally, for (X)WEAT T2, which defines a universally accepted bias (Instruments vs. Weapons), tr and hr exhibit the smallest effect sizes, while the highest bias is observed for en and it.', '1904.11783-1-29-7': 'We measure the highest gender bias, according to (X)WEAT T6, for tr and ru, and the lowest for de.', '1904.11783-1-30-0': 'Biases in Cross-Lingual Embeddings.', '1904.11783-1-30-1': 'We report bias effects for all 21 bilingual embedding spaces in Table [REF].', '1904.11783-1-30-2': 'For brevity, here we report the bias effects averaged over all six XWEAT tests (we provide results detailing bias effects for each of the tests separately in the supplementary materials).', '1904.11783-1-30-3': 'Generally, the bias effects of bilingual spaces are in between the bias effects of the two corresponding monolingual spaces (cf. Table [REF]): this means that we can roughly predict the amount of bias in a cross-lingual embedding space from the same bias effects of corresponding monolingual spaces.', '1904.11783-1-30-4': 'For example, effects in cross-lingual spaces increase over monolingual effects for low-bias languages (hr and tr), and decrease for high-bias languages (en and es).', '1904.11783-1-31-0': '# Conclusion'}

{'1904.11783-2-0-0': 'Word embeddings have recently been shown to reflect many of the pronounced societal biases (e.g., gender bias or racial bias).', '1904.11783-2-0-1': 'Existing studies are, however, limited in scope and do not investigate the consistency of biases across relevant dimensions like embedding models, types of texts, and different languages.', '1904.11783-2-0-2': 'In this work, we present a systematic study of biases encoded in distributional word vector spaces: we analyze how consistent the bias effects are across languages, corpora, and embedding models.', '1904.11783-2-0-3': 'Furthermore, we analyze the cross-lingual biases encoded in bilingual embedding spaces, indicative of the effects of bias transfer encompassed in cross-lingual transfer of NLP models.', '1904.11783-2-0-4': 'Our study yields some unexpected findings, e.g., that biases can be emphasized or downplayed by different embedding models or that user-generated content may be less biased than encyclopedic text.', '1904.11783-2-0-5': 'We hope our work catalyzes bias research in NLP and informs the development of bias reduction techniques.', '1904.11783-2-1-0': '# Introduction', '1904.11783-2-2-0': '# Data for Measuring Biases', '1904.11783-2-3-0': 'We first introduce the WEAT dataset [CITATION] and then describe XWEAT, our multilingual and cross-lingual extension of WEAT designed for comparative bias analyses across languages and in cross-lingual embedding spaces.', '1904.11783-2-4-0': '## WEAT', '1904.11783-2-5-0': 'The Word Embedding Association Test (WEAT) [CITATION] is an adaptation of the Implicit Association Test (IAT) [CITATION].', '1904.11783-2-5-1': 'Whereas IAT measures biases based on response times of human subjects to provided stimuli, WEAT quantifies the biases using semantic similarities between word embeddings of the same stimuli.', '1904.11783-2-5-2': 'For each bias test, WEAT specifies four stimuli sets: two sets of target words and two sets of attribute words.', '1904.11783-2-5-3': 'The sets of target words represent stimuli between which we want to measure the bias (e.g., for gender biases, one target set could contain male names and the other females names).', '1904.11783-2-5-4': 'The attribute words, on the other hand, represent stimuli towards which the bias should be measured (e.g., one list could contain pleasant stimuli like health and love and the other negative war and death).', '1904.11783-2-5-5': 'The WEAT dataset defines ten bias tests, each containing two target and two attribute sets.', '1904.11783-2-5-6': 'Table [REF] enumerates the WEAT tests and provides examples of the respective target and attribute words.', '1904.11783-2-6-0': '## Multilingual and Cross-Lingual WEAT', '1904.11783-2-7-0': 'We port the WEAT tests to the multilingual and cross-lingual settings by translating the test vocabularies consisting of attribute and target terms from English to six other languages: German (de), Spanish (es), Italian (it), Russian (ru), Croatian (hr), and Turkish (tr).', '1904.11783-2-7-1': 'We first automatically translate the vocabularies and then let native speakers of the respective languages (also fluent in English) fix the incorrect automatic translations (or introduce better fitting ones).', '1904.11783-2-7-2': 'Our aim was to translate WEAT vocabularies to languages from diverse language families for which we also had access to native speakers.', '1904.11783-2-7-3': 'Whenever the translation of an English term indicated the gender in a target language (e.g., Freund vs. Freundin in de), we asked the translator to provide both male and female forms and included both forms in the respective test vocabularies.', '1904.11783-2-7-4': 'This helps avoiding artificially amplifying the gender bias stemming from the grammatically masculine or feminine word forms.', '1904.11783-2-8-0': 'The monolingual tests in other languages are created by simply using the corresponding translations of target and attribute sets in those languages.', '1904.11783-2-8-1': 'For every two languages, L1 and L2 (e.g., de and it), we create two cross-lingual bias tests: we pair (1) target translations in L1 with L2 translations of attributes (e.g., for T2 we combine de target sets Klavier, Cello, Gitarre, and Gewehr, Schwert, Schleuder, with it attribute sets salute, amore, pace, and abuso, omicidio, tragedia, ), and vice versa, (2) target translations in L2 with attribute translations in L1 (e.g., for T2, it target sets pianoforte, violoncello, chitarra, and fucile, spada, fionda, with de attribute sets Gesundheit, Liebe, Frieden, and Missbrauch, Mord, Tragödie, ).', '1904.11783-2-8-2': 'We did not translate or modify proper names from WEAT sets 3-6.', '1904.11783-2-8-3': 'In our multilingual and cross-lingual experiments we, however, discard the (translations of) WEAT tests for which we cannot find more than 20% of words from some target or attribute set in the embedding vocabulary of the respective language.', '1904.11783-2-8-4': 'This strategy eliminates tests 3-5 and 10 which include proper American names, majority of which can not be found in distributional vocabularies of other languages.', '1904.11783-2-8-5': 'The exception to this is test 6, containing frequent English first names (e.g., Paul, Lisa), which we do find in distributional vocabularies of other languages as well.', '1904.11783-2-8-6': 'In summary, for languages other than en and for cross-lingual settings, we execute six bias tests (T1, T2, T6-T9).', '1904.11783-2-9-0': '# Methodology', '1904.11783-2-10-0': 'We adopt the general bias-testing framework from Caliskan183, but we span our study over multiple dimensions: (1) corpora - we analyze the consistency of biases across distributional vectors induced from different types of text; (2) embedding models - we compare biases across distributional vectors induced by different embedding models (on the same corpora); and (3) languages - we measure biases for word embeddings of different languages, trained from comparable corpora.', '1904.11783-2-10-1': 'Furthermore, unlike Caliskan183, we test whether biases depend on the selection of the similarity metric.', '1904.11783-2-10-2': 'Finally, given the ubiquitous adoption of cross-lingual embeddings [CITATION], we investigate biases in a variety of bilingual embedding spaces.', '1904.11783-2-11-0': 'Bias-Testing Framework.', '1904.11783-2-11-1': 'We first describe the WEAT framework [CITATION].', '1904.11783-2-11-2': 'Let [MATH] and [MATH] be two sets of targets, and [MATH] and [MATH] two sets of attributes (see [REF]).', '1904.11783-2-11-3': 'The tested statistic is the difference between [MATH] and [MATH] in average similarity of their terms with terms from [MATH] and [MATH]: -1em', '1904.11783-2-12-0': '[EQUATION]', '1904.11783-2-13-0': '-1em', '1904.11783-2-14-0': 'with association difference for term [MATH] computed as: -2.5em', '1904.11783-2-15-0': '[EQUATION]', '1904.11783-2-16-0': '-1em', '1904.11783-2-17-0': 'where [MATH] is the distributional vector of term [MATH] and [MATH] is a similarity or distance metric, fixed to cosine similarity in the original work [CITATION].', '1904.11783-2-17-1': 'The significance of the statistic is validated by comparing the score [MATH] with the scores [MATH] obtained for different equally sized partitions [MATH] of the set [MATH].', '1904.11783-2-17-2': 'The [MATH]-value of this permutation test is then measured as the probability of [MATH] computed over all permutations [MATH].', '1904.11783-2-17-3': 'The effect size, that is, the "amount of bias", is computed as the normalized measure of separation between association distributions: -1em', '1904.11783-2-18-0': '[EQUATION]', '1904.11783-2-19-0': '-1em where [MATH] denotes the mean and [MATH] standard deviation.', '1904.11783-2-20-0': 'Dimensions of Bias Analysis.', '1904.11783-2-20-1': 'We analyze the bias effects across multiple dimensions.', '1904.11783-2-20-2': 'First, we analyze the effect that different embedding models have: we compare biases of distributional spaces induced from English Wikipedia, using CBOW [CITATION], GloVe [CITATION], fastText [CITATION], and Dict2Vec algorithms [CITATION].', '1904.11783-2-20-3': 'Secondly, we investigate the effects of biases in different corpora: we compare biases between embeddings trained on the Common Crawl, Wikipedia, and a corpus of tweets.', '1904.11783-2-20-4': 'Finally, and (arguably) most interestingly, we test the consistency of biases across seven languages (see [REF]).', '1904.11783-2-20-5': 'To this end, we test for biases in seven monolingual fastText spaces trained on Wikipedia dumps of the respective languages.', '1904.11783-2-21-0': 'Biases in Cross-Lingual Embeddings.', '1904.11783-2-21-1': 'Cross-lingual embeddings (CLEs) are widely used in multilingual NLP and cross-lingual transfer of NLP models.', '1904.11783-2-21-2': 'Despite the ubiquitous usage of CLEs, the biases they potentially encode have not been analyzed so far.', '1904.11783-2-21-3': 'We analyze projection-based CLEs [CITATION], induced through post-hoc linear projections between monolingual embedding spaces [CITATION].', '1904.11783-2-21-4': 'The projection is commonly learned through supervision with few thousand word translation pairs.', '1904.11783-2-21-5': 'Most recently, however, a number of models have been proposed that learn the projection without any bilingual signal [CITATION]]artetxe2018robust,conneau2018word,hoshen2018nonadversarial,alvarez2018gromov.', '1904.11783-2-21-6': 'Let [MATH] and [MATH] be, respectively, the distributional spaces of the source (S) and target (T) language and let [MATH] be the word translation dictionary.', '1904.11783-2-21-7': 'Let [MATH] be the aligned subsets of monolingual embeddings, corresponding to word-aligned pairs from [MATH].', '1904.11783-2-21-8': 'We then compute the orthogonal matrix [MATH] that minimizes the Euclidean distance between [MATH] and [MATH] [CITATION]: [MATH], where [MATH].', '1904.11783-2-21-9': 'We create comparable bilingual dictionaries [MATH] by translating 5K most frequent en words to other six languages and induce a bilingual space for all [MATH] language pairs.', '1904.11783-2-22-0': '# Findings', '1904.11783-2-23-0': 'Here, we report and discuss the results of our multi-dimensional analysis.', '1904.11783-2-23-1': 'Table [REF] shows the effect sizes for WEAT T1-T10 based on Euclidean or cosine similarity between word vector representations trained on the en Wikipedia using fastText.', '1904.11783-2-24-0': 'We observe the highest bias effects for T6 (Male/Female - Career/Family), T9 (Physical/Mental deseases - Long-term/Short-term), and T1 (Insects/Flowera - Positive/Negative).', '1904.11783-2-24-1': 'Importantly, the results show that biases do not depend on the similarity metric.', '1904.11783-2-24-2': 'We observe nearly identical effects for cosine similarity and Euclidean distance for all WEAT tests.', '1904.11783-2-24-3': 'In the following experiments we thus analyze biases only for cosine similarity.', '1904.11783-2-25-0': 'Word Embedding Models.', '1904.11783-2-25-1': 'Table [REF] compares biases in embedding spaces induced with different models: CBOW, GloVe, fastText, and Dict2Vec.', '1904.11783-2-25-2': 'While the first three embedding methods are trained on Wikipedia only, Dict2Vec employs definitions from dictionaries (e.g., Oxford dictionary) as additional resources for identifying strongly related terms.', '1904.11783-2-25-3': "We only report WEAT test results T1, T2, and T7-T9 for Dict2Vec, as the Dict2Vec's vocabulary does not cover most of the proper names from the remaining tests.", '1904.11783-2-26-0': 'Somewhat surprisingly, the bias effects seem to vary greatly across embedding models.', '1904.11783-2-26-1': 'While GloVe embeddings are biased according to all tests, fastText and especially CBOW exhibit significant biases only for a subset of tests.', '1904.11783-2-26-2': 'We hypothesize that the bias effects reflected in the distributional space depend on the preprocessing steps of the embedding model.', '1904.11783-2-26-3': 'fastText, for instance, relies on embedding subword information, in order to avoid issues with representations of out-of-vocabulary and underrepresented terms: additional reliance on morpho-syntactic signal may make fastText more resilient to biases stemming from distributional signal (i.e., word co-occurrences).', '1904.11783-2-26-4': 'The fact that the embedding space induced with Dict2Vec exhibits larger bias effects may seem counterintuitive at first, since the dictionaries used for vector training should be more objective and therefore less biased than encyclopedic text.', '1904.11783-2-26-5': 'We believe, however, that the additional dictionary-based training objective only propagates the distributional biases across definitionally related words.', '1904.11783-2-26-6': 'Generally, we find these results to be important as they indicate that embedding models may accentuate or diminish biases expressed in text.', '1904.11783-2-27-0': 'Corpora.', '1904.11783-2-27-1': 'In Table [REF] we compare the biases of embeddings trained with the same model (GloVe) but on different corpora: Common Crawl (i.e., noisy web content), Wikipedia (i.e., encyclopedic text) and a corpus of tweets (i.e., user-generated content).', '1904.11783-2-28-0': 'Expectedly, the biases are slightly more pronounced for embeddings trained on the Common Crawl than for those obtained on encyclopedic texts (Wikipedia).', '1904.11783-2-28-1': 'Countering our intuition, the corpus of tweets seems to be consistently less biased (across all tests) than Wikipedia.', '1904.11783-2-28-2': 'In fact, the biases covered by tests T7-T10 are not even significantly present in the vectors trained on tweets.', '1904.11783-2-28-3': 'This finding is indeed surprising and the phenomenon warrants further investigation.', '1904.11783-2-29-0': 'Multilingual Comparison.', '1904.11783-2-29-1': 'Table [REF] compares the bias effects across the seven different languages.', '1904.11783-2-29-2': 'Whereas many of the biases are significant in all languages, de, hr, and tr consistently display smaller effect sizes.', '1904.11783-2-29-3': 'Intuitively, the amount of bias should be proportional to the size of the corpus.', '1904.11783-2-29-4': 'Wikipedias in tr and hr are the two smallest ones - thus they are expected to contain least biased statements.', '1904.11783-2-29-5': 'de Wikipedia, on the other hand, is the second largest and low bias effects here suggest that German texts are indeed less biased than texts in other languages.', '1904.11783-2-29-6': 'Additionally, for (X)WEAT T2, which defines a universally accepted bias (Instruments vs. Weapons), tr and hr exhibit the smallest effect sizes, while the highest bias is observed for en and it.', '1904.11783-2-29-7': 'We measure the highest gender bias, according to (X)WEAT T6, for tr and ru, and the lowest for de.', '1904.11783-2-30-0': 'Biases in Cross-Lingual Embeddings.', '1904.11783-2-30-1': 'We report bias effects for all 21 bilingual embedding spaces in Table [REF].', '1904.11783-2-30-2': 'For brevity, here we report the bias effects averaged over all six XWEAT tests (we provide results detailing bias effects for each of the tests separately in the supplementary materials).', '1904.11783-2-30-3': 'Generally, the bias effects of bilingual spaces are in between the bias effects of the two corresponding monolingual spaces (cf. Table [REF]): this means that we can roughly predict the amount of bias in a cross-lingual embedding space from the same bias effects of corresponding monolingual spaces.', '1904.11783-2-30-4': 'For example, effects in cross-lingual spaces increase over monolingual effects for low-bias languages (hr and tr), and decrease for high-bias languages (en and es).', '1904.11783-2-31-0': '# Conclusion'}

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[]

[]

[]

[]

['1904.11783-1-11-0', '1904.11783-1-12-0', '1904.11783-1-13-0', '1904.11783-1-14-0', '1904.11783-1-15-0', '1904.11783-1-16-0', '1904.11783-1-18-0', '1904.11783-1-19-0', '1904.11783-1-25-0', '1904.11783-1-27-0', '1904.11783-1-29-0', '1904.11783-2-11-0', '1904.11783-2-12-0', '1904.11783-2-13-0', '1904.11783-2-14-0', '1904.11783-2-15-0', '1904.11783-2-16-0', '1904.11783-2-18-0', '1904.11783-2-19-0', '1904.11783-2-25-0', '1904.11783-2-27-0', '1904.11783-2-29-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1904.11783

1512.07837

{'1512.07837-1-0-0': 'In models with extended Higgs sector and additional matter fields, the decay modes of heavy Higgs bosons can be dominated by cascade decays through the new fermions rendering present search strategies ineffective.', '1512.07837-1-0-1': 'We investigate new decay topologies of heavy neutral Higgses in two Higgs doublet model with vectorlike leptons.', '1512.07837-1-0-2': 'We also discuss constraints from existing searches and discovery prospects.', '1512.07837-1-0-3': 'Among the most interesting signatures are monojet, mono Z, mono Higgs, and Z and Higgs bosons produced with a pair of charged leptons.', '1512.07837-1-1-0': '# Introduction', '1512.07837-1-2-0': 'In models with extended Higgs sector and additional matter fields, the decay patterns of heavy Higgs bosons can be dramatically altered limiting the potential of present search strategies or rendering them nugatory.', '1512.07837-1-2-1': 'However, new search strategies can be designed that could lead to a simultaneous discovery of heavy Higgs bosons and matter particles, and that are more potent than separate searches for such particles.', '1512.07837-1-2-2': 'Such a situation occurs already in a two Higgs doublet model with vectorlike leptons.', '1512.07837-1-3-0': 'We consider an extension of the two Higgs doublet model type-II by vectorlike pairs of new leptons introduced in Ref. [CITATION].', '1512.07837-1-3-1': 'In this model, the new leptons mix only with one family of standard model (SM) leptons and we will use the second family as an example.', '1512.07837-1-3-2': 'As a result of the mixing of new vectorlike leptons with leptons in the SM the flavor changing couplings of [MATH], [MATH] and Higgs bosons between heavy and light leptons are generated.', '1512.07837-1-3-3': 'These couplings allow new decay modes for heavy CP even (or CP odd) Higgs boson: [MATH] and [MATH], where [MATH] and [MATH] are the lightest new charged and neutral leptons.', '1512.07837-1-3-4': 'These decay modes can be very large when the mass of the heavy Higgs boson is below the [MATH] threshold and the light Higgs boson ([MATH]) is SM-like so that [MATH] are suppressed or not present.', '1512.07837-1-3-5': 'In this case, flavor violating decays [MATH] or [MATH] compete only with [MATH] and for sufficiently heavy [MATH] also with [MATH].', '1512.07837-1-3-6': 'Subsequent decay modes of [MATH] and [MATH]: [MATH], [MATH], [MATH] and [MATH], [MATH], [MATH] lead to the following 6 decay chains of the heavy Higgs boson: [EQUATION] which are also depicted in Fig. [REF].', '1512.07837-1-3-7': 'In addition, [MATH] could also decay into pairs of vectorlike leptons.', '1512.07837-1-3-8': 'This is however limited to smaller ranges for masses in which these decays are kinematically open.', '1512.07837-1-3-9': 'Moreover, the final states are the same as in pair production of vectorlike leptons.', '1512.07837-1-3-10': 'We will not consider these possibilities here.', '1512.07837-1-3-11': 'Finally, although we focus on the second family of SM leptons in final states, the modification for a different family of leptons or quarks is straightforward.', '1512.07837-1-4-0': 'We show that in a large range of the parameter space branching ratios for the decay modes ([REF]) and ([REF]) can be sizable or even dominant while satisfying constraints from searches for heavy Higgs bosons, pair production of vectorlike leptons [CITATION] obtained from searches for anomalous production of multilepton events and constraints from precision electroweak observables [CITATION].', '1512.07837-1-4-1': 'Since the Higgs production cross section can be very large, for example the cross section for a 200 GeV Higgs boson at 8 TeV (13 TeV) LHC for [MATH] is 7pb (18pb) [CITATION], the final states above can be produced in large numbers.', '1512.07837-1-4-2': 'Thus searching for these processes could lead to the simultaneous discovery of a new Higgs boson and a new lepton if they exist.', '1512.07837-1-4-3': 'Some of the decay modes in Fig. [REF] also allow for full reconstruction of the masses of both new particles in the decay chain.', '1512.07837-1-5-0': 'The final states of the processes ([REF]) and ([REF]) are the same as final states of [MATH] production or [MATH] decays with one of the gauge bosons decaying into second generation of leptons.', '1512.07837-1-5-1': 'Since searching for leptons in final states is typically advantageous, our processes contribute to a variety of existing searches.', '1512.07837-1-5-2': 'Even searches for processes with fairly large cross sections can be significantly affected.', '1512.07837-1-5-3': 'For example, the contribution of [MATH] to [MATH] can be close to current limits while satisfying the constraints from [MATH].', '1512.07837-1-5-4': 'This has been recently studied in Ref. [CITATION] in the two Higgs doublet model we consider here, and also in a more model independent way in Ref. [CITATION].', '1512.07837-1-5-5': 'However, the processes with tiny SM rates would be the best place to look for this scenario and here we will focus on such signatures.', '1512.07837-1-5-6': 'Examples of almost background free processes include [MATH] and [MATH] with [MATH].', '1512.07837-1-6-0': 'Vectorlike quarks and leptons near the electroweak scale provide a very rich phenomenology.', '1512.07837-1-6-1': 'For example, similar processes to ([REF]) and ([REF]) involving SM-like Higgs boson decaying into [MATH] or [MATH] through a new lepton were previously studied in Ref. [CITATION] and the [MATH] case also in Ref. [CITATION].', '1512.07837-1-6-2': 'An explanation of the muon g-2 anomaly with vectorlike leptons was studied in [CITATION].', '1512.07837-1-6-3': 'Vectorlike quarks and possibly [MATH] offer possibilities to explain anomalies in Z-pole observables [CITATION].', '1512.07837-1-6-4': 'Extensions of the SM with complete vectorlike families were considered to provide an understanding of values of gauge couplings from IR fixed point behavior and threshold effects of vectorlike fermions [CITATION].', '1512.07837-1-6-5': 'Vectorlike quarks and leptons were also considered in supersymmetric framework, see for example refs. [CITATION].', '1512.07837-1-6-6': 'Further discussion and references can be found in a recent review [CITATION].', '1512.07837-1-7-0': 'This paper is organized as follows.', '1512.07837-1-7-1': 'In section [REF] we briefly summarize the model, discuss constraints and present result for branching ratios of the heavy Higgs boson and new leptons.', '1512.07837-1-7-2': 'In section [REF] we discuss relevant existing searches and the most promising search strategies for each of the six processes.', '1512.07837-1-7-3': 'We summarize and present concluding remarks in section [REF].', '1512.07837-1-8-0': '# Two Higgs doublet model with vectorlike leptons', '1512.07837-1-9-0': 'In Ref. [CITATION] we introduced an explicit model consisting of a type-II two Higgs doublet model augmented by vectorlike pairs of new leptons: SU(2) doublets [MATH], SU(2) singlets [MATH] and SM singlets [MATH], where the [MATH] and [MATH] have the same hypercharges as leptons in the SM.', '1512.07837-1-9-1': 'In order to avoid dangerous rates of lepton flavor violating transitions between the light leptons we further assume that the new leptons mix only with one family of SM leptons and we consider the mixing with the second family as an example.', '1512.07837-1-9-2': 'This can be achieved by requiring that the individual lepton number is an approximate symmetry (violated only by light neutrino masses).', '1512.07837-1-9-3': 'With these assumptions, one can write the most general renormalizable Lagrangian containing Yukawa and mass terms for the second generation of SM leptons and new vectorlike leptons.', '1512.07837-1-10-0': 'After spontaneous symmetry breaking, [MATH] and [MATH] with [MATH] GeV (we also define [MATH]), the model can be summarized by mass matrices in the charged lepton sector, with left-handed fields [MATH] on the left and right-handed fields [MATH] on the right [CITATION], [EQUATION] and in the neutral lepton sector, with left-handed fields [MATH] on the left and right-handed fields [MATH] on the right [CITATION], [EQUATION]', '1512.07837-1-10-1': 'The superscripts on vectorlike fields represent the charged and the neutral components (we inserted [MATH] for the right-handed neutrino which is absent in our framework in order to keep the mass matrix [MATH] in complete analogy with the charged sector).', '1512.07837-1-10-2': 'The usual SM Yukawa coupling of the muon is denoted by [MATH], the Yukawa couplings to [MATH] are denoted by various [MATH]s, the Yukawa couplings to [MATH] are denoted by various [MATH]s and finally the explicite mass terms for vectorlike leptons are given by [MATH].', '1512.07837-1-10-3': 'These mass matrices can be diagonalized by bi-unitary transformations and we label the two new charged and neutral mass eigenstates by [MATH] and [MATH] respectively: [EQUATION]', '1512.07837-1-10-4': 'Since SU(2) singlets mix with SU(2) doublets, the couplings of all involved particles to the [MATH], [MATH] and Higgs bosons are in general modified.', '1512.07837-1-10-5': 'The flavor conserving couplings receive corrections and flavor violating couplings between the muon (or muon neutrino) and heavy leptons are generated.', '1512.07837-1-10-6': 'The relevant formulas for these couplings in terms of diagonalization matrices defined above can be found in Refs. [CITATION].', '1512.07837-1-10-7': 'In the limit of small mixing, approximate analytic expressions for diagonalization matrices can be obtained which are often useful for understanding of numerical results.', '1512.07837-1-10-8': 'These are also given in Refs. [CITATION].', '1512.07837-1-11-0': '## Branching ratios of the heavy Higgs boson', '1512.07837-1-12-0': 'In order to focus on decay modes ([REF]) or ([REF]) we either allow mixing only in the neutral sector, [MATH] couplings, or only in the charged sector, [MATH] couplings.', '1512.07837-1-12-1': 'We further assume that the relevant lighter new lepton, [MATH] or [MATH], is heavier that [MATH] to avoid decays into pairs of new leptons and the heavier new lepton, [MATH] or [MATH], is heavier than [MATH].', '1512.07837-1-12-2': 'Finally, we work in the limit with the light Higgs boson being fully SM-like and thus the heavy CP even Higgs [MATH] has no direct couplings to gauge bosons.', '1512.07837-1-12-3': 'We apply these requirements on randomly generated points in the parameter space specified by following ranges: [EQUATION] where we focus on [MATH] in order to avoid [MATH] and on the small [MATH] region where the heavy Higgs production cross section is the largest.', '1512.07837-1-13-0': 'We impose constraints from precision EW data related to the muon and muon neutrino: muon lifetime, [MATH]-pole observables ([MATH] partial width to [MATH], the invisible width, forward-backward asymmetry, left-right asymmetry) and the [MATH] partial width; constraints from oblique corrections, namely from S and T parameters; and the LEP limit on the mass of a new charged lepton, 105 GeV.', '1512.07837-1-13-1': 'These constraints are obtained from Ref. [CITATION].', '1512.07837-1-13-2': 'We also impose constraints on pair production of vectorlike leptons [CITATION] obtained from searches for anomalous production of multilepton events.', '1512.07837-1-14-0': 'In addition to constraints on new leptons we also impose constraints from searches for new Higgses: [MATH] [CITATION] and [MATH] [CITATION].', '1512.07837-1-14-1': 'Although the processes in ([REF]) or ([REF]) do not contribute to [MATH] directly, they contribute to the same final states as obtained from decays of [MATH] bosons.', '1512.07837-1-14-2': 'In applying the constraints from [MATH] we follow the analyses presented in Refs. [CITATION].', '1512.07837-1-14-3': 'We implement the cut-based analysis using the data for [MATH] final state in [CITATION] but the results using [MATH] would be similar.', '1512.07837-1-14-4': 'Since our [MATH] has no direct coupling to the [MATH] boson, it is only the top quark and new charged leptons in loops that contribute to [MATH].', '1512.07837-1-14-5': 'The top quark contribution to [MATH] scales as [MATH] and thus this process is highly constraining the parameter space at very small [MATH].', '1512.07837-1-14-6': 'In the usual two Higgs doublet model this was studied in Ref. [CITATION].', '1512.07837-1-14-7': 'However, new charged leptons can enhance or partially cancel the contribution from the top quark depending on the signs of the new Yukawa couplings.', '1512.07837-1-14-8': 'Thus the allowed range of [MATH] is expected to extend to lower values compared to the usual two Higgs doublet model.', '1512.07837-1-14-9': 'Since new leptons also couple to the SM-like Higgs boson, the constraints from [MATH] have to be also satisfied.', '1512.07837-1-14-10': 'These at present still allow for a sizable new physics contribution [CITATION].', '1512.07837-1-15-0': 'Results of the scan in the case of mixing in the neutral lepton sector, allowing for [MATH], are depicted in Fig. [REF] in various planes of relevant branching ratios and [MATH].', '1512.07837-1-15-1': 'The dark colored points satisfy all the limits summarized above.', '1512.07837-1-15-2': 'The gray points satisfy all the limits on new leptons but are excluded by [MATH] and [MATH].', '1512.07837-1-15-3': 'We clearly see the lower bound on [MATH].', '1512.07837-1-15-4': 'Finally, the light colors represent the points which are phenomenologically viable and satisfy [MATH] and [MATH] limits only if mixing in the charged lepton sector is simultaneously allowed which partially cancels the contribution from the top quark.', '1512.07837-1-15-5': 'In this case, for simplicity, we allow [MATH] and [MATH] to mix but not with the muon and we conservatively extend the ranges for [MATH] and [MATH] to [-1,1].', '1512.07837-1-15-6': 'Without the mixing in the charged sector the lower limit on [MATH] moves to about 0.7.', '1512.07837-1-16-0': 'Similarly, results of the scan in the case of mixing only in the charged lepton sector, allowing for [MATH], are given in Fig. [REF] in the same planes and color scheme.', '1512.07837-1-16-1': 'In this case, the [MATH] dependence is not so significant because the new Yukawa coupling inducing [MATH] scales with [MATH] in the same way as the [MATH] coupling.', '1512.07837-1-16-2': 'The lowest possible value of [MATH] is 0.7 as it was in the [MATH] case.', '1512.07837-1-16-3': 'Even extending the ranges for [MATH] and [MATH] to [-1,1], as indicated by light colors, only allows [MATH].', '1512.07837-1-17-0': 'For completeness we plot the same points in the plane of Higgs branching ratios for [MATH] and [MATH] versus the Higgs mass in Fig. [REF].', '1512.07837-1-17-1': 'Finally, although we focussed on the CP even Higgs, the results for CP odd Higgs would be qualitatively similar.', '1512.07837-1-18-0': '## Branching ratios of the lightest neutral and charged leptons', '1512.07837-1-19-0': 'The branching ratios of new lightest neutral and charged leptons decaying through [MATH] and [MATH] bosons are shown in Fig. [REF].', '1512.07837-1-19-1': 'Although not explicitly shown, the remaining branching ratio of the decay through the SM Higgs boson can be easily read out since only three decay modes are possible.', '1512.07837-1-19-2': 'In figures on the left only constraints from EW precision data and direct searches are imposed.', '1512.07837-1-19-3': 'The figures on the right show the impact of constraints from searches for anomalous production of multilepton events [CITATION].', '1512.07837-1-19-4': 'In these plots, the colors indicate the doublet fractions of [MATH] or [MATH].', '1512.07837-1-19-5': 'The blue, cyan, magenta, and red points have doublet fractions in the ranges [95,100]%, [50,95]%, [5, 50]%, and [0, 5]% respectively.', '1512.07837-1-19-6': 'The doublet fraction the [MATH] is defined as [EQUATION] and the doublet fraction of [MATH] is obtained by replacing the [MATH] matrices by [MATH] matrices in the formula above.', '1512.07837-1-19-7': 'Singlet fractions are given by [MATH] - (doublet fraction).', '1512.07837-1-20-0': 'We see that the multilepton searches are very constraining for doublet-like new leptons and do not even allow a doublet-like charged lepton in the mass range considered.', '1512.07837-1-20-1': 'Note however that in the case of charged leptons with large [MATH] the constraints come from the pair production of [MATH] that accompanies doublet-like [MATH] or from [MATH] production; the [MATH] pair production is not directly constrained by multilepton searches.', '1512.07837-1-20-2': 'Without mixing in the neutral sector, [MATH] and this decay mode is highly constrained [CITATION].', '1512.07837-1-20-3': 'Allowing simultaneously full mixing in both charged and neutral sectors would relax this constraint somewhat.', '1512.07837-1-21-0': 'The main features of plots in Fig. [REF] can be understood from analytic formulas for couplings that can be obtained in the limit of small mixing [CITATION].', '1512.07837-1-21-1': 'For singlet-like [MATH] or [MATH], the flavor changing couplings to W and Z have, in the leading order, the same dependence on parameters controlling the mixing.', '1512.07837-1-21-2': 'Thus this dependence disappears in the ratio and we find [MATH] to be approximately 2:1 leading to the red bands with this slope.', '1512.07837-1-22-0': 'For doublet-like [MATH], the flavor changing couplings to [MATH] have the form [MATH] where [MATH] and [MATH] are functions of [MATH], [MATH] and [MATH].', '1512.07837-1-22-1': 'This implies immediately that, for fixed [MATH], [MATH] and [MATH], a scan over the three couplings [MATH], [MATH] and [MATH] will result in an ellipse in the [MATH] plane.', '1512.07837-1-22-2': 'The major axis of these ellipses is almost horizontal for [MATH]; for smaller [MATH] the ellipses collapse onto the diagonal, which corresponds to [MATH].', '1512.07837-1-22-3': 'This behavior can be seen in Figs. 3, 4 and 7 of Ref. [CITATION] for various choices of [MATH].', '1512.07837-1-23-0': 'Finally, for the doublet-like [MATH], there are two couplings, [MATH] and [MATH], connecting the muon to vectorlike fermions and thus controlling the overall strengths of flavor changing couplings to [MATH], [MATH], and [MATH].', '1512.07837-1-23-1': 'In result, there is significantly more freedom and generating couplings to [MATH], [MATH], and [MATH] are less correlated.', '1512.07837-1-24-0': 'In order to illuminate more subtle features of the scenario, we plot the same points in the planes of branching ratios versus the masses of [MATH] and [MATH] in Fig. [REF].', '1512.07837-1-25-0': '# Signatures', '1512.07837-1-26-0': 'In this section we discuss each of the novel heavy Higgs decay modes with details of the main features of each channel, of existing experimental searches to which these new process contribute, and of possible new searches.', '1512.07837-1-26-1': 'Considering the variety of possible final states, detailed Monte Carlo studies of these signatures are beyond the scope of this paper.', '1512.07837-1-27-0': 'For estimates of rates of various processes we will use, as a reference point, the production cross section of a 200 GeV Higgs boson at 8 TeV LHC for [MATH] which is 7pb.', '1512.07837-1-27-1': 'The corresponding cross section at 13 TeV LHC is 18pb, and for different values of [MATH] these numbers should be divided by [MATH].', '1512.07837-1-27-2': 'Furthermore, we will assume [MATH] or [MATH] to be 50% and branching ratios of [MATH] or [MATH] relevant for a given process to be 100%.', '1512.07837-1-27-3': 'These branching ratios are close to the upper possible values allowed as we saw in the previous section.', '1512.07837-1-28-0': '## [MATH]', '1512.07837-1-29-0': 'The diagrams in Figs. [REF]a and [REF]d yield the [MATH] final state.', '1512.07837-1-29-1': 'A detailed study of these topologies with [MATH] has been presented in Refs. [CITATION], where with focus on contributions to the existing [MATH] and [MATH] measurements.', '1512.07837-1-30-0': 'A crucial feature of the process in Fig. [REF]a is that the intermediate [MATH] decay, with a hadronically decaying [MATH], allows the kinematic reconstruction of the neutral fermion [MATH].', '1512.07837-1-30-1': 'Experimental studies of this mode can be affected by searches for semileptonic decays of a heavy Higgs ([MATH]) which have been presented in Refs. [CITATION].', '1512.07837-1-30-2': 'In these searches, the assumption that the observed missing transverse momentum is caused by a neutrino emitted in the [MATH] decay, is used to reconstruct the complete four-momentum of the neutrino; thus allowing the reconstruction of the Higgs mass ([MATH]).', '1512.07837-1-30-3': 'The efficiency loss due to this reconstruction procedure is about 50%.', '1512.07837-1-30-4': 'However, in our case, the neutrino does not originate from a [MATH] and this procedure does not reconstruct the correct Higgs mass.', '1512.07837-1-30-5': 'An alternative approach is to consider the transverse mass [MATH] that is expected to have an edge at [MATH].', '1512.07837-1-30-6': 'Moreover the [MATH] distribution of the neutrino is expected to be different because the latter does not originate from a [MATH].', '1512.07837-1-30-7': "Finally, for Higgs masses above 200 GeV our signal is potentially enhanced, with respect to the one studied in Ref. [CITATION], by the ratio of [MATH] (where the maximal [MATH] and [MATH] branching ratios are taken from Fig. [REF] and [REF]) to [MATH] (where we included a combinatoric factor of 2 for the [MATH]'s decays).", '1512.07837-1-30-8': 'For [MATH] our process is [MATH] and is constrained by searches for [MATH] (with a leptonic tau) and [MATH].', '1512.07837-1-31-0': '## [MATH]', '1512.07837-1-32-0': 'The diagram in Fig. [REF]b leads to the [MATH] final state.', '1512.07837-1-32-1': 'For [MATH] our process contributes to the jet plus missing [MATH] signature, that is common to monojet searches for dark matter pair production or for invisible Higgs boson decays.', '1512.07837-1-32-2': 'For instance, in Refs. [CITATION] the ATLAS and CMS collaborations placed upper limits on the visible cross section defined as the product of cross section, Monte Carlo acceptances and detector efficiencies, i.e. the observed number of events divided by the integrated luminosity.', '1512.07837-1-32-3': 'Requiring [MATH], the ATLAS limit is [MATH] which is larger than the typical values that we can obtain.', '1512.07837-1-32-4': 'For our reference point, even before requiring an extra high-[MATH] jet and including acceptances and detector efficiencies, the total cross section is [MATH].', '1512.07837-1-33-0': 'For leptonic [MATH] decays, [MATH], our signal contributes to the [MATH] measurement [CITATION] and to searches for [MATH] [CITATION], mono-[MATH] with missing energy, [MATH] [CITATION] and [MATH] [CITATION].', '1512.07837-1-33-1': 'For instance, the CMS search for [MATH] [CITATION] finds a limit on the visible cross section (for various [MATH] cuts) at the 1-2 fb level; in our case the [MATH] GeV total cross section is up to [MATH], before acceptances and efficiencies.', '1512.07837-1-33-2': 'Therefore, this channel is likely to offer the strongest constraint.', '1512.07837-1-34-0': 'The hadronic [MATH] mode is less clean and has been studied in the context of the [MATH] presented in Ref. [CITATION].', '1512.07837-1-35-0': '## [MATH]', '1512.07837-1-36-0': 'The diagram in Fig. [REF]e leads to a [MATH] final state with two resonances corresponding to the Higgs, [MATH], and charged vectorlike lepton, [MATH] (up to a dilution due to picking the wrong muon).', '1512.07837-1-37-0': 'The most promising channel is [MATH].', '1512.07837-1-37-1': 'A recent ATLAS search for [MATH] with the [MATH] produced with another vectorlike lepton in a Drell-Yan process is presented in Ref. [CITATION] where the visible cross section into [MATH], [MATH] and [MATH] final states is found to be smaller than 1 fb.', '1512.07837-1-37-2': 'For our reference point, we obtain [MATH].', '1512.07837-1-37-3': 'Since our signal has a different underlying process and kinematics with respect to vectorlike leptons pair production, we expect our acceptance for the ATLAS search to be somewhat smaller than the quoted [MATH] acceptance.', '1512.07837-1-37-4': 'In addition, one could additionally search for a fourth lepton and require one lepton pair to form a [MATH] while imposing a [MATH] veto on the second pair thus suppressing [MATH] backgrounds; the four leptons invariant mass is also expected to peak at the heavy Higgs mass.', '1512.07837-1-38-0': 'In Ref. [CITATION], CMS presented a study of a heavy SM-like Higgs decaying to [MATH] and found that for Higgs masses smaller than about 500 GeV the 95% upper limit on the total cross section into four leptons is [MATH].', '1512.07837-1-38-1': 'For [MATH] this corresponds to about [MATH] (the factor of 3 takes into account that only the [MATH] and [MATH] final states should be included), which has to be compared to our reference cross section of about 210 fb discussed above.', '1512.07837-1-38-2': 'Since these searches require two on-shell [MATH] bosons compared to the single [MATH] in our signal, we expect the Monte Carlo level acceptances for this process to be very small.', '1512.07837-1-38-3': 'A similar argument applies to [MATH] [CITATION].', '1512.07837-1-38-4': 'However, one can again impose a [MATH] veto on two of the charged leptons, almost completely removing the SM background.', '1512.07837-1-39-0': 'The [MATH] mode is also interesting but it yields experimental bounds that are roughly an order of magnitude weaker than in the di-lepton case [CITATION].', '1512.07837-1-39-1': 'The invisible [MATH] channel is problematic because it leads to a dimuon plus missing energy final state in which the two muons do not reconstruct a [MATH].', '1512.07837-1-39-2': 'This final state would also contribute to a [MATH] search.', '1512.07837-1-40-0': '## [MATH]', '1512.07837-1-41-0': 'This mode stems from the diagram in Fig. [REF]f and leads to large contributions to several very promising final states: [MATH], [MATH], [MATH] and [MATH] (the subscript indicates the particles whose invariant mass reconstruct the SM Higgs).', '1512.07837-1-41-1': 'Standard Model backgrounds to the first three modes are essentially absent making them golden channels to discover this process.', '1512.07837-1-41-2': 'In fact, the dominant SM backgrounds are [MATH]; moreover, the latter can be further suppressed by requiring the [MATH] to be virtual by vetoing di-leptons with invariant mass close to [MATH].', '1512.07837-1-42-0': 'For our reference point we find [MATH] and [MATH].', '1512.07837-1-42-1': 'A large number of expected events (over almost zero background) could be already present in the existing data set.', '1512.07837-1-43-0': '## [MATH]', '1512.07837-1-44-0': 'The SM Higgs plus missing energy signal, depicted in Fig. [REF]c, is also very interesting because it overlaps with dark matter searches.', '1512.07837-1-45-0': 'The clear golden mode is the [MATH] final state.', '1512.07837-1-45-1': 'A search for this signal has been performed in Ref. [CITATION] where a small excess of 3 events has been observed over (essentially) no background.', '1512.07837-1-45-2': 'In fact, the [MATH] process has a total cross section of about 0.2 fb and, with a 10% acceptance, leads to about 0.5 events at 20 fb[MATH].', '1512.07837-1-45-3': 'The cross section for our reference point is about [MATH].', '1512.07837-1-45-4': 'Assuming that our signal acceptance is identical to the acceptance used in this search (10%), we expect about 21 events with 20 fb[MATH] of integrated luminosity.', '1512.07837-1-45-5': 'Future updates of this search will certainly place interesting constraints on our model.', '1512.07837-1-46-0': 'In Ref. [CITATION] the [MATH] was studied.', '1512.07837-1-46-1': 'Interestingly a small excess of 20 events is observed for [MATH] that would be compatible with our signal with [MATH].', '1512.07837-1-46-2': 'For our reference point and considering the [MATH] decay of the SM Higgs we get a cross section of up to [MATH].', '1512.07837-1-46-3': 'This is further reduced by the [MATH] tagging efficiency ([MATH]) and by the acceptance.', '1512.07837-1-46-4': 'In Ref. [CITATION], CMS presents a similar study but the di-jet invariant mass distribution after b-tagging is not presented; hence we do not know whether an excess at [MATH] is seen.', '1512.07837-1-47-0': 'In Refs. [CITATION] ATLAS performed a dedicated search for [MATH].', '1512.07837-1-47-1': 'Events are classified according to the number of leptons in the final state: [MATH] (0-leptons), [MATH] (1-lepton), [MATH] (2-leptons).', '1512.07837-1-47-2': 'The 0-lepton channel shares the final state with the search presented in Ref. [CITATION] and discussed above and a small excess compatible with the one observed in that search has also been observed.', '1512.07837-1-47-3': 'A similar search has been performed by CMS in Ref. [CITATION].', '1512.07837-1-48-0': 'A search for [MATH] produced in association with dark matter has been presented in Ref. [CITATION] where, unfortunately, the [MATH] invariant mass distribution is not shown.', '1512.07837-1-49-0': 'Other interesting final states are [MATH].', '1512.07837-1-49-1': 'These modes are similar to the corresponding SM [MATH] ones, albeit with sizable extra missing energy that further reduces all backgrounds.', '1512.07837-1-50-0': '# Conclusions', '1512.07837-1-51-0': 'In two Higgs doublet model type II with vectorlike leptons, the decay modes of heavy Higgs bosons can be dominated by cascade decays through the new leptons into [MATH], [MATH] and Higgs bosons and SM leptons.', '1512.07837-1-51-1': 'These processes are listed in eqs. ([REF]) and ([REF]) and corresponding Feynman diagrams are in Fig. [REF].', '1512.07837-1-52-0': 'After applying constraints from precision electroweak observables, searches for heavy Higgs bosons and constraints on pair production of vectorlike leptons obtained from searches for anomalous production of multilepton events we found that branching ratios of [MATH] and [MATH], where [MATH] and [MATH] are the lightest new charged and neutral leptons can be as large as 50%.', '1512.07837-1-52-1': 'These decay modes are especially relevant below the [MATH] threshold and when the light Higgs boson ([MATH]) is SM-like so that [MATH] are suppressed or not present, competing only with [MATH] and for sufficiently heavy [MATH] also with [MATH].', '1512.07837-1-53-0': 'Furthermore, we found that each of the subsequent decay modes of [MATH] and [MATH]: [MATH], [MATH], [MATH] and [MATH], [MATH], [MATH] can be close to 100% providing many possible search opportunities.', '1512.07837-1-53-1': 'Among the most interesting signatures are monojet, mono Z, mono Higgs, and Z and Higgs bosons produced with a pair of charged leptons.', '1512.07837-1-53-2': 'Some of these signatures are almost background free.', '1512.07837-1-53-3': 'Combining this with potentially large production cross section for these processes presents great discover prospects at the LHC.', '1512.07837-1-54-0': 'The work of RD was supported in part by the U.S. Department of Energy under grant number DE-SC0010120 and by the Ministry of Science, ICT and Planning (MSIP), South Korea, through the Brain Pool Program.', '1512.07837-1-54-1': 'RD also thanks the Galileo Galilei Institute for Theoretical Physics for hospitality and support during part of this work.'}

{'1512.07837-2-0-0': '# Introduction', '1512.07837-2-1-0': 'In models with extended Higgs sector and additional matter fields, the decay patterns of heavy Higgs bosons can be dramatically altered limiting the potential of present search strategies or rendering them nugatory.', '1512.07837-2-1-1': 'However, new search strategies can be designed that could lead to a simultaneous discovery of heavy Higgs bosons and matter particles, and that are more potent than separate searches for such particles.', '1512.07837-2-1-2': 'Such a situation occurs already in a two Higgs doublet model with vectorlike leptons.', '1512.07837-2-2-0': 'We consider an extension of the two Higgs doublet model type-II by vectorlike pairs of new leptons introduced in ref. [CITATION].', '1512.07837-2-2-1': 'In this model, the new leptons mix only with one family of standard model (SM) leptons and we will use the second family as an example.', '1512.07837-2-2-2': 'As a result of the mixing of new vectorlike leptons with leptons in the SM the flavor changing couplings of [MATH], [MATH] and Higgs bosons between heavy and light leptons are generated.', '1512.07837-2-2-3': 'These couplings allow new decay modes for heavy CP even (or CP odd) Higgs boson: [MATH] and [MATH], where [MATH] and [MATH] are the lightest new charged and neutral leptons.', '1512.07837-2-2-4': 'These decay modes can be very large when the mass of the heavy Higgs boson is below the [MATH] threshold and the light Higgs boson ([MATH]) is SM-like so that [MATH] are suppressed or not present.', '1512.07837-2-2-5': 'In this case, flavor changing decays [MATH] or [MATH] compete only with [MATH] and for sufficiently heavy [MATH] also with [MATH].', '1512.07837-2-2-6': 'Subsequent decay modes of [MATH] and [MATH]: [MATH], [MATH], [MATH] and [MATH], [MATH], [MATH] lead to the following 6 decay chains of the heavy Higgs boson: [EQUATION] which are also depicted in figure [REF].', '1512.07837-2-2-7': 'In addition, [MATH] could also decay into pairs of vectorlike leptons.', '1512.07837-2-2-8': 'This is however limited to smaller ranges for masses in which these decays are kinematically open.', '1512.07837-2-2-9': 'Moreover, the final states are the same as in pair production of vectorlike leptons.', '1512.07837-2-2-10': 'We will not consider these possibilities here.', '1512.07837-2-2-11': 'Finally, although we focus on the second family of SM leptons in final states, the modification for a different family of leptons or quarks is straightforward.', '1512.07837-2-3-0': 'We show that in a large range of the parameter space branching ratios for the decay modes ([REF]) and ([REF]) can be sizable or even dominant while satisfying constraints from searches for heavy Higgs bosons, pair production of vectorlike leptons [CITATION] obtained from searches for anomalous production of multilepton events and constraints from precision electroweak (EW) observables [CITATION].', '1512.07837-2-3-1': 'Since the Higgs production cross section can be very large, for example the cross section for a 200 GeV Higgs boson at 8 TeV (13 TeV) LHC for [MATH] is 7pb (18pb) [CITATION], the final states above can be produced in large numbers.', '1512.07837-2-3-2': 'Thus searching for these processes could lead to the simultaneous discovery of a new Higgs boson and a new lepton if they exist.', '1512.07837-2-3-3': 'Some of the decay modes in figure [REF] also allow for full reconstruction of the masses of both new particles in the decay chain.', '1512.07837-2-4-0': 'The final states of the processes ([REF]) and ([REF]) are the same as final states of [MATH] production or [MATH] decays with one of the gauge bosons decaying into second generation of leptons.', '1512.07837-2-4-1': 'Since searching for leptons in final states is typically advantageous, our processes contribute to a variety of existing searches.', '1512.07837-2-4-2': 'Even searches for processes with fairly large cross sections can be significantly affected.', '1512.07837-2-4-3': 'For example, the contribution of [MATH] to [MATH] can be close to current limits while satisfying the constraints from [MATH].', '1512.07837-2-4-4': 'This has been recently studied in ref. [CITATION] in the two Higgs doublet model we consider here, and also in a more model independent way in ref. [CITATION].', '1512.07837-2-4-5': 'However, the processes with tiny SM rates would be the best place to look for this scenario and here we will focus on such signatures.', '1512.07837-2-4-6': 'Examples of almost background free processes include [MATH] and [MATH] with [MATH].', '1512.07837-2-5-0': 'Vectorlike quarks and leptons near the electroweak scale provide a very rich phenomenology.', '1512.07837-2-5-1': 'For example, similar processes to ([REF]) and ([REF]) involving SM-like Higgs boson decaying into [MATH] or [MATH] through a new lepton were previously studied in ref. [CITATION] and the [MATH] case also in ref. [CITATION].', '1512.07837-2-5-2': 'An explanation of the muon g-2 anomaly with vectorlike leptons was studied in [CITATION].', '1512.07837-2-5-3': 'Vectorlike quarks and possibly [MATH] offer possibilities to explain anomalies in [MATH]-pole observables [CITATION].', '1512.07837-2-5-4': 'Extensions of the SM with complete vectorlike families were considered to provide an understanding of values of gauge couplings from IR fixed point behavior and threshold effects of vectorlike fermions [CITATION].', '1512.07837-2-5-5': 'Vectorlike quarks and leptons were also considered in supersymmetric framework, see for example refs. [CITATION].', '1512.07837-2-5-6': 'Further discussion and references can be found in a recent review [CITATION].', '1512.07837-2-6-0': 'This paper is organized as follows.', '1512.07837-2-6-1': 'In section [REF] we briefly summarize the model, discuss constraints and present result for branching ratios of the heavy Higgs boson and new leptons.', '1512.07837-2-6-2': 'In section [REF] we discuss relevant existing searches and the most promising search strategies for each of the six processes.', '1512.07837-2-6-3': 'We summarize and present concluding remarks in section [REF].', '1512.07837-2-7-0': '# Two Higgs doublet model with vectorlike leptons', '1512.07837-2-8-0': 'In ref. [CITATION] we introduced an explicit model consisting of a type-II two Higgs doublet model augmented by vectorlike pairs of new leptons: SU(2) doublets [MATH], SU(2) singlets [MATH] and SM singlets [MATH], where the [MATH] and [MATH] have the same hypercharges as leptons in the SM.', '1512.07837-2-8-1': 'In order to avoid dangerous rates of lepton flavor changing transitions between the light leptons we further assume that the new leptons mix only with one family of SM leptons and we consider the mixing with the second family as an example.', '1512.07837-2-8-2': 'This can be achieved by requiring that the individual lepton number is an approximate symmetry (violated only by light neutrino masses).', '1512.07837-2-8-3': 'With these assumptions, one can write the most general renormalizable Lagrangian containing Yukawa and mass terms for the second generation of SM leptons and new vectorlike leptons.', '1512.07837-2-9-0': 'After spontaneous symmetry breaking, [MATH] and [MATH] with [MATH] GeV (we also define [MATH]), the model can be summarized by mass matrices in the charged lepton sector, with left-handed fields [MATH] on the left and right-handed fields [MATH] on the right [CITATION], [EQUATION] and in the neutral lepton sector, with left-handed fields [MATH] on the left and right-handed fields [MATH] on the right [CITATION], [EQUATION]', '1512.07837-2-9-1': 'The superscripts on vectorlike fields represent the charged and the neutral components (we inserted [MATH] for the right-handed neutrino which is absent in our framework in order to keep the mass matrix [MATH] in complete analogy with the charged sector).', '1512.07837-2-9-2': 'The usual SM Yukawa coupling of the muon is denoted by [MATH], the Yukawa couplings to [MATH] are denoted by various [MATH]s, the Yukawa couplings to [MATH] are denoted by various [MATH]s and finally the explicit mass terms for vectorlike leptons are given by [MATH].', '1512.07837-2-9-3': 'Note that explicit mass terms between SM and vectorlike fields (i.e. [MATH] and [MATH]) can be rotated away.', '1512.07837-2-9-4': 'These mass matrices can be diagonalized by bi-unitary transformations and we label the two new charged and neutral mass eigenstates by [MATH] and [MATH] respectively: [EQUATION]', '1512.07837-2-9-5': 'Since SU(2) singlets mix with SU(2) doublets, the couplings of all involved particles to the [MATH], [MATH] and Higgs bosons are in general modified.', '1512.07837-2-9-6': 'The flavor conserving couplings receive corrections and flavor changing couplings between the muon (or muon neutrino) and heavy leptons are generated.', '1512.07837-2-9-7': 'The relevant formulas for these couplings in terms of diagonalization matrices defined above can be found in Refs. [CITATION].', '1512.07837-2-9-8': 'In the limit of small mixing, approximate analytic expressions for diagonalization matrices can be obtained which are often useful for understanding of numerical results.', '1512.07837-2-9-9': 'These are also given in Refs. [CITATION].', '1512.07837-2-10-0': '## Branching ratios of the heavy Higgs boson', '1512.07837-2-11-0': 'In order to focus on decay modes ([REF]) or ([REF]) we either allow mixing only in the neutral sector, [MATH] couplings, or only in the charged sector, [MATH] couplings.', '1512.07837-2-11-1': 'We further assume that the relevant lighter new lepton, [MATH] or [MATH], is heavier that [MATH] to avoid decays into pairs of new leptons and the heavier new lepton, [MATH] or [MATH], is heavier than [MATH].', '1512.07837-2-11-2': 'Finally, we work in the limit with the light Higgs boson being fully SM-like and thus the heavy CP even Higgs [MATH] has no direct couplings to gauge bosons.', '1512.07837-2-11-3': 'We apply these requirements on randomly generated points in the parameter space specified by following ranges: [EQUATION] where we focus on [MATH] in order to avoid [MATH] and on the small [MATH] region where the heavy Higgs production cross section is the largest.', '1512.07837-2-12-0': 'We impose constraints from precision EW data related to the muon and muon neutrino: muon lifetime, [MATH]-pole observables ([MATH] partial width to [MATH], the invisible width, forward-backward asymmetry, left-right asymmetry) and the [MATH] partial width; constraints from oblique corrections, namely from S and T parameters; and the LEP limit on the mass of a new charged lepton, 105 GeV.', '1512.07837-2-12-1': 'These constraints are obtained from ref. [CITATION].', '1512.07837-2-12-2': 'We also impose constraints on pair production of vectorlike leptons [CITATION] obtained from searches for anomalous production of multilepton events.', '1512.07837-2-13-0': 'In addition to constraints on new leptons we also impose constraints from searches for new Higgses: [MATH] [CITATION] and [MATH] [CITATION].', '1512.07837-2-13-1': 'Although the processes in ([REF]) or ([REF]) do not contribute to [MATH] directly, they contribute to the same final states as obtained from decays of [MATH] bosons.', '1512.07837-2-13-2': 'In applying the constraints from [MATH] we follow the analyses presented in Refs. [CITATION].', '1512.07837-2-13-3': 'We implement the cut-based analysis using the data for [MATH] final state in [CITATION] but the results using [MATH] would be similar.', '1512.07837-2-13-4': 'Since our [MATH] has no direct coupling to the [MATH] boson, it is only the top quark and new charged leptons in loops that contribute to [MATH].', '1512.07837-2-13-5': 'The top quark contribution to [MATH] scales as [MATH] and thus this process is highly constraining the parameter space at very small [MATH].', '1512.07837-2-13-6': 'In the usual two Higgs doublet model this was studied in ref. [CITATION].', '1512.07837-2-13-7': 'However, new charged leptons can enhance or partially cancel the contribution from the top quark depending on the signs of the new Yukawa couplings.', '1512.07837-2-13-8': 'Thus the allowed range of [MATH] is expected to extend to lower values compared to the usual two Higgs doublet model.', '1512.07837-2-13-9': 'Since new leptons also couple to the SM-like Higgs boson, the constraints from [MATH] have to be also satisfied.', '1512.07837-2-13-10': 'These at present still allow for a sizable new physics contribution [CITATION].', '1512.07837-2-14-0': 'Results of the scan in the case of mixing in the neutral lepton sector, allowing for [MATH], are depicted in figure [REF] in various planes of relevant branching ratios and [MATH].', '1512.07837-2-14-1': 'The dark colored points satisfy all the limits summarized above.', '1512.07837-2-14-2': 'The gray points satisfy all the limits on new leptons but are excluded by [MATH] and [MATH].', '1512.07837-2-14-3': 'Finally, the light colors represent the points which are phenomenologically viable and satisfy [MATH] and [MATH] limits only if mixing in the charged lepton sector is simultaneously allowed which partially cancels the contribution from the top quark.', '1512.07837-2-14-4': 'In this case, for simplicity, we allow [MATH] and [MATH] to mix but not with the muon and we conservatively extend the ranges for [MATH] and [MATH] to [-1,1].', '1512.07837-2-14-5': 'We clearly see the lower bound on [MATH].', '1512.07837-2-14-6': 'Without the mixing in the charged sector the lower limit on [MATH] moves to about 0.7.', '1512.07837-2-15-0': 'Similarly, results of the scan in the case of mixing only in the charged lepton sector, allowing for [MATH], are given in figure [REF] in the same planes and color scheme.', '1512.07837-2-15-1': 'In this case, the [MATH] dependence is not so significant because the new Yukawa coupling inducing [MATH] scales with [MATH] in the same way as the [MATH] coupling.', '1512.07837-2-15-2': 'The lowest possible value of [MATH] is 0.7 as it was in the [MATH] case.', '1512.07837-2-15-3': 'Even extending the ranges for [MATH] and [MATH] to [-1,1], as indicated by light colors, only allows [MATH].', '1512.07837-2-16-0': 'For completeness we plot the same points in the plane of Higgs branching ratios for [MATH] and [MATH] versus the Higgs mass in figure [REF].', '1512.07837-2-16-1': 'Finally, although we focussed on the CP even Higgs, the results for CP odd Higgs would be qualitatively similar.', '1512.07837-2-17-0': '## Branching ratios of the lightest neutral and charged leptons', '1512.07837-2-18-0': 'The branching ratios of new lightest neutral and charged leptons decaying through [MATH] and [MATH] bosons are shown in figure [REF].', '1512.07837-2-18-1': 'Although not explicitly shown, the remaining branching ratio of the decay through the SM Higgs boson can be easily read out since only three decay modes are possible.', '1512.07837-2-18-2': 'In the left panels only constraints from EW precision data and direct searches are imposed.', '1512.07837-2-18-3': 'In the right panels we include the impact of constraints from searches for anomalous production of multilepton events [CITATION].', '1512.07837-2-18-4': 'In these plots, the colors indicate the doublet fractions of [MATH] or [MATH].', '1512.07837-2-18-5': 'The blue, cyan, magenta, and red points have doublet fractions in the ranges [95,100]%, [50,95]%, [5, 50]%, and [0, 5]% respectively.', '1512.07837-2-18-6': 'The doublet fraction the [MATH] is defined as [EQUATION] and the doublet fraction of [MATH] is obtained by replacing the [MATH] matrices by [MATH] matrices in the formula above.', '1512.07837-2-18-7': 'Singlet fractions are given by [MATH] - (doublet fraction).', '1512.07837-2-19-0': 'We see that the multilepton searches are very constraining for doublet-like new leptons and do not even allow a doublet-like charged lepton in the mass range considered.', '1512.07837-2-19-1': 'Note however that in the case of charged leptons with large [MATH] the constraints come from the pair production of [MATH] that accompanies doublet-like [MATH] or from [MATH] production; the [MATH] pair production is not directly constrained by multilepton searches.', '1512.07837-2-19-2': 'Without mixing in the neutral sector, [MATH] and this decay mode is highly constrained [CITATION].', '1512.07837-2-19-3': 'Allowing simultaneously full mixing in both charged and neutral sectors would relax this constraint somewhat.', '1512.07837-2-20-0': 'The main features of plots in figure [REF] can be understood from analytic formulas for couplings that can be obtained in the limit of small mixing [CITATION].', '1512.07837-2-20-1': 'For singlet-like [MATH] or [MATH], the flavor changing couplings to W and Z have, in the leading order, the same dependence on parameters controlling the mixing.', '1512.07837-2-20-2': 'Thus this dependence disappears in the ratio and we find [MATH] to be approximately 2:1 leading to the red bands with this slope.', '1512.07837-2-21-0': 'For doublet-like [MATH], the flavor changing couplings to [MATH] have the form [MATH] where [MATH] and [MATH] are functions of [MATH], [MATH] and [MATH].', '1512.07837-2-21-1': 'This implies immediately that, for fixed [MATH], [MATH] and [MATH], a scan over the three couplings [MATH], [MATH] and [MATH] will result in an ellipse in the [MATH] plane.', '1512.07837-2-21-2': 'The major axis of these ellipses is almost horizontal for [MATH]; for smaller [MATH] the ellipses collapse onto the diagonal, which corresponds to [MATH].', '1512.07837-2-21-3': 'This behavior can be seen in figures 3, 4 and 7 of ref. [CITATION] for various choices of [MATH].', '1512.07837-2-22-0': 'Finally, for the doublet-like [MATH], there are two couplings, [MATH] and [MATH], connecting the muon to vectorlike fermions and thus controlling the overall strengths of flavor changing couplings to [MATH], [MATH], and [MATH].', '1512.07837-2-22-1': 'In result, there is significantly more freedom and generating couplings to [MATH], [MATH], and [MATH] are less correlated.', '1512.07837-2-23-0': 'In order to illuminate more subtle features of the scenario, we plot the same points in the planes of branching ratios versus the masses of [MATH] and [MATH] in figure [REF].', '1512.07837-2-24-0': '# Signatures', '1512.07837-2-25-0': 'In this section we discuss each of the novel heavy Higgs decay modes with details of the main features of each channel, of existing experimental searches to which these new process contribute, and of possible new searches.', '1512.07837-2-25-1': 'Considering the variety of possible final states, detailed Monte Carlo studies of these signatures are beyond the scope of this paper.', '1512.07837-2-26-0': 'For estimates of rates of various processes we will use, as a reference point, the production cross section of a 200 GeV Higgs boson at 8 TeV LHC for [MATH] which is 7pb.', '1512.07837-2-26-1': 'The corresponding cross section at 13 TeV LHC is 18pb, and for different values of [MATH] these numbers should be divided by [MATH].', '1512.07837-2-26-2': 'Furthermore, we will assume [MATH] or [MATH] to be 50% and branching ratios of [MATH] or [MATH] relevant for a given process to be 100%.', '1512.07837-2-26-3': 'These branching ratios are close to the upper possible values allowed as we saw in the previous section.', '1512.07837-2-26-4': 'Note however that after imposing all constraints [MATH] branching ratios above 40%, while possible, are difficult to achieve.', '1512.07837-2-27-0': '## [MATH]', '1512.07837-2-28-0': 'The diagrams in figures [REF]a and [REF]d yield the [MATH] final state.', '1512.07837-2-28-1': 'A detailed study of these topologies with [MATH] has been presented in Refs. [CITATION], where with focus on contributions to the existing [MATH] and [MATH] measurements.', '1512.07837-2-29-0': 'A crucial feature of the process in figure [REF]a is that the intermediate [MATH] decay, with a hadronically decaying [MATH], allows the kinematic reconstruction of the neutral fermion [MATH].', '1512.07837-2-29-1': 'Experimental studies of this mode can be affected by searches for semileptonic decays of a heavy Higgs ([MATH]) which have been presented in Refs. [CITATION].', '1512.07837-2-29-2': 'In these searches, the assumption that the observed missing transverse momentum is caused by a neutrino emitted in the [MATH] decay, is used to reconstruct the complete four-momentum of the neutrino; thus allowing the reconstruction of the Higgs mass ([MATH]).', '1512.07837-2-29-3': 'The efficiency loss due to this reconstruction procedure is about 50%.', '1512.07837-2-29-4': 'However, in our case, the neutrino does not originate from a [MATH] and this procedure does not reconstruct the correct Higgs mass.', '1512.07837-2-29-5': 'An alternative approach is to consider the transverse mass [MATH] that is expected to have an edge at [MATH].', '1512.07837-2-29-6': 'Moreover the [MATH] distribution of the neutrino is expected to be different because the latter does not originate from a [MATH].', '1512.07837-2-29-7': "Finally, for Higgs masses above 200 GeV our signal is potentially enhanced, with respect to the one studied in ref. [CITATION], by the ratio of [MATH] (where the maximal [MATH] and [MATH] branching ratios are taken from figure [REF] and [REF]) to [MATH] (where we included a combinatoric factor of 2 for the [MATH]'s decays).", '1512.07837-2-29-8': 'For [MATH] our process is [MATH] and is constrained by searches for [MATH] (with a leptonic tau) and [MATH].', '1512.07837-2-30-0': '## [MATH]', '1512.07837-2-31-0': 'The diagram in figure [REF]b leads to the [MATH] final state.', '1512.07837-2-31-1': 'For [MATH] our process contributes to the jet plus missing [MATH] signature, that is common to monojet searches for dark matter pair production or for invisible Higgs boson decays.', '1512.07837-2-31-2': 'For instance, in Refs. [CITATION] the ATLAS and CMS collaborations placed upper limits on the visible cross section defined as the product of cross section, Monte Carlo acceptances and detector efficiencies, i.e. the observed number of events divided by the integrated luminosity.', '1512.07837-2-31-3': 'Requiring [MATH], the ATLAS limit is [MATH] which is larger than the typical values that we can obtain.', '1512.07837-2-31-4': 'For our reference point, even before requiring an extra high-[MATH] jet and including acceptances and detector efficiencies, the total cross section is [MATH].', '1512.07837-2-32-0': 'For leptonic [MATH] decays, [MATH], our signal contributes to the [MATH] measurement [CITATION] and to searches for [MATH] [CITATION], mono-[MATH] with missing energy, [MATH] [CITATION] and [MATH] [CITATION].', '1512.07837-2-32-1': 'For instance, the CMS search for [MATH] [CITATION] finds a limit on the visible cross section (for various [MATH] cuts) at the 1-2 fb level; in our case the [MATH] GeV total cross section is up to [MATH], before acceptances and efficiencies.', '1512.07837-2-32-2': 'Therefore, this channel is likely to offer the strongest constraint.', '1512.07837-2-33-0': 'The hadronic [MATH] mode is less clean and has been studied in the context of the [MATH] presented in ref. [CITATION].', '1512.07837-2-34-0': '## [MATH]', '1512.07837-2-35-0': 'The diagram in figure [REF]e leads to a [MATH] final state with two resonances corresponding to the Higgs, [MATH], and charged vectorlike lepton, [MATH] (up to a dilution due to picking the wrong muon).', '1512.07837-2-36-0': 'The most promising channel is [MATH].', '1512.07837-2-36-1': 'A recent ATLAS search for [MATH] with the [MATH] produced with another vectorlike lepton in a Drell-Yan process is presented in ref. [CITATION] where the visible cross section into [MATH], [MATH] and [MATH] final states is found to be smaller than 1 fb.', '1512.07837-2-36-2': 'For our reference point, we obtain [MATH].', '1512.07837-2-36-3': 'Since our signal has a different underlying process and kinematics with respect to vectorlike leptons pair production, we expect our acceptance for the ATLAS search to be somewhat smaller than the quoted [MATH] acceptance.', '1512.07837-2-36-4': 'In addition, one could additionally search for a fourth lepton and require one lepton pair to form a [MATH] while imposing a [MATH] veto on the second pair thus suppressing [MATH] backgrounds; the four leptons invariant mass is also expected to peak at the heavy Higgs mass.', '1512.07837-2-37-0': 'In ref. [CITATION], CMS presented a study of a heavy SM-like Higgs decaying to [MATH] and found that for Higgs masses smaller than about 500 GeV the 95% upper limit on the total cross section into four leptons is [MATH].', '1512.07837-2-37-1': 'For [MATH] this corresponds to about [MATH] (the factor of 3 takes into account that only the [MATH] and [MATH] final states should be included), which has to be compared to our reference cross section of about 210 fb discussed above.', '1512.07837-2-37-2': 'Since these searches require two on-shell [MATH] bosons compared to the single [MATH] in our signal, we expect the Monte Carlo level acceptances for this process to be very small.', '1512.07837-2-37-3': 'A similar argument applies to [MATH] [CITATION].', '1512.07837-2-37-4': 'However, one can again impose a [MATH] veto on two of the charged leptons, almost completely removing the SM background.', '1512.07837-2-38-0': 'The [MATH] mode is also interesting but it yields experimental bounds that are roughly an order of magnitude weaker than in the di-lepton case [CITATION].', '1512.07837-2-38-1': 'The invisible [MATH] channel is problematic because it leads to a dimuon plus missing energy final state in which the two muons do not reconstruct a [MATH].', '1512.07837-2-38-2': 'This final state would also contribute to a [MATH] search.', '1512.07837-2-39-0': '## [MATH]', '1512.07837-2-40-0': 'This mode stems from the diagram in figure [REF]f and leads to large contributions to several very promising final states: [MATH], [MATH], [MATH] and [MATH] (the subscript indicates the particles whose invariant mass reconstruct the SM Higgs).', '1512.07837-2-40-1': 'Standard Model backgrounds to the first three modes are essentially absent making them golden channels to discover this process.', '1512.07837-2-40-2': 'In fact, the dominant SM backgrounds are [MATH]; moreover, the latter can be further suppressed by requiring the [MATH] to be virtual by vetoing di-leptons with invariant mass close to [MATH].', '1512.07837-2-41-0': 'For our reference point we find [MATH] and [MATH].', '1512.07837-2-41-1': 'A large number of expected events (over almost zero background) could be already present in the existing data set.', '1512.07837-2-42-0': '## [MATH]', '1512.07837-2-43-0': 'The SM Higgs plus missing energy signal, depicted in figure [REF]c, is also very interesting because it overlaps with dark matter searches.', '1512.07837-2-44-0': 'The clear golden mode is the [MATH] final state.', '1512.07837-2-44-1': 'A search for this signal has been performed in ref. [CITATION] where a small excess of 3 events has been observed over (essentially) no background.', '1512.07837-2-44-2': 'In fact, the [MATH] process has a total cross section of about 0.2 fb and, with a 10% acceptance, leads to about 0.5 events at 20 fb[MATH].', '1512.07837-2-44-3': 'The cross section for our reference point is about [MATH].', '1512.07837-2-44-4': 'Assuming that our signal acceptance is identical to the acceptance used in this search (10%), we expect about 21 events with 20 fb[MATH] of integrated luminosity.', '1512.07837-2-44-5': 'Future updates of this search will certainly place interesting constraints on our model.', '1512.07837-2-45-0': 'In ref. [CITATION] the [MATH] was studied.', '1512.07837-2-45-1': 'Interestingly a small excess of 20 events is observed for [MATH] that would be compatible with our signal with [MATH].', '1512.07837-2-45-2': 'For our reference point and considering the [MATH] decay of the SM Higgs we get a cross section of up to [MATH].', '1512.07837-2-45-3': 'This is further reduced by the [MATH]-tagging efficiency ([MATH]) and by the acceptance.', '1512.07837-2-45-4': 'In ref. [CITATION], CMS presents a similar study but the di-jet invariant mass distribution after [MATH]-tagging is not presented; hence we do not know whether an excess at [MATH] is seen.', '1512.07837-2-46-0': 'In Refs. [CITATION] ATLAS performed a dedicated search for [MATH].', '1512.07837-2-46-1': 'Events are classified according to the number of leptons in the final state: [MATH] (0-leptons), [MATH] (1-lepton), [MATH] (2-leptons).', '1512.07837-2-46-2': 'The 0-lepton channel shares the final state with the search presented in ref. [CITATION] (that we discussed above) and a small excess compatible with the one observed in that search has also been observed.', '1512.07837-2-46-3': 'A similar search has been performed by CMS in ref. [CITATION].', '1512.07837-2-47-0': 'A search for [MATH] produced in association with dark matter has been presented in ref. [CITATION] where, unfortunately, the [MATH] invariant mass distribution is not shown.', '1512.07837-2-48-0': 'Other interesting final states are [MATH].', '1512.07837-2-48-1': 'These modes are similar to the corresponding SM [MATH] ones, albeit with sizable extra missing energy that further reduces all backgrounds.', '1512.07837-2-49-0': '# Conclusions', '1512.07837-2-50-0': 'In two Higgs doublet model type II with vectorlike leptons, the decay modes of heavy Higgs bosons can be dominated by cascade decays through the new leptons into [MATH], [MATH] and Higgs bosons and SM leptons.', '1512.07837-2-50-1': 'These processes are listed in eqs. ([REF]) and ([REF]) and corresponding Feynman diagrams are in figure [REF].', '1512.07837-2-51-0': 'After applying constraints from precision electroweak observables, searches for heavy Higgs bosons and constraints on pair production of vectorlike leptons obtained from searches for anomalous production of multilepton events we found that branching ratios of [MATH] and [MATH], where [MATH] and [MATH] are the lightest new charged and neutral leptons can be as large as 50%.', '1512.07837-2-51-1': 'These decay modes are especially relevant below the [MATH] threshold and when the light Higgs boson ([MATH]) is SM-like so that [MATH] are suppressed or not present, competing only with [MATH] and for sufficiently heavy [MATH] also with [MATH].', '1512.07837-2-52-0': 'Furthermore, we found that each of the subsequent decay modes of [MATH] and [MATH]: [MATH], [MATH], [MATH] and [MATH], [MATH], [MATH] can be close to 100% providing many possible search opportunities.', '1512.07837-2-52-1': 'Among the most interesting signatures are monojet, mono [MATH], mono Higgs, and [MATH] and Higgs bosons produced with a pair of charged leptons.', '1512.07837-2-52-2': 'Some of these signatures are almost background free.', '1512.07837-2-52-3': 'Combining this with potentially large production cross section for these processes presents great discover prospects at the LHC.', '1512.07837-2-53-0': 'The work of RD and EL was supported in part by the U.S. Department of Energy under grant number DE-SC0010120.', '1512.07837-2-53-1': 'RD was supported in part by the Ministry of Science, ICT and Planning (MSIP), South Korea, through the Brain Pool Program.', '1512.07837-2-53-2': 'RD also thanks the Galileo Galilei Institute for Theoretical Physics for hospitality and support during part of this work.'}

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['1512.07837-1-6-0', '1512.07837-2-5-0'], ['1512.07837-1-6-2', '1512.07837-2-5-2'], ['1512.07837-1-6-4', '1512.07837-2-5-4'], ['1512.07837-1-6-5', '1512.07837-2-5-5'], ['1512.07837-1-6-6', '1512.07837-2-5-6'], ['1512.07837-1-38-1', '1512.07837-2-37-1'], ['1512.07837-1-38-2', '1512.07837-2-37-2'], ['1512.07837-1-38-3', '1512.07837-2-37-3'], ['1512.07837-1-38-4', '1512.07837-2-37-4'], ['1512.07837-1-45-0', '1512.07837-2-44-0'], ['1512.07837-1-45-2', '1512.07837-2-44-2'], ['1512.07837-1-45-3', '1512.07837-2-44-3'], ['1512.07837-1-45-4', '1512.07837-2-44-4'], ['1512.07837-1-45-5', '1512.07837-2-44-5'], ['1512.07837-1-26-0', '1512.07837-2-25-0'], ['1512.07837-1-26-1', '1512.07837-2-25-1'], ['1512.07837-1-37-0', '1512.07837-2-36-0'], ['1512.07837-1-37-2', '1512.07837-2-36-2'], ['1512.07837-1-37-3', '1512.07837-2-36-3'], ['1512.07837-1-37-4', '1512.07837-2-36-4'], ['1512.07837-1-54-1', '1512.07837-2-53-2'], ['1512.07837-1-19-1', '1512.07837-2-18-1'], ['1512.07837-1-19-4', '1512.07837-2-18-4'], ['1512.07837-1-19-6', '1512.07837-2-18-6'], ['1512.07837-1-19-7', '1512.07837-2-18-7'], ['1512.07837-1-9-2', '1512.07837-2-8-2'], ['1512.07837-1-9-3', '1512.07837-2-8-3'], ['1512.07837-1-27-0', '1512.07837-2-26-0'], ['1512.07837-1-27-1', '1512.07837-2-26-1'], ['1512.07837-1-27-2', '1512.07837-2-26-2'], ['1512.07837-1-27-3', '1512.07837-2-26-3'], ['1512.07837-1-41-1', '1512.07837-2-40-1'], ['1512.07837-1-41-2', '1512.07837-2-40-2'], ['1512.07837-1-12-0', '1512.07837-2-11-0'], ['1512.07837-1-12-1', '1512.07837-2-11-1'], ['1512.07837-1-12-2', '1512.07837-2-11-2'], ['1512.07837-1-12-3', '1512.07837-2-11-3'], ['1512.07837-1-46-1', '1512.07837-2-45-1'], ['1512.07837-1-46-2', '1512.07837-2-45-2'], ['1512.07837-1-22-0', '1512.07837-2-21-0'], ['1512.07837-1-22-1', '1512.07837-2-21-1'], ['1512.07837-1-22-2', '1512.07837-2-21-2']]

[['1512.07837-1-51-1', '1512.07837-2-50-1'], ['1512.07837-1-16-0', '1512.07837-2-15-0'], ['1512.07837-1-47-2', '1512.07837-2-46-2'], ['1512.07837-1-47-3', '1512.07837-2-46-3'], ['1512.07837-1-3-0', '1512.07837-2-2-0'], ['1512.07837-1-3-5', '1512.07837-2-2-5'], ['1512.07837-1-3-6', '1512.07837-2-2-6'], ['1512.07837-1-13-1', '1512.07837-2-12-1'], ['1512.07837-1-32-0', '1512.07837-2-31-0'], ['1512.07837-1-30-0', '1512.07837-2-29-0'], ['1512.07837-1-30-7', '1512.07837-2-29-7'], ['1512.07837-1-10-2', '1512.07837-2-9-2'], ['1512.07837-1-10-5', '1512.07837-2-9-6'], ['1512.07837-1-14-6', '1512.07837-2-13-6'], ['1512.07837-1-15-0', '1512.07837-2-14-0'], ['1512.07837-1-24-0', '1512.07837-2-23-0'], ['1512.07837-1-21-0', '1512.07837-2-20-0'], ['1512.07837-1-5-4', '1512.07837-2-4-4'], ['1512.07837-1-36-0', '1512.07837-2-35-0'], ['1512.07837-1-34-0', '1512.07837-2-33-0'], ['1512.07837-1-44-0', '1512.07837-2-43-0'], ['1512.07837-1-48-0', '1512.07837-2-47-0'], ['1512.07837-1-53-1', '1512.07837-2-52-1'], ['1512.07837-1-17-0', '1512.07837-2-16-0'], ['1512.07837-1-4-0', '1512.07837-2-3-0'], ['1512.07837-1-4-3', '1512.07837-2-3-3'], ['1512.07837-1-29-0', '1512.07837-2-28-0'], ['1512.07837-1-6-1', '1512.07837-2-5-1'], ['1512.07837-1-6-3', '1512.07837-2-5-3'], ['1512.07837-1-38-0', '1512.07837-2-37-0'], ['1512.07837-1-45-1', '1512.07837-2-44-1'], ['1512.07837-1-37-1', '1512.07837-2-36-1'], ['1512.07837-1-19-0', '1512.07837-2-18-0'], ['1512.07837-1-19-2', '1512.07837-2-18-2'], ['1512.07837-1-9-0', '1512.07837-2-8-0'], ['1512.07837-1-9-1', '1512.07837-2-8-1'], ['1512.07837-1-41-0', '1512.07837-2-40-0'], ['1512.07837-1-46-0', '1512.07837-2-45-0'], ['1512.07837-1-46-3', '1512.07837-2-45-3'], ['1512.07837-1-46-4', '1512.07837-2-45-4'], ['1512.07837-1-22-3', '1512.07837-2-21-3']]

[]

[['1512.07837-1-54-0', '1512.07837-2-53-0'], ['1512.07837-1-54-0', '1512.07837-2-53-1'], ['1512.07837-1-19-3', '1512.07837-2-18-3']]

[]

['1512.07837-1-19-5', '1512.07837-2-18-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1512.07837

hep-th-0203094

{'hep-th-0203094-1-0-0': 'We show that a formalism for analyzing the near-horizon conformal symmetry of Schwarzschild black holes using a scalar field probe is capable of describing black hole decay.', 'hep-th-0203094-1-0-1': 'The decay rate is shown to be correctly described by geodesic motion in the space of black hole masses.', 'hep-th-0203094-1-0-2': 'This provides a novel geometric interpretation for the decay of black holes.', 'hep-th-0203094-1-0-3': 'We also show that the near-horizon conformal symmetry predicts a precise correction term to the usual expression for the decay rate of black holes.', 'hep-th-0203094-1-0-4': 'The results obtained here are a consequence of the holographic nature of the system.', 'hep-th-0203094-1-1-0': 'March 2002', 'hep-th-0203094-1-2-0': 'PACS : 04.70.', 'hep-th-0203094-1-2-1': 'Dy, 04.60.-m', 'hep-th-0203094-1-3-0': 'The discovery of a conformal group structure in the near-horizon region of a black hole has led to interesting developments in quantum gravity [CITATION].', 'hep-th-0203094-1-3-1': 'The emergence of such a structure is closely related to the holographic nature of the system [CITATION].', 'hep-th-0203094-1-3-2': 'Recently we have shown that by studying the algebraic structure of an operator that governs the near-horizon dynamics of a scalar probe in the background of a massive Schwarzschild black hole, the underlying conformal group structure can be explicitly revealed [CITATION].', 'hep-th-0203094-1-3-3': 'Our analysis was based on the requirement of unitarity of the conformal theory and self-adjointness [CITATION] of the associated near-horizon Hamiltonian [CITATION].', 'hep-th-0203094-1-3-4': 'This led to the introduction of the self-adjoint extension of the near-horizon Hamiltonian, described by a real parameter [MATH].', 'hep-th-0203094-1-3-5': 'In our approach, the holographic nature of the black hole is thus realized by the self-adjoint extension.', 'hep-th-0203094-1-3-6': 'It was also shown that for the formalism to be self-consistent, [MATH] must be a small positive number.', 'hep-th-0203094-1-3-7': 'This constraint on [MATH] essentially incorporates the correct boundary conditions required for holography.', 'hep-th-0203094-1-4-0': 'In a subsequent paper [CITATION] we showed that further confirmation of the near-horizon conformal structure emerged when [MATH] was related to the mass [MATH] of the Schwarzschild black hole as [EQUATION] where [MATH] is a nonzero positive constant.', 'hep-th-0203094-1-4-1': 'It is a special feature of this system that the self-adjoint extension parameter is related to the mass of the black hole.', 'hep-th-0203094-1-4-2': 'For the Schwarzschild background it was found that [MATH].', 'hep-th-0203094-1-4-3': 'For the moment we consider a general positive value of [MATH].', 'hep-th-0203094-1-4-4': 'It may be mentioned that our analysis is valid only for massive black holes.', 'hep-th-0203094-1-4-5': 'For any such black hole, Eqn. (1) gives a value of [MATH] which is small and positive.', 'hep-th-0203094-1-4-6': 'The identification proposed in Eqn. (1) is thus consistent with the constraints on [MATH].', 'hep-th-0203094-1-4-7': 'Eqn. (1) is also consistent with the Bekenstein-Hawking entropy formula.', 'hep-th-0203094-1-4-8': 'As shown in Ref. [CITATION], the above relation between [MATH] and [MATH] naturally produces the characteristic [MATH] correction to the Bekenstein-Hawking entropy [CITATION].', 'hep-th-0203094-1-4-9': 'It is known that such a term is universally present in any calculation of black hole entropy within a conformal field theory framework [CITATION].', 'hep-th-0203094-1-4-10': 'The natural appearance of this term in this formalism provided evidence for the underlying near-horizon conformal structure present in this system.', 'hep-th-0203094-1-5-0': 'In this Letter we demonstrate that our formalism [CITATION] for analyzing the near-horizon conformal structure is also capable of describing black hole decay.', 'hep-th-0203094-1-5-1': 'For this purpose, we consider the space of all self-adjoint extensions which is denoted by [MATH].', 'hep-th-0203094-1-5-2': 'As indicated in Eqn. (1), the self-adjoint extension parameter [MATH] is related to the mass [MATH] of the black hole.', 'hep-th-0203094-1-5-3': '[MATH] can therefore be taken as a good coordinate to describe [MATH].', 'hep-th-0203094-1-5-4': 'The central idea in this Letter is to study the geodesic motion in the space of self-adjoint extensions.', 'hep-th-0203094-1-5-5': 'We find that within our formalism, there is a natural way to associate a metric to the space [MATH].', 'hep-th-0203094-1-5-6': 'It is shown that the geodesic motion in [MATH] calculated using this metric naturally leads to the rate of change of the black hole mass, i.e. it describes the black hole decay [CITATION].', 'hep-th-0203094-1-5-7': 'This analysis does not depend on the specific value of the constant [MATH] and is thus independent of the details of our formalism.', 'hep-th-0203094-1-5-8': 'We also show that the near-horizon conformal symmetry predicts a precise correction term in the standard equation governing the decay of black holes.', 'hep-th-0203094-1-5-9': 'The results obtained here are consequences of properly implementing the principle of holography.', 'hep-th-0203094-1-6-0': 'We start by recalling some of the basic results obtained in references [CITATION] and [CITATION] which shall be useful for our analysis.', 'hep-th-0203094-1-6-1': 'The action [MATH] for a massless scalar field coupled to a Schwarzschild black hole of mass [MATH] in 3+1 dimensions is given by [EQUATION] where the Schwarzschild metric in the spherical polar coordinates has the form [EQUATION]', 'hep-th-0203094-1-6-2': 'In the near-horizon limit, the corresponding Klein-Gordon operator for the time-independent and zero angular momentum modes of a massless scalar field is given by [CITATION] [EQUATION] where [MATH] is the near-horizon coordinate.', 'hep-th-0203094-1-6-3': 'In the quantum theory, [MATH] obeys the equation [EQUATION] where [MATH] is the eigenvalue and [MATH] is the associated wavefunction.', 'hep-th-0203094-1-6-4': '[MATH] belongs to the class of unbounded linear operators on a Hilbert space [CITATION] and it admits a one-parameter family of self-adjoint extensions labelled by a [MATH] parameter [MATH], where [MATH] is real.', 'hep-th-0203094-1-6-5': 'Moreover, there is an infinite number of bound states for a given self-adjoint extension [MATH].', 'hep-th-0203094-1-6-6': 'The normalized bound state eigenfunctions and eigenvalues of Eqn. (5) are given by [CITATION] [EQUATION] and [EQUATION] respectively, where [MATH] is zero or a positive integer and [MATH] is the modified Bessel function.', 'hep-th-0203094-1-6-7': 'In our formalism, these solutions are interpreted as bound state excitations of the black hole due to the capture of the scalar field.', 'hep-th-0203094-1-6-8': 'From the above equations we see that the wavefunctions with [MATH] are exponentially suppressed compared to [MATH].', 'hep-th-0203094-1-6-9': 'In what follows we shall therefore focus our attention to the ground state wavefunction [MATH].', 'hep-th-0203094-1-7-0': 'As mentioned before, the requirement of near-horizon conformal symmetry places important constraint on [MATH] [CITATION].', 'hep-th-0203094-1-7-1': 'To see this, consider a band-like region [MATH], where [MATH] and [MATH] is real and positive.', 'hep-th-0203094-1-7-2': 'When [MATH] and satisfies the condition [MATH], we see that [MATH].', 'hep-th-0203094-1-7-3': 'Under this condition, [MATH] belongs to the near-horizon region of the black hole.', 'hep-th-0203094-1-7-4': "At any point [MATH] the leading behaviour of the ground state wavefunction is given by [CITATION] [EQUATION] where [MATH] and [MATH] is Euler's constant.", 'hep-th-0203094-1-7-5': 'Taking a typical value of [MATH], we can write the ground state wavefunction as [EQUATION] for [MATH].', 'hep-th-0203094-1-7-6': 'Substituting the value of [MATH] from Eqn. (1) in Eqn. (9), the ground state wavefunction can be written as a function of [MATH] as [EQUATION]', 'hep-th-0203094-1-7-7': 'The function [MATH] captures the [MATH] dependence of the wavefunction and [MATH] is a positive constant.', 'hep-th-0203094-1-7-8': 'For the Schwarzschild background we have [MATH].', 'hep-th-0203094-1-8-0': 'The wave function of a system is a natural object to examine to understand any symmetry present in the system.', 'hep-th-0203094-1-8-1': 'To this end, consider the set [MATH].', 'hep-th-0203094-1-8-2': 'The elements of [MATH] are the functions [MATH] defined in Eqn. (10) with different values of [MATH] corresponding to different elements of [MATH].', 'hep-th-0203094-1-8-3': 'For any two elements of [MATH] given by [MATH] and [MATH], we can define a composition law as [MATH].', 'hep-th-0203094-1-8-4': 'Similarly, for any [MATH], we can define the inverse element as [MATH].', 'hep-th-0203094-1-8-5': 'With respect to the composition law defined above, the set [MATH] has the structure of a continuous abelian group.', 'hep-th-0203094-1-9-0': 'The presence of the continuous abelian group [MATH] allows us to describe the way the black hole mass changes in a geometric fashion.', 'hep-th-0203094-1-9-1': 'To do this we need to construct a group invariant metric on the space [MATH].', 'hep-th-0203094-1-9-2': 'There is a well known procedure for doing this.', 'hep-th-0203094-1-9-3': 'We begin with the observation that the group [MATH] has a natural action on the space [MATH].', 'hep-th-0203094-1-9-4': 'Under the action of [MATH], a point [MATH] transforms as [MATH].', 'hep-th-0203094-1-9-5': '[MATH] therefore acts as a group of transformations on the space [MATH].', 'hep-th-0203094-1-9-6': 'On a continuous group [MATH], the group invariant metric can be written as [CITATION] [EQUATION]', 'hep-th-0203094-1-9-7': 'In our case, [MATH] is abelian and using Eqns. (10) and (11), we obtain the expression of the metric on [MATH] as [EQUATION] where [MATH].', 'hep-th-0203094-1-10-0': 'We now have all the ingredients to calculate the geodesic equation of motion in [MATH].', 'hep-th-0203094-1-10-1': 'For this purpose, consider a parametrised curve [MATH] where [MATH] is taken as time.', 'hep-th-0203094-1-10-2': 'Using the metric in Eqn. (12), the geodesic equation of motion in [MATH] can be written as [EQUATION] where [EQUATION] and [MATH].', 'hep-th-0203094-1-10-3': 'In terms of the variable [MATH], Eqn. (13) can be expressed as [EQUATION]', 'hep-th-0203094-1-10-4': 'Integrating Eqn. (15) we get, [EQUATION]', 'hep-th-0203094-1-10-5': 'Eqn. (16) describes how the mass of the black hole changes with time upto an overall undetermined constant.', 'hep-th-0203094-1-10-6': 'It is known that in the quantization of a scalar field in the Schwarzschild background, energy is radiated to infinity by particle creation via the Hawking effect [CITATION].', 'hep-th-0203094-1-10-7': 'From physical grounds it is expected that the back-reaction effects will cause the mass of the black hole to decrease in order to compensate for this energy loss [CITATION].', 'hep-th-0203094-1-10-8': 'Following this standard argument we take the constant in Eqn. (16) to be negative.', 'hep-th-0203094-1-10-9': 'Thus the rate of change of black hole mass is obtained as [EQUATION] where [MATH] is a constant.', 'hep-th-0203094-1-10-10': 'Eqn. (17) agrees with the standard result for the decay of black holes.', 'hep-th-0203094-1-10-11': 'The assumption of large black hole mass used in our formalism is a feature present in the usual approach to black hole decay as well [CITATION].', 'hep-th-0203094-1-10-12': 'It may be noted that the decay rate obtained above is independent of the constant [MATH] which appears in the relation between [MATH] and [MATH] in Eqn. (1).', 'hep-th-0203094-1-10-13': 'We thus have a robust description of the decay of black holes which is independent of the microscopic details of our formalism.', 'hep-th-0203094-1-10-14': 'The idea outlined above can be applied elsewhere.', 'hep-th-0203094-1-10-15': 'There is, for instance, a well known example in non-relativistic quantum mechanics where the mass [MATH] appears as a parameter in the projective representation of the Galilean Group [CITATION].', 'hep-th-0203094-1-10-16': 'Interpreting the set of Galilean transformations as an abelian group on the space of masses and following the steps outlined above we get [MATH], i.e. conservation of mass.', 'hep-th-0203094-1-10-17': 'This is a simple example which reinforces the approach suggested in this Letter.', 'hep-th-0203094-1-11-0': 'We have mentioned before that the near-horizon conformal structure leads to a logarithmic correction to the Bekenstein-Hawking entropy.', 'hep-th-0203094-1-11-1': 'We would now demonstrate that this logarithmic contribution to the entropy generates a corresponding correction term for decay rate of black holes.', 'hep-th-0203094-1-11-2': 'In our formalism, density of states for the black hole was related to the modulus square of the ground state wavefunction as [MATH] [CITATION].', 'hep-th-0203094-1-11-3': 'As mentioned before, the logarithmic correction to the Bekenstein-Hawking entropy is given by [MATH].', 'hep-th-0203094-1-11-4': 'A change in the entropy due to this term would lead to a corresponding change in the density of states.', 'hep-th-0203094-1-11-5': 'Let [MATH] denote the effective wavefunction associated with this new density of states.', 'hep-th-0203094-1-11-6': 'In our formalism, we then have [EQUATION] which gives [EQUATION]', 'hep-th-0203094-1-11-7': 'Next we observe that [MATH] can be written as [EQUATION] where [MATH].', 'hep-th-0203094-1-11-8': 'The set [MATH] forms an abelian group with respect to the composition law [MATH] with [MATH].', 'hep-th-0203094-1-11-9': 'The wavefunction [MATH] thus belongs to the direct product [MATH] which is again an abelian group.', 'hep-th-0203094-1-11-10': 'Following the analysis presented above, we can write the metric on [MATH] as [EQUATION] where [MATH] and [MATH].', 'hep-th-0203094-1-11-11': 'In the limit of large black hole mass, this metric has the form [EQUATION] where we have scaled the black hole mass as [MATH].', 'hep-th-0203094-1-11-12': 'This is compatible with the assumption of large black hole mass.', 'hep-th-0203094-1-11-13': 'The corresponding geodesic equation of motion gives [EQUATION] where [MATH].', 'hep-th-0203094-1-11-14': 'Eqn. (23) can be integrated to give [EQUATION]', 'hep-th-0203094-1-11-15': 'Following the same logic as discussed before, we again take the constant in Eqn. (24) to be negative.', 'hep-th-0203094-1-11-16': 'For large black hole mass, the decay rate is then given by [EQUATION] where [MATH] is a constant.', 'hep-th-0203094-1-11-17': 'Both terms in the r.h.s. of Eqn. (25) contribute to the decay with the same sign.', 'hep-th-0203094-1-11-18': 'Eqn. (25) is again independent of the constant [MATH] and provides a robust expression of the decay rate independent of the details of the formalism.', 'hep-th-0203094-1-11-19': 'The first term on the r.h.s. of Eqn. (25) is identical to what was obtained before while the second term therein is determined solely by the logarithmic correction to the Bekenstein-Hawking entropy.', 'hep-th-0203094-1-11-20': 'Such a logarithmic terms appears universally as a correction to the Bekenstein-Hawking entropy whenever the same is calculated within a conformal field theory framework [CITATION].', 'hep-th-0203094-1-11-21': 'Thus the correction term appearing in the expression for the decay rate of black holes should also be universal.', 'hep-th-0203094-1-12-0': 'To summarize we note that presence of a conformal structure in the near-horizon region of a black hole is a consequence of the holographic principle.', 'hep-th-0203094-1-12-1': 'In our formalism, the appropriate condition for realizing holography is encoded in the self-adjoint extension parameter [MATH].', 'hep-th-0203094-1-12-2': 'On the other hand, [MATH], or equivalently the black hole mass [MATH], has the natural interpretation as a moduli.', 'hep-th-0203094-1-12-3': 'It is known that the geodesic motion in the moduli space of certain physical system provides an appropriate description for the corresponding dynamics [CITATION].', 'hep-th-0203094-1-12-4': 'The same idea applied to our formalism describes the geodesic motion on the space of black hole mass, i.e. it describes black hole decay.', 'hep-th-0203094-1-12-5': 'We are thus led to the surprising conclusion that even the decay of black hole can be given a novel geometric interpretation within the context of the holographic principle.', 'hep-th-0203094-1-13-0': 'It has been claimed that the near-horizon conformal symmetry is associated with a large class of black holes in arbitrary dimensions [CITATION].', 'hep-th-0203094-1-13-1': 'It seems plausible that probing the near-horizon geometry of these black holes would lead to an operator of the form of [MATH], possibly with different coefficients of the inverse square term [CITATION].', 'hep-th-0203094-1-13-2': 'It is thus likely that the analysis presented above for the massive Schwarzschild black hole could be generalized to include other cases as well.'}

{'hep-th-0203094-2-0-0': 'We show that a formalism for analyzing the near-horizon conformal symmetry of Schwarzschild black holes using a scalar field probe is capable of describing black hole decay.', 'hep-th-0203094-2-0-1': 'The equation governing black hole decay can be identified as the geodesic equation in the space of black hole masses.', 'hep-th-0203094-2-0-2': 'This provides a novel geometric interpretation for the decay of black holes.', 'hep-th-0203094-2-0-3': 'Moreover, this approach predicts a precise correction term to the usual expression for the decay rate of black holes.', 'hep-th-0203094-2-1-0': 'August 2003', 'hep-th-0203094-2-2-0': 'PACS : 04.70.', 'hep-th-0203094-2-2-1': 'Dy, 04.60.-m', 'hep-th-0203094-2-3-0': 'The discovery of a conformal group structure in the near-horizon region of a black hole has led to interesting developments in quantum gravity [CITATION].', 'hep-th-0203094-2-3-1': 'The emergence of such a structure is closely related to the holographic nature of the system [CITATION].', 'hep-th-0203094-2-3-2': 'Recently we have shown that by studying the algebraic structure of an operator that governs the near-horizon dynamics of a scalar probe in the background of a massive Schwarzschild black hole, the underlying conformal group structure can be explicitly revealed [CITATION].', 'hep-th-0203094-2-3-3': 'Our analysis was based on the requirement of unitarity of the conformal theory and self-adjointness [CITATION] of the associated near-horizon Hamiltonian [CITATION].', 'hep-th-0203094-2-3-4': 'This led to the introduction of the self-adjoint extension of the near-horizon Hamiltonian, described by a real parameter [MATH].', 'hep-th-0203094-2-3-5': 'In this approach, the holographic nature of the black hole is thus realized by the self-adjoint extension.', 'hep-th-0203094-2-3-6': 'It was also shown that for the formalism to be self-consistent, [MATH] must be a small positive number.', 'hep-th-0203094-2-3-7': 'This constraint on [MATH] essentially incorporates the correct boundary conditions required for holography.', 'hep-th-0203094-2-4-0': 'In a subsequent paper [CITATION] we showed that further confirmation of the near-horizon conformal structure emerged when [MATH] was related to the mass [MATH] of the Schwarzschild black hole as [EQUATION] where [MATH] is a nonzero positive constant.', 'hep-th-0203094-2-4-1': 'For the Schwarzschild background it was found that [MATH].', 'hep-th-0203094-2-4-2': 'For the moment we consider a general positive value of [MATH].', 'hep-th-0203094-2-4-3': 'It should be stressed that our analysis is valid only for massive black holes, i.e. for small values of [MATH].', 'hep-th-0203094-2-4-4': 'Once the identification given by Eqn. (1) is made, it was shown in Ref. [CITATION] that our approach naturally leads to the characteristic logarithmic correction to the Bekenstein-Hawking entropy [CITATION].', 'hep-th-0203094-2-4-5': 'It is known that such a logarithmic correction term is universally present in any calculation of black hole entropy within a conformal field theory framework [CITATION].', 'hep-th-0203094-2-5-0': 'In this Letter we demonstrate that our formalism [CITATION] for analyzing the near-horizon conformal structure is also capable of describing black hole decay.', 'hep-th-0203094-2-5-1': 'For this purpose, we consider the space of all self-adjoint extensions which we denote by [MATH].', 'hep-th-0203094-2-5-2': 'As indicated in Eqn. (1), the self-adjoint extension parameter [MATH] is related to the mass [MATH] of the black hole.', 'hep-th-0203094-2-5-3': '[MATH] can therefore be taken as a good coordinate to describe [MATH].', 'hep-th-0203094-2-5-4': 'The central idea here is to study geodesic motion in the space of self-adjoint extensions.', 'hep-th-0203094-2-5-5': 'We find that within our formalism, there is a natural way to give the space [MATH] a metric.', 'hep-th-0203094-2-5-6': 'The geodesic motion in [MATH] is then calculated using this metric.', 'hep-th-0203094-2-5-7': 'We find that the equation for geodesic motion in [MATH] agrees with the equation for black hole decay [CITATION] for a suitable choice of the affine parameter.', 'hep-th-0203094-2-5-8': 'Once the affine parameter has been so chosen, our approach predicts a precise correction term in the equation for black hole decay arising from the logarithmic correction to the black hole entropy.', 'hep-th-0203094-2-6-0': 'We start by recalling some of the basic results obtained in references [CITATION] and [CITATION] which shall be useful for our analysis.', 'hep-th-0203094-2-6-1': 'The action [MATH] for a massless scalar field coupled to a Schwarzschild black hole of mass [MATH] in 3+1 dimensions is given by [EQUATION] where the Schwarzschild metric in the spherical polar coordinates has the form [EQUATION]', 'hep-th-0203094-2-6-2': 'In the near-horizon limit, the corresponding Klein-Gordon operator for the time-independent, zero frequency and zero angular momentum modes of a massless scalar field is given by [CITATION] [EQUATION] where [MATH] is the near-horizon coordinate.', 'hep-th-0203094-2-6-3': 'In order to study the quantum properties of [MATH], we consider the equation [EQUATION] where [MATH] is the eigenvalue and [MATH] is the associated wavefunction.', 'hep-th-0203094-2-6-4': '[MATH] belongs to the class of unbounded linear operators on a Hilbert space [CITATION] and it admits a one-parameter family of self-adjoint extensions labelled by a [MATH] parameter [MATH], where [MATH] is real.', 'hep-th-0203094-2-6-5': 'The normalized bound state eigenfunction and eigenvalue of Eqn. (5) are given by [CITATION] [EQUATION] and [EQUATION] respectively, where [MATH] is the modified Bessel function.', 'hep-th-0203094-2-6-6': 'In our formalism, this solution is interpreted as bound state excitation of the black hole due to the capture of the scalar field.', 'hep-th-0203094-2-7-0': 'As mentioned before, the requirement of near-horizon conformal symmetry places an important constraint on [MATH] [CITATION].', 'hep-th-0203094-2-7-1': 'To see this, consider a band-like region [MATH], where [MATH] and [MATH] is real and positive.', 'hep-th-0203094-2-7-2': 'When [MATH] and satisfies the condition [MATH], we see that [MATH].', 'hep-th-0203094-2-7-3': 'Under this condition, [MATH] belongs to the near-horizon region of the black hole.', 'hep-th-0203094-2-7-4': "At any point [MATH] the leading behaviour of the wavefunction is given by [CITATION] [EQUATION] where [MATH] and [MATH] is Euler's constant.", 'hep-th-0203094-2-7-5': 'Taking a typical value of [MATH], we can write the wavefunction as [EQUATION] for [MATH].', 'hep-th-0203094-2-7-6': 'Substituting the value of [MATH] from Eqn. (1) in Eqn. (9), the wavefunction can be written as a function of [MATH] as [EQUATION]', 'hep-th-0203094-2-7-7': 'The function [MATH] captures the [MATH] dependence of the wavefunction and [MATH] is a positive constant.', 'hep-th-0203094-2-7-8': 'For the Schwarzschild background we have [MATH].', 'hep-th-0203094-2-8-0': 'The wave function of a system is a natural object to examine in order to understand any symmetry that might be present in the system.', 'hep-th-0203094-2-8-1': 'To this end, consider the set [MATH].', 'hep-th-0203094-2-8-2': 'The elements of [MATH] are the functions [MATH] defined in Eqn. (10) with different values of [MATH] corresponding to different elements of [MATH].', 'hep-th-0203094-2-8-3': 'For any two elements of [MATH] given by [MATH] and [MATH], we can define a composition law as [MATH].', 'hep-th-0203094-2-8-4': 'Similarly, for any [MATH], we can define the inverse element as [MATH].', 'hep-th-0203094-2-8-5': 'With respect to the composition law defined above, the set [MATH] has the structure of a continuous abelian group.', 'hep-th-0203094-2-9-0': 'The presence of the continuous abelian group [MATH] allows us to describe the way the black hole mass changes in a geometric fashion.', 'hep-th-0203094-2-9-1': 'To do this we need to construct a group invariant metric on the space [MATH].', 'hep-th-0203094-2-9-2': 'There is a well known procedure for doing this.', 'hep-th-0203094-2-9-3': 'We begin with the observation that the group [MATH] has a natural action on the space [MATH].', 'hep-th-0203094-2-9-4': 'Under the action of [MATH], a point [MATH] transforms as [MATH].', 'hep-th-0203094-2-9-5': '[MATH] therefore acts as a group of transformations on the space [MATH].', 'hep-th-0203094-2-9-6': 'On a continuous group [MATH], the group invariant metric can be written as [CITATION] [EQUATION]', 'hep-th-0203094-2-9-7': 'In our case, [MATH] is abelian and using Eqns. (10) and (11), we obtain the expression of the metric on [MATH] as [EQUATION] where [MATH].', 'hep-th-0203094-2-10-0': 'We now have all the ingredients to calculate the geodesic equation of motion in [MATH].', 'hep-th-0203094-2-10-1': 'For this purpose, consider a parametrised curve [MATH] where [MATH] is taken as the affine parameter.', 'hep-th-0203094-2-10-2': 'Using the metric in Eqn. (12), the geodesic equation of motion in [MATH] can be written as [EQUATION] where [EQUATION] and [MATH].', 'hep-th-0203094-2-10-3': 'In terms of the variable [MATH], Eqn. (13) can be expressed as [EQUATION]', 'hep-th-0203094-2-10-4': 'Integrating Eqn. (15) we get, [EQUATION]', 'hep-th-0203094-2-10-5': 'Eqn. (16) describes how the mass of the black hole changes with respect to the affine parameter [MATH] upto an overall undetermined constant.', 'hep-th-0203094-2-10-6': 'It is chosen to be given by [MATH] where [MATH] is time and [MATH] is a constant, Eqn. (16) reduces to [EQUATION] where [MATH] is a constant.', 'hep-th-0203094-2-10-7': 'The sign of the constant [MATH] has been chosen based on the expectation that the back-reaction effects will cause the mass of the black hole to decrease in order to compensate for this energy loss [CITATION].', 'hep-th-0203094-2-10-8': 'With these identifications, Eqn. (17) agrees with the standard result of black hole decay.', 'hep-th-0203094-2-10-9': 'The assumption of large black hole mass used in our formalism is a feature present in the usual approach to black hole decay as well [CITATION].', 'hep-th-0203094-2-10-10': 'It may be noted that the decay rate obtained above is independent of the constant [MATH] which appears in the relation between [MATH] and [MATH] in Eqn. (1).', 'hep-th-0203094-2-10-11': 'We thus have a robust description of the decay of black holes which is independent of the microscopic details of our formalism.', 'hep-th-0203094-2-11-0': 'The idea outlined above can be applied elsewhere as well.', 'hep-th-0203094-2-11-1': 'There is, for instance, a well known example in non-relativistic quantum mechanics where the mass [MATH] appears as a parameter in the projective representation of the Galilean Group [CITATION].', 'hep-th-0203094-2-11-2': 'Interpreting the set of Galilean transformations as an abelian group on the space of masses and following the steps outlined above we get [MATH], where [MATH] is a constant.', 'hep-th-0203094-2-11-3': 'Again, with the choice of [MATH] where [MATH] is time and [MATH] are constants leads to [MATH].', 'hep-th-0203094-2-11-4': 'Choosing [MATH] we see that this is consistent with conservation of mass.', 'hep-th-0203094-2-12-0': 'In ref. [CITATION] we showed that the near-horizon conformal structure leads to a logarithmic correction to the Bekenstein-Hawking entropy.', 'hep-th-0203094-2-12-1': 'We would now demonstrate that this logarithmic contribution to the entropy generates a corresponding correction term for decay rate of black holes.', 'hep-th-0203094-2-12-2': 'In our formalism, density of states for the black hole was related to the modulus square of the wavefunction as [MATH] [CITATION].', 'hep-th-0203094-2-12-3': 'The logarithmic correction to the Bekenstein-Hawking entropy is given by [MATH].', 'hep-th-0203094-2-12-4': 'A change in the entropy due to this term would lead to a corresponding change in the density of states.', 'hep-th-0203094-2-12-5': 'Let [MATH] denote the effective wavefunction associated with this new density of states.', 'hep-th-0203094-2-12-6': 'In our formalism, we then have [EQUATION] which gives [EQUATION]', 'hep-th-0203094-2-12-7': 'Next we observe that [MATH] can be written as [EQUATION] where [MATH].', 'hep-th-0203094-2-12-8': 'The set [MATH] forms an abelian group with respect to the composition law [MATH] with [MATH].', 'hep-th-0203094-2-12-9': 'The wavefunction [MATH] thus belongs to the direct product [MATH] which is again an abelian group.', 'hep-th-0203094-2-12-10': 'Following the analysis presented above, we can write the metric on [MATH] as [EQUATION] where [MATH] and [MATH].', 'hep-th-0203094-2-12-11': 'In the limit of large black hole mass, this metric has the form [EQUATION]', 'hep-th-0203094-2-12-12': 'The corresponding geodesic equation of motion gives [EQUATION] where [MATH].', 'hep-th-0203094-2-12-13': 'Eqn. (23) can be integrated to give [EQUATION]', 'hep-th-0203094-2-12-14': 'Recall that the affine parameter [MATH] has already been identified with [MATH].', 'hep-th-0203094-2-12-15': 'With this identification, and for large black hole mass, the decay rate of black hole is obtained as [EQUATION]', 'hep-th-0203094-2-12-16': 'Note that using the same logic as discussed before, the constant in Eqn. (24) has been written as [MATH] with [MATH].', 'hep-th-0203094-2-12-17': 'Both terms in the r.h.s. of Eqn. (25) contribute to the decay with the same sign.', 'hep-th-0203094-2-12-18': 'The first term on the r.h.s. of Eqn. (25) is identical to what was already obtained before through appropriate identification of the affine parameter [MATH].', 'hep-th-0203094-2-12-19': 'The main result of this Letter is that once [MATH] is chosen in the fashion indicated here, the formalism naturally leads to a precise correction term for the black hole decay rate, determined solely by the logarithmic correction to the Bekenstein-Hawking entropy.', 'hep-th-0203094-2-12-20': 'Such a logarithmic term appears universally whenever the Bekenstein-Hawking entropy is calculated within a conformal field theory framework [CITATION].', 'hep-th-0203094-2-12-21': 'It is thus expected that the correction obtained for the black hole decay should share this same universal property as well.', 'hep-th-0203094-2-13-0': 'To summarize we note that the presence of a conformal structure in the near-horizon region of a black hole is a consequence of the holographic principle.', 'hep-th-0203094-2-13-1': 'In our formalism, the appropriate condition for realizing holography is encoded in the self-adjoint extension parameter [MATH].', 'hep-th-0203094-2-13-2': 'The parameter [MATH], or equivalently the black hole mass [MATH], then has the natural interpretation as a moduli.', 'hep-th-0203094-2-13-3': 'It is known that the geodesic motion in the moduli space of certain physical system provides an appropriate description for the corresponding dynamics [CITATION].', 'hep-th-0203094-2-13-4': 'Implementing this idea with an appropriate identification of the affine parameter is consistent with the standard description of black hole decay.', 'hep-th-0203094-2-13-5': 'Moreover, the approach then predicts a correction term to the black hole decay rate arising from the logarithmic correction to the Bekenstein-Hawking entropy.', 'hep-th-0203094-2-13-6': 'We are thus led to the surprising conclusion that even the decay of black hole can be given a novel geometric interpretation within the context of the holographic principle.', 'hep-th-0203094-2-14-0': 'It has been claimed that the near-horizon conformal symmetry is associated with a large class of black holes in arbitrary dimensions [CITATION].', 'hep-th-0203094-2-14-1': 'It seems plausible that probing the near-horizon geometry of these black holes would lead to an operator of the form of [MATH], possibly with a different coefficient for the inverse square term [CITATION].', 'hep-th-0203094-2-14-2': 'It is thus likely that the analysis presented above for the massive Schwarzschild black hole could be generalized to include other cases as well.'}

[['hep-th-0203094-1-12-1', 'hep-th-0203094-2-13-1'], ['hep-th-0203094-1-12-3', 'hep-th-0203094-2-13-3'], ['hep-th-0203094-1-12-5', 'hep-th-0203094-2-13-6'], ['hep-th-0203094-1-13-0', 'hep-th-0203094-2-14-0'], ['hep-th-0203094-1-13-2', 'hep-th-0203094-2-14-2'], ['hep-th-0203094-1-7-1', 'hep-th-0203094-2-7-1'], ['hep-th-0203094-1-7-2', 'hep-th-0203094-2-7-2'], ['hep-th-0203094-1-7-3', 'hep-th-0203094-2-7-3'], ['hep-th-0203094-1-7-7', 'hep-th-0203094-2-7-7'], ['hep-th-0203094-1-7-8', 'hep-th-0203094-2-7-8'], ['hep-th-0203094-1-8-1', 'hep-th-0203094-2-8-1'], ['hep-th-0203094-1-8-2', 'hep-th-0203094-2-8-2'], ['hep-th-0203094-1-8-3', 'hep-th-0203094-2-8-3'], ['hep-th-0203094-1-8-4', 'hep-th-0203094-2-8-4'], ['hep-th-0203094-1-8-5', 'hep-th-0203094-2-8-5'], ['hep-th-0203094-1-9-0', 'hep-th-0203094-2-9-0'], ['hep-th-0203094-1-9-1', 'hep-th-0203094-2-9-1'], ['hep-th-0203094-1-9-2', 'hep-th-0203094-2-9-2'], ['hep-th-0203094-1-9-3', 'hep-th-0203094-2-9-3'], ['hep-th-0203094-1-9-4', 'hep-th-0203094-2-9-4'], ['hep-th-0203094-1-9-5', 'hep-th-0203094-2-9-5'], ['hep-th-0203094-1-9-6', 'hep-th-0203094-2-9-6'], ['hep-th-0203094-1-9-7', 'hep-th-0203094-2-9-7'], ['hep-th-0203094-1-6-0', 'hep-th-0203094-2-6-0'], ['hep-th-0203094-1-6-1', 'hep-th-0203094-2-6-1'], ['hep-th-0203094-1-6-4', 'hep-th-0203094-2-6-4'], ['hep-th-0203094-1-4-0', 'hep-th-0203094-2-4-0'], ['hep-th-0203094-1-4-2', 'hep-th-0203094-2-4-1'], ['hep-th-0203094-1-4-3', 'hep-th-0203094-2-4-2'], ['hep-th-0203094-1-3-0', 'hep-th-0203094-2-3-0'], ['hep-th-0203094-1-3-1', 'hep-th-0203094-2-3-1'], ['hep-th-0203094-1-3-2', 'hep-th-0203094-2-3-2'], ['hep-th-0203094-1-3-3', 'hep-th-0203094-2-3-3'], ['hep-th-0203094-1-3-4', 'hep-th-0203094-2-3-4'], ['hep-th-0203094-1-3-6', 'hep-th-0203094-2-3-6'], ['hep-th-0203094-1-3-7', 'hep-th-0203094-2-3-7'], ['hep-th-0203094-1-0-0', 'hep-th-0203094-2-0-0'], ['hep-th-0203094-1-0-2', 'hep-th-0203094-2-0-2'], ['hep-th-0203094-1-5-0', 'hep-th-0203094-2-5-0'], ['hep-th-0203094-1-5-2', 'hep-th-0203094-2-5-2'], ['hep-th-0203094-1-5-3', 'hep-th-0203094-2-5-3'], ['hep-th-0203094-1-11-1', 'hep-th-0203094-2-12-1'], ['hep-th-0203094-1-11-4', 'hep-th-0203094-2-12-4'], ['hep-th-0203094-1-11-5', 'hep-th-0203094-2-12-5'], ['hep-th-0203094-1-11-6', 'hep-th-0203094-2-12-6'], ['hep-th-0203094-1-11-7', 'hep-th-0203094-2-12-7'], ['hep-th-0203094-1-11-8', 'hep-th-0203094-2-12-8'], ['hep-th-0203094-1-11-9', 'hep-th-0203094-2-12-9'], ['hep-th-0203094-1-11-10', 'hep-th-0203094-2-12-10'], ['hep-th-0203094-1-11-13', 'hep-th-0203094-2-12-12'], ['hep-th-0203094-1-11-14', 'hep-th-0203094-2-12-13'], ['hep-th-0203094-1-11-17', 'hep-th-0203094-2-12-17'], ['hep-th-0203094-1-10-0', 'hep-th-0203094-2-10-0'], ['hep-th-0203094-1-10-2', 'hep-th-0203094-2-10-2'], ['hep-th-0203094-1-10-3', 'hep-th-0203094-2-10-3'], ['hep-th-0203094-1-10-4', 'hep-th-0203094-2-10-4'], ['hep-th-0203094-1-10-11', 'hep-th-0203094-2-10-9'], ['hep-th-0203094-1-10-12', 'hep-th-0203094-2-10-10'], ['hep-th-0203094-1-10-13', 'hep-th-0203094-2-10-11'], ['hep-th-0203094-1-10-15', 'hep-th-0203094-2-11-1'], ['hep-th-0203094-1-12-0', 'hep-th-0203094-2-13-0'], ['hep-th-0203094-1-12-2', 'hep-th-0203094-2-13-2'], ['hep-th-0203094-1-13-1', 'hep-th-0203094-2-14-1'], ['hep-th-0203094-1-7-0', 'hep-th-0203094-2-7-0'], ['hep-th-0203094-1-7-4', 'hep-th-0203094-2-7-4'], ['hep-th-0203094-1-7-5', 'hep-th-0203094-2-7-5'], ['hep-th-0203094-1-7-6', 'hep-th-0203094-2-7-6'], ['hep-th-0203094-1-8-0', 'hep-th-0203094-2-8-0'], ['hep-th-0203094-1-6-2', 'hep-th-0203094-2-6-2'], ['hep-th-0203094-1-6-6', 'hep-th-0203094-2-6-5'], ['hep-th-0203094-1-6-7', 'hep-th-0203094-2-6-6'], ['hep-th-0203094-1-4-9', 'hep-th-0203094-2-4-5'], ['hep-th-0203094-1-3-5', 'hep-th-0203094-2-3-5'], ['hep-th-0203094-1-5-1', 'hep-th-0203094-2-5-1'], ['hep-th-0203094-1-5-4', 'hep-th-0203094-2-5-4'], ['hep-th-0203094-1-11-2', 'hep-th-0203094-2-12-2'], ['hep-th-0203094-1-11-3', 'hep-th-0203094-2-12-3'], ['hep-th-0203094-1-11-20', 'hep-th-0203094-2-12-20'], ['hep-th-0203094-1-10-1', 'hep-th-0203094-2-10-1'], ['hep-th-0203094-1-10-5', 'hep-th-0203094-2-10-5'], ['hep-th-0203094-1-10-16', 'hep-th-0203094-2-11-2'], ['hep-th-0203094-1-6-3', 'hep-th-0203094-2-6-3'], ['hep-th-0203094-1-4-4', 'hep-th-0203094-2-4-3'], ['hep-th-0203094-1-4-8', 'hep-th-0203094-2-4-4'], ['hep-th-0203094-1-0-1', 'hep-th-0203094-2-0-1'], ['hep-th-0203094-1-0-1', 'hep-th-0203094-2-0-3'], ['hep-th-0203094-1-0-3', 'hep-th-0203094-2-0-3'], ['hep-th-0203094-1-5-5', 'hep-th-0203094-2-5-5'], ['hep-th-0203094-1-5-6', 'hep-th-0203094-2-5-6'], ['hep-th-0203094-1-5-6', 'hep-th-0203094-2-5-7'], ['hep-th-0203094-1-11-0', 'hep-th-0203094-2-12-0'], ['hep-th-0203094-1-11-0', 'hep-th-0203094-2-12-19'], ['hep-th-0203094-1-11-11', 'hep-th-0203094-2-12-11'], ['hep-th-0203094-1-11-11', 'hep-th-0203094-2-12-15'], ['hep-th-0203094-1-11-12', 'hep-th-0203094-2-12-15'], ['hep-th-0203094-1-11-15', 'hep-th-0203094-2-12-16'], ['hep-th-0203094-1-11-16', 'hep-th-0203094-2-12-15'], ['hep-th-0203094-1-11-19', 'hep-th-0203094-2-12-18'], ['hep-th-0203094-1-11-21', 'hep-th-0203094-2-12-19'], ['hep-th-0203094-1-11-21', 'hep-th-0203094-2-12-21'], ['hep-th-0203094-1-10-7', 'hep-th-0203094-2-10-7'], ['hep-th-0203094-1-10-10', 'hep-th-0203094-2-10-8'], ['hep-th-0203094-1-10-14', 'hep-th-0203094-2-11-0']]

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[]

[['hep-th-0203094-1-6-3', 'hep-th-0203094-2-6-3'], ['hep-th-0203094-1-4-4', 'hep-th-0203094-2-4-3'], ['hep-th-0203094-1-4-8', 'hep-th-0203094-2-4-4'], ['hep-th-0203094-1-0-1', 'hep-th-0203094-2-0-1'], ['hep-th-0203094-1-0-1', 'hep-th-0203094-2-0-3'], ['hep-th-0203094-1-0-3', 'hep-th-0203094-2-0-3'], ['hep-th-0203094-1-5-5', 'hep-th-0203094-2-5-5'], ['hep-th-0203094-1-5-6', 'hep-th-0203094-2-5-6'], ['hep-th-0203094-1-5-6', 'hep-th-0203094-2-5-7'], ['hep-th-0203094-1-11-0', 'hep-th-0203094-2-12-0'], ['hep-th-0203094-1-11-0', 'hep-th-0203094-2-12-19'], ['hep-th-0203094-1-11-11', 'hep-th-0203094-2-12-11'], ['hep-th-0203094-1-11-11', 'hep-th-0203094-2-12-15'], ['hep-th-0203094-1-11-12', 'hep-th-0203094-2-12-15'], ['hep-th-0203094-1-11-15', 'hep-th-0203094-2-12-16'], ['hep-th-0203094-1-11-16', 'hep-th-0203094-2-12-15'], ['hep-th-0203094-1-11-19', 'hep-th-0203094-2-12-18'], ['hep-th-0203094-1-11-21', 'hep-th-0203094-2-12-19'], ['hep-th-0203094-1-11-21', 'hep-th-0203094-2-12-21'], ['hep-th-0203094-1-10-7', 'hep-th-0203094-2-10-7'], ['hep-th-0203094-1-10-10', 'hep-th-0203094-2-10-8'], ['hep-th-0203094-1-10-14', 'hep-th-0203094-2-11-0']]

[]

['hep-th-0203094-1-1-0', 'hep-th-0203094-1-2-0', 'hep-th-0203094-1-2-1', 'hep-th-0203094-2-1-0', 'hep-th-0203094-2-2-0', 'hep-th-0203094-2-2-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0203094

1808.04807

{'1808.04807-1-0-0': 'We consider the problem of testing graph cluster structure: given access to a graph [MATH], can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph?', '1808.04807-1-0-1': 'This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish between the graph having good expansion (i.e. being a good single cluster) and the graph having a sparse cut (i.e. being a union of at least two clusters).', '1808.04807-1-0-2': "A recent work of Czumaj, Peng, and Sohler (STOC'15) gave an ingenious sublinear time algorithm for testing [MATH]-clusterability in time [MATH]: their algorithm implicitly embeds a random sample of vertices of the graph into Euclidean space, and then clusters the samples based on estimates of Euclidean distances between the points.", '1808.04807-1-0-3': 'This yields a very efficient testing algorithm, but only works if the cluster structure is very strong: it is necessary to assume that the gap between conductances of accepted and rejected graphs is at least logarithmic in the size of the graph [MATH].', '1808.04807-1-0-4': 'In this paper we show how one can leverage more refined geometric information, namely angles as opposed to distances, to obtain a sublinear time tester that works even when the gap is a sufficiently large constant.', '1808.04807-1-0-5': 'Our tester is based on the singular value decomposition of a natural matrix derived from random walk transition probabilities from a small sample of seed nodes.', '1808.04807-1-1-0': 'We complement our algorithm with a matching lower bound on the query complexity of testing clusterability.', '1808.04807-1-1-1': 'Our lower bound is based on a novel property testing problem, which we analyze using Fourier analytic tools.', '1808.04807-1-1-2': 'As a byproduct of our techniques, we also achieve new lower bounds for the problem of approximating MAX-CUT value in sublinear time.', '1808.04807-1-2-0': '# Preliminaries', '1808.04807-1-3-0': 'Let [MATH] be a graph and let [MATH] be its adjacency matrix.', '1808.04807-1-4-0': 'The normalized adjacency matrix [MATH] of [MATH] is [MATH], where [MATH] is the diagonal matrix of the degrees.', '1808.04807-1-4-1': 'The normalized Laplacian of [MATH] is [MATH].', '1808.04807-1-5-0': '0 The internal and external conductance parameters of (subsets of vertices of) [MATH] are defined as follows.', '1808.04807-1-6-0': 'Let [MATH] be the degree of vertex [MATH].', '1808.04807-1-6-1': 'For a set [MATH], let [MATH] denote the volume of set [MATH].', '1808.04807-1-6-2': 'For a set [MATH], the conductance of [MATH] within [MATH], denoted by [MATH], is the number of edges with one endpoint in [MATH] and the other in [MATH] divided by [MATH].', '1808.04807-1-6-3': 'Equivalently, [MATH] is the probability that a uniformly random neighbor of a vertex in [MATH], that is selected with probability proportional to degree is in [MATH].', '1808.04807-1-7-0': 'The internal conductance of [MATH], denoted by [MATH], is defined to be [MATH] if [MATH] and one otherwise.', '1808.04807-1-7-1': 'The conductance of [MATH] is [MATH].', '1808.04807-1-7-2': 'The external conductance of [MATH] is defined to be [MATH].', '1808.04807-1-8-0': 'Based on the conductance parameters, clusterability and unclusterability of [MATH] is defined as follows.', '1808.04807-1-9-0': 'Graph [MATH] is defined to be [MATH]-clusterable if [MATH] can be partitioned into [MATH] for some [MATH] such that for all [MATH], [MATH].', '1808.04807-1-10-0': 'Graph [MATH] is defined to be [MATH]-unclusterable if its vertex set contains [MATH] pairwise disjoint subsets [MATH] such that for all [MATH], [MATH], and [MATH].', '1808.04807-1-11-0': 'Our main focus is to get an algorithm which distinguishes between the following types of graphs.', '1808.04807-1-12-0': 'The YES case: graphs which are [MATH]-clusterable The NO case: graphs which are [MATH]-unclusterable', '1808.04807-1-13-0': 'under certain conditions on [MATH], [MATH], and [MATH].', '1808.04807-1-13-1': 'We then use this algorithm to test whether a graph is clusterable or far from clusterable.', '1808.04807-1-14-0': 'To design an algorithm for the above problem, we use the following notion of a random walk assocated with [MATH].', '1808.04807-1-15-0': 'The random walk associated with [MATH] is defined to be the random walk with transition matrix [MATH].', '1808.04807-1-15-1': 'Equivalently, from any vertex [MATH], this random walk takes every edge of [MATH] incident on [MATH] with probability [MATH], and stays on [MATH] with probability [MATH].', '1808.04807-1-15-2': 'We can write the transition matrix as [MATH], where [MATH].', '1808.04807-1-16-0': 'To see the equivalence of the two definitions of [MATH] above, observe that the transition matrix is [MATH] and [MATH].', '1808.04807-1-16-1': 'Hence, [MATH].', '1808.04807-1-17-0': 'Our algorithm and analysis use spectral techniques, and therefore, we setup the following notation.', '1808.04807-1-18-0': 'Let [MATH] be the matrix whose columns are the orthonormal eigenvectors of [MATH] arranged in descending order of eigenvalues.', '1808.04807-1-18-1': 'Let [MATH] denote the diagonal matrix of the eigenvalues of [MATH].', '1808.04807-1-18-2': 'Then [EQUATION]', '1808.04807-1-18-3': 'Notice that for each [MATH], we can write [MATH] as [MATH].', '1808.04807-1-18-4': 'Hence, [MATH] is a left eigenvector of [MATH] with eigenvalue [MATH].', '1808.04807-1-18-5': 'Similarly, [MATH] is a right eigenvector of [MATH] with eigenvalue [MATH].', '1808.04807-1-18-6': 'Then we have [MATH] and [MATH].', '1808.04807-1-19-0': 'We will use the following standard results on matrix norms and eigenvalues.', '1808.04807-1-20-0': 'Frobenius norm [MATH] (resp. spectral norm [MATH]) is submultiplicative on all (resp. positive semidefinite) matrices.', '1808.04807-1-20-1': 'That is, for any two [MATH] (positive semidefinite) matrices [MATH] and [MATH], [MATH] (resp. [MATH]).', '1808.04807-1-21-0': 'The following is a result from [CITATION] (Theorem 1.3.20 on page 53).', '1808.04807-1-22-0': 'For any [MATH] matrix [MATH] and any [MATH] matrix [MATH], the multisets of nonzero eigenvalues of [MATH] and [MATH] are equal.', '1808.04807-1-22-1': 'In particular, if one of [MATH] and [MATH] is positive semidefinite, then [MATH].', '1808.04807-1-23-0': "[Weyl's Inequality] Let [MATH] and [MATH] be symmetric [MATH] matrices.", '1808.04807-1-23-1': 'Then for all [MATH], [MATH].', '1808.04807-1-24-0': 'The next linear algebraic lemma will be useful in our analysis.', '1808.04807-1-24-1': 'The (simple) proof is given in Appendix [REF].', '1808.04807-1-25-0': 'Let [MATH] be an [MATH] matrix, [MATH] be a [MATH] matrix with orthonormal columns, and [MATH] be a [MATH] matrix with orthonormal columns.', '1808.04807-1-25-1': 'Then for all [MATH],', '1808.04807-1-26-0': '[MATH].', '1808.04807-1-26-1': '[MATH].', '1808.04807-1-27-0': '[Courant-Fischer]', '1808.04807-1-28-0': 'Let [MATH] be a symmetric [MATH] matrix with eigenvalues [MATH] If [MATH] denotes the set of subspaces of [MATH] of dimension [MATH], then [EQUATION] [Gershgorin Circle Theorem] Let [MATH] be a [MATH] matrix, with entries [MATH].', '1808.04807-1-28-1': 'For [MATH], let [MATH] be the sum of the absolute values of the non-diagonal entries in the [MATH]-th row.', '1808.04807-1-28-2': 'Let [MATH] be the closed disc centered at [MATH] with radius [MATH].', '1808.04807-1-28-3': 'Such a disc is called a Gershgorin disc.', '1808.04807-1-28-4': 'Every eigenvalue of [MATH] lies within at least one of the Gershgorin discs [MATH].', '1808.04807-1-29-0': 'Let [MATH] be a symmetric PSD matrix with eigenvalues [MATH] and eigenvectors [MATH].', '1808.04807-1-29-1': 'Suppose [MATH] is a matrix such that each column of [MATH] lies in the span of [MATH].', '1808.04807-1-29-2': 'Then for any [MATH], [MATH].', '1808.04807-1-30-0': '# Proof Lemma [REF]', '1808.04807-1-31-0': '[Proof of Lemma [REF]] For the first part, let [MATH] matrix whose columns complete the columns of [MATH] to an orthonormal basis.', '1808.04807-1-31-1': 'Then [MATH] projects a vector onto the column space of [MATH], and [MATH] projects onto the columns space of [MATH], which is also the orthogonal complement of the column space of [MATH].', '1808.04807-1-31-2': 'Therefore, [MATH].', '1808.04807-1-31-3': "Thus, [EQUATION] where the first inequality follows from Weyl's inequality, and the second one holds because [MATH] is positive semidefinite.", '1808.04807-1-32-0': 'The second part follows from the first part by observing that [MATH] and [MATH].', '1808.04807-1-33-0': 'of Lemma [REF] We first express [MATH] in the eigenbasis of [MATH], writing [MATH] for some [MATH], where [MATH] is the matrix with columns [MATH].', '1808.04807-1-33-1': 'We then have, [EQUATION]', '1808.04807-1-33-2': 'At the same time, [EQUATION] where [MATH] is a diagonal matrix with [MATH] for every [MATH].', '1808.04807-1-33-3': 'By Courant–Fischer min-max principle we have, [EQUATION].', '1808.04807-1-33-4': 'Let [MATH] be the subspace with [MATH] which maximizes [MATH].', '1808.04807-1-33-5': 'Since [MATH] for every [MATH] by assumption of the claim, we get [MATH] for every [MATH] with unit [MATH] norm.', '1808.04807-1-33-6': 'Thus by Courant–Fischer min-max principle we get, [EQUATION] as required.'}

{'1808.04807-2-0-0': 'We consider the problem of testing graph cluster structure: given access to a graph [MATH], can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph?', '1808.04807-2-0-1': 'This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish between the graph having good expansion (i.e. being a good single cluster) and the graph having a sparse cut (i.e. being a union of at least two clusters).', '1808.04807-2-0-2': "A recent work of Czumaj, Peng, and Sohler (STOC'15) gave an ingenious sublinear time algorithm for testing [MATH]-clusterability in time [MATH]: their algorithm implicitly embeds a random sample of vertices of the graph into Euclidean space, and then clusters the samples based on estimates of Euclidean distances between the points.", '1808.04807-2-0-3': 'This yields a very efficient testing algorithm, but only works if the cluster structure is very strong: it is necessary to assume that the gap between conductances of accepted and rejected graphs is at least logarithmic in the size of the graph [MATH].', '1808.04807-2-0-4': 'In this paper we show how one can leverage more refined geometric information, namely angles as opposed to distances, to obtain a sublinear time tester that works even when the gap is a sufficiently large constant.', '1808.04807-2-0-5': 'Our tester is based on the singular value decomposition of a natural matrix derived from random walk transition probabilities from a small sample of seed nodes.', '1808.04807-2-1-0': 'We complement our algorithm with a matching lower bound on the query complexity of testing clusterability.', '1808.04807-2-1-1': 'Our lower bound is based on a novel property testing problem, which we analyze using Fourier analytic tools.', '1808.04807-2-1-2': 'As a byproduct of our techniques, we also achieve new lower bounds for the problem of approximating MAX-CUT value in sublinear time.', '1808.04807-2-2-0': '# Preliminaries', '1808.04807-2-3-0': 'Let [MATH] be a graph and let [MATH] be its adjacency matrix.', '1808.04807-2-4-0': 'The normalized adjacency matrix [MATH] of [MATH] is [MATH], where [MATH] is the diagonal matrix of the degrees.', '1808.04807-2-4-1': 'The normalized Laplacian of [MATH] is [MATH].', '1808.04807-2-5-0': '0 The internal and external conductance parameters of (subsets of vertices of) [MATH] are defined as follows.', '1808.04807-2-6-0': 'Let [MATH] be the degree of vertex [MATH].', '1808.04807-2-6-1': 'For a set [MATH], let [MATH] denote the volume of set [MATH].', '1808.04807-2-6-2': 'For a set [MATH], the conductance of [MATH] within [MATH], denoted by [MATH], is the number of edges with one endpoint in [MATH] and the other in [MATH] divided by [MATH].', '1808.04807-2-6-3': 'Equivalently, [MATH] is the probability that a uniformly random neighbor of a vertex in [MATH], that is selected with probability proportional to degree is in [MATH].', '1808.04807-2-7-0': 'The internal conductance of [MATH], denoted by [MATH], is defined to be [MATH] if [MATH] and one otherwise.', '1808.04807-2-7-1': 'The conductance of [MATH] is [MATH].', '1808.04807-2-7-2': 'The external conductance of [MATH] is defined to be [MATH].', '1808.04807-2-8-0': 'Based on the conductance parameters, clusterability and unclusterability of [MATH] is defined as follows.', '1808.04807-2-9-0': 'Graph [MATH] is defined to be [MATH]-clusterable if [MATH] can be partitioned into [MATH] for some [MATH] such that for all [MATH], [MATH].', '1808.04807-2-10-0': 'Graph [MATH] is defined to be [MATH]-unclusterable if its vertex set contains [MATH] pairwise disjoint subsets [MATH] such that for all [MATH], [MATH], and [MATH].', '1808.04807-2-11-0': 'Our main focus is to get an algorithm which distinguishes between the following types of graphs.', '1808.04807-2-12-0': 'The YES case: graphs which are [MATH]-clusterable The NO case: graphs which are [MATH]-unclusterable', '1808.04807-2-13-0': 'under certain conditions on [MATH], [MATH], and [MATH].', '1808.04807-2-13-1': 'We then use this algorithm to test whether a graph is clusterable or far from clusterable.', '1808.04807-2-14-0': 'To design an algorithm for the above problem, we use the following notion of a random walk assocated with [MATH].', '1808.04807-2-15-0': 'The random walk associated with [MATH] is defined to be the random walk with transition matrix [MATH].', '1808.04807-2-15-1': 'Equivalently, from any vertex [MATH], this random walk takes every edge of [MATH] incident on [MATH] with probability [MATH], and stays on [MATH] with probability [MATH].', '1808.04807-2-15-2': 'We can write the transition matrix as [MATH], where [MATH].', '1808.04807-2-16-0': 'To see the equivalence of the two definitions of [MATH] above, observe that the transition matrix is [MATH] and [MATH].', '1808.04807-2-16-1': 'Hence, [MATH].', '1808.04807-2-17-0': 'Our algorithm and analysis use spectral techniques, and therefore, we setup the following notation.', '1808.04807-2-18-0': 'Let [MATH] be the matrix whose columns are the orthonormal eigenvectors of [MATH] arranged in descending order of eigenvalues.', '1808.04807-2-18-1': 'Let [MATH] denote the diagonal matrix of the eigenvalues of [MATH].', '1808.04807-2-18-2': 'Then [EQUATION]', '1808.04807-2-18-3': 'Notice that for each [MATH], we can write [MATH] as [MATH].', '1808.04807-2-18-4': 'Hence, [MATH] is a left eigenvector of [MATH] with eigenvalue [MATH].', '1808.04807-2-18-5': 'Similarly, [MATH] is a right eigenvector of [MATH] with eigenvalue [MATH].', '1808.04807-2-18-6': 'Then we have [MATH] and [MATH].', '1808.04807-2-19-0': 'We will use the following standard results on matrix norms and eigenvalues.', '1808.04807-2-20-0': 'Frobenius norm [MATH] (resp. spectral norm [MATH]) is submultiplicative on all (resp. positive semidefinite) matrices.', '1808.04807-2-20-1': 'That is, for any two [MATH] (positive semidefinite) matrices [MATH] and [MATH], [MATH] (resp. [MATH]).', '1808.04807-2-21-0': 'The following is a result from [CITATION] (Theorem 1.3.20 on page 53).', '1808.04807-2-22-0': 'For any [MATH] matrix [MATH] and any [MATH] matrix [MATH], the multisets of nonzero eigenvalues of [MATH] and [MATH] are equal.', '1808.04807-2-22-1': 'In particular, if one of [MATH] and [MATH] is positive semidefinite, then [MATH].', '1808.04807-2-23-0': "[Weyl's Inequality] Let [MATH] and [MATH] be symmetric [MATH] matrices.", '1808.04807-2-23-1': 'Then for all [MATH], [MATH].', '1808.04807-2-24-0': 'The next linear algebraic lemma will be useful in our analysis.', '1808.04807-2-24-1': 'The (simple) proof is given in Appendix [REF].', '1808.04807-2-25-0': 'Let [MATH] be an [MATH] matrix, [MATH] be a [MATH] matrix with orthonormal columns, and [MATH] be a [MATH] matrix with orthonormal columns.', '1808.04807-2-25-1': 'Then for all [MATH],', '1808.04807-2-26-0': '[MATH].', '1808.04807-2-26-1': '[MATH].', '1808.04807-2-27-0': '[Courant-Fischer]', '1808.04807-2-28-0': 'Let [MATH] be a symmetric [MATH] matrix with eigenvalues [MATH] If [MATH] denotes the set of subspaces of [MATH] of dimension [MATH], then [EQUATION] [Gershgorin Circle Theorem] Let [MATH] be a [MATH] matrix, with entries [MATH].', '1808.04807-2-28-1': 'For [MATH], let [MATH] be the sum of the absolute values of the non-diagonal entries in the [MATH]-th row.', '1808.04807-2-28-2': 'Let [MATH] be the closed disc centered at [MATH] with radius [MATH].', '1808.04807-2-28-3': 'Such a disc is called a Gershgorin disc.', '1808.04807-2-28-4': 'Every eigenvalue of [MATH] lies within at least one of the Gershgorin discs [MATH].', '1808.04807-2-29-0': 'Let [MATH] be a symmetric PSD matrix with eigenvalues [MATH] and eigenvectors [MATH].', '1808.04807-2-29-1': 'Suppose [MATH] is a matrix such that each column of [MATH] lies in the span of [MATH].', '1808.04807-2-29-2': 'Then for any [MATH], [MATH].', '1808.04807-2-30-0': '# Proof Lemma [REF]', '1808.04807-2-31-0': '[Proof of Lemma [REF]] For the first part, let [MATH] matrix whose columns complete the columns of [MATH] to an orthonormal basis.', '1808.04807-2-31-1': 'Then [MATH] projects a vector onto the column space of [MATH], and [MATH] projects onto the columns space of [MATH], which is also the orthogonal complement of the column space of [MATH].', '1808.04807-2-31-2': 'Therefore, [MATH].', '1808.04807-2-31-3': "Thus, [EQUATION] where the first inequality follows from Weyl's inequality, and the second one holds because [MATH] is positive semidefinite.", '1808.04807-2-32-0': 'The second part follows from the first part by observing that [MATH] and [MATH].', '1808.04807-2-33-0': 'of Lemma [REF] We first express [MATH] in the eigenbasis of [MATH], writing [MATH] for some [MATH], where [MATH] is the matrix with columns [MATH].', '1808.04807-2-33-1': 'We then have, [EQUATION]', '1808.04807-2-33-2': 'At the same time, [EQUATION] where [MATH] is a diagonal matrix with [MATH] for every [MATH].', '1808.04807-2-33-3': 'By Courant–Fischer min-max principle we have, [EQUATION].', '1808.04807-2-33-4': 'Let [MATH] be the subspace with [MATH] which maximizes [MATH].', '1808.04807-2-33-5': 'Since [MATH] for every [MATH] by assumption of the claim, we get [MATH] for every [MATH] with unit [MATH] norm.', '1808.04807-2-33-6': 'Thus by Courant–Fischer min-max principle we get, [EQUATION] as required.'}

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['1808.04807-1-9-0', '1808.04807-1-16-1', '1808.04807-1-18-2', '1808.04807-1-23-0', '1808.04807-1-23-1', '1808.04807-1-25-1', '1808.04807-1-26-0', '1808.04807-1-26-1', '1808.04807-1-27-0', '1808.04807-1-31-2', '1808.04807-2-9-0', '1808.04807-2-16-1', '1808.04807-2-18-2', '1808.04807-2-23-0', '1808.04807-2-23-1', '1808.04807-2-25-1', '1808.04807-2-26-0', '1808.04807-2-26-1', '1808.04807-2-27-0', '1808.04807-2-31-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1808.04807

1201.2485

{'1201.2485-1-0-0': 'We report a dynamical formation of a (Mott) gap from holographic fermions.', '1201.2485-1-0-1': 'By coupling a fermion field with dipole action to the charged dilaton black branes with a Lifshitz like IR geometry and [MATH] boundary, we find that when the dipole interaction is large enough, spectral weight is transferred between bands, and beyond a critical dipole interaction, a gap emerges in the fermion density of states.', '1201.2485-1-0-2': 'The value of the gap becomes larger as the strength of the interaction keeps increasing.', '1201.2485-1-1-0': 'Introduction', '1201.2485-1-2-0': 'As a new method to study strong coupled quantum field theory, AdS/CFT correspondence [CITATION] has been used to study the strong coupled phenomena in quantum many-body systems.', '1201.2485-1-2-1': 'One of the most successful progress is that by coupling a free Fermion field to the [MATH] (RN) black hole, a class of non-Fermion Liquid at zero temperature was found holographically[CITATION].', '1201.2485-1-2-2': 'The properties of this non-fermion liquid in the presence of a magnetic field have been studied in[CITATION].', '1201.2485-1-2-3': 'The extension to finite temperature was investigated in[CITATION] etc.', '1201.2485-1-2-4': 'In [CITATION], free fermions in AdS BTZ black hole and Gauss-Bonnet black hole have also been studied respectively.', '1201.2485-1-2-5': 'We can refer to the lecture [CITATION] for an excellent latest review on this subjects.', '1201.2485-1-3-0': 'In [CITATION], the authors studied the holographic fermion system with RN black hole background when a dipole interaction is added, it is found that the strength of the interaction [MATH] is similar to [MATH] in the fermions Hubbard model.', '1201.2485-1-3-1': 'When [MATH] is large enough, a gap is open just like the Mott insulator, in which a large repulsive interaction [MATH] will change the structure of the bands.', '1201.2485-1-3-2': 'Then the fermion energy is in the middle of the gap, which is an insulator.', '1201.2485-1-4-0': 'However, the RN black hole background has nonzero ground state entropy density, which seems to be inconsistent with our intuition that a system of degenerate fermions has a unique ground state.', '1201.2485-1-4-1': 'Therefore, a systematic exploration of the system that has zero extremal entropy will be important and valuable.', '1201.2485-1-4-2': 'Such models have been proposed in Refs.[CITATION].', '1201.2485-1-4-3': 'Furthermore, in Ref.[CITATION], the author investigated the free fermionic response in the background proposed by Gubser and Rocha [CITATION].', '1201.2485-1-4-4': 'They find that the dispersion relation is linear, just like a Fermi liquid.', '1201.2485-1-4-5': 'It is very different from that found in RN black hole [CITATION].', '1201.2485-1-5-0': 'Among these models of zero ground state entropy density, a very important and valuable model is that proposed by Goldstein etc. [CITATION], which near horizon geometry is Lifshitz-like.', '1201.2485-1-5-1': 'Therefore, in this paper, we will study the characteristics of the fermionic response with dipole interaction in this background.', '1201.2485-1-5-2': 'The interesting results we find is that, similar to the fermions in the RN black hole background, the dipole interaction will open a gap for the strong coupled fermions system when the value of the interaction [MATH] is large enough.', '1201.2485-1-6-0': 'The organization of this paper is as follows.', '1201.2485-1-6-1': 'In section II, we first review the charged dilaton black branes with zero entropy at zero temperature.', '1201.2485-1-6-2': 'In section III, we derive the equations of motion for the fermion field with a dipole action.', '1201.2485-1-6-3': 'The main results of the emergence of Gap is in section IV.', '1201.2485-1-6-4': 'Finally, the discussion and conclusions are presented in section V.', '1201.2485-1-7-0': '# An Extremal charged Dilatonic black brane solutions', '1201.2485-1-8-0': '## Einstein-Maxwell-Dilaton model', '1201.2485-1-9-0': 'For completeness let us also review the the Dilaton gravity system [CITATION].', '1201.2485-1-9-1': 'Our discussion follows [CITATION] closely and some details are omitted.', '1201.2485-1-9-2': 'The system we consider is as follows: [EQUATION] where [MATH] is Ricci scalar, [MATH] is the dilaton field, [MATH] is the field strength, and [MATH] is the AdS radius.', '1201.2485-1-9-3': 'In this paper we will focus ourself on electrically charge black branes solution, so the metric and gauge fields can then be written as, [EQUATION]', '1201.2485-1-9-4': 'Such a gauge field satisfies the gauge field equation of motion automatically.', '1201.2485-1-9-5': 'With the above ansatz, the remaining equations of motion can be expressed as follow: [EQUATION] where the prime denote the differentiation with respect to the coordinate [MATH].', '1201.2485-1-9-6': 'In the subsequent subsections, we will construct such solutions that the near-horizon geometry has a Lifshitz-like symmetry but at infinity the geometry asymptotes to AdS.', '1201.2485-1-9-7': 'For convenience, we shall set [MATH] and [MATH] in the following.', '1201.2485-1-10-0': '## Scaling solution near the horizon', '1201.2485-1-11-0': 'In order to construct the scaling solution near the horizon [MATH], we consider the following ansatz: [EQUATION] where [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] are all constants and [MATH] and [MATH] are set to be positive in the following, without loss of generality.', '1201.2485-1-11-1': 'Note that for convenience, one has introduced the variable [MATH] in the above ansatz.', '1201.2485-1-11-2': 'With this ansatz and the equations of motion, we have [EQUATION]', '1201.2485-1-11-3': 'Such a solution has a Lifshitz-like symmetry in the metric, with a dynamical critical exponent [MATH], although such a symmetry is broken by the logarithmic dependence of the dilaton on [MATH].', '1201.2485-1-11-4': 'Also we note that in such a solution, the metric component [MATH] has a double zero at the horizon where [MATH] also vanishes and therefore corresponds to an extremal brane with vanishing ground state entropy density.', '1201.2485-1-11-5': 'Finally, we also point out that for [MATH], such a solution corresponds to an [MATH] geometry, which is also the near horizon geometry of the extremal RN black hole.', '1201.2485-1-11-6': 'The Fermionic response in RN black hole has been explore carefully in Refs. [CITATION].', '1201.2485-1-11-7': 'Therefore we shall assume [MATH] and focus on the Lifshitz like near horizon geometry in the following.', '1201.2485-1-12-0': '## The solution with asymptotes to AdS at infinity', '1201.2485-1-13-0': 'To obtain a solution which asymptotes to AdS at infinity, we can add a perturbation to the scaling solution which is irrelevant in the IR scaling region but relevant in the UV region.', '1201.2485-1-13-1': 'In order to set the constant [MATH] to unity and [MATH] to vanishing, we carry out a coordinate transformation under which, [EQUATION] with [MATH] and [MATH].', '1201.2485-1-13-2': 'Under such rescaling, we have [EQUATION] and in terms of [MATH], the charge can be expressed as [EQUATION]', '1201.2485-1-13-3': 'Subsequently, we add the perturbation to the Eqs. ([REF]).', '1201.2485-1-13-4': 'The resulting functions satisfied the equations of motion at the leading order are [EQUATION] where [MATH], [MATH], and [MATH] keep unchanged.', '1201.2485-1-13-5': 'The perturbation is characterized by the exponent [MATH] and the three constants [MATH], [MATH] and [MATH].', '1201.2485-1-13-6': 'Also, [MATH] and [MATH] are determined by [MATH] and they have the following relations: [MATH], [MATH].', '1201.2485-1-13-7': 'In addition, the perturbation should die out at small [MATH].', '1201.2485-1-13-8': 'Therefore, we must require [MATH], which gives rise to a unique allowed value for this exponent: [MATH].', '1201.2485-1-13-9': 'Thus there are two parameters [MATH] and [MATH] for this perturbated solution.', '1201.2485-1-13-10': 'For simplicity, we will set [MATH].', '1201.2485-1-14-0': 'The initial data for numerical integration is taken from the above perturbated solution near the horizon.', '1201.2485-1-14-1': 'As depicted in Fig. [REF], for the perturbation with [MATH], [MATH] as [MATH], that is to say, the solution near the boundary takes the standard asymptotics of AdS for the perturbation with [MATH] .', '1201.2485-1-15-0': 'In the next section, we shall investigate the holographic fermions in such a background.', '1201.2485-1-15-1': 'Without loss of generality, we shall take [MATH] in the following.', '1201.2485-1-15-2': 'In this case, the chemical potential can be obtained as [MATH] numerically.', '1201.2485-1-16-0': '# Holographic fermion with bulk dipole coupling', '1201.2485-1-17-0': 'In order to know the effect of magnetic dipole coupling on the structure of the spectral functions of the dual fermionic operator, we consider the following bulk fermion action [EQUATION] where [MATH] is related to the usual flat space gamma matrix by a factor of the vielbein, [MATH], [MATH] is the covariant derivative with [MATH] the spin connection 1-forms and [MATH].', '1201.2485-1-17-1': 'Before taking the particular background (Eqs. ([REF]) and ([REF])), we will derive the flow equation as Refs. [CITATION] in a more general static background, i.e., [EQUATION]', '1201.2485-1-17-2': 'Firstly, the Dirac equation can be derived from the action [MATH] [EQUATION]', '1201.2485-1-17-3': 'Making a transformation [MATH] to remove the spin connection and expanding [MATH] as [MATH] in Fourier space, the Dirac equation ([REF]) turns out to be [EQUATION] where due to rotational symmetry in the spatial directions, we set [MATH] and [MATH] without losing generality.', '1201.2485-1-17-4': 'Notice that Eq. ([REF]) only depends on three Gamma matrices [MATH].', '1201.2485-1-17-5': 'So it is convenient to split the spinors [MATH] into [MATH] and choose the following basis for our gamma matrices as in [CITATION]: [EQUATION]', '1201.2485-1-17-6': 'So, we have a new version of the Dirac equation as [EQUATION]', '1201.2485-1-17-7': 'Furthermore, according to eigenvalues of [MATH], we make such a decomposition [MATH].', '1201.2485-1-17-8': 'Then [EQUATION]', '1201.2485-1-17-9': 'Under such decomposition, the Dirac equation ([REF]) can be rewritten as [EQUATION]', '1201.2485-1-17-10': 'Introducing the ratios [MATH], one can package the Dirac equation ([REF]) and ([REF]) into the evolution equation of [MATH], [EQUATION] where [MATH].', '1201.2485-1-17-11': 'The above flow equation will be more convenient to impose the boundary conditions at the horizon and read off the boundary Green functions.', '1201.2485-1-18-0': 'Now, we take the particular background (Eqs. ([REF]) and ([REF])).', '1201.2485-1-18-1': 'Because near the boundary, the geometry is an [MATH], the solution of the Dirac equation ([REF]) can be expressed as [EQUATION]', '1201.2485-1-18-2': "If [MATH] and [MATH] are related by [MATH], then the boundary Green's functions [MATH] is given by [MATH] [CITATION].", '1201.2485-1-18-3': 'Therefore [EQUATION]', '1201.2485-1-18-4': 'Near the horizon, we take the scaling solution near the horizon ([REF]).', '1201.2485-1-18-5': 'We find that the requirement that the solutions of Eqs. ([REF]) and ([REF]) near the horizon be in-falling implies [EQUATION]', '1201.2485-1-19-0': '# Emergence of the gap', '1201.2485-1-20-0': 'In this section, to study the spectral function, we will numerically solve the flow equation ([REF]) with the initial condition ([REF]).', '1201.2485-1-20-1': 'Up to normalization, the spectral function is given by [MATH].', '1201.2485-1-20-2': 'However, due to the relation [MATH], [MATH] can be recovered by [MATH].', '1201.2485-1-20-3': 'Therefore, in the following 3D plots and density plots, to make numerical calculation easier, we will only show [MATH].', '1201.2485-1-21-0': 'From the plots above in FIG. [REF], one can see that for [MATH], a sharp quasiparticle-like peak occurs near [MATH] and [MATH], indicating a Fermi surface .', '1201.2485-1-21-1': 'In addition, the band occurs both positive and negative frequency regions .', '1201.2485-1-21-2': 'It is different from the case in RN black hole, in which the band mainly distributes at the positive frequency region .', '1201.2485-1-21-3': 'When we turn on [MATH] to the value of [MATH], a gap emerges and there are two bands located at the positive frequency and negative frequency regions, respectively (the plots below in FIG. [REF]).', '1201.2485-1-21-4': 'Evidently, the strength of the lower band is bigger than the upper band.', '1201.2485-1-21-5': "We also show the 3D plot and density plot of the Green's function [MATH] for [MATH] and [MATH] in FIG. [REF].", '1201.2485-1-21-6': 'We can see that with the increase of [MATH], the width of the gap becomes larger and the band in the negative frequency region switch gradually to the positive frequency region.', '1201.2485-1-21-7': 'As [MATH] increases further, the lower band disperses and the upper band becomes stronger and sharper.', '1201.2485-1-22-0': 'In order to explore further the characteristics of the spectral function, We show the spectral function [MATH] for [MATH], [MATH], [MATH] and [MATH] for sample values of [MATH].', '1201.2485-1-22-1': 'From the left plot above in FIG. [REF], we find that some peaks distribute at both positive and negative frequency regions.', '1201.2485-1-22-2': 'When the peak approaches [MATH], its height goes to infinity, and its width goes to zero, indicating that there is a Fermi surface at [MATH].', '1201.2485-1-22-3': 'When we amplify [MATH], the strength of the quasiparticle-like peak at [MATH] degrades, and vanishes at a critical value [MATH].', '1201.2485-1-22-4': 'For [MATH], the dipole interaction will open a gap as the case in RN black hole [CITATION].', '1201.2485-1-22-5': 'Differ from the case of [MATH], the height of the peaks degrade and the width becomes larger when [MATH] is approached, and the peak vanishes around [MATH].', '1201.2485-1-22-6': 'We also note that the spectral density mainly appears at negative frequency regions for [MATH] (right plot above in FIG.[', '1201.2485-1-22-7': 'As [MATH] increases, the spectral density begins to distribute the positive frequency and negative frequency regions, respectively (left plot below in FIG.[', '1201.2485-1-22-8': 'When [MATH] increases further, the spectral density switches gradually to the positive frequency regions (right plot below in FIG.[', '1201.2485-1-23-0': 'The another important quantity of interest for us is the density of states [MATH], which is defined as the integral of [MATH] over [MATH].', '1201.2485-1-23-1': 'It is the total spectral weight.', '1201.2485-1-23-2': 'By careful numerical computations, we find that the onset of the gap is near [MATH], which is different from the onset value of [MATH] in the case of RN background.', '1201.2485-1-23-3': 'Consistent with the above observations (FIG. [REF], FIG. [REF] and FIG. [REF]), the total spectral weight is also mainly distribute at negative frequency regions for small [MATH], left plot in FIG. [REF]), and switches to the positive frequency regions as the increase of [MATH].', '1201.2485-1-24-0': 'Finally, we show the relation between the width of the gap [MATH] and [MATH].', '1201.2485-1-24-1': 'Evidently, the gap becomes wider as [MATH] increases.', '1201.2485-1-25-0': '# Conclusions and discussion', '1201.2485-1-26-0': 'In this paper, we have explored the fermionic response in the presence of a bulk dipole interaction term with strength [MATH] in the dilatonic black brane with a Lifshitz like IR geometry and [MATH] boundary.', '1201.2485-1-26-1': 'When [MATH], a sharp quasiparticle-like peak occurs near [MATH] and [MATH], indicating a Fermi surface.', '1201.2485-1-26-2': 'As [MATH] increases, the peak become wide and its strength degrades gradually, and the spectral weight is transferred between bands.', '1201.2485-1-26-3': 'Furthermore, when [MATH] goes beyond a critical dipole interaction [MATH], the Fermi sea disappears and a gap emerges as the case in RN black hole [CITATION].', '1201.2485-1-26-4': 'By studying the another important quantity: the density of states [MATH], we clearly find that the onset of the gap is near [MATH] and the width of the gap becomes bigger as [MATH] increases.', '1201.2485-1-27-0': 'So far, we have found a (Mott) gap from holographic fermions in the background of the dilatonic black brane with a Lifshitz like IR geometry when a dipole interaction term are added as that found in RN black hole.', '1201.2485-1-27-1': 'However, for more general dilatonic black brane, which also have vanishing ground entropy density, for example, a class of systems proposed in Refs.[CITATION], is the emergence of gap robust?', '1201.2485-1-27-2': 'The fermionic response without dipole interaction in this class backgrounds has been investigated in Refs.[CITATION].', '1201.2485-1-27-3': 'Therefore, it is valuable to test the robust of the emergence of gap in the backgrounds of more general dilatonic black brane.', '1201.2485-1-27-4': 'In addition, it is also interesting to extend our investigations in this paper to the case of the non-relativistic fermionic fixed point.', '1201.2485-1-27-5': 'We will address them in the near future.', '1201.2485-1-28-0': 'We especially thank Professor Robert Leigh for his very valuable correspondence.', '1201.2485-1-28-1': 'J.P. Wu would also like to thank Hongbao Zhang and Wei-Jia Li for their collaboration in the related project.', '1201.2485-1-28-2': 'J.P. Wu is partly supported by NSFC(No.10975017) and the Fundamental Research Funds for the central Universities.', '1201.2485-1-28-3': 'H.B. Zeng is supported by the Fundamental Research Funds for the Central Universities (Grant No.1107020117) and the China Postdoctoral Science Foundation (Grant No. 20100481120).'}

{'1201.2485-2-0-0': 'We report a dynamical formation of a (Mott) gap from holographic fermions.', '1201.2485-2-0-1': 'By coupling a fermion field with dipole action to the charged dilaton black branes with a Lifshitz like IR geometry and [MATH] boundary, we find that when the dipole interaction is large enough, spectral weight is transferred between bands, and beyond a critical dipole interaction, a gap emerges in the fermion density of states.', '1201.2485-2-0-2': 'The value of the gap becomes larger as the strength of the interaction keeps increasing.', '1201.2485-2-1-0': 'Introduction', '1201.2485-2-2-0': 'As a new method to study strongly coupled quantum field theory, AdS/CFT correspondence [CITATION] has been used to study the strongly coupled phenomena in quantum many-body systems.', '1201.2485-2-2-1': 'One of the most successful progress is that by coupling a free Fermion field to the [MATH] (RN) black hole, a class of non-Fermi liquid at zero temperature was found holographically[CITATION].', '1201.2485-2-2-2': 'The properties of this non-Fermi liquid in the presence of a magnetic field have been studied in[CITATION].', '1201.2485-2-2-3': 'The extension to finite temperature was investigated in[CITATION].', '1201.2485-2-2-4': 'In [CITATION], free fermions in AdS BTZ black hole and Gauss-Bonnet black hole have also been studied respectively.', '1201.2485-2-2-5': 'In addition, the fermions on Lifshitz Background have been explored in Refs. [CITATION].', '1201.2485-2-2-6': 'We can refer to the lecture [CITATION] for an excellent latest review on this subjects.', '1201.2485-2-3-0': 'In [CITATION], the authors studied the holographic fermion system with RN black hole background when a dipole interaction is added, it is found that the strength of the interaction [MATH] is similar to [MATH] in the fermions Hubbard model.', '1201.2485-2-3-1': 'When [MATH] is large enough, a gap opens just like in a Mott insulator, in which a large repulsive interaction [MATH] will change the structure of the bands.', '1201.2485-2-3-2': 'Then the fermion energy is in the middle of the gap, which makes an insulator.', '1201.2485-2-4-0': 'However, the RN black hole background has nonzero ground state entropy density, which seems to be inconsistent with our intuition that a system of degenerate fermions has a unique ground state.', '1201.2485-2-4-1': 'Therefore, a systematic exploration of the system that has zero extremal entropy will be important and valuable.', '1201.2485-2-4-2': 'Such models have been proposed in Refs.[CITATION].', '1201.2485-2-4-3': 'Furthermore, in Ref.[CITATION], the author investigated the free fermionic response in the background proposed by Gubser and Rocha [CITATION].', '1201.2485-2-4-4': 'They find that the dispersion relation is linear, just like a Fermi liquid.', '1201.2485-2-4-5': 'It is very different from that found in RN black hole [CITATION].', '1201.2485-2-5-0': 'Among these models of zero ground state entropy density, a very important and valuable model is that proposed by Goldstein et al. [CITATION].', '1201.2485-2-5-1': 'Here the near horizon geometry is Lifshitz-like.', '1201.2485-2-5-2': 'Therefore, in this paper, we will study the characteristics of the fermionic response with dipole interaction in this background.', '1201.2485-2-5-3': 'The interesting results we find is that, similar to the fermions in the RN black hole background, the dipole interaction will open a gap for the strongly coupled fermions system when the value of the interaction [MATH] is large enough.', '1201.2485-2-6-0': 'The organization of this paper is as follows.', '1201.2485-2-6-1': 'In section II, we first review the charged dilaton black branes with zero entropy at zero temperature.', '1201.2485-2-6-2': 'In section III, we derive the equations of motion for the fermion field with a dipole action.', '1201.2485-2-6-3': 'The main results of the emergence of gap is in section IV.', '1201.2485-2-6-4': 'Finally, the discussion and conclusions are presented in section V.', '1201.2485-2-7-0': '# An extremal charged dilatonic black brane solutions', '1201.2485-2-8-0': '## Einstein-Maxwell-dilaton model', '1201.2485-2-9-0': 'For completeness let us also review the dilaton gravity system [CITATION].', '1201.2485-2-9-1': 'Our discussion follows [CITATION] closely and some details are omitted.', '1201.2485-2-9-2': 'The system we consider is as follows: [EQUATION] where [MATH] is Ricci scalar, [MATH] is the dilaton field, [MATH] is the field strength, and [MATH] is the AdS radius.', '1201.2485-2-9-3': 'In this paper we will focus on an electrically charged black brane solution.', '1201.2485-2-9-4': 'The metric and gauge fields can be written as, [EQUATION]', '1201.2485-2-9-5': 'Such a gauge field satisfies the gauge field equation of motion automatically.', '1201.2485-2-9-6': 'With the above ansatz, the remaining equations of motion can be expressed as follow: [EQUATION] where the prime denote the differentiation with respect to the coordinate [MATH].', '1201.2485-2-9-7': 'In the subsequent subsections, we will construct such solutions that the near-horizon geometry has a Lifshitz-like symmetry but at infinity the geometry asymptotes to AdS.', '1201.2485-2-9-8': 'For convenience, we shall set [MATH] and [MATH] in the following.', '1201.2485-2-10-0': '## Scaling solution near the horizon', '1201.2485-2-11-0': 'In order to construct the scaling solution near the horizon [MATH], we consider the following ansatz: [EQUATION] where [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] are all constants and [MATH] and [MATH] are set to be positive in the following, without loss of generality.', '1201.2485-2-11-1': 'Note that for convenience, one has introduced the variable [MATH] in the above ansatz.', '1201.2485-2-11-2': 'With this ansatz and the equations of motion, we have [EQUATION]', '1201.2485-2-11-3': 'Such a solution has a Lifshitz-like symmetry in the metric, with a dynamical critical exponent [MATH], although such a symmetry is broken by the logarithmic dependence of the dilaton on [MATH].', '1201.2485-2-11-4': 'Also we note that in the above solution, the metric component [MATH] has a double zero at the horizon where [MATH] also vanishes and therefore corresponds to an extremal brane with vanishing ground state entropy density.', '1201.2485-2-11-5': 'Finally, we also point out that for [MATH], such a solution corresponds to an [MATH] geometry, which is also the near horizon geometry of the extremal RN black hole.', '1201.2485-2-11-6': 'The fermionic response in RN black hole has been explore carefully in Refs. [CITATION].', '1201.2485-2-11-7': 'Therefore we shall assume [MATH] and focus on the Lifshitz like near horizon geometry in the following.', '1201.2485-2-12-0': '## The solution with asymptotes to AdS at infinity', '1201.2485-2-13-0': 'To obtain a solution which asymptotes to AdS at infinity, we can add a perturbation to the scaling solution which is irrelevant in the IR scaling region but relevant in the UV region.', '1201.2485-2-13-1': 'In order to set the constant [MATH] to unity and [MATH] to vanishing, we carry out a coordinate transformation under which, [EQUATION] with [MATH] and [MATH].', '1201.2485-2-13-2': 'Under such rescaling, we have [EQUATION] and in terms of [MATH], the charge can be expressed as [EQUATION]', '1201.2485-2-13-3': 'Subsequently, we add the perturbation to the Eqs. ([REF]).', '1201.2485-2-13-4': 'The resulting functions satisfied the equations of motion at the leading order are [EQUATION] where [MATH], [MATH], and [MATH] keep unchanged.', '1201.2485-2-13-5': 'The perturbation is characterized by the exponent [MATH] and the three constants [MATH], [MATH] and [MATH].', '1201.2485-2-13-6': 'Also, [MATH] and [MATH] are determined by [MATH] and they have the following relations: [MATH], [MATH].', '1201.2485-2-13-7': 'In addition, the perturbation should die out at small [MATH].', '1201.2485-2-13-8': 'Therefore, we must require [MATH], which gives rise to a unique allowed value for this exponent: [MATH].', '1201.2485-2-13-9': 'Thus there are two parameters [MATH] and [MATH] for this perturbated solution.', '1201.2485-2-13-10': 'For simplicity, we will set [MATH].', '1201.2485-2-14-0': 'The initial data for numerical integration is taken from the above perturbated solution near the horizon.', '1201.2485-2-14-1': 'As depicted in Fig. [REF], for the perturbation with [MATH], [MATH] as [MATH], that is to say, the solution near the boundary takes the standard asymptotics of AdS for the perturbation with [MATH] .', '1201.2485-2-14-2': 'In the next section, we shall investigate the holographic fermions with dipole term in such a background.', '1201.2485-2-15-0': '# Holographic fermion with bulk dipole coupling', '1201.2485-2-16-0': 'In order to know the effect of magnetic dipole coupling on the structure of the spectral functions of the dual fermionic operator, we consider the following bulk fermion action [EQUATION] where [MATH] is related to the usual flat space gamma matrix by a factor of the vielbein, [MATH], [MATH] is the covariant derivative with [MATH] the spin connection 1-forms and [MATH].', '1201.2485-2-16-1': 'Before taking the particular background (Eqs. ([REF]) and ([REF])), we will derive the flow equation as Refs. [CITATION] in a more general static background, i.e., [EQUATION]', '1201.2485-2-16-2': 'Firstly, the Dirac equation can be derived from the action [MATH] [EQUATION]', '1201.2485-2-16-3': 'Making a transformation [MATH] to remove the spin connection and expanding [MATH] as [MATH] in Fourier space, the Dirac equation ([REF]) turns out to be [EQUATION] where due to rotational symmetry in the spatial directions, we set [MATH] and [MATH] without losing generality.', '1201.2485-2-16-4': 'Notice that Eq. ([REF]) only depends on three Gamma matrices [MATH].', '1201.2485-2-16-5': 'So it is convenient to split the spinors [MATH] into [MATH] and choose the following basis for our gamma matrices as in [CITATION]: [EQUATION]', '1201.2485-2-16-6': 'So, we have a new version of the Dirac equation as [EQUATION]', '1201.2485-2-16-7': 'Furthermore, according to eigenvalues of [MATH], we make such a decomposition [MATH].', '1201.2485-2-16-8': 'Then [EQUATION]', '1201.2485-2-16-9': 'Under such decomposition, the Dirac equation ([REF]) can be rewritten as [EQUATION]', '1201.2485-2-16-10': 'Introducing the ratios [MATH], one can package the Dirac equation ([REF]) and ([REF]) into the evolution equation of [MATH], [EQUATION] where [MATH].', '1201.2485-2-16-11': 'The above flow equation will be more convenient to impose the boundary conditions at the horizon and read off the boundary Green functions.', '1201.2485-2-17-0': 'Now, we take the particular background (Eqs. ([REF]) and ([REF])).', '1201.2485-2-17-1': 'Because near the boundary, the geometry is an [MATH], the solution of the Dirac equation ([REF]) can be expressed as [EQUATION]', '1201.2485-2-17-2': "If [MATH] and [MATH] are related by [MATH], then the boundary Green's functions [MATH] is given by [MATH] [CITATION].", '1201.2485-2-17-3': 'Therefore [EQUATION]', '1201.2485-2-17-4': 'Near the horizon, we take the scaling solution near the horizon ([REF]).', '1201.2485-2-17-5': 'We find that the requirement that the solutions of Eqs. ([REF]) and ([REF]) near the horizon be in-falling implies [EQUATION]', '1201.2485-2-18-0': '# Emergence of the gap', '1201.2485-2-19-0': 'In this section, to study the spectral function, we will numerically solve the flow equation ([REF]) with the initial condition ([REF]).', '1201.2485-2-19-1': 'Up to normalization, the spectral function is given by [MATH].', '1201.2485-2-19-2': 'However, due to the relation [MATH], [MATH] can be recovered from [MATH].', '1201.2485-2-19-3': 'Therefore, in the following 3D plots and density plots, to make numerical calculation easier, we will only show [MATH].', '1201.2485-2-20-0': '## The Fermi momentum [MATH]', '1201.2485-2-21-0': 'Before discussing the spectral function for the larger [MATH], we would like to give a simple discussion on the Fermi momentum [MATH] for [MATH].', '1201.2485-2-21-1': 'From the plots above in FIG. [REF], one can see that for [MATH], a sharp quasiparticle-like peak occurs near [MATH] and [MATH], indicating a Fermi surface.', '1201.2485-2-21-2': 'When [MATH] is increased, the Fermi momentum [MATH] increases (Table I).', '1201.2485-2-21-3': 'As [MATH] is further increased, several Fermi surfaces emerge.', '1201.2485-2-21-4': 'For example, for [MATH], the another Fermi surface begin to occur at [MATH].', '1201.2485-2-21-5': 'In fact, the case that with the increase of [MATH], several Fermi surfaces produce also occurs in RN-AdS background.', '1201.2485-2-21-6': 'It seems that the gravity duals with larger charge [MATH] posses more branches of Fermi surfaces.', '1201.2485-2-21-7': 'Conversely, with the decrease of [MATH], the Fermi sea gradually disappears.', '1201.2485-2-22-0': 'In addition, we would also like to point out that for fixed [MATH] and [MATH], [MATH], which recover the results of Ref. [CITATION].', '1201.2485-2-22-1': 'For more discussions on the Fermi momentum [MATH] and the low energy behaviors in this case, we will explore it in the another companion paper.', '1201.2485-2-23-0': '## Emergence of the gap', '1201.2485-2-24-0': 'When we turn on [MATH] to the value of [MATH], a gap emerges and there are two bands located at the positive frequency and negative frequency regions, respectively (the plots below in FIG. [REF]).', '1201.2485-2-24-1': 'Evidently, the strength of the lower band is bigger than the upper band.', '1201.2485-2-24-2': "We also show the 3D plot and density plot of the Green's function [MATH] for [MATH] and [MATH] in FIG. [REF].", '1201.2485-2-24-3': 'We can see that with the increase of [MATH], the width of the gap becomes larger and the band in the negative frequency region switch gradually to the positive frequency region.', '1201.2485-2-24-4': 'As [MATH] increases further, the lower band disperses and the upper band becomes stronger and sharper.', '1201.2485-2-25-0': 'In order to explore further the characteristics of the spectral function, We show the spectral function [MATH] for [MATH], [MATH], [MATH] and [MATH] for sample values of [MATH].', '1201.2485-2-25-1': 'From the left plot above in FIG. [REF], we find that some peaks distribute at both positive and negative frequency regions.', '1201.2485-2-25-2': 'When the peak approaches [MATH], its height goes to infinity, and its width goes to zero, indicating that there is a Fermi surface at [MATH].', '1201.2485-2-25-3': 'When we amplify [MATH], the strength of the quasiparticle-like peak at [MATH] degrades, and vanishes at a critical value [MATH].', '1201.2485-2-25-4': 'For [MATH], the dipole interaction will open a gap as the case in RN black hole [CITATION].', '1201.2485-2-25-5': 'Differ from the case of [MATH], the height of the peaks degrade and the width becomes larger when [MATH] is approached, and the peak vanishes around [MATH].', '1201.2485-2-25-6': 'We also note that the spectral density mainly appears at negative frequency regions for [MATH] (the right plot above in FIG.[', '1201.2485-2-25-7': 'As [MATH] increases, the spectral density begins to distribute the positive frequency and negative frequency regions, respectively (the left plot below in FIG.[', '1201.2485-2-25-8': 'When [MATH] increases further, the spectral density switches gradually to the positive frequency regions (the right plot below in FIG.[', '1201.2485-2-26-0': 'The another important quantity of interest for us is the density of states [MATH], which is defined as the integral of [MATH] over [MATH].', '1201.2485-2-26-1': 'It is the total spectral weight.', '1201.2485-2-26-2': 'By careful numerical computations, we find that the onset of the gap is near [MATH], which is different from the onset value of [MATH] in the case of RN background.', '1201.2485-2-26-3': 'Consistent with the above observations (FIG. [REF], FIG. [REF] and FIG. [REF]), the total spectral weight is also mainly distribute at negative frequency regions for small [MATH], left plot in FIG. [REF]), and switches to the positive frequency regions as the increase of [MATH].', '1201.2485-2-27-0': 'Finally, we show the relation between the width of the gap [MATH] and [MATH].', '1201.2485-2-27-1': 'Evidently, the gap becomes wider as [MATH] increases.', '1201.2485-2-28-0': '# Conclusions and discussion', '1201.2485-2-29-0': 'In this paper, we have explored the fermionic response in the presence of a bulk dipole interaction term with strength [MATH] in the dilatonic black brane with a Lifshitz like IR geometry and [MATH] boundary.', '1201.2485-2-29-1': 'When [MATH], a sharp quasiparticle-like peak occurs near [MATH] and [MATH], indicating a Fermi surface.', '1201.2485-2-29-2': 'As [MATH] increases, the peak become wide and its strength degrades gradually, and the spectral weight is transferred between bands.', '1201.2485-2-29-3': 'Furthermore, when [MATH] goes beyond a critical dipole interaction [MATH], the Fermi sea disappears and a gap emerges as the case in RN black hole [CITATION].', '1201.2485-2-29-4': 'By studying the another important quantity: the density of states [MATH], we clearly find that the onset of the gap is near [MATH] and the width of the gap becomes bigger as [MATH] increases.', '1201.2485-2-30-0': 'So far, we have found a (Mott) gap from holographic fermions in the background of the dilatonic black brane with a Lifshitz like IR geometry when a dipole interaction term are added as that found in RN black hole.', '1201.2485-2-30-1': 'However, for more general dilatonic black brane, which also have vanishing ground entropy density, for example, a class of systems proposed in Refs.[CITATION], is the emergence of gap robust?', '1201.2485-2-30-2': 'The fermionic response without dipole interaction in this class backgrounds has been investigated in Refs.[CITATION].', '1201.2485-2-30-3': 'Therefore, it is valuable to test the robustness of the emergence of the gap in the backgrounds of more general dilatonic black branes.', '1201.2485-2-30-4': 'In addition, it is also interesting to extend our investigations in this paper to the case of the non-relativistic fermionic fixed point.', '1201.2485-2-30-5': 'We will address them in the near future.', '1201.2485-2-31-0': 'We especially thank Professor Robert Leigh for his very valuable correspondence.', '1201.2485-2-31-1': 'J.P. Wu would also like to thank Hongbao Zhang and Wei-Jia Li for their collaboration in the related project.', '1201.2485-2-31-2': 'We also thank the referee for many valuable comments and correcting some English language typos.', '1201.2485-2-31-3': 'J.P. Wu is partly supported by NSFC(No.10975017) and the Fundamental Research Funds for the central Universities.', '1201.2485-2-31-4': 'H.B. Zeng is supported by the Fundamental Research Funds for the Central Universities (Grant No.1107020117) and the China Postdoctoral Science Foundation (Grant No. 20100481120).'}

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'1201.2485-3-27-1'], ['1201.2485-2-19-0', '1201.2485-3-19-0'], ['1201.2485-2-19-1', '1201.2485-3-19-1'], ['1201.2485-2-19-2', '1201.2485-3-19-2'], ['1201.2485-2-19-3', '1201.2485-3-19-3'], ['1201.2485-2-11-0', '1201.2485-3-11-0'], ['1201.2485-2-11-1', '1201.2485-3-11-1'], ['1201.2485-2-11-2', '1201.2485-3-11-2'], ['1201.2485-2-11-3', '1201.2485-3-11-3'], ['1201.2485-2-11-4', '1201.2485-3-11-4'], ['1201.2485-2-11-5', '1201.2485-3-11-5'], ['1201.2485-2-11-6', '1201.2485-3-11-6'], ['1201.2485-2-11-7', '1201.2485-3-11-7'], ['1201.2485-2-25-0', '1201.2485-3-25-0'], ['1201.2485-2-25-1', '1201.2485-3-25-1'], ['1201.2485-2-25-2', '1201.2485-3-25-2'], ['1201.2485-2-25-3', '1201.2485-3-25-3'], ['1201.2485-2-25-4', '1201.2485-3-25-4'], ['1201.2485-2-25-5', '1201.2485-3-25-5'], ['1201.2485-2-25-6', '1201.2485-3-25-6'], ['1201.2485-2-25-7', '1201.2485-3-25-7'], ['1201.2485-2-25-8', '1201.2485-3-25-8'], ['1201.2485-2-6-0', '1201.2485-3-6-0'], ['1201.2485-2-6-1', '1201.2485-3-6-1'], ['1201.2485-2-6-2', '1201.2485-3-6-2'], ['1201.2485-2-6-3', '1201.2485-3-6-3'], ['1201.2485-2-6-4', '1201.2485-3-6-4'], ['1201.2485-2-24-0', '1201.2485-3-24-0'], ['1201.2485-2-24-1', '1201.2485-3-24-1'], ['1201.2485-2-24-2', '1201.2485-3-24-2'], ['1201.2485-2-24-3', '1201.2485-3-24-3'], ['1201.2485-2-24-4', '1201.2485-3-24-4'], ['1201.2485-2-0-0', '1201.2485-3-0-0'], ['1201.2485-2-0-1', '1201.2485-3-0-1'], ['1201.2485-2-0-2', '1201.2485-3-0-2'], ['1201.2485-2-14-0', '1201.2485-3-14-0'], ['1201.2485-2-14-1', '1201.2485-3-14-1'], ['1201.2485-2-14-2', '1201.2485-3-14-2'], ['1201.2485-2-29-0', '1201.2485-3-29-0'], ['1201.2485-2-29-1', '1201.2485-3-29-1'], ['1201.2485-2-29-2', '1201.2485-3-29-2'], ['1201.2485-2-29-3', '1201.2485-3-29-3'], ['1201.2485-2-29-4', '1201.2485-3-29-4'], ['1201.2485-1-21-3', '1201.2485-2-24-0'], ['1201.2485-1-21-4', '1201.2485-2-24-1'], ['1201.2485-1-21-5', '1201.2485-2-24-2'], ['1201.2485-1-21-6', '1201.2485-2-24-3'], ['1201.2485-1-21-7', '1201.2485-2-24-4'], ['1201.2485-1-14-0', '1201.2485-2-14-0'], ['1201.2485-1-14-1', '1201.2485-2-14-1']]

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[]

[['1201.2485-1-5-0', '1201.2485-2-5-0'], ['1201.2485-1-5-0', '1201.2485-2-5-1'], ['1201.2485-1-9-3', '1201.2485-2-9-3'], ['1201.2485-1-9-3', '1201.2485-2-9-4'], ['1201.2485-1-2-3', '1201.2485-2-2-3']]

[]

['1201.2485-1-1-0', '1201.2485-1-17-8', '1201.2485-1-18-3', '1201.2485-2-1-0', '1201.2485-2-16-8', '1201.2485-2-17-3', '1201.2485-3-1-0', '1201.2485-3-16-8', '1201.2485-3-17-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1201.2485

{'1201.2485-3-0-0': 'We report a dynamical formation of a (Mott) gap from holographic fermions.', '1201.2485-3-0-1': 'By coupling a fermion field with dipole action to the charged dilaton black branes with a Lifshitz like IR geometry and [MATH] boundary, we find that when the dipole interaction is large enough, spectral weight is transferred between bands, and beyond a critical dipole interaction, a gap emerges in the fermion density of states.', '1201.2485-3-0-2': 'The value of the gap becomes larger as the strength of the interaction keeps increasing.', '1201.2485-3-1-0': 'Introduction', '1201.2485-3-2-0': 'As a new method to study strongly coupled quantum field theory, AdS/CFT correspondence [CITATION] has been used to study the strongly coupled phenomena in quantum many-body systems.', '1201.2485-3-2-1': 'One of the most successful progress is that by coupling a free Fermion field to the [MATH] (RN) black hole, a class of non-Fermi liquid at zero temperature was found holographically[CITATION].', '1201.2485-3-2-2': 'The properties of this non-Fermi liquid in the presence of a magnetic field have been studied in[CITATION].', '1201.2485-3-2-3': 'The extension to finite temperature was investigated in[CITATION].', '1201.2485-3-2-4': 'In [CITATION], free fermions in AdS BTZ black hole and Gauss-Bonnet black hole have also been studied respectively.', '1201.2485-3-2-5': 'In addition, the fermions on Lifshitz Background have been explored in Refs. [CITATION].', '1201.2485-3-2-6': 'We can refer to the lecture [CITATION] for an excellent latest review on this subjects.', '1201.2485-3-3-0': 'In [CITATION], the authors studied the holographic fermion system with RN black hole background when a dipole interaction is added, it is found that the strength of the interaction [MATH] is similar to [MATH] in the fermions Hubbard model.', '1201.2485-3-3-1': 'When [MATH] is large enough, a gap opens just like in a Mott insulator, in which a large repulsive interaction [MATH] will change the structure of the bands.', '1201.2485-3-3-2': 'Then the fermion energy is in the middle of the gap, which makes an insulator.', '1201.2485-3-4-0': 'However, the RN black hole background has nonzero ground state entropy density, which seems to be inconsistent with our intuition that a system of degenerate fermions has a unique ground state.', '1201.2485-3-4-1': 'Therefore, a systematic exploration of the system that has zero extremal entropy will be important and valuable.', '1201.2485-3-4-2': 'Such models have been proposed in Refs.[CITATION].', '1201.2485-3-4-3': 'Furthermore, in Ref.[CITATION], the author investigated the free fermionic response in the background proposed by Gubser and Rocha [CITATION].', '1201.2485-3-4-4': 'They find that the dispersion relation is linear, just like a Fermi liquid.', '1201.2485-3-4-5': 'It is very different from that found in RN black hole [CITATION].', '1201.2485-3-5-0': 'Among these models of zero ground state entropy density, a very important and valuable model is that proposed by Goldstein et al. [CITATION].', '1201.2485-3-5-1': 'Here the near horizon geometry is Lifshitz-like.', '1201.2485-3-5-2': 'Therefore, in this paper, we will study the characteristics of the fermionic response with dipole interaction in this background.', '1201.2485-3-5-3': 'The interesting results we find is that, similar to the fermions in the RN black hole background, the dipole interaction will open a gap for the strongly coupled fermions system when the value of the interaction [MATH] is large enough.', '1201.2485-3-6-0': 'The organization of this paper is as follows.', '1201.2485-3-6-1': 'In section II, we first review the charged dilaton black branes with zero entropy at zero temperature.', '1201.2485-3-6-2': 'In section III, we derive the equations of motion for the fermion field with a dipole action.', '1201.2485-3-6-3': 'The main results of the emergence of gap is in section IV.', '1201.2485-3-6-4': 'Finally, the discussion and conclusions are presented in section V.', '1201.2485-3-7-0': '# An extremal charged dilatonic black brane solutions', '1201.2485-3-8-0': '## Einstein-Maxwell-dilaton model', '1201.2485-3-9-0': 'For completeness let us also review the dilaton gravity system [CITATION].', '1201.2485-3-9-1': 'Our discussion follows [CITATION] closely and some details are omitted.', '1201.2485-3-9-2': 'The system we consider is as follows: [EQUATION] where [MATH] is Ricci scalar, [MATH] is the dilaton field, [MATH] is the field strength, and [MATH] is the AdS radius.', '1201.2485-3-9-3': 'In this paper we will focus on an electrically charged black brane solution.', '1201.2485-3-9-4': 'The metric and gauge fields can be written as, [EQUATION]', '1201.2485-3-9-5': 'Such a gauge field satisfies the gauge field equation of motion automatically.', '1201.2485-3-9-6': 'With the above ansatz, the remaining equations of motion can be expressed as follow: [EQUATION] where the prime denote the differentiation with respect to the coordinate [MATH].', '1201.2485-3-9-7': 'In the subsequent subsections, we will construct such solutions that the near-horizon geometry has a Lifshitz-like symmetry but at infinity the geometry asymptotes to AdS.', '1201.2485-3-9-8': 'For convenience, we shall set [MATH] and [MATH] in the following.', '1201.2485-3-10-0': '## Scaling solution near the horizon', '1201.2485-3-11-0': 'In order to construct the scaling solution near the horizon [MATH], we consider the following ansatz: [EQUATION] where [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] are all constants and [MATH] and [MATH] are set to be positive in the following, without loss of generality.', '1201.2485-3-11-1': 'Note that for convenience, one has introduced the variable [MATH] in the above ansatz.', '1201.2485-3-11-2': 'With this ansatz and the equations of motion, we have [EQUATION]', '1201.2485-3-11-3': 'Such a solution has a Lifshitz-like symmetry in the metric, with a dynamical critical exponent [MATH], although such a symmetry is broken by the logarithmic dependence of the dilaton on [MATH].', '1201.2485-3-11-4': 'Also we note that in the above solution, the metric component [MATH] has a double zero at the horizon where [MATH] also vanishes and therefore corresponds to an extremal brane with vanishing ground state entropy density.', '1201.2485-3-11-5': 'Finally, we also point out that for [MATH], such a solution corresponds to an [MATH] geometry, which is also the near horizon geometry of the extremal RN black hole.', '1201.2485-3-11-6': 'The fermionic response in RN black hole has been explore carefully in Refs. [CITATION].', '1201.2485-3-11-7': 'Therefore we shall assume [MATH] and focus on the Lifshitz like near horizon geometry in the following.', '1201.2485-3-12-0': '## The solution with asymptotes to AdS at infinity', '1201.2485-3-13-0': 'To obtain a solution which asymptotes to AdS at infinity, we can add a perturbation to the scaling solution which is irrelevant in the IR scaling region but relevant in the UV region.', '1201.2485-3-13-1': 'In order to set the constant [MATH] to unity and [MATH] to vanishing, we carry out a coordinate transformation under which, [EQUATION] with [MATH] and [MATH].', '1201.2485-3-13-2': 'Under such rescaling, we have [EQUATION] and in terms of [MATH], the charge can be expressed as [EQUATION]', '1201.2485-3-13-3': 'Subsequently, we add the perturbation to the Eqs. ([REF]).', '1201.2485-3-13-4': 'The resulting functions satisfied the equations of motion at the leading order are [EQUATION] where [MATH], [MATH], and [MATH] keep unchanged.', '1201.2485-3-13-5': 'The perturbation is characterized by the exponent [MATH] and the three constants [MATH], [MATH] and [MATH].', '1201.2485-3-13-6': 'Also, [MATH] and [MATH] are determined by [MATH] and they have the following relations: [MATH], [MATH].', '1201.2485-3-13-7': 'In addition, the perturbation should die out at small [MATH].', '1201.2485-3-13-8': 'Therefore, we must require [MATH], which gives rise to a unique allowed value for this exponent: [MATH].', '1201.2485-3-13-9': 'Thus there are two parameters [MATH] and [MATH] for this perturbated solution.', '1201.2485-3-13-10': 'For simplicity, we will set [MATH].', '1201.2485-3-14-0': 'The initial data for numerical integration is taken from the above perturbated solution near the horizon.', '1201.2485-3-14-1': 'As depicted in Fig. [REF], for the perturbation with [MATH], [MATH] as [MATH], that is to say, the solution near the boundary takes the standard asymptotics of AdS for the perturbation with [MATH] .', '1201.2485-3-14-2': 'In the next section, we shall investigate the holographic fermions with dipole term in such a background.', '1201.2485-3-15-0': '# Holographic fermion with bulk dipole coupling', '1201.2485-3-16-0': 'In order to know the effect of magnetic dipole coupling on the structure of the spectral functions of the dual fermionic operator, we consider the following bulk fermion action [EQUATION] where [MATH] is related to the usual flat space gamma matrix by a factor of the vielbein, [MATH], [MATH] is the covariant derivative with [MATH] the spin connection 1-forms and [MATH].', '1201.2485-3-16-1': 'Before taking the particular background (Eqs. ([REF]) and ([REF])), we will derive the flow equation as Refs. [CITATION] in a more general static background, i.e., [EQUATION]', '1201.2485-3-16-2': 'Firstly, the Dirac equation can be derived from the action [MATH] [EQUATION]', '1201.2485-3-16-3': 'Making a transformation [MATH] to remove the spin connection and expanding [MATH] as [MATH] in Fourier space, the Dirac equation ([REF]) turns out to be [EQUATION] where due to rotational symmetry in the spatial directions, we set [MATH] and [MATH] without losing generality.', '1201.2485-3-16-4': 'Notice that Eq. ([REF]) only depends on three Gamma matrices [MATH].', '1201.2485-3-16-5': 'So it is convenient to split the spinors [MATH] into [MATH] and choose the following basis for our gamma matrices as in [CITATION]: [EQUATION]', '1201.2485-3-16-6': 'So, we have a new version of the Dirac equation as [EQUATION]', '1201.2485-3-16-7': 'Furthermore, according to eigenvalues of [MATH], we make such a decomposition [MATH].', '1201.2485-3-16-8': 'Then [EQUATION]', '1201.2485-3-16-9': 'Under such decomposition, the Dirac equation ([REF]) can be rewritten as [EQUATION]', '1201.2485-3-16-10': 'Introducing the ratios [MATH], one can package the Dirac equation ([REF]) and ([REF]) into the evolution equation of [MATH], [EQUATION] where [MATH].', '1201.2485-3-16-11': 'The above flow equation will be more convenient to impose the boundary conditions at the horizon and read off the boundary Green functions.', '1201.2485-3-17-0': 'Now, we take the particular background (Eqs. ([REF]) and ([REF])).', '1201.2485-3-17-1': 'Because near the boundary, the geometry is an [MATH], the solution of the Dirac equation ([REF]) can be expressed as [EQUATION]', '1201.2485-3-17-2': "If [MATH] and [MATH] are related by [MATH], then the boundary Green's functions [MATH] is given by [MATH] [CITATION].", '1201.2485-3-17-3': 'Therefore [EQUATION]', '1201.2485-3-17-4': 'Near the horizon, we take the scaling solution near the horizon ([REF]).', '1201.2485-3-17-5': 'We find that the requirement that the solutions of Eqs. ([REF]) and ([REF]) near the horizon be in-falling implies [EQUATION]', '1201.2485-3-18-0': '# Emergence of the gap', '1201.2485-3-19-0': 'In this section, to study the spectral function, we will numerically solve the flow equation ([REF]) with the initial condition ([REF]).', '1201.2485-3-19-1': 'Up to normalization, the spectral function is given by [MATH].', '1201.2485-3-19-2': 'However, due to the relation [MATH], [MATH] can be recovered from [MATH].', '1201.2485-3-19-3': 'Therefore, in the following 3D plots and density plots, to make numerical calculation easier, we will only show [MATH].', '1201.2485-3-20-0': '## The Fermi momentum [MATH]', '1201.2485-3-21-0': 'Before discussing the spectral function for the larger [MATH], we would like to give a simple discussion on the Fermi momentum [MATH] for [MATH].', '1201.2485-3-21-1': 'From the plots above in FIG. [REF], one can see that for [MATH], a sharp quasiparticle-like peak occurs near [MATH] and [MATH], indicating a Fermi surface.', '1201.2485-3-21-2': 'When [MATH] is increased, the Fermi momentum [MATH] increases (Table I).', '1201.2485-3-21-3': 'As [MATH] is further increased, several Fermi surfaces emerge.', '1201.2485-3-21-4': 'For example, for [MATH], the another Fermi surface begin to occur at [MATH].', '1201.2485-3-21-5': 'In fact, the case that with the increase of [MATH], several Fermi surfaces produce also occurs in RN-AdS background.', '1201.2485-3-21-6': 'It seems that the gravity duals with larger charge [MATH] posses more branches of Fermi surfaces.', '1201.2485-3-21-7': 'Conversely, with the decrease of [MATH], the Fermi sea gradually disappears.', '1201.2485-3-22-0': 'In addition, we would also like to point out that for fixed [MATH] and [MATH], [MATH], which recover the results of Ref. [CITATION].', '1201.2485-3-22-1': 'For more discussions on the Fermi momentum [MATH] and the low energy behaviors in this case, we will explore it in the another companion paper.', '1201.2485-3-23-0': '## Emergence of the gap', '1201.2485-3-24-0': 'When we turn on [MATH] to the value of [MATH], a gap emerges and there are two bands located at the positive frequency and negative frequency regions, respectively (the plots below in FIG. [REF]).', '1201.2485-3-24-1': 'Evidently, the strength of the lower band is bigger than the upper band.', '1201.2485-3-24-2': "We also show the 3D plot and density plot of the Green's function [MATH] for [MATH] and [MATH] in FIG. [REF].", '1201.2485-3-24-3': 'We can see that with the increase of [MATH], the width of the gap becomes larger and the band in the negative frequency region switch gradually to the positive frequency region.', '1201.2485-3-24-4': 'As [MATH] increases further, the lower band disperses and the upper band becomes stronger and sharper.', '1201.2485-3-25-0': 'In order to explore further the characteristics of the spectral function, We show the spectral function [MATH] for [MATH], [MATH], [MATH] and [MATH] for sample values of [MATH].', '1201.2485-3-25-1': 'From the left plot above in FIG. [REF], we find that some peaks distribute at both positive and negative frequency regions.', '1201.2485-3-25-2': 'When the peak approaches [MATH], its height goes to infinity, and its width goes to zero, indicating that there is a Fermi surface at [MATH].', '1201.2485-3-25-3': 'When we amplify [MATH], the strength of the quasiparticle-like peak at [MATH] degrades, and vanishes at a critical value [MATH].', '1201.2485-3-25-4': 'For [MATH], the dipole interaction will open a gap as the case in RN black hole [CITATION].', '1201.2485-3-25-5': 'Differ from the case of [MATH], the height of the peaks degrade and the width becomes larger when [MATH] is approached, and the peak vanishes around [MATH].', '1201.2485-3-25-6': 'We also note that the spectral density mainly appears at negative frequency regions for [MATH] (the right plot above in FIG.[', '1201.2485-3-25-7': 'As [MATH] increases, the spectral density begins to distribute the positive frequency and negative frequency regions, respectively (the left plot below in FIG.[', '1201.2485-3-25-8': 'When [MATH] increases further, the spectral density switches gradually to the positive frequency regions (the right plot below in FIG.[', '1201.2485-3-26-0': 'The another important quantity of interest for us is the density of states [MATH], which is defined as the integral of [MATH] over [MATH].', '1201.2485-3-26-1': 'It is the total spectral weight.', '1201.2485-3-26-2': 'By careful numerical computations, we find that the onset of the gap is near [MATH], which is different from the onset value of [MATH] in the case of RN background.', '1201.2485-3-26-3': 'Consistent with the above observations (FIG. [REF], FIG. [REF] and FIG. [REF]), the total spectral weight is also mainly distribute at negative frequency regions for small [MATH], left plot in FIG. [REF]), and switches to the positive frequency regions as the increase of [MATH].', '1201.2485-3-27-0': 'Finally, we show the relation between the width of the gap [MATH] and [MATH].', '1201.2485-3-27-1': 'Evidently, the gap becomes wider as [MATH] increases.', '1201.2485-3-28-0': '# Conclusions and discussion', '1201.2485-3-29-0': 'In this paper, we have explored the fermionic response in the presence of a bulk dipole interaction term with strength [MATH] in the dilatonic black brane with a Lifshitz like IR geometry and [MATH] boundary.', '1201.2485-3-29-1': 'When [MATH], a sharp quasiparticle-like peak occurs near [MATH] and [MATH], indicating a Fermi surface.', '1201.2485-3-29-2': 'As [MATH] increases, the peak become wide and its strength degrades gradually, and the spectral weight is transferred between bands.', '1201.2485-3-29-3': 'Furthermore, when [MATH] goes beyond a critical dipole interaction [MATH], the Fermi sea disappears and a gap emerges as the case in RN black hole [CITATION].', '1201.2485-3-29-4': 'By studying the another important quantity: the density of states [MATH], we clearly find that the onset of the gap is near [MATH] and the width of the gap becomes bigger as [MATH] increases.', '1201.2485-3-30-0': 'So far, we have found a (Mott) gap from holographic fermions in the background of the dilatonic black brane with a Lifshitz like IR geometry when a dipole interaction term are added as that found in RN black hole.', '1201.2485-3-30-1': 'However, for more general dilatonic black brane, which also have vanishing ground entropy density, for example, a class of systems proposed in Refs.[CITATION], is the emergence of gap robust?', '1201.2485-3-30-2': 'The fermionic response without dipole interaction in this class backgrounds has been investigated in Refs.[CITATION].', '1201.2485-3-30-3': 'Therefore, it is valuable to test the robustness of the emergence of the gap in the backgrounds of more general dilatonic black branes.', '1201.2485-3-30-4': 'In addition, it is also interesting to extend our investigations in this paper to the case of the non-relativistic fermionic fixed point.', '1201.2485-3-30-5': 'We will address them in the near future.', '1201.2485-3-31-0': 'We especially thank Professor Robert Leigh for his very valuable correspondence.', '1201.2485-3-31-1': 'J.P. Wu would also like to thank Hongbao Zhang and Wei-Jia Li for their collaboration in the related project.', '1201.2485-3-31-2': 'We also thank the referee for many valuable comments and correcting some English language typos.', '1201.2485-3-31-3': 'J.P. Wu is partly supported by NSFC(No.10975017) and the Fundamental Research Funds for the central Universities.', '1201.2485-3-31-4': 'H.B. Zeng is supported by the Fundamental Research Funds for the Central Universities (Grant No.1107020117) and the China Postdoctoral Science Foundation (Grant No. 20100481120).'}

1701.01370

{'1701.01370-1-0-0': 'Estimating human pose, shape, and motion from images and videos are fundamental challenges with many applications.', '1701.01370-1-0-1': 'Recent advances in 2D human pose estimation use large amounts of manually-labeled training data for learning convolutional neural networks (CNNs).', '1701.01370-1-0-2': 'Such data is time consuming to acquire and difficult to extend.', '1701.01370-1-0-3': 'Moreover, manual labeling of 3D pose, depth and motion is impractical.', '1701.01370-1-0-4': 'In this work we present SURREAL (Synthetic hUmans foR REAL tasks): a new large-scale dataset with synthetically-generated but realistic images of people rendered from 3D sequences of human motion capture data.', '1701.01370-1-0-5': 'We generate more than 6 million frames together with ground truth pose, depth maps, and segmentation masks.', '1701.01370-1-0-6': 'We show that CNNs trained on our synthetic dataset allow for accurate human depth estimation and human part segmentation in real RGB images.', '1701.01370-1-0-7': 'Our results and the new dataset open up new possibilities for advancing person analysis using cheap and large-scale synthetic data.', '1701.01370-1-1-0': '# Introduction', '1701.01370-1-2-0': 'Convolutional Neural Networks provide significant gains to problems with large amounts of training data.', '1701.01370-1-2-1': 'In the field of human analysis, recent datasets [CITATION] now gather a sufficient number of annotated images to train networks for 2D human pose estimation [CITATION].', '1701.01370-1-2-2': 'Other tasks such as accurate estimation of human motion, depth and body-part segmentation are lagging behind as manual supervision for such problems at large scale is prohibitively expensive.', '1701.01370-1-3-0': 'Images of people have rich variation in poses, clothing, hair styles, body shapes, occlusions, viewpoints, motion blur and other factors.', '1701.01370-1-3-1': 'Much of these variations, however, can be synthesized using existing 3D motion capture (MoCap) data [CITATION] and modern tools for realistic rendering.', '1701.01370-1-3-2': 'Provided sufficient realism, such an approach would be highly useful for many tasks as it can generate rich ground truth in terms of depth, motion, body-part segmentation and occlusions to mention a few.', '1701.01370-1-4-0': 'In this work we present SURREAL: a new large-scale dataset with synthetically-generated but realistic images of people.', '1701.01370-1-4-1': 'Images are rendered from 3D sequences of MoCap data.', '1701.01370-1-4-2': 'To ensure realism, the synthetic bodies are created using the SMPL body model [CITATION], whose parameters are fit by the MoSh [CITATION] method given raw 3D MoCap marker data.', '1701.01370-1-4-3': 'We randomly sample a large variety of viewpoints, clothing and lighting.', '1701.01370-1-4-4': 'SURREAL contains more than 6 million frames together with ground truth pose, depth maps, and segmentation masks.', '1701.01370-1-4-5': 'We show that CNNs trained on synthetic data allow for accurate human depth estimation and human part segmentation in real RGB images, see Figure [REF].', '1701.01370-1-4-6': 'SURREAL dataset will become publicly available [CITATION].', '1701.01370-1-5-0': 'The rest of this paper is organized as follows.', '1701.01370-1-5-1': 'Section [REF] reviews previous work on the use of synthetic datasets in computer vision.', '1701.01370-1-5-2': 'Section [REF] presents our approach for generating realistic synthetic videos of people.', '1701.01370-1-5-3': 'In Section [REF] we describe our convolution architecture for human body-part segmentation and depth estimation.', '1701.01370-1-5-4': 'Section [REF] reports experiments.', '1701.01370-1-5-5': 'We conclude in Section [REF].', '1701.01370-1-6-0': '# Related work', '1701.01370-1-7-0': 'Knowledge transfer from synthetic to real images has been recently studied with deep neural networks.', '1701.01370-1-7-1': 'Dosovitskiy [CITATION] learn a CNN for optical flow estimation using synthetically generated images of rendered 3D moving chairs.', '1701.01370-1-7-2': 'Peng [CITATION] study the effect of different visual cues such as object/background texture and color when rendering synthetic 3D objects for object detection task.', '1701.01370-1-7-3': 'Similarly, [CITATION] explores rendering 3D objects to perform viewpoint estimation.', '1701.01370-1-7-4': 'Recently, Gaidon [CITATION] have released the Virtual KITTI dataset with synthetically generated videos of cars to study multi-object tracking.', '1701.01370-1-8-0': 'Several works focused on creating synthetic images of humans for learning 2D pose estimation [CITATION], 3D pose estimation [CITATION], pedestrian detection [CITATION], and action recognition [CITATION].', '1701.01370-1-8-1': 'Pishchulin [CITATION] generate synthetic images with a game engine.', '1701.01370-1-8-2': 'In [CITATION], they deform 2D images with a 3D model.', '1701.01370-1-8-3': 'More recently, Rogez and Schmid [CITATION] used image-based synthesis engine to augment existing real images.', '1701.01370-1-8-4': 'Ghezelghieh [CITATION] render synthetic images with 10 simple body models with an emphasis on upright people; however, the main challenge using existing MoCap data for training is to generalize to poses that are not upright.', '1701.01370-1-9-0': 'A similar direction has been explored in [CITATION].', '1701.01370-1-9-1': 'In [CITATION], action recognition problem is addressed with synthetic human trajectories from MoCap data.', '1701.01370-1-9-2': '[CITATION] trains CNNs with synthetic depth images.', '1701.01370-1-10-0': 'The closest work to this paper is [CITATION], where the authors render large-scale synthetic images for predicting 3D pose with CNNs.', '1701.01370-1-10-1': 'Our dataset differs from [CITATION] by having a richer, per-pixel ground truth, thus allowing to train for pixel-wise predictions and multi-task scenarios.', '1701.01370-1-10-2': 'In addition, we argue that the realism in our synthetic images is better (see [CITATION]), thus resulting in a smaller gap between features learned from synthetic and real images.', '1701.01370-1-10-3': 'The method in [CITATION] heavily relies on real images as input in their training with domain adaptation.', '1701.01370-1-10-4': 'This is not the case for our synthetic training.', '1701.01370-1-10-5': 'Moreover, we render video sequences which can be used for temporal modeling.', '1701.01370-1-11-0': 'In this paper, we show that photo-realistic renderings of people under large variations in shape, texture, viewpoint and pose can help to solve pixel-wise human labeling tasks.', '1701.01370-1-11-1': 'We use synthesized data to learn models of people and demonstrate improvements of body part segmentation and body depth estimation in real images.', '1701.01370-1-12-0': '# Data generation', '1701.01370-1-13-0': 'This section presents our SURREAL (Synthetic hUmans foR REAL tasks) dataset and describes key steps for its generation (Section [REF]).', '1701.01370-1-13-1': 'We also describe how we obtain ground truth data for real MoCap sequences (Section [REF]).', '1701.01370-1-14-0': '## Synthetic humans', '1701.01370-1-15-0': 'Our pipeline for generating synthetic data is illustrated in Figure [REF].', '1701.01370-1-15-1': 'A human body with a random 3D pose, random shape and random texture is rendered from a random view-point for some random lighting and a random background image.', '1701.01370-1-15-2': 'Below we define what "random" means in all these cases.', '1701.01370-1-15-3': 'Since the data is synthetic, we also generate ground truth depth maps, optical flow, surface normals, human part segmentations and joint locations (both 2D and 3D).', '1701.01370-1-15-4': 'As result, we obtain 6.5 million frames grouped into [MATH] continuous image sequences.', '1701.01370-1-15-5': 'See Table [REF] for more statistics and Figure [REF] for samples from the SURREAL dataset.', '1701.01370-1-16-0': 'Body model.', '1701.01370-1-16-1': 'Synthetic bodies are created using the SMPL body model [CITATION].', '1701.01370-1-16-2': 'SMPL is a realistic articulated model of the body created from thousands of high-quality 3D scans, which decomposes body deformations into pose (kinematic deformations due to skeletal posture) and shape (body deformations intrinsic to a particular person that make them different from others).', '1701.01370-1-16-3': 'SMPL is compatible with most animation packages like Blender [CITATION].', '1701.01370-1-16-4': 'SMPL deformations are modeled as a combination of linear blend skinning and linear blendshapes defined by principal components of body shape variation.', '1701.01370-1-16-5': 'SMPL pose and shape parameters are converted to a triangulated mesh using Blender, which then applies texture, shading and adds a background to generate the final RGB output.', '1701.01370-1-17-0': 'Body shape.', '1701.01370-1-17-1': 'In order to render varied, but realistic, body shapes we make use of the CAESAR dataset [CITATION], which was used to train SMPL.', '1701.01370-1-17-2': 'To create a body shape, we select one of the CAESAR subjects at random and approximate their shape with the first 10 SMPL shape principal components.', '1701.01370-1-17-3': 'Ten shape components explain more than [MATH] of the shape variance in CAESAR (at the resolution of our mesh) and produce quite realistic body shapes.', '1701.01370-1-18-0': 'Body pose.', '1701.01370-1-18-1': 'To generate images of people in realistic poses, we take motion capture data from the CMU MoCap database [CITATION].', '1701.01370-1-18-2': 'CMU MoCap contains more than 2000 sequences of 23 high-level action categories, resulting in more than 10 hours of recorded 3D locations of body markers.', '1701.01370-1-19-0': 'It is often challenging to realistically and automatically retarget MoCap skeleton data to a new model.', '1701.01370-1-19-1': 'Consequently we do not use the skeleton data but rather use MoSh [CITATION] to fit the SMPL parameters that best explain raw 3D MoCap marker locations.', '1701.01370-1-19-2': 'This gives both the 3D shape of the subject and the articulated pose parameters of SMPL.', '1701.01370-1-19-3': 'To increase the diversity, we replace the estimated 3D body shape with a set of randomly sampled body shapes.', '1701.01370-1-20-0': 'We render each CMU MoCap sequence three times using different random view-points.', '1701.01370-1-20-1': 'Moreover, we divide the sequences into chunks of 100 frames with 30%, 50% and 70% overlaps for these three renderings.', '1701.01370-1-20-2': 'Every pose of the sequence is rendered with consistent parameters (i.e. body shape, clothing, light, background etc.) within each 100-frame chunk.', '1701.01370-1-21-0': 'Human texture.', '1701.01370-1-21-1': 'We texture the body model with texture maps extracted from two types of real scans.', '1701.01370-1-21-2': 'The first one uses CAESAR scans, which come with a color texture per 3D point, and we extract SMPL texture maps from the scans.', '1701.01370-1-21-3': 'These texture maps are varied in terms of skin color and identity.', '1701.01370-1-21-4': 'However, their quality is not ideal due to low resolution, uniform tight-fitting clothing, and visible markers placed on the face and body.', '1701.01370-1-21-5': 'Anthropometric markers are automatically removed from the texture images and inpainted.', '1701.01370-1-22-0': 'In order to provide more variety, we extract a second set of textures from 3D scans of [MATH] different subjects with normal clothing from a total of 98 different capture sessions.', '1701.01370-1-22-1': 'These scans are registered with 4Cap as in [CITATION].', '1701.01370-1-22-2': 'The texture of real clothing substantially increases the realism of generated images, even though SMPL does not model 3D deformations of clothes.', '1701.01370-1-23-0': '[MATH] of our data is rendered with the first set ([MATH] CAESAR textures randomly sampled from [MATH]), and the rest with the second set ([MATH] clothed textures).', '1701.01370-1-23-1': 'To preserve the anonymity of subjects, we replace all faces in the texture maps by the average CAESAR face.', '1701.01370-1-23-2': 'The skin color of this average face is corrected to fit the face skin color of the original texture map.', '1701.01370-1-23-3': 'This corrected average face is blended smoothly with the original map, resulting in a realistic and anonymized body texture.', '1701.01370-1-24-0': 'Light.', '1701.01370-1-24-1': 'The body is illuminated using Spherical Harmonics with [MATH] coefficients [CITATION].', '1701.01370-1-24-2': 'The coefficients are randomly sampled from a uniform distribution between [MATH] and [MATH], apart from the ambient illumination coefficient (which has a minimum value of [MATH]) and the vertical illumination component, which is biased to encourage the illumination from above.', '1701.01370-1-24-3': 'Since Blender does not provide Spherical Harmonics illumination, a spherical harmonic shader for the body material was implemented in Open Shading Language (OSL).', '1701.01370-1-25-0': 'Camera.', '1701.01370-1-25-1': 'The projective camera has a resolution of [MATH], focal length of [MATH]mm and sensor size of [MATH]mm.', '1701.01370-1-25-2': 'To generate images of the body in a wide range of positions, we take 100-frame MoCap sub-sequences and, in the first frame, render the body so that the center of the viewport points to the pelvis of the body, at a random distance (sampled from a normal distribution with [MATH] meters mean, [MATH] meter deviation) with a random yaw angle.', '1701.01370-1-25-3': 'The remainder of the sequence then effectively produces bodies in a range of locations relative to the camera.', '1701.01370-1-26-0': 'Background.', '1701.01370-1-26-1': 'We render the person on top of a static background image.', '1701.01370-1-26-2': 'To ensure that the backgrounds are reasonably realistic and do not include other people, we sample from a subset of LSUN dataset [CITATION] that includes total of 400K images from the categories kitchen, living room, bedroom and dining room.', '1701.01370-1-27-0': 'Ground truth.', '1701.01370-1-27-1': 'We perform multiple rendering passes in Blender to generate different types of per-pixel ground truth.', '1701.01370-1-27-2': 'The material pass generates pixel-wise segmentation of rendered body parts, given different material indices assigned to different parts of our body model.', '1701.01370-1-27-3': 'The velocity pass, typically used to simulate motion blur, provides us with a render simulating ground truth optical flow.', '1701.01370-1-27-4': 'The depth and normal passes, used for emulating effects like fog, bokeh or for performing shading, produces per-pixel depth maps and normal maps.', '1701.01370-1-27-5': 'The final texture rendering pass composites the shaded, textured body over the random background.', '1701.01370-1-27-6': 'Together with this data we also save camera parameters, lighting parameters as well as the 2D and 3D positions of body joints.', '1701.01370-1-28-0': '## Generating ground truth for real human data', '1701.01370-1-29-0': 'Human3.6M dataset [CITATION] provides ground truth for 2D and 3D human poses.', '1701.01370-1-29-1': 'We complement this ground truth and generate predicted body-part segmentation and depth maps for people in Human3.6M.', '1701.01370-1-29-2': 'Here again we use MoSh [CITATION] to fit the SMPL body shape and pose to the raw MoCap marker data.', '1701.01370-1-29-3': 'This provides a good fit of the model to the shape and the pose of real bodies.', '1701.01370-1-29-4': 'Given the provided camera calibration, we project models to images.', '1701.01370-1-29-5': 'We then render the ground truth segmentation, depth, and 2D/3D joints just as above.', '1701.01370-1-29-6': 'The only difference here is that these now are in correspondence with real pixel values in the dataset.', '1701.01370-1-29-7': 'As MoSh provides almost perfect fits of the model, we consider this data to be "ground truth".', '1701.01370-1-29-8': 'See Figures [REF] and [REF] for generated examples.', '1701.01370-1-30-0': '# Approach', '1701.01370-1-31-0': 'In this section, we present our approach for human body part segmentation [CITATION] and human depth estimation [CITATION], which we train with synthetic and/or real data, see Section [REF] for an evaluation.', '1701.01370-1-32-0': 'Our approach builds on the stacked hourglass network architecture introduced originally for 2D pose estimation problem [CITATION].', '1701.01370-1-32-1': 'This network involves several repetitions of contraction followed by expansion layers which have skip connections to implicitly model spatial relations from different resolutions that allows bottom-up and top-down structured prediction.', '1701.01370-1-32-2': "The convolutional layers with residual connections and 8 'hourglass' modules are stacked on top of each other, each successive stack taking the previous stack's prediction as input.", '1701.01370-1-32-3': 'The reader is referred to [CITATION] for more details.', '1701.01370-1-32-4': 'A variant of this network is used for scene depth estimation [CITATION].', '1701.01370-1-32-5': 'We choose this architecture because it can infer pixel-wise output by taking into account human body structure.', '1701.01370-1-33-0': 'Our network input is a 3-channel RGB image of size [MATH] cropped and scaled to fit a human bounding box using the ground truth.', '1701.01370-1-33-1': 'The network output for each stack has dimensions [MATH] in the case of segmentation (14 classes plus the background) and [MATH] for depth (19 depth classes plus the background).', '1701.01370-1-33-2': 'We use cross-entropy loss defined on all pixels for both segmentation and depth.', '1701.01370-1-33-3': 'The final loss of the network is the sum over 8 stacks.', '1701.01370-1-33-4': 'We train for 50K iterations for synthetic pre-training using the RMSprop algorithm with mini-batches of size 6 and a learning rate of [MATH].', '1701.01370-1-33-5': 'Our data augmentation during training includes random rotations, scaling and color jittering.', '1701.01370-1-34-0': 'We formulate the problem as pixel-wise classification task for both segmentation and depth.', '1701.01370-1-34-1': 'When addressing segmentation, each pixel is assigned to one of the pre-defined 14 human parts, namely head, torso, upper legs, lower legs, upper arms, lower arms, hands, feet (separately for right and left) or to the background class.', '1701.01370-1-34-2': 'Regarding the depth, we align ground-truth depth maps on the z-axis by the depth of the pelvis joint, and then quantize depth values into 19 bins (9 behind and 9 in front of the pelvis) We set the quantization constant to 45mm to roughly cover the depth extent of common human poses.', '1701.01370-1-34-3': 'The network is trained to classify each pixel into one of the 19 depth bins or background.', '1701.01370-1-34-4': 'At test time, we first upsample feature maps of each class with bilinear interpolation by a factor of 4 to output the original resolution.', '1701.01370-1-34-5': 'Then, each pixel is assigned to the class for which the corresponding channel has the maximum activation.', '1701.01370-1-35-0': '# Experiments', '1701.01370-1-36-0': 'We test our approach on several datasets.', '1701.01370-1-36-1': 'First, we validate whether our models for segmentation and depth estimation perform well on the synthetic test set of our SURREAL dataset.', '1701.01370-1-36-2': "This gives an idea about the model's ability to learn the given task for the same domain.", '1701.01370-1-36-3': 'Second, we evaluate the segmentation task on real images from the Freiburg Sitting People dataset [CITATION].', '1701.01370-1-36-4': 'Next, we measure segmentation and depth estimation performance on videos from the Human3.6M dataset [CITATION], where the 3D information is also available.', '1701.01370-1-36-5': 'Finally, we qualitatively evaluate our approach on the MPII Human Pose dataset [CITATION] to see success and failure cases in a challenging domain adaptation scenario.', '1701.01370-1-37-0': '## Evaluation metrics', '1701.01370-1-38-0': 'We choose intersection over union (IOU) and pixel accuracy metrics for evaluating the segmentation approach.', '1701.01370-1-38-1': 'The final measure is the average over 14 human parts as in [CITATION].', '1701.01370-1-38-2': 'Depth estimation is formulated as a classification problem, but we take into account the continuity when we evaluate.', '1701.01370-1-38-3': 'We use root-mean-squared-error (RMSE) that is computed between the predicted quantized depth value (class) and the ground truth quantized depth on the human pixels.', '1701.01370-1-38-4': 'However, to be able to interpret the error in terms of real world coordinates, we multiply the error by the quantization constant that we used, which is 45mm.', '1701.01370-1-38-5': 'Furthermore, we report a scale and translation invariant RMSE (st-RMSE) by solving for the best translation and scaling in z-axis to fit the prediction to the ground truth.', '1701.01370-1-38-6': 'Since inferring depth from RGB is ambiguous, this is a common technique used in evaluations [CITATION].', '1701.01370-1-39-0': '## Validation on synthetic images', '1701.01370-1-40-0': 'Train/test split.', '1701.01370-1-40-1': 'To evaluate our methods on synthetic images, we separate [MATH] of the synthetic frames for the test set and train all our networks on the remaining training set.', '1701.01370-1-40-2': 'The split is constructed such that a given CMU MoCap subject is assigned as either train or test.', '1701.01370-1-40-3': 'However some subjects have a large number of instances, some subjects have unique actions, and some actions are very common (walk, run, jump).', '1701.01370-1-40-4': 'Overall, 30 subjects out of 145 are assigned as test.', '1701.01370-1-40-5': '28 test subjects cover all common actions, and 2 test subjects have unique actions.', '1701.01370-1-40-6': 'Remaining subjects are used for training.', '1701.01370-1-40-7': 'Although our synthetic images have different body shape and appearance than the subject in the originating MoCap sequence, we still found it appropriate to split by subjects.', '1701.01370-1-40-8': 'We separate a subset of our body shapes, clothing and background images for the test set.', '1701.01370-1-40-9': 'This ensures that our tests are unbiased with regards to appearance, yet are still representative of all actions.', '1701.01370-1-40-10': 'Table [REF] summarizes the number of frames, clips and MoCap sequences in each split.', '1701.01370-1-40-11': 'Clips are the continuous 100-frame image sequences where we have the same random body shape, background, clothing, camera and lighting.', '1701.01370-1-40-12': 'A new random set is picked at every clip.', '1701.01370-1-40-13': 'Note that a few sequences have less than 100 frames, therefore not all the clips have 100-frames.', '1701.01370-1-41-0': 'Results on synthetic test set.', '1701.01370-1-41-1': 'The evaluation is performed on the middle frame of each 100-frame clip on the aforementioned held-out synthetic test set, totaling in 12,528 images.', '1701.01370-1-41-2': 'For segmentation, the IOU and pixel accuracy are 69.13% and 80.61%, respectively.', '1701.01370-1-41-3': 'Evaluation of depth estimation gives 72.9mm and 56.3mm for RMSE and st-RMSE errors, respectively.', '1701.01370-1-41-4': 'Figure [REF] shows sample predictions.', '1701.01370-1-41-5': 'For both tasks, the results are mostly accurate on synthetic test images.', '1701.01370-1-41-6': 'However, there exist a few challenging poses (e.g. crawling), test samples with extreme close-up views, and fine details of the hands that are causing errors.', '1701.01370-1-41-7': 'In the following sections, we investigate if similar conclusions can be made for real images.', '1701.01370-1-42-0': '## Segmentation on Freiburg Sitting People', '1701.01370-1-43-0': 'Freiburg Sitting People (FSitting) dataset [CITATION] is composed of 200 high resolution (300x300 pixels) front view images of 6 subjects sitting on a wheel chair.', '1701.01370-1-43-1': 'There are 14 human part annotations available.', '1701.01370-1-43-2': 'See Figure [REF] for sample test images and corresponding ground truth (GT) annotation.', '1701.01370-1-43-3': 'We use the same train/test split as [CITATION], 2 subjects for training and 4 subjects for test.', '1701.01370-1-43-4': 'The amount of data is limited for training deep networks.', '1701.01370-1-43-5': 'We show that our network pre-trained only on synthetic images is already able to segment human body parts.', '1701.01370-1-43-6': 'This shows that the human renderings in the synthetic dataset are representative of the real images, such that networks trained exclusively on synthetic data can generalize quite well to real data.', '1701.01370-1-44-0': 'Table [REF] summarizes segmentation results on FSitting.', '1701.01370-1-44-1': 'We carry out several experiments to understand the gain from synthetic pre-training.', '1701.01370-1-44-2': "For the 'Real' baseline, we train the same network from scratch using 2 training subjects.", '1701.01370-1-44-3': 'This network quickly overfits as there are few subjects to learn from and the performance is quite low.', '1701.01370-1-44-4': "Our 'Synthetic' result is obtained using the network pre-trained on synthetic images without fine-tuning.", '1701.01370-1-44-5': 'We get 51.88% pixel accuracy and 40.1% IOU with this method and clearly outperform training from real images.', '1701.01370-1-44-6': "Furthermore, fine-tuning ('Synthetic+Real') with 2 training subjects helps significantly.", '1701.01370-1-44-7': 'See Figure [REF] for qualitative results.', '1701.01370-1-44-8': 'Given the little amount for training in FSitting, the fine-tuning quickly converges after 200 iterations.', '1701.01370-1-45-0': 'In [CITATION], the authors introduce a network that outputs a high-resolution segmentation after several layers of upconvolutions.', '1701.01370-1-45-1': 'For a fair comparison, we modify our network to output full resolution by adding one bilinear upsampling layer followed by nonlinearity (ReLU) and a convolutional layer with [MATH] filters that outputs [MATH] instead of [MATH] as explained in Section [REF].', '1701.01370-1-45-2': "If we fine-tune this network ('Synth+Real+up') on FSitting, we improve performance and outperform [CITATION] by a large margin.", '1701.01370-1-45-3': 'Note that [CITATION] trains on the same FSitting training images, but added around 2,800 Pascal images.', '1701.01370-1-45-4': 'Hence they use significantly more manual annotation than our method.', '1701.01370-1-46-0': '## Segmentation and depth on Human3.6M', '1701.01370-1-47-0': 'Showcasing our approach requires expensive annotation if applied on real data.', '1701.01370-1-47-1': 'This is the reason for existing real datasets being small and not suitable for training deep networks.', '1701.01370-1-47-2': 'As a result, there is no available dataset to carry out a clean evaluation.', '1701.01370-1-47-3': 'For this reason, we came up with generating nearly perfect ground truth for images recorded with a calibrated camera and given their MoCap data.', '1701.01370-1-47-4': 'Human3.6M is currently the largest dataset where such information is available.', '1701.01370-1-47-5': 'There are 3.6 million frames from 4 cameras.', '1701.01370-1-47-6': 'We use subjects 1, 5, 6, 7, 8 for training, subject 9 for validation and subject 11 for testing as in [CITATION].', '1701.01370-1-47-7': 'We use all frames from one repetition of each action for training, and every 64[MATH] frame from all repetitions at test.', '1701.01370-1-47-8': 'The frames have resolution [MATH] pixels, we assume a [MATH] cropped human bounding box is given to reduce computational complexity.', '1701.01370-1-47-9': 'We evaluate the performance of both segmentation and depth, and compare with the baseline for which we train a network on real images only.', '1701.01370-1-48-0': '### Segmentation', '1701.01370-1-49-0': 'Table [REF] summarizes the parts segmentation results on Human3.6M.', '1701.01370-1-49-1': 'We report both the mean over 14 human parts (fg) and the mean together with the background class (fg+bg).', '1701.01370-1-49-2': 'Training on real images instead of synthetic images increases IOU by 3.4% and pixel accuracy by 2.14%.', '1701.01370-1-49-3': 'This is expected because the training distribution matches the test distribution in terms of background, camera position and action categories (i.e. poses).', '1701.01370-1-49-4': 'The amount of real data is sufficient to perform CNN training.', '1701.01370-1-49-5': "However, since there are very few subjects available, we see that the network doesn't generalize to different clothing.", '1701.01370-1-49-6': "In Figure [REF], the 'Real' baseline has the border between shoulders and upper arms exactly on the t-shirt boundaries.", '1701.01370-1-49-7': 'This reveals that the network learns about skin color rather than actual body parts.', '1701.01370-1-49-8': 'Our pre-trained network (Synth) performs reasonably well, even though the pose distribution in our MoCap is quite different than that of Human3.6M.', '1701.01370-1-49-9': 'When we fine-tune the network with real images from Human3.6M (Real+Synth), the segmentations look very similar to the ground truth and we outperform the baseline by a large margin.', '1701.01370-1-49-10': 'Moreover, our model is capable of distinguishing left and right most of the time on all 4 views since it has been trained with randomly sampled views.', '1701.01370-1-50-0': '### Depth estimation', '1701.01370-1-51-0': 'Depth estimation results on Human3.6M for various poses and viewpoints are illustrated in Figure [REF].', '1701.01370-1-51-1': 'Here, the pre-trained network fails at the very challenging poses, although it still captures partly correct estimates (left column, third, fourth rows).', '1701.01370-1-51-2': 'Fine-tuning on real data compensates for these errors and refines estimations.', '1701.01370-1-51-3': 'In Table [REF], we show RMSE error measured on foreground pixels, together with the scale-translation invariant version (see Section [REF]).', '1701.01370-1-51-4': 'We also report the error only on known 2D joints (PoseRMSE) to have an idea of how well a 3D pose estimation model would work based on the depth predictions.', '1701.01370-1-51-5': 'One would need to handle occluded joints to infer 3D locations of all joints, and this is beyond the scope of the current paper.', '1701.01370-1-52-0': '## Qualitative results on MPII Human Pose', '1701.01370-1-53-0': 'The datasets we evaluated our approach with are relatively simple, i.e. no background clutter, few subjects, single person per image, full body visible.', '1701.01370-1-53-1': 'Except the first two properties, this setup matches our synthetic training data.', '1701.01370-1-53-2': 'Therefore, we test the generalization of our model on more challenging images.', '1701.01370-1-53-3': 'Currently, MPII Human Pose [CITATION] is one of the biggest datasets with diverse viewpoints and clutter.', '1701.01370-1-53-4': 'However, this dataset has no annotation for part segmentation nor depth.', '1701.01370-1-53-5': 'For this reason, we qualitatively show our predictions.', '1701.01370-1-53-6': 'In Figure [REF], several success and failure cases are illustrated.', '1701.01370-1-53-7': 'Our model generalizes reasonably well, except when there are multiple people close to each other, and in extreme viewpoints and occlusions, which have not appeared during training.', '1701.01370-1-54-0': '# Conclusions', '1701.01370-1-55-0': 'In this study, we have shown successful large-scale training of CNNs from synthetically generated images or people.', '1701.01370-1-55-1': 'We have addressed two tasks, namely, human body parts segmentation and depth estimation, for which large-scale manual annotation is infeasible.', '1701.01370-1-55-2': 'Our generated synthetic dataset comes with rich pixel-wise ground truth information and can potentially be used for other tasks than considered here.', '1701.01370-1-55-3': 'Unlike many existing synthetic datasets, the focus of SURREAL is on the realistic rendering of people, which is a challenging task.', '1701.01370-1-55-4': 'In our future work, we plan to integrate the person into the background in a more realistic way by taking into account the lighting and the 3D scene layout.', '1701.01370-1-55-5': 'We also plan to augment the data with more challenging scenarios by creating occlusions and rendering multiple people in the same scene.'}

{'1701.01370-2-0-0': 'Estimating human pose, shape, and motion from images and videos are fundamental challenges with many applications.', '1701.01370-2-0-1': 'Recent advances in 2D human pose estimation use large amounts of manually-labeled training data for learning convolutional neural networks (CNNs).', '1701.01370-2-0-2': 'Such data is time consuming to acquire and difficult to extend.', '1701.01370-2-0-3': 'Moreover, manual labeling of 3D pose, depth and motion is impractical.', '1701.01370-2-0-4': 'In this work we present SURREAL (Synthetic hUmans foR REAL tasks): a new large-scale dataset with synthetically-generated but realistic images of people rendered from 3D sequences of human motion capture data.', '1701.01370-2-0-5': 'We generate more than 6 million frames together with ground truth pose, depth maps, and segmentation masks.', '1701.01370-2-0-6': 'We show that CNNs trained on our synthetic dataset allow for accurate human depth estimation and human part segmentation in real RGB images.', '1701.01370-2-0-7': 'Our results and the new dataset open up new possibilities for advancing person analysis using cheap and large-scale synthetic data.', '1701.01370-2-1-0': '# Introduction', '1701.01370-2-2-0': 'Convolutional Neural Networks provide significant gains to problems with large amounts of training data.', '1701.01370-2-2-1': 'In the field of human analysis, recent datasets [CITATION] now gather a sufficient number of annotated images to train networks for 2D human pose estimation [CITATION].', '1701.01370-2-2-2': 'Other tasks such as accurate estimation of human motion, depth and body-part segmentation are lagging behind as manual supervision for such problems at large scale is prohibitively expensive.', '1701.01370-2-3-0': 'Images of people have rich variation in poses, clothing, hair styles, body shapes, occlusions, viewpoints, motion blur and other factors.', '1701.01370-2-3-1': 'Many of these variations, however, can be synthesized using existing 3D motion capture (MoCap) data [CITATION] and modern tools for realistic rendering.', '1701.01370-2-3-2': 'Provided sufficient realism, such an approach would be highly useful for many tasks as it can generate rich ground truth in terms of depth, motion, body-part segmentation and occlusions.', '1701.01370-2-4-0': 'Although synthetic data has been used for many years, realism has been limited.', '1701.01370-2-4-1': 'In this work we present SURREAL: a new large-scale dataset with synthetically-generated but realistic images of people.', '1701.01370-2-4-2': 'Images are rendered from 3D sequences of MoCap data.', '1701.01370-2-4-3': 'To ensure realism, the synthetic bodies are created using the SMPL body model [CITATION], whose parameters are fit by the MoSh [CITATION] method given raw 3D MoCap marker data.', '1701.01370-2-4-4': 'We randomly sample a large variety of viewpoints, clothing and lighting.', '1701.01370-2-4-5': 'SURREAL contains more than 6 million frames together with ground truth pose, depth maps, and segmentation masks.', '1701.01370-2-4-6': 'We show that CNNs trained on synthetic data allow for accurate human depth estimation and human part segmentation in real RGB images, see Figure [REF].', '1701.01370-2-4-7': 'Here, we demonstrate that our dataset, while being synthetic, reaches the level of realism necessary to support training for multiple complex tasks.', '1701.01370-2-4-8': 'This opens up opportunities for training deep networks using graphics techniques available now.', '1701.01370-2-4-9': 'SURREAL dataset is publicly available together with the code to generate synthetic data and to train models for body part segmentation and depth estimation [CITATION].', '1701.01370-2-5-0': 'The rest of this paper is organized as follows.', '1701.01370-2-5-1': 'Section [REF] reviews related work.', '1701.01370-2-5-2': 'Section [REF] presents our approach for generating realistic synthetic videos of people.', '1701.01370-2-5-3': 'In Section [REF] we describe our CNN architecture for human body part segmentation and depth estimation.', '1701.01370-2-5-4': 'Section [REF] reports experiments.', '1701.01370-2-5-5': 'We conclude in Section [REF].', '1701.01370-2-6-0': '# Related work', '1701.01370-2-7-0': 'Knowledge transfer from synthetic to real images has been recently studied with deep neural networks.', '1701.01370-2-7-1': 'Dosovitskiy [CITATION] learn a CNN for optical flow estimation using synthetically generated images of rendered 3D moving chairs.', '1701.01370-2-7-2': 'Peng [CITATION] study the effect of different visual cues such as object/background texture and color when rendering synthetic 3D objects for object detection task.', '1701.01370-2-7-3': 'Similarly, [CITATION] explores rendering 3D objects to perform viewpoint estimation.', '1701.01370-2-7-4': 'Fanello [CITATION] render synthetic infrared images of hands and faces to predict depth and parts.', '1701.01370-2-7-5': 'Recently, Gaidon [CITATION] have released the Virtual KITTI dataset with synthetically generated videos of cars to study multi-object tracking.', '1701.01370-2-8-0': 'Several works focused on creating synthetic images of human bodies for learning 2D pose estimation [CITATION], 3D pose estimation [CITATION], pedestrian detection [CITATION], and action recognition [CITATION].', '1701.01370-2-8-1': 'Pishchulin [CITATION] generate synthetic images with a game engine.', '1701.01370-2-8-2': 'In [CITATION], they deform 2D images with a 3D model.', '1701.01370-2-8-3': 'More recently, Rogez and Schmid [CITATION] use an image-based synthesis engine to augment existing real images.', '1701.01370-2-8-4': 'Ghezelghieh [CITATION] render synthetic images with 10 simple body models with an emphasis on upright people; however, the main challenge using existing MoCap data for training is to generalize to poses that are not upright.', '1701.01370-2-9-0': 'A similar direction has been explored in [CITATION].', '1701.01370-2-9-1': 'In [CITATION], action recognition is addressed with synthetic human trajectories from MoCap data.', '1701.01370-2-9-2': '[CITATION] train CNNs with synthetic depth images.', '1701.01370-2-9-3': 'EgoCap [CITATION] creates a dataset by augmenting egocentric sequences with background.', '1701.01370-2-10-0': 'The closest work to this paper is [CITATION], where the authors render large-scale synthetic images for predicting 3D pose with CNNs.', '1701.01370-2-10-1': 'Our dataset differs from [CITATION] by having a richer, per-pixel ground truth, thus allowing to train for pixel-wise predictions and multi-task scenarios.', '1701.01370-2-10-2': 'In addition, we argue that the realism in our synthetic images is better (see sample videos in [CITATION]), thus resulting in a smaller gap between features learned from synthetic and real images.', '1701.01370-2-10-3': 'The method in [CITATION] heavily relies on real images as input in their training with domain adaptation.', '1701.01370-2-10-4': 'This is not the case for our synthetic training.', '1701.01370-2-10-5': 'Moreover, we render video sequences which can be used for temporal modeling.', '1701.01370-2-11-0': 'Our dataset presents several differences with existing synthetic datasets.', '1701.01370-2-11-1': 'It is the first large-scale person dataset providing depth, part segmentation and flow ground truth for synthetic RGB frames.', '1701.01370-2-11-2': 'Other existing datasets are used either for taking RGB image as input and training only for 2D/3D pose, or for taking depth/infrared images as input and training for depth/parts segmentation.', '1701.01370-2-11-3': 'In this paper, we show that photo-realistic renderings of people under large variations in shape, texture, viewpoint and pose can help solving pixel-wise human labeling tasks.', '1701.01370-2-12-0': '# Data generation', '1701.01370-2-13-0': 'This section presents our SURREAL (Synthetic hUmans foR REAL tasks) dataset and describes key steps for its generation (Section [REF]).', '1701.01370-2-13-1': 'We also describe how we obtain ground truth data for real MoCap sequences (Section [REF]).', '1701.01370-2-14-0': '## Synthetic humans', '1701.01370-2-15-0': 'Our pipeline for generating synthetic data is illustrated in Figure [REF].', '1701.01370-2-15-1': 'A human body with a random 3D pose, random shape and random texture is rendered from a random view-point for some random lighting and a random background image.', '1701.01370-2-15-2': 'Below we define what "random" means in all these cases.', '1701.01370-2-15-3': 'Since the data is synthetic, we also generate ground truth depth maps, optical flow, surface normals, human part segmentations and joint locations (both 2D and 3D).', '1701.01370-2-15-4': 'As a result, we obtain 6.5 million frames grouped into [MATH] continuous image sequences.', '1701.01370-2-15-5': 'See Table [REF] for more statistics, Section [REF] for the description of the synthetic train/test split, and Figure [REF] for samples from the SURREAL dataset.', '1701.01370-2-16-0': 'Body model.', '1701.01370-2-16-1': 'Synthetic bodies are created using the SMPL body model [CITATION].', '1701.01370-2-16-2': 'SMPL is a realistic articulated model of the body created from thousands of high-quality 3D scans, which decomposes body deformations into pose (kinematic deformations due to skeletal posture) and shape (body deformations intrinsic to a particular person that make them different from others).', '1701.01370-2-16-3': 'SMPL is compatible with most animation packages like Blender [CITATION].', '1701.01370-2-16-4': 'SMPL deformations are modeled as a combination of linear blend skinning and linear blendshapes defined by principal components of body shape variation.', '1701.01370-2-16-5': 'SMPL pose and shape parameters are converted to a triangulated mesh using Blender, which then applies texture, shading and adds a background to generate the final RGB output.', '1701.01370-2-17-0': 'Body shape.', '1701.01370-2-17-1': 'In order to render varied, but realistic, body shapes we make use of the CAESAR dataset [CITATION], which was used to train SMPL.', '1701.01370-2-17-2': 'To create a body shape, we select one of the CAESAR subjects at random and approximate their shape with the first 10 SMPL shape principal components.', '1701.01370-2-17-3': 'Ten shape components explain more than [MATH] of the shape variance in CAESAR (at the resolution of our mesh) and produce quite realistic body shapes.', '1701.01370-2-18-0': 'Body pose.', '1701.01370-2-18-1': 'To generate images of people in realistic poses, we take motion capture data from the CMU MoCap database [CITATION].', '1701.01370-2-18-2': 'CMU MoCap contains more than 2000 sequences of 23 high-level action categories, resulting in more than 10 hours of recorded 3D locations of body markers.', '1701.01370-2-19-0': 'It is often challenging to realistically and automatically retarget MoCap skeleton data to a new model.', '1701.01370-2-19-1': 'For this reason we do not use the skeleton data but rather use MoSh [CITATION] to fit the SMPL parameters that best explain raw 3D MoCap marker locations.', '1701.01370-2-19-2': 'This gives both the 3D shape of the subject and the articulated pose parameters of SMPL.', '1701.01370-2-19-3': 'To increase the diversity, we replace the estimated 3D body shape with a set of randomly sampled body shapes.', '1701.01370-2-20-0': 'We render each CMU MoCap sequence three times using different random parameters.', '1701.01370-2-20-1': 'Moreover, we divide the sequences into clips of 100 frames with 30%, 50% and 70% overlaps for these three renderings.', '1701.01370-2-20-2': 'Every pose of the sequence is rendered with consistent parameters (i.e. body shape, clothing, light, background etc.) within each clip.', '1701.01370-2-21-0': 'Human texture.', '1701.01370-2-21-1': 'We use two types of real scans for the texture of body models.', '1701.01370-2-21-2': 'First, we extract SMPL texture maps from CAESAR scans, which come with a color texture per 3D point.', '1701.01370-2-21-3': 'These maps vary in skin color and person identities, however, their quality is often low due to the low resolution, uniform tight-fitting clothing, and visible markers placed on the face and the body.', '1701.01370-2-21-4': 'Anthropometric markers are automatically removed from the texture images and inpainted.', '1701.01370-2-21-5': 'To provide more variety, we extract a second set of textures obtained from 3D scans of subjects with normal clothing.', '1701.01370-2-21-6': 'These scans are registered with 4Cap as in [CITATION].', '1701.01370-2-21-7': 'The texture of real clothing substantially increases the realism of generated images, even though SMPL does not model 3D deformations of clothes.', '1701.01370-2-22-0': '[MATH] of our data is rendered with the first set ([MATH] CAESAR textures randomly sampled from [MATH]), and the rest with the second set ([MATH] clothed textures).', '1701.01370-2-22-1': 'To preserve the anonymity of subjects, we replace all faces in the texture maps by the average CAESAR face.', '1701.01370-2-22-2': 'The skin color of this average face is corrected to fit the face skin color of the original texture map.', '1701.01370-2-22-3': 'This corrected average face is blended smoothly with the original map, resulting in a realistic and anonymized body texture.', '1701.01370-2-23-0': 'Light.', '1701.01370-2-23-1': 'The body is illuminated using Spherical Harmonics with [MATH] coefficients [CITATION].', '1701.01370-2-23-2': 'The coefficients are randomly sampled from a uniform distribution between [MATH] and [MATH], apart from the ambient illumination coefficient (which has a minimum value of [MATH]) and the vertical illumination component, which is biased to encourage the illumination from above.', '1701.01370-2-23-3': 'Since Blender does not provide Spherical Harmonics illumination, a spherical harmonic shader for the body material was implemented in Open Shading Language.', '1701.01370-2-24-0': 'Camera.', '1701.01370-2-24-1': 'The projective camera has a resolution of [MATH], focal length of [MATH]mm and sensor size of [MATH]mm.', '1701.01370-2-24-2': 'To generate images of the body in a wide range of positions, we take 100-frame MoCap sub-sequences and, in the first frame, render the body so that the center of the viewport points to the pelvis of the body, at a random distance (sampled from a normal distribution with [MATH] meters mean, [MATH] meter deviation) with a random yaw angle.', '1701.01370-2-24-3': 'The remainder of the sequence then effectively produces bodies in a range of locations relative to the static camera.', '1701.01370-2-25-0': 'Background.', '1701.01370-2-25-1': 'We render the person on top of a static background image.', '1701.01370-2-25-2': 'To ensure that the backgrounds are reasonably realistic and do not include other people, we sample from a subset of LSUN dataset [CITATION] that includes total of 400K images from the categories kitchen, living room, bedroom and dining room.', '1701.01370-2-26-0': 'Ground truth.', '1701.01370-2-26-1': 'We perform multiple rendering passes in Blender to generate different types of per-pixel ground truth.', '1701.01370-2-26-2': 'The material pass generates pixel-wise segmentation of rendered body parts, given different material indices assigned to different parts of our body model.', '1701.01370-2-26-3': 'The velocity pass, typically used to simulate motion blur, provides us with a render simulating optical flow.', '1701.01370-2-26-4': 'The depth and normal passes, used for emulating effects like fog, bokeh or for performing shading, produce per-pixel depth maps and normal maps.', '1701.01370-2-26-5': 'The final texture rendering pass overlays the shaded, textured body over the random background.', '1701.01370-2-26-6': 'Together with this data we save camera and lighting parameters as well as the 2D/3D positions of body joints.', '1701.01370-2-27-0': '## Generating ground truth for real human data', '1701.01370-2-28-0': 'Human3.6M dataset [CITATION] provides ground truth for 2D and 3D human poses.', '1701.01370-2-28-1': 'We complement this ground truth and generate predicted body-part segmentation and depth maps for people in Human3.6M.', '1701.01370-2-28-2': 'Here again we use MoSh [CITATION] to fit the SMPL body shape and pose to the raw MoCap marker data.', '1701.01370-2-28-3': 'This provides a good fit of the model to the shape and the pose of real bodies.', '1701.01370-2-28-4': 'Given the provided camera calibration, we project models to images.', '1701.01370-2-28-5': 'We then render the ground truth segmentation, depth, and 2D/3D joints as above, while ensuring correspondence with real pixel values in the dataset.', '1701.01370-2-28-6': 'As MoSh provides almost perfect fits of the model, we consider this data to be "ground truth".', '1701.01370-2-28-7': 'See Figures [REF] and [REF] for generated examples.', '1701.01370-2-28-8': 'We use this ground truth for the baseline where we train only on real data, and also for fine-tuning our models pre-trained on synthetic data.', '1701.01370-2-28-9': 'In the rest of the paper, all frames from the synthetic training set are used for synthetic pre-training.', '1701.01370-2-29-0': '# Approach', '1701.01370-2-30-0': 'In this section, we present our approach for human body part segmentation [CITATION] and human depth estimation [CITATION], which we train with synthetic and/or real data, see Section [REF] for the evaluation.', '1701.01370-2-31-0': 'Our approach builds on the stacked hourglass network architecture introduced originally for 2D pose estimation problem [CITATION].', '1701.01370-2-31-1': 'This network involves several repetitions of contraction followed by expansion layers which have skip connections to implicitly model spatial relations from different resolutions that allows bottom-up and top-down structured prediction.', '1701.01370-2-31-2': "The convolutional layers with residual connections and 8 'hourglass' modules are stacked on top of each other, each successive stack taking the previous stack's prediction as input.", '1701.01370-2-31-3': 'The reader is referred to [CITATION] for more details.', '1701.01370-2-31-4': 'A variant of this network has been used for scene depth estimation [CITATION].', '1701.01370-2-31-5': 'We choose this architecture because it can infer pixel-wise output by taking into account human body structure.', '1701.01370-2-32-0': 'Our network input is a 3-channel RGB image of size [MATH] cropped and scaled to fit a human bounding box using the ground truth.', '1701.01370-2-32-1': 'The network output for each stack has dimensions [MATH] in the case of segmentation (14 classes plus the background) and [MATH] for depth (19 depth classes plus the background).', '1701.01370-2-32-2': 'We use cross-entropy loss defined on all pixels for both segmentation and depth.', '1701.01370-2-32-3': 'The final loss of the network is the sum over 8 stacks.', '1701.01370-2-32-4': 'We train for 50K iterations for synthetic pre-training using the RMSprop algorithm with mini-batches of size 6 and a learning rate of [MATH].', '1701.01370-2-32-5': 'Our data augmentation during training includes random rotations, scaling and color jittering.', '1701.01370-2-33-0': 'We formulate the problem as pixel-wise classification task for both segmentation and depth.', '1701.01370-2-33-1': 'When addressing segmentation, each pixel is assigned to one of the pre-defined 14 human parts, namely head, torso, upper legs, lower legs, upper arms, lower arms, hands, feet (separately for right and left) or to the background class.', '1701.01370-2-33-2': 'Regarding the depth, we align ground-truth depth maps on the z-axis by the depth of the pelvis joint, and then quantize depth values into 19 bins (9 behind and 9 in front of the pelvis).', '1701.01370-2-33-3': 'We set the quantization constant to 45mm to roughly cover the depth extent of common human poses.', '1701.01370-2-33-4': 'The network is trained to classify each pixel into one of the 19 depth bins or background.', '1701.01370-2-33-5': 'At test time, we first upsample feature maps of each class with bilinear interpolation by a factor of 4 to output the original resolution.', '1701.01370-2-33-6': 'Then, each pixel is assigned to the class for which the corresponding channel has the maximum activation.', '1701.01370-2-34-0': '# Experiments', '1701.01370-2-35-0': 'We test our approach on several datasets.', '1701.01370-2-35-1': 'First, we evaluate the segmentation and depth estimation on the test set of our synthetic SURREAL dataset.', '1701.01370-2-35-2': 'Second, we test the performance of segmentation on real images from the Freiburg Sitting People dataset [CITATION].', '1701.01370-2-35-3': 'Next, we evaluate segmentation and depth estimation on real videos from the Human3.6M dataset [CITATION] with available 3D information.', '1701.01370-2-35-4': 'Then, we qualitatively evaluate our approach on the more challenging MPII Human Pose dataset [CITATION].', '1701.01370-2-35-5': 'Finally, we experiment and discuss design choices of the SURREAL dataset.', '1701.01370-2-36-0': '## Evaluation measures', '1701.01370-2-37-0': 'We use intersection over union (IOU) and pixel accuracy measures for evaluating the segmentation approach.', '1701.01370-2-37-1': 'The final measure is the average over 14 human parts as in [CITATION].', '1701.01370-2-37-2': 'Depth estimation is formulated as a classification problem, but we take into account the continuity when we evaluate.', '1701.01370-2-37-3': 'We compute root-mean-squared-error (RMSE) between the predicted quantized depth value (class) and the ground truth quantized depth on the human pixels.', '1701.01370-2-37-4': 'To interpret the error in real world coordinates, we multiply it by the quantization constant (45mm).', '1701.01370-2-37-5': 'We also report a scale and translation invariant RMSE (st-RMSE) by solving for the best translation and scaling in z-axis to fit the prediction to the ground truth.', '1701.01370-2-37-6': 'Since inferring depth from RGB is ambiguous, this is a common technique used in evaluations [CITATION].', '1701.01370-2-38-0': '## Validation on synthetic images', '1701.01370-2-39-0': 'Train/test split.', '1701.01370-2-39-1': 'To evaluate our methods on synthetic images, we separate [MATH] of the synthetic frames for the test set and train all our networks on the remaining training set.', '1701.01370-2-39-2': 'The split is constructed such that a given CMU MoCap subject is assigned as either train or test.', '1701.01370-2-39-3': 'Whereas some subjects have a large number of instances, some subjects have unique actions, and some actions are very common (walk, run, jump).', '1701.01370-2-39-4': 'Overall, 30 subjects out of 145 are assigned as test.', '1701.01370-2-39-5': '28 test subjects cover all common actions, and 2 have unique actions.', '1701.01370-2-39-6': 'Remaining subjects are used for training.', '1701.01370-2-39-7': 'Although our synthetic images have different body shape and appearance than the subject in the originating MoCap sequence, we still found it appropriate to split by subjects.', '1701.01370-2-39-8': 'We separate a subset of our body shapes, clothing and background images for the test set.', '1701.01370-2-39-9': 'This ensures that our tests are unbiased with regards to appearance, yet are still representative of all actions.', '1701.01370-2-39-10': 'Table [REF] summarizes the number of frames, clips and MoCap sequences in each split.', '1701.01370-2-39-11': 'Clips are the continuous 100-frame sequences where we have the same random body shape, background, clothing, camera and lighting.', '1701.01370-2-39-12': 'A new random set is picked at every clip.', '1701.01370-2-39-13': 'Note that a few sequences have less than 100 frames.', '1701.01370-2-40-0': 'Results on synthetic test set.', '1701.01370-2-40-1': 'The evaluation is performed on the middle frame of each 100-frame clip on the aforementioned held-out synthetic test set, totaling in 12,528 images.', '1701.01370-2-40-2': 'For segmentation, the IOU and pixel accuracy are 69.13% and 80.61%, respectively.', '1701.01370-2-40-3': 'Evaluation of depth estimation gives 72.9mm and 56.3mm for RMSE and st-RMSE errors, respectively.', '1701.01370-2-40-4': 'Figure [REF] shows sample predictions.', '1701.01370-2-40-5': 'For both tasks, the results are mostly accurate on synthetic test images.', '1701.01370-2-40-6': 'However, there exist a few challenging poses (e.g. crawling), test samples with extreme close-up views, and fine details of the hands that are causing errors.', '1701.01370-2-40-7': 'In the following sections, we investigate if similar conclusions can be made for real images.', '1701.01370-2-41-0': '## Segmentation on Freiburg Sitting People', '1701.01370-2-42-0': 'Freiburg Sitting People (FSitting) dataset [CITATION] is composed of 200 high resolution (300x300 pixels) front view images of 6 subjects sitting on a wheel chair.', '1701.01370-2-42-1': 'There are 14 human part annotations available.', '1701.01370-2-42-2': 'See Figure [REF] for sample test images and corresponding ground truth (GT) annotation.', '1701.01370-2-42-3': 'We use the same train/test split as [CITATION], 2 subjects for training and 4 subjects for test.', '1701.01370-2-42-4': 'The amount of data is limited for training deep networks.', '1701.01370-2-42-5': 'We show that our network pre-trained only on synthetic images is already able to segment human body parts.', '1701.01370-2-42-6': 'This shows that the human renderings in the synthetic dataset are representative of the real images, such that networks trained exclusively on synthetic data can generalize quite well to real data.', '1701.01370-2-43-0': 'Table [REF] summarizes segmentation results on FSitting.', '1701.01370-2-43-1': 'We carry out several experiments to understand the gain from synthetic pre-training.', '1701.01370-2-43-2': "For the 'Real' baseline, we train a network from scratch using 2 training subjects.", '1701.01370-2-43-3': 'This network overfits as there are few subjects to learn from and the performance is quite low.', '1701.01370-2-43-4': "Our 'Synth' result is obtained using the network pre-trained on synthetic images without fine-tuning.", '1701.01370-2-43-5': 'We get 51.88% pixel accuracy and 40.1% IOU with this method and clearly outperform training from real images.', '1701.01370-2-43-6': 'Furthermore, fine-tuning (Synth+Real) with 2 training subjects helps significantly.', '1701.01370-2-43-7': 'See Figure [REF] for qualitative results.', '1701.01370-2-43-8': 'Given the little amount for training in FSitting, the fine-tuning converges after 200 iterations.', '1701.01370-2-44-0': 'In [CITATION], the authors introduce a network that outputs a high-resolution segmentation after several layers of upconvolutions.', '1701.01370-2-44-1': 'For a fair comparison, we modify our network to output full resolution by adding one bilinear upsampling layer followed by nonlinearity (ReLU) and a convolutional layer with [MATH] filters that outputs [MATH] instead of [MATH] as explained in Section [REF].', '1701.01370-2-44-2': 'If we fine-tune this network (Synth+Real+up) on FSitting, we improve performance and outperform [CITATION] by a large margin.', '1701.01370-2-44-3': 'Note that [CITATION] trains on the same FSitting training images, but added around 2,800 Pascal images.', '1701.01370-2-44-4': 'Hence they use significantly more manual annotation than our method.', '1701.01370-2-45-0': '## Segmentation and depth on Human3.6M', '1701.01370-2-46-0': 'To evaluate our approach, we need sufficient real data with ground truth annotations.', '1701.01370-2-46-1': 'Such data is expensive to obtain and currently not available.', '1701.01370-2-46-2': 'For this reason, we generate nearly perfect ground truth for images recorded with a calibrated camera and given their MoCap data.', '1701.01370-2-46-3': 'Human3.6M is currently the largest dataset where such information is available.', '1701.01370-2-46-4': 'There are 3.6 million frames from 4 cameras.', '1701.01370-2-46-5': 'We use subjects S1, S5, S6, S7, S8 for training, S9 for validation and S11 for testing as in [CITATION].', '1701.01370-2-46-6': 'Each subject performs each of the 15 actions twice.', '1701.01370-2-46-7': 'We use all frames from one of the two instances of each action for training, and every 64[MATH] frame from all instances for testing.', '1701.01370-2-46-8': 'The frames have resolution [MATH] pixels, we assume a [MATH] cropped human bounding box is given to reduce computational complexity.', '1701.01370-2-46-9': 'We evaluate the performance of both segmentation and depth, and compare with the baseline for which we train a network on real images only.', '1701.01370-2-47-0': 'Segmentation.', '1701.01370-2-47-1': 'Table [REF] summarizes the parts segmentation results on Human3.6M.', '1701.01370-2-47-2': 'We report both the mean over 14 human parts (fg) and the mean together with the background class (fg+bg).', '1701.01370-2-47-3': 'Training on real images instead of synthetic images increases IOU by 3.4% and pixel accuracy by 2.14%.', '1701.01370-2-47-4': 'This is expected because the training distribution matches the test distribution in terms of background, camera position and action categories (i.e. poses).', '1701.01370-2-47-5': 'Furthermore, the amount of real data is sufficient to perform CNN training.', '1701.01370-2-47-6': "However, since there are very few subjects available, we see that the network doesn't generalize to different clothing.", '1701.01370-2-47-7': "In Figure [REF], the 'Real' baseline has the border between shoulders and upper arms exactly on the T-shirt boundaries.", '1701.01370-2-47-8': 'This reveals that the network learns about skin color rather than actual body parts.', '1701.01370-2-47-9': 'Our pre-trained network (Synth) performs reasonably well, even though the pose distribution in our MoCap is quite different than that of Human3.6M.', '1701.01370-2-47-10': "When we fine-tune the network with real images from Human3.6M (Synth+Real), the model predicts very accurate segmentations and outperforms the 'Real' baseline by a large margin.", '1701.01370-2-47-11': 'Moreover, our model is capable of distinguishing left and right most of the time on all 4 views since it has been trained with randomly sampled views.', '1701.01370-2-48-0': 'Depth estimation.', '1701.01370-2-48-1': 'Depth estimation results on Human3.6M for various poses and viewpoints are illustrated in Figure [REF].', '1701.01370-2-48-2': 'Here, the pre-trained network fails at the very challenging poses, although it still captures partly correct estimates (first row).', '1701.01370-2-48-3': 'Fine-tuning on real data compensates for these errors and refines estimations.', '1701.01370-2-48-4': 'In Table [REF], we show RMSE error measured on foreground pixels, together with the scale-translation invariant version (see Section [REF]).', '1701.01370-2-48-5': 'We also report the error only on known 2D joints (PoseRMSE) to have an idea of how well a 3D pose estimation model would work based on the depth predictions.', '1701.01370-2-48-6': 'One would need to handle occluded joints to infer 3D locations of all joints, and this is beyond the scope of the current paper.', '1701.01370-2-49-0': '## Qualitative results on MPII Human Pose', '1701.01370-2-50-0': 'FSitting and Human3.6M are relatively simple datasets with limited background clutter, few subjects, single person per image, full body visible.', '1701.01370-2-50-1': 'In this section, we test the generalization of our model on more challenging images.', '1701.01370-2-50-2': 'MPII Human Pose [CITATION] is one of the largest datasets with diverse viewpoints and clutter.', '1701.01370-2-50-3': 'However, this dataset has no ground truth for part segmentation nor depth.', '1701.01370-2-50-4': 'Therefore, we qualitatively show our predictions.', '1701.01370-2-50-5': 'Figure [REF] illustrates several success and failure cases.', '1701.01370-2-50-6': 'Our model generalizes reasonably well, except when there are multiple people close to each other and extreme viewpoints, which have not appeared during training.', '1701.01370-2-50-7': 'It is interesting to note that although lower body occlusions and cloth shapes are not present in synthetic training, the models perform accurately in such cases, see Figure [REF] caption.', '1701.01370-2-51-0': '## Design choices', '1701.01370-2-52-0': "We did several experiments to answer questions such as 'How much data should we synthesize?'", '1701.01370-2-52-1': ", 'Is CMU MoCap enough?'", '1701.01370-2-52-2': ", 'What's the effect having clothing variation?'", '1701.01370-2-53-0': 'Amount of data.', '1701.01370-2-53-1': 'We plot the performance as a function of training data size.', '1701.01370-2-53-2': 'We train with a random subset of [MATH], [MATH], [MATH], [MATH]% of the 55K training clips using all frames of the selected clips, i.e., [MATH]% corresponds to 550 clips with a total of 55k frames.', '1701.01370-2-53-3': 'Figure [REF] (left) shows the increase in performance for both segmentation and depth as we increase training data.', '1701.01370-2-53-4': 'Results are plotted on synthetic and Human3.6M test sets with and without fine-tuning.', '1701.01370-2-53-5': 'The performance gain is higher at the beginning of all curves.', '1701.01370-2-53-6': 'There is some saturation, training with 55k frames is sufficient, and it is more evident on Human3.6M after a certain point.', '1701.01370-2-53-7': 'We explain this by the lack of diversity in Human3.6M test set and the redundancy of MoCap poses.', '1701.01370-2-54-0': 'Clothing variation.', '1701.01370-2-54-1': 'Similarly, we study what happens when we add more clothing.', '1701.01370-2-54-2': 'We train with a subset of 100 clips containing only 1, 10 or 100 different clothings (out of a total of 930), because the dataset has maximum 100 clips for a given clothing and we want to use same number of training clips, i.e., 1 clothing with 100 clips, 10 clothings with 10 clips each and 100 clothings with 1 clip each.', '1701.01370-2-54-3': 'Figure [REF] (right) shows the increase in performance for both tasks as we increase clothing variation.', '1701.01370-2-54-4': 'In the case of fine-tuning, the impact gets less prominent because training and test images of Human3.6M are recorded in the same room.', '1701.01370-2-54-5': 'Moreover, there is only one subject in the test set, ideally such experiment should be evaluated on more diverse data.', '1701.01370-2-55-0': 'MoCap variation.', '1701.01370-2-55-1': 'Pose distribution depends on the MoCap source.', '1701.01370-2-55-2': 'To experiment with the effect of having similar poses in training as in test, we rendered synthetic data using Human3.6M MoCap.', '1701.01370-2-55-3': 'Segmentation and depth networks pre-trained on this data (IOU: 48.11%, RMSE: 2.44) outperform the ones pre-trained on CMU MoCap (42.82%, 2.57) when tested on real Human3.6M.', '1701.01370-2-55-4': 'It is important to have diverse MoCap and to match the target distribution.', '1701.01370-2-55-5': 'Note that we exclude the Human3.6M synthetic data in Section [REF] to address the more generic case where there is no dataset specific MoCap data available.', '1701.01370-2-56-0': '# Conclusions', '1701.01370-2-57-0': 'In this study, we have shown successful large-scale training of CNNs from synthetically generated images of people.', '1701.01370-2-57-1': 'We have addressed two tasks, namely, human body part segmentation and depth estimation, for which large-scale manual annotation is infeasible.', '1701.01370-2-57-2': 'Our generated synthetic dataset comes with rich pixel-wise ground truth information and can potentially be used for other tasks than considered here.', '1701.01370-2-57-3': 'Unlike many existing synthetic datasets, the focus of SURREAL is on the realistic rendering of people, which is a challenging task.', '1701.01370-2-57-4': 'In our future work, we plan to integrate the person into the background in a more realistic way by taking into account the lighting and the 3D scene layout.', '1701.01370-2-57-5': 'We also plan to augment the data with more challenging scenarios such as occlusions and multiple people.'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1701.01370

{'1701.01370-3-0-0': 'Estimating human pose, shape, and motion from images and videos are fundamental challenges with many applications.', '1701.01370-3-0-1': 'Recent advances in 2D human pose estimation use large amounts of manually-labeled training data for learning convolutional neural networks (CNNs).', '1701.01370-3-0-2': 'Such data is time consuming to acquire and difficult to extend.', '1701.01370-3-0-3': 'Moreover, manual labeling of 3D pose, depth and motion is impractical.', '1701.01370-3-0-4': 'In this work we present SURREAL (Synthetic hUmans foR REAL tasks): a new large-scale dataset with synthetically-generated but realistic images of people rendered from 3D sequences of human motion capture data.', '1701.01370-3-0-5': 'We generate more than 6 million frames together with ground truth pose, depth maps, and segmentation masks.', '1701.01370-3-0-6': 'We show that CNNs trained on our synthetic dataset allow for accurate human depth estimation and human part segmentation in real RGB images.', '1701.01370-3-0-7': 'Our results and the new dataset open up new possibilities for advancing person analysis using cheap and large-scale synthetic data.', '1701.01370-3-1-0': '# Introduction', '1701.01370-3-2-0': 'Convolutional Neural Networks provide significant gains to problems with large amounts of training data.', '1701.01370-3-2-1': 'In the field of human analysis, recent datasets [CITATION] now gather a sufficient number of annotated images to train networks for 2D human pose estimation [CITATION].', '1701.01370-3-2-2': 'Other tasks such as accurate estimation of human motion, depth and body-part segmentation are lagging behind as manual supervision for such problems at large scale is prohibitively expensive.', '1701.01370-3-3-0': 'Images of people have rich variation in poses, clothing, hair styles, body shapes, occlusions, viewpoints, motion blur and other factors.', '1701.01370-3-3-1': 'Many of these variations, however, can be synthesized using existing 3D motion capture (MoCap) data [CITATION] and modern tools for realistic rendering.', '1701.01370-3-3-2': 'Provided sufficient realism, such an approach would be highly useful for many tasks as it can generate rich ground truth in terms of depth, motion, body-part segmentation and occlusions.', '1701.01370-3-4-0': 'Although synthetic data has been used for many years, realism has been limited.', '1701.01370-3-4-1': 'In this work we present SURREAL: a new large-scale dataset with synthetically-generated but realistic images of people.', '1701.01370-3-4-2': 'Images are rendered from 3D sequences of MoCap data.', '1701.01370-3-4-3': 'To ensure realism, the synthetic bodies are created using the SMPL body model [CITATION], whose parameters are fit by the MoSh [CITATION] method given raw 3D MoCap marker data.', '1701.01370-3-4-4': 'We randomly sample a large variety of viewpoints, clothing and lighting.', '1701.01370-3-4-5': 'SURREAL contains more than 6 million frames together with ground truth pose, depth maps, and segmentation masks.', '1701.01370-3-4-6': 'We show that CNNs trained on synthetic data allow for accurate human depth estimation and human part segmentation in real RGB images, see Figure [REF].', '1701.01370-3-4-7': 'Here, we demonstrate that our dataset, while being synthetic, reaches the level of realism necessary to support training for multiple complex tasks.', '1701.01370-3-4-8': 'This opens up opportunities for training deep networks using graphics techniques available now.', '1701.01370-3-4-9': 'SURREAL dataset is publicly available together with the code to generate synthetic data and to train models for body part segmentation and depth estimation [CITATION].', '1701.01370-3-5-0': 'The rest of this paper is organized as follows.', '1701.01370-3-5-1': 'Section [REF] reviews related work.', '1701.01370-3-5-2': 'Section [REF] presents our approach for generating realistic synthetic videos of people.', '1701.01370-3-5-3': 'In Section [REF] we describe our CNN architecture for human body part segmentation and depth estimation.', '1701.01370-3-5-4': 'Section [REF] reports experiments.', '1701.01370-3-5-5': 'We conclude in Section [REF].', '1701.01370-3-6-0': '# Related work', '1701.01370-3-7-0': 'Knowledge transfer from synthetic to real images has been recently studied with deep neural networks.', '1701.01370-3-7-1': 'Dosovitskiy [CITATION] learn a CNN for optical flow estimation using synthetically generated images of rendered 3D moving chairs.', '1701.01370-3-7-2': 'Peng [CITATION] study the effect of different visual cues such as object/background texture and color when rendering synthetic 3D objects for object detection task.', '1701.01370-3-7-3': 'Similarly, [CITATION] explores rendering 3D objects to perform viewpoint estimation.', '1701.01370-3-7-4': 'Fanello [CITATION] render synthetic infrared images of hands and faces to predict depth and parts.', '1701.01370-3-7-5': 'Recently, Gaidon [CITATION] have released the Virtual KITTI dataset with synthetically generated videos of cars to study multi-object tracking.', '1701.01370-3-8-0': 'Several works focused on creating synthetic images of human bodies for learning 2D pose estimation [CITATION], 3D pose estimation [CITATION], pedestrian detection [CITATION], and action recognition [CITATION].', '1701.01370-3-8-1': 'Pishchulin [CITATION] generate synthetic images with a game engine.', '1701.01370-3-8-2': 'In [CITATION], they deform 2D images with a 3D model.', '1701.01370-3-8-3': 'More recently, Rogez and Schmid [CITATION] use an image-based synthesis engine to augment existing real images.', '1701.01370-3-8-4': 'Ghezelghieh [CITATION] render synthetic images with 10 simple body models with an emphasis on upright people; however, the main challenge using existing MoCap data for training is to generalize to poses that are not upright.', '1701.01370-3-8-5': 'Human3.6M dataset [CITATION] presents realistic rendering of people in mixed reality settings; however, the approach to create these is expensive.', '1701.01370-3-9-0': 'A similar direction has been explored in [CITATION].', '1701.01370-3-9-1': 'In [CITATION], action recognition is addressed with synthetic human trajectories from MoCap data.', '1701.01370-3-9-2': '[CITATION] train CNNs with synthetic depth images.', '1701.01370-3-9-3': 'EgoCap [CITATION] creates a dataset by augmenting egocentric sequences with background.', '1701.01370-3-10-0': 'The closest work to this paper is [CITATION], where the authors render large-scale synthetic images for predicting 3D pose with CNNs.', '1701.01370-3-10-1': 'Our dataset differs from [CITATION] by having a richer, per-pixel ground truth, thus allowing to train for pixel-wise predictions and multi-task scenarios.', '1701.01370-3-10-2': 'In addition, we argue that the realism in our synthetic images is better (see sample videos in [CITATION]), thus resulting in a smaller gap between features learned from synthetic and real images.', '1701.01370-3-10-3': 'The method in [CITATION] heavily relies on real images as input in their training with domain adaptation.', '1701.01370-3-10-4': 'This is not the case for our synthetic training.', '1701.01370-3-10-5': 'Moreover, we render video sequences which can be used for temporal modeling.', '1701.01370-3-11-0': 'Our dataset presents several differences with existing synthetic datasets.', '1701.01370-3-11-1': 'It is the first large-scale person dataset providing depth, part segmentation and flow ground truth for synthetic RGB frames.', '1701.01370-3-11-2': 'Other existing datasets are used either for taking RGB image as input and training only for 2D/3D pose, or for taking depth/infrared images as input and training for depth/parts segmentation.', '1701.01370-3-11-3': 'In this paper, we show that photo-realistic renderings of people under large variations in shape, texture, viewpoint and pose can help solving pixel-wise human labeling tasks.', '1701.01370-3-12-0': '# Data generation', '1701.01370-3-13-0': 'This section presents our SURREAL (Synthetic hUmans foR REAL tasks) dataset and describes key steps for its generation (Section [REF]).', '1701.01370-3-13-1': 'We also describe how we obtain ground truth data for real MoCap sequences (Section [REF]).', '1701.01370-3-14-0': '## Synthetic humans', '1701.01370-3-15-0': 'Our pipeline for generating synthetic data is illustrated in Figure [REF].', '1701.01370-3-15-1': 'A human body with a random 3D pose, random shape and random texture is rendered from a random view-point for some random lighting and a random background image.', '1701.01370-3-15-2': 'Below we define what "random" means in all these cases.', '1701.01370-3-15-3': 'Since the data is synthetic, we also generate ground truth depth maps, optical flow, surface normals, human part segmentations and joint locations (both 2D and 3D).', '1701.01370-3-15-4': 'As a result, we obtain 6.5 million frames grouped into [MATH] continuous image sequences.', '1701.01370-3-15-5': 'See Table [REF] for more statistics, Section [REF] for the description of the synthetic train/test split, and Figure [REF] for samples from the SURREAL dataset.', '1701.01370-3-16-0': 'Body model.', '1701.01370-3-16-1': 'Synthetic bodies are created using the SMPL body model [CITATION].', '1701.01370-3-16-2': 'SMPL is a realistic articulated model of the body created from thousands of high-quality 3D scans, which decomposes body deformations into pose (kinematic deformations due to skeletal posture) and shape (body deformations intrinsic to a particular person that make them different from others).', '1701.01370-3-16-3': 'SMPL is compatible with most animation packages like Blender [CITATION].', '1701.01370-3-16-4': 'SMPL deformations are modeled as a combination of linear blend skinning and linear blendshapes defined by principal components of body shape variation.', '1701.01370-3-16-5': 'SMPL pose and shape parameters are converted to a triangulated mesh using Blender, which then applies texture, shading and adds a background to generate the final RGB output.', '1701.01370-3-17-0': 'Body shape.', '1701.01370-3-17-1': 'In order to render varied, but realistic, body shapes we make use of the CAESAR dataset [CITATION], which was used to train SMPL.', '1701.01370-3-17-2': 'To create a body shape, we select one of the CAESAR subjects at random and approximate their shape with the first 10 SMPL shape principal components.', '1701.01370-3-17-3': 'Ten shape components explain more than [MATH] of the shape variance in CAESAR (at the resolution of our mesh) and produce quite realistic body shapes.', '1701.01370-3-18-0': 'Body pose.', '1701.01370-3-18-1': 'To generate images of people in realistic poses, we take motion capture data from the CMU MoCap database [CITATION].', '1701.01370-3-18-2': 'CMU MoCap contains more than 2000 sequences of 23 high-level action categories, resulting in more than 10 hours of recorded 3D locations of body markers.', '1701.01370-3-19-0': 'It is often challenging to realistically and automatically retarget MoCap skeleton data to a new model.', '1701.01370-3-19-1': 'For this reason we do not use the skeleton data but rather use MoSh [CITATION] to fit the SMPL parameters that best explain raw 3D MoCap marker locations.', '1701.01370-3-19-2': 'This gives both the 3D shape of the subject and the articulated pose parameters of SMPL.', '1701.01370-3-19-3': 'To increase the diversity, we replace the estimated 3D body shape with a set of randomly sampled body shapes.', '1701.01370-3-20-0': 'We render each CMU MoCap sequence three times using different random parameters.', '1701.01370-3-20-1': 'Moreover, we divide the sequences into clips of 100 frames with 30%, 50% and 70% overlaps for these three renderings.', '1701.01370-3-20-2': 'Every pose of the sequence is rendered with consistent parameters (i.e. body shape, clothing, light, background etc.) within each clip.', '1701.01370-3-21-0': 'Human texture.', '1701.01370-3-21-1': 'We use two types of real scans for the texture of body models.', '1701.01370-3-21-2': 'First, we extract SMPL texture maps from CAESAR scans, which come with a color texture per 3D point.', '1701.01370-3-21-3': 'These maps vary in skin color and person identities, however, their quality is often low due to the low resolution, uniform tight-fitting clothing, and visible markers placed on the face and the body.', '1701.01370-3-21-4': 'Anthropometric markers are automatically removed from the texture images and inpainted.', '1701.01370-3-21-5': 'To provide more variety, we extract a second set of textures obtained from 3D scans of subjects with normal clothing.', '1701.01370-3-21-6': 'These scans are registered with 4Cap as in [CITATION].', '1701.01370-3-21-7': 'The texture of real clothing substantially increases the realism of generated images, even though SMPL does not model 3D deformations of clothes.', '1701.01370-3-22-0': '[MATH] of our data is rendered with the first set ([MATH] CAESAR textures randomly sampled from [MATH]), and the rest with the second set ([MATH] clothed textures).', '1701.01370-3-22-1': 'To preserve the anonymity of subjects, we replace all faces in the texture maps by the average CAESAR face.', '1701.01370-3-22-2': 'The skin color of this average face is corrected to fit the face skin color of the original texture map.', '1701.01370-3-22-3': 'This corrected average face is blended smoothly with the original map, resulting in a realistic and anonymized body texture.', '1701.01370-3-23-0': 'Light.', '1701.01370-3-23-1': 'The body is illuminated using Spherical Harmonics with [MATH] coefficients [CITATION].', '1701.01370-3-23-2': 'The coefficients are randomly sampled from a uniform distribution between [MATH] and [MATH], apart from the ambient illumination coefficient (which has a minimum value of [MATH]) and the vertical illumination component, which is biased to encourage the illumination from above.', '1701.01370-3-23-3': 'Since Blender does not provide Spherical Harmonics illumination, a spherical harmonic shader for the body material was implemented in Open Shading Language.', '1701.01370-3-24-0': 'Camera.', '1701.01370-3-24-1': 'The projective camera has a resolution of [MATH], focal length of [MATH]mm and sensor size of [MATH]mm.', '1701.01370-3-24-2': 'To generate images of the body in a wide range of positions, we take 100-frame MoCap sub-sequences and, in the first frame, render the body so that the center of the viewport points to the pelvis of the body, at a random distance (sampled from a normal distribution with [MATH] meters mean, [MATH] meter deviation) with a random yaw angle.', '1701.01370-3-24-3': 'The remainder of the sequence then effectively produces bodies in a range of locations relative to the static camera.', '1701.01370-3-25-0': 'Background.', '1701.01370-3-25-1': 'We render the person on top of a static background image.', '1701.01370-3-25-2': 'To ensure that the backgrounds are reasonably realistic and do not include other people, we sample from a subset of LSUN dataset [CITATION] that includes total of 400K images from the categories kitchen, living room, bedroom and dining room.', '1701.01370-3-26-0': 'Ground truth.', '1701.01370-3-26-1': 'We perform multiple rendering passes in Blender to generate different types of per-pixel ground truth.', '1701.01370-3-26-2': 'The material pass generates pixel-wise segmentation of rendered body parts, given different material indices assigned to different parts of our body model.', '1701.01370-3-26-3': 'The velocity pass, typically used to simulate motion blur, provides us with a render simulating optical flow.', '1701.01370-3-26-4': 'The depth and normal passes, used for emulating effects like fog, bokeh or for performing shading, produce per-pixel depth maps and normal maps.', '1701.01370-3-26-5': 'The final texture rendering pass overlays the shaded, textured body over the random background.', '1701.01370-3-26-6': 'Together with this data we save camera and lighting parameters as well as the 2D/3D positions of body joints.', '1701.01370-3-27-0': '## Generating ground truth for real human data', '1701.01370-3-28-0': 'Human3.6M dataset [CITATION] provides ground truth for 2D and 3D human poses.', '1701.01370-3-28-1': 'Additionally, a subset of the dataset (H80K) [CITATION] has segmentation annotation, but the definition of parts is different from the SMPL body parts used for our training.', '1701.01370-3-28-2': 'We complement this ground truth and generate predicted SMPL body-part segmentation and depth maps for people in Human3.6M for all frames.', '1701.01370-3-28-3': 'Here again we use MoSh [CITATION] to fit the SMPL body shape and pose to the raw MoCap marker data.', '1701.01370-3-28-4': 'This provides a good fit of the model to the shape and the pose of real bodies.', '1701.01370-3-28-5': 'Given the provided camera calibration, we project models to images.', '1701.01370-3-28-6': 'We then render the ground truth segmentation, depth, and 2D/3D joints as above, while ensuring correspondence with real pixel values in the dataset.', '1701.01370-3-28-7': 'The depth is different from the time-of-flight (depth) data provided by the official dataset.', '1701.01370-3-28-8': 'These MoSh fits provide a form of approximate "ground truth".', '1701.01370-3-28-9': 'See Figures [REF] and [REF] for generated examples.', '1701.01370-3-28-10': 'We use this for evaluation on the test set as well as for the baseline where we train only on real data, and also for fine-tuning our models pre-trained on synthetic data.', '1701.01370-3-28-11': 'In the rest of the paper, all frames from the synthetic training set are used for synthetic pre-training.', '1701.01370-3-29-0': '# Approach', '1701.01370-3-30-0': 'In this section, we present our approach for human body part segmentation [CITATION] and human depth estimation [CITATION], which we train with synthetic and/or real data, see Section [REF] for the evaluation.', '1701.01370-3-31-0': 'Our approach builds on the stacked hourglass network architecture introduced originally for 2D pose estimation problem [CITATION].', '1701.01370-3-31-1': 'This network involves several repetitions of contraction followed by expansion layers which have skip connections to implicitly model spatial relations from different resolutions that allows bottom-up and top-down structured prediction.', '1701.01370-3-31-2': "The convolutional layers with residual connections and 8 'hourglass' modules are stacked on top of each other, each successive stack taking the previous stack's prediction as input.", '1701.01370-3-31-3': 'The reader is referred to [CITATION] for more details.', '1701.01370-3-31-4': 'A variant of this network has been used for scene depth estimation [CITATION].', '1701.01370-3-31-5': 'We choose this architecture because it can infer pixel-wise output by taking into account human body structure.', '1701.01370-3-32-0': 'Our network input is a 3-channel RGB image of size [MATH] cropped and scaled to fit a human bounding box using the ground truth.', '1701.01370-3-32-1': 'The network output for each stack has dimensions [MATH] in the case of segmentation (14 classes plus the background) and [MATH] for depth (19 depth classes plus the background).', '1701.01370-3-32-2': 'We use cross-entropy loss defined on all pixels for both segmentation and depth.', '1701.01370-3-32-3': 'The final loss of the network is the sum over 8 stacks.', '1701.01370-3-32-4': 'We train for 50K iterations for synthetic pre-training using the RMSprop algorithm with mini-batches of size 6 and a learning rate of [MATH].', '1701.01370-3-32-5': 'Our data augmentation during training includes random rotations, scaling and color jittering.', '1701.01370-3-33-0': 'We formulate the problem as pixel-wise classification task for both segmentation and depth.', '1701.01370-3-33-1': 'When addressing segmentation, each pixel is assigned to one of the pre-defined 14 human parts, namely head, torso, upper legs, lower legs, upper arms, lower arms, hands, feet (separately for right and left) or to the background class.', '1701.01370-3-33-2': 'Regarding the depth, we align ground-truth depth maps on the z-axis by the depth of the pelvis joint, and then quantize depth values into 19 bins (9 behind and 9 in front of the pelvis).', '1701.01370-3-33-3': 'We set the quantization constant to 45mm to roughly cover the depth extent of common human poses.', '1701.01370-3-33-4': 'The network is trained to classify each pixel into one of the 19 depth bins or background.', '1701.01370-3-33-5': 'At test time, we first upsample feature maps of each class with bilinear interpolation by a factor of 4 to output the original resolution.', '1701.01370-3-33-6': 'Then, each pixel is assigned to the class for which the corresponding channel has the maximum activation.', '1701.01370-3-34-0': '# Experiments', '1701.01370-3-35-0': 'We test our approach on several datasets.', '1701.01370-3-35-1': 'First, we evaluate the segmentation and depth estimation on the test set of our synthetic SURREAL dataset.', '1701.01370-3-35-2': 'Second, we test the performance of segmentation on real images from the Freiburg Sitting People dataset [CITATION].', '1701.01370-3-35-3': 'Next, we evaluate segmentation and depth estimation on real videos from the Human3.6M dataset [CITATION] with available 3D information.', '1701.01370-3-35-4': 'Then, we qualitatively evaluate our approach on the more challenging MPII Human Pose dataset [CITATION].', '1701.01370-3-35-5': 'Finally, we experiment and discuss design choices of the SURREAL dataset.', '1701.01370-3-36-0': '## Evaluation measures', '1701.01370-3-37-0': 'We use intersection over union (IOU) and pixel accuracy measures for evaluating the segmentation approach.', '1701.01370-3-37-1': 'The final measure is the average over 14 human parts as in [CITATION].', '1701.01370-3-37-2': 'Depth estimation is formulated as a classification problem, but we take into account the continuity when we evaluate.', '1701.01370-3-37-3': 'We compute root-mean-squared-error (RMSE) between the predicted quantized depth value (class) and the ground truth quantized depth on the human pixels.', '1701.01370-3-37-4': 'To interpret the error in real world coordinates, we multiply it by the quantization constant (45mm).', '1701.01370-3-37-5': 'We also report a scale and translation invariant RMSE (st-RMSE) by solving for the best translation and scaling in z-axis to fit the prediction to the ground truth.', '1701.01370-3-37-6': 'Since inferring depth from RGB is ambiguous, this is a common technique used in evaluations [CITATION].', '1701.01370-3-38-0': '## Validation on synthetic images', '1701.01370-3-39-0': 'Train/test split.', '1701.01370-3-39-1': 'To evaluate our methods on synthetic images, we separate [MATH] of the synthetic frames for the test set and train all our networks on the remaining training set.', '1701.01370-3-39-2': 'The split is constructed such that a given CMU MoCap subject is assigned as either train or test.', '1701.01370-3-39-3': 'Whereas some subjects have a large number of instances, some subjects have unique actions, and some actions are very common (walk, run, jump).', '1701.01370-3-39-4': 'Overall, 30 subjects out of 145 are assigned as test.', '1701.01370-3-39-5': '28 test subjects cover all common actions, and 2 have unique actions.', '1701.01370-3-39-6': 'Remaining subjects are used for training.', '1701.01370-3-39-7': 'Although our synthetic images have different body shape and appearance than the subject in the originating MoCap sequence, we still found it appropriate to split by subjects.', '1701.01370-3-39-8': 'We separate a subset of our body shapes, clothing and background images for the test set.', '1701.01370-3-39-9': 'This ensures that our tests are unbiased with regards to appearance, yet are still representative of all actions.', '1701.01370-3-39-10': 'Table [REF] summarizes the number of frames, clips and MoCap sequences in each split.', '1701.01370-3-39-11': 'Clips are the continuous 100-frame sequences where we have the same random body shape, background, clothing, camera and lighting.', '1701.01370-3-39-12': 'A new random set is picked at every clip.', '1701.01370-3-39-13': 'Note that a few sequences have less than 100 frames.', '1701.01370-3-40-0': 'Results on synthetic test set.', '1701.01370-3-40-1': 'The evaluation is performed on the middle frame of each 100-frame clip on the aforementioned held-out synthetic test set, totaling in 12,528 images.', '1701.01370-3-40-2': 'For segmentation, the IOU and pixel accuracy are 69.13% and 80.61%, respectively.', '1701.01370-3-40-3': 'Evaluation of depth estimation gives 72.9mm and 56.3mm for RMSE and st-RMSE errors, respectively.', '1701.01370-3-40-4': 'Figure [REF] shows sample predictions.', '1701.01370-3-40-5': 'For both tasks, the results are mostly accurate on synthetic test images.', '1701.01370-3-40-6': 'However, there exist a few challenging poses (e.g. crawling), test samples with extreme close-up views, and fine details of the hands that are causing errors.', '1701.01370-3-40-7': 'In the following sections, we investigate if similar conclusions can be made for real images.', '1701.01370-3-41-0': '## Segmentation on Freiburg Sitting People', '1701.01370-3-42-0': 'Freiburg Sitting People (FSitting) dataset [CITATION] is composed of 200 high resolution (300x300 pixels) front view images of 6 subjects sitting on a wheel chair.', '1701.01370-3-42-1': 'There are 14 human part annotations available.', '1701.01370-3-42-2': 'See Figure [REF] for sample test images and corresponding ground truth (GT) annotation.', '1701.01370-3-42-3': 'We use the same train/test split as [CITATION], 2 subjects for training and 4 subjects for test.', '1701.01370-3-42-4': 'The amount of data is limited for training deep networks.', '1701.01370-3-42-5': 'We show that our network pre-trained only on synthetic images is already able to segment human body parts.', '1701.01370-3-42-6': 'This shows that the human renderings in the synthetic dataset are representative of the real images, such that networks trained exclusively on synthetic data can generalize quite well to real data.', '1701.01370-3-43-0': 'Table [REF] summarizes segmentation results on FSitting.', '1701.01370-3-43-1': 'We carry out several experiments to understand the gain from synthetic pre-training.', '1701.01370-3-43-2': "For the 'Real' baseline, we train a network from scratch using 2 training subjects.", '1701.01370-3-43-3': 'This network overfits as there are few subjects to learn from and the performance is quite low.', '1701.01370-3-43-4': "Our 'Synth' result is obtained using the network pre-trained on synthetic images without fine-tuning.", '1701.01370-3-43-5': 'We get 51.88% pixel accuracy and 40.1% IOU with this method and clearly outperform training from real images.', '1701.01370-3-43-6': 'Furthermore, fine-tuning (Synth+Real) with 2 training subjects helps significantly.', '1701.01370-3-43-7': 'See Figure [REF] for qualitative results.', '1701.01370-3-43-8': 'Given the little amount for training in FSitting, the fine-tuning converges after 200 iterations.', '1701.01370-3-44-0': 'In [CITATION], the authors introduce a network that outputs a high-resolution segmentation after several layers of upconvolutions.', '1701.01370-3-44-1': 'For a fair comparison, we modify our network to output full resolution by adding one bilinear upsampling layer followed by nonlinearity (ReLU) and a convolutional layer with [MATH] filters that outputs [MATH] instead of [MATH] as explained in Section [REF].', '1701.01370-3-44-2': 'If we fine-tune this network (Synth+Real+up) on FSitting, we improve performance and outperform [CITATION] by a large margin.', '1701.01370-3-44-3': 'Note that [CITATION] trains on the same FSitting training images, but added around 2,800 Pascal images.', '1701.01370-3-44-4': 'Hence they use significantly more manual annotation than our method.', '1701.01370-3-45-0': '## Segmentation and depth on Human3.6M', '1701.01370-3-46-0': 'To evaluate our approach, we need sufficient real data with ground truth annotations.', '1701.01370-3-46-1': 'Such data is expensive to obtain and currently not available.', '1701.01370-3-46-2': 'For this reason, we generate nearly perfect ground truth for images recorded with a calibrated camera and given their MoCap data.', '1701.01370-3-46-3': 'Human3.6M [CITATION] is currently the largest dataset where such information is available.', '1701.01370-3-46-4': 'There are 3.6 million frames from 4 cameras.', '1701.01370-3-46-5': 'We use subjects S1, S5, S6, S7, S8 for training, S9 for validation and S11 for testing as in [CITATION], but from all 4 cameras.', '1701.01370-3-46-6': 'Note that this is different from the official train/test split [CITATION].', '1701.01370-3-46-7': 'Each subject performs each of the 15 actions twice.', '1701.01370-3-46-8': 'We use all frames from one of the two instances of each action for training, and every 64[MATH] frame from all instances for testing.', '1701.01370-3-46-9': 'The frames have resolution [MATH] pixels, we assume a [MATH] cropped human bounding box is given to reduce computational complexity.', '1701.01370-3-46-10': 'We evaluate the performance of both segmentation and depth, and compare with the baseline for which we train a network on real images only.', '1701.01370-3-47-0': 'Segmentation.', '1701.01370-3-47-1': 'Table [REF] summarizes the parts segmentation results on Human3.6M. Note that these are not comparable to the results in [CITATION] both because they assume the background segment is given and our ground truth segmentation data is not part of the official release (see Section [REF]).', '1701.01370-3-47-2': 'We report both the mean over 14 human parts (fg) and the mean together with the background class (fg+bg).', '1701.01370-3-47-3': 'Training on real images instead of synthetic images increases IOU by 3.4% and pixel accuracy by 2.14%.', '1701.01370-3-47-4': 'This is expected because the training distribution matches the test distribution in terms of background, camera position and action categories (i.e. poses).', '1701.01370-3-47-5': 'Furthermore, the amount of real data is sufficient to perform CNN training.', '1701.01370-3-47-6': "However, since there are very few subjects available, we see that the network doesn't generalize to different clothing.", '1701.01370-3-47-7': "In Figure [REF], the 'Real' baseline has the border between shoulders and upper arms exactly on the T-shirt boundaries.", '1701.01370-3-47-8': 'This reveals that the network learns about skin color rather than actual body parts.', '1701.01370-3-47-9': 'Our pre-trained network (Synth) performs reasonably well, even though the pose distribution in our MoCap is quite different than that of Human3.6M.', '1701.01370-3-47-10': "When we fine-tune the network with real images from Human3.6M (Synth+Real), the model predicts very accurate segmentations and outperforms the 'Real' baseline by a large margin.", '1701.01370-3-47-11': 'Moreover, our model is capable of distinguishing left and right most of the time on all 4 views since it has been trained with randomly sampled views.', '1701.01370-3-48-0': 'Depth estimation.', '1701.01370-3-48-1': 'Depth estimation results on Human3.6M for various poses and viewpoints are illustrated in Figure [REF].', '1701.01370-3-48-2': 'Here, the pre-trained network fails at the very challenging poses, although it still captures partly correct estimates (first row).', '1701.01370-3-48-3': 'Fine-tuning on real data compensates for these errors and refines estimations.', '1701.01370-3-48-4': 'In Table [REF], we show RMSE error measured on foreground pixels, together with the scale-translation invariant version (see Section [REF]).', '1701.01370-3-48-5': 'We also report the error only on known 2D joints (PoseRMSE) to have an idea of how well a 3D pose estimation model would work based on the depth predictions.', '1701.01370-3-48-6': 'One would need to handle occluded joints to infer 3D locations of all joints, and this is beyond the scope of the current paper.', '1701.01370-3-49-0': '## Qualitative results on MPII Human Pose', '1701.01370-3-50-0': 'FSitting and Human3.6M are relatively simple datasets with limited background clutter, few subjects, single person per image, full body visible.', '1701.01370-3-50-1': 'In this section, we test the generalization of our model on more challenging images.', '1701.01370-3-50-2': 'MPII Human Pose [CITATION] is one of the largest datasets with diverse viewpoints and clutter.', '1701.01370-3-50-3': 'However, this dataset has no ground truth for part segmentation nor depth.', '1701.01370-3-50-4': 'Therefore, we qualitatively show our predictions.', '1701.01370-3-50-5': 'Figure [REF] illustrates several success and failure cases.', '1701.01370-3-50-6': 'Our model generalizes reasonably well, except when there are multiple people close to each other and extreme viewpoints, which have not appeared during training.', '1701.01370-3-50-7': 'It is interesting to note that although lower body occlusions and cloth shapes are not present in synthetic training, the models perform accurately in such cases, see Figure [REF] caption.', '1701.01370-3-51-0': '## Design choices', '1701.01370-3-52-0': "We did several experiments to answer questions such as 'How much data should we synthesize?'", '1701.01370-3-52-1': ", 'Is CMU MoCap enough?'", '1701.01370-3-52-2': ", 'What's the effect having clothing variation?'", '1701.01370-3-53-0': 'Amount of data.', '1701.01370-3-53-1': 'We plot the performance as a function of training data size.', '1701.01370-3-53-2': 'We train with a random subset of [MATH], [MATH], [MATH], [MATH]% of the 55K training clips using all frames of the selected clips, i.e., [MATH]% corresponds to 550 clips with a total of 55k frames.', '1701.01370-3-53-3': 'Figure [REF] (left) shows the increase in performance for both segmentation and depth as we increase training data.', '1701.01370-3-53-4': 'Results are plotted on synthetic and Human3.6M test sets with and without fine-tuning.', '1701.01370-3-53-5': 'The performance gain is higher at the beginning of all curves.', '1701.01370-3-53-6': 'There is some saturation, training with 55k frames is sufficient, and it is more evident on Human3.6M after a certain point.', '1701.01370-3-53-7': 'We explain this by the lack of diversity in Human3.6M test set and the redundancy of MoCap poses.', '1701.01370-3-54-0': 'Clothing variation.', '1701.01370-3-54-1': 'Similarly, we study what happens when we add more clothing.', '1701.01370-3-54-2': 'We train with a subset of 100 clips containing only 1, 10 or 100 different clothings (out of a total of 930), because the dataset has maximum 100 clips for a given clothing and we want to use same number of training clips, i.e., 1 clothing with 100 clips, 10 clothings with 10 clips each and 100 clothings with 1 clip each.', '1701.01370-3-54-3': 'Figure [REF] (right) shows the increase in performance for both tasks as we increase clothing variation.', '1701.01370-3-54-4': 'In the case of fine-tuning, the impact gets less prominent because training and test images of Human3.6M are recorded in the same room.', '1701.01370-3-54-5': 'Moreover, there is only one subject in our test set, ideally such experiment should be evaluated on more diverse data.', '1701.01370-3-55-0': 'MoCap variation.', '1701.01370-3-55-1': 'Pose distribution depends on the MoCap source.', '1701.01370-3-55-2': 'To experiment with the effect of having similar poses in training as in test, we rendered synthetic data using Human3.6M MoCap.', '1701.01370-3-55-3': 'Segmentation and depth networks pre-trained on this data (IOU: 48.11%, RMSE: 2.44) outperform the ones pre-trained on CMU MoCap (42.82%, 2.57) when tested on real Human3.6M.', '1701.01370-3-55-4': 'It is important to have diverse MoCap and to match the target distribution.', '1701.01370-3-55-5': 'Note that we exclude the Human3.6M synthetic data in Section [REF] to address the more generic case where there is no dataset specific MoCap data available.', '1701.01370-3-56-0': '# Conclusions', '1701.01370-3-57-0': 'In this study, we have shown successful large-scale training of CNNs from synthetically generated images of people.', '1701.01370-3-57-1': 'We have addressed two tasks, namely, human body part segmentation and depth estimation, for which large-scale manual annotation is infeasible.', '1701.01370-3-57-2': 'Our generated synthetic dataset comes with rich pixel-wise ground truth information and can potentially be used for other tasks than considered here.', '1701.01370-3-57-3': 'Unlike many existing synthetic datasets, the focus of SURREAL is on the realistic rendering of people, which is a challenging task.', '1701.01370-3-57-4': 'In our future work, we plan to integrate the person into the background in a more realistic way by taking into account the lighting and the 3D scene layout.', '1701.01370-3-57-5': 'We also plan to augment the data with more challenging scenarios such as occlusions and multiple people.'}

1512.09005

{'1512.09005-1-0-0': 'We compute the facets of the effective cones of divisors on the blow-up of [MATH] up to five lines in general position.', '1512.09005-1-0-1': 'We prove that up to four lines these threefolds are log Fano.', '1512.09005-1-1-0': '# Introduction', '1512.09005-1-2-0': 'In classical algebraic geometry, the study of linear systems in [MATH] of hypersurfaces of degree [MATH] with prescribed multiplicities at a collection of [MATH] points in general position was investigated by many authors for over a century (see [CITATION] for an overview).', '1512.09005-1-3-0': 'We will briefly recall the most important results in this area.', '1512.09005-1-3-1': 'The well-known Alexander-Hirschowitz theorem [CITATION] classifies completely the dimensionality problem for linear systems with double points (see [CITATION] for more recent and simplified proofs).', '1512.09005-1-3-2': 'Besides this theorem general results about linear systems are rare and few things are known.', '1512.09005-1-3-3': 'For arbitrary number of points in the projective space [MATH] the dimensionality problem for linear was analysed in the following papers [CITATION], [CITATION], or [CITATION].', '1512.09005-1-3-4': 'On the one hand, linear systems can be studied via techniques of commutative algebra.', '1512.09005-1-3-5': 'In fact, the dimensionality problem is related via apolarity to the Froberg-Iarrobino Weak and Strong conjectures [CITATION].', '1512.09005-1-3-6': 'These conjectures give a predicted value for the Hilbert series of an ideal generated by [MATH] general powers of linear forms in the polynomial ring with [MATH] variables.', '1512.09005-1-3-7': 'On the other hand, linear systems correspond in birational geometry to the space of global sections of line bundles on the blown-up space [MATH] in points.', '1512.09005-1-3-8': 'Form this perspective, the dimensions of such linear systems reduces to vanishing theorems of divisors on blown-up spaces in points and along higher dimensional cycles of the base locus.', '1512.09005-1-3-9': 'Vanishing theorems are important in algebraic geometry because they are related to positivity properties of divisors.', '1512.09005-1-3-10': 'In this direction, there are positivity conjectures in birational geometry that involve complete understanding of cohomology groups of divisors (see [CITATION] for an introduction).', '1512.09005-1-4-0': 'For a small number of points, namely whenever [MATH], understanding cohomology of divisors on the blown-up [MATH] in [MATH] points is easier since this space is a Mori dream space.', '1512.09005-1-4-1': 'A projective variety [MATH] is called a Mori dream space if its Cox ring is finitely generated.', '1512.09005-1-4-2': 'The Cox ring of an algebraic variety naturally generalizes the homogeneous coordinate ring of the projective space.', '1512.09005-1-4-3': 'Mori dream spaces are generalizations of toric varieties that have a polynomial Cox ring and their geometry can be encoded by combinatorial data.', '1512.09005-1-4-4': 'For example, Mori dream spaces have rational polyhedral effective cone.', '1512.09005-1-4-5': 'In other words, the cone of effective divisors can be described as intersection of half-spaces in [MATH], the Neron-Severi group of [MATH] tensored with the real numbers.', '1512.09005-1-5-0': 'Motivated by the study of the classical interpolation problems in [MATH] started by the Italian school of algebraic geometry, in this article we study the blown-up projective space [MATH] in a collection of [MATH] lines in general position [MATH].', '1512.09005-1-5-1': 'We give a description of the cone of effective divisors whenever [MATH] is bounded above by five.', '1512.09005-1-5-2': 'The technique for finding the facets of the effective cone was first developed for blown-up [MATH] in [MATH] points in [CITATION].', '1512.09005-1-5-3': 'Moreover in this article we prove that these threefolds are log Fano, hence Mori dream spaces, for a number of lines bounded above by four.', '1512.09005-1-6-0': 'These notes are organized as follows.', '1512.09005-1-6-1': 'In Section [REF] we introduce the notation that will be used throughout.', '1512.09005-1-6-2': 'In Section [REF] we describe cycles of the base locus of linear systems of divisors on the blown-up space in a collection of lines in general position.', '1512.09005-1-6-3': 'Section [REF] contains the main results of these notes.'}

{'1512.09005-2-0-0': 'We compute the facets of the effective cones of divisors on the blow-up of [MATH] in up to five lines in general position.', '1512.09005-2-0-1': 'We prove that up to six lines these threefolds are weak Fano and hence Mori Dream Spaces.', '1512.09005-2-1-0': '# Introduction', '1512.09005-2-2-0': 'In classical algebraic geometry, the study of linear systems in [MATH] of hypersurfaces of degree [MATH] with prescribed multiplicities at a collection of [MATH] points in general position was investigated by many authors for over a century (see [CITATION] for an overview).', '1512.09005-2-3-0': 'We will briefly recall the most important results in this area.', '1512.09005-2-3-1': 'The well-known Alexander-Hirschowitz theorem [CITATION] classifies completely the dimensionality problem for linear systems with double points (see [CITATION] for more recent and simplified proofs).', '1512.09005-2-3-2': 'Besides this theorem, general results about linear systems are rare and few things are known.', '1512.09005-2-3-3': 'For arbitrary number of points in the projective space [MATH], the dimensionality problem for linear systems was analysed in the articles [CITATION].', '1512.09005-2-3-4': 'On the one hand, linear systems can be studied via techniques of commutative algebra.', '1512.09005-2-3-5': 'In fact, the dimensionality problem is related via apolarity to the Froberg-Iarrobino Weak and Strong conjectures [CITATION].', '1512.09005-2-3-6': 'These conjectures give a predicted value for the Hilbert series of an ideal generated by [MATH] general powers of linear forms in the polynomial ring with [MATH] variables.', '1512.09005-2-3-7': 'On the other hand, linear systems correspond in birational geometry to the space of global sections of line bundles on the blown-up space [MATH] in points.', '1512.09005-2-3-8': 'Form this perspective, the dimensionality problem of such linear systems reduces to vanishing theorems of divisors on blown-up spaces in points and along higher dimensional cycles of the base locus.', '1512.09005-2-3-9': 'Vanishing theorems are important in algebraic geometry because they are related to positivity properties of divisors.', '1512.09005-2-3-10': 'In this direction, there are positivity conjectures in birational geometry that involve complete understanding of cohomology groups of divisors (see [CITATION] for an introduction).', '1512.09005-2-4-0': 'For a small number of points, namely whenever [MATH], understanding the cohomology of divisors on the blown-up [MATH] in [MATH] points is easier since this space is a Mori dream space, see [CITATION].', '1512.09005-2-4-1': 'A projective variety [MATH] is called a Mori dream space if its Cox ring is finitely generated.', '1512.09005-2-4-2': 'The Cox ring of an algebraic variety naturally generalizes the homogeneous coordinate ring of the projective space.', '1512.09005-2-4-3': 'Mori dream spaces are generalizations of toric varieties that have a polynomial Cox ring and their geometry can be encoded by combinatorial data.', '1512.09005-2-4-4': 'For example, Mori dream spaces have rational polyhedral effective cone.', '1512.09005-2-4-5': 'In other words, the cone of effective divisors can be described as intersection of half-spaces in [MATH], the Neron-Severi group of [MATH] tensored with the real numbers.', '1512.09005-2-5-0': 'Motivated by the study of the classical interpolation problems in [MATH] started by the Italian school of algebraic geometry, in this article we study the blown-up projective space [MATH] in a collection of [MATH] lines in general position [MATH].', '1512.09005-2-5-1': 'We give a description of the cone of effective divisors whenever [MATH] is bounded above by five.', '1512.09005-2-5-2': 'The technique for finding the facets of the effective cone was first developed for blown-up [MATH] in [MATH] points in [CITATION].', '1512.09005-2-6-0': 'Moreover, in this article we prove that these threefolds are weak Fano, hence Mori dream spaces, for a number of lines bounded above by six.', '1512.09005-2-6-1': 'We also prove that for [MATH] these threefolds are not weak Fano.', '1512.09005-2-7-0': 'These notes are organized as follows.', '1512.09005-2-7-1': 'In Section [REF] we introduce the notation that will be used throughout.', '1512.09005-2-7-2': 'In Section [REF] we describe cycles of the base locus of linear systems of divisors on the blown-up space in a collection of lines in general position.', '1512.09005-2-7-3': 'In section [REF] we prove the first main result of these notes, i.e. we bound the number of blown-up general lines to obtain a weak Fano threefolds.', '1512.09005-2-7-4': 'Section [REF] contains the second main result of these notes, a complete description of the effective cone of the blown-up projective space at up to five general lines.'}

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[]

[['1512.09005-1-0-1', '1512.09005-2-0-1'], ['1512.09005-1-3-3', '1512.09005-2-3-3'], ['1512.09005-1-6-3', '1512.09005-2-7-3'], ['1512.09005-1-6-3', '1512.09005-2-7-4']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1512.09005

1505.04226

{'1505.04226-1-0-0': 'We generalize the Gubser solution of viscous hydrodynamics by including the finite density effect and analytically compute the flow harmonics [MATH].', '1505.04226-1-0-1': 'We explicitly show how [MATH] and their viscous corrections depend on the chemical potential.', '1505.04226-1-0-2': 'The difference in [MATH] between particles and antiparticles is also analytically computed and shown to be proportional to various chemical potentials and the viscosity.', '1505.04226-1-0-3': 'Excellent agreement is obtained between the results and the available experimental data from the SPS, RHIC and the LHC.', '1505.04226-1-1-0': '# Introduction', '1505.04226-1-2-0': 'Relativistic hydrodynamics is a general theoretical framework to describe the collective dynamics of high-energy systems near local thermal equilibrium.', '1505.04226-1-2-1': "Its first application to hadron physics dates back to Landau's attempt to describe multi-particle production in hadron-hadron collisions [CITATION].", '1505.04226-1-2-2': 'It has become a topic of great interest since the discovery of the quark-gluon plasma (QGP) as a nearly-perfect fluid in the "Little Bangs" at BNL Relativistic Heavy Ion Collider (RHIC) [CITATION] and CERN Large Hadron collider (LHC) [CITATION].', '1505.04226-1-2-3': 'This is supported by the observations that the azimuthal momentum anisotropy of hadronic distribution [CITATION], characterized by flow harmonics [MATH], are found to reflect the geometrical anisotropy [MATH] of the overlapping region of two colliding nuclei, and that they are in good quantitative agreement with theoretical estimations.', '1505.04226-1-2-4': 'Nowadays the viscous hydrodynamic modeling is considered as one of the most powerful tools to quantify the QGP medium near the crossover phase transition [CITATION].', '1505.04226-1-3-0': 'The recent Beam Energy Scan (BES) experiments at RHIC pose us intriguing challenges to study the properties of the medium at finite density and to explore the QCD phase diagram to find signs of a critical point [CITATION].', '1505.04226-1-3-1': 'Conserved charges such as net baryon number, strangeness and isospin would play important roles in the collisions with lower energies, as the differences between particle and antiparticle yields are clearly seen [CITATION].', '1505.04226-1-3-2': 'Historically, it had long been speculated based on several idealized calculations that the strong coupling limit is achieved only at highest energies of RHIC experiments.', '1505.04226-1-3-3': 'On the other hand, recent improvements in off-equilibrium hydrodynamic modeling motivates us to reexamine the validity of hydrodynamics in exploring the dense quark matter created at mid-low energies, especially since the differential elliptic flow [MATH] is found to remain large in Phase I of the BES experiments.', '1505.04226-1-3-4': 'The applicability of hydrodynamic models is closely related to the origin of fluidity, about which little is known, and thus its verification would be a very important step towards a full understanding of the hot QCD medium.', '1505.04226-1-4-0': 'So far many hydrodynamic analyses have been performed numerically because it is generally quite nontrivial to solve the partial differential equations involved.', '1505.04226-1-4-1': 'Analytical solutions of relativistic hydrodynamics, on the other hand, can be obtained with certain symmetry conditions and they are very instructive in understanding the essence of heavy-ion dynamics.', '1505.04226-1-4-2': 'The boost-invariant Bjorken flow [CITATION] is one such classic example.', '1505.04226-1-4-3': 'More recently, Gubser found an exact boost-invariant solution of the Navier-Stokes equation which has a nontrivial dependence on the transverse coordinate [CITATION].', '1505.04226-1-4-4': 'The latter solution has the advantage that one can add azimuthally anisotropic perturbations [CITATION] and analytically compute the corresponding flow harmonics [MATH] including the viscosity effects [CITATION] (see, also, [CITATION]).', '1505.04226-1-5-0': 'In this study, we investigate [MATH] at finite density by analytically solving the viscous hydrodynamic equations coupled with conserved currents assuming conformal and boost-invariant symmetries.', '1505.04226-1-5-1': 'Aside from the fact that the solution itself is new and of theoretical importance, it gives us a theoretical guidance about the behavior of [MATH] over a wide range of the beam energy for which there are not many numerical simulations [CITATION] and the previous knowledge obtained through the precision analyses in the RHIC-LHC energy regime, such as the value of the shear viscosity [MATH], are no longer fully applicable.', '1505.04226-1-5-2': 'We discuss extensively the nature of flow in the presence of currents and estimate the beam energy (or chemical potential) dependence of [MATH].', '1505.04226-1-5-3': 'The difference in [MATH] between particles and antiparticles is also analytically computed.', '1505.04226-1-5-4': 'The results are compared with the experimental data from SPS, RHIC and the LHC [CITATION].', '1505.04226-1-5-5': 'We see that they are in qualitative agreement, which suggests that a reasonable description of the low-energy experimental data might be possible within a hydrodynamic framework.', '1505.04226-1-6-0': 'The paper is organized as follows.', '1505.04226-1-6-1': 'The basic setup of relativistic hydrodynamics is outlined in Section [REF].', '1505.04226-1-6-2': 'We then present analytical formulas of the flow harmonics [MATH] in the ideal and viscous cases in Sections [REF] and [REF], respectively.', '1505.04226-1-6-3': 'Phenomenological inputs for our model are summarized in Section [REF].', '1505.04226-1-6-4': 'Using these formulas and input parameters, we compare our results with the experimental data in Section [REF].', '1505.04226-1-6-5': 'Section VII is devoted to summary and conclusions.', '1505.04226-1-7-0': '# Hydrodynamic equations', '1505.04226-1-8-0': '## Setup', '1505.04226-1-9-0': 'We shall consider hydrodynamics of a conformal theory.', '1505.04226-1-9-1': 'The system is characterized by the local temperature [MATH] and a set of local chemical potentials [MATH] where the subscript [MATH] labels various conserved charges of the theory.', '1505.04226-1-9-2': 'The flow velocity is denoted by [MATH] with the normalization [MATH].', '1505.04226-1-9-3': 'The energy-momentum tensor in the Navier-Stokes approximation takes the form [EQUATION] where [MATH] is the shear tensor and [MATH] is the shear viscosity.', '1505.04226-1-9-4': 'In ([REF]), the conformal equation of state [MATH] between the energy density [MATH] and the pressure [MATH] has been used.', '1505.04226-1-9-5': 'The conserved current [MATH] associated with the chemical potential [MATH] can be written as [EQUATION] where [MATH] is the charge density and [MATH] is the charge conductivity.', '1505.04226-1-9-6': 'The hydrodynamic equations consist of the conservation equations for [MATH] and [MATH] [EQUATION] where [MATH] is the covariant derivative.', '1505.04226-1-10-0': 'Since there is no intrinsic mass scale in a conformal theory, the energy density [MATH] and the charge densities [MATH] can be generically written as [EQUATION]', '1505.04226-1-10-1': 'With a view to applying to heavy-ion collisions, we shall focus on the following representative situation.', '1505.04226-1-10-2': "We assume that there is the leading current [MATH] ('baryon number current') and the corresponding chemical potential [MATH] is treated to all orders.", '1505.04226-1-10-3': "In addition, there is one subleading current [MATH] ('isospin number current') whose chemical potential [MATH] is small and treated only to linear order.", '1505.04226-1-10-4': "We take [MATH] to be 'orthogonal' to [MATH], in that [MATH] is invariant under a sign flip [MATH] (i.e., cross terms like [MATH] are absent).", '1505.04226-1-10-5': 'With these assumptions, we can parameterize [EQUATION]', '1505.04226-1-10-6': 'The last equation may be written as [MATH] where [MATH] is the susceptibility.', '1505.04226-1-11-0': '## Gubser flow', '1505.04226-1-12-0': 'We shall solve the hydrodynamic equations ([REF]) for a given flow velocity [EQUATION] and [MATH].', '1505.04226-1-12-1': 'The parameter [MATH] is the characteristic length scale of the system.', '1505.04226-1-12-2': 'In heavy-ion collisions, it is roughly the transverse size of the colliding nuclei.', '1505.04226-1-12-3': 'Eq. ([REF]) is called Gubser flow [CITATION] expressed in the coordinate system [EQUATION] where [MATH] is the proper time, [MATH] is the spacetime rapidity and [MATH] is the transverse coordinate.', '1505.04226-1-12-4': 'The condition [MATH] means that the flow is boost invariant along the beam ([MATH]) direction.', '1505.04226-1-13-0': 'Gubser flow takes a very simple form in a cleverly chosen coordinate system [MATH] which is related to the Minkowski coordinates via a Weyl rescaling of the metric.', '1505.04226-1-13-1': '[EQUATION] where [EQUATION]', '1505.04226-1-13-2': 'In this coordinate system, the flow velocity is simply [MATH].', '1505.04226-1-13-3': "In addition to the boost invariance, the flow respects the [MATH] symmetry with respect to the 'polar' angles [MATH].", '1505.04226-1-13-4': "Variables in this coordinate system will be denoted with a 'hat', e.g., [MATH], [MATH].", '1505.04226-1-14-0': '# Inviscid case', '1505.04226-1-15-0': 'In this section, we solve the hydrodynamic equations ([REF]) in the ideal case [MATH].', '1505.04226-1-15-1': 'We then deform the solution in the azimuthal direction [MATH] and compute flow harmonics [MATH].', '1505.04226-1-16-0': '## Isotropic ideal solution', '1505.04226-1-17-0': 'The isotropic solution (i.e., independent of [MATH]) has been obtained already in [CITATION] in the presence of a current [MATH].', '1505.04226-1-17-1': 'Assuming that all the quantities depend only on [MATH], we can readily solve the hydrodynamic equations for [MATH] and [MATH] in the coordinates ([REF]).', '1505.04226-1-17-2': 'We then perform the Weyl transformation back to the Minkowski space [MATH], [MATH] to get [EQUATION]', '1505.04226-1-17-3': 'These equations can be solved for [MATH] and [MATH].', '1505.04226-1-17-4': 'It is consistent to look for the solution where [MATH] and [MATH] have the same [MATH]-dependence such that the ratios [MATH], [MATH] are independent of [MATH].', '1505.04226-1-17-5': 'We find [EQUATION] and therefore, [EQUATION]', '1505.04226-1-17-6': 'The parameter [MATH] is related to the particle multiplicity to be extracted from the experimental data.', '1505.04226-1-17-7': "For a massless particle species [MATH] ('pion'), the relation is [CITATION] [EQUATION] where [MATH] is the momentum rapidity and [MATH] is the degeneracy factor.", '1505.04226-1-18-0': '## Anisotropic ideal solution', '1505.04226-1-19-0': 'We now perturb the solution anisotropically to introduce the [MATH] dependence.', '1505.04226-1-19-1': 'In doing so, we shall focus on the early time regime [MATH] (or [MATH], see ([REF])).', '1505.04226-1-19-2': 'As observed in [CITATION], in this regime the perturbed solution is fully under analytical control including the viscous case to be discussed in the next section.', '1505.04226-1-20-0': 'Following [CITATION], we consider the following deformation of the isotropic solution [EQUATION] where [EQUATION] is proportional to the spherical harmonics [MATH] in the early time regime [MATH].', '1505.04226-1-20-1': 'Note that we preserve boost invariance [MATH] in this paper, but the case [MATH] was also considered in [CITATION].', '1505.04226-1-20-2': '[MATH] is the eccentricity which we assume to be small [MATH] and keep only linear terms in [MATH].', '1505.04226-1-20-3': '[MATH] and [MATH] have to be determined by solving the hydrodynamic equations linearized around the isotropic solution.', '1505.04226-1-20-4': 'Plugging ([REF]) into ([REF]), we find the following equation for [MATH] [EQUATION]', '1505.04226-1-20-5': 'This turns out to be exactly the same as the equation satisfied by [MATH] [CITATION].', '1505.04226-1-20-6': 'Therefore, in the ideal case we have [MATH], which means that [MATH] and [MATH] are rescaled by the same factor [MATH], [MATH] and [MATH].', '1505.04226-1-20-7': 'The ratios [MATH] and [MATH] are thus unchanged.', '1505.04226-1-20-8': 'At early times [MATH], the right hand side of ([REF]) is negligible and we can set [MATH] [CITATION].', '1505.04226-1-21-0': '## [MATH] at finite [MATH]', '1505.04226-1-22-0': 'In order to compute flow harmonics [MATH], we use the Cooper-Frye formula [CITATION] [EQUATION] where we assumed the Boltzmann distribution and [MATH] is the deviation from the equilibrium distribution.', '1505.04226-1-22-1': 'The use of the Boltzmann distribution (rather than the Fermi/Bose distributions) may be justified for the purpose of computing the integrated [MATH] [CITATION].', '1505.04226-1-22-2': '[MATH] generically represents a set of chemical potentials for net baryon number, isospin and strangeness.', '1505.04226-1-22-3': 'We assign [MATH] for particles with positive/negative quantum numbers mentioned above, and [MATH] for neutral particles with respect to the corresponding quantum number.', '1505.04226-1-22-4': 'The integral in ([REF]) is taken over the hypersurface [MATH] of constant energy density where the kinetic freezeout occurs.', '1505.04226-1-22-5': 'In the ideal case, constant [MATH] means constant [MATH] since [MATH] is a constant.', '1505.04226-1-22-6': 'Let us write the condition of constant energy density as [EQUATION]', '1505.04226-1-22-7': 'Typically, [MATH] is of the order of the critical energy density of the QCD phase transition.', '1505.04226-1-22-8': 'We take [MATH] in this paper.', '1505.04226-1-22-9': 'Following [CITATION], we assume that the condition ([REF]) is reached within the early time regime [MATH] where we can use the approximate solution ([REF]).', '1505.04226-1-22-10': 'The parameter [MATH] in ([REF]) is then related to the (position-dependent) freezeout time [MATH] as [EQUATION]', '1505.04226-1-22-11': 'For consistency with our early freezeout scenario, we must have [MATH].', '1505.04226-1-23-0': 'Under these assumptions, the integral ([REF]) can be performed analytically and the integrated [MATH] is obtained from the formula [EQUATION]', '1505.04226-1-23-1': 'In the ideal case [MATH], [MATH] does not depend on [MATH] since the factor [MATH] cancels in the ratio ([REF]).', '1505.04226-1-23-2': 'The result is [CITATION] [EQUATION]', '1505.04226-1-23-3': 'This determines the [MATH] dependence of [MATH].', '1505.04226-1-23-4': 'Quite generally, [MATH] is an increasing function [MATH].', '1505.04226-1-23-5': 'On the other hand, [MATH] is a decreasing function of [MATH].', '1505.04226-1-23-6': 'We shall see that, in heavy-ion collisions, the latter dependence is stronger, and as a result ([REF]) is a decreasing function of [MATH], or equivalently, an increasing function of the collision energy [MATH].', '1505.04226-1-23-7': 'Incidentally, we note that the directed flow [MATH] vanishes, consistently with our assumption of boost-invariance.', '1505.04226-1-24-0': '# Viscous case', '1505.04226-1-25-0': 'We now turn to the viscous case [MATH].', '1505.04226-1-25-1': 'Although the system is out of equilibrium, from the Landau matching condition we can define the local [MATH] and [MATH] using the same relations as in equilibrium [EQUATION] but now [MATH] cannot be a constant.', '1505.04226-1-26-0': '## Isotropic viscous solution', '1505.04226-1-27-0': 'First consider the isotropic case [MATH].', '1505.04226-1-27-1': 'Although [MATH] in ([REF]) is not a constant anymore, it depends only on [MATH] (see below).', '1505.04226-1-27-2': 'Then we still have [MATH] so that [EQUATION] is the same as in the ideal case [CITATION].', '1505.04226-1-27-3': 'However, the solution of the Navier-Stokes equation [MATH] has an extra [MATH]-dependence proportional to the shear viscosity [MATH].', '1505.04226-1-27-4': 'In the case of vanishing chemical potentials, this [MATH]-dependence can be obtained exactly [CITATION] [EQUATION] where [MATH] is independent of [MATH].', '1505.04226-1-28-0': 'However, at finite density, [MATH] will depend on [MATH], and this makes it difficult to find an exact solution.', '1505.04226-1-28-1': 'Related to this, [MATH] can now depend on both [MATH] and [MATH], and this relation can be model-dependent.', '1505.04226-1-28-2': 'We can get around this problem by assuming that [MATH] is small.', '1505.04226-1-28-3': 'Specifically, we rescale [MATH] by the entropy density [MATH] [EQUATION] as is often done in hydrodynamic simulations.', '1505.04226-1-28-4': 'We then regard [MATH] as a small parameter ([MATH]) and keep only terms linear in [MATH].', '1505.04226-1-28-5': 'In this approximation, we may replace [MATH] and [MATH] in ([REF]) by their equilibrium values at [MATH], namely, [MATH] and [EQUATION]', '1505.04226-1-28-6': 'We then find the solution valid to [MATH] [EQUATION] where in the second line of ([REF]) we have taken the early-time limit [MATH].', '1505.04226-1-29-0': 'Using ([REF]) and ([REF]), we can eliminate [MATH] [EQUATION]', '1505.04226-1-29-1': 'Writing [EQUATION] we find the deviation from constancy due to the viscosity [EQUATION] where [EQUATION]', '1505.04226-1-29-2': 'Note that [MATH] as [MATH].', '1505.04226-1-29-3': 'At the freezeout time [MATH], we have the relation [EQUATION]', '1505.04226-1-29-4': 'Finally, we can solve for [MATH] and [MATH] using ([REF]).', '1505.04226-1-29-5': 'The result is [EQUATION]', '1505.04226-1-30-0': '## Anisotropic viscous solution', '1505.04226-1-31-0': 'We now perturb the solution as in ([REF]).', '1505.04226-1-31-1': 'First consider the current in ([REF]).', '1505.04226-1-31-2': '[MATH] now depends not only on [MATH], but also on [MATH] and [MATH].', '1505.04226-1-31-3': 'However, the dependence is of order [MATH].', '1505.04226-1-31-4': '(See ([REF]) below.', '1505.04226-1-31-5': 'Remember that for the ideal solution [MATH] is constant even in the anisotropic case.)', '1505.04226-1-31-6': 'Therefore, if we neglect terms of order [MATH], we can approximate [MATH].', '1505.04226-1-31-7': 'Then ([REF]) is still valid and we get [EQUATION]', '1505.04226-1-31-8': 'As for the energy density, we find [EQUATION] where [CITATION] [EQUATION]', '1505.04226-1-31-9': "From the constant energy condition [EQUATION] we can determine the freezeout surface [MATH] in the viscous case [CITATION] [EQUATION] where the 'Knudsen number' is proportional to the shear viscosity [EQUATION] ([REF]) and ([REF]) are also modified as [MATH] where [EQUATION] and [EQUATION]", '1505.04226-1-32-0': '## [MATH] at finite [MATH] and [MATH]', '1505.04226-1-33-0': 'The computation of [MATH] is more complicated than the [MATH] case.', '1505.04226-1-33-1': 'This is because [MATH] does not mean [MATH], and therefore one cannot treat [MATH] in the Boltzmann factor ([REF]) as a constant when integrating over the hypersurface of constant energy.', '1505.04226-1-33-2': 'In order to cope with this, we write ([REF]) as [EQUATION] where [EQUATION] is constant by virtue of ([REF]).', '1505.04226-1-33-3': 'We then expand the Boltzmann factor as [EQUATION] where we approximated [MATH] in the [MATH] term.', '1505.04226-1-34-0': 'The first term in ([REF]), proportional to unity, gives the same result as in [CITATION] [EQUATION]', '1505.04226-1-34-1': 'Note that [MATH] for [MATH].', '1505.04226-1-34-2': 'The second term in ([REF]) leads to a new order [MATH] contribution to [MATH].', '1505.04226-1-34-3': 'To compute it, we borrow some results from [CITATION].', '1505.04226-1-34-4': 'First, the perturbed flow velocity [MATH] on the freezeout surface has the following components in the coordinates ([REF]) [EQUATION] where [EQUATION] (The viscosity can be neglected here.)', '1505.04226-1-34-5': 'The exponential factor in the Boltzmann distribution reads [EQUATION] where [MATH] is the transverse mass.', '1505.04226-1-34-6': 'The volume element of the constant energy hypersurface is [EQUATION] where [MATH] is given by ([REF]) with the viscous term set to zero.', '1505.04226-1-34-7': 'Finally, we need the more precise version of ([REF]) [EQUATION]', '1505.04226-1-34-8': 'Armed with these formulas, let us decompose the contribution from the second term in ([REF]) as [EQUATION] corresponding to the three terms in ([REF]).', '1505.04226-1-34-9': 'Consider [MATH] first.', '1505.04226-1-34-10': 'To [MATH] we have to evaluate [EQUATION]', '1505.04226-1-34-11': 'This can be efficiently evaluated using the trick introduced in [CITATION] (see Eq. (73) there).', '1505.04226-1-34-12': 'The [MATH]-integral gives Bessel functions [MATH] where [EQUATION]', '1505.04226-1-34-13': 'This can be expanded as [MATH] anticipating that the subsequent [MATH]-integral is dominated by the region [MATH].', '1505.04226-1-34-14': 'We thus find [EQUATION]', '1505.04226-1-34-15': 'The correction to [MATH] can be calculated from the formula (cf. ([REF])) [EQUATION] where [MATH] is the azimuthally symmetric part (cf. Eq. (45) of [CITATION]) [EQUATION]', '1505.04226-1-34-16': 'In the massless case [MATH], the integral can be done exactly and we find [EQUATION] and from ([REF]), [EQUATION]', '1505.04226-1-34-17': 'It is important to emphasize that ([REF]) is induced by the combined effect of the chemical potential and the viscosity.', '1505.04226-1-34-18': 'It vanishes when [MATH] or [MATH] because [MATH] (cf. ([REF])).', '1505.04226-1-34-19': 'Compared with ([REF]) which schematically reads [MATH], we notice that ([REF]) is not enhanced by a factor of [MATH], hence subleading at large [MATH].', '1505.04226-1-34-20': 'However, it is the leading contribution to the difference in [MATH] between particles ([MATH]) and antiparticles ([MATH]).', '1505.04226-1-34-21': 'If [MATH] is the baryon chemical potential, the protons have larger [MATH] than the antiprotons.', '1505.04226-1-34-22': 'We shall study this effect in detail later.', '1505.04226-1-35-0': 'In fact, for protons the approximation [MATH] is not valid.', '1505.04226-1-35-1': 'Instead, we now assume [MATH] and reevaluate [MATH].', '1505.04226-1-35-2': 'Note that when [MATH], [MATH] is parametrically larger than [MATH], so it is enough to consider only [MATH].', '1505.04226-1-36-0': 'When [MATH], the Bessel function is independent of the order [EQUATION] so that ([REF]) becomes [EQUATION]', '1505.04226-1-36-1': 'On the other hand, from Eq. (47) of [CITATION], [EQUATION]', '1505.04226-1-36-2': 'The [MATH]-integral can be evaluated by the saddle point at [MATH] for [MATH] and we obtain [EQUATION]', '1505.04226-1-36-3': 'The [MATH]-independent part is order [MATH], but we shall see later that it is numerically small for realistic values of [MATH] because the factor [MATH] is small.', '1505.04226-1-36-4': 'The [MATH]-dependent term is again of order [MATH] without an enhancement by a factor of [MATH].', '1505.04226-1-37-0': 'The evaluation of [MATH] in ([REF]) can be done similarly, though it is considerably more tedious.', '1505.04226-1-37-1': 'Here we only show the final result in the massless case [MATH], relegating the details to Appendix [EQUATION]', '1505.04226-1-37-2': 'Comparing with ([REF]), we notice that [MATH].', '1505.04226-1-37-3': 'Actually, this relation was repeatedly observed in [CITATION] when computing other contributions to [MATH].', '1505.04226-1-37-4': 'We do not have a simple explanation for this.', '1505.04226-1-38-0': 'Summing all the contributions including the previously computed term [CITATION], our final result of the viscous correction [MATH] in the massless case is [EQUATION]', '1505.04226-1-38-1': 'The second term in the curly brackets is the new contribution at finite density.', '1505.04226-1-38-2': 'It is subleading in [MATH], and actually the factor [MATH] is also numerically small.', '1505.04226-1-38-3': 'However, it gives the leading contribution to the difference in [MATH] between particles and antiparticles.', '1505.04226-1-39-0': '## Isospin chemical potential', '1505.04226-1-40-0': "In the previous subsection, we computed [MATH] of particles which couple to the 'large' chemical potential [MATH].", '1505.04226-1-40-1': 'Here let us compute [MATH] of particles neutral under [MATH] but charged under [MATH].', '1505.04226-1-40-2': 'We have in mind the charged pions [MATH] in the presence of the isospin chemical potential.', '1505.04226-1-40-3': 'We start with the formula (cf. ([REF])) [EQUATION]', '1505.04226-1-40-4': 'We treat [MATH] as a small parameter and keep only terms linear in [MATH].', '1505.04226-1-40-5': 'Dividing by [MATH] from ([REF]) and using [MATH], we find [EQUATION]', '1505.04226-1-40-6': 'The fugacity factor thus becomes [EQUATION]', '1505.04226-1-40-7': 'As before, the factor [MATH] drops out in the computation of [MATH].', '1505.04226-1-40-8': 'We see that the only difference from the previous case ([REF]) is that [MATH] is replaced by [EQUATION]', '1505.04226-1-40-9': 'Thus the final result is the same as ([REF]) except that [MATH] is replaced by ([REF]).', '1505.04226-1-40-10': '([REF]) may look singular when [MATH], but in the above derivation we assumed [MATH].', '1505.04226-1-41-0': '# Phenomenological inputs', '1505.04226-1-42-0': 'This section serves as a preparation for the next section where we compare our results with the experimental data.', '1505.04226-1-43-0': '## Models', '1505.04226-1-44-0': 'In order to make quantitative predictions, we need models for the functions [MATH], [MATH], [MATH] defined in ([REF]).', '1505.04226-1-44-1': 'Here we consider two extreme scenarios in terms of the interaction strength.', '1505.04226-1-45-0': '### Free quark-gluon gas', '1505.04226-1-46-0': 'The energy density of free, massless three flavor QCD is [EQUATION] where [MATH], [MATH] and [MATH].', '1505.04226-1-46-1': '[MATH], [MATH] and [MATH] are the baryon, isospin and strangeness chemical potentials, respectively.', '1505.04226-1-46-2': 'Since the net strangeness is zero in heavy-ion collisions, we set [MATH] and obtain ([MATH]) [EQUATION]', '1505.04226-1-46-3': 'It turns out that, due to the large denominators [MATH] or [MATH], the effect of [MATH] and [MATH] on [MATH] is numerically small.', '1505.04226-1-47-0': '### [MATH] SYM at finite [MATH]-charge chemical potential', '1505.04226-1-48-0': 'Next we consider strongly coupled [MATH] supersymmetric Yang-Mills theory at finite [MATH]-charge chemical potential [MATH].', '1505.04226-1-48-1': 'This theory is conformal, and in the limit of strong coupling and at large [MATH], the thermodynamic quantities can be computed from the AdS/CFT correspondence.', '1505.04226-1-48-2': 'The results are [CITATION] [EQUATION] where [MATH] is the [MATH]-charge density.', '1505.04226-1-48-3': 'The shear viscosity is given by [MATH].', '1505.04226-1-49-0': 'There are uncertainties when treating this model as a proxy of strongly coupled QCD, such as the value of [MATH] and the proportionality constant between [MATH] and [MATH].', '1505.04226-1-49-1': 'However, in practical fits, the normalization of [MATH] can be absorbed by a change in [MATH] (cf. Eq. ([REF])).', '1505.04226-1-49-2': 'Moreover, as long as [MATH], the two functions ([REF]) and ([REF]) are qualitatively not so different in shape for [MATH].', '1505.04226-1-49-3': 'As a result, the quality of fits is similar in the two cases despite the huge differences in the underlying dynamics.', '1505.04226-1-49-4': 'Therefore, in the next section we show only the results based on ([REF]) and ([REF]).', '1505.04226-1-50-0': '## Freezeout conditions', '1505.04226-1-51-0': 'We employ the following phenomenological parametrization [CITATION] of the freezeout temperature [MATH] and chemical potential [MATH] (in units of GeV) as a function of the collision energy [MATH] (per nucleon, in units of GeV) [EQUATION] with [MATH], [MATH], [MATH], [MATH], [MATH].', '1505.04226-1-51-1': 'This gives [MATH] as a function of [MATH] as shown in Fig. [REF].', '1505.04226-1-51-2': 'The curve is well approximated by [MATH].', '1505.04226-1-51-3': 'Actually, [MATH] and [MATH] here are the chemical freezeout parameters which are in general different from those entering the Cooper-Frye formula ([REF]) used at the kinetic freezeout.', '1505.04226-1-51-4': 'However, in our model only the ratio [MATH] matters, and this ratio is roughly constant as we have seen.', '1505.04226-1-51-5': 'We thus use the relation in Fig. [REF] for the evaluation of [MATH].', '1505.04226-1-52-0': 'The parameter [MATH] also depends on [MATH] via ([REF]).', '1505.04226-1-52-1': 'We use the following empirical formula for the charged particle multiplicity [CITATION] [EQUATION] where the factor of 2 counts the degeneracy between [MATH] and [MATH].', '1505.04226-1-52-2': 'From ([REF]) and ([REF]), we see that the Knudsen number ([REF]) behaves as [EQUATION] where [MATH] is in units of fermi and [MATH] is in units of GeV/fm[MATH].', '1505.04226-1-52-3': 'Putting aside the potential [MATH]-dependence of [MATH], we see that [MATH] is an increasing function of [MATH] (up to [MATH] in our model) or a decreasing function of [MATH].', '1505.04226-1-53-0': '# Comparison with the experimental data', '1505.04226-1-54-0': 'In this section, we compare our results with three different experimental data: (i) the [MATH]-dependence of [MATH] measured at the LHC; (ii) the collision energy dependence of [MATH] measured at the SPS; (iii) the difference in [MATH] between particles and antiparticles measured at RHIC.', '1505.04226-1-55-0': '## Higher harmonics [MATH]', '1505.04226-1-56-0': 'The CMS collaboration at the LHC has measured [MATH] in lead-lead collisions at [MATH] TeV up to rather high orders ([MATH]) [CITATION].', '1505.04226-1-56-1': 'Using ([REF]) and ([REF]) together with the phenomenological inputs in the previous section, we can evaluate [MATH] and compare with the CMS data.', '1505.04226-1-56-2': 'The result is shown in Fig. [REF].', '1505.04226-1-56-3': 'Here we set [MATH] for all different values of [MATH].', '1505.04226-1-56-4': 'Taking [MATH] to be independent of [MATH] may be a good approximation for the very central (0-0.2% centrality) nucleus collisions.', '1505.04226-1-56-5': 'The parameter [MATH] is set to [MATH].', '1505.04226-1-56-6': 'The corresponding value of [MATH] in ([REF]) is [MATH] which is consistent with the assumption [MATH].', '1505.04226-1-57-0': 'As a matter of fact, since [MATH] at the LHC, the new term at [MATH] (the term proportional to [MATH] in ([REF])) is negligibly small, and the present fit could have been done in [CITATION] treating [MATH] as a fitting parameter.', '1505.04226-1-57-1': 'By expressing [MATH] in terms of observables as we have done here, we can test our result at lower energies or higher chemical potentials [MATH].', '1505.04226-1-57-2': 'Note that since [MATH] is larger at lower energies, [MATH] [CITATION] decreases faster with [MATH], and this will make the measurement of higher harmonics difficult at low energies [CITATION].', '1505.04226-1-58-0': '## Energy dependence of [MATH]', '1505.04226-1-59-0': 'Next we turn to the energy dependence of the elliptic flow [MATH] for which there are already a wealth of experimental data from the SPS and the RHIC BES program [CITATION].', '1505.04226-1-59-1': 'We decided to compare our formulas ([REF]) and ([REF]) (with [MATH]) for [MATH] with the SPS, mid-central data collected in the low energy region [MATH] [CITATION].', '1505.04226-1-59-2': 'The result with three different values of [MATH] is shown in Fig. [REF] where we set [MATH]fm and [MATH].', '1505.04226-1-59-3': 'The value of [MATH] here is slightly smaller than the one ([MATH]fm) used in Fig. [REF].', '1505.04226-1-59-4': 'This is consistent with the perception that the QGP droplet is larger at higher energies at the time of thermalization.', '1505.04226-1-59-5': 'The rise of [MATH] with energy is nicely reproduced by our formula and attributed mostly to the rise of [MATH] as seen in ([REF]).', '1505.04226-1-59-6': 'It turns out that the newly calculated viscous correction in Section [REF] (the last term in ([REF])) is numerically very small (about an order of magnitude smaller than the first term in ([REF])) even in the highest density region.', '1505.04226-1-60-0': 'Unfortunately, this fit, which agrees reasonably well with the low energy data, overshoots the high energy RHIC data at [MATH] [CITATION] in similar centrality bins by a factor of 2 (assuming that [MATH] is independent of energy).', '1505.04226-1-60-1': 'This is because the rise of [MATH] ([REF]) with energy is too steep.', '1505.04226-1-60-2': 'If we artificially reduce the exponent in ([REF]) as [MATH], for example, we get a decent description of [MATH] over a broader range in [MATH].', '1505.04226-1-60-3': 'Alternatively, the dependence [MATH] from ([REF]) may be too strong, and the experimental data actually suggest a weaker [MATH]-dependence [CITATION].', '1505.04226-1-60-4': 'While we do not have a resolution of this problem in the present framework, it seems qualitatively correct that [MATH] is directly proportional to the multiplicity to some positive power, and therefore it is an increasing function of [MATH] (see, also, Section VII of [CITATION]).', '1505.04226-1-61-0': 'Incidentally, the decrease of [MATH] with increasing [MATH] in the very low energy region [MATH]GeV is due to the following reason.', '1505.04226-1-61-1': 'From ([REF]), ([REF]) and ([REF]) we have that [EQUATION]', '1505.04226-1-61-2': 'Since [MATH] (see Fig. [REF]), the [MATH]-dependence of [MATH] is not strong except in the lowest energy regime where it is dominant and changes the slope of [MATH].', '1505.04226-1-61-3': 'This is however a model-dependent effect and may not be present in more realistic models.', '1505.04226-1-61-4': 'The fact that the experimental [MATH] is still rising in the low energy regime [MATH]GeV may thus provide a constraint on the function [MATH].', '1505.04226-1-62-0': '## Difference in [MATH] between particles and antiparticles', '1505.04226-1-63-0': 'Finally, we investigate the difference in [MATH] between particles and antiparticles which has been measured by the STAR collaboration at RHIC [CITATION] and attracted some attention from theoretical viewpoints [CITATION].', '1505.04226-1-63-1': 'For a hadron with the quantum numbers [MATH] (baryon number, isospin, strangeness), we assign the fugacity factor [EQUATION] ([MATH] for the strange quark.)', '1505.04226-1-63-2': 'In heavy-ion collisions, [MATH] since the colliding nuclei are neutron rich, and [MATH] since the net strangeness vanishes.', '1505.04226-1-63-3': 'The latter condition implies that we should not treat [MATH] as a small perturbation.', '1505.04226-1-63-4': 'Indeed, various estimates of [MATH] based on the SPS [CITATION] and RHIC [CITATION] data, and also from lattice QCD [CITATION] all found similar values within the range [MATH].', '1505.04226-1-63-5': 'We thus regard [MATH] as a shift of [MATH] for strange hadrons and treat it as a fitting parameter, anticipating that the value of [MATH] should come out in the window [MATH].', '1505.04226-1-63-6': 'On the other hand, we regard [MATH] as a small parameter compared to unity and use the result obtained in Section [REF].', '1505.04226-1-64-0': 'Let us define the difference in [MATH] between hadrons [MATH] and antihadrons [MATH] as [EQUATION]', '1505.04226-1-64-1': 'This can be evaluated from ([REF]) and ([REF]).', '1505.04226-1-64-2': "Focusing now on the elliptic flow case [MATH], we can immediately write down the following 'master formula' [EQUATION]", '1505.04226-1-64-3': 'By construction, ([REF]) has been derived for massless particles.', '1505.04226-1-64-4': 'In the massive case, we observe that the following ratio [EQUATION] is exactly independent of [MATH].', '1505.04226-1-64-5': 'This is due to the nontrivial cancelation of [MATH]-integrals such as ([REF]) in the ratio for the [MATH]-dependent part.', '1505.04226-1-64-6': 'In order to get [MATH] itself, we must multiply ([REF]) by [CITATION] [EQUATION]', '1505.04226-1-64-7': 'The [MATH]-dependence of ([REF]) is sensitive to the cutoffs of the [MATH]-integral, but overall the dependence is not very strong.', '1505.04226-1-64-8': 'For simplicity, in this study we ignore the [MATH]-dependence of [MATH] and use ([REF]) for all hadron species.', '1505.04226-1-64-9': 'It is not difficult to implement this mass effect, but there are other subtleties which are not taken into account, either.', '1505.04226-1-64-10': 'Clearly, it is desirable that the experimental results are plotted in the form ([REF]) in order to avoid various systematic uncertainties.', '1505.04226-1-65-0': 'The most important feature of ([REF]) or ([REF]) is that [MATH] is proportional to both the shear viscosity [MATH] and the chemical potentials.', '1505.04226-1-65-1': '(Remember that [MATH] as defined in ([REF]) is roughly proportional to [MATH].)', '1505.04226-1-65-2': 'This in particular means that [MATH] can be nonzero in viscous hydrodynamics in the presence of the isospin chemical potential.', '1505.04226-1-66-0': 'Let us confront ([REF]) with the data.', '1505.04226-1-66-1': 'The STAR collaboration has measured [MATH] for [MATH] [CITATION].', '1505.04226-1-66-2': 'This is plotted in Fig. [REF] together with our fit based on ([REF]) with [MATH].', '1505.04226-1-66-3': 'We have used [MATH] and [MATH] as in Fig. [REF], and used the fit parameters [MATH] and [MATH], the former is consistent with our expectation mentioned above.', '1505.04226-1-66-4': 'The steep rise of [MATH] for baryons towards the low-[MATH] region is due to the rough proportionality [MATH].', '1505.04226-1-66-5': 'Compared to this, the [MATH]-dependence of the factor [MATH] is subleading.', '1505.04226-1-66-6': 'Since [MATH] have [MATH], [MATH], [MATH], [MATH], respectively, we expect the ordering [MATH] for reasonable values of [MATH] and [MATH].', '1505.04226-1-66-7': 'This tendency is obeyed by most data points except a few in the low energy region.', '1505.04226-1-66-8': "We note that the [MATH] data point at [MATH] GeV should not be taken seriously because, according to the STAR collaboration [CITATION], this data point is afflicted with 'additional systematic effects which are not included in the error bars'.", '1505.04226-1-67-0': 'Concerning the pions, the negative [MATH] can be naturally explained by the negative isospin chemical potential.', '1505.04226-1-67-1': 'However, the magnitude is problematic.', '1505.04226-1-67-2': 'Our choice [MATH], which describes the pion data very well, is too large compared with the value [MATH] extracted from the SPS data [CITATION].', '1505.04226-1-67-3': 'We may dial [MATH] down to, say, [MATH] without spoiling much the quality of the [MATH] fit, but not further down.', '1505.04226-1-67-4': 'On the other hand, the other hadrons ([MATH]) are more or less unaffected by [MATH] and can be well fitted even with [MATH] and [MATH].', '1505.04226-1-67-5': 'This may be an indication that there are other mechanisms to generate the difference [MATH] which predominantly act on the pions.', '1505.04226-1-68-0': 'In the large-[MATH] region, our result tends to slightly overestimate [MATH].', '1505.04226-1-68-1': 'This is partly due to the too fast rise of [MATH] with energy as mentioned before.', '1505.04226-1-68-2': 'However, in Fig. [REF] we assumed that [MATH] is independent of [MATH].', '1505.04226-1-68-3': 'A recent hydrodynamic simulation suggests that [MATH] is a decreasing function of [MATH] [CITATION], and this could alleviate the (small) discrepancy in the large-[MATH] region (remember that [MATH]).', '1505.04226-1-69-0': '# Summary and conclusions', '1505.04226-1-70-0': 'In this paper, we have revealed, in a completely analytical manner, a number of interesting features about the nature of hydrodynamics in the presence of conserved currents as well as the chemical potential (collision energy) dependence of the flow harmonics [MATH].', '1505.04226-1-70-1': 'Let us summarize the main findings.', '1505.04226-1-71-0': 'Presumably some of the above features are empirically well known to the experts of hydrodynamic simulations.', '1505.04226-1-71-1': 'However, they have not been systematically derived with the level of analytical detail presented in this work.', '1505.04226-1-72-0': 'There are a number of directions for future work.', '1505.04226-1-72-1': 'Admittedly, the assumptions of boost invariance and conformal invariance are too simplistic, especially at high density.', '1505.04226-1-72-2': 'One has to relax these approximations to be more realistic.', '1505.04226-1-72-3': 'Related to this, we only considered the conformal equation of state [MATH] where the function [MATH] does not carry any information about the crossover and possibly first order phase transitions at finite density.', '1505.04226-1-72-4': '(Nevertheless it is remarkable that we can explain many features of [MATH] measured at different energies without such information.)', '1505.04226-1-72-5': 'It is important to figure out how the presence of phase transitions in [MATH] is encoded in the observed behavior of [MATH].', '1505.04226-1-72-6': 'Including the effects of anomaly (see, e.g., [CITATION]) is also interesting.', '1505.04226-1-72-7': 'We hope to address these questions in future work.'}

{'1505.04226-2-0-0': 'We investigate the Gubser solution of viscous hydrodynamics at finite density and analytically compute the flow harmonics [MATH].', '1505.04226-2-0-1': 'We explicitly show how [MATH] and their viscous corrections depend on the chemical potential.', '1505.04226-2-0-2': 'The difference in [MATH] between particles and antiparticles is also analytically computed and shown to be proportional to various chemical potentials and the viscosity.', '1505.04226-2-0-3': 'Excellent agreement is obtained between the results and the available experimental data from the SPS, RHIC and the LHC.', '1505.04226-2-1-0': '# Introduction', '1505.04226-2-2-0': 'Relativistic hydrodynamics is a general theoretical framework to describe the collective dynamics of high-energy systems near local thermal equilibrium.', '1505.04226-2-2-1': "Its first application to hadron physics dates back to Landau's attempt to describe multi-particle production in hadron-hadron collisions [CITATION].", '1505.04226-2-2-2': 'It has become a topic of great interest since the discovery of the quark-gluon plasma (QGP) as a nearly-perfect fluid in the "Little Bangs" at BNL Relativistic Heavy Ion Collider (RHIC) [CITATION] and CERN Large Hadron collider (LHC) [CITATION].', '1505.04226-2-2-3': 'This is supported by the observations that the azimuthal momentum anisotropy of hadronic distribution [CITATION], characterized by flow harmonics [MATH], are found to reflect the geometrical anisotropy [MATH] of the overlapping region of two colliding nuclei, and that they are in good quantitative agreement with theoretical estimations.', '1505.04226-2-2-4': 'Nowadays the viscous hydrodynamic modeling is considered as one of the most powerful tools to quantify the QGP medium near the crossover phase transition [CITATION].', '1505.04226-2-3-0': 'The recent Beam Energy Scan (BES) experiments at RHIC pose us intriguing challenges to study the properties of the medium at finite density and to explore the QCD phase diagram to find signs of a critical point [CITATION].', '1505.04226-2-3-1': 'Conserved charges such as net baryon number, strangeness and isospin would play important roles in the collisions with lower energies, as the differences between particle and antiparticle yields are clearly seen [CITATION].', '1505.04226-2-3-2': 'Historically, it had long been speculated based on several idealized calculations that the strong coupling limit is achieved only at highest energies of RHIC experiments.', '1505.04226-2-3-3': 'On the other hand, recent improvements in off-equilibrium hydrodynamic modeling motivates us to reexamine the validity of hydrodynamics in exploring the dense quark matter created at mid-low energies, especially since the differential elliptic flow [MATH] is found to remain large in Phase I of the BES experiments.', '1505.04226-2-3-4': 'The applicability of hydrodynamic models is closely related to the origin of fluidity, about which little is known, and thus its verification would be a very important step towards a full understanding of the hot QCD medium.', '1505.04226-2-4-0': 'So far many hydrodynamic analyses have been performed numerically because it is generally quite nontrivial to solve the partial differential equations involved.', '1505.04226-2-4-1': 'Analytical solutions of relativistic hydrodynamics, on the other hand, can be obtained with certain symmetry conditions and they are very instructive in understanding the essence of heavy-ion dynamics.', '1505.04226-2-4-2': 'The boost-invariant Bjorken flow [CITATION] is one such classic example.', '1505.04226-2-4-3': 'More recently, Gubser found an exact boost-invariant solution of the Navier-Stokes equation which has a nontrivial dependence on the transverse coordinate [CITATION].', '1505.04226-2-4-4': 'The latter solution has the advantage that one can add azimuthally anisotropic perturbations [CITATION] and analytically compute the corresponding flow harmonics [MATH] including the viscosity effects [CITATION] (see, also, [CITATION]).', '1505.04226-2-5-0': 'In this study, we investigate [MATH] at finite density by analytically solving the viscous hydrodynamic equations coupled with conserved currents assuming conformal and boost-invariant symmetries.', '1505.04226-2-5-1': 'Aside from the fact that the solution itself is new and of theoretical importance, it gives us a theoretical guidance about the behavior of [MATH] over a wide range of the beam energy for which there are not many numerical simulations [CITATION] and the previous knowledge obtained through the precision analyses in the RHIC-LHC energy regime, such as the value of the shear viscosity [MATH], are no longer fully applicable.', '1505.04226-2-5-2': 'We discuss extensively the nature of flow in the presence of currents and estimate the beam energy (or chemical potential) dependence of [MATH].', '1505.04226-2-5-3': 'The difference in [MATH] between particles and antiparticles is also analytically computed.', '1505.04226-2-5-4': 'The results are compared with the experimental data from SPS, RHIC and the LHC [CITATION].', '1505.04226-2-5-5': 'We see that they are in qualitative agreement, which suggests that a reasonable description of the low-energy experimental data might be possible within a hydrodynamic framework.', '1505.04226-2-6-0': 'The paper is organized as follows.', '1505.04226-2-6-1': 'The basic setup of relativistic hydrodynamics is outlined in Section [REF].', '1505.04226-2-6-2': 'We then present analytical formulas of the flow harmonics [MATH] in the ideal and viscous cases in Sections [REF] and [REF], respectively.', '1505.04226-2-6-3': 'Phenomenological inputs for our model are summarized in Section [REF].', '1505.04226-2-6-4': 'Using these formulas and input parameters, we compare our results with the experimental data in Section [REF].', '1505.04226-2-6-5': 'Section VII is devoted to summary and conclusions.', '1505.04226-2-7-0': '# Hydrodynamic equations', '1505.04226-2-8-0': '## Setup', '1505.04226-2-9-0': 'We shall consider hydrodynamics of a conformal theory.', '1505.04226-2-9-1': 'The system is characterized by the local temperature [MATH] and a set of local chemical potentials [MATH] where the subscript [MATH] labels various conserved charges of the theory.', '1505.04226-2-9-2': 'The flow velocity is denoted by [MATH] with the normalization [MATH].', '1505.04226-2-9-3': 'The energy-momentum tensor in the Navier-Stokes approximation takes the form [EQUATION] where [MATH] is the shear tensor and [MATH] is the shear viscosity.', '1505.04226-2-9-4': 'In ([REF]), the conformal equation of state [MATH] between the energy density [MATH] and the pressure [MATH] has been used.', '1505.04226-2-9-5': 'The conserved current [MATH] associated with the chemical potential [MATH] can be written as [EQUATION] where [MATH] is the charge density and [MATH] is the charge conductivity.', '1505.04226-2-9-6': 'The hydrodynamic equations consist of the conservation equations for [MATH] and [MATH] [EQUATION] where [MATH] is the covariant derivative.', '1505.04226-2-10-0': 'Since there is no intrinsic mass scale in a conformal theory, the energy density [MATH] and the charge densities [MATH] can be generically written as [EQUATION]', '1505.04226-2-10-1': 'With a view to applying to heavy-ion collisions, we shall focus on the following representative situation.', '1505.04226-2-10-2': "We assume that there is the leading current [MATH] ('baryon number current') and the corresponding chemical potential [MATH] is treated to all orders.", '1505.04226-2-10-3': "In addition, there is one subleading current [MATH] ('isospin number current') whose chemical potential [MATH] is small and treated only to linear order.", '1505.04226-2-10-4': "We take [MATH] to be 'orthogonal' to [MATH], in that [MATH] is invariant under a sign flip [MATH] (i.e., cross terms like [MATH] are absent).", '1505.04226-2-10-5': 'With these assumptions, we can parameterize [EQUATION]', '1505.04226-2-10-6': 'The last equation may be written as [MATH] where [MATH] is the susceptibility.', '1505.04226-2-11-0': '## Gubser flow', '1505.04226-2-12-0': 'We shall solve the hydrodynamic equations ([REF]) for a given flow velocity [EQUATION] and [MATH].', '1505.04226-2-12-1': 'The parameter [MATH] is the characteristic length scale of the system.', '1505.04226-2-12-2': 'In heavy-ion collisions, it is roughly the transverse size of the colliding nuclei.', '1505.04226-2-12-3': 'Eq. ([REF]) is called Gubser flow [CITATION] expressed in the coordinate system [EQUATION] where [MATH] is the proper time, [MATH] is the spacetime rapidity and [MATH] is the transverse coordinate.', '1505.04226-2-12-4': 'The condition [MATH] means that the flow is boost invariant along the beam ([MATH]) direction.', '1505.04226-2-13-0': 'Gubser flow takes a very simple form in a cleverly chosen coordinate system [MATH] which is related to the Minkowski coordinates via a Weyl rescaling of the metric.', '1505.04226-2-13-1': '[EQUATION] where [EQUATION]', '1505.04226-2-13-2': 'In this coordinate system, the flow velocity is simply [MATH].', '1505.04226-2-13-3': "In addition to the boost invariance, the flow respects the [MATH] symmetry with respect to the 'polar' angles [MATH].", '1505.04226-2-13-4': "Variables in this coordinate system will be denoted with a 'hat', e.g., [MATH], [MATH].", '1505.04226-2-14-0': '# Inviscid case', '1505.04226-2-15-0': 'In this section, we solve the hydrodynamic equations ([REF]) in the ideal case [MATH].', '1505.04226-2-15-1': 'We then deform the solution in the azimuthal direction [MATH] and compute flow harmonics [MATH].', '1505.04226-2-16-0': '## Isotropic ideal solution', '1505.04226-2-17-0': 'The isotropic solution (i.e., independent of [MATH]) has been obtained already in [CITATION] in the presence of a current [MATH].', '1505.04226-2-17-1': 'Assuming that all the quantities depend only on [MATH], we can readily solve the hydrodynamic equations for [MATH] and [MATH] in the coordinates ([REF]).', '1505.04226-2-17-2': 'We then perform the Weyl transformation back to the Minkowski space [MATH], [MATH] to get [EQUATION]', '1505.04226-2-17-3': 'These equations can be solved for [MATH] and [MATH].', '1505.04226-2-17-4': 'It is consistent to look for the solution where [MATH] and [MATH] have the same [MATH]-dependence such that the ratios [MATH], [MATH] are independent of [MATH].', '1505.04226-2-17-5': 'We find [EQUATION] and therefore, [EQUATION]', '1505.04226-2-17-6': 'The parameter [MATH] is related to the particle multiplicity to be extracted from the experimental data.', '1505.04226-2-17-7': "For a massless particle species [MATH] ('pion'), the relation is [CITATION] [EQUATION] where [MATH] is the momentum rapidity and [MATH] is the degeneracy factor.", '1505.04226-2-18-0': '## Anisotropic ideal solution', '1505.04226-2-19-0': 'We now perturb the solution anisotropically to introduce the [MATH] dependence.', '1505.04226-2-19-1': 'In doing so, we shall focus on the early time regime [MATH] (or [MATH], see ([REF])).', '1505.04226-2-19-2': 'As observed in [CITATION], in this regime the perturbed solution is fully under analytical control including the viscous case to be discussed in the next section.', '1505.04226-2-20-0': 'Following [CITATION], we consider the following deformation of the isotropic solution [EQUATION] where [EQUATION] is proportional to the spherical harmonics [MATH] in the early time regime [MATH].', '1505.04226-2-20-1': 'Note that we preserve boost invariance [MATH] in this paper, but the case [MATH] was also considered in [CITATION].', '1505.04226-2-20-2': '[MATH] is the eccentricity which we assume to be small [MATH] and keep only linear terms in [MATH].', '1505.04226-2-20-3': '[MATH] and [MATH] have to be determined by solving the hydrodynamic equations linearized around the isotropic solution.', '1505.04226-2-20-4': 'Plugging ([REF]) into ([REF]), we find the following equation for [MATH] [EQUATION]', '1505.04226-2-20-5': 'This turns out to be exactly the same as the equation satisfied by [MATH] [CITATION].', '1505.04226-2-20-6': 'Therefore, in the ideal case we have [MATH], which means that [MATH] and [MATH] are rescaled by the same factor [MATH], [MATH] and [MATH].', '1505.04226-2-20-7': 'The ratios [MATH] and [MATH] are thus unchanged.', '1505.04226-2-20-8': 'At early times [MATH], the right hand side of ([REF]) is negligible and we can set [MATH] [CITATION].', '1505.04226-2-21-0': '## [MATH] at finite [MATH]', '1505.04226-2-22-0': 'In order to compute flow harmonics [MATH], we use the Cooper-Frye formula [CITATION] [EQUATION] where we assumed the Boltzmann distribution and [MATH] is the deviation from the equilibrium distribution.', '1505.04226-2-22-1': 'The use of the Boltzmann distribution (rather than the Fermi/Bose distributions) may be justified for the purpose of computing the integrated [MATH] [CITATION].', '1505.04226-2-22-2': '[MATH] generically represents a set of chemical potentials for net baryon number, isospin and strangeness.', '1505.04226-2-22-3': 'We assign [MATH] for particles with positive/negative quantum numbers mentioned above, and [MATH] for neutral particles with respect to the corresponding quantum number.', '1505.04226-2-22-4': 'In principle, since we are assuming conformal symmetry, the formula ([REF]) should be used only for massless particles, or particles that can be approximately treated as massless (i.e., pions).', '1505.04226-2-22-5': "However, for the sake of discussion in Section [REF], we shall later introduce massive particles and compute their [MATH] in the 'probe approximation', namely, by neglecting their backreaction to the flow velocity.", '1505.04226-2-22-6': 'Since we add in particles in the final state that do not exist in the fluid, the total energy is not conserved at freezeout.', '1505.04226-2-22-7': 'But the fraction of the change [MATH] is exponentially suppressed by the mass [MATH] and will be neglected.', '1505.04226-2-23-0': 'The integral in ([REF]) is taken over the hypersurface [MATH] of constant energy density where the kinetic freezeout occurs.', '1505.04226-2-23-1': 'In the ideal case, constant [MATH] means constant [MATH] since [MATH] is a constant.', '1505.04226-2-23-2': 'Let us write the condition of constant energy density as [EQUATION]', '1505.04226-2-23-3': 'Typically, [MATH] is of the order of the critical energy density of the QCD phase transition.', '1505.04226-2-23-4': 'We take [MATH] in this paper.', '1505.04226-2-23-5': 'Following [CITATION], we assume that the condition ([REF]) is reached within the early time regime [MATH] where we can use the approximate solution ([REF]).', '1505.04226-2-23-6': 'The parameter [MATH] in ([REF]) is then related to the (position-dependent) freezeout time [MATH] as [EQUATION]', '1505.04226-2-23-7': 'For consistency with our early freezeout scenario, we must have [MATH].', '1505.04226-2-24-0': 'Under these assumptions, the integral ([REF]) can be performed analytically and the integrated [MATH] is obtained from the formula [EQUATION]', '1505.04226-2-24-1': 'In the ideal case [MATH], [MATH] does not depend on [MATH] since the factor [MATH] cancels in the ratio ([REF]).', '1505.04226-2-24-2': 'The result is [CITATION] [EQUATION]', '1505.04226-2-24-3': 'This determines the [MATH] dependence of [MATH].', '1505.04226-2-24-4': 'Quite generally, [MATH] is an increasing function [MATH].', '1505.04226-2-24-5': 'On the other hand, [MATH] is a decreasing function of [MATH].', '1505.04226-2-24-6': 'We shall see that, in heavy-ion collisions, the latter dependence is stronger, and as a result ([REF]) is a decreasing function of [MATH], or equivalently, an increasing function of the collision energy [MATH].', '1505.04226-2-24-7': 'Incidentally, we note that the directed flow [MATH] vanishes, consistently with our assumption of boost-invariance.', '1505.04226-2-25-0': '# Viscous case', '1505.04226-2-26-0': 'We now turn to the viscous case [MATH].', '1505.04226-2-26-1': 'Although the system is out of equilibrium, from the Landau matching condition we can define the local [MATH] and [MATH] using the same relations as in equilibrium [EQUATION] but now [MATH] cannot be a constant.', '1505.04226-2-27-0': '## Isotropic viscous solution', '1505.04226-2-28-0': 'First consider the isotropic case [MATH].', '1505.04226-2-28-1': 'Although [MATH] in ([REF]) is not a constant anymore, it depends only on [MATH] (see below).', '1505.04226-2-28-2': 'Then we still have [MATH] so that [EQUATION] is the same as in the ideal case [CITATION].', '1505.04226-2-28-3': 'However, the solution of the Navier-Stokes equation [MATH] has an extra [MATH]-dependence proportional to the shear viscosity [MATH].', '1505.04226-2-28-4': 'In the case of vanishing chemical potentials, this [MATH]-dependence can be obtained exactly [CITATION] [EQUATION] where [MATH] is independent of [MATH].', '1505.04226-2-29-0': 'However, at finite density, [MATH] will depend on [MATH], and this makes it difficult to find an exact solution.', '1505.04226-2-29-1': 'Related to this, [MATH] can now depend on both [MATH] and [MATH], and this relation can be model-dependent.', '1505.04226-2-29-2': 'We can get around this problem by assuming that [MATH] is small.', '1505.04226-2-29-3': 'Specifically, we rescale [MATH] by the entropy density [MATH] [EQUATION] as is often done in hydrodynamic simulations.', '1505.04226-2-29-4': 'We then regard [MATH] as a small parameter ([MATH]) and keep only terms linear in [MATH].', '1505.04226-2-29-5': 'In this approximation, we may replace [MATH] and [MATH] in ([REF]) by their equilibrium values at [MATH], namely, [MATH] and [EQUATION]', '1505.04226-2-29-6': 'We then find the solution valid to [MATH] [EQUATION] where in the second line of ([REF]) we focus on the early-time regime where [MATH] is negative and large.', '1505.04226-2-30-0': 'Using ([REF]) and ([REF]), we can eliminate [MATH] [EQUATION]', '1505.04226-2-30-1': 'Writing [EQUATION] we find the deviation from constancy due to the viscosity [EQUATION] where [EQUATION]', '1505.04226-2-30-2': 'Note that [MATH] as [MATH].', '1505.04226-2-30-3': 'At the freezeout time [MATH], we have the relation [EQUATION]', '1505.04226-2-30-4': 'Finally, we can solve for [MATH] and [MATH] using ([REF]).', '1505.04226-2-30-5': 'The result is [EQUATION]', '1505.04226-2-31-0': '## Anisotropic viscous solution', '1505.04226-2-32-0': 'We now perturb the solution as in ([REF]).', '1505.04226-2-32-1': 'First consider the current in ([REF]).', '1505.04226-2-32-2': '[MATH] now depends not only on [MATH], but also on [MATH] and [MATH].', '1505.04226-2-32-3': 'However, the dependence is of order [MATH].', '1505.04226-2-32-4': '(See ([REF]) below.', '1505.04226-2-32-5': 'Remember that for the ideal solution [MATH] is constant even in the anisotropic case.)', '1505.04226-2-32-6': 'Therefore, if we neglect terms of order [MATH], we can approximate [MATH].', '1505.04226-2-32-7': 'Then ([REF]) is still valid and we get [EQUATION]', '1505.04226-2-32-8': 'As for the energy density, we find [EQUATION] where [CITATION] [EQUATION]', '1505.04226-2-32-9': "From the constant energy condition [EQUATION] we can determine the freezeout surface [MATH] in the viscous case [CITATION] [EQUATION] where the 'Knudsen number' is proportional to the shear viscosity [EQUATION] ([REF]) and ([REF]) are also modified as [MATH] where [EQUATION] and [EQUATION]", '1505.04226-2-33-0': '## [MATH] at finite [MATH] and [MATH]', '1505.04226-2-34-0': 'The computation of [MATH] is more complicated than the [MATH] case.', '1505.04226-2-34-1': 'This is because [MATH] does not mean [MATH], and therefore one cannot treat [MATH] in the Boltzmann factor ([REF]) as a constant when integrating over the hypersurface of constant energy.', '1505.04226-2-34-2': 'In order to cope with this, we write ([REF]) as [EQUATION] where [EQUATION] is constant by virtue of ([REF]).', '1505.04226-2-34-3': 'We then expand the Boltzmann factor as [EQUATION] where we approximated [MATH] in the [MATH] term.', '1505.04226-2-35-0': 'The first term in ([REF]), proportional to unity, gives the same result as in [CITATION] [EQUATION]', '1505.04226-2-35-1': 'Note that [MATH] for [MATH] (see, however, [CITATION]).', '1505.04226-2-35-2': 'The second term in ([REF]) leads to a new order [MATH] contribution to [MATH].', '1505.04226-2-35-3': 'To compute it, we borrow some results from [CITATION].', '1505.04226-2-35-4': 'First, the perturbed flow velocity [MATH] on the freezeout surface has the following components in the coordinates ([REF]) [EQUATION] where [EQUATION] (The viscosity can be neglected here.)', '1505.04226-2-35-5': 'The exponential factor in the Boltzmann distribution reads [EQUATION] where [MATH] is the transverse mass.', '1505.04226-2-35-6': 'The volume element of the constant energy hypersurface is [EQUATION] where [MATH] is given by ([REF]) with the viscous term set to zero.', '1505.04226-2-35-7': 'Finally, we need the more precise version of ([REF]) [EQUATION]', '1505.04226-2-35-8': 'Armed with these formulas, let us decompose the contribution from the second term in ([REF]) as [EQUATION] corresponding to the three terms in ([REF]).', '1505.04226-2-35-9': 'Consider [MATH] first.', '1505.04226-2-35-10': 'To [MATH] we have to evaluate [EQUATION]', '1505.04226-2-35-11': 'This can be efficiently evaluated using the trick introduced in [CITATION] (see Eq. (73) there).', '1505.04226-2-35-12': 'The [MATH]-integral gives Bessel functions [MATH] where [EQUATION]', '1505.04226-2-35-13': 'This can be expanded as [MATH] anticipating that the subsequent [MATH]-integral is dominated by the region [MATH].', '1505.04226-2-35-14': 'We thus find [EQUATION]', '1505.04226-2-35-15': 'The correction to [MATH] can be calculated from the formula (cf. ([REF])) [EQUATION] where [MATH] is the azimuthally symmetric part (cf. Eq. (45) of [CITATION]) [EQUATION]', '1505.04226-2-35-16': 'In the massless case [MATH], the integral can be done exactly and we find [EQUATION] and from ([REF]), [EQUATION]', '1505.04226-2-35-17': 'It is important to emphasize that ([REF]) is induced by the combined effect of the chemical potential and the viscosity.', '1505.04226-2-35-18': 'It vanishes when [MATH] or [MATH] because [MATH] (cf. ([REF])).', '1505.04226-2-35-19': 'Compared with ([REF]) which schematically reads [MATH], we notice that ([REF]) is not enhanced by a factor of [MATH], hence subleading at large [MATH].', '1505.04226-2-35-20': 'However, it is the leading contribution to the difference in [MATH] between particles ([MATH]) and antiparticles ([MATH]).', '1505.04226-2-35-21': 'If [MATH] is the baryon chemical potential, the protons have larger [MATH] than the antiprotons.', '1505.04226-2-35-22': 'We shall study this effect in detail later.', '1505.04226-2-36-0': 'In fact, for protons the approximation [MATH] is not valid.', '1505.04226-2-36-1': 'Instead, we now assume [MATH] and reevaluate [MATH].', '1505.04226-2-36-2': 'Note that when [MATH], [MATH] is parametrically larger than [MATH], so it is enough to consider only [MATH].', '1505.04226-2-37-0': 'When [MATH], the Bessel function is independent of the order [EQUATION] so that ([REF]) becomes [EQUATION]', '1505.04226-2-37-1': 'On the other hand, from Eq. (47) of [CITATION], [EQUATION]', '1505.04226-2-37-2': 'The [MATH]-integral can be evaluated by the saddle point at [MATH] for [MATH] and we obtain [EQUATION]', '1505.04226-2-37-3': 'The [MATH]-independent part is order [MATH], but we shall see later that it is numerically small for realistic values of [MATH] because the factor [MATH] is small.', '1505.04226-2-37-4': 'The [MATH]-dependent term is again of order [MATH] without an enhancement by a factor of [MATH].', '1505.04226-2-38-0': 'The evaluation of [MATH] in ([REF]) can be done similarly, though it is considerably more tedious.', '1505.04226-2-38-1': 'Here we only show the final result in the massless case [MATH], relegating the details to Appendix [EQUATION]', '1505.04226-2-38-2': 'Comparing with ([REF]), we notice that [MATH].', '1505.04226-2-38-3': 'Actually, this relation was repeatedly observed in [CITATION] when computing other contributions to [MATH].', '1505.04226-2-38-4': 'We do not have a simple explanation for this.', '1505.04226-2-39-0': 'Summing all the contributions including the previously computed term [CITATION], our final result of the viscous correction [MATH] in the massless case is [EQUATION]', '1505.04226-2-39-1': 'The second term in the curly brackets is the new contribution at finite density.', '1505.04226-2-39-2': 'It is subleading in [MATH], and actually the factor [MATH] is also numerically small.', '1505.04226-2-39-3': 'However, it gives the leading contribution to the difference in [MATH] between particles and antiparticles.', '1505.04226-2-40-0': '## Isospin chemical potential', '1505.04226-2-41-0': "In the previous subsection, we computed [MATH] of particles which couple to the 'large' chemical potential [MATH].", '1505.04226-2-41-1': 'Here let us compute [MATH] of particles neutral under [MATH] but charged under [MATH].', '1505.04226-2-41-2': 'We have in mind the charged pions [MATH] in the presence of the isospin chemical potential.', '1505.04226-2-41-3': 'We start with the formula (cf. ([REF])) [EQUATION]', '1505.04226-2-41-4': 'We treat [MATH] as a small parameter and keep only terms linear in [MATH].', '1505.04226-2-41-5': 'Dividing by [MATH] from ([REF]) and using [MATH], we find [EQUATION]', '1505.04226-2-41-6': 'The fugacity factor thus becomes [EQUATION]', '1505.04226-2-41-7': 'As before, the factor [MATH] drops out in the computation of [MATH].', '1505.04226-2-41-8': 'We see that the only difference from the previous case ([REF]) is that [MATH] is replaced by [EQUATION]', '1505.04226-2-41-9': 'Thus the final result is the same as ([REF]) except that [MATH] is replaced by ([REF]).', '1505.04226-2-42-0': '# Phenomenological inputs', '1505.04226-2-43-0': 'This section serves as a preparation for the next section where we compare our results with the experimental data.', '1505.04226-2-44-0': '## Models', '1505.04226-2-45-0': 'In order to make quantitative predictions, we need models for the functions [MATH], [MATH], [MATH] defined in ([REF]).', '1505.04226-2-45-1': 'Here we consider two extreme scenarios in terms of the interaction strength.', '1505.04226-2-46-0': '### Free quark-gluon gas', '1505.04226-2-47-0': 'The energy density of free, massless three flavor QCD is [EQUATION] where [MATH], [MATH] and [MATH].', '1505.04226-2-47-1': '[MATH], [MATH] and [MATH] are the baryon, isospin and strangeness chemical potentials, respectively.', '1505.04226-2-47-2': 'Since the net strangeness is zero in heavy-ion collisions, we set [MATH] and obtain ([MATH]) [EQUATION]', '1505.04226-2-47-3': 'It turns out that, due to the large denominators [MATH] or [MATH], the effect of [MATH] and [MATH] on [MATH] is numerically small.', '1505.04226-2-48-0': '### [MATH] SYM at finite [MATH]-charge chemical potential', '1505.04226-2-49-0': 'Next we consider strongly coupled [MATH] supersymmetric Yang-Mills theory at finite [MATH]-charge chemical potential [MATH].', '1505.04226-2-49-1': 'This theory is conformal, and in the limit of strong coupling and at large [MATH], the thermodynamic quantities can be computed from the AdS/CFT correspondence.', '1505.04226-2-49-2': 'The results are [CITATION] [EQUATION] where [MATH] is the [MATH]-charge density.', '1505.04226-2-49-3': 'The shear viscosity is given by [MATH].', '1505.04226-2-50-0': 'There are uncertainties when treating this model as a proxy of strongly coupled QCD, such as the value of [MATH] and the proportionality constant between [MATH] and [MATH].', '1505.04226-2-50-1': 'However, in practical fits, the normalization of [MATH] can be absorbed by a change in [MATH] (cf. Eq. ([REF])).', '1505.04226-2-50-2': 'Moreover, as long as [MATH], the two functions ([REF]) and ([REF]) are qualitatively not so different in shape for [MATH].', '1505.04226-2-50-3': 'As a result, the quality of fits is similar in the two cases despite the huge differences in the underlying dynamics.', '1505.04226-2-50-4': 'Therefore, in the next section we show only the results based on ([REF]) and ([REF]).', '1505.04226-2-51-0': '## Freezeout conditions', '1505.04226-2-52-0': 'We employ the following phenomenological parametrization [CITATION] of the freezeout temperature [MATH] and chemical potential [MATH] (in units of GeV) as a function of the collision energy [MATH] (per nucleon, in units of GeV) [EQUATION] with [MATH], [MATH], [MATH], [MATH], [MATH].', '1505.04226-2-52-1': 'This gives [MATH] as a function of [MATH] as shown in Fig. [REF].', '1505.04226-2-52-2': 'The curve is well approximated by [MATH].', '1505.04226-2-52-3': 'Actually, [MATH] and [MATH] here are the chemical freezeout parameters which are in general different from those entering the Cooper-Frye formula ([REF]) used at the kinetic freezeout.', '1505.04226-2-52-4': 'However, in our model only the ratio [MATH] matters, and this ratio is roughly constant as we have seen.', '1505.04226-2-52-5': 'We thus use the relation in Fig. [REF] for the evaluation of [MATH].', '1505.04226-2-53-0': 'The parameter [MATH] also depends on [MATH] via ([REF]).', '1505.04226-2-53-1': 'We use the following empirical formula for the charged particle multiplicity [CITATION] [EQUATION] where the factor of 2 counts the degeneracy between [MATH] and [MATH].', '1505.04226-2-53-2': 'From ([REF]) and ([REF]), we see that the Knudsen number ([REF]) behaves as [EQUATION] where [MATH] is in units of fermi and [MATH] is in units of GeV/fm[MATH].', '1505.04226-2-53-3': 'Putting aside the potential [MATH]-dependence of [MATH], we see that [MATH] is an increasing function of [MATH] (up to [MATH] in our model) or a decreasing function of [MATH].', '1505.04226-2-54-0': 'In fact, up to the RHIC energy, we find that the following parametrization also gives a good description of the data [CITATION] [EQUATION]', '1505.04226-2-54-1': 'We shall also use this in Section [REF].', '1505.04226-2-55-0': '# Comparison with the experimental data', '1505.04226-2-56-0': 'In this section, we compare our results with three different experimental data: (i) the [MATH]-dependence of [MATH] measured at the LHC; (ii) the collision energy dependence of [MATH] measured at the SPS; (iii) the difference in [MATH] between particles and antiparticles measured at RHIC.', '1505.04226-2-57-0': '## Higher harmonics [MATH]', '1505.04226-2-58-0': 'The CMS collaboration at the LHC has measured the [MATH]-integrated [MATH] in lead-lead collisions at [MATH] TeV up to rather high orders ([MATH]) [CITATION].', '1505.04226-2-58-1': 'Using ([REF]) and ([REF]) together with the phenomenological inputs in the previous section, we can evaluate [MATH] and compare with the CMS data.', '1505.04226-2-58-2': 'The result is shown in Fig. [REF].', '1505.04226-2-58-3': 'Here we set [MATH] for all different values of [MATH].', '1505.04226-2-58-4': 'Taking [MATH] to be independent of [MATH] may be a good approximation for the very central (0-0.2% centrality) nucleus collisions.', '1505.04226-2-58-5': 'The parameter [MATH] is set to [MATH].', '1505.04226-2-58-6': 'The corresponding value of [MATH] in ([REF]) is [MATH] which is consistent with the assumption [MATH].', '1505.04226-2-59-0': 'As a matter of fact, since [MATH] at the LHC, the new term at [MATH] (the term proportional to [MATH] in ([REF])) is negligibly small, and the present fit could have been done in [CITATION] treating [MATH] as a fitting parameter.', '1505.04226-2-59-1': 'By expressing [MATH] in terms of observables as we have done here, we can test our result at lower energies or higher chemical potentials [MATH].', '1505.04226-2-59-2': 'Note that since [MATH] is larger at lower energies, [MATH] [CITATION] decreases faster with [MATH], and this will make the measurement of higher harmonics difficult at low energies [CITATION].', '1505.04226-2-60-0': '## Energy dependence of [MATH]', '1505.04226-2-61-0': 'Next we turn to the energy dependence of the elliptic flow [MATH] for which there are already a wealth of experimental data from the SPS and the RHIC BES program [CITATION].', '1505.04226-2-61-1': 'We compare our formulas ([REF]) and ([REF]) (with [MATH]) for [MATH] with the SPS, mid-central data collected in the low energy region [MATH] [CITATION].', '1505.04226-2-61-2': 'The result with three different values of [MATH] is shown in Fig. [REF] where we tried both ([REF]) and ([REF]), the latter actually gives a better description of [MATH] in this low energy region.', '1505.04226-2-61-3': 'The other parameters are chosen as [MATH]fm and [MATH].', '1505.04226-2-61-4': 'The value of [MATH] here is slightly smaller than the one ([MATH]fm) used in Fig. [REF].', '1505.04226-2-61-5': 'This is consistent with the perception that the QGP droplet is larger at higher energies at the time of thermalization.', '1505.04226-2-61-6': 'The rise of [MATH] with energy is nicely reproduced by our formula and attributed to the rise of [MATH].', '1505.04226-2-61-7': 'It turns out that the newly calculated viscous correction in Section [REF] (the last term in ([REF])) is numerically very small (about an order of magnitude smaller than the first term in ([REF])) even in the highest density region.', '1505.04226-2-62-0': 'Unfortunately, this fit, which agrees reasonably well with the low energy data, overshoots the high energy RHIC data at [MATH] [CITATION] in similar centrality bins by a factor of 2 (assuming that [MATH] is independent of energy).', '1505.04226-2-62-1': 'This is because the rise of [MATH] with energy is too steep.', '1505.04226-2-62-2': 'If we artificially reduce the exponent in ([REF]) as [MATH], for example, we get a decent description of [MATH] over a broader range in [MATH].', '1505.04226-2-62-3': 'Alternatively, the dependence [MATH] from ([REF]) may be too strong, and the experimental data actually suggest a weaker [MATH]-dependence [CITATION].', '1505.04226-2-62-4': 'While we do not have a resolution of this problem in the present framework, it seems qualitatively correct that [MATH] is directly proportional to the multiplicity to some positive power, and therefore it is an increasing function of [MATH] (see, also, Section VII of [CITATION]).', '1505.04226-2-63-0': '## Difference in [MATH] between particles and antiparticles', '1505.04226-2-64-0': 'Finally, we investigate the difference in [MATH] between particles and antiparticles which has been measured by the STAR collaboration at RHIC [CITATION] and attracted some attention from theoretical viewpoints [CITATION].', '1505.04226-2-64-1': 'For a hadron with the quantum numbers [MATH] (baryon number, isospin, strangeness), we assign the fugacity factor [EQUATION] ([MATH] for the strange quark.)', '1505.04226-2-64-2': 'In heavy-ion collisions, [MATH] since the colliding nuclei are neutron rich, and [MATH] since the net strangeness vanishes.', '1505.04226-2-64-3': 'The latter condition implies that we should not treat [MATH] as a small perturbation.', '1505.04226-2-64-4': 'Indeed, various estimates of [MATH] based on the SPS [CITATION] and RHIC [CITATION] data, and also from lattice QCD [CITATION] all found similar values within the range [MATH].', '1505.04226-2-64-5': 'We thus regard [MATH] as a shift of [MATH] for strange hadrons and treat it as a fitting parameter, anticipating that the value of [MATH] should come out in the window [MATH].', '1505.04226-2-64-6': 'On the other hand, we regard [MATH] as a small parameter compared to unity and use the result obtained in Section [REF].', '1505.04226-2-65-0': 'Let us define the difference in [MATH] between hadrons [MATH] and antihadrons [MATH] as [EQUATION]', '1505.04226-2-65-1': 'This can be evaluated from ([REF]) and ([REF]).', '1505.04226-2-65-2': "Focusing now on the elliptic flow case [MATH], we can immediately write down the following 'master formula' [EQUATION]", '1505.04226-2-65-3': 'By construction, ([REF]) has been derived for massless particles.', '1505.04226-2-65-4': 'In the massive case, we observe that the following ratio [EQUATION] is exactly independent of [MATH].', '1505.04226-2-65-5': 'This is due to the nontrivial cancelation of [MATH]-integrals such as ([REF]) in the ratio for the [MATH]-dependent part.', '1505.04226-2-65-6': 'In order to get [MATH] itself, we must multiply ([REF]) by [CITATION] [EQUATION]', '1505.04226-2-65-7': 'The [MATH]-dependence of ([REF]) is sensitive to the cutoffs of the [MATH]-integral, but overall the dependence is not very strong.', '1505.04226-2-65-8': 'For simplicity, in this study we ignore the [MATH]-dependence of [MATH] and use ([REF]) for all hadron species.', '1505.04226-2-65-9': 'It is not difficult to implement this mass effect, but there are other subtleties which are not taken into account, either.', '1505.04226-2-65-10': 'Clearly, it is desirable that the experimental results are plotted in the form ([REF]) in order to avoid various systematic uncertainties.', '1505.04226-2-66-0': 'The most important feature of ([REF]) or ([REF]) is that [MATH] is proportional to both the shear viscosity [MATH] and the chemical potentials.', '1505.04226-2-66-1': '(Remember that [MATH] as defined in ([REF]) is roughly proportional to [MATH].)', '1505.04226-2-66-2': 'This in particular means that [MATH] can be nonzero in viscous hydrodynamics in the presence of the isospin chemical potential.', '1505.04226-2-67-0': 'Let us confront ([REF]) with the data.', '1505.04226-2-67-1': 'The STAR collaboration has measured [MATH] for [MATH] [CITATION].', '1505.04226-2-67-2': 'This is plotted in Fig. [REF] together with our fit based on ([REF]) with [MATH].', '1505.04226-2-67-3': 'We have used [MATH] and [MATH] as in Fig. [REF], and used the fit parameters [MATH] and [MATH], the former is consistent with our expectation mentioned above.', '1505.04226-2-67-4': 'The steep rise of [MATH] for baryons towards the low-[MATH] region is due to the rough proportionality [MATH].', '1505.04226-2-67-5': 'Compared to this, the [MATH]-dependence of the factor [MATH] is subleading.', '1505.04226-2-67-6': 'Since [MATH] have [MATH], [MATH], [MATH], [MATH], respectively, we expect the ordering [MATH] for reasonable values of [MATH] and [MATH].', '1505.04226-2-67-7': 'This tendency is obeyed by most data points except a few in the low energy region.', '1505.04226-2-67-8': "We note that the [MATH] data point at [MATH] GeV should not be taken seriously because, according to the STAR collaboration [CITATION], this data point is afflicted with 'additional systematic effects which are not included in the error bars'.", '1505.04226-2-67-9': 'In Fig. [REF], we have also included our prediction for the [MATH]-baryon.', '1505.04226-2-67-10': 'Since [MATH] has [MATH], we expect that [MATH] is smaller than other baryons.', '1505.04226-2-68-0': 'Concerning the pions, the negative [MATH] can be naturally explained by the negative isospin chemical potential.', '1505.04226-2-68-1': 'However, the magnitude is problematic.', '1505.04226-2-68-2': 'Our choice [MATH], which describes the pion data very well, is too large compared with the value [MATH] extracted from the SPS data [CITATION].', '1505.04226-2-68-3': 'We may dial [MATH] down to, say, [MATH] without spoiling much the quality of the [MATH] fit, but not further down.', '1505.04226-2-68-4': 'On the other hand, the other hadrons ([MATH]) are more or less unaffected by [MATH] and can be well fitted even with [MATH] and [MATH].', '1505.04226-2-68-5': 'This may be an indication that there are other mechanisms to generate the difference [MATH] which predominantly act on the pions.', '1505.04226-2-69-0': 'In the large-[MATH] region, our result tends to slightly overestimate [MATH].', '1505.04226-2-69-1': 'This is partly due to the too fast rise of [MATH] with energy as mentioned before.', '1505.04226-2-69-2': 'However, in Fig. [REF] we assumed that [MATH] is independent of [MATH].', '1505.04226-2-69-3': 'A recent hydrodynamic simulation suggests that [MATH] is a decreasing function of [MATH] [CITATION], and this could alleviate the (small) discrepancy in the large-[MATH] region (remember that [MATH]).', '1505.04226-2-70-0': '# Summary and conclusions', '1505.04226-2-71-0': 'In this paper, we have revealed, in a completely analytical manner, a number of interesting features about the nature of hydrodynamics in the presence of conserved currents as well as the chemical potential (collision energy) dependence of the flow harmonics [MATH].', '1505.04226-2-71-1': 'Let us summarize the main findings.', '1505.04226-2-72-0': 'Presumably some of the above features are empirically well known to the experts of hydrodynamic simulations.', '1505.04226-2-72-1': 'However, they have not been systematically derived with the level of analytical detail presented in this work.', '1505.04226-2-73-0': 'There are a number of directions for future work.', '1505.04226-2-73-1': 'Admittedly, the assumptions of boost invariance and conformal invariance are too simplistic, especially at high density.', '1505.04226-2-73-2': 'One has to relax these approximations to be more realistic.', '1505.04226-2-73-3': 'Related to this, we only considered the conformal equation of state [MATH] where the function [MATH] does not carry any information about the crossover and possibly first order phase transitions at finite density.', '1505.04226-2-73-4': '(Nevertheless it is remarkable that we can explain many features of [MATH] measured at different energies without such information.)', '1505.04226-2-73-5': 'It is important to figure out how the presence of phase transitions in [MATH] is encoded in the observed behavior of [MATH].', '1505.04226-2-73-6': 'Including the effects of anomaly (see, e.g., [CITATION]) is also interesting.', '1505.04226-2-73-7': 'We hope to address these questions in future work.'}

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['1505.04226-1-64-1', '1505.04226-2-65-1'], ['1505.04226-1-64-2', '1505.04226-2-65-2'], ['1505.04226-1-64-3', '1505.04226-2-65-3'], ['1505.04226-1-64-4', '1505.04226-2-65-4'], ['1505.04226-1-64-5', '1505.04226-2-65-5'], ['1505.04226-1-64-6', '1505.04226-2-65-6'], ['1505.04226-1-64-7', '1505.04226-2-65-7'], ['1505.04226-1-64-8', '1505.04226-2-65-8'], ['1505.04226-1-64-9', '1505.04226-2-65-9'], ['1505.04226-1-64-10', '1505.04226-2-65-10'], ['1505.04226-1-68-0', '1505.04226-2-69-0'], ['1505.04226-1-68-1', '1505.04226-2-69-1'], ['1505.04226-1-68-2', '1505.04226-2-69-2'], ['1505.04226-1-68-3', '1505.04226-2-69-3'], ['1505.04226-1-27-0', '1505.04226-2-28-0'], ['1505.04226-1-27-1', '1505.04226-2-28-1'], ['1505.04226-1-27-2', '1505.04226-2-28-2'], ['1505.04226-1-27-3', '1505.04226-2-28-3'], ['1505.04226-1-27-4', '1505.04226-2-28-4'], ['1505.04226-1-12-0', '1505.04226-2-12-0'], ['1505.04226-1-12-1', '1505.04226-2-12-1'], ['1505.04226-1-12-2', '1505.04226-2-12-2'], ['1505.04226-1-12-3', '1505.04226-2-12-3'], ['1505.04226-1-12-4', '1505.04226-2-12-4'], ['1505.04226-1-34-0', '1505.04226-2-35-0'], ['1505.04226-1-34-2', '1505.04226-2-35-2'], ['1505.04226-1-34-3', '1505.04226-2-35-3'], ['1505.04226-1-34-4', '1505.04226-2-35-4'], ['1505.04226-1-34-5', '1505.04226-2-35-5'], ['1505.04226-1-34-6', '1505.04226-2-35-6'], ['1505.04226-1-34-7', '1505.04226-2-35-7'], ['1505.04226-1-34-8', '1505.04226-2-35-8'], ['1505.04226-1-34-10', '1505.04226-2-35-10'], ['1505.04226-1-34-11', '1505.04226-2-35-11'], ['1505.04226-1-34-12', '1505.04226-2-35-12'], ['1505.04226-1-34-13', '1505.04226-2-35-13'], ['1505.04226-1-34-14', '1505.04226-2-35-14'], ['1505.04226-1-34-15', '1505.04226-2-35-15'], ['1505.04226-1-34-16', '1505.04226-2-35-16'], ['1505.04226-1-34-17', '1505.04226-2-35-17'], ['1505.04226-1-34-18', '1505.04226-2-35-18'], ['1505.04226-1-34-19', '1505.04226-2-35-19'], ['1505.04226-1-34-20', '1505.04226-2-35-20'], ['1505.04226-1-34-21', '1505.04226-2-35-21'], ['1505.04226-1-34-22', '1505.04226-2-35-22'], ['1505.04226-1-36-0', '1505.04226-2-37-0'], ['1505.04226-1-36-1', '1505.04226-2-37-1'], ['1505.04226-1-36-2', '1505.04226-2-37-2'], ['1505.04226-1-36-3', '1505.04226-2-37-3'], ['1505.04226-1-36-4', '1505.04226-2-37-4'], ['1505.04226-1-48-0', '1505.04226-2-49-0'], ['1505.04226-1-48-1', '1505.04226-2-49-1'], ['1505.04226-1-48-2', '1505.04226-2-49-2'], ['1505.04226-1-48-3', '1505.04226-2-49-3'], ['1505.04226-1-40-0', '1505.04226-2-41-0'], ['1505.04226-1-40-1', '1505.04226-2-41-1'], ['1505.04226-1-40-2', '1505.04226-2-41-2'], ['1505.04226-1-40-3', '1505.04226-2-41-3'], ['1505.04226-1-40-4', '1505.04226-2-41-4'], ['1505.04226-1-40-5', '1505.04226-2-41-5'], ['1505.04226-1-40-6', '1505.04226-2-41-6'], ['1505.04226-1-40-7', '1505.04226-2-41-7'], ['1505.04226-1-40-8', '1505.04226-2-41-8'], ['1505.04226-1-40-9', '1505.04226-2-41-9'], ['1505.04226-1-13-0', '1505.04226-2-13-0'], ['1505.04226-1-13-2', '1505.04226-2-13-2'], ['1505.04226-1-13-3', '1505.04226-2-13-3'], ['1505.04226-1-13-4', '1505.04226-2-13-4'], ['1505.04226-1-10-0', '1505.04226-2-10-0'], ['1505.04226-1-10-1', '1505.04226-2-10-1'], ['1505.04226-1-10-2', '1505.04226-2-10-2'], ['1505.04226-1-10-3', '1505.04226-2-10-3'], ['1505.04226-1-10-4', '1505.04226-2-10-4'], ['1505.04226-1-10-5', '1505.04226-2-10-5'], ['1505.04226-1-10-6', '1505.04226-2-10-6'], ['1505.04226-1-6-0', '1505.04226-2-6-0'], ['1505.04226-1-6-1', '1505.04226-2-6-1'], ['1505.04226-1-6-2', '1505.04226-2-6-2'], ['1505.04226-1-6-3', '1505.04226-2-6-3'], ['1505.04226-1-6-4', '1505.04226-2-6-4'], ['1505.04226-1-6-5', '1505.04226-2-6-5'], ['1505.04226-1-38-0', '1505.04226-2-39-0'], ['1505.04226-1-38-1', '1505.04226-2-39-1'], ['1505.04226-1-38-2', '1505.04226-2-39-2'], ['1505.04226-1-38-3', '1505.04226-2-39-3'], ['1505.04226-1-17-0', '1505.04226-2-17-0'], ['1505.04226-1-17-1', '1505.04226-2-17-1'], ['1505.04226-1-17-2', '1505.04226-2-17-2'], ['1505.04226-1-17-3', '1505.04226-2-17-3'], ['1505.04226-1-17-4', '1505.04226-2-17-4'], ['1505.04226-1-17-5', '1505.04226-2-17-5'], ['1505.04226-1-17-6', '1505.04226-2-17-6'], ['1505.04226-1-17-7', '1505.04226-2-17-7'], ['1505.04226-1-49-0', '1505.04226-2-50-0'], ['1505.04226-1-49-1', '1505.04226-2-50-1'], ['1505.04226-1-49-2', '1505.04226-2-50-2'], ['1505.04226-1-49-3', '1505.04226-2-50-3'], ['1505.04226-1-49-4', '1505.04226-2-50-4'], ['1505.04226-1-52-0', '1505.04226-2-53-0'], ['1505.04226-1-52-1', '1505.04226-2-53-1'], ['1505.04226-1-52-2', '1505.04226-2-53-2'], ['1505.04226-1-52-3', '1505.04226-2-53-3'], ['1505.04226-1-20-0', '1505.04226-2-20-0'], ['1505.04226-1-20-1', '1505.04226-2-20-1'], ['1505.04226-1-20-2', '1505.04226-2-20-2'], ['1505.04226-1-20-3', '1505.04226-2-20-3'], ['1505.04226-1-20-4', '1505.04226-2-20-4'], ['1505.04226-1-20-5', '1505.04226-2-20-5'], ['1505.04226-1-20-6', '1505.04226-2-20-6'], ['1505.04226-1-20-7', '1505.04226-2-20-7'], ['1505.04226-1-20-8', '1505.04226-2-20-8'], ['1505.04226-1-56-1', '1505.04226-2-58-1'], ['1505.04226-1-56-2', '1505.04226-2-58-2'], ['1505.04226-1-56-3', '1505.04226-2-58-3'], ['1505.04226-1-56-4', '1505.04226-2-58-4'], ['1505.04226-1-56-5', '1505.04226-2-58-5'], ['1505.04226-1-56-6', '1505.04226-2-58-6'], ['1505.04226-1-0-1', '1505.04226-2-0-1'], ['1505.04226-1-0-2', '1505.04226-2-0-2'], ['1505.04226-1-0-3', '1505.04226-2-0-3'], ['1505.04226-1-25-0', '1505.04226-2-26-0'], ['1505.04226-1-25-1', '1505.04226-2-26-1'], ['1505.04226-1-70-0', '1505.04226-2-71-0'], ['1505.04226-1-70-1', '1505.04226-2-71-1'], ['1505.04226-1-3-0', '1505.04226-2-3-0'], ['1505.04226-1-3-1', '1505.04226-2-3-1'], ['1505.04226-1-3-2', '1505.04226-2-3-2'], ['1505.04226-1-3-3', '1505.04226-2-3-3'], ['1505.04226-1-3-4', '1505.04226-2-3-4'], ['1505.04226-1-15-0', '1505.04226-2-15-0'], ['1505.04226-1-15-1', '1505.04226-2-15-1'], ['1505.04226-1-4-0', '1505.04226-2-4-0'], ['1505.04226-1-4-1', '1505.04226-2-4-1'], ['1505.04226-1-4-2', '1505.04226-2-4-2'], ['1505.04226-1-4-3', '1505.04226-2-4-3'], ['1505.04226-1-4-4', 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'1505.04226-2-58-2'], ['1505.04226-1-56-3', '1505.04226-2-58-3'], ['1505.04226-1-56-4', '1505.04226-2-58-4'], ['1505.04226-1-56-5', '1505.04226-2-58-5'], ['1505.04226-1-56-6', '1505.04226-2-58-6'], ['1505.04226-1-0-1', '1505.04226-2-0-1'], ['1505.04226-1-0-2', '1505.04226-2-0-2'], ['1505.04226-1-0-3', '1505.04226-2-0-3'], ['1505.04226-1-25-0', '1505.04226-2-26-0'], ['1505.04226-1-25-1', '1505.04226-2-26-1'], ['1505.04226-1-70-0', '1505.04226-2-71-0'], ['1505.04226-1-70-1', '1505.04226-2-71-1'], ['1505.04226-1-3-0', '1505.04226-2-3-0'], ['1505.04226-1-3-1', '1505.04226-2-3-1'], ['1505.04226-1-3-2', '1505.04226-2-3-2'], ['1505.04226-1-3-3', '1505.04226-2-3-3'], ['1505.04226-1-3-4', '1505.04226-2-3-4'], ['1505.04226-1-15-0', '1505.04226-2-15-0'], ['1505.04226-1-15-1', '1505.04226-2-15-1'], ['1505.04226-1-4-0', '1505.04226-2-4-0'], ['1505.04226-1-4-1', '1505.04226-2-4-1'], ['1505.04226-1-4-2', '1505.04226-2-4-2'], ['1505.04226-1-4-3', '1505.04226-2-4-3'], ['1505.04226-1-4-4', '1505.04226-2-4-4'], ['1505.04226-1-19-0', '1505.04226-2-19-0'], ['1505.04226-1-19-1', '1505.04226-2-19-1'], ['1505.04226-1-19-2', '1505.04226-2-19-2'], ['1505.04226-1-51-0', '1505.04226-2-52-0'], ['1505.04226-1-51-1', '1505.04226-2-52-1'], ['1505.04226-1-51-2', '1505.04226-2-52-2'], ['1505.04226-1-51-3', '1505.04226-2-52-3'], ['1505.04226-1-51-4', '1505.04226-2-52-4'], ['1505.04226-1-51-5', '1505.04226-2-52-5'], ['1505.04226-1-9-0', '1505.04226-2-9-0'], ['1505.04226-1-9-1', '1505.04226-2-9-1'], ['1505.04226-1-9-2', '1505.04226-2-9-2'], ['1505.04226-1-9-3', '1505.04226-2-9-3'], ['1505.04226-1-9-4', '1505.04226-2-9-4'], ['1505.04226-1-9-5', '1505.04226-2-9-5'], ['1505.04226-1-9-6', '1505.04226-2-9-6'], ['1505.04226-1-60-0', '1505.04226-2-62-0'], ['1505.04226-1-60-2', '1505.04226-2-62-2'], ['1505.04226-1-60-3', '1505.04226-2-62-3'], ['1505.04226-1-60-4', '1505.04226-2-62-4'], ['1505.04226-1-72-0', '1505.04226-2-73-0'], ['1505.04226-1-72-1', '1505.04226-2-73-1'], ['1505.04226-1-72-2', '1505.04226-2-73-2'], ['1505.04226-1-72-3', '1505.04226-2-73-3'], ['1505.04226-1-72-4', '1505.04226-2-73-4'], ['1505.04226-1-72-5', '1505.04226-2-73-5'], ['1505.04226-1-72-6', '1505.04226-2-73-6'], ['1505.04226-1-72-7', '1505.04226-2-73-7'], ['1505.04226-1-37-0', '1505.04226-2-38-0'], ['1505.04226-1-37-1', '1505.04226-2-38-1'], ['1505.04226-1-37-2', '1505.04226-2-38-2'], ['1505.04226-1-37-3', '1505.04226-2-38-3'], ['1505.04226-1-37-4', '1505.04226-2-38-4'], ['1505.04226-1-28-0', '1505.04226-2-29-0'], ['1505.04226-1-28-1', '1505.04226-2-29-1'], ['1505.04226-1-28-2', '1505.04226-2-29-2'], ['1505.04226-1-28-3', '1505.04226-2-29-3'], ['1505.04226-1-28-4', '1505.04226-2-29-4'], ['1505.04226-1-28-5', '1505.04226-2-29-5'], ['1505.04226-1-31-0', '1505.04226-2-32-0'], ['1505.04226-1-31-1', '1505.04226-2-32-1'], ['1505.04226-1-31-2', '1505.04226-2-32-2'], ['1505.04226-1-31-3', '1505.04226-2-32-3'], ['1505.04226-1-31-5', '1505.04226-2-32-5'], ['1505.04226-1-31-6', '1505.04226-2-32-6'], ['1505.04226-1-31-7', '1505.04226-2-32-7'], ['1505.04226-1-31-8', '1505.04226-2-32-8'], ['1505.04226-1-31-9', '1505.04226-2-32-9'], ['1505.04226-1-63-0', '1505.04226-2-64-0'], ['1505.04226-1-63-1', '1505.04226-2-64-1'], ['1505.04226-1-63-2', '1505.04226-2-64-2'], ['1505.04226-1-63-3', '1505.04226-2-64-3'], ['1505.04226-1-63-4', '1505.04226-2-64-4'], ['1505.04226-1-63-5', '1505.04226-2-64-5'], ['1505.04226-1-63-6', '1505.04226-2-64-6'], ['1505.04226-1-22-0', '1505.04226-2-22-0'], ['1505.04226-1-22-1', '1505.04226-2-22-1'], ['1505.04226-1-22-2', '1505.04226-2-22-2'], ['1505.04226-1-22-3', '1505.04226-2-22-3'], ['1505.04226-1-22-4', '1505.04226-2-23-0'], ['1505.04226-1-22-5', '1505.04226-2-23-1'], ['1505.04226-1-22-6', '1505.04226-2-23-2'], ['1505.04226-1-22-7', '1505.04226-2-23-3'], ['1505.04226-1-22-8', '1505.04226-2-23-4'], ['1505.04226-1-22-9', '1505.04226-2-23-5'], ['1505.04226-1-22-10', '1505.04226-2-23-6'], ['1505.04226-1-22-11', '1505.04226-2-23-7']]

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[]

[['1505.04226-1-59-2', '1505.04226-2-61-2'], ['1505.04226-1-34-1', '1505.04226-2-35-1'], ['1505.04226-1-28-6', '1505.04226-2-29-6']]

[]

['1505.04226-1-13-1', '1505.04226-1-31-4', '1505.04226-1-34-9', '1505.04226-1-61-1', '1505.04226-2-13-1', '1505.04226-2-32-4', '1505.04226-2-35-9']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1505.04226

1602.08739

{'1602.08739-1-0-0': 'It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra.', '1602.08739-1-0-1': 'In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.', '1602.08739-1-1-0': '# Introduction', '1602.08739-1-2-0': 'A Frobenius algebra, or more generally, a Frobenius object in a monoidal category [CITATION], is special when the comultiplication followed by the multiplication is the identity map.', '1602.08739-1-2-1': 'In the categorical approach to quantum communication initiated by Abramsky and Coecke [CITATION], and developed extensively by the Oxford school, the special commutative Frobenius objects play the role of classical information channels in quantum protocols, with the comultiplication and counit modeling the possibility of duplicating and discarding information [CITATION].', '1602.08739-1-3-0': 'A symmetric monoidal category containing the universal special commutative Frobenius algebra was exhibited by Lack [CITATION], and independently by Rosebrugh, Sabadini and Walters [CITATION] in their study of general processes: it is the category whose objects are finite sets and whose morphisms are cospans; the composition of cospans is given by pushout.', '1602.08739-1-3-1': 'Special commutative Frobenius objects and cospan categories have also been studied recently by Baez and Fong [CITATION] in the context of electrical network theory.', '1602.08739-1-4-0': 'In this note we observe that "specialness" is a result of the simple-minded nature of pushouts in the category of sets: if one replaces pushouts by homotopy pushouts, the resulting cospan category contains the universal commutative Frobenius object, rather than the special one.', '1602.08739-1-4-1': 'To make sense of the homotopy pushout, a homotopical setting is required.', '1602.08739-1-4-2': 'The minimal setting in which the construction works is that of finite [MATH]-dimensional CW complexes up to homotopy, i.e. topological graphs, the endpoints of the cospans being still just finite sets.', '1602.08739-1-4-3': 'Graph cospans like this have a long history in the theory of graph rewriting (an early explicit reference being [CITATION]), but hitherto, as far as we know, only set-theoretical pushouts have been considered.', '1602.08739-1-5-0': 'In light of the growing importance of homotopy viewpoints in modern mathematical sciences, we wonder whether the finer composition of cospans, provided by the homotopy pushout, may be of relevance in the various application areas of the cospan model.', '1602.08739-1-5-1': 'We leave the investigation of these issues to the experts in the respective fields.', '1602.08739-1-6-0': '# Cobordisms versus cospans', '1602.08739-1-7-0': 'To put the cospan construction in perspective, and to explain its homotopy version, we exploit the standard result that the universal commutative Frobenius algebra lives in the symmetric monoidal category [MATH] of 2-dimensional cobordisms [CITATION].', '1602.08739-1-7-1': '(A corollary is the notorious equivalence between 2d topological quantum field theories and commutative Frobenius algebras.)', '1602.08739-1-8-0': 'The result essentially amounts to writing the axioms for commutative Frobenius algebras in graphical language, and observing that they precisely express evident topological properties of cobordisms.', '1602.08739-1-8-1': 'We shall not repeat the arguments (nor even the axioms) here, but content ourselves to list the graphical building blocks corresponding to multiplication and unit, comultiplication and counit, and the corresponding cospans: [EQUATION]', '1602.08739-1-8-2': 'The analogy shown in [REF] between 2d cobordisms and cospans [MATH] of finite sets is clear: [MATH] corresponds to the set of in-boundaries, [MATH] corresponds to the set of out-boundaries, and [MATH] corresponds to the set of connected components of a cobordism.', '1602.08739-1-8-3': 'Composition of cobordisms is given by gluing at the boundaries (in fact a pushout operation); cospans compose by pushout.', '1602.08739-1-9-0': 'However, the analogy breaks down quickly in more complicated situations, because the set [MATH] does not contain information about the genus of each component of a cobordism.', '1602.08739-1-9-1': 'The crucial difference can be pinpointed to the following basic composition of cospans of sets, expressing precisely the axiom characterizing special Frobenius algebras among all Frobenius algebras: [EQUATION]', '1602.08739-1-9-2': 'We see that cospans (in the category of sets) cannot render the idea of a "hole" (as resulting from the corresponding composition of cobordisms,', '1602.08739-1-10-0': '0.35 (0 0) 2.5 6.25 (0 0) rx: ry: (40 -15) rx: ry: (40 15) rx: ry: 0 0 - 15 15 - -15 -15 - (40 ) (15 )(20 )(0 ) (40 ) (22 )(22 )(40 ) (40 ) (15 )(20 )(0 ) (40 -15)(*ex *ey) (*ex *ey) 2.5 6.25 (0 0) rx: ry: (0 30) rx: ry: (40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey) (120 0)[MATH] (160 0) 2.5 6.25 (40 -15) (-40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (-25 )(-20 )(-40 ) (0 ) (-18 )(-18 )(0 ) (0 ) (-25 )(-20 )(-40 ) (0 0)(*ex *ey) (*ex *ey)', '1602.08739-1-11-0': '(40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey)', '1602.08739-1-12-0': '(*ex *ey) (280 0)[MATH] (320 0) 2.5 6.25 (0 0) rx: ry: (40 0) rx: ry: 0 0 - (0 ) (40 0) (0 ) (40 0) (40 0)(*ex *ey) (*ex *ey)', '1602.08739-1-13-0': ').', '1602.08739-1-14-0': '# Homotopy pushouts and finite [MATH]-dimensional CW-complexes', '1602.08739-1-15-0': 'The general idea of homotopy pushouts and homotopy quotients, which has become an essential slogan in higher category theory [CITATION], and more recently in type theory [CITATION], is that instead of equating elements, one sews in a path between them, so as to keep track of the fact that two elements could be equal in more than one way.', '1602.08739-1-15-1': 'When applied to the composition of cospans, the homotopy-pushout operation matches well with the various interpretations of cospans in applications: to connect two "devices", draw "cables" from the outputs of the first to the inputs of the second.', '1602.08739-1-16-0': 'Let [MATH] and [MATH] be topological spaces, and let [MATH] be a finite set, considered as a discrete space.', '1602.08739-1-16-1': 'The homotopy pushout [MATH] of a span [MATH], [EQUATION] is obtained by sewing a path in [MATH] from [MATH] to [MATH], for each point [MATH].', '1602.08739-1-16-2': 'For example, the homotopical version of the pushout in [REF] is shown below, clearly corresponding to 0.35 2.5 6.25 (40 -15) (-40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (-25 )(-20 )(-40 ) (0 ) (-18 )(-18 )(0 ) (0 ) (-25 )(-20 )(-40 ) (0 0)(*ex *ey) (*ex *ey)', '1602.08739-1-17-0': '(40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey)', '1602.08739-1-18-0': '(*ex *ey) : [EQUATION]', '1602.08739-1-19-0': 'We see in particular, that even when the spaces [MATH], [MATH], and [MATH] are all discrete, the homotopy pushout [MATH] will not in general be discrete.', '1602.08739-1-19-1': 'To accommodate the homotopy pushout, we need a category of cospans with discrete endpoints, but with apex at least a homotopy [MATH]-type.', '1602.08739-1-19-2': 'Recall that a finite [MATH]-dimensional CW-complex is a space obtained from a finite set of points by attaching a finite set of segments by their endpoints.', '1602.08739-1-19-3': 'The homotopy pushout described above is a paradigmatic example.', '1602.08739-1-19-4': 'A finite [MATH]-dimensional CW-complex is thus a topological graph; if it is connected it is characterized up to homotopy by its first Betti number, the number of independent cycles.', '1602.08739-1-19-5': 'Any connected finite [MATH]-dimensional CW-complex is therefore homotopy equivalent to a bouquet of circles.', '1602.08739-1-20-0': 'Let [MATH] denote the symmetric monoidal category whose objects are finite sets, and whose arrows are homotopy classes of cospans [MATH], where [MATH] is a finite [MATH]-dimensional CW-complex.', '1602.08739-1-20-1': 'The monoidal structure is given by disjoint union.', '1602.08739-1-20-2': 'It is easy to see that such cospans are closed under composition given by homotopy pushout, since this operation attaches [MATH]-cells, but nothing higher-dimensional.', '1602.08739-1-20-3': 'Since we only consider cospans up to homotopy, it is also clear that composition is associative and unital.', '1602.08739-1-21-0': '# Homotopy cospans and cobordisms', '1602.08739-1-22-0': 'Theorem.', '1602.08739-1-22-1': 'The symmetric monoidal category [MATH], whose objects are finite sets and whose morphisms are homotopy classes of cospans, is the free symmetric monoidal category on a commutative Frobenius object.', '1602.08739-1-23-0': 'The universal property can be established directly, without reference to cobordisms, by copying over the proof in [CITATION] almost verbatim.', '1602.08739-1-23-1': 'This is possible since the generators and relations of [MATH] as a symmetric monoidal category are readily seen to correspond exactly to the axioms for commutative Frobenius algebras.', '1602.08739-1-23-2': 'The key point here is that the invariants detecting whether two connected cospans are homotopic are the same as those classifying connected [MATH]-cobordisms, as used in [CITATION].', '1602.08739-1-24-0': 'It is also possible to establish this result without mention of generators and relations, by means of a geometric formalization of the analogy between cospans and cobordisms, establishing an equivalence of symmetric monoidal categories [MATH].', '1602.08739-1-24-1': 'This provides an alternative proof of the Theorem, since [MATH] is already known to have the universal property ([CITATION], Theorem 3.6.19).', '1602.08739-1-25-0': 'We first need to enhance the geometric realization of a cospan [MATH], by separating out its input and output points.', '1602.08739-1-25-1': 'We do that systematically by replacing [MATH] by the mapping cylinder of [MATH]; this amounts to attaching a [MATH]-cell from each point in [MATH] to its image point in [MATH], and produces thus a homotopy equivalent graph whose input and output points are all pairwise disjoint.', '1602.08739-1-26-0': 'As an example, here is the replacement for the cospan representing multiplication: [EQUATION] already considerably strengthening the analogy with', '1602.08739-1-27-0': '0.35 (0 0) 2.5 6.25 (0 0) rx: ry: (0 30) rx: ry: (40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey) .', '1602.08739-1-27-1': '(At this point it is worth recalling that a cobordism is also in fact a kind of cospan (cf. [CITATION], 1.2): it is the surface with boundary, together with specific inclusions [MATH] from closed [MATH]-manifolds onto the boundary.)', '1602.08739-1-28-0': 'With this preparatory mapping-cylinder step, we can describe the correspondence geometrically in Euclidean [MATH]-space.', '1602.08739-1-28-1': 'Given a cospan we choose an embedding of the mapping-cylinder graph into [MATH], with its input points at level [MATH] and output points at level [MATH].', '1602.08739-1-28-2': '(It is not important how exactly the embedding is arranged-even the isotopy class does not matter.)', '1602.08739-1-28-3': 'In order to construct a cobordism from this, we first take the boundary of a tubular neighborhood of the embedded graph as a subset of [MATH]; specifically we can take the set of points in [MATH] at fixed small distance [MATH] from the graph.', '1602.08739-1-28-4': 'This defines a topological surface, which we intersect with [MATH] to obtain a cobordism, whose boundary consists of small circles around the input and output points of the graph.', '1602.08739-1-29-0': 'Conversely, any cobordism can be embedded in [MATH] with in-boundaries at level [MATH] and out-boundaries at level [MATH].', '1602.08739-1-29-1': 'Its interior admits a deformation retract onto a graph, commuting (at the boundary levels [MATH]) with the deformation retract of the open disks onto their center points; the result is unique up to homotopy.', '1602.08739-1-29-2': 'With boundary points determined as those at levels [MATH] and [MATH], this defines a cospan.', '1602.08739-1-29-3': 'It is clear that these two constructions are mutually inverse (up to the equivalences in question: homotopy equivalence of graph cospans, and homeomorphism rel the boundary for cobordisms).', '1602.08739-1-30-0': 'Finally, it is clear from the geometry that composition of cospans corresponds precisely to composition of cobordisms: Given composable cospans [MATH] and [MATH], the composite given by the homotopy pushout can be realized geometrically by embedding (the mapping cylinders of) [MATH] into [MATH] and [MATH] into [MATH] and sewing in [MATH]-cells [MATH] in [MATH].', '1602.08739-1-30-1': 'Cobordisms can be composed in exactly the same way, by sewing in cylinders in [MATH] between the boundary circles at level [MATH] and [MATH].'}

{'1602.08739-2-0-0': 'It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra.', '1602.08739-2-0-1': 'In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.', '1602.08739-2-1-0': '# Introduction', '1602.08739-2-2-0': 'A Frobenius algebra, or more generally, a Frobenius object in a monoidal category [CITATION], is special when the comultiplication followed by the multiplication is the identity map.', '1602.08739-2-2-1': 'A symmetric monoidal category containing the universal special commutative Frobenius algebra was exhibited by Lack [CITATION], and independently by Rosebrugh, Sabadini and Walters [CITATION] in their study of general processes: it is the category whose objects are finite sets and whose morphisms are cospans; the composition of cospans is given by pushout.', '1602.08739-2-2-2': 'Special commutative Frobenius objects and cospan categories have also been studied recently by Baez and Fong [CITATION] in the context of electrical network theory, and Coya and Fong [CITATION] have shown that the category of jointly surjective cospans contains the universal extraspecial commutative Frobenius algebra, meaning that also unit-followed-by-counit is required to be the identity.', '1602.08739-2-3-0': 'In this note we observe that "specialness" is a result of the simple-minded nature of pushouts in the category of sets: if one replaces pushouts by homotopy pushouts, the resulting cospan category contains the universal commutative Frobenius object, rather than the special one.', '1602.08739-2-3-1': 'To make sense of the homotopy pushout, a homotopical setting is required.', '1602.08739-2-3-2': 'The minimal setting in which the construction works is that of finite [MATH]-dimensional CW complexes up to homotopy, i.e. topological graphs, the endpoints of the cospans being still just finite sets.', '1602.08739-2-3-3': 'Graph cospans like this have a long history in the theory of graph rewriting (an early explicit reference being [CITATION]), but hitherto, as far as we know, only set-theoretical pushouts have been considered.', '1602.08739-2-4-0': 'In light of the growing importance of homotopy viewpoints in modern mathematical sciences, we wonder whether the finer composition of cospans, provided by the homotopy pushout, may be of relevance in the various application areas of the cospan model.', '1602.08739-2-4-1': 'We leave the investigation of these issues to the experts in the respective fields.', '1602.08739-2-5-0': '# Cobordisms versus cospans', '1602.08739-2-6-0': 'To put the cospan construction in perspective, and to explain its homotopy version, we exploit the standard result that the universal commutative Frobenius algebra lives in the symmetric monoidal category [MATH] of 2-dimensional cobordisms [CITATION].', '1602.08739-2-6-1': '(A corollary is the notorious equivalence between 2d topological quantum field theories and commutative Frobenius algebras.)', '1602.08739-2-7-0': 'The result essentially amounts to writing the axioms for commutative Frobenius algebras in graphical language, and observing that they precisely express evident topological properties of cobordisms.', '1602.08739-2-7-1': 'We shall not repeat the arguments (nor even the axioms) here, but content ourselves to list the graphical building blocks corresponding to multiplication and unit, comultiplication and counit, and the corresponding cospans: [EQUATION]', '1602.08739-2-7-2': 'The analogy shown in [REF] between 2d cobordisms and cospans [MATH] of finite sets is clear: [MATH] corresponds to the set of in-boundaries, [MATH] corresponds to the set of out-boundaries, and [MATH] corresponds to the set of connected components of a cobordism.', '1602.08739-2-7-3': 'Composition of cobordisms is given by gluing at the boundaries (in fact a pushout operation); cospans compose by pushout.', '1602.08739-2-8-0': 'However, the analogy breaks down quickly in more complicated situations, because the set [MATH] does not contain information about the genus of each component of a cobordism.', '1602.08739-2-8-1': 'The crucial difference can be pinpointed to the following basic composition of cospans of sets, expressing precisely the axiom characterizing special Frobenius algebras among all Frobenius algebras: [EQUATION]', '1602.08739-2-8-2': 'We see that cospans (in the category of sets) cannot render the idea of a "hole" (as resulting from the corresponding composition of cobordisms,', '1602.08739-2-9-0': '0.35 (0 0) 2.5 6.25 (0 0) rx: ry: (40 -15) rx: ry: (40 15) rx: ry: 0 0 - 15 15 - -15 -15 - (40 ) (15 )(20 )(0 ) (40 ) (22 )(22 )(40 ) (40 ) (15 )(20 )(0 ) (40 -15)(*ex *ey) (*ex *ey) 2.5 6.25 (0 0) rx: ry: (0 30) rx: ry: (40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey) (120 0)[MATH] (160 0) 2.5 6.25 (40 -15) (-40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (-25 )(-20 )(-40 ) (0 ) (-18 )(-18 )(0 ) (0 ) (-25 )(-20 )(-40 ) (0 0)(*ex *ey) (*ex *ey)', '1602.08739-2-10-0': '(40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey)', '1602.08739-2-11-0': '(*ex *ey) (280 0)[MATH] (320 0) 2.5 6.25 (0 0) rx: ry: (40 0) rx: ry: 0 0 - (0 ) (40 0) (0 ) (40 0) (40 0)(*ex *ey) (*ex *ey)', '1602.08739-2-12-0': ').', '1602.08739-2-13-0': '# Homotopy pushouts and finite [MATH]-dimensional CW-complexes', '1602.08739-2-14-0': 'The general idea of homotopy pushouts and homotopy quotients, which has become an essential slogan in higher category theory [CITATION], and more recently in type theory [CITATION], is that instead of equating elements, one sews in a path between them, so as to keep track of the fact that two elements could be equal in more than one way.', '1602.08739-2-14-1': 'When applied to the composition of cospans, the homotopy-pushout operation matches well with the various interpretations of cospans in applications: to connect two "devices", draw "cables" from the outputs of the first to the inputs of the second.', '1602.08739-2-15-0': 'Let [MATH] and [MATH] be topological spaces, and let [MATH] be a finite set, considered as a discrete space.', '1602.08739-2-15-1': 'The homotopy pushout [MATH] of a span [MATH], [EQUATION] is obtained by attaching an interval in [MATH] from [MATH] to [MATH], for each point [MATH].', '1602.08739-2-15-2': 'For example, the homotopical version of the pushout in [REF] is shown below, clearly corresponding to 0.35 2.5 6.25 (40 -15) (-40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (-25 )(-20 )(-40 ) (0 ) (-18 )(-18 )(0 ) (0 ) (-25 )(-20 )(-40 ) (0 0)(*ex *ey) (*ex *ey)', '1602.08739-2-16-0': '(40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey)', '1602.08739-2-17-0': '(*ex *ey) : [EQUATION]', '1602.08739-2-18-0': 'We see in particular, that even when the spaces [MATH], [MATH], and [MATH] are all discrete, the homotopy pushout [MATH] will not in general be discrete.', '1602.08739-2-18-1': 'To accommodate the homotopy pushout, we need a category of cospans with discrete endpoints, but with apex at least a homotopy [MATH]-type, say a [MATH]-dimensional CW complex.', '1602.08739-2-18-2': 'Recall that a finite [MATH]-dimensional CW-complex is a space obtained from a finite set of points by attaching a finite set of segments by their endpoints.', '1602.08739-2-18-3': 'The homotopy pushout described above is a paradigmatic example.', '1602.08739-2-18-4': 'A finite [MATH]-dimensional CW-complex is thus a topological graph; if it is connected it is characterized up to homotopy by its first Betti number, the number of independent cycles.', '1602.08739-2-18-5': 'Any connected finite [MATH]-dimensional CW-complex is therefore homotopy equivalent to a bouquet of circles.', '1602.08739-2-18-6': 'Note that the homotopy pushout is a homotopy-invariant construction: replacing any of the three objects in the span [MATH] by something homotopy equivalent will not change the homotopy class of the homotopy pushout.', '1602.08739-2-19-0': 'Let [MATH] denote the symmetric monoidal category whose objects are finite sets, and whose arrows are homotopy classes of cospans [MATH], where [MATH] is a finite [MATH]-dimensional CW-complex.', '1602.08739-2-19-1': 'The monoidal structure is given by disjoint union.', '1602.08739-2-19-2': 'It is easy to see that such cospans are closed under composition given by homotopy pushout, since this operation attaches [MATH]-cells, but nothing higher-dimensional.', '1602.08739-2-19-3': 'Since we only consider cospans up to homotopy, it is also clear that composition is associative and unital.', '1602.08739-2-20-0': '# Homotopy cospans and cobordisms', '1602.08739-2-21-0': 'Theorem.', '1602.08739-2-21-1': 'The symmetric monoidal category [MATH], whose objects are finite sets and whose morphisms are homotopy classes of cospans, is the free symmetric monoidal category on a commutative Frobenius object.', '1602.08739-2-22-0': 'The universal property can be established directly, without reference to cobordisms, by copying over the proof in [CITATION] almost verbatim.', '1602.08739-2-22-1': 'This is possible since the generators and relations of [MATH] as a symmetric monoidal category are readily seen to correspond exactly to the axioms for commutative Frobenius algebras.', '1602.08739-2-22-2': 'The key point here is that the invariants detecting whether two connected cospans are homotopic are the same as those classifying connected [MATH]-cobordisms, as used in [CITATION].', '1602.08739-2-23-0': 'It is also possible to establish this result without mention of generators and relations, by means of a geometric formalization of the analogy between cospans and cobordisms, establishing an equivalence of symmetric monoidal categories [MATH].', '1602.08739-2-23-1': 'This provides an alternative proof of the Theorem, since [MATH] is already known to have the universal property ([CITATION], Theorem 3.6.19).', '1602.08739-2-24-0': 'We first need to enhance the geometric realization of a cospan [MATH], by separating out its input and output points.', '1602.08739-2-24-1': 'We do that systematically by replacing [MATH] by the mapping cylinder of [MATH]; this amounts to attaching a [MATH]-cell from each point in [MATH] to its image point in [MATH], and produces thus a homotopy equivalent graph whose input and output points are all pairwise disjoint.', '1602.08739-2-25-0': 'As an example, here is the replacement for the cospan representing multiplication: [EQUATION] already considerably strengthening the analogy with', '1602.08739-2-26-0': '0.35 (0 0) 2.5 6.25 (0 0) rx: ry: (0 30) rx: ry: (40 15) rx: ry: 15 15 - 30 30 - 0 0 - (0 ) (25 )(20 )(40 ) (0 ) (18 )(18 )(0 ) (0 ) (25 )(20 )(40 ) (40 15)(*ex *ey) (*ex *ey) .', '1602.08739-2-26-1': '(At this point it is worth recalling that a cobordism is also in fact a kind of cospan (cf. [CITATION], 1.2): it is the surface with boundary, together with specific inclusions [MATH] from closed [MATH]-manifolds onto the boundary.)', '1602.08739-2-27-0': 'With this preparatory mapping-cylinder step, we can describe the correspondence geometrically in Euclidean [MATH]-space.', '1602.08739-2-27-1': 'Given a cospan we choose a suitable embedding of the mapping-cylinder graph into [MATH], with its input points at level [MATH] and output points at level [MATH].', '1602.08739-2-27-2': 'By suitable, we mean: smooth on each [MATH]-cell, and at each vertex the adjacent [MATH]-cells should not share tangent directions.', '1602.08739-2-27-3': 'Other than these requirements, it is not important how exactly the embedding is arranged-even the isotopy class does not matter.', '1602.08739-2-27-4': 'In order to construct a cobordism from this, we first take the boundary of a tubular neighborhood of the embedded graph as a subset of [MATH]; specifically we can take the set of points in [MATH] at fixed distance [MATH] from the graph.', '1602.08739-2-27-5': 'If [MATH] is chosen small enough, which is possible by compactness of the finitely many closed cells, the resulting subset will be a topological surface.', '1602.08739-2-27-6': 'We intersect this surface with [MATH] to obtain a cobordism, whose boundary consists of small circles around the input and output points of the graph.', '1602.08739-2-28-0': 'Conversely, any cobordism can be embedded in [MATH] with in-boundaries at level [MATH] and out-boundaries at level [MATH].', '1602.08739-2-28-1': 'Its interior admits a deformation retract onto a graph, commuting (at the boundary levels [MATH]) with the deformation retract of the open disks onto their center points; the result is unique up to homotopy.', '1602.08739-2-28-2': 'With boundary points determined as those at levels [MATH] and [MATH], this defines a cospan.', '1602.08739-2-28-3': 'It is clear that these two constructions are mutually inverse (up to the equivalences in question: homotopy equivalence of graph cospans, and homeomorphism rel the boundary for cobordisms).', '1602.08739-2-29-0': 'Finally, it is clear from the geometry that composition of cospans corresponds precisely to composition of cobordisms: Given composable cospans [MATH] and [MATH], the composite given by the homotopy pushout can be realized geometrically by embedding (the mapping cylinders of) [MATH] into [MATH] and [MATH] into [MATH] and sewing in [MATH]-cells [MATH] in [MATH].', '1602.08739-2-29-1': 'Cobordisms can be composed in exactly the same way, by sewing in cylinders in [MATH] between the boundary circles at level [MATH] and [MATH].'}

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[]

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[]

['1602.08739-1-9-2', '1602.08739-1-10-0', '1602.08739-1-11-0', '1602.08739-1-12-0', '1602.08739-1-13-0', '1602.08739-1-16-2', '1602.08739-1-17-0', '1602.08739-1-18-0', '1602.08739-1-22-0', '1602.08739-1-27-0', '1602.08739-2-8-2', '1602.08739-2-9-0', '1602.08739-2-10-0', '1602.08739-2-11-0', '1602.08739-2-12-0', '1602.08739-2-15-2', '1602.08739-2-16-0', '1602.08739-2-17-0', '1602.08739-2-21-0', '1602.08739-2-26-0']

{'1': 'http://creativecommons.org/licenses/by/4.0/', '2': 'http://creativecommons.org/licenses/by/4.0/'}

https://arxiv.org/abs/1602.08739

cond-mat-0402534

{'cond-mat-0402534-1-0-0': 'The low temperature magnetoconductance of a large array of quantum coherent loops exhibits Altshuler-Aronov-Spivak oscillations which periodicity corresponds to 1/2 flux quantum per loop.', 'cond-mat-0402534-1-0-1': 'We show that the measurement of the harmonics content in a square network provides an accurate way to determine the electron phase coherence length [MATH] in units of the lattice length without any adjustable parameters.', 'cond-mat-0402534-1-0-2': 'We use this method to determine [MATH] in a network realised from a 2D electron gas (2DEG) in a GaAS/GaAlAs heterojunction.', 'cond-mat-0402534-1-0-3': 'The temperature dependence follows a power law [MATH] from 1.3 K to 25 mK with no saturation, as expected for 1D diffusive electronic motion and electron-electron scattering as the main decoherence mechanism.', 'cond-mat-0402534-1-1-0': 'The characteristic scale on which quantum interference can occur in a conductor, the phase coherence length [MATH], is the key parameter of quantum transport.', 'cond-mat-0402534-1-1-1': 'In particular, the dependence of [MATH] on temperature can discriminate between the various scattering mechanisms which limit phase coherence: electron-electron (e-e), electron-phonon or electron-magnetic impurity interactions.', 'cond-mat-0402534-1-1-2': 'Interference on the scale of [MATH] gives rise to two different types of contributions to the conductance in a transport experiment.', 'cond-mat-0402534-1-1-3': 'Some are sample specific and depend on the particular disorder configuration.', 'cond-mat-0402534-1-1-4': 'These are conductance fluctuations (magnetofingerprints) and the [MATH] periodic Aharonov-Bohm (AB) oscillations ([MATH] is the flux quantum).', 'cond-mat-0402534-1-1-5': 'Their amplitudes are governed both by [MATH] and the thermal length [MATH] which in most cases is smaller than [MATH].', 'cond-mat-0402534-1-1-6': 'This makes an accurate determination of [MATH] difficult [CITATION].', 'cond-mat-0402534-1-1-7': 'The second type of contribution, called the weak localisation (WL) correction, is obtained after ensemble averaging of quantum interferences on many configurations of disorder.', 'cond-mat-0402534-1-1-8': 'It originates from interferences between time reversed electronic trajectories, which are the only ones surviving the disorder average.', 'cond-mat-0402534-1-1-9': 'It is also observed in samples of size [MATH] and only depends on [MATH] since it involves trajectories at the same energy.', 'cond-mat-0402534-1-1-10': 'Manifestations of WL are the magnetoconductance (MC) of large connex samples [CITATION] and the Altshuler-Aronov-Spivak (AAS) [MATH] periodic oscillations resulting from the ensemble average of AB oscillations in a long cylinder or large arrays of connected phase coherent rings [CITATION].', 'cond-mat-0402534-1-1-11': 'The WL provides thus in general a much more direct measurement of [MATH] than sample specific corrections.', 'cond-mat-0402534-1-2-0': 'The analysis of the MC in 1D diffusive metallic wires (with transverse dimensions smaller than [MATH]) has led to accurate determinations of [MATH].', 'cond-mat-0402534-1-2-1': 'It was found that the dominant phase breaking mechanism at very low temperature, in the absence of magnetic impurities, is due to e-e scattering and is well described by the Altshuler-Aronov-Khmelnitskii (AAK) theory [CITATION] yielding [MATH] with no saturation down to [MATH]mK [CITATION].', 'cond-mat-0402534-1-2-2': 'This analysis requires a good knowledge of transport parameters such as the diffusion coefficient ([MATH]), and the width of the wire ([MATH]) or the electronic density ([MATH]).', 'cond-mat-0402534-1-2-3': 'On the other hand these parameters are not straightforwardly determined in wires fabricated from GaAs/GaAlAs 2DEG, and therefore much less evaluations of [MATH] exist [CITATION].', 'cond-mat-0402534-1-3-0': 'Here we show that [MATH] can be determined without adjustable parameters from the analysis of the AAS oscillations in a periodic network (Fig. [REF]), when the circumference [MATH] of the elementary loop is of the order of [MATH].', 'cond-mat-0402534-1-3-1': 'It is indeed known that the amplitude of AAS oscillations decreases exponentially with [MATH].', 'cond-mat-0402534-1-3-2': 'In this letter, following [CITATION], we explain how to calculate the harmonics content of the oscillations.', 'cond-mat-0402534-1-3-3': 'Since it depends only on [MATH] it is possible to determine unambiguously [MATH] as a function of temperature, in a 2DEG square lattice containing [MATH] loops in the diffusive transport regime.', 'cond-mat-0402534-1-3-4': 'Moreover once [MATH] is determined it is also possible from the analysis of the high field positive MC to deduce the elastic mean free path ([MATH]), the width of the wires, and to make a detailed comparison with theoretical predictions of the AAK theory on dephasing by e-e interactions.', 'cond-mat-0402534-1-3-5': 'We find a very good agreement in this regime, never explored before, of very few conducting channels.', 'cond-mat-0402534-1-4-0': 'In the weakly disordered diffusive regime corresponding to [MATH], the WL correction is directly related to the Cooperon, which can be computed from the time integrated return probability [MATH] for a diffusive electron.', 'cond-mat-0402534-1-4-1': 'In a cylinder or an array of connected loops the contribution to the Cooperon of trajectories enlacing at least one loop oscillates with a flux periodicity of [MATH] giving rise to the AAS oscillations.', 'cond-mat-0402534-1-4-2': 'A systematic way of calculating WL in a mesoscopic network of diffusive wires was derived by Doucot Rammal [CITATION].', 'cond-mat-0402534-1-4-3': 'More recently Pascaud Montambaux [CITATION] found a relation between the WL correction and the spectral determinant [MATH] of the Laplace operator [MATH] defined on the network : [EQUATION] where [MATH].', 'cond-mat-0402534-1-4-4': 'This approach, which is meaningful only for regular networks, is particularly efficient because [MATH] can be computed systematically for any given network in terms of the determinant of a finite size matrix encoding the information about the network (topology, lengths of the wires, magnetic flux).', 'cond-mat-0402534-1-4-5': 'It can also be shown that the WL can be expressed, in the small [MATH] limit, as a trace expansion over periodic orbits, denoted [MATH], [EQUATION] where [MATH], [MATH], is the total number of vertices, bonds respectively.', 'cond-mat-0402534-1-4-6': '[MATH] is the primitive orbit related to [MATH].', 'cond-mat-0402534-1-4-7': '[MATH] is the total length.', 'cond-mat-0402534-1-4-8': 'We explain briefly this formula, demonstrated in [CITATION] and discussed in detail in [CITATION].', 'cond-mat-0402534-1-4-9': 'Each orbit contributes to the MC with a phase factor which depends on the enclosed flux: [MATH].', 'cond-mat-0402534-1-4-10': 'It is also characterised by its length [MATH] and by a geometrical weight [MATH].', 'cond-mat-0402534-1-4-11': 'In the case of a square lattice of periodicity [MATH], the periodically oscillating conductance can be decomposed in the Fourier space as a sum of harmonics of the fundamental periodicity corresponding to [MATH] per elementary cell.', 'cond-mat-0402534-1-4-12': 'The first terms of this expansion in [MATH] read: [EQUATION] where [MATH], [MATH] being the flux per elementary cell.', 'cond-mat-0402534-1-4-13': 'The amplitude of the [MATH]th harmonics is evaluated by counting the paths enclosing [MATH] fluxes [MATH].', 'cond-mat-0402534-1-4-14': 'The counting becomes rapidly cumbersome, for instance 156 orbits are involved in the last term, but the crucial point is that the coefficient of each term depends only on the lattice geometry.', 'cond-mat-0402534-1-5-0': 'More generally, the WL correction can be obtained for all values of [MATH] from the numerical computation of the determinant in Eq. ([REF]).', 'cond-mat-0402534-1-5-1': 'The numerical FFT of the computed MC yields the ratio [MATH] of the 2 first harmonics as a function of [MATH] (Fig. [REF]a).', 'cond-mat-0402534-1-5-2': 'It appears that the small orbit expansion ([REF]) is a good approximation up to [MATH].', 'cond-mat-0402534-1-5-3': 'In any case the ratio of two harmonics is completely determined by [MATH] and provides a method for a direct evaluation of [MATH] without any adjustable parameter.', 'cond-mat-0402534-1-5-4': 'The square lattice is particularly appropriate for such a determination of [MATH] due to its large harmonics content: the second harmonic is dominated by orbits of length [MATH] instead of [MATH] for a statistical ensemble of single rings or a necklace of identical rings, for example.', 'cond-mat-0402534-1-6-0': 'We now use this method to determine the phase coherence length of square networks etched in a 2DEG of a GaAs/AlGaAs heterostructure.', 'cond-mat-0402534-1-6-1': 'The networks consist of [MATH] square loops of side [MATH]m and nominal width [MATH]m and cover a total area of [MATH]mm[MATH].', 'cond-mat-0402534-1-6-2': 'A metallic (Au) gate deposited [MATH]nm above the 2DEG offers the possibility to increase or decrease the number of electrons in the network.', 'cond-mat-0402534-1-6-3': 'Measurements were done on two identical networks (A and B), giving the same results.', 'cond-mat-0402534-1-6-4': 'Except when specified, figures show the data for sample A.', 'cond-mat-0402534-1-6-5': 'We have measured the MC up to [MATH]T between [MATH]mK and [MATH]K, using a standard lock-in technique (ac current of [MATH]nA at [MATH]Hz).', 'cond-mat-0402534-1-7-0': 'The samples were in general strongly depleted at low temperature because of the etching.', 'cond-mat-0402534-1-7-1': 'The intrinsic electron density of the 2DEG, [MATH]m[MATH], was recovered after illuminating the samples during several minutes at [MATH]K.', 'cond-mat-0402534-1-7-2': 'This density was determined from the period of Shubnikov-de Haas oscillations visible above [MATH]T.', 'cond-mat-0402534-1-7-3': 'It was found to vary only by less than 30% when the gate voltage was varied between [MATH]V and [MATH]V, whereas the network resistance varied by a factor [MATH].', 'cond-mat-0402534-1-7-4': 'This shows that the main effect of the gate voltage is to change the width of the wires and disconnect some bonds.', 'cond-mat-0402534-1-7-5': 'Because of depletion after etching of the 2DEG, it is difficult to estimate the real width of the wires and the elastic mean free path.', 'cond-mat-0402534-1-7-6': 'The mobility of the carriers was estimated to be [MATH]m[MATH]V[MATH]s[MATH] i.e. 10 times smaller than the mobility of the original 2DEG .', 'cond-mat-0402534-1-8-0': 'In the following we focus on the MC data at low magnetic field (Fig. [REF]a).', 'cond-mat-0402534-1-8-1': 'The data exhibit large AAS oscillations with a period [MATH]G corresponding to a flux [MATH] in a square cell of area [MATH].', 'cond-mat-0402534-1-8-2': 'The oscillations are clearly not purely sinusoidal.', 'cond-mat-0402534-1-8-3': 'At the lowest temperature, [MATH]mK, three harmonics are visible in the Fourier spectrum of the MC (Fig. [REF]b).', 'cond-mat-0402534-1-8-4': 'Moreover, as shown in Fig. [REF] which represents the conductance for a wider range of magnetic field, the oscillations disappear above [MATH]G but the WL magnetoconductance due to the penetration of the magnetic field through the finite width of the wires constituting the network is still clearly visible.', 'cond-mat-0402534-1-8-5': 'At high temperature, above [MATH]mK, the AAS oscillations disappear even at low field.', 'cond-mat-0402534-1-8-6': 'Only the positive MC remains with a smaller amplitude.', 'cond-mat-0402534-1-8-7': 'The same experiments for different gate voltages were also performed.', 'cond-mat-0402534-1-9-0': 'We first concentrate on the AAS oscillations (Fig. [REF]a).', 'cond-mat-0402534-1-9-1': 'The Fourier spectrum of the MC exhibits a series of peaks corresponding to successive harmonics of the [MATH] periodicity.', 'cond-mat-0402534-1-9-2': 'The finite width of the peaks (Fig. [REF]b) is due to the penetration of the magnetic field in the wires which damps the AAS oscillations at high field.', 'cond-mat-0402534-1-9-3': 'We have checked that the mechanisms responsible for the broadening of the Fourier components do not affect their integrated values which we have computed in order to compare with the theory depicted above.', 'cond-mat-0402534-1-9-4': 'From the ratio [MATH] of integrated peaks of the two first harmonics it is thus possible to directly determine [MATH] via the theoretical relation between [MATH] and [MATH].', 'cond-mat-0402534-1-9-5': 'The temperature dependence is deduced this way between [MATH]mK and [MATH]mK as shown in Fig. [REF].', 'cond-mat-0402534-1-9-6': 'We find that [MATH] follows a power law [MATH], where [MATH].', 'cond-mat-0402534-1-9-7': 'The coherence length reaches almost [MATH]m at [MATH]mK and there is no sign of saturation.', 'cond-mat-0402534-1-10-0': 'Once [MATH] is determined, the sample parameters ([MATH],[MATH]) can be deduced from the WL envelope.', 'cond-mat-0402534-1-10-1': 'The magnetic field appears as an additional effective phase breaking rate for the time reversed trajectories responsible of the WL leading to an effective [MATH] given by [CITATION]: [EQUATION] where [MATH] can be understood as a renormalized width which appears in the WL correction for a semi ballistic wire in the regime [MATH] due to the phenomenon of flux cancellation.', 'cond-mat-0402534-1-10-2': 'The coefficient [MATH] depends on the specific boundary conditions.', 'cond-mat-0402534-1-10-3': 'The samples under consideration are close to the case of specular boundary scattering [CITATION] for which [MATH].', 'cond-mat-0402534-1-10-4': 'The envelope of the MC, given by [MATH], can be analytically computed for the square lattice geometry and is given by: [EQUATION] where [MATH] is a complete elliptic integral.', 'cond-mat-0402534-1-10-5': 'We can then fit the measured [MATH] where [MATH] and [MATH] are the Drude conductivity and conductance.', 'cond-mat-0402534-1-10-6': '[MATH] is deduced from the high field measurements when the WL vanishes, and the parameters of the fit are [MATH] and [MATH].', 'cond-mat-0402534-1-10-7': 'By fitting the WL envelope at different temperatures we extract the parameters [MATH] and [MATH] which are found independent of temperature: this confirms the consistency of our analysis.', 'cond-mat-0402534-1-10-8': 'We obtain [MATH]nm and [MATH]nm, yielding [MATH]nm.', 'cond-mat-0402534-1-10-9': 'This shows that the networks are indeed in the diffusive regime.', 'cond-mat-0402534-1-10-10': 'Knowing [MATH]nm from the Shubnikov-de Haas measurement, these numbers correspond to [MATH] transverse channels per wire and the number of effective conducting channels [MATH] on the scale of [MATH] is of the order of unity.', 'cond-mat-0402534-1-10-11': 'At higher temperature where no oscillations are visible, we can nevertheless deduce [MATH] and [MATH] by fitting the envelope of the MC, knowing the values of [MATH] and [MATH] (Fig. [REF]).', 'cond-mat-0402534-1-10-12': 'The knowledge of these parameters makes possible a quantitative comparison of [MATH] with the theoretical prediction of AAK [CITATION] [MATH] written for a 2D wire.', 'cond-mat-0402534-1-10-13': 'This theory applies to diffusive metallic wires with a large number of conducting channels in a limit where e-e interaction is treated perturbatively.', 'cond-mat-0402534-1-10-14': 'We find a very good agreement (see Fig. [REF]b) which is surprising for two reasons: (i) we are confronted here with a small number of conducting channels where strong interaction effects could be expected, (ii) the result of AAK was not extended to network geometry.', 'cond-mat-0402534-1-10-15': 'Recently it was found in [CITATION] that [MATH] extracted from the AB or AAS oscillations in a single ring of perimeter [MATH] behaves rather like [MATH].', 'cond-mat-0402534-1-10-16': 'This behaviour is not observed in our experiment.', 'cond-mat-0402534-1-10-17': 'For the second sample we have applied gate voltages between [MATH]V and [MATH]V changing the resistance from [MATH]k[MATH] to [MATH]k[MATH].', 'cond-mat-0402534-1-10-18': 'A good filtering of the gate voltage line is needed to avoid saturation of the coherence length.', 'cond-mat-0402534-1-10-19': 'We find again a [MATH] law for [MATH].', 'cond-mat-0402534-1-10-20': 'It is also possible to estimate for each gate voltage the values of [MATH] and [MATH].', 'cond-mat-0402534-1-10-21': 'The width [MATH] is not found to change within our experimental accuracy but [MATH] decreases with the number of electrons by a factor [MATH] when the gate voltage varies between [MATH]V and [MATH]V; at the same time [MATH] decreases by a factor [MATH].', 'cond-mat-0402534-1-10-22': 'This shows that the effect of the gate is also to disconnect bonds of the network.', 'cond-mat-0402534-1-11-0': 'As a consistency check we have computed numerically the oscillating part of the MC with formula ([REF]) et ([REF]) using the value of [MATH] determined above from the WL envelope of the MC curves.', 'cond-mat-0402534-1-11-1': 'We find that this value also precisely describes the damping of the AAS oscillations, if the oscillations amplitude is multiplied by a factor ranging from 1.6 to 2 depending on the gate voltage.', 'cond-mat-0402534-1-11-2': 'This can be explained by the existence of broken bonds in the network which influences envelope and oscillations differently.', 'cond-mat-0402534-1-11-3': 'We obtain a very good agreement between theory and experiments (Fig. [REF]).', 'cond-mat-0402534-1-12-0': 'In conclusion we have shown that magnetoconductance experiments in GaAs/GaAlAs networks can be described very accurately by the diagrammatic theory of quantum transport in diffusive networks.', 'cond-mat-0402534-1-12-1': 'It is remarkable that this agreement is achieved in a limit where the conductance on the scale of the period of the network, [MATH], is of the order of the quantum conductance, in contrast with metallic wires where the number of channels is of order 1000.', 'cond-mat-0402534-1-12-2': 'Moreover we extracted from the AAS oscillations the temperature dependance of the phase coherence length [MATH] that agrees with AAK theory down to [MATH]mK.', 'cond-mat-0402534-1-13-0': 'We thank B. Etienne for the fabrication of the heterojunctions.', 'cond-mat-0402534-1-13-1': 'We acknowledge R. Deblock, B. Reulet and A. Rowe for fruitful discussions.'}

{'cond-mat-0402534-2-0-0': 'The low temperature magnetoconductance of a large array of quantum coherent loops exhibits Altshuler-Aronov-Spivak oscillations which periodicity corresponds to 1/2 flux quantum per loop.', 'cond-mat-0402534-2-0-1': 'We show that the measurement of the harmonics content provides an accurate way to determine the electron phase coherence length [MATH] in units of the lattice length with no adjustable parameters.', 'cond-mat-0402534-2-0-2': 'We use this method to determine [MATH] in a square network realised from a 2D electron gas (2DEG) in a GaAS/GaAlAs heterojunction, with only a few conducting channels.', 'cond-mat-0402534-2-0-3': 'The temperature dependence follows a power law [MATH] from 1.3 K to 25 mK with no saturation, as expected for 1D diffusive electronic motion and electron-electron scattering as the main decoherence mechanism.', 'cond-mat-0402534-2-1-0': 'The characteristic scale on which quantum interference can occur in a conductor, the phase coherence length [MATH], is the key parameter of quantum transport.', 'cond-mat-0402534-2-1-1': 'In particular, the dependence of [MATH] on temperature can discriminate between the various scattering mechanisms which limit phase coherence: electron-electron (e-e), electron-phonon or electron-magnetic impurity interactions.', 'cond-mat-0402534-2-1-2': 'Interference on the scale of [MATH] gives rise to two different types of contributions to the conductance in a transport experiment.', 'cond-mat-0402534-2-1-3': 'Some are sample specific and depend on the particular disorder configuration.', 'cond-mat-0402534-2-1-4': 'These are conductance fluctuations (magnetofingerprints) and the [MATH] periodic Aharonov-Bohm (AB) oscillations ([MATH] is the flux quantum).', 'cond-mat-0402534-2-1-5': 'Their amplitudes are governed both by [MATH] and the thermal length [MATH], in general smaller than [MATH].', 'cond-mat-0402534-2-1-6': 'This makes an accurate determination of [MATH] difficult [CITATION].', 'cond-mat-0402534-2-1-7': 'The second type of contribution, called the weak localisation (WL) correction, is obtained after ensemble averaging of quantum interferences on many configurations of disorder.', 'cond-mat-0402534-2-1-8': 'It originates from interferences between time reversed electronic trajectories, which are the only ones surviving the disorder average.', 'cond-mat-0402534-2-1-9': 'It is also observed in samples of size [MATH] and only depends on [MATH] since it involves trajectories at the same energy.', 'cond-mat-0402534-2-1-10': 'Manifestations of WL are the magnetoconductance (MC) of large connex samples [CITATION] and the Altshuler-Aronov-Spivak (AAS) [MATH] periodic oscillations resulting from the ensemble average of AB oscillations in a long cylinder or large arrays of connected phase coherent rings [CITATION].', 'cond-mat-0402534-2-1-11': 'The WL provides thus in general a much more direct measurement of [MATH] than sample specific corrections.', 'cond-mat-0402534-2-2-0': 'The analysis of the MC in 1D diffusive metallic wires (with transverse dimensions smaller than [MATH]) has led to accurate determinations of [MATH].', 'cond-mat-0402534-2-2-1': 'It was found that the dominant phase breaking mechanism at very low temperature, in the absence of magnetic impurities, is due to e-e scattering and is well described by the Altshuler-Aronov-Khmelnitskii (AAK) theory [CITATION] yielding [MATH] with no saturation down to [MATH]mK [CITATION].', 'cond-mat-0402534-2-2-2': 'Such a remarkable agreement between theory and experiment has not been established for semiconducting wires, where most WL experiments have been performed only above [MATH]K or with insufficient ensemble averaging [CITATION].', 'cond-mat-0402534-2-2-3': 'It is however essential to check the validity of the AAK theory for these systems which correspond to radically different physical parameters: fewer conducting channels and larger screening lengths.', 'cond-mat-0402534-2-2-4': 'In this Letter we present MC data down to 25 mK of networks fabricated from a GaAs/GaAlAs 2DEG, which contain [MATH] square loops in the diffusive transport regime, and determine [MATH] without adjustable parameters from the analysis of the AAS oscillations (Fig. [REF]).', 'cond-mat-0402534-2-2-5': 'Following [CITATION], we explain how to calculate the harmonics content of these oscillations and show that it depends only on [MATH]where [MATH] is the circumference of the elementary loop.', 'cond-mat-0402534-2-2-6': 'It is then possible to determine [MATH] and its temperature dependence exclusively from geometrical parameters of the network.', 'cond-mat-0402534-2-2-7': 'This new method of determinating [MATH] is especially interesting in these 2DEG wires for which basic transport parameters such as the electron density and wire width ([MATH]) are not straightforwardly determined, unlike metals.', 'cond-mat-0402534-2-3-0': 'Moreover once [MATH] is determined, we deduce from the analysis of the high field positive MC the elastic mean free path ([MATH]), W, and make a detailed comparison with theoretical predictions of the AAK theory on dephasing by e-e interactions.', 'cond-mat-0402534-2-3-1': 'We find a very good quantitative agreement in the regime, never explored before, of very few conducting channels.', 'cond-mat-0402534-2-4-0': 'In the weakly disordered diffusive regime ([MATH]), the WL correction is directly related to the Cooperon, which can be computed from the time integrated return probability [MATH] for a diffusive electron.', 'cond-mat-0402534-2-4-1': 'In a cylinder or an array of connected loops the contribution to the Cooperon of trajectories enlacing at least one loop oscillates with a flux periodicity of [MATH] giving rise to the AAS oscillations.', 'cond-mat-0402534-2-4-2': 'A systematic way of calculating WL in a mesoscopic network of diffusive wires was derived in [CITATION].', 'cond-mat-0402534-2-4-3': 'More recently [CITATION], a relation was found between the WL correction and the spectral determinant [MATH] of the Laplace operator [MATH] defined on the network.', 'cond-mat-0402534-2-4-4': 'If we write [MATH], then [EQUATION] where [MATH].', 'cond-mat-0402534-2-4-5': 'Eq. ([REF]) assumes an exponential relaxation of phase coherence.', 'cond-mat-0402534-2-4-6': "This approach, which is meaningful only for regular networks, is particularly efficient because [MATH] can be computed systematically for any given network in terms of the determinant of a finite size matrix encoding the network's characteristics (topology, length of the wires, magnetic flux).", 'cond-mat-0402534-2-4-7': 'It can also be shown that the WL can be expressed, in the small [MATH] limit, as a trace expansion over periodic orbits, denoted [MATH], [EQUATION] where [MATH]), is the total number of vertices (bonds).', 'cond-mat-0402534-2-4-8': '[MATH] is the primitive orbit related to [MATH].', 'cond-mat-0402534-2-4-9': '[MATH] is the total length.', 'cond-mat-0402534-2-4-10': 'We explain briefly this formula, demonstrated in [CITATION] and discussed in detail in [CITATION].', 'cond-mat-0402534-2-4-11': 'Each orbit contributes to the MC with a phase factor which depends on the enclosed flux: [MATH].', 'cond-mat-0402534-2-4-12': 'It is also characterised by its length [MATH] and by a geometrical weight [MATH].', 'cond-mat-0402534-2-4-13': 'In the case of a square lattice of periodicity [MATH], the periodically oscillating conductance can be decomposed in Fourier space as a sum of harmonics of the fundamental periodicity corresponding to [MATH] per elementary cell.', 'cond-mat-0402534-2-4-14': 'The first terms of this expansion in [MATH] read: [EQUATION]', 'cond-mat-0402534-2-4-15': 'Here [MATH], and [MATH] is the flux per elementary cell.', 'cond-mat-0402534-2-4-16': 'The amplitude of the [MATH]th harmonics is evaluated by counting the paths enclosing [MATH] fluxes [MATH].', 'cond-mat-0402534-2-4-17': 'The counting is rapidly cumbersome (156 orbits are involved in the last term), but the crucial point is that the coefficient of each term depends only on the lattice geometry.', 'cond-mat-0402534-2-5-0': 'More generally, the WL correction can be obtained for all values of [MATH] from the numerical computation of the determinant in Eq. ([REF]).', 'cond-mat-0402534-2-5-1': 'The numerical FFT of the computed MC yields the ratio [MATH] of the 2 first harmonics as a function of [MATH] (Fig. [REF]a).', 'cond-mat-0402534-2-5-2': 'It appears that the small orbit expansion ([REF]) is a good approximation up to [MATH].', 'cond-mat-0402534-2-5-3': 'In any case the ratio of two harmonics is completely determined by [MATH] and provides a method for a direct evaluation of [MATH] without any adjustable parameter.', 'cond-mat-0402534-2-5-4': 'The square lattice is particularly appropriate for such a determination of [MATH] due to its large harmonics content: the second harmonic is dominated by orbits of length [MATH] instead of [MATH] for a statistical ensemble of single rings or a necklace of identical rings, for example.', 'cond-mat-0402534-2-6-0': 'We now use this method to determine the phase coherence length of square networks etched in a 2DEG of a GaAs/AlGaAs heterostructure.', 'cond-mat-0402534-2-6-1': 'The networks consist of [MATH] square loops of side [MATH]m and nominal width [MATH]m and cover a total area of [MATH]mm[MATH].', 'cond-mat-0402534-2-6-2': 'A gold gate deposited [MATH]nm above the 2DEG offers the possibility to change the number of electrons in the network.', 'cond-mat-0402534-2-6-3': 'Measurements were done on three networks (A,B with gate, C without ), giving the same results.', 'cond-mat-0402534-2-6-4': 'Except when specified, figures show the data for sample A.', 'cond-mat-0402534-2-6-5': 'We have measured the MC up to [MATH]T between [MATH]mK and [MATH]K, using a standard lock-in technique (ac current of [MATH]nA at [MATH]Hz).', 'cond-mat-0402534-2-6-6': 'The samples were in general strongly depleted at low temperature because of the etching.', 'cond-mat-0402534-2-6-7': 'The intrinsic electron density of the 2DEG, [MATH]m[MATH], was recovered after illuminating the samples during several minutes at [MATH]K.', 'cond-mat-0402534-2-6-8': 'This density was determined from Shubnikov-de Haas oscillations visible above [MATH]T. Because of depletion after etching of the 2DEG, it is difficult to estimate the real width of the wires (W) and [MATH].', 'cond-mat-0402534-2-7-0': 'At low magnetic field (Fig. [REF]a), the MC exhibit large AAS oscillations with a period [MATH]G corresponding to a flux [MATH] in a square cell of area [MATH].', 'cond-mat-0402534-2-7-1': 'The oscillations are clearly not purely sinusoidal.', 'cond-mat-0402534-2-7-2': 'At the lowest temperature, [MATH]mK, three harmonics are visible in the Fourier spectrum of the MC (Fig. [REF]b).', 'cond-mat-0402534-2-7-3': 'Moreover, as shown in Fig. [REF] which represents the MC for a wider range of field, the oscillations disappear above [MATH]G but the WL magnetoconductance due to the penetration of the field through the finite width of the wires constituting the network is still clearly visible.', 'cond-mat-0402534-2-7-4': 'At high temperature, above [MATH]mK, the AAS oscillations disappear even at low field.', 'cond-mat-0402534-2-7-5': 'Only the positive MC remains with a smaller amplitude.', 'cond-mat-0402534-2-7-6': 'In sample B, the same experiments for different gate voltages were also performed.', 'cond-mat-0402534-2-8-0': 'We first concentrate on the AAS oscillations (Fig. [REF]a).', 'cond-mat-0402534-2-8-1': 'The Fourier spectrum of the MC exhibits a series of peaks corresponding to successive harmonics of the [MATH] periodicity.', 'cond-mat-0402534-2-8-2': 'The finite width of the peaks (Fig. [REF]b) is due to the penetration of the magnetic field in the wires which damps the AAS oscillations at high field.', 'cond-mat-0402534-2-8-3': 'It can be shown that this broadening does not affect the integral of the peak.', 'cond-mat-0402534-2-8-4': 'A first rough analysis shows that the ratio [MATH] of integrated peaks of the two first harmonics behaves like [MATH].', 'cond-mat-0402534-2-8-5': 'We now use the theory described above to quantitatively determine [MATH] via the relation between [MATH] and [MATH].', 'cond-mat-0402534-2-8-6': 'We deduce its temperature dependence between [MATH]mK and [MATH]mK as shown in Fig. [REF].', 'cond-mat-0402534-2-8-7': 'We find that [MATH] follows a power law [MATH], where [MATH].', 'cond-mat-0402534-2-8-8': 'The coherence length reaches almost [MATH]m at [MATH]mK and there is no sign of saturation.', 'cond-mat-0402534-2-9-0': 'Once [MATH] is determined, the sample parameters ([MATH], [MATH]) can be deduced from the WL envelope.', 'cond-mat-0402534-2-9-1': 'The magnetic field appears as an additional effective phase breaking rate for the time reversed trajectories responsible for the WL leading to an effective [MATH] given by [CITATION]: [EQUATION] where [MATH] is a renormalized width which appears in the WL correction for a semi ballistic wire ([MATH]) due to the phenomenon of flux cancellation.', 'cond-mat-0402534-2-9-2': 'The coefficient [MATH] depends on the specific boundary conditions.', 'cond-mat-0402534-2-9-3': 'The samples under consideration are close to the case of specular boundary scattering [CITATION] for which [MATH].', 'cond-mat-0402534-2-9-4': 'The MC envelope, given by [MATH], can be analytically computed for the square lattice geometry and is given by: [EQUATION] where [MATH] is a complete elliptic integral.', 'cond-mat-0402534-2-9-5': 'This expression is used in a 2-parameter ([MATH], [MATH]) fit of [MATH] where [MATH] and [MATH] are the Drude conductivity and conductance.', 'cond-mat-0402534-2-9-6': 'Since [MATH] is determined independently from Shubnikov-de Haas measurements, [MATH] determines [MATH].', 'cond-mat-0402534-2-9-7': 'The above expression for [MATH] can then be used to find [MATH].', 'cond-mat-0402534-2-9-8': 'For sample A/B/C, [MATH]nm and [MATH]nm, independent of temperature as expected.', 'cond-mat-0402534-2-9-9': 'This also shows that the networks are indeed in the diffusive regime.', 'cond-mat-0402534-2-9-10': 'In sample A the number of transverse channels per wire is [MATH], and the number of effective conducting channels on the scale of [MATH] is only [MATH].', 'cond-mat-0402534-2-9-11': 'Results for samples B and C are similar.', 'cond-mat-0402534-2-10-0': 'At higher temperature where no AAS oscillations are visible, we can nevertheless deduce [MATH] and [MATH] by fitting the MC, knowing the temperature independent values of [MATH] and [MATH] (Fig. [REF]).', 'cond-mat-0402534-2-10-1': 'Thus a quantitative comparison of [MATH] with the theoretical prediction of AAK [CITATION] [MATH] written for a 2DEG wire is possible.', 'cond-mat-0402534-2-10-2': 'This theory applies to diffusive metallic wires with a large number of conducting channels in a limit where e-e interactions are treated perturbatively.', 'cond-mat-0402534-2-10-3': 'We find a very good agreement (see Fig. [REF]b) which is remarquable for two reasons: (i) we are confronted here with a small number of conducting channels where strong interaction effects could be expected, (ii) the result of AAK was not extended to network geometry.', 'cond-mat-0402534-2-10-4': 'Recently it was predicted in [CITATION] that [MATH] extracted from the AB or AAS oscillations in a single ring of perimeter [MATH] should behave like [MATH] corresponding to [MATH].', 'cond-mat-0402534-2-10-5': 'This behaviour is not observed in our experiment.', 'cond-mat-0402534-2-11-0': 'For sample B gate voltages between [MATH]V and [MATH]V changed the resistance from [MATH]k[MATH] to [MATH]k[MATH].', 'cond-mat-0402534-2-11-1': 'A good filtering of the gate voltage line is needed to avoid saturation of [MATH].', 'cond-mat-0402534-2-11-2': 'Within our experimental accuracy, we find that [MATH] is not changed and still varies as [MATH].', 'cond-mat-0402534-2-11-3': 'We estimated for each gate voltage [MATH] and [MATH].', 'cond-mat-0402534-2-11-4': 'When the gate voltage varies between [MATH]V and [MATH]V, [MATH] is unchanged but [MATH] decreases by a factor [MATH] and [MATH] by [MATH]; [MATH] decreases by a factor [MATH].', 'cond-mat-0402534-2-11-5': 'This shows that the effect of the gate is mainly to disconnect bonds of the network.', 'cond-mat-0402534-2-12-0': 'As a consistency check we have computed numerically the oscillating part of the MC with formula ([REF]) and ([REF]) using the value of [MATH] determined above from the WL envelope of the MC curves.', 'cond-mat-0402534-2-12-1': 'We find that this value also precisely describes the damping of the AAS oscillations, if the oscillations amplitude is multiplied by a factor ranging from 1.6 to 2 depending on the gate voltage.', 'cond-mat-0402534-2-12-2': 'This can be explained by the existence of broken bonds in the network which influences envelope and oscillations differently [CITATION].', 'cond-mat-0402534-2-12-3': 'We obtain a very good agreement between theory and experiments (Fig. [REF]).', 'cond-mat-0402534-2-13-0': 'In conclusion we have shown that magnetoconductance experiments in GaAs/GaAlAs networks can be described very accurately by the diagrammatic theory of quantum transport in diffusive networks.', 'cond-mat-0402534-2-13-1': 'It is remarkable that this agreement is achieved in a limit where the dimensionless conductance on the scale of the period of the network, [MATH], is of the order 1, and down to temperatures corresponding to [MATH] i.e. close to the limit of validity of AAK theory.', 'cond-mat-0402534-2-13-2': 'In contrast, metallic wires deep in the diffusive regime have a number of conducting channels of order 1000 and [MATH] is always fulfilled.', 'cond-mat-0402534-2-13-3': 'Moreover we extracted from the AAS oscillations the temperature dependance of the phase coherence length [MATH] that agrees with AAK theory down to [MATH]mK.', 'cond-mat-0402534-2-14-0': 'We thank B. Etienne for the heterojunctions, and R. Deblock and B. Reulet for fruitful discussions.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0402534

1310.3349

{'1310.3349-1-0-0': 'The central cD galaxy of the Hydra A cluster has one of the most powerful active galactic nuclei (AGNs) in the nearby Universe ([MATH]).', '1310.3349-1-0-1': 'We report on the discovery of a dust lane in the cD galaxy using Subaru telescope.', '1310.3349-1-0-2': 'The [MATH]-band image shows the existence of a dark band of the size of [MATH]), which appears to be quite similar to the dust lane observed in Centaurus A.', '1310.3349-1-0-3': 'The morphology indicates that the cold disk that seen as the dust lane is almost edge-on and rotates around the AGN.', '1310.3349-1-0-4': 'Since the minor axis of the dust lane is nearly parallel to the radio jets emerging from the AGN, the disk is probably feeding its gas into the central black hole.', '1310.3349-1-0-5': 'From the absorption, we estimate the hydrogen column density of the lane is [MATH], and the mass of the disk is [MATH].', '1310.3349-1-0-6': 'The column density is consistent with constraints obtained from Chandra X-ray observations.', '1310.3349-1-0-7': 'The age of the disk is [MATH] yr.', '1310.3349-1-0-8': "The position angle of the disk and the galaxy's photometric axis are misaligned, which may imply that the cold gas in the disk is brought via galaxy mergers.", '1310.3349-1-0-9': 'Our observations may indicate that the supply of cold gas by galaxy mergers is required for the most intensive feedback from AGNs.', '1310.3349-1-1-0': '# Introduction', '1310.3349-1-2-0': 'The AGNs in the central cD galaxies of clusters sometimes show violent activities.', '1310.3349-1-2-1': 'Such examples are MS 0735.6[MATH]7421 , Hercules A , and Hydra A .', '1310.3349-1-2-2': 'In these clusters, cluster-scale shock waves have been observed, which indicates that the central AGNs ejected enormous energy ([MATH]-[MATH] erg) in a short time ([MATH] yr).', '1310.3349-1-2-3': 'The intracluster medium (ICM) of these clusters should have been strongly affected by the AGN activities.', '1310.3349-1-2-4': 'It seems that Bondi accretion from hot gas is not sufficient and fueling of cold gas is required for the most powerful feedback from AGNs .', '1310.3349-1-3-0': 'The cD galaxy of the Hydra A cluster has been studied by many researchers.', '1310.3349-1-3-1': 'It harbors a well known radio source classified as a Fanaroff-Riley type-I (FR-I) source.', '1310.3349-1-3-2': 'A pair of jets is ejected from the AGN (3C 218; [CITATION]), and the huge amount of cosmic-rays contained in the jets may be stably heating the cool core of the cluster .', '1310.3349-1-3-3': 'The mass of the central supermassive black holes (SMBHs) is estimated to be [MATH] .', '1310.3349-1-3-4': 'Although the galaxy is an elliptical, star formation activities ([MATH]) have been discovered within [MATH] kpc from the center of the galaxy .', '1310.3349-1-3-5': 'This may mean that some amount of gas is cooling in the galaxy, although the existence of a massive cooling flow has been denied with the ASCA satellite .', '1310.3349-1-3-6': 'Previous observations have indicated that there is a disk-like structure in the galaxy.', '1310.3349-1-3-7': 'In the [MATH]-band, the surface brightness profile is elongated at the galaxy center .', '1310.3349-1-3-8': 'Observations of optical emission-lines ([OIII], H[MATH], and H[MATH]) show that the galaxy has a rotating component with a velocity of [MATH] .', '1310.3349-1-3-9': 'HI absorption is seen toward the AGN and can be interpreted as a disk .', '1310.3349-1-3-10': 'However, the morphology of the disk has not been well understood.', '1310.3349-1-3-11': 'This is mainly because the cluster is relatively distant ([MATH]) and the angular resolution of [MATH] is required to resolve fine structures.', '1310.3349-1-4-0': 'In this letter, we present a Subaru [MATH]-band image of the cD galaxy of the Hydra A cluster with a superb spatial resolution.', '1310.3349-1-4-1': 'We show that the galaxy has a spectacular dust lane.', '1310.3349-1-4-2': 'Throughout this letter, we adopt cosmological parameters of [MATH], [MATH], and [MATH].', '1310.3349-1-4-3': 'We use a redshift for the Hydra A cluster of 0.054878, which gives a scale of [MATH] kpc per arcsec. The angular diameter distance and the luminosity distance to Hydra A are [MATH] Mpc and 245 Mpc, respectively.', '1310.3349-1-5-0': '# Observations and Results', '1310.3349-1-6-0': 'We observed Hydra A with the Prime Focus Camera (Suprime-Cam; [CITATION]) on the Subaru telescope in Hawaii on January 7, 2013.', '1310.3349-1-6-1': 'We use the standard pipeline reduction software for Suprime-Cam, SDFRED , for flat-fielding, instrumental distortion correction, differential refraction, PSF matching, and sky subtraction and stacking.', '1310.3349-1-6-2': 'The seeing was [MATH].', '1310.3349-1-6-3': 'The [MATH]-band image of the cD galaxy is shown in figure [REF]a.', '1310.3349-1-6-4': 'A dust lane (dark band) can be clearly seen in the central region of the cD galaxy.', '1310.3349-1-6-5': 'The size of the lane is [MATH] (4 kpc[MATH]0.8 kpc).', '1310.3349-1-6-6': 'The position angle of the minor axis is [MATH] from the north to the east.', '1310.3349-1-6-7': 'We fit the galaxy image with a Seric profile using galfit , taking PSF size into account.', '1310.3349-1-6-8': 'The residual from the best-fits is shown in figure [REF]b.', '1310.3349-1-6-9': 'The position angle of the minor axis of the Seric profile is [MATH], which is close to the result by [CITATION] for the [MATH]-band image ([MATH]), but is not parallel to the minor axis of the disk ([MATH]).', '1310.3349-1-6-10': 'This may indicate that the disk formed after the overall structures of the cD galaxy had formed.', '1310.3349-1-6-11': 'The extinction is most effective at the center (40%), and it is [MATH].', '1310.3349-1-6-12': 'Since the extinction curves of elliptical galaxies are not much different from the one for the Milky-Way , we adopt the Galactic conversion factor ([MATH]; [CITATION]) and obtain [MATH] and the column density of [MATH] from [MATH] .', '1310.3349-1-6-13': 'The value of [MATH] is almost the same as HI absorption ([MATH]) obtained by [CITATION].', '1310.3349-1-6-14': 'If the gas is distributed as a uniform disk with a radius of [MATH] kpc and a height of [MATH] kpc, the disk mass is [MATH].', '1310.3349-1-6-15': 'Figure [REF] shows the relation of the galaxy to the radio source.', '1310.3349-1-6-16': 'A pair of jets emerges with a position angle of [MATH] and is almost parallel to the minor axis of the dust lane.', '1310.3349-1-6-17': 'The jet-dust lane configuration is quite similar to that of Centaurus A.', '1310.3349-1-7-0': '# Discussion', '1310.3349-1-8-0': '## X-ray Absorption', '1310.3349-1-9-0': 'We have checked X-ray data obtained with the Chandra X-ray telescope.', '1310.3349-1-9-1': 'Hydra A has been observed with Chandra for total 250 ksec. Figure [REF] shows the exposure-corrected X-ray image of Chandra ACIS-S for 200 ksec data in the 0.5-5 keV band.', '1310.3349-1-9-2': 'While the central AGN is prominent, the disk-structure is not seen in the figure.', '1310.3349-1-9-3': 'X-ray spectral analysis for Chandra ACIS-S and ACIS-I data of the total of 250 ksec was performed to constrain an absorption by the disk with response files appropriate for each observation in the 0.4-7.0 keV band.', '1310.3349-1-10-0': 'We study the spectrum of a [MATH] region of the disk (region "D" in figure [REF]).', '1310.3349-1-10-1': 'We exclude the nuclear region "N", because the emission from it can be affected by the absorption in the vicinity of the SMBH.', '1310.3349-1-10-2': 'In fact, we find that the absorption associated with the AGN is [MATH], which is consistent with the previous result ([MATH] .', '1310.3349-1-10-3': 'Owing to the finite size of pixels ([MATH]), the pixels we used actually cover the optical disk ([MATH]).', '1310.3349-1-10-4': 'First, we extract a spectrum within a [MATH] circle centered on the AGN excluding the AGN and disk regions ("N" and "D") in order to estimate the fore- and/or background emissions.', '1310.3349-1-10-5': 'The spectrum is represented by two-temperature (cD galaxy + ICM) thermal models (apec) with a common metal abundance multiplied by absorption (phabs): [MATH].', '1310.3349-1-10-6': 'The results are [MATH] and [MATH] keV with 0.7[MATH] solar abundance, and [MATH] cm[MATH] (errors of 90% confidence).', '1310.3349-1-10-7': 'The absorption value is almost consistent with the Galactic absorption ([MATH] cm[MATH], [CITATION]) in the direction of Hydra A. Fixing these fore- and/or background parameters, we investigate the upper limit of the absorption by the disk.', '1310.3349-1-10-8': 'Here, we assume that the emission from the disk region is represented by two-temperature thermal models (cD galaxy + ICM) and only the cooler component (cD galaxy) is absorbed by the disk: [MATH].', '1310.3349-1-10-9': 'We employ an increment of [MATH] as the measure for the 90% upper limit of the absorption.', '1310.3349-1-10-10': 'We find that the upper limit is [MATH] cm[MATH], which is consistent with the disk absorption of [MATH] (section [REF]).', '1310.3349-1-10-11': 'Note that the absorption by the dust lane on the both sides of the AGN should be smaller than that in the direction of the AGN (region "N").', '1310.3349-1-11-0': '## Nature of the Disk', '1310.3349-1-12-0': 'Optical emission-lines have been discovered from the cD galaxy of Hydra A.', '1310.3349-1-12-1': 'If the rotation of the emission-line component corresponds to that of the dust lane, the rotation velocity is [MATH] , which gives the rotation period of [MATH] yr.', '1310.3349-1-12-2': 'The age of the cold disk should be larger than this period.', '1310.3349-1-12-3': 'From the observations of a shock wave in the ICM, [CITATION] indicated that there was a strong outburst of the AGN with an energy of [MATH] erg [MATH] yrs ago.', '1310.3349-1-12-4': 'This outburst may be related to the formation of the disk.', '1310.3349-1-12-5': 'Note that if all the disk gas at present ([MATH]) flowed into the SMBH, energy comparable to the previous outburst might be released ([MATH]).', '1310.3349-1-13-0': 'The angular distance between the X-ray AGN and the center line of the dust lane is [MATH] or [MATH] kpc (figure [REF]).', '1310.3349-1-13-1': 'If the AGN is located at the center of the disk, the inclination angle of the disk is [MATH], and the disk is almost edge-on.', '1310.3349-1-13-2': 'This may also mean that the jets from the AGN are almost on the plane of the sky at present.', '1310.3349-1-13-3': 'On the other hand, the positions of pairs of outer lobes show that the jets were not being injected on the plane of the sky and were oriented [MATH] about the plane in the past .', '1310.3349-1-13-4': 'The change of the jet direction may be caused by precession .', '1310.3349-1-14-0': '## Origin of the Dust Lane', '1310.3349-1-15-0': 'The cD galaxy is a harsh environment for dust.', '1310.3349-1-15-1': 'In the presence of the hot ICM of [MATH] K, collision with hot particles destroys the dust.', '1310.3349-1-15-2': 'The destruction time is [EQUATION] where [MATH] is the ICM density and [MATH] is the radius of dust grains .', '1310.3349-1-15-3': 'If we adopt [MATH] and [MATH], the destruction time is [MATH] yr.', '1310.3349-1-15-4': 'The mass loss rate from evolved stars in the galaxy is [EQUATION] where [MATH] is the distance to the galaxy, and [MATH] is the flux at [MATH] .', '1310.3349-1-15-5': 'For Hydra A, the luminosity distance is [MATH] Mpc, and the flux is [MATH] .', '1310.3349-1-15-6': 'Since we are interested in the upper limit of the dust mass [MATH] and since we avoid the complications caused by the difference of apertures at different wave lengths, we assume that [MATH].', '1310.3349-1-15-7': 'For these values, we obtain [MATH].', '1310.3349-1-15-8': 'Assuming that the gas-to-dust ratio is [MATH], the rate of accumulation of dust is [EQUATION] .', '1310.3349-1-15-9': 'This equation can be solved easily; [MATH] approaches its final value [MATH] in a time-scale of a few [MATH].', '1310.3349-1-15-10': 'Since [MATH], this may indicate that the dust is externally provided in the galaxy.', '1310.3349-1-16-0': 'If the dust is not provided internally, the dust may be brought by merged galaxies.', '1310.3349-1-16-1': 'That is the case for Centaurus A .', '1310.3349-1-16-2': "The misalignment between the disk and the galaxy's photometric axis (section [REF]) is favorable to the merger scenario.", '1310.3349-1-16-3': 'Based on a semi-analytic model, [CITATION] indicated that the fraction of cD galaxies that have cold gas brought through galaxy mergers induced by dynamical friction is small.', '1310.3349-1-16-4': 'However, their criterion for existence of cold gas in a cD galaxy is [MATH], and thus more cD galaxies should have cold gas of [MATH].', '1310.3349-1-16-5': 'Alternatively, a complicated solution may be required.', '1310.3349-1-16-6': 'For example, while dust provided by evolved stars in a cD galaxy is shielded from the surrounding hot gas , it is mixed with cold gas provided by a weak cooling flow.', '1310.3349-1-17-0': '## Comparison with Other cD Galaxies', '1310.3349-1-18-0': 'Among cD galaxies with outstanding dust lanes and disks, those of Cygnus A and Hydra A are known for having extremely active radio sources .', '1310.3349-1-18-1': 'The cD galaxy of Cygnus A has a well-developed cold disk rotating around the AGN and the minor axis of the disk is parallel to the jets like Hydra A .', '1310.3349-1-18-2': 'On the other hand, the cD galaxy of Hercules A does not have a disk-structure, although it has prominent twin jets .', '1310.3349-1-18-3': 'This may be because the cold disk has disappeared by now, while it caused the strong outburst in the past.', '1310.3349-1-18-4': 'Note that in some non-cD radio galaxies, dust lanes that resemble the one we found in Hydra A have been observed .', '1310.3349-1-19-0': '# Conclusions', '1310.3349-1-20-0': 'We observed the cD galaxy of the Hydra A cluster with Subaru telescope and found an outstanding dust lane.', '1310.3349-1-20-1': 'The disk seen as the dust lane is edge-on and the minor axis is parallel to radio jets, but it is not parallel to the minor axis of the host galaxy.', '1310.3349-1-20-2': 'The disk mass and the rotation period are [MATH] and [MATH] yr, respectively.', '1310.3349-1-20-3': 'The column density of the dust lane ([MATH]) is consistent with the constraints obtained by X-ray observations.', '1310.3349-1-20-4': 'Our results may indicate that galaxy mergers and the supply of cold gas to the SMBHs is essential to the most powerful activities of the AGNs.', '1310.3349-1-21-0': 'This work was supported by KAKENHI (YF: 23540308, KS: 25800112).'}

{'1310.3349-2-0-0': 'The central cD galaxy of the Hydra A cluster has one of the most powerful active galactic nuclei (AGNs) in the nearby Universe ([MATH]).', '1310.3349-2-0-1': 'We report on the discovery of a dust lane in the cD galaxy using Subaru telescope.', '1310.3349-2-0-2': 'The [MATH]-band image shows the existence of a dark band of the size of [MATH]), which appears to be quite similar to the dust lane observed in Centaurus A.', '1310.3349-2-0-3': 'The morphology indicates that the cold disk that seen as the dust lane is almost edge-on and rotates around the AGN.', '1310.3349-2-0-4': 'Since the minor axis of the dust lane is nearly parallel to the radio jets emerging from the AGN, the disk is probably feeding its gas into the central black hole.', '1310.3349-2-0-5': 'From the absorption, we estimate the hydrogen column density of the lane is [MATH], and the mass of the disk is [MATH].', '1310.3349-2-0-6': 'The column density is consistent with constraints obtained from Chandra X-ray observations.', '1310.3349-2-0-7': 'The age of the disk is [MATH] yr.', '1310.3349-2-0-8': "The position angle of the disk and the galaxy's photometric axis are misaligned, which may imply that the cold gas in the disk is brought via galaxy mergers.", '1310.3349-2-0-9': 'Our observations may indicate that the supply of cold gas by galaxy mergers is required for the most intensive feedback from AGNs.', '1310.3349-2-1-0': '# Introduction', '1310.3349-2-2-0': 'The AGNs in the central cD galaxies of clusters sometimes show violent activities.', '1310.3349-2-2-1': 'Such examples are MS 0735.6[MATH]7421 , Hercules A , and Hydra A .', '1310.3349-2-2-2': 'In these clusters, cluster-scale shock waves have been observed, which indicates that the central AGNs ejected enormous energy ([MATH]-[MATH] erg) in a short time ([MATH] yr).', '1310.3349-2-2-3': 'The intracluster medium (ICM) of these clusters should have been strongly affected by the AGN activities.', '1310.3349-2-2-4': 'It seems that Bondi accretion from hot gas is not sufficient and fueling of cold gas is required for the most powerful feedback from AGNs .', '1310.3349-2-3-0': 'The cD galaxy of the Hydra A cluster has been studied by many researchers.', '1310.3349-2-3-1': 'It harbors a well known radio source classified as a Fanaroff-Riley type-I (FR-I) source.', '1310.3349-2-3-2': 'A pair of jets is ejected from the AGN (3C 218; [CITATION]), and the huge amount of cosmic-rays contained in the jets may be stably heating the cool core of the cluster .', '1310.3349-2-3-3': 'The mass of the central supermassive black holes (SMBHs) is estimated to be [MATH] .', '1310.3349-2-3-4': 'Although the galaxy is an elliptical, star formation activities ([MATH]) have been discovered within [MATH] kpc from the center of the galaxy .', '1310.3349-2-3-5': 'This may mean that some amount of gas is cooling in the galaxy, although the existence of a massive cooling flow has been denied with the ASCA satellite .', '1310.3349-2-3-6': 'Previous observations have indicated that there is a disk-like structure in the galaxy.', '1310.3349-2-3-7': 'In the [MATH]-band, the surface brightness profile is elongated at the galaxy center .', '1310.3349-2-3-8': 'Observations of optical emission-lines ([OIII], H[MATH], and H[MATH]) show that the galaxy has a rotating component with a velocity of [MATH] .', '1310.3349-2-3-9': 'HI absorption is seen toward the AGN and can be interpreted as a disk .', '1310.3349-2-3-10': 'However, the morphology of the disk has not been well understood.', '1310.3349-2-3-11': 'This is mainly because the cluster is relatively distant ([MATH]) and the angular resolution of [MATH] is required to resolve fine structures.', '1310.3349-2-4-0': 'In this letter, we present a Subaru [MATH]-band image of the cD galaxy of the Hydra A cluster with a superb spatial resolution.', '1310.3349-2-4-1': 'We show that the galaxy has a spectacular dust lane.', '1310.3349-2-4-2': 'Throughout this letter, we adopt cosmological parameters of [MATH], [MATH], and [MATH].', '1310.3349-2-4-3': 'We use a redshift for the Hydra A cluster of 0.054878, which gives a scale of [MATH] kpc per arcsec. The angular diameter distance and the luminosity distance to Hydra A are [MATH] Mpc and 245 Mpc, respectively.', '1310.3349-2-5-0': '# Observations and Results', '1310.3349-2-6-0': 'We observed Hydra A with the Prime Focus Camera (Suprime-Cam; [CITATION]) on the Subaru telescope in Hawaii on January 7, 2013.', '1310.3349-2-6-1': 'We use the standard pipeline reduction software for Suprime-Cam, SDFRED , for flat-fielding, instrumental distortion correction, differential refraction, PSF matching, and sky subtraction and stacking.', '1310.3349-2-6-2': 'The seeing was [MATH].', '1310.3349-2-6-3': 'The [MATH]-band image of the cD galaxy is shown in figure [REF]a.', '1310.3349-2-6-4': 'A dust lane (dark band) can be clearly seen in the central region of the cD galaxy.', '1310.3349-2-6-5': 'The size of the lane is [MATH] (4 kpc[MATH]0.8 kpc).', '1310.3349-2-6-6': 'The position angle of the minor axis is [MATH] from the north to the east.', '1310.3349-2-6-7': 'We fit the galaxy image with a Seric profile using galfit , taking PSF size into account.', '1310.3349-2-6-8': 'The residual from the best-fits is shown in figure [REF]b.', '1310.3349-2-6-9': 'The position angle of the minor axis of the Seric profile is [MATH], which is close to the result by [CITATION] for the [MATH]-band image ([MATH]), but is not parallel to the minor axis of the disk ([MATH]).', '1310.3349-2-6-10': 'This may indicate that the disk formed after the overall structures of the cD galaxy had formed.', '1310.3349-2-6-11': 'The extinction is most effective at the center (40%), and it is [MATH].', '1310.3349-2-6-12': 'Since the extinction curves of elliptical galaxies are not much different from the one for the Milky-Way , we adopt the Galactic conversion factor ([MATH]; [CITATION]) and obtain [MATH] and the column density of [MATH] from [MATH] .', '1310.3349-2-6-13': 'The value of [MATH] is almost the same as HI absorption ([MATH]) obtained by [CITATION].', '1310.3349-2-6-14': 'If the gas is distributed as a uniform disk with a radius of [MATH] kpc and a height of [MATH] kpc, the disk mass is [MATH].', '1310.3349-2-6-15': 'Figure [REF] shows the relation of the galaxy to the radio source.', '1310.3349-2-6-16': 'A pair of jets emerges with a position angle of [MATH] and is almost parallel to the minor axis of the dust lane.', '1310.3349-2-6-17': 'The jet-dust lane configuration is quite similar to that of Centaurus A.', '1310.3349-2-7-0': '# Discussion', '1310.3349-2-8-0': '## X-ray Absorption', '1310.3349-2-9-0': 'We have checked X-ray data obtained with the Chandra X-ray telescope.', '1310.3349-2-9-1': 'Hydra A has been observed with Chandra for total 250 ksec. Figure [REF] shows the exposure-corrected X-ray image of Chandra ACIS-S for 200 ksec data in the 0.5-5 keV band.', '1310.3349-2-9-2': 'While the central AGN is prominent, the disk-structure is not seen in the figure.', '1310.3349-2-9-3': 'X-ray spectral analysis for Chandra ACIS-S and ACIS-I data of the total of 250 ksec was performed to constrain an absorption by the disk with response files appropriate for each observation in the 0.4-7.0 keV band.', '1310.3349-2-10-0': 'We study the spectrum of a [MATH] region of the disk (region "D" in figure [REF]).', '1310.3349-2-10-1': 'We exclude the nuclear region "N", because the emission from it can be affected by the absorption in the vicinity of the SMBH.', '1310.3349-2-10-2': 'In fact, we find that the absorption associated with the AGN is [MATH], which is consistent with the previous result ([MATH] .', '1310.3349-2-10-3': 'Owing to the finite size of pixels ([MATH]), the pixels we used actually cover the optical disk ([MATH]).', '1310.3349-2-10-4': 'First, we extract a spectrum within a [MATH] circle centered on the AGN excluding the AGN and disk regions ("N" and "D") in order to estimate the fore- and/or background emissions.', '1310.3349-2-10-5': 'The spectrum is represented by two-temperature (cD galaxy + ICM) thermal models (apec) with a common metal abundance multiplied by absorption (phabs): [MATH].', '1310.3349-2-10-6': 'The results are [MATH] and [MATH] keV with 0.7[MATH] solar abundance, and [MATH] cm[MATH] (errors of 90% confidence).', '1310.3349-2-10-7': 'The absorption value is almost consistent with the Galactic absorption ([MATH] cm[MATH], [CITATION]) in the direction of Hydra A. Fixing these fore- and/or background parameters, we investigate the upper limit of the absorption by the disk.', '1310.3349-2-10-8': 'Here, we assume that the emission from the disk region is represented by two-temperature thermal models (cD galaxy + ICM) and only the cooler component (cD galaxy) is absorbed by the disk: [MATH].', '1310.3349-2-10-9': 'We employ an increment of [MATH] as the measure for the 90% upper limit of the absorption.', '1310.3349-2-10-10': 'We find that the upper limit is [MATH] cm[MATH], which is consistent with the disk absorption of [MATH] (section [REF]).', '1310.3349-2-10-11': 'Note that the absorption by the dust lane on the both sides of the AGN should be smaller than that in the direction of the AGN (region "N").', '1310.3349-2-11-0': '## Nature of the Disk', '1310.3349-2-12-0': 'Optical emission-lines have been discovered from the cD galaxy of Hydra A.', '1310.3349-2-12-1': 'If the rotation of the emission-line component corresponds to that of the dust lane, the rotation velocity is [MATH] , which gives the rotation period of [MATH] yr.', '1310.3349-2-12-2': 'The age of the cold disk should be larger than this period.', '1310.3349-2-12-3': 'From the observations of a shock wave in the ICM, [CITATION] indicated that there was a strong outburst of the AGN with an energy of [MATH] erg [MATH] yrs ago.', '1310.3349-2-12-4': 'This outburst may be related to the formation of the disk.', '1310.3349-2-12-5': 'Note that if all the disk gas at present ([MATH]) flowed into the SMBH, energy comparable to the previous outburst might be released ([MATH]).', '1310.3349-2-13-0': 'The angular distance between the X-ray AGN and the center line of the dust lane is [MATH] or [MATH] kpc (figure [REF]).', '1310.3349-2-13-1': 'If the AGN is located at the center of the disk, the inclination angle of the disk is [MATH], and the disk is almost edge-on.', '1310.3349-2-13-2': 'This may also mean that the jets from the AGN are almost on the plane of the sky at present.', '1310.3349-2-13-3': 'On the other hand, the positions of pairs of outer lobes show that the jets were not being injected on the plane of the sky and were oriented [MATH] about the plane in the past .', '1310.3349-2-13-4': 'The change of the jet direction may be caused by precession .', '1310.3349-2-14-0': '## Origin of the Dust Lane', '1310.3349-2-15-0': 'The cD galaxy is a harsh environment for dust.', '1310.3349-2-15-1': 'In the presence of the hot ICM of [MATH] K, collision with hot particles destroys the dust.', '1310.3349-2-15-2': 'The destruction time is [EQUATION] where [MATH] is the ICM density and [MATH] is the radius of dust grains .', '1310.3349-2-15-3': 'If we adopt [MATH] and [MATH], the destruction time is [MATH] yr.', '1310.3349-2-15-4': 'The mass loss rate from evolved stars in the galaxy is [EQUATION] where [MATH] is the distance to the galaxy, and [MATH] is the flux at [MATH] .', '1310.3349-2-15-5': 'For Hydra A, the luminosity distance is [MATH] Mpc, and the flux is [MATH] .', '1310.3349-2-15-6': 'Since we are interested in the upper limit of the dust mass [MATH] and since we avoid the complications caused by the difference of apertures at different wave lengths, we assume that [MATH].', '1310.3349-2-15-7': 'For these values, we obtain [MATH].', '1310.3349-2-15-8': 'Assuming that the gas-to-dust ratio is [MATH], the rate of accumulation of dust is [EQUATION] .', '1310.3349-2-15-9': 'This equation can be solved easily; [MATH] approaches its final value [MATH] in a time-scale of a few [MATH].', '1310.3349-2-15-10': 'Since [MATH], this may indicate that the dust is externally provided in the galaxy.', '1310.3349-2-16-0': 'If the dust is not provided internally, the dust may be brought by merged galaxies.', '1310.3349-2-16-1': 'That is the case for Centaurus A .', '1310.3349-2-16-2': "The misalignment between the disk and the galaxy's photometric axis (section [REF]) is favorable to the merger scenario.", '1310.3349-2-16-3': 'Based on a semi-analytic model, [CITATION] indicated that the fraction of cD galaxies that have cold gas brought through galaxy mergers induced by dynamical friction is small.', '1310.3349-2-16-4': 'However, their criterion for existence of cold gas in a cD galaxy is [MATH], and thus more cD galaxies should have cold gas of [MATH] (see [CITATION]).', '1310.3349-2-16-5': 'Alternatively, a complicated solution may be required.', '1310.3349-2-16-6': 'For example, while dust provided by evolved stars in a cD galaxy is shielded from the surrounding hot gas , it is mixed with cold gas provided by a weak cooling flow .', '1310.3349-2-17-0': '## Comparison with Other cD Galaxies', '1310.3349-2-18-0': 'Among cD galaxies with outstanding dust lanes and disks, those of Cygnus A and Hydra A are known for having extremely active radio sources .', '1310.3349-2-18-1': 'The cD galaxy of Cygnus A has a well-developed cold disk rotating around the AGN and the minor axis of the disk is parallel to the jets like Hydra A .', '1310.3349-2-18-2': 'On the other hand, the cD galaxy of Hercules A does not have a disk-structure, although it has prominent twin jets .', '1310.3349-2-18-3': 'This may be because the cold disk has disappeared by now, while it caused the strong outburst in the past.', '1310.3349-2-18-4': 'Note that in some non-cD radio galaxies, dust lanes that resemble the one we found in Hydra A have been observed .', '1310.3349-2-19-0': '# Conclusions', '1310.3349-2-20-0': 'We observed the cD galaxy of the Hydra A cluster with Subaru telescope and found an outstanding dust lane.', '1310.3349-2-20-1': 'The disk seen as the dust lane is edge-on and the minor axis is parallel to radio jets, but it is not parallel to the minor axis of the host galaxy.', '1310.3349-2-20-2': 'The disk mass and the rotation period are [MATH] and [MATH] yr, respectively.', '1310.3349-2-20-3': 'The column density of the dust lane ([MATH]) is consistent with the constraints obtained by X-ray observations.', '1310.3349-2-20-4': 'Our results may indicate that galaxy mergers and the supply of cold gas to the SMBHs is essential to the most powerful activities of the AGNs.', '1310.3349-2-21-0': 'This work was supported by KAKENHI (YF: 23540308, KS: 25800112), and World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan'}

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[['1310.3349-1-16-4', '1310.3349-2-16-4'], ['1310.3349-1-16-6', '1310.3349-2-16-6']]

[]

[]

[]

['1310.3349-1-21-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1310.3349

1004.1626

{'1004.1626-1-0-0': 'Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method.', '1004.1626-1-0-1': 'We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map.', '1004.1626-1-0-2': 'In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers.', '1004.1626-1-0-3': 'We show that the infinite-dimensional transfer-operator (Dittes F-M, Doron E and Smilansky U 1994 Phys.', '1004.1626-1-0-4': 'Rev. E 49 963) emerges as the continuum limit of the transfer matrix.', '1004.1626-1-0-5': 'The possibility of using suitable truncation schemes for calculating long-time traces is considered.', '1004.1626-1-0-6': 'We also discuss some aspects of the long-time divergence of the semiclassical traces.', '1004.1626-1-1-0': '# Introduction', '1004.1626-1-2-0': "The quantum baker's map [CITATION] is a very useful toy model for investigating quantum-classical correspondence issues in a variety of settings.", '1004.1626-1-2-1': "Conceived as a test bench for studying the semiclassical limit of closed chaotic systems, it was first applied to analyze the random-matrix conjecture, the scarring phenomenon, Gutwiller's trace formula and the long-time validity of semiclassical approximations (see, e.g., [CITATION]).", '1004.1626-1-2-2': 'In the last years "open" quantum bakers were constructed and employed for studying semiclassical aspects of the scattering problem [CITATION], e.g., the fractal Weyl law for the distribution of resonances [CITATION].', '1004.1626-1-3-0': 'The quantum baker also appeared in a variety of problems of Quantum Information, Quantum Computation and Quantum Open Systems.', '1004.1626-1-3-1': 'It was noted that the quantum baker could be efficiently realized in terms of quantum gates [CITATION].', '1004.1626-1-3-2': 'A three qubit Nuclear Magnetic Resonance experiment was proposed [CITATION] and then implemented (with some simplifications) [CITATION].', '1004.1626-1-3-3': 'On the theoretical side, Schack and Caves [CITATION] showed that the Balazs-Voros-Saraceno quantum baker [CITATION] can be seen as a shift on a string of quantum bits -in full analogy with the classical case- and exhibited a family of alternative quantizations.', '1004.1626-1-3-4': 'This family of bakers was the subject of several studies [CITATION].', '1004.1626-1-3-5': 'The ability of the baker family to generate entanglement was first studied by Scott and Caves [CITATION] (see also [CITATION]).', '1004.1626-1-3-6': 'Decoherent variants of the baker map were constructed by including mechanisms of dissipation and/or diffusion [CITATION].', '1004.1626-1-4-0': '>From the point of view of the structure of the quantum baker map, important results were recently obtained by Ermann and Saraceno [CITATION], who, building upon previous work by Lakshminarayan and Meenakshisundaram [CITATION], generalized the Schack-Caves family, and showed that all quantum bakers are perturbations of a common kernel (the "essential" baker).', '1004.1626-1-5-0': 'The present paper focuses on an almost unexplored aspect of the semiclassical theory of the quantum baker map.', '1004.1626-1-5-1': "Gutzwiller's approximate formula for the traces of the baker map is formally equivalent to the partition function of a finite Ising chain (with exponentially decaying interactions and imaginary temperature).", '1004.1626-1-5-2': 'Thus, it can be evaluated with the standard transfer-matrix method.', '1004.1626-1-5-3': 'We know of two previous studies of the baker map which used the transfer matrix method.', '1004.1626-1-5-4': 'Heller and Kaplan applied this scheme in their work on semiclassical long-time propagation of wave-packets [CITATION], but gave no details of their calculations.', '1004.1626-1-5-5': 'Smilansky and Verdene [CITATION] wrote down the transfer matrix and did some analytical calculations for the few-neighbor case.', '1004.1626-1-5-6': 'However, their focus was on periodic-orbit action correlations [CITATION], and did not examine the transfer matrices themselves.', '1004.1626-1-6-0': 'Our purpose is to present an extensive analysis of the transfer matrices associated with the semiclassical traces of the baker map.', '1004.1626-1-6-1': 'First we discuss the general structure of the transfer matrices.', '1004.1626-1-6-2': 'We show that they have a simple quantum-circuit representation.', '1004.1626-1-6-3': 'Though they are not unitary, nounitarity is restricted to a one-qubit "gate".', '1004.1626-1-6-4': 'Except for this fact, the transfer matrices resemble qubit quantum bakers (Sec. [REF]).', '1004.1626-1-7-0': 'Section [REF] is devoted to spectral properties.', '1004.1626-1-7-1': 'We show that all eigenvalues lie inside the unit circle in the complex plane, or very close to it.', '1004.1626-1-7-2': 'The number of eigenvalues close to the unit circle coincides approximately with the quantum dimension (for a suitable parameter range).', '1004.1626-1-7-3': 'This situation is very similar to that found in the infinite-dimensional transfer-operator method [CITATION].', '1004.1626-1-8-0': 'We evaluate the possibility of using truncation schemes for calculating long-time traces in Sec. [REF].', '1004.1626-1-8-1': 'Some aspects of the long-time divergence of the semiclassical traces are assessed in Sec. [REF].', '1004.1626-1-9-0': 'The baker map possesses a spatial symmetry.', '1004.1626-1-9-1': 'If one is interested in separating the spectrum/traces into symmetry classes, then one must consider also the symmetry-related transfer matrices.', '1004.1626-1-9-2': 'Though similar to the matrices of Sec. [REF] in many respects, these matrices happen to be unitary.', '1004.1626-1-9-3': 'They are studied in Sec. [REF].', '1004.1626-1-10-0': 'Concluding remarks are presented in Sec. [REF].', '1004.1626-1-11-0': '# Baker map: Transfer matrix', '1004.1626-1-12-0': 'In order to make the paper self-contained we start by summarizing some basic information about the baker map.', '1004.1626-1-13-0': 'The dynamics of the classical baker map is schematically depicted in Fig. [REF].', '1004.1626-1-14-0': 'It acts on the unit square piecewise-linearly.', '1004.1626-1-14-1': 'Points with [MATH] are governed by the hyperbolic point at the origin.', '1004.1626-1-14-2': 'If [MATH] the fixed point at [MATH] rules.', '1004.1626-1-14-3': 'In any case the coordinates [MATH] relative to the fixed point are compressed/stretched by a factor of two: [EQUATION] where [MATH], the integer part of [MATH].', '1004.1626-1-15-0': 'The quantum analogue of the classical mapping is a unitary operator acting on an even-dimensional Hilbert space [CITATION].', '1004.1626-1-15-1': 'In the [MATH]-representation its matrix reads [CITATION]: [EQUATION] where [MATH] is the [MATH]-dimensional antiperiodic Fourier matrix: [EQUATION] with [MATH].', '1004.1626-1-15-2': 'For this abstract system the Planck constant coincides with the inverse of the dimension, i.e., [MATH].', '1004.1626-1-16-0': 'Our main concern are the traces of the quantum baker, [MATH], for [MATH].', '1004.1626-1-16-1': 'In the semiclassical regime of [MATH] large enough (for a given [MATH]) the traces can be approximated by the Gutzwiller formula [CITATION] [EQUATION] where the sum runs over all periodic trajectories of length [MATH] of the classical map, indexed by the binary column vectors [MATH], with [MATH].', '1004.1626-1-16-2': 'The corresponding actions [MATH] are quadratic functions of the binary symbols, [EQUATION]', '1004.1626-1-16-3': 'The "coupling" matrix [MATH] is suitably expressed in terms of the matrix [MATH] of a cyclic shift, [MATH] [CITATION]: [EQUATION]', '1004.1626-1-16-4': 'The trace formula for the baker map ([REF]) can be derived as follows.', '1004.1626-1-16-5': 'Write [MATH] as a sum of products of Fourier matrix elements using Eq. ([REF]).', '1004.1626-1-16-6': 'Approximate sums by integrals.', '1004.1626-1-16-7': 'Extend the limits of integration (which in principle are finite) to [MATH].', '1004.1626-1-16-8': 'The remaining integrals, being Gaussian, can be done exactly, giving the Gutzwiller sum [CITATION].', '1004.1626-1-17-0': 'Equation ([REF]) is a particular case of the general Gutzwiller trace formula for systems with a chaotic classical limit.', '1004.1626-1-17-1': 'For such systems the trace formula lays a bridge between the quantum spectrum and the set of classical periodic orbits.', '1004.1626-1-17-2': 'It constitutes the core of all semiclassical schemes for relating energy levels to classical information, from the pioneer attempts [CITATION], to the most recent, highly sophisticated developments [CITATION].', '1004.1626-1-18-0': 'The baker is very special in that all its periodic orbits, and properties thereof, are known analytically.', '1004.1626-1-18-1': 'So, in principle, the periodic-orbit sum ([REF]) can be calculated for any [MATH].', '1004.1626-1-18-2': 'However, because of the exponential growth of the number of periodic orbits with period, the brute-force summation is restricted to, say, [MATH].', '1004.1626-1-18-3': 'Remarkably, the method developed by Dittes, Doron and Smilansky [CITATION] does not suffer from such a limitation.', '1004.1626-1-18-4': 'These authors defined an infinite-dimensional integral operator, [EQUATION] [MATH], whose traces give exactly the periodic-orbit sum of Eq. ([REF]).', '1004.1626-1-18-5': 'They showed that [MATH] can be efficiently calculated from the eigenvalues of the matrix of [MATH] in the Fourier basis, after suitable truncation [CITATION].', '1004.1626-1-19-0': 'An alternative to the infinite-dimensional operator method relies on the formal identification of the periodic-orbit sum ([REF]) with the partition function of an Ising chain, for a purely imaginary temperature.', '1004.1626-1-19-1': 'The cyclical nature of the coupling matrix [MATH] says that this is a circular chain, consisting of [MATH] spins.', '1004.1626-1-19-2': 'All spins interact among themselves, but, as interactions decay exponentially with distance, one may expect some computational benefit (without much loss of accuracy) by truncating the interaction to some given number [MATH] of closest neighbors, i.e., [EQUATION]', '1004.1626-1-19-3': 'The exact periodic-orbit sum corresponds to setting [MATH] (no truncation at all), but, in principle, one can consider any truncation, even to first neighbors ([MATH]).', '1004.1626-1-20-0': 'Once one has established the equivalence between the baker periodic-orbit sum and the Ising partition function, the transfer-matrix method can be invoked.', '1004.1626-1-20-1': 'The first-neighbor case is explained in textbooks [CITATION].', '1004.1626-1-20-2': 'It seems that the many-neighbor case has not been explicitly worked out in the literature, but some discussions exist [CITATION].', '1004.1626-1-20-3': 'Anyway, even if not trivial, the generalization is relatively simple and, in the case of the baker-Ising, leads to [EQUATION] the explicit expression for [MATH] being [CITATION]: [EQUATION]', '1004.1626-1-20-4': 'The elements [MATH] are given by [EQUATION] with [MATH], and [EQUATION]', '1004.1626-1-20-5': 'Note that [MATH] is a complex matrix depending on the three parameters [MATH].', '1004.1626-1-20-6': 'Its dimension is [MATH], with [MATH].', '1004.1626-1-20-7': 'When [MATH] one recovers the exact semiclassical traces, i.e., Eq. ([REF]) becomes an equality.', '1004.1626-1-21-0': 'A prefactor [MATH] has been absorbed into the definition of [MATH] because in this way most of the spectrum of [MATH] lies close to the unit circle or inside it (see Sec. [REF] below).', '1004.1626-1-21-1': 'So, the transfer matrix [MATH] becomes qualitatively similar to the semiclassical transfer operator of Refs. [CITATION] (we come back to this point later).', '1004.1626-1-22-0': 'The transfer matrix approach transforms the calculation of the Gutzwiller sum into the problem of obtaining the trace of the [MATH]-th power of the finite matrix [MATH]. >', '1004.1626-1-22-1': 'From a numerical point of view this is advantageous only if truncation of [MATH] is admissible.', '1004.1626-1-22-2': 'We defer the analysis of this question until Sec. [REF].', '1004.1626-1-22-3': 'Independently of its numerical usefulness, the matrices generated in the transfer matrix approach exhibit interesting properties, as we now proceed to show.', '1004.1626-1-23-0': 'The transfer matrix ([REF]) possesses the structure of a tensor product which is conveniently displayed by switching to a qubit representation.', '1004.1626-1-23-1': 'This consists in identifying the "spatial" degree of freedom [MATH] with the tensor product of [MATH] two-level systems: [EQUATION]', '1004.1626-1-23-2': 'Here [MATH] indexes the states of the [MATH]-basis and [MATH] is a label for the qubit basis states; they are related through the binary expansion [EQUATION]', '1004.1626-1-23-3': 'In the qubit picture the matrix [MATH] can be broken up into the elementary gates shown in the quantum circuit of Fig. [REF].', '1004.1626-1-24-0': 'The circuit acts on [MATH] qubits, starting with a full downwards shift of all qubits, i.e., [EQUATION]', '1004.1626-1-24-1': 'A nonunitary gate acting on the first qubit follows, which bears some resemblance with a Hadamard gate, [EQUATION] with [EQUATION]', '1004.1626-1-24-2': 'It must be noted that, as the dimension [MATH] is an even number, we always have [MATH], and [MATH] is never unitary.', '1004.1626-1-24-3': 'After the [MATH] gate one finds a single qubit phase gate, [EQUATION]', '1004.1626-1-24-4': 'Finally one has a sequence of two-qubit phase gates, acting symmetrically between the first and the [MATH]-th qubit, [EQUATION] with exponentially decreasing phases, [MATH].', '1004.1626-1-25-0': 'Writing [MATH] as a circuit has helped us in identifying its basic constituents.', '1004.1626-1-25-1': 'In particular we recognize the one-qubit gate [MATH] as responsible for the deviation from unitarity.', '1004.1626-1-25-2': 'If we substitute [MATH] by a Hadamard gate, i.e., set [MATH] in ([REF]), then the circuit of Fig. [REF] acquires the typical structure of the baker family [CITATION]: an "essential-baker" block (formed by a shift [MATH] and a single-qubit Fourier transform) followed by a "diffraction kernel" (phase gates).', '1004.1626-1-25-3': 'In this case the diffraction kernel happens to be exactly diagonal in the [MATH]-representation, while for typical quantum bakers it is almost diagonal [CITATION].', '1004.1626-1-26-0': '# Spectral Properties', '1004.1626-1-27-0': 'Figure [REF] displays a typical transfer-matrix spectrum in the case that the dimension of the transfer matrix is much larger than the quantum dimension, i.e., [MATH].', '1004.1626-1-28-0': 'The spectrum can be coarsely divided into three parts.', '1004.1626-1-28-1': 'Approximately [MATH] eigenvalues are localized close to the unit circle.', '1004.1626-1-28-2': 'Most of the remaining ones are concentrated in a disk of smaller radius, and there are some transitional eigenvalues spiraling out from the inner disk to the unit circle.', '1004.1626-1-28-3': 'The eigenvalues lying close to the unit circle can be put in almost one-to-one correspondence with the exact eigenvalues of the quantum baker ([REF]).', '1004.1626-1-29-0': 'Increasing [MATH] while keeping [MATH] fixed does not significantly change the almost unitary part of spectrum, but populates the region of small moduli (see Fig. [REF]).', '1004.1626-1-30-0': "This fact, when combined with the existence of eigenvalues with moduli larger than one, leads to the conclusion that Gutzwiller's traces for the baker are divergent in the limit [MATH] (fixed [MATH]).", '1004.1626-1-30-1': 'This result has been confirmed previously using the operator-transfer method [CITATION].', '1004.1626-1-31-0': 'Indeed, the overall features of the transfer-matrix spectrum described above can also be found in the transfer-operator spectra discussed in Refs. [CITATION].', '1004.1626-1-31-1': 'Looking for an explanation for this similarity, we recall that the transfer operator is infinite-dimensional and independent of [MATH].', '1004.1626-1-31-2': 'Thus, one may be tempted to compare Eq. ([REF]) with the infinite-[MATH] limit of the transfer matrix ([REF]).', '1004.1626-1-31-3': 'In this limit, the transfer matrix becomes an integral kernel, indices becoming continuous variables: [EQUATION]', '1004.1626-1-31-4': 'A simple calculation verifies that the transfer matrix tends to the transfer operator, i.e., [EQUATION] with [MATH] precisely that given in Eq. ([REF])!', '1004.1626-1-31-5': 'So, the transfer operator is formally recovered as the continuum limit of the transfer matrix.', '1004.1626-1-32-0': '# Truncation Schemes', '1004.1626-1-33-0': 'Here we discuss the numerical calculation of long-time Gutzwiller traces using the transfer-matrix formula ([REF]).', '1004.1626-1-33-1': 'The most natural procedure for calculating [MATH] requires the diagonalization of [MATH] to obtain its eigenvalues.', '1004.1626-1-33-2': 'This scheme is limited to very small values of [MATH], e.g., to [MATH] in a personal computer.', '1004.1626-1-33-3': 'So, truncation becomes essential.', '1004.1626-1-33-4': 'The error introduced in truncating the interactions can be estimated from Eq. ([REF]).', '1004.1626-1-33-5': 'A back-of-the-envelope calculation leads us to the following estimate for the validity of the truncation to [MATH] neighbors: [EQUATION]', '1004.1626-1-33-6': 'However, even after truncating the transfer matrix, we may need to use values of [MATH] which make diagonalization prohibitive, e.g., for [MATH], [MATH].', '1004.1626-1-33-7': 'Thinking of such cases, we note that the sparsity of [MATH] makes it possible to implement an alternative, more efficient method.', '1004.1626-1-33-8': 'Consider the following identity: [EQUATION] where overline means average over random complex vectors [MATH] uniformly distributed over the corresponding unit sphere [CITATION].', '1004.1626-1-33-9': '(The notation [MATH] has been used to indicate the truncation of [MATH] to [MATH] qubits.)', '1004.1626-1-33-10': 'The idea behind ([REF]) is to calculate [MATH] by applying iteratively the matrix [MATH] to [MATH], and then averaging over [MATH].', '1004.1626-1-33-11': 'This method allows one, in principle, to consider [MATH] as large as 20.', '1004.1626-1-34-0': 'Table [REF] displays some examples.', '1004.1626-1-35-0': 'The first rows (a-e) illustrate the improvement of the results as the truncation level [MATH] is increased, for fixed [MATH].', '1004.1626-1-35-1': 'Comparison with the exact results is satisfactory for [MATH], corresponding to [MATH].', '1004.1626-1-35-2': 'In rows (f-i) we consider [MATH] fixed and large values of [MATH] which can in no way be reached by direct computation of the periodic orbit summation, i.e., exact results are not known for such [MATH].', '1004.1626-1-35-3': 'In these cases we compare with results obtained using the transfer-operator method, which is not limited to small [MATH] [CITATION].', '1004.1626-1-35-4': 'Now we see good agreement for [MATH], i.e., [MATH].', '1004.1626-1-36-0': 'So, we have checked that the transfer-matrix method works satisfactorily in the parameter regime specified by ([REF]).', '1004.1626-1-37-0': '# Divergence of long-time traces', '1004.1626-1-38-0': 'For chaotic systems, the Gutzwiller trace formula is only the first term of an expansion in powers of [MATH].', '1004.1626-1-38-1': 'Because of this, the semiclassical eigenvalues obtained from the Gutwiller traces deviate in general from the unit circle in the complex plane.', '1004.1626-1-38-2': 'For typical chaotic systems it is expected that the distance of the largest eigenvalue to the unit circle scales like [MATH], i.e., [MATH] for quantum maps [CITATION].', '1004.1626-1-38-3': 'In the case of the baker map, an anomalous behavior [MATH] is observed which is due to diffraction effects originating in the discontinuities of the mapping [CITATION].', '1004.1626-1-38-4': "In any case, this lack of unitarity makes Gutzwiller's traces exponentially growing with [MATH], a behavior that may be guessed from Table [REF].", '1004.1626-1-39-0': 'The question we want to pose is: can the [MATH] law be associated with the particular structure of the baker transfer matrix?', '1004.1626-1-39-1': 'In the circuit representation (Fig. [REF]) nonunitarity arises from the single-qubit gate [MATH].', '1004.1626-1-39-2': 'So, we ask: what is typically the largest eigenvalue (in modulus) of a map obtained by tensoring [MATH] with a generic [MATH]-dimensional unitary gate [MATH]?', '1004.1626-1-39-3': 'On the other side, the [MATH]-representation ([REF]) offers a different point of view of the nonunitarity: the transfer matrix can be split as [MATH], with [MATH] unitary matrices of dimension [MATH] [see Eq. ([REF])]; how does the leading eigenvalue of such a matrix [MATH] scale with [MATH]?', '1004.1626-1-40-0': 'In order to answer the questions above we resorted to numerical calculations.', '1004.1626-1-40-1': 'First we analyzed the matrices [MATH] with [MATH] a random matrix belonging to the Circular Unitary Ensemble (CUE).', '1004.1626-1-40-2': 'The matrix [MATH] acts on the least significant qubit and is defined by Eq. ([REF]) with [MATH], i.e., [MATH] is chosen to be the large-[MATH] limit of [MATH].', '1004.1626-1-40-3': 'Then we considered the ensemble [MATH] with [MATH] independent random matrices belonging to CUE [CITATION].', '1004.1626-1-40-4': 'In order to check any possible ensemble dependence we also calculated the case [MATH] with [MATH] in the Circular Orthogonal Ensemble (COE).', '1004.1626-1-40-5': 'Our results are exhibited in Fig. [REF].', '1004.1626-1-41-0': 'In the three cases considered we observe clearly that the decay laws are not [MATH] but something closer to [MATH].', '1004.1626-1-41-1': 'Thus, we must conclude that the minimum information models considered above lack some essential ingredient which is responsible for the [MATH] scaling.', '1004.1626-1-42-0': '# Reflected baker map', '1004.1626-1-43-0': 'The quantum baker map [MATH] is invariant under the parity symmetry [MATH], its action on the [MATH]-basis being just [MATH].', '1004.1626-1-43-1': 'This is the quantum counterpart of the classical reflection symmetry [MATH].', '1004.1626-1-43-2': 'When analyzing spectral properties it is convenient to separate eigenstates and eigenvalues of [MATH] according to their parity.', '1004.1626-1-43-3': 'Thus, one is led to consider the parity-projected bakers [MATH].', '1004.1626-1-43-4': 'In the semiclassical domain the traces of [MATH] can be approximated by [CITATION] [EQUATION]', '1004.1626-1-43-5': 'The first summation above correspond to the baker traces ([REF]).', '1004.1626-1-43-6': 'In the second sum [MATH] stands for half the action [Eq. ([REF])] of a periodic orbit of length [MATH] with symbolic code [MATH], where [MATH].', '1004.1626-1-43-7': 'These are precisely the orbits invariant under the parity transformation.', '1004.1626-1-44-0': 'For the sake of completeness we shall exhibit the transfer matrix associated to the reflected traces, pointing out its most important features.', '1004.1626-1-44-1': 'However, we shall not discuss its spectral properties, truncation schemes, or long-time divergence.', '1004.1626-1-45-0': 'Following the same procedure as in Sec. [REF] we have constructed a transfer matrix [MATH] such that [EQUATION]', '1004.1626-1-45-1': 'After eliminating the null subspace that appears as a consequence of considering just the parity-invariant trajectories of length [MATH], the matrix [MATH] is reduced to dimension [MATH]: [EQUATION] where the matrix elements [MATH] are given by [EQUATION] with [EQUATION]', '1004.1626-1-45-2': 'The last undefined ingredient is the integer [MATH].', '1004.1626-1-45-3': 'It is best expressed in terms of the binary digits of the integer [MATH] [see Eq. ([REF])].', '1004.1626-1-45-4': 'If [MATH], then [EQUATION]', '1004.1626-1-45-5': "The main difference between the baker's transfer matrix ([REF]) and the matrix above, is that [MATH] is exactly unitary.", '1004.1626-1-45-6': 'Except for this fact, both [MATH] and [MATH] are structurally very similar.', '1004.1626-1-45-7': 'Perhaps this can be better appreciated by looking at the quantum circuit for [MATH] in Fig. [REF].', '1004.1626-1-46-0': 'The circuit acts on [MATH] qubits (we are not considering truncation).', '1004.1626-1-46-1': "It starts like the baker's, with a downwards shift of all qubits.", '1004.1626-1-46-2': 'Then we have a NOT gate acting on the first qubit given by [EQUATION]', '1004.1626-1-46-3': 'A phase gate [MATH] follows that acts on the first qubit, with [EQUATION]', '1004.1626-1-46-4': 'Finally there is a sequence of [MATH] symmetrical two-qubit phase gates [MATH] acting between the first and the [MATH]-th qubit with exponentially decreasing phases, [MATH], where [MATH].', '1004.1626-1-47-0': 'The unitarity of [MATH] implies that its spectrum lies on the unit circle.', '1004.1626-1-47-1': 'The following observation gives us some hint about its nature.', '1004.1626-1-47-2': 'Consider the [MATH]-th power of the transfer matrix [MATH], i.e., the [MATH]-th iteration of the circuit above.', '1004.1626-1-47-3': 'It is easy to see that the shifts and NOT gates cancel out and only phase gates remain.', '1004.1626-1-47-4': 'This means that [MATH] is a diagonal matrix (the phase gates are diagonal).', '1004.1626-1-47-5': 'It turns out that, as expected, the diagonal elements are exactly the Gutwiller phases [MATH] in Eq. ([REF]) (we verified this numerically for some cases).', '1004.1626-1-47-6': 'Thus, the spectrum of [MATH] is formed by roots of the Gutzwiller phases.', '1004.1626-1-48-0': '# Conclusions', '1004.1626-1-49-0': 'We presented a study of the transfer matrix approach to the semiclassical traces of the baker map.', '1004.1626-1-49-1': 'Spectral properties of the resulting transfer matrices were analysed, showing that there exists a close similitude with the transfer operator of Ref. [CITATION].', '1004.1626-1-49-2': 'In fact, we (heuristically) proved that the transfer operator arises as a suitable asymptotic limit of the transfer matrix formalism.', '1004.1626-1-50-0': 'Truncation schemes were discussed which, even if not as efficient as the operator method, permit the numerical calculation of long-time traces well inside a domain where direct summation of the Gutzwiller formula is impossible.', '1004.1626-1-51-0': 'Remarkably the transfer matrices are structurally very similar to qubit quantum bakers, i.e., they exhibit an essential-baker core perturbed by a diffraction kernel [CITATION].', '1004.1626-1-51-1': 'It is tempting to think of the transfer matrices as maps for which the Gutzwiller formula is exact.', '1004.1626-1-51-2': 'However, this point of view has some limitations because there is a map for each value of [MATH].', '1004.1626-1-51-3': 'It is only in the large-[MATH] limit that the transfer matrices become independent of [MATH] and define a quantum map.', '1004.1626-1-51-4': 'Even so, such a map in nonunitary.', '1004.1626-1-51-5': 'Curiously, the simmetry-reflected case produces a unitary transfer matrix for all values of [MATH].', '1004.1626-1-52-0': 'The divergence of the Gutzwiller trace formula for the baker map for large [MATH] prohibits the existence of a quantum map whose traces are exactly given by the Gutzwiller formula.', '1004.1626-1-52-1': 'Anyway, our results suggest that we may be not far from finding a close relative to the baker map having an exact periodic-orbit trace formula.'}

{'1004.1626-2-0-0': 'Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method.', '1004.1626-2-0-1': 'We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not).', '1004.1626-2-0-2': 'In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers.', '1004.1626-2-0-3': 'In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some parameter values).', '1004.1626-2-0-4': 'In a suitable continuum limit we recover the already known infinite-dimensional transfer-operator.', '1004.1626-2-0-5': "We devise truncation schemes allowing the calculation of long-time traces in regimes where the direct summation of Gutzwiller's formula is impossible.", '1004.1626-2-0-6': 'Some aspects of the long-time divergence of the semiclassical traces are also discussed.', '1004.1626-2-1-0': '# Introduction', '1004.1626-2-2-0': "The quantum baker's map [CITATION] is a very useful test-bench for investigating quantum-classical correspondence issues in a variety of settings.", '1004.1626-2-2-1': "Conceived as a model for studying the semiclassical limit of closed chaotic systems, it was first applied to analyze the random-matrix conjecture, the scarring phenomenon, Gutwiller's trace formula and the long-time validity of semiclassical approximations (see, e.g., [CITATION]).", '1004.1626-2-2-2': 'In the last years "open" quantum bakers were constructed and employed for studying semiclassical aspects of the scattering problem [CITATION], e.g., the fractal Weyl law for the distribution of resonances [CITATION].', '1004.1626-2-3-0': 'The quantum baker also appeared in a variety of problems of Quantum Information, Quantum Computation and Quantum Open Systems.', '1004.1626-2-3-1': 'It was noted that the quantum baker could be efficiently realized in terms of quantum gates [CITATION].', '1004.1626-2-3-2': 'A three qubit Nuclear Magnetic Resonance experiment was proposed [CITATION] and then implemented (with some simplifications) [CITATION].', '1004.1626-2-4-0': 'On the theoretical side, Schack and Caves [CITATION] showed that the Balazs-Voros-Saraceno quantum baker [CITATION] can be seen as a shift on a string of quantum bits (in full analogy with the classical case), thus taking an important step towards generalizing the method of symbolic dynamics to the quantum case.', '1004.1626-2-4-1': 'At the same time, their circuit representation led naturally to a family of alternative quantizations.', '1004.1626-2-4-2': 'This family of bakers was the subject of several studies [CITATION].', '1004.1626-2-4-3': 'The ability of the baker family to generate entanglement was first studied by Scott and Caves [CITATION] (see also [CITATION]).', '1004.1626-2-4-4': 'Decoherent variants of the baker map were constructed by including mechanisms of dissipation and/or diffusion [CITATION].', '1004.1626-2-5-0': 'From the point of view of the structure of the quantum baker map, important results were recently obtained by Ermann and Saraceno [CITATION], who, building upon previous work by Lakshminarayan and Meenakshisundaram [CITATION], generalized the Schack-Caves family, and showed that all quantum bakers are perturbations of a common kernel (the "essential" baker).', '1004.1626-2-6-0': 'The present paper focuses on an almost unexplored aspect of the semiclassical theory of the quantum baker map.', '1004.1626-2-6-1': "Gutzwiller's approximate formula for the traces of the baker map is formally equivalent to the partition function of a finite Ising chain (with exponentially decaying interactions and imaginary temperature).", '1004.1626-2-6-2': 'Thus, it can be evaluated with the standard transfer-matrix method.', '1004.1626-2-6-3': 'We have studied extensively the transfer matrices associated with the semiclassical traces of the baker map and the symmetry-reflected baker map.', '1004.1626-2-6-4': 'Our most remarkable finding is the existence of a qubit structure hidden in the semiclassical traces: the transfer matrices admit a quantum-circuit representation that is very similar to that found by Schack-Caves for the quantum baker.', '1004.1626-2-7-0': 'We know of two studies of the baker map which applied the transfer matrix method (to the semiclassical long-time propagation of wave-packets [CITATION], to the analysis of periodic-orbit action correlations [CITATION]).', '1004.1626-2-7-1': 'Different in spirit, the present work aims at analyzing the transfer-matrix method in itself.', '1004.1626-2-8-0': 'The paper is organized as follows.', '1004.1626-2-8-1': 'First we discuss the general structure of the transfer matrices for the baker map, exhibiting their quantum-circuit representation.', '1004.1626-2-8-2': 'Though the transfer matrices are not unitary, nonunitarity is restricted to a one-qubit "gate".', '1004.1626-2-8-3': 'Except for this fact, the transfer matrices resemble qubit quantum bakers (Sec. [REF]).', '1004.1626-2-9-0': 'Section [REF] is devoted to spectral properties.', '1004.1626-2-9-1': 'We show that all eigenvalues lie inside the unit circle in the complex plane, or very close to it.', '1004.1626-2-9-2': 'The number of eigenvalues close to the unit circle coincides approximately with the quantum dimension (for a suitable parameter range).', '1004.1626-2-9-3': 'This situation is very similar to that found in the infinite-dimensional transfer-operator method [CITATION].', '1004.1626-2-9-4': 'Indeed we show that the transfer operator arises as a suitable continuum limit of the transfer matrix.', '1004.1626-2-10-0': 'In Sec. [REF] we exhibit truncation schemes which permit the calculation of long-time traces.', '1004.1626-2-11-0': 'Some aspects of the long-time divergence of the semiclassical traces are assessed in Sec. [REF].', '1004.1626-2-12-0': 'The baker map possesses a spatial symmetry.', '1004.1626-2-12-1': 'If one is interested in separating the spectrum/traces into symmetry classes, then one must consider also the symmetry-related transfer matrices.', '1004.1626-2-12-2': 'Though similar to the matrices of Sec. [REF] in many respects, these matrices happen to be exactly unitary.', '1004.1626-2-12-3': 'They are studied in Sec. [REF].', '1004.1626-2-13-0': 'Concluding remarks are presented in Sec. [REF].', '1004.1626-2-14-0': '# Baker map: Transfer matrix', '1004.1626-2-15-0': 'In order to make the paper self-contained we start by summarizing some basic information about the baker map.', '1004.1626-2-16-0': 'The dynamics of the classical baker map is schematically depicted in Fig. [REF].', '1004.1626-2-17-0': 'It acts on the unit square piecewise-linearly.', '1004.1626-2-17-1': 'Points with [MATH] are governed by the hyperbolic point at the origin.', '1004.1626-2-17-2': 'If [MATH] the fixed point at [MATH] rules.', '1004.1626-2-17-3': 'In any case the coordinates [MATH] relative to the fixed point are compressed/stretched by a factor of two: [EQUATION] where [MATH], the integer part of [MATH].', '1004.1626-2-18-0': 'The quantum analogue of the classical mapping is a unitary operator acting on an even-dimensional Hilbert space [CITATION].', '1004.1626-2-18-1': 'In the [MATH]-representation its matrix reads [CITATION]: [EQUATION] where [MATH] is the [MATH]-dimensional antiperiodic Fourier matrix: [EQUATION] with [MATH].', '1004.1626-2-18-2': 'For this abstract system the Planck constant coincides with the inverse of the dimension, i.e., [MATH].', '1004.1626-2-19-0': 'Our main concern are the traces of the quantum baker, [MATH], for [MATH].', '1004.1626-2-19-1': 'In the semiclassical regime of [MATH] large enough (for a given [MATH]) the traces can be approximated by the Gutzwiller formula [CITATION] [EQUATION] where the sum runs over all periodic trajectories of length [MATH] of the classical map, indexed by the binary column vectors [MATH], with [MATH].', '1004.1626-2-19-2': 'The corresponding actions [MATH] are quadratic functions of the binary symbols, [EQUATION]', '1004.1626-2-19-3': 'The "coupling" matrix [MATH] is suitably expressed in terms of the matrix [MATH] of a cyclic shift, [MATH] [CITATION]: [EQUATION]', '1004.1626-2-19-4': 'The trace formula for the baker map ([REF]) can be derived as follows.', '1004.1626-2-19-5': 'Write [MATH] as a sum of products of Fourier matrix elements using Eq. ([REF]).', '1004.1626-2-19-6': 'Approximate sums by integrals.', '1004.1626-2-19-7': 'Extend the limits of integration (which in principle are finite) to [MATH].', '1004.1626-2-19-8': 'The remaining integrals, being Gaussian, can be done exactly, giving the Gutzwiller sum [CITATION].', '1004.1626-2-20-0': 'Equation ([REF]) is a particular case of the general Gutzwiller trace formula for systems with a chaotic classical limit.', '1004.1626-2-20-1': 'For such systems the trace formula lays a bridge between the quantum spectrum and the set of classical periodic orbits.', '1004.1626-2-20-2': 'It constitutes the core of all semiclassical schemes for relating energy levels to classical information, from the pioneer attempts [CITATION], to the most recent, highly sophisticated developments [CITATION].', '1004.1626-2-21-0': 'The baker is very special in that all its periodic orbits, and properties thereof, are known analytically.', '1004.1626-2-21-1': 'So, in principle, the periodic-orbit sum ([REF]) can be calculated for any [MATH].', '1004.1626-2-21-2': 'However, because of the exponential growth of the number of periodic orbits with period, the brute-force summation is restricted to, say, [MATH].', '1004.1626-2-21-3': 'Remarkably, the method developed by Dittes, Doron and Smilansky [CITATION] does not suffer from such a limitation.', '1004.1626-2-21-4': 'These authors defined an infinite-dimensional integral operator, [EQUATION] [MATH], whose traces give exactly the periodic-orbit sum of Eq. ([REF]).', '1004.1626-2-21-5': 'They showed that [MATH] can be efficiently calculated from the eigenvalues of the matrix of [MATH] in the Fourier basis, after suitable truncation [CITATION].', '1004.1626-2-22-0': 'An alternative to the infinite-dimensional operator method relies on the formal identification of the periodic-orbit sum ([REF]) with the partition function of an Ising chain, for a purely imaginary temperature.', '1004.1626-2-22-1': 'The cyclical nature of the coupling matrix [MATH] says that this is a circular chain, consisting of [MATH] spins.', '1004.1626-2-22-2': 'All spins interact among themselves, but, as interactions decay exponentially with distance, one may expect some computational benefit (without much loss of accuracy) by truncating the interaction to some given number [MATH] of closest neighbors, i.e., [EQUATION]', '1004.1626-2-22-3': 'The exact periodic-orbit sum corresponds to setting [MATH] (no truncation at all), but, in principle, one can consider any truncation, even to first neighbors ([MATH]).', '1004.1626-2-23-0': 'Once one has established the equivalence between the baker periodic-orbit sum and the Ising partition function, the transfer-matrix method can be invoked.', '1004.1626-2-23-1': 'The first-neighbor case is explained in textbooks [CITATION].', '1004.1626-2-23-2': 'It seems that the many-neighbor case has not been explicitly worked out in the literature, but some discussions exist [CITATION].', '1004.1626-2-23-3': 'Anyway, even if not trivial, the generalization can be carried out following the spirit of the one-neighbor case and, in the case of the baker-Ising, leads to [EQUATION] the explicit expression for [MATH] being [CITATION]: [EQUATION]', '1004.1626-2-23-4': 'The elements [MATH] are given by [EQUATION] with [MATH], and [EQUATION]', '1004.1626-2-23-5': 'Note that [MATH] is a complex matrix depending on the three parameters [MATH].', '1004.1626-2-23-6': 'Its dimension is [MATH], with [MATH].', '1004.1626-2-23-7': 'When [MATH] one recovers the exact semiclassical traces, i.e., Eq. ([REF]) becomes an equality.', '1004.1626-2-24-0': 'A prefactor [MATH] has been absorbed into the definition of [MATH] because in this way most of the spectrum of [MATH] lies close to the unit circle or inside it (see Sec. [REF] below).', '1004.1626-2-24-1': 'So, the transfer matrix [MATH] becomes qualitatively similar to the semiclassical transfer operator of Refs. [CITATION] (we come back to this point later).', '1004.1626-2-25-0': 'The transfer matrix approach transforms the calculation of the Gutzwiller sum into the problem of obtaining the trace of the [MATH]-th power of the finite matrix [MATH].', '1004.1626-2-25-1': 'From a numerical point of view this is advantageous only if truncation of [MATH] is admissible.', '1004.1626-2-25-2': 'We defer the analysis of this question until Sec. [REF].', '1004.1626-2-25-3': 'Now we concentrate on the transfer matrices themselves which, as we shall show, possess very interesting properties.', '1004.1626-2-26-0': 'The transfer matrix ([REF]) exhibits the structure of a tensor product which is conveniently displayed by switching to a qubit representation.', '1004.1626-2-26-1': 'This consists in identifying the "spatial" degree of freedom [MATH] with the tensor product of [MATH] two-level systems: [EQUATION]', '1004.1626-2-26-2': 'Here [MATH] indexes the states of the [MATH]-basis and [MATH] is a label for the qubit basis states; they are related through the binary expansion [EQUATION]', '1004.1626-2-26-3': 'In the qubit picture the matrix [MATH] can be broken up into the elementary gates shown in the quantum circuit of Fig. [REF].', '1004.1626-2-27-0': 'The circuit acts on [MATH] qubits, starting with a downwards cyclic shift of all qubits, i.e., [EQUATION]', '1004.1626-2-27-1': 'A nonunitary gate acting on the first qubit follows, which bears some resemblance with a Hadamard gate, [EQUATION] with [EQUATION]', '1004.1626-2-27-2': 'It must be noted that, as the dimension [MATH] is an even number, we always have [MATH], and [MATH] is never unitary.', '1004.1626-2-27-3': 'After the [MATH] gate one finds a single qubit phase gate, [EQUATION]', '1004.1626-2-27-4': 'Finally one has a sequence of two-qubit phase gates, acting symmetrically between the first and the [MATH]-th qubit, [EQUATION] with exponentially decreasing phases, [MATH].', '1004.1626-2-28-0': 'Writing [MATH] as a circuit has helped us in identifying its basic constituents.', '1004.1626-2-28-1': 'In particular we recognize the one-qubit gate [MATH] as responsible for the deviation from unitarity.', '1004.1626-2-28-2': 'If we substitute [MATH] by a Hadamard gate, i.e., set [MATH] in ([REF]), then the circuit of Fig. [REF] acquires the typical structure of the baker family [CITATION]: an "essential-baker" block (formed by a shift [MATH] and a single-qubit Fourier transform) followed by a "diffraction kernel" (phase gates).', '1004.1626-2-29-0': '# Spectral Properties', '1004.1626-2-30-0': 'Figure [REF] displays a typical transfer-matrix spectrum in the case that the dimension of the transfer matrix is much larger than the quantum dimension, i.e., [MATH].', '1004.1626-2-31-0': 'The spectrum can be coarsely divided into three parts.', '1004.1626-2-31-1': 'Approximately [MATH] eigenvalues are localized close to the unit circle.', '1004.1626-2-31-2': 'Most of the remaining ones are concentrated in a disk of smaller radius, and there are some transitional eigenvalues spiraling out from the inner disk to the unit circle.', '1004.1626-2-31-3': 'The eigenvalues lying close to the unit circle can be put in almost one-to-one correspondence with the exact eigenvalues of the quantum baker ([REF]).', '1004.1626-2-32-0': 'Increasing [MATH] while keeping [MATH] fixed does not significantly change the almost unitary part of spectrum, but populates the region of small moduli (see Fig. [REF]).', '1004.1626-2-33-0': "This fact, when combined with the existence of eigenvalues with moduli larger than one, leads to the conclusion that Gutzwiller's traces for the baker are divergent in the limit [MATH] (fixed [MATH]).", '1004.1626-2-33-1': 'This result has been confirmed previously using the transfer-operator method [CITATION].', '1004.1626-2-34-0': 'Indeed, the overall features of the transfer-matrix spectrum described above can also be found in the transfer-operator spectra discussed in Refs. [CITATION].', '1004.1626-2-34-1': 'Looking for an explanation for this similarity, we recall that the transfer operator is infinite-dimensional and independent of [MATH].', '1004.1626-2-34-2': 'Thus, one may be tempted to compare Eq. ([REF]) with the infinite-[MATH] limit of the transfer matrix ([REF]).', '1004.1626-2-34-3': 'In this limit, the transfer matrix becomes an integral kernel, indices becoming continuous variables: [EQUATION]', '1004.1626-2-34-4': 'A careful calculation verifies that the transfer matrix tends to the transfer operator, i.e., [EQUATION] with [MATH] precisely that given in Eq. ([REF])!', '1004.1626-2-34-5': 'So, the transfer operator is formally recovered as the continuum limit of the transfer matrix.', '1004.1626-2-35-0': '# Truncation Schemes', '1004.1626-2-36-0': 'Here we discuss the numerical calculation of long-time Gutzwiller traces using the transfer-matrix formula ([REF]).', '1004.1626-2-36-1': 'The most natural procedure for calculating [MATH] requires the diagonalization of [MATH] to obtain its eigenvalues.', '1004.1626-2-36-2': 'This scheme is limited to very small values of [MATH], e.g., to [MATH] in a personal computer.', '1004.1626-2-36-3': 'So, truncation becomes essential.', '1004.1626-2-37-0': 'The error introduced in truncating the interactions to [MATH] neighbors can be roughly estimated from Eq. ([REF]) and the basic definitions ([REF],[REF],[REF]).', '1004.1626-2-37-1': 'Such a truncation produces an error in the actions ([REF]) approximately given by [EQUATION]', '1004.1626-2-37-2': 'For typical vectors [MATH], containing randomly distributed 0/1 bits, we have [EQUATION]', '1004.1626-2-37-3': 'This implies that the corresponding error in the Gutzwiller phases amounts to [EQUATION]', '1004.1626-2-37-4': 'Finally, if we assume that this is the error committed in most phases in the Gutzwiller sum ([REF]), then [MATH] amounts to the relative error in the semiclassical traces, for [EQUATION]', '1004.1626-2-37-5': 'Thus, we arrive at the following condition for the validity of the truncation to [MATH] neighbors: [EQUATION]', '1004.1626-2-37-6': 'However, even after truncating the transfer matrix, we may need to use values of [MATH] which make diagonalization prohibitive, e.g., for [MATH], [MATH].', '1004.1626-2-37-7': 'Remarkably there is an alternative to diagonalization which arises from the sparsity of [MATH].', '1004.1626-2-37-8': 'For, even if the special type of sparsity [MATH] exhibits is not helpful in speeding-up its diagonalization, it permits us to implement an alternative, much more efficient method for calculating the traces of [MATH].', '1004.1626-2-38-0': 'Consider the following identity: [EQUATION] where overline means average over random complex vectors [MATH] uniformly distributed over the corresponding unit sphere [CITATION].', '1004.1626-2-38-1': '(The notation [MATH] has been used to indicate the truncation of [MATH] to [MATH] qubits.)', '1004.1626-2-38-2': 'The idea behind ([REF]) is to calculate [MATH] by applying iteratively the matrix [MATH] to [MATH], and then averaging over [MATH].', '1004.1626-2-38-3': 'This method allows one, in principle, to consider [MATH] as large as 20.', '1004.1626-2-39-0': 'Table [REF] displays some examples.', '1004.1626-2-40-0': 'The first rows (a-e) illustrate the improvement of the results as the truncation level [MATH] is increased, for fixed [MATH].', '1004.1626-2-40-1': 'Agreement with the exact results is met, within the specified statistical errors, for [MATH], corresponding to [MATH].', '1004.1626-2-40-2': 'In rows (f-i) we consider [MATH] fixed and large values of [MATH] which can in no way be reached by direct computation of the periodic orbit summation, so that exact results are not known for such [MATH].', '1004.1626-2-40-3': 'In these cases we compare with the approximate results obtained using the transfer-operator method [CITATION].', '1004.1626-2-41-0': 'We see good agreement for [MATH], i.e., we find the condition [MATH] again.', '1004.1626-2-42-0': 'So, we have checked that the transfer-matrix method works satisfactorily in the parameter regime specified by ([REF]).', '1004.1626-2-43-0': '# Divergence of long-time traces', '1004.1626-2-44-0': 'For chaotic systems, the Gutzwiller trace formula is only the first term of an expansion in powers of [MATH].', '1004.1626-2-44-1': 'Because of this, the semiclassical eigenvalues obtained from the Gutwiller traces deviate in general from the unit circle in the complex plane.', '1004.1626-2-44-2': 'For typical chaotic systems it is expected that the distance of the largest semiclassical eigenvalue to the unit circle scales like [MATH], i.e., [MATH] for quantum maps [CITATION].', '1004.1626-2-44-3': 'In the case of the baker map, an anomalous behavior [MATH] is observed which is due to diffraction effects originating in the discontinuities of the mapping [CITATION].', '1004.1626-2-44-4': "In any case, this lack of unitarity makes Gutzwiller's traces exponentially growing with [MATH], a behavior that may be guessed from Table [REF].", '1004.1626-2-45-0': 'The following question arises naturally: can the [MATH] law be associated with the particular structure of the baker transfer matrix?', '1004.1626-2-45-1': 'In the circuit representation (Fig. [REF]) nonunitarity arises from the single-qubit gate [MATH].', '1004.1626-2-45-2': 'So, one may ask: what is typically the largest eigenvalue (in modulus) of a map obtained by tensoring [MATH] with a generic [MATH]-dimensional unitary gate [MATH]?', '1004.1626-2-46-0': 'On the other side, the [MATH]-representation ([REF]) offers a different point of view of the nonunitarity: the transfer matrix can be split as [MATH], with [MATH] unitary matrices of dimension [MATH] [see Eq. ([REF])].', '1004.1626-2-46-1': 'How does the leading eigenvalue of such a matrix [MATH] scale with [MATH]?', '1004.1626-2-47-0': 'In order to answer the questions above we resorted to numerical calculations.', '1004.1626-2-47-1': 'First we analyzed the matrices [MATH] with [MATH] a random matrix belonging to the Circular Unitary Ensemble (CUE).', '1004.1626-2-47-2': 'The matrix [MATH] acts on the least significant qubit and is defined by Eq. ([REF]) with [MATH], i.e., [MATH] is chosen to be the large-[MATH] limit of [MATH].', '1004.1626-2-47-3': 'Then we considered the ensemble [MATH] with [MATH] independent random matrices belonging to CUE [CITATION].', '1004.1626-2-47-4': 'In order to check any possible ensemble dependence we also calculated the case [MATH] with [MATH] in the Circular Orthogonal Ensemble (COE).', '1004.1626-2-48-0': 'Our results are exhibited in Fig. [REF].', '1004.1626-2-49-0': 'In the three considered cases we observe very similar decay laws.', '1004.1626-2-49-1': 'These decays are much closer to the diffractive scaling [MATH] than to the universal semiclassical behavior [MATH], meaning that the random-matrix modeling has partially captured the essence of the semiclassical baker.', '1004.1626-2-49-2': 'However, there is still some noticeable departure from the [MATH] decay (rather, our numerical results seem to follow a [MATH] law).', '1004.1626-2-49-3': 'Thus, we conclude that the minimum-information models we have constructed still have to be complemented with some ingredient to properly describe the [MATH] baker scaling.', '1004.1626-2-49-4': 'Further research is necessary to discover what such an additional information should be (e.g., some particular correlation between [MATH] and [MATH]?)', '1004.1626-2-50-0': '# Reflected baker map', '1004.1626-2-51-0': 'The quantum baker map [MATH] is invariant under the parity symmetry [MATH], its action on the [MATH]-basis being just [MATH].', '1004.1626-2-51-1': 'This is the quantum counterpart of the classical reflection symmetry [MATH].', '1004.1626-2-51-2': 'When analyzing spectral properties it is convenient to separate eigenstates and eigenvalues of [MATH] according to their parity.', '1004.1626-2-51-3': 'Thus, one is led to consider the parity-projected bakers [MATH].', '1004.1626-2-51-4': 'In the semiclassical domain the traces of [MATH] can be approximated by [CITATION] [EQUATION]', '1004.1626-2-51-5': 'The first summation above correspond to the baker traces ([REF]).', '1004.1626-2-51-6': 'In the second sum [MATH] stands for half the action [Eq. ([REF])] of a periodic orbit of length [MATH] with symbolic code [MATH], where [MATH].', '1004.1626-2-51-7': 'These are precisely the orbits invariant under the parity transformation.', '1004.1626-2-52-0': 'We shall exhibit the transfer matrix associated to the reflected traces, pointing out the most important features.', '1004.1626-2-52-1': 'Following the same procedure as in Sec. [REF] we constructed a transfer matrix [MATH] such that [EQUATION]', '1004.1626-2-52-2': 'After eliminating the null subspace that appears as a consequence of considering just the parity-invariant trajectories of length [MATH], the matrix [MATH] is reduced to dimension [MATH]: [EQUATION] where the matrix elements [MATH] are given by [EQUATION] with [EQUATION]', '1004.1626-2-52-3': 'The last undefined ingredient is the integer [MATH].', '1004.1626-2-52-4': 'It is best expressed in terms of the binary digits of the integer [MATH] [see Eq. ([REF])].', '1004.1626-2-52-5': 'If [MATH], then [EQUATION]', '1004.1626-2-52-6': "The main difference between the baker's transfer matrix ([REF]) and the matrix above, is that [MATH] is exactly unitary (thus, its spectrum lies on the unit circle).", '1004.1626-2-52-7': 'Except for this fact, both [MATH] and [MATH] are structurally very similar.', '1004.1626-2-52-8': 'Perhaps this can be better appreciated by looking at the quantum circuit for [MATH] in Fig. [REF].', '1004.1626-2-53-0': 'The circuit acts on [MATH] qubits (without truncation).', '1004.1626-2-53-1': "It starts like the baker's, with a downwards shift of all qubits.", '1004.1626-2-53-2': 'Then we have a NOT gate acting on the first qubit given by [EQUATION]', '1004.1626-2-53-3': 'A phase gate [MATH] follows that acts on the first qubit, with [EQUATION]', '1004.1626-2-53-4': 'Finally there is a sequence of [MATH] symmetrical two-qubit phase gates [MATH] acting between the first and the [MATH]-th qubit with exponentially decreasing phases, [MATH], where [MATH].', '1004.1626-2-54-0': 'A complete study of the spectral properties of the reflected-baker transfer matrix, truncation schemes, etc, is deferred to a future publication because, as [MATH] is unitary, such analyses would take us in directions very different than those followed in the case of the baker map.', '1004.1626-2-54-1': 'However, we would like to anticipate one result which gives a hint about the nature of the spectrum of [MATH].', '1004.1626-2-55-0': 'Consider the [MATH]-th power of the transfer matrix [MATH], i.e., the [MATH]-th iteration of the circuit above.', '1004.1626-2-55-1': 'We verified that the shifts and NOT gates cancel out and only phase gates remain.', '1004.1626-2-55-2': 'This means that [MATH] is a diagonal matrix (given that the phase gates are diagonal).', '1004.1626-2-55-3': 'It turns out that the diagonal elements are exactly the Gutwiller phases [MATH] in Eq. ([REF]) (we tested this numerically for some cases).', '1004.1626-2-55-4': 'Thus, the spectrum of [MATH] is formed by roots of the Gutzwiller phases!', '1004.1626-2-56-0': '# Conclusions', '1004.1626-2-57-0': 'We presented a study of the transfer matrix approach to the semiclassical traces of the baker map and its reflected version.', '1004.1626-2-57-1': 'We found that the transfer matrices admit a tensor product decomposition leading to simple circuit representations, similar to those obtained by Schack and Caves for the quantum baker.', '1004.1626-2-57-2': 'Remarkably, in the case of the baker, the corresponding circuit clearly isolates the source of nonunitarity of the semiclassical traces in the form of a one-qubit Hadamard-like gate.', '1004.1626-2-57-3': 'Given that both exact and semiclassical bakers can now be written as circuits, this representation appears as a promising tool for studying the corrections to the Gutzwiller trace formula.', '1004.1626-2-58-0': 'In the case of the baker, spectral properties were analyzed, showing that there exists a close similitude with the transfer operator of Ref. [CITATION].', '1004.1626-2-58-1': 'In fact, we proved that the transfer operator arises as a suitable asymptotic limit of the transfer matrix formalism.', '1004.1626-2-58-2': 'Truncation schemes were discussed which permit the numerical calculation of long-time traces well inside a domain where direct summation of the Gutzwiller formula is impossible.', '1004.1626-2-59-0': 'We would like to conclude by mentioning a very exciting (though rather speculative) possibility.', '1004.1626-2-59-1': 'It is tempting to think of the transfer matrices as maps for which the Gutzwiller formula is exact.', '1004.1626-2-59-2': 'Admittedly this point of view has some limitations because there is a map for each value of [MATH].', '1004.1626-2-59-3': 'Even so, our results suggest that we may be not far from finding a close relative to the baker map having an exact periodic-orbit trace formula.', '1004.1626-2-59-4': 'The search for such a map constitutes a very attractive and challenging project which, however, exceeds the scope of the present paper.'}

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'1004.1626-2-18-1'], ['1004.1626-1-15-2', '1004.1626-2-18-2'], ['1004.1626-1-19-0', '1004.1626-2-22-0'], ['1004.1626-1-19-1', '1004.1626-2-22-1'], ['1004.1626-1-19-2', '1004.1626-2-22-2'], ['1004.1626-1-19-3', '1004.1626-2-22-3'], ['1004.1626-1-6-4', '1004.1626-2-8-3'], ['1004.1626-1-23-1', '1004.1626-2-26-1'], ['1004.1626-1-23-2', '1004.1626-2-26-2'], ['1004.1626-1-23-3', '1004.1626-2-26-3'], ['1004.1626-1-7-0', '1004.1626-2-9-0'], ['1004.1626-1-7-1', '1004.1626-2-9-1'], ['1004.1626-1-7-2', '1004.1626-2-9-2'], ['1004.1626-1-7-3', '1004.1626-2-9-3'], ['1004.1626-1-29-0', '1004.1626-2-32-0'], ['1004.1626-1-14-0', '1004.1626-2-17-0'], ['1004.1626-1-14-1', '1004.1626-2-17-1'], ['1004.1626-1-14-2', '1004.1626-2-17-2'], ['1004.1626-1-14-3', '1004.1626-2-17-3'], ['1004.1626-1-13-0', '1004.1626-2-16-0'], ['1004.1626-1-22-1', '1004.1626-2-25-1'], ['1004.1626-1-22-2', '1004.1626-2-25-2'], ['1004.1626-1-12-0', '1004.1626-2-15-0'], ['1004.1626-1-18-0', '1004.1626-2-21-0'], ['1004.1626-1-18-1', '1004.1626-2-21-1'], ['1004.1626-1-18-2', '1004.1626-2-21-2'], ['1004.1626-1-18-3', '1004.1626-2-21-3'], ['1004.1626-1-18-4', '1004.1626-2-21-4'], ['1004.1626-1-18-5', '1004.1626-2-21-5'], ['1004.1626-1-40-0', '1004.1626-2-47-0'], ['1004.1626-1-40-1', '1004.1626-2-47-1'], ['1004.1626-1-40-2', '1004.1626-2-47-2'], ['1004.1626-1-40-3', '1004.1626-2-47-3'], ['1004.1626-1-40-4', '1004.1626-2-47-4'], ['1004.1626-1-30-0', '1004.1626-2-33-0'], ['1004.1626-1-16-0', '1004.1626-2-19-0'], ['1004.1626-1-16-1', '1004.1626-2-19-1'], ['1004.1626-1-16-2', '1004.1626-2-19-2'], ['1004.1626-1-16-3', '1004.1626-2-19-3'], ['1004.1626-1-16-4', '1004.1626-2-19-4'], ['1004.1626-1-16-5', '1004.1626-2-19-5'], ['1004.1626-1-16-6', '1004.1626-2-19-6'], ['1004.1626-1-16-7', '1004.1626-2-19-7'], ['1004.1626-1-16-8', '1004.1626-2-19-8'], ['1004.1626-1-46-1', '1004.1626-2-53-1'], ['1004.1626-1-46-2', '1004.1626-2-53-2'], ['1004.1626-1-46-3', '1004.1626-2-53-3'], ['1004.1626-1-46-4', '1004.1626-2-53-4'], ['1004.1626-1-36-0', '1004.1626-2-42-0'], ['1004.1626-1-17-0', '1004.1626-2-20-0'], ['1004.1626-1-17-1', '1004.1626-2-20-1'], ['1004.1626-1-17-2', '1004.1626-2-20-2'], ['1004.1626-1-28-0', '1004.1626-2-31-0'], ['1004.1626-1-28-1', '1004.1626-2-31-1'], ['1004.1626-1-28-2', '1004.1626-2-31-2'], ['1004.1626-1-28-3', '1004.1626-2-31-3'], ['1004.1626-1-9-0', '1004.1626-2-12-0'], ['1004.1626-1-9-1', '1004.1626-2-12-1'], ['1004.1626-1-9-3', '1004.1626-2-12-3'], ['1004.1626-1-31-0', '1004.1626-2-34-0'], ['1004.1626-1-31-1', '1004.1626-2-34-1'], ['1004.1626-1-31-2', '1004.1626-2-34-2'], ['1004.1626-1-31-3', '1004.1626-2-34-3'], ['1004.1626-1-31-5', '1004.1626-2-34-5'], ['1004.1626-1-0-0', '1004.1626-2-0-0'], ['1004.1626-1-0-2', '1004.1626-2-0-2'], ['1004.1626-1-38-0', '1004.1626-2-44-0'], ['1004.1626-1-38-1', '1004.1626-2-44-1'], ['1004.1626-1-38-3', '1004.1626-2-44-3'], ['1004.1626-1-38-4', '1004.1626-2-44-4'], ['1004.1626-1-25-0', '1004.1626-2-28-0'], ['1004.1626-1-25-1', '1004.1626-2-28-1'], ['1004.1626-1-25-2', '1004.1626-2-28-2'], ['1004.1626-1-5-0', '1004.1626-2-6-0'], ['1004.1626-1-5-1', '1004.1626-2-6-1'], ['1004.1626-1-5-2', '1004.1626-2-6-2'], ['1004.1626-1-47-2', '1004.1626-2-55-0'], ['1004.1626-1-27-0', '1004.1626-2-30-0'], ['1004.1626-1-3-0', '1004.1626-2-3-0'], ['1004.1626-1-3-1', '1004.1626-2-3-1'], ['1004.1626-1-3-2', '1004.1626-2-3-2'], ['1004.1626-1-3-4', '1004.1626-2-4-2'], ['1004.1626-1-3-5', '1004.1626-2-4-3'], ['1004.1626-1-3-6', '1004.1626-2-4-4'], ['1004.1626-1-35-0', '1004.1626-2-40-0'], ['1004.1626-1-33-0', '1004.1626-2-36-0'], ['1004.1626-1-33-1', '1004.1626-2-36-1'], ['1004.1626-1-33-2', '1004.1626-2-36-2'], ['1004.1626-1-33-3', '1004.1626-2-36-3'], ['1004.1626-1-33-6', '1004.1626-2-37-6'], ['1004.1626-1-33-8', '1004.1626-2-38-0'], ['1004.1626-1-33-9', '1004.1626-2-38-1'], ['1004.1626-1-33-10', '1004.1626-2-38-2'], ['1004.1626-1-33-11', '1004.1626-2-38-3'], ['1004.1626-1-39-1', '1004.1626-2-45-1'], ['1004.1626-1-8-1', '1004.1626-2-11-0'], ['1004.1626-1-45-1', '1004.1626-2-52-2'], ['1004.1626-1-45-2', '1004.1626-2-52-3'], ['1004.1626-1-45-3', '1004.1626-2-52-4'], ['1004.1626-1-45-6', '1004.1626-2-52-7'], ['1004.1626-1-45-7', '1004.1626-2-52-8'], ['1004.1626-1-51-1', '1004.1626-2-59-1'], ['1004.1626-1-2-0', '1004.1626-2-2-0'], ['1004.1626-1-2-1', '1004.1626-2-2-1'], ['1004.1626-1-20-3', '1004.1626-2-23-3'], ['1004.1626-1-24-0', '1004.1626-2-27-0'], ['1004.1626-1-23-0', '1004.1626-2-26-0'], ['1004.1626-1-22-0', '1004.1626-2-25-0'], ['1004.1626-1-30-1', '1004.1626-2-33-1'], ['1004.1626-1-9-2', '1004.1626-2-12-2'], ['1004.1626-1-31-4', '1004.1626-2-34-4'], ['1004.1626-1-38-2', '1004.1626-2-44-2'], ['1004.1626-1-4-0', '1004.1626-2-5-0'], ['1004.1626-1-47-4', '1004.1626-2-55-2'], ['1004.1626-1-47-5', '1004.1626-2-55-3'], ['1004.1626-1-47-6', '1004.1626-2-55-4'], ['1004.1626-1-35-2', '1004.1626-2-40-2'], ['1004.1626-1-39-2', '1004.1626-2-45-2'], ['1004.1626-1-39-3', '1004.1626-2-46-0'], ['1004.1626-1-44-0', '1004.1626-2-52-0'], ['1004.1626-1-45-0', '1004.1626-2-52-1'], ['1004.1626-1-51-2', '1004.1626-2-59-2'], ['1004.1626-1-52-1', '1004.1626-2-59-3'], ['1004.1626-1-49-1', '1004.1626-2-58-0'], ['1004.1626-1-49-2', '1004.1626-2-58-1'], ['1004.1626-1-50-0', '1004.1626-2-58-2'], ['1004.1626-1-41-0', '1004.1626-2-49-0'], ['1004.1626-1-41-1', '1004.1626-2-49-3'], ['1004.1626-1-6-1', '1004.1626-2-8-1'], ['1004.1626-1-6-3', '1004.1626-2-8-2'], ['1004.1626-1-22-3', '1004.1626-2-25-3'], ['1004.1626-1-46-0', '1004.1626-2-53-0'], ['1004.1626-1-0-1', '1004.1626-2-0-1'], ['1004.1626-1-0-6', '1004.1626-2-0-6'], ['1004.1626-1-47-3', '1004.1626-2-55-1'], ['1004.1626-1-3-3', '1004.1626-2-4-0'], ['1004.1626-1-35-3', '1004.1626-2-40-3'], ['1004.1626-1-35-4', '1004.1626-2-41-0'], ['1004.1626-1-33-4', '1004.1626-2-37-0'], ['1004.1626-1-33-5', '1004.1626-2-37-5'], ['1004.1626-1-39-0', '1004.1626-2-45-0'], ['1004.1626-1-45-5', '1004.1626-2-52-6'], ['1004.1626-1-49-0', '1004.1626-2-57-0']]

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'1004.1626-2-18-1'], ['1004.1626-1-15-2', '1004.1626-2-18-2'], ['1004.1626-1-19-0', '1004.1626-2-22-0'], ['1004.1626-1-19-1', '1004.1626-2-22-1'], ['1004.1626-1-19-2', '1004.1626-2-22-2'], ['1004.1626-1-19-3', '1004.1626-2-22-3'], ['1004.1626-1-6-4', '1004.1626-2-8-3'], ['1004.1626-1-23-1', '1004.1626-2-26-1'], ['1004.1626-1-23-2', '1004.1626-2-26-2'], ['1004.1626-1-23-3', '1004.1626-2-26-3'], ['1004.1626-1-7-0', '1004.1626-2-9-0'], ['1004.1626-1-7-1', '1004.1626-2-9-1'], ['1004.1626-1-7-2', '1004.1626-2-9-2'], ['1004.1626-1-7-3', '1004.1626-2-9-3'], ['1004.1626-1-29-0', '1004.1626-2-32-0'], ['1004.1626-1-14-0', '1004.1626-2-17-0'], ['1004.1626-1-14-1', '1004.1626-2-17-1'], ['1004.1626-1-14-2', '1004.1626-2-17-2'], ['1004.1626-1-14-3', '1004.1626-2-17-3'], ['1004.1626-1-13-0', '1004.1626-2-16-0'], ['1004.1626-1-22-1', '1004.1626-2-25-1'], ['1004.1626-1-22-2', '1004.1626-2-25-2'], ['1004.1626-1-12-0', '1004.1626-2-15-0'], ['1004.1626-1-18-0', '1004.1626-2-21-0'], ['1004.1626-1-18-1', 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[]

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[]

['1004.1626-1-10-0', '1004.1626-1-34-0', '1004.1626-1-45-4', '1004.1626-2-13-0', '1004.1626-2-39-0', '1004.1626-2-48-0', '1004.1626-2-52-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1004.1626

cond-mat-0602062

{'cond-mat-0602062-1-0-0': 'We apply the truncated Wigner method to the process of three-body recombination in ultracold Bose gases.', 'cond-mat-0602062-1-0-1': 'We find that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be treated using a set of coupled stochastic differential equations that include diffusion terms, and can be simulated using known numerical methods.', 'cond-mat-0602062-1-0-2': 'As an example we investigate the behaviour of a simple homogeneous Bose gas.', 'cond-mat-0602062-1-1-0': '# Introduction', 'cond-mat-0602062-1-2-0': 'The dominant loss process affecting ultracold gaseous alkali metal systems is inelastic three-body recombination [CITATION], a process characterised by collisional events involving three atoms leading to the creation of a single two-atom molecule (a dimer).', 'cond-mat-0602062-1-2-1': 'The binding energy released by the molecule formation is retained by the particles as kinetic energy.', 'cond-mat-0602062-1-2-2': 'Typically this results in the loss of all three atoms from the system as the molecule is not trapped by any applied external potential and the energy of the free atom is high enough to overcome any confinement barrier.', 'cond-mat-0602062-1-2-3': 'Indeed it is this process that limits the lifetime of experimentally produced alkali metal Bose-Einstein condensates, due to the large increase in density once the temperature is lowered past the critical point [CITATION].', 'cond-mat-0602062-1-3-0': 'In a previous paper [CITATION] we presented a comprehensive treatment of the truncated Wigner approach for ultracold Bose gases including elastic two-body interactions.', 'cond-mat-0602062-1-3-1': 'In this paper we extend that treatment to include three-body recombination events, which modifies the ensemble differential equations describing the evolution of a single realisation of the field.', 'cond-mat-0602062-1-3-2': 'These modified differential equations are explicitly stochastic, including dynamic noise sources arising from the action of three-body recombination on the virtual particle background field.', 'cond-mat-0602062-1-4-0': 'To provide a demonstration of our extended formalism we examine the evolution of a simple homogeneous system, starting from a zero-temperature state where the particle population is initially confined to a single (condensate) mode.', 'cond-mat-0602062-1-5-0': '## Three-body recombination in ultracold gases', 'cond-mat-0602062-1-6-0': 'Assuming that three-body recombination is the only particle loss mechanism affecting the system, it can be shown that the rate of change of total particle number is [CITATION] [EQUATION] where [MATH] is the total number density of atoms and [MATH] is the three-body recombination event rate constant.', 'cond-mat-0602062-1-6-1': 'We have assumed that all the particles involved in the recombination process are lost from the system, hence the prefactor of 3 in Eq. ([REF]), which describes the number of particles lost from the system.', 'cond-mat-0602062-1-6-2': 'The third-order normalised equiposition correlation function [MATH] measures the statistics of the field, being unity for a fully coherent system, i.e. a zero-temperature condensate, and [MATH] for a purely thermal system.', 'cond-mat-0602062-1-6-3': 'The factor of 6 increase in the loss rate of thermal over coherent systems for similar densities has been observed experimentally [CITATION].', 'cond-mat-0602062-1-7-0': '# Truncated Wigner treatment', 'cond-mat-0602062-1-8-0': '## The restricted field', 'cond-mat-0602062-1-9-0': 'As in our previous work [CITATION], we describe the many-body system of identical bosons using the Schrodinger picture bosonic field operator [EQUATION]', 'cond-mat-0602062-1-9-1': 'Here the mode operators [MATH] annihilate a single boson from the [MATH]th mode, and obey the commutation relations [EQUATION] while the coordinate space functions [MATH] form an infinite orthonormal basis set where [EQUATION] where [MATH] is the applied external potential.', 'cond-mat-0602062-1-10-0': 'We now divide mode space into two subspaces, a low-energy space ([MATH]) consisting of all those modes whose eigenenergies are less than the boundary energy [MATH], and a high-energy space ([MATH]) that includes all remaining modes.', 'cond-mat-0602062-1-10-1': 'For this work our interest lies with the dynamics of the low-energy subspace.', 'cond-mat-0602062-1-10-2': 'Using these subspaces we define the field operators [EQUATION]', 'cond-mat-0602062-1-10-3': 'Of most importance for this paper is the low-energy restricted basis field operator [MATH], which can be obtained from the total field operator [EQUATION] using the projector [EQUATION] as [MATH].', 'cond-mat-0602062-1-10-4': 'The restricted basis field operator obeys the commutation relations [EQUATION] where the restricted delta function is defined by [EQUATION]', 'cond-mat-0602062-1-10-5': 'The conjugate projector [MATH] can be obtained using the complementarity relation [MATH].', 'cond-mat-0602062-1-11-0': '## Master equation', 'cond-mat-0602062-1-12-0': 'Our previous paper [CITATION] assumed that only two atoms participate in any single scattering event.', 'cond-mat-0602062-1-12-1': 'In this way the particle interactions are described using a simple [MATH]-wave contact potential as an approximation to the full two-body T-matrix.', 'cond-mat-0602062-1-12-2': 'Obviously such a description does not include three-body scattering events.', 'cond-mat-0602062-1-12-3': 'Full theoretical treatments of three-body scattering including all possible collisional channels are extremely complicated, and we do not attempt such an approach here.', 'cond-mat-0602062-1-12-4': 'Instead we adopt a quantum-optical approach, starting from a phenomenologically appropriate Hamiltonian including inelastic three-body recombination events, to which we apply the truncated Wigner method.', 'cond-mat-0602062-1-13-0': 'We assume that within the dilute limit the characteristic range of the three-body recombination potential [MATH] is much smaller than the average interparticle spacing.', 'cond-mat-0602062-1-13-1': 'Thus, following thematically the approach for pairwise scattering, we replace this scattering potential by an effective zero-range three-body scattering T-operator, whose interaction strength is essentially a free parameter that will be chosen to satisfy experimentally observed loss rates.', 'cond-mat-0602062-1-13-2': 'Within this approach then, in order to include the effects of three-body recombination the Schrodinger picture effective Hamiltonian is modified to include the term [CITATION] [EQUATION] where the molecule field operator [MATH] annihilates a dimer from the field and [MATH] is a measure of the energy associated with the three-body process.', 'cond-mat-0602062-1-13-3': 'We have assumed in formulating Eq. ([REF]) that the binding energy associated with the molecule formation is large enough that the unpaired atom generated by a recombination event is created within the high-energy subspace [MATH], rather than the low-energy (system) subspace [MATH], and is described by the high-energy field operator [MATH].', 'cond-mat-0602062-1-14-0': 'Jack [CITATION] has considered this partial Hamiltonian, and has shown that by eliminating both the molecular and high-energy atomic fields from the evolution using a standard interaction picture approach for initially uncoupled fields [CITATION], one obtains the master equation term [EQUATION] for the low-energy atomic subspace (system) density operator [MATH].', 'cond-mat-0602062-1-14-1': 'The quantity [MATH] governs the rate of recombination events, and its relationship to [MATH] we consider later.', 'cond-mat-0602062-1-14-2': 'In arriving at this master equation term it has been assumed that the output products of the recombination events immediately exit the coordinate space region containing the system, such that they play no further role in the evolution.', 'cond-mat-0602062-1-14-3': 'To describe the full master equation for the system density operator one combines Eq. ([REF]) with the von Neumann equation calculated in [CITATION].', 'cond-mat-0602062-1-15-0': '## Functional Wigner function correspondences', 'cond-mat-0602062-1-16-0': 'The master equation term given by Eq. ([REF]) can be used to calculate the evolution of the corresponding multimode Wigner function [MATH] using appropriate operator correspondences [CITATION].', 'cond-mat-0602062-1-16-1': 'However, rather than using the mode operator correspondences that were used in [CITATION], here we perform this step using functional operator correspondences.', 'cond-mat-0602062-1-17-0': 'We define, similar to the restricted basis field operator [MATH], the restricted basis wavefunctions [EQUATION] and the related functional derivatives [EQUATION]', 'cond-mat-0602062-1-17-1': 'Using these definitions together with the Wigner function mode operator correpondences [CITATION], we find that the actions of the restricted basis field operator on the system density operator [MATH] can be expressed as actions on the corresponding Wigner function using [EQUATION]', 'cond-mat-0602062-1-17-2': 'Such functional Wigner function operator correspondences have been previously used by Steel et al. [CITATION].', 'cond-mat-0602062-1-18-0': '## Wigner function evolution', 'cond-mat-0602062-1-19-0': 'Applying the functional Wigner function operator correspondences, Eq. ([REF]), to the master equation term describing three-body recombination, Eq. ([REF]), we obtain, after some manipulation, the Wigner function evolution term [EQUATION]', 'cond-mat-0602062-1-19-1': 'Eq. ([REF]) is a rather complex equation of motion, including derivative terms up to sixth order.', 'cond-mat-0602062-1-19-2': 'However, we can only write differential equations describing the evolution of a single ensemble member for Wigner function evolutions containing derivative terms up to second order.', 'cond-mat-0602062-1-19-3': 'Thus to proceed we must truncate the higher order terms in Eq. ([REF]), a process that is also required for the pairwise scattering [CITATION].', 'cond-mat-0602062-1-20-0': '### Wigner truncation', 'cond-mat-0602062-1-21-0': 'To justify the truncation of the higher order terms in the Wigner function evolution, we follow a similar method to that given in [CITATION].', 'cond-mat-0602062-1-21-1': 'Let us assume that at some time [MATH] the Wigner function of the (low-energy) system has the factorisable form [EQUATION]', 'cond-mat-0602062-1-21-2': 'Here [MATH] is the expectation value amplitude of the [MATH]th mode, and [MATH] is proportional to the inverse width of the Wigner function for that mode.', 'cond-mat-0602062-1-21-3': 'This type of function describes both coherent (where [MATH]) and thermally distributed modes, but does not describe number states or other more exotic states.', 'cond-mat-0602062-1-21-4': 'The factorisability of this Wigner function indicates that number fluctuations between disparate modes are uncorrelated.', 'cond-mat-0602062-1-22-0': 'Evaluating the Wigner function evolution given by Eq. ([REF]) using the Wigner function given by Eq. ([REF]) returns a rather complicated expression, which we give in full in the appendix, Eq. ([REF]).', 'cond-mat-0602062-1-22-1': 'Essentially we find that for increasing order in [MATH], the leading order term in [MATH] decreases.', 'cond-mat-0602062-1-22-2': 'In performing the Wigner truncation for the two-body scattering in [CITATION], we required that in the coordinate space regions of high real particle density that [MATH].', 'cond-mat-0602062-1-22-3': 'Given that [MATH] and that all remaining terms scale as [MATH], and assuming that there is significant real particle density in the regions where three-body recombination is important compared to the local density of modefunctions, only the first few terms in [MATH] are important.', 'cond-mat-0602062-1-22-4': 'By keeping only those terms of 4th and 5th order in [MATH] we find that, within the same validity regime for the two-body elastic scattering, the Wigner function equation of motion can be accurately described by [EQUATION]', 'cond-mat-0602062-1-22-5': 'While it is possible to directly convert this equation of motion into a set of coupled differential equations, the functional nature of the derivative operators can obscure some of the details.', 'cond-mat-0602062-1-22-6': 'Instead we choose to perform the conversion using an explicit mode representation.', 'cond-mat-0602062-1-22-7': 'Using our definitions of the functional derivatives, Eq. ([REF]), we find that the Wigner function evolution due to three-body recombination can be expressed as [EQUATION]', 'cond-mat-0602062-1-23-0': '## Stochastic differential equations', 'cond-mat-0602062-1-24-0': 'It is important to remember when converting Eq. ([REF]) to its equivalent differential equations that we have two sets of independent variables, [MATH] and [MATH].', 'cond-mat-0602062-1-24-1': 'Thus while the drift terms are straightforward, and we find by using the relations given in [CITATION] for the Ito calculus that [EQUATION] where [MATH] as required, the diffusion terms are not so easily obtained.', 'cond-mat-0602062-1-25-0': 'To obtain the terms in the stochastic differential equations corresponding to the diffusion terms in Eq. ([REF]), we first find it necessary to rewrite the coefficient of that diffusive part as [EQUATION] where it is important to note that the summation over the index [MATH] runs over the complete mode space [MATH].', 'cond-mat-0602062-1-25-1': 'Including basis modes that are not part of the system subspace into the formalism in this way may appear to be cause for concern, as the master equation term from which we are working, Eq. ([REF]) contains no reference to these high-energy states.', 'cond-mat-0602062-1-25-2': 'However, we note that we are free to choose any set of ensemble differential equations that can be shown to be mathematically equivalent to the Fokker-Planck equation [CITATION], such that our inclusion of the high-energy modes in Eq. ([REF]) is certainly mathematically accurate.', 'cond-mat-0602062-1-26-0': 'It can be shown, either by rewriting Eq. ([REF]) in terms of explicitly real quantities (including the mode amplitude quadratures) and directly using the conversion relations given in [CITATION], or by working backwards using complex Ito calculus, that the ensemble differential equations corresponding to the diffusive part of the Wigner function evolution are given by [EQUATION] for all those modes [MATH].', 'cond-mat-0602062-1-26-1': 'Here the complex Wiener processes [MATH] obey the relations [EQUATION]', 'cond-mat-0602062-1-26-2': 'In fact, given the local nature of the recombination process in coordinate space, a more useful form of the total Wiener process is given by [EQUATION] which can be straightforwardly shown to obey [EQUATION]', 'cond-mat-0602062-1-26-3': 'Inserting the spatial Wiener process [MATH] as appropriate into the diffusive mode evolution given by Eq. ([REF]), using the drift mode evolutions of Eq. ([REF]) and including the evolution in the absence of three-body recombination given in [CITATION], gives the total evolution of the low-energy system mode amplitudes as [EQUATION]', 'cond-mat-0602062-1-26-4': 'Using the definition of the system wavefunction given by Eq. ([REF]) we find that the corresponding evolution of the coordinate space field is [EQUATION] where we have recognised the low-energy projector [MATH], Eq. ([REF]).', 'cond-mat-0602062-1-27-0': '### Rate of population change', 'cond-mat-0602062-1-28-0': 'The total (real) particle population of the field is defined as [EQUATION]', 'cond-mat-0602062-1-28-1': 'Using the correspondence of moments of the Wigner function to symmetrically ordered products of quantum operators [CITATION], we find that [MATH] can be calculated using Wigner function averages as [EQUATION]', 'cond-mat-0602062-1-28-2': "Using this result, we find the rate of change of particle number for a single trajectory to be given by Ito's formula [CITATION] [EQUATION]", 'cond-mat-0602062-1-28-3': 'The terms [MATH] are included here because, unlike ordinary deterministic calculus, the presence of the Wiener processes in Eq. ([REF]) give these terms a non-zero value in the limit [MATH].', 'cond-mat-0602062-1-29-0': 'Taking expectation values, using the properties of the spatially-dependent Wiener process, Eq. ([REF]), we find the ensemble averaged rate of normalisation change to be [EQUATION] where we have written [MATH] for [MATH] for compactness, as we also do below.', 'cond-mat-0602062-1-29-1': 'It may appear from Eq. ([REF]) that our truncated Wigner treatment of three-body recombination introduces a small correction to the rate of particle loss, apparently creating particles (in the average).', 'cond-mat-0602062-1-29-2': 'However, a clearer understanding can be obtained by expressing the moments of the Wigner function as physically significant quantities.', 'cond-mat-0602062-1-30-0': 'Using the properties of the Wigner function moments we find, for example, that [EQUATION] where we have again suppressed the spatial dependences.', 'cond-mat-0602062-1-30-1': 'Replacing the Wigner function moments in Eq. ([REF]) in this way returns [EQUATION]', 'cond-mat-0602062-1-30-2': 'Thus, rather than reducing the rate of particle loss, the truncated Wigner treatment leads to an increased rate of particle loss.', 'cond-mat-0602062-1-30-3': 'However, given that we have required that [MATH] to perform the Wigner truncation, this correction should be small.', 'cond-mat-0602062-1-30-4': 'Note that Eq. ([REF]) also shows that particle loss only occurs in those regions where there is real particle density, such that those coordinate space regions solely occupied by virtual particles will exhibit zero particle loss.', 'cond-mat-0602062-1-31-0': 'Comparing Eq. ([REF]) to Eq. ([REF]) shows that [MATH].', 'cond-mat-0602062-1-31-1': 'Thus while [MATH] is the rate constant for three-body recombination events, [MATH] is the number loss rate constant.', 'cond-mat-0602062-1-32-0': 'It is worthwhile discussing a possible point of confusion when using these classical field methods.', 'cond-mat-0602062-1-32-1': 'As the field for a single trajectory is represented by a single wavefunction, it could be considered that the field is therefore uniformly coherent at all points.', 'cond-mat-0602062-1-32-2': 'In such a case those behaviours that depend upon the statistics of the field, such as three-body recombination, would be improperly treated.', 'cond-mat-0602062-1-32-3': 'However, this view is incorrect, as such statistics only obtain physical meaning when considering ensembles of trajectories.', 'cond-mat-0602062-1-32-4': 'As an example, while direct inclusion of the statistics is relevant when considering three-body recombination using the mean (either spatial or temporal) particle density, in those regions where the system exhibits thermal statistics, the trajectory wavefunction will exhibit density fluctuations both spatially and temporally.', 'cond-mat-0602062-1-32-5': 'Thus the (spatial and temporal) mean of [MATH] will be larger than the mean density cubed, leading to the increased rate of loss observed experimentally.', 'cond-mat-0602062-1-32-6': 'Indeed, given that a single trajectory is entirely analogous to a single experimental run, the fact that here the wavefunction contains the full behaviour of the field is, if not obvious, at least eminently reasonable.', 'cond-mat-0602062-1-32-7': 'This result also provides for the factor of two increase in nonlinear interaction strength between the condensate and the thermal particles due to the exchange energy [CITATION].', 'cond-mat-0602062-1-33-0': '## Plane wave basis', 'cond-mat-0602062-1-34-0': 'While our formalism is applicable to any orthonormal single-particle basis [MATH], the most useful set of mode-functions for many situations, including the simple system we consider in this paper, is the plane-wave modes.', 'cond-mat-0602062-1-34-1': 'For this basis the modes are eigenstates of the curvature operator only, such that [MATH] and [MATH] where [MATH], and [EQUATION]', 'cond-mat-0602062-1-34-2': 'Within a periodically bounded volume of extent [MATH] the orthonormal plane-wave modes are arranged in momentum space such that [EQUATION] where [MATH], [MATH] and [MATH] are integers.', 'cond-mat-0602062-1-34-3': 'Using this plane-wave basis, the energy cutoff that defines the low-energy mode subspace becomes a spherical cutoff in momentum space, with the boundary defined by [MATH].', 'cond-mat-0602062-1-35-0': '# Numerical simulations', 'cond-mat-0602062-1-36-0': 'To demonstrate our truncated Wigner treatment of three-body recombination we have numerically simulated a (relatively) simple zero-temperature homogeneous gas of [MATH]Na atoms.', 'cond-mat-0602062-1-36-1': 'We describe the system using a set of plane-wave modes, with the initial real particle population confined to the ground ([MATH]) mode.', 'cond-mat-0602062-1-37-0': 'Determination of the three-body recombination event rate constant [MATH] for various alkali metals has been performed both theoretically [CITATION] and experimentally [CITATION].', 'cond-mat-0602062-1-37-1': 'For this paper we take as a best estimate of the relevant [MATH] the value measured at MIT [CITATION] for a fully condensed gas [EQUATION]', 'cond-mat-0602062-1-37-2': 'In that work it was reported that optical confinement was used to produce a rather large particle density of [MATH]Na of [MATH] cm[MATH] at the centre of the trap.', 'cond-mat-0602062-1-37-3': 'Thus, given that the rate of particle loss scales as [MATH] and the correction due to the dynamic noise sources as [MATH], such a large particle density should provide information on a parameter regime where three-body recombination is significant.', 'cond-mat-0602062-1-38-0': 'We use a simulation volume of [MATH], such that to achieve an initial (uniform) density of [MATH] cm[MATH] we use [MATH].', 'cond-mat-0602062-1-38-1': 'The mode spacing (in velocity space) along each of the cartesian directions, Eq. ([REF]), is determined by the volume to be 3.7 mms[MATH].', 'cond-mat-0602062-1-38-2': 'To characterise the strength of the interactions, we use [MATH] nm.', 'cond-mat-0602062-1-39-0': 'We have performed simulations using two distinct boundaries to the low-energy subspace: [MATH] mms[MATH], for which the number of modes [MATH]; and [MATH] mms[MATH], for which [MATH].', 'cond-mat-0602062-1-39-1': 'In both cases the number of modes is significantly less than the number of real particles, and we therefore expect that both cutoffs will return valid results.', 'cond-mat-0602062-1-40-0': '## Initial states', 'cond-mat-0602062-1-41-0': 'For our zero-temperature homogeneous system, the appropriate initial state for a single trajectory is described by [EQUATION]', 'cond-mat-0602062-1-41-1': 'Here [MATH] and [MATH] are Gaussian random variables of zero mean and unit variance, such that [EQUATION]', 'cond-mat-0602062-1-41-2': 'This initial state, Eq. ([REF]) satisfies the assumed Wigner function used to justify the Wigner truncation, Eq. ([REF]), with [MATH] for all [MATH], [MATH] and [MATH].', 'cond-mat-0602062-1-42-0': '## Evolution algorithm', 'cond-mat-0602062-1-43-0': 'The dynamic noise term present in the Eq. ([REF]) means that we cannot directly apply the deterministic projected RK4IP algorithm, which was used to obtain the results of [CITATION].', 'cond-mat-0602062-1-43-1': 'Rather we must employ an algorithm that explicitly allows for such time-dependent random processes.', 'cond-mat-0602062-1-44-0': 'The simplest such method is the Euler algorithm [CITATION], in which the drift terms are calculated at the start of each time step and the continuous Wiener processes [MATH] are replaced by a single discrete Wiener process [MATH].', 'cond-mat-0602062-1-44-1': 'However, for any reasonable accuracy the time step for an Euler algorithm must be very small, thus requiring very long calculation times.', 'cond-mat-0602062-1-44-2': 'Milstein and Tretyakov [CITATION] have considered various more sophisticated algorithms for propagating stochastic differential equation s with dynamic random processes.', 'cond-mat-0602062-1-44-3': 'In particular, they have shown that for situations where the influence of the dynamic noise on the system is very much less than the deterministic evolution (the small noise limit), one can accurately describe the total evolution using a relatively simple modification to the fourth-order Runge-Kutta (RK4) algorithm.', 'cond-mat-0602062-1-44-4': 'Essentially, in this method one calculates the deterministic evolution using the RK4 algorithm, while the dynamic noise is calculated using an Euler type derivative calculation based on the state of the system at the start of each time step.', 'cond-mat-0602062-1-44-5': 'This result therefore allows us to use a slightly modified version of the projected RK4IP algorithm to propagate Eq. ([REF]).', 'cond-mat-0602062-1-45-0': 'Importantly, any numerical propagation method requires a discrete coordinate space, such that the relations given for the spatial Wiener process, Eq. ([REF], do not apply.', 'cond-mat-0602062-1-45-1': 'Rather, we use noise sources that obey [EQUATION] where [MATH] is the time-dependent Wiener process at the [MATH]th point on the coordinate space simulation grid and [MATH] is the volume space increment about that grid point.', 'cond-mat-0602062-1-45-2': 'The algorithm advances the field in time by the increment [MATH] with each application, and we use [MATH] ns for all our simulations.', 'cond-mat-0602062-1-46-0': 'The total number of real particles within the low-energy subspace is given by Eq. ([REF]), where the subtraction of [MATH] can be understood as removing the virtual particles introduced into the initial state of the field, Eq. ([REF]).', 'cond-mat-0602062-1-46-1': 'For systems with a large number of modes [MATH], an excellent estimate of the total particle number can be made using [EQUATION]', 'cond-mat-0602062-1-46-2': 'In Fig. [REF] we plot the estimated total (real) particle numbers calculated using Eq. ([REF]) for single trajectories of the system described above using cutoffs of [MATH] mms[MATH] and 59.1 mms[MATH].', 'cond-mat-0602062-1-46-3': 'From these curves we observe that, over the larger time-scale, the total particle populations of the systems decrease, apparently monotonically, with the [MATH] mms[MATH] trajectory showing greater particle loss.', 'cond-mat-0602062-1-46-4': 'On the smaller time-scale however, as shown by the inset, the total particle populations fluctuate rapidly, on a scale of roughly 1-10 particles per time-step.', 'cond-mat-0602062-1-47-0': 'To provide a comparison with our truncated Wigner results, consider a simple model.', 'cond-mat-0602062-1-47-1': 'For a homogeneous system, and assuming that the third-order correlation function is both spatially and temporally invariant, such that [MATH], Eq. ([REF]) can be integrated to return the time-dependent total particle number [EQUATION]', 'cond-mat-0602062-1-47-2': 'For our system, using [MATH], the model shows a smaller rate of loss than that observed for either of our trajectories, as shown by the dashed line in Fig. [REF].', 'cond-mat-0602062-1-47-3': 'This result is predicted by Eq. ([REF]), as is the difference in the two trajectory populations.', 'cond-mat-0602062-1-48-0': '## Results', 'cond-mat-0602062-1-49-0': '# Conclusion', 'cond-mat-0602062-1-50-0': 'The truncated Wigner description of ultracold Bose gases has many significant advantages over more traditional approaches, such as the Gross-Pitaevskii equation, and the extension outlined in this paper allows for the inclusion of three-body recombination processes.', 'cond-mat-0602062-1-50-1': 'We have shown that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be described using stochastic differential equations describing the evolution of a single trajectory, which can be solved using numerical techniques.', 'cond-mat-0602062-1-51-0': 'We wish to thank Dr. P. B. Blakie for informative discussions.', 'cond-mat-0602062-1-52-0': '# Mathematical details', 'cond-mat-0602062-1-53-0': 'Using the particular Wigner function given by Eq. ([REF]) in the full three-body recombination Wigner function equation of motion, Eq. ([REF]), and evaluating all [MATH] operators returns [EQUATION]', 'cond-mat-0602062-1-53-1': 'Here the terms arising from the separate derivative orders are kept separated, as indicated on the left-hand side (first order derivative terms are given on the first and second lines).', 'cond-mat-0602062-1-53-2': 'For compactness we have written [MATH].'}

{'cond-mat-0602062-2-0-0': 'We apply the truncated Wigner method to the process of three-body recombination in ultracold Bose gases.', 'cond-mat-0602062-2-0-1': 'We find that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be treated using a set of coupled stochastic differential equations that include diffusion terms, and can be simulated using known numerical methods.', 'cond-mat-0602062-2-0-2': 'As an example we investigate the behaviour of a simple homogeneous Bose gas.', 'cond-mat-0602062-2-1-0': '# Introduction', 'cond-mat-0602062-2-2-0': 'The dominant loss process affecting ultracold gaseous alkali metal systems is inelastic three-body recombination [CITATION], a process characterised by collisional events involving three atoms leading to the creation of a single two-atom molecule (a dimer).', 'cond-mat-0602062-2-2-1': 'The binding energy released by the molecule formation is retained by the particles as kinetic energy.', 'cond-mat-0602062-2-2-2': 'Typically this results in the loss of all three atoms from the system as the molecule is not trapped by any applied external potential and the energy of the free atom is high enough to overcome any confinement barrier.', 'cond-mat-0602062-2-2-3': 'Indeed it is this process that limits the lifetime of experimentally produced alkali metal Bose-Einstein condensates, due to the large increase in density once the temperature is lowered past the critical point [CITATION].', 'cond-mat-0602062-2-3-0': 'In a previous paper [CITATION] we presented a comprehensive treatment of the truncated Wigner approach for ultracold Bose gases including elastic two-body interactions.', 'cond-mat-0602062-2-3-1': 'In this paper we extend that treatment to include three-body recombination events, which modifies the ensemble differential equations describing the evolution of a single realisation of the field.', 'cond-mat-0602062-2-3-2': 'These modified differential equations are explicitly stochastic, including dynamic noise sources arising from the action of three-body recombination on the virtual particle background field.', 'cond-mat-0602062-2-4-0': 'To provide a demonstration of our extended formalism we examine the evolution of a simple homogeneous system, starting from a zero-temperature state where the particle population is initially confined to a single (condensate) mode.', 'cond-mat-0602062-2-5-0': '## Three-body recombination in ultracold gases', 'cond-mat-0602062-2-6-0': 'Assuming that three-body recombination is the only particle loss mechanism affecting the system, it can be shown that the rate of change of total particle number is [CITATION] [EQUATION] where [MATH] is the total number density of atoms and [MATH] is the three-body recombination event rate constant.', 'cond-mat-0602062-2-6-1': 'We have assumed that all the particles involved in the recombination process are lost from the system, hence the prefactor of 3 in Eq. ([REF]), which describes the number of particles lost from the system.', 'cond-mat-0602062-2-6-2': 'The third-order normalised equiposition correlation function [MATH] measures the statistics of the field, being unity for a fully coherent system, i.e. a zero-temperature condensate, and [MATH] for a purely thermal system.', 'cond-mat-0602062-2-6-3': 'The factor of 6 increase in the loss rate of thermal over coherent systems for similar densities has been observed experimentally [CITATION].', 'cond-mat-0602062-2-7-0': '# Truncated Wigner treatment', 'cond-mat-0602062-2-8-0': '## The restricted field', 'cond-mat-0602062-2-9-0': 'As in our previous work [CITATION], we describe the many-body system of identical bosons using the Schrodinger picture bosonic field operator [EQUATION]', 'cond-mat-0602062-2-9-1': 'Here the mode operators [MATH] annihilate a single boson from the [MATH]th mode, and obey the commutation relations [EQUATION] while the coordinate space functions [MATH] form an infinite orthonormal basis set where [EQUATION] where [MATH] is the applied external potential.', 'cond-mat-0602062-2-10-0': 'We now divide mode space into two subspaces, a low-energy space ([MATH]) consisting of all those modes whose eigenenergies are less than the boundary energy [MATH], and a high-energy space ([MATH]) that includes all remaining modes.', 'cond-mat-0602062-2-10-1': 'For this work our interest lies with the dynamics of the low-energy subspace.', 'cond-mat-0602062-2-10-2': 'Using these subspaces we define the field operators [EQUATION]', 'cond-mat-0602062-2-10-3': 'Of most importance for this paper is the low-energy restricted basis field operator [MATH], which can be obtained from the total field operator [EQUATION] using the projector [EQUATION] as [MATH].', 'cond-mat-0602062-2-10-4': 'The restricted basis field operator obeys the commutation relations [EQUATION] where the restricted delta function is defined by [EQUATION]', 'cond-mat-0602062-2-10-5': 'The conjugate projector [MATH] can be obtained using the complementarity relation [MATH].', 'cond-mat-0602062-2-11-0': '## Master equation', 'cond-mat-0602062-2-12-0': 'Our previous paper [CITATION] assumed that only two atoms participate in any single scattering event.', 'cond-mat-0602062-2-12-1': 'In this way the particle interactions are described using a simple [MATH]-wave contact potential as an approximation to the full two-body T-matrix.', 'cond-mat-0602062-2-12-2': 'Obviously such a description does not include three-body scattering events.', 'cond-mat-0602062-2-12-3': 'Full theoretical treatments of three-body scattering including all possible collisional channels are extremely complicated, and we do not attempt such an approach here.', 'cond-mat-0602062-2-12-4': 'Instead we adopt a quantum-optical approach, starting from a phenomenologically appropriate Hamiltonian including inelastic three-body recombination events, to which we apply the truncated Wigner method.', 'cond-mat-0602062-2-13-0': 'We assume that within the dilute limit the characteristic range of the three-body recombination potential [MATH] is much smaller than the average interparticle spacing.', 'cond-mat-0602062-2-13-1': 'Thus, following thematically the approach for pairwise scattering, we replace this scattering potential by an effective zero-range three-body scattering T-operator, whose interaction strength is essentially a free parameter that will be chosen to satisfy experimentally observed loss rates.', 'cond-mat-0602062-2-13-2': 'Within this approach then, in order to include the effects of three-body recombination the Schrodinger picture effective Hamiltonian is modified to include the term [CITATION] [EQUATION] where the molecule field operator [MATH] annihilates a dimer from the field and [MATH] is a measure of the energy associated with the three-body process.', 'cond-mat-0602062-2-13-3': 'We have assumed in formulating Eq. ([REF]) that the binding energy associated with the molecule formation is large enough that the unpaired atom generated by a recombination event is created within the high-energy subspace [MATH], rather than the low-energy (system) subspace [MATH], and is described by the high-energy field operator [MATH].', 'cond-mat-0602062-2-14-0': 'Jack [CITATION] has considered this partial Hamiltonian, and has shown that by eliminating both the molecular and high-energy atomic fields from the evolution using a standard interaction picture approach for initially uncoupled fields [CITATION], one obtains the master equation term [EQUATION] for the low-energy atomic subspace (system) density operator [MATH].', 'cond-mat-0602062-2-14-1': 'The quantity [MATH] governs the rate of recombination events, and its relationship to [MATH] we consider later.', 'cond-mat-0602062-2-14-2': 'In arriving at this master equation term it has been assumed that the output products of the recombination events immediately exit the coordinate space region containing the system, such that they play no further role in the evolution.', 'cond-mat-0602062-2-14-3': 'To describe the full master equation for the system density operator one combines Eq. ([REF]) with the von Neumann equation calculated in [CITATION].', 'cond-mat-0602062-2-15-0': '## Functional Wigner function correspondences', 'cond-mat-0602062-2-16-0': 'The master equation term given by Eq. ([REF]) can be used to calculate the evolution of the corresponding multimode Wigner function [MATH] using appropriate operator correspondences [CITATION].', 'cond-mat-0602062-2-16-1': 'However, rather than using the mode operator correspondences that were used in [CITATION], here we perform this step using functional operator correspondences.', 'cond-mat-0602062-2-17-0': 'We define, similar to the restricted basis field operator [MATH], the restricted basis wavefunctions [EQUATION] and the related functional derivatives [EQUATION]', 'cond-mat-0602062-2-17-1': 'Using these definitions together with the Wigner function mode operator correpondences [CITATION], we find that the actions of the restricted basis field operator on the system density operator [MATH] can be expressed as actions on the corresponding Wigner function using [EQUATION]', 'cond-mat-0602062-2-17-2': 'Such functional Wigner function operator correspondences have been previously used by Steel et al. [CITATION].', 'cond-mat-0602062-2-18-0': '## Wigner function evolution', 'cond-mat-0602062-2-19-0': 'Applying the functional Wigner function operator correspondences, Eq. ([REF]), to the master equation term describing three-body recombination, Eq. ([REF]), we obtain, after some manipulation, the Wigner function evolution term [EQUATION]', 'cond-mat-0602062-2-19-1': 'Eq. ([REF]) is a rather complex equation of motion, including derivative terms up to sixth order.', 'cond-mat-0602062-2-19-2': 'However, we can only write differential equations describing the evolution of a single ensemble member for Wigner function evolutions containing derivative terms up to second order.', 'cond-mat-0602062-2-19-3': 'Thus to proceed we must truncate the higher order terms in Eq. ([REF]), a process that is also required for the pairwise scattering [CITATION].', 'cond-mat-0602062-2-20-0': '### Wigner truncation', 'cond-mat-0602062-2-21-0': 'To justify the truncation of the higher order terms in the Wigner function evolution, we follow a similar method to that given in [CITATION].', 'cond-mat-0602062-2-21-1': 'Let us assume that at some time [MATH] the Wigner function of the (low-energy) system has the factorisable form [EQUATION]', 'cond-mat-0602062-2-21-2': 'Here [MATH] is the expectation value amplitude of the [MATH]th mode, and [MATH] is proportional to the inverse width of the Wigner function for that mode.', 'cond-mat-0602062-2-21-3': 'This type of function describes both coherent (where [MATH]) and thermally distributed modes, but does not describe number states or other more exotic states.', 'cond-mat-0602062-2-21-4': 'The factorisability of this Wigner function indicates that number fluctuations between disparate modes are uncorrelated.', 'cond-mat-0602062-2-22-0': 'Evaluating the Wigner function evolution given by Eq. ([REF]) using the Wigner function given by Eq. ([REF]) returns a rather complicated expression, which we give in full in the appendix, Eq. ([REF]).', 'cond-mat-0602062-2-22-1': 'Essentially we find that for increasing order in [MATH], the leading order term in [MATH] decreases.', 'cond-mat-0602062-2-22-2': 'In performing the Wigner truncation for the two-body scattering in [CITATION], we required that in the coordinate space regions of high real particle density that [MATH].', 'cond-mat-0602062-2-22-3': 'Given that [MATH] and that all remaining terms scale as [MATH], and assuming that there is significant real particle density in the regions where three-body recombination is important compared to the local density of modefunctions, only the first few terms in [MATH] are important.', 'cond-mat-0602062-2-22-4': 'By keeping only those terms of 4th and 5th order in [MATH] we find that, within the same validity regime for the two-body elastic scattering, the Wigner function equation of motion can be accurately described by [EQUATION]', 'cond-mat-0602062-2-22-5': 'While it is possible to directly convert this equation of motion into a set of coupled differential equations, the functional nature of the derivative operators can obscure some of the details.', 'cond-mat-0602062-2-22-6': 'Instead we choose to perform the conversion using an explicit mode representation.', 'cond-mat-0602062-2-22-7': 'Using our definitions of the functional derivatives, Eq. ([REF]), we find that the Wigner function evolution due to three-body recombination can be expressed as [EQUATION]', 'cond-mat-0602062-2-23-0': '## Stochastic differential equations', 'cond-mat-0602062-2-24-0': 'It is important to remember when converting Eq. ([REF]) to its equivalent differential equations that we have two sets of independent variables, [MATH] and [MATH].', 'cond-mat-0602062-2-24-1': 'Thus while the drift terms are straightforward, and we find by using the relations given in [CITATION] for the Ito calculus that [EQUATION] where [MATH] as required, the diffusion terms are not so easily obtained.', 'cond-mat-0602062-2-25-0': 'To obtain the terms in the stochastic differential equations corresponding to the diffusion terms in Eq. ([REF]), we first find it necessary to rewrite the coefficient of that diffusive part as [EQUATION] where it is important to note that the summation over the index [MATH] runs over the complete mode space [MATH].', 'cond-mat-0602062-2-25-1': 'Including basis modes that are not part of the system subspace into the formalism in this way may appear to be cause for concern, as the master equation term from which we are working, Eq. ([REF]) contains no reference to these high-energy states.', 'cond-mat-0602062-2-25-2': 'However, we note that we are free to choose any set of ensemble differential equations that can be shown to be mathematically equivalent to the Fokker-Planck equation [CITATION], such that our inclusion of the high-energy modes in Eq. ([REF]) is certainly mathematically accurate.', 'cond-mat-0602062-2-26-0': 'It can be shown, either by rewriting Eq. ([REF]) in terms of explicitly real quantities (including the mode amplitude quadratures) and directly using the conversion relations given in [CITATION], or by working backwards using complex Ito calculus, that the ensemble differential equations corresponding to the diffusive part of the Wigner function evolution are given by [EQUATION] for all those modes [MATH].', 'cond-mat-0602062-2-26-1': 'Here the complex Wiener processes [MATH] obey the relations [EQUATION]', 'cond-mat-0602062-2-26-2': 'In fact, given the local nature of the recombination process in coordinate space, a more useful form of the total Wiener process is given by [EQUATION] which can be straightforwardly shown to obey [EQUATION]', 'cond-mat-0602062-2-26-3': 'Inserting the spatial Wiener process [MATH] as appropriate into the diffusive mode evolution given by Eq. ([REF]), using the drift mode evolutions of Eq. ([REF]) and including the evolution in the absence of three-body recombination given in [CITATION], gives the total evolution of the low-energy system mode amplitudes as [EQUATION]', 'cond-mat-0602062-2-26-4': 'Using the definition of the system wavefunction given by Eq. ([REF]) we find that the corresponding evolution of the coordinate space field is [EQUATION] where we have recognised the low-energy projector [MATH], Eq. ([REF]).', 'cond-mat-0602062-2-27-0': '### Rate of population change', 'cond-mat-0602062-2-28-0': 'The total (real) particle population of the field is defined as [EQUATION]', 'cond-mat-0602062-2-28-1': 'Using the correspondence of moments of the Wigner function to symmetrically ordered products of quantum operators [CITATION], we find that [MATH] can be calculated using Wigner function averages as [EQUATION]', 'cond-mat-0602062-2-28-2': "Using this result, we find the rate of change of particle number for a single trajectory to be given by Ito's formula [CITATION] [EQUATION]", 'cond-mat-0602062-2-28-3': 'The terms [MATH] are included here because, unlike ordinary deterministic calculus, the presence of the Wiener processes in Eq. ([REF]) give these terms a non-zero value in the limit [MATH].', 'cond-mat-0602062-2-29-0': 'Taking expectation values, using the properties of the spatially-dependent Wiener process, Eq. ([REF]), we find the ensemble averaged rate of normalisation change to be [EQUATION] where we have written [MATH] for [MATH] for compactness, as we also do below.', 'cond-mat-0602062-2-29-1': 'It may appear from Eq. ([REF]) that our truncated Wigner treatment of three-body recombination introduces a small correction to the rate of particle loss, apparently creating particles (in the average).', 'cond-mat-0602062-2-29-2': 'However, a clearer understanding can be obtained by expressing the moments of the Wigner function as physically significant quantities.', 'cond-mat-0602062-2-30-0': 'Using the properties of the Wigner function moments we find, for example, that [EQUATION] where we have again suppressed the spatial dependences.', 'cond-mat-0602062-2-30-1': 'Replacing the Wigner function moments in Eq. ([REF]) in this way returns [EQUATION]', 'cond-mat-0602062-2-30-2': 'Thus, rather than reducing the rate of particle loss, the truncated Wigner treatment leads to an increased rate of particle loss.', 'cond-mat-0602062-2-30-3': 'However, given that we have required that [MATH] to perform the Wigner truncation, this correction should be small.', 'cond-mat-0602062-2-30-4': 'Note that Eq. ([REF]) also shows that particle loss only occurs in those regions where there is real particle density, such that those coordinate space regions solely occupied by virtual particles will exhibit zero particle loss.', 'cond-mat-0602062-2-31-0': 'Comparing Eq. ([REF]) to Eq. ([REF]) shows that [MATH].', 'cond-mat-0602062-2-31-1': 'Thus while [MATH] is the rate constant for three-body recombination events, [MATH] is the number loss rate constant.', 'cond-mat-0602062-2-32-0': 'It is worthwhile discussing a possible point of confusion when using these classical field methods.', 'cond-mat-0602062-2-32-1': 'As the field for a single trajectory is represented by a single wavefunction, it could be considered that the field is therefore uniformly coherent at all points.', 'cond-mat-0602062-2-32-2': 'In such a case those behaviours that depend upon the statistics of the field, such as three-body recombination, would be improperly treated.', 'cond-mat-0602062-2-32-3': 'However, this view is incorrect, as such statistics only obtain physical meaning when considering ensembles of trajectories.', 'cond-mat-0602062-2-32-4': 'As an example, while direct inclusion of the statistics is relevant when considering three-body recombination using the mean (either spatial or temporal) particle density, in those regions where the system exhibits thermal statistics, the trajectory wavefunction will exhibit density fluctuations both spatially and temporally.', 'cond-mat-0602062-2-32-5': 'Thus the (spatial and temporal) mean of [MATH] will be larger than the mean density cubed, leading to the increased rate of loss observed experimentally.', 'cond-mat-0602062-2-32-6': 'Indeed, given that a single trajectory is entirely analogous to a single experimental run, the fact that here the wavefunction contains the full behaviour of the field is, if not obvious, at least eminently reasonable.', 'cond-mat-0602062-2-32-7': 'This result also provides for the factor of two increase in nonlinear interaction strength between the condensate and the thermal particles due to the exchange energy [CITATION].', 'cond-mat-0602062-2-33-0': '## Plane wave basis', 'cond-mat-0602062-2-34-0': 'While our formalism is applicable to any orthonormal single-particle basis [MATH], the most useful set of mode-functions for many situations, including the simple system we consider in this paper, is the plane-wave modes.', 'cond-mat-0602062-2-34-1': 'For this basis the modes are eigenstates of the curvature operator only, such that [MATH] and [MATH] where [MATH], and [EQUATION]', 'cond-mat-0602062-2-34-2': 'Within a periodically bounded volume of extent [MATH] the orthonormal plane-wave modes are arranged in momentum space such that [EQUATION] where [MATH], [MATH] and [MATH] are integers.', 'cond-mat-0602062-2-34-3': 'Using this plane-wave basis, the energy cutoff that defines the low-energy mode subspace becomes a spherical cutoff in momentum space, with the boundary defined by [MATH].', 'cond-mat-0602062-2-35-0': '# Numerical simulations', 'cond-mat-0602062-2-36-0': 'To demonstrate our truncated Wigner treatment of three-body recombination we have numerically simulated a (relatively) simple zero-temperature homogeneous gas of [MATH]Na atoms.', 'cond-mat-0602062-2-36-1': 'We describe the system using a set of plane-wave modes, with the initial real particle population confined to the ground ([MATH]) mode.', 'cond-mat-0602062-2-37-0': 'Determination of the three-body recombination event rate constant [MATH] for various alkali metals has been performed both theoretically [CITATION] and experimentally [CITATION].', 'cond-mat-0602062-2-37-1': 'For this paper we take as a best estimate of the relevant [MATH] the value measured at MIT [CITATION] for a fully condensed gas [EQUATION]', 'cond-mat-0602062-2-37-2': 'In that work it was reported that optical confinement was used to produce a rather large particle density of [MATH]Na of [MATH] cm[MATH] at the centre of the trap.', 'cond-mat-0602062-2-37-3': 'Thus, given that the rate of particle loss scales as [MATH] and the correction due to the dynamic noise sources as [MATH], such a large particle density should provide information on a parameter regime where three-body recombination is significant.', 'cond-mat-0602062-2-38-0': 'We use a simulation volume of [MATH], such that to achieve an initial (uniform) density of [MATH] cm[MATH] we use [MATH].', 'cond-mat-0602062-2-38-1': 'The mode spacing (in velocity space) along each of the cartesian directions, Eq. ([REF]), is determined by the volume to be 3.7 mms[MATH].', 'cond-mat-0602062-2-38-2': 'To characterise the strength of the interactions, we use [MATH] nm.', 'cond-mat-0602062-2-39-0': 'We have performed simulations using two distinct boundaries to the low-energy subspace: [MATH] mms[MATH], for which the number of modes [MATH]; and [MATH] mms[MATH], for which [MATH].', 'cond-mat-0602062-2-39-1': 'In both cases the number of modes is significantly less than the number of real particles, and we therefore expect that both cutoffs will return valid results.', 'cond-mat-0602062-2-40-0': '## Initial states', 'cond-mat-0602062-2-41-0': 'For our zero-temperature homogeneous system, the appropriate initial state for a single trajectory is described by [EQUATION]', 'cond-mat-0602062-2-41-1': 'Here [MATH] and [MATH] are Gaussian random variables of zero mean and unit variance, such that [EQUATION]', 'cond-mat-0602062-2-41-2': 'This initial state, Eq. ([REF]) satisfies the assumed Wigner function used to justify the Wigner truncation, Eq. ([REF]), with [MATH] for all [MATH], [MATH] and [MATH].', 'cond-mat-0602062-2-42-0': '## Evolution algorithm', 'cond-mat-0602062-2-43-0': 'The dynamic noise term present in the Eq. ([REF]) means that we cannot directly apply the deterministic projected RK4IP algorithm, which was used to obtain the results of [CITATION].', 'cond-mat-0602062-2-43-1': 'Rather we must employ an algorithm that explicitly allows for such time-dependent random processes.', 'cond-mat-0602062-2-44-0': 'The simplest such method is the Euler algorithm [CITATION], in which the drift terms are calculated at the start of each time step and the continuous Wiener processes [MATH] are replaced by a single discrete Wiener process [MATH].', 'cond-mat-0602062-2-44-1': 'However, for any reasonable accuracy the time step for an Euler algorithm must be very small, thus requiring very long calculation times.', 'cond-mat-0602062-2-44-2': 'Milstein and Tretyakov [CITATION] have considered various more sophisticated algorithms for propagating stochastic differential equation s with dynamic random processes.', 'cond-mat-0602062-2-44-3': 'In particular, they have shown that for situations where the influence of the dynamic noise on the system is very much less than the deterministic evolution (the small noise limit), one can accurately describe the total evolution using a relatively simple modification to the fourth-order Runge-Kutta (RK4) algorithm.', 'cond-mat-0602062-2-44-4': 'Essentially, in this method one calculates the deterministic evolution using the RK4 algorithm, while the dynamic noise is calculated using an Euler type derivative calculation based on the state of the system at the start of each time step.', 'cond-mat-0602062-2-44-5': 'This result therefore allows us to use a slightly modified version of the projected RK4IP algorithm to propagate Eq. ([REF]).', 'cond-mat-0602062-2-45-0': 'Importantly, any numerical propagation method requires a discrete coordinate space, such that the relations given for the spatial Wiener process, Eq. ([REF], do not apply.', 'cond-mat-0602062-2-45-1': 'Rather, we use noise sources that obey [EQUATION] where [MATH] is the time-dependent Wiener process at the [MATH]th point on the coordinate space simulation grid and [MATH] is the volume space increment about that grid point.', 'cond-mat-0602062-2-45-2': 'The algorithm advances the field in time by the increment [MATH] with each application, and we use [MATH] ns for all our simulations.', 'cond-mat-0602062-2-46-0': 'The total number of real particles within the low-energy subspace is given by Eq. ([REF]), where the subtraction of [MATH] can be understood as removing the virtual particles introduced into the initial state of the field, Eq. ([REF]).', 'cond-mat-0602062-2-46-1': 'For systems with a large number of modes [MATH], an excellent estimate of the total particle number can be made using [EQUATION]', 'cond-mat-0602062-2-46-2': 'In Fig. [REF] we plot the estimated total (real) particle numbers calculated using Eq. ([REF]) for single trajectories of the system described above using cutoffs of [MATH] mms[MATH] and 59.1 mms[MATH].', 'cond-mat-0602062-2-46-3': 'From these curves we observe that, over the larger time-scale, the total particle populations of the systems decrease, apparently monotonically, with the [MATH] mms[MATH] trajectory showing greater particle loss.', 'cond-mat-0602062-2-46-4': 'On the smaller time-scale however, as shown by the inset, the total particle populations fluctuate rapidly, on a scale of roughly 1-10 particles per time-step.', 'cond-mat-0602062-2-47-0': 'To provide a comparison with our truncated Wigner results, consider a simple model.', 'cond-mat-0602062-2-47-1': 'For a homogeneous system, and assuming that the third-order correlation function is both spatially and temporally invariant, such that [MATH], Eq. ([REF]) can be integrated to return the time-dependent total particle number [EQUATION]', 'cond-mat-0602062-2-47-2': 'For our system, using [MATH], the model shows a smaller rate of loss than that observed for either of our trajectories, as shown by the dashed line in Fig. [REF].', 'cond-mat-0602062-2-47-3': 'This result is predicted by Eq. ([REF]), as is the difference in the two trajectory populations.', 'cond-mat-0602062-2-48-0': '## Results', 'cond-mat-0602062-2-49-0': '# Conclusion', 'cond-mat-0602062-2-50-0': 'The truncated Wigner description of ultracold Bose gases has many significant advantages over more traditional approaches, such as the Gross-Pitaevskii equation, and the extension outlined in this paper allows for the inclusion of three-body recombination processes.', 'cond-mat-0602062-2-50-1': 'We have shown that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be described using stochastic differential equations describing the evolution of a single trajectory, which can be solved using numerical techniques.', 'cond-mat-0602062-2-51-0': 'We wish to thank Dr. P. B. Blakie for informative discussions.', 'cond-mat-0602062-2-52-0': '# Mathematical details', 'cond-mat-0602062-2-53-0': 'Using the particular Wigner function given by Eq. ([REF]) in the full three-body recombination Wigner function equation of motion, Eq. ([REF]), and evaluating all [MATH] operators returns [EQUATION]', 'cond-mat-0602062-2-53-1': 'Here the terms arising from the separate derivative orders are kept separated, as indicated on the left-hand side (first order derivative terms are given on the first and second lines).', 'cond-mat-0602062-2-53-2': 'For compactness we have written [MATH].'}

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['cond-mat-0602062-1-45-2', 'cond-mat-0602062-2-45-2'], ['cond-mat-0602062-1-13-0', 'cond-mat-0602062-2-13-0'], ['cond-mat-0602062-1-13-1', 'cond-mat-0602062-2-13-1'], ['cond-mat-0602062-1-13-2', 'cond-mat-0602062-2-13-2'], ['cond-mat-0602062-1-13-3', 'cond-mat-0602062-2-13-3'], ['cond-mat-0602062-1-26-0', 'cond-mat-0602062-2-26-0'], ['cond-mat-0602062-1-26-1', 'cond-mat-0602062-2-26-1'], ['cond-mat-0602062-1-26-2', 'cond-mat-0602062-2-26-2'], ['cond-mat-0602062-1-26-3', 'cond-mat-0602062-2-26-3'], ['cond-mat-0602062-1-26-4', 'cond-mat-0602062-2-26-4'], ['cond-mat-0602062-1-32-0', 'cond-mat-0602062-2-32-0'], ['cond-mat-0602062-1-32-1', 'cond-mat-0602062-2-32-1'], ['cond-mat-0602062-1-32-2', 'cond-mat-0602062-2-32-2'], ['cond-mat-0602062-1-32-3', 'cond-mat-0602062-2-32-3'], ['cond-mat-0602062-1-32-4', 'cond-mat-0602062-2-32-4'], ['cond-mat-0602062-1-32-5', 'cond-mat-0602062-2-32-5'], ['cond-mat-0602062-1-32-6', 'cond-mat-0602062-2-32-6'], ['cond-mat-0602062-1-32-7', 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'cond-mat-0602062-2-47-0'], ['cond-mat-0602062-1-47-1', 'cond-mat-0602062-2-47-1'], ['cond-mat-0602062-1-47-2', 'cond-mat-0602062-2-47-2'], ['cond-mat-0602062-1-47-3', 'cond-mat-0602062-2-47-3'], ['cond-mat-0602062-1-28-0', 'cond-mat-0602062-2-28-0'], ['cond-mat-0602062-1-28-1', 'cond-mat-0602062-2-28-1'], ['cond-mat-0602062-1-28-2', 'cond-mat-0602062-2-28-2'], ['cond-mat-0602062-1-28-3', 'cond-mat-0602062-2-28-3'], ['cond-mat-0602062-1-21-0', 'cond-mat-0602062-2-21-0'], ['cond-mat-0602062-1-21-1', 'cond-mat-0602062-2-21-1'], ['cond-mat-0602062-1-21-2', 'cond-mat-0602062-2-21-2'], ['cond-mat-0602062-1-21-3', 'cond-mat-0602062-2-21-3'], ['cond-mat-0602062-1-21-4', 'cond-mat-0602062-2-21-4'], ['cond-mat-0602062-1-50-0', 'cond-mat-0602062-2-50-0'], ['cond-mat-0602062-1-50-1', 'cond-mat-0602062-2-50-1'], ['cond-mat-0602062-1-25-0', 'cond-mat-0602062-2-25-0'], ['cond-mat-0602062-1-25-1', 'cond-mat-0602062-2-25-1'], ['cond-mat-0602062-1-25-2', 'cond-mat-0602062-2-25-2'], ['cond-mat-0602062-1-2-0', 'cond-mat-0602062-2-2-0'], ['cond-mat-0602062-1-2-1', 'cond-mat-0602062-2-2-1'], ['cond-mat-0602062-1-2-2', 'cond-mat-0602062-2-2-2'], ['cond-mat-0602062-1-2-3', 'cond-mat-0602062-2-2-3'], ['cond-mat-0602062-1-30-0', 'cond-mat-0602062-2-30-0'], ['cond-mat-0602062-1-30-1', 'cond-mat-0602062-2-30-1'], ['cond-mat-0602062-1-30-2', 'cond-mat-0602062-2-30-2'], ['cond-mat-0602062-1-30-3', 'cond-mat-0602062-2-30-3'], ['cond-mat-0602062-1-30-4', 'cond-mat-0602062-2-30-4'], ['cond-mat-0602062-1-16-0', 'cond-mat-0602062-2-16-0'], ['cond-mat-0602062-1-16-1', 'cond-mat-0602062-2-16-1'], ['cond-mat-0602062-1-4-0', 'cond-mat-0602062-2-4-0'], ['cond-mat-0602062-1-19-0', 'cond-mat-0602062-2-19-0'], ['cond-mat-0602062-1-19-1', 'cond-mat-0602062-2-19-1'], ['cond-mat-0602062-1-19-2', 'cond-mat-0602062-2-19-2'], ['cond-mat-0602062-1-19-3', 'cond-mat-0602062-2-19-3'], ['cond-mat-0602062-1-45-0', 'cond-mat-0602062-2-45-0'], ['cond-mat-0602062-1-45-1', 'cond-mat-0602062-2-45-1'], ['cond-mat-0602062-1-45-2', 'cond-mat-0602062-2-45-2'], ['cond-mat-0602062-1-13-0', 'cond-mat-0602062-2-13-0'], ['cond-mat-0602062-1-13-1', 'cond-mat-0602062-2-13-1'], ['cond-mat-0602062-1-13-2', 'cond-mat-0602062-2-13-2'], ['cond-mat-0602062-1-13-3', 'cond-mat-0602062-2-13-3'], ['cond-mat-0602062-1-26-0', 'cond-mat-0602062-2-26-0'], ['cond-mat-0602062-1-26-1', 'cond-mat-0602062-2-26-1'], ['cond-mat-0602062-1-26-2', 'cond-mat-0602062-2-26-2'], ['cond-mat-0602062-1-26-3', 'cond-mat-0602062-2-26-3'], ['cond-mat-0602062-1-26-4', 'cond-mat-0602062-2-26-4'], ['cond-mat-0602062-1-32-0', 'cond-mat-0602062-2-32-0'], ['cond-mat-0602062-1-32-1', 'cond-mat-0602062-2-32-1'], ['cond-mat-0602062-1-32-2', 'cond-mat-0602062-2-32-2'], ['cond-mat-0602062-1-32-3', 'cond-mat-0602062-2-32-3'], ['cond-mat-0602062-1-32-4', 'cond-mat-0602062-2-32-4'], ['cond-mat-0602062-1-32-5', 'cond-mat-0602062-2-32-5'], ['cond-mat-0602062-1-32-6', 'cond-mat-0602062-2-32-6'], ['cond-mat-0602062-1-32-7', 'cond-mat-0602062-2-32-7'], ['cond-mat-0602062-1-43-0', 'cond-mat-0602062-2-43-0'], ['cond-mat-0602062-1-43-1', 'cond-mat-0602062-2-43-1'], ['cond-mat-0602062-1-34-0', 'cond-mat-0602062-2-34-0'], ['cond-mat-0602062-1-34-1', 'cond-mat-0602062-2-34-1'], ['cond-mat-0602062-1-34-2', 'cond-mat-0602062-2-34-2'], ['cond-mat-0602062-1-34-3', 'cond-mat-0602062-2-34-3'], ['cond-mat-0602062-1-6-0', 'cond-mat-0602062-2-6-0'], ['cond-mat-0602062-1-6-1', 'cond-mat-0602062-2-6-1'], ['cond-mat-0602062-1-6-2', 'cond-mat-0602062-2-6-2'], ['cond-mat-0602062-1-6-3', 'cond-mat-0602062-2-6-3']]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0602062

1904.02990

{'1904.02990-1-0-0': 'We show that forward mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives.', '1904.02990-1-0-1': 'This is in stark contrast to the common claim that they are substantially different.', '1904.02990-1-0-2': 'The difference is often illustrated by claiming that symbolic differentiation suffers from "expression swell" whereas automatic differentiation does not.', '1904.02990-1-0-3': 'Here, we show that this statement is not true.', '1904.02990-1-0-4': '"Expression swell" refers to the phenomenon of a much larger representation of the derivative as opposed to the representation of the original function.', '1904.02990-1-1-0': '# Common Myths', '1904.02990-1-2-0': 'When distinguishing automatic differentiation from symbolic differentiation one frequently encounters the following myths:', '1904.02990-1-3-0': 'Symbolic differentiation leads to "expression swell", i.e., careless symbolic differentiation can easily produce exponentially large symbolic expressions.', '1904.02990-1-3-1': 'Hence, automatic differentiation is more efficient than symbolic differentiation.', '1904.02990-1-3-2': 'Symbolic and automatic differentiation are very different, especially in the presence of control flow statements like if or while loops.', '1904.02990-1-3-3': 'Symbolic differentiation faces the problem of converting a computer program into a single expression.', '1904.02990-1-3-4': 'After computing a derivative using symbolic differentiation, which might lead to an exponential growth, one can reduce the size of the resulting expressions by using common subexpressions.', '1904.02990-1-3-5': 'However, this is has to be done very carefully and is not straightforward and the intermediate representation will still see an exponential growth.', '1904.02990-1-4-0': 'The purpose of this paper is to shed some light onto these issues and show, that they are not true.', '1904.02990-1-4-1': 'In fact, we will show that forward mode automatic differentiation and symbolic differentiation are equivalent.', '1904.02990-1-5-0': '# Introduction', '1904.02990-1-6-0': 'Computing derivatives is a fundamental task in computer science, especially in optimization and machine learning.', '1904.02990-1-6-1': 'Most optimization schemes rely on derivative information when minimizing a function.', '1904.02990-1-6-2': 'However, computing derivatives by hand is error prone and can be a time consuming task, especially when the function to be differentiated is more complex.', '1904.02990-1-6-3': 'Hence, methods for automatically computing derivatives have been designed.', '1904.02990-1-6-4': 'The two major approaches that can be found in the literature are automatic differentiation and symbolic differentiation.', '1904.02990-1-6-5': 'Symbolic differentiation is basically what one knows from high school.', '1904.02990-1-6-6': 'Automatic differentiation refers to the fact, that one can compute derivatives even for computer programs that compute functions.', '1904.02990-1-6-7': 'Automatic differentiation is often also referred to as algorithmic differentiation.', '1904.02990-1-6-8': 'The literature and tools on automatic differentiation is extensive, see, e.g., [CITATION].', '1904.02990-1-6-9': 'Its popularity increased significantly over the last few years, especially in the area of machine learning/deep learning [CITATION] where it is necessary to compute gradients of loss functions of deep nets.', '1904.02990-1-7-0': 'It is often claimed in the literature that automatic differentiation is not symbolic differentiation.', '1904.02990-1-7-1': 'To tell them apart, the phenomenon of "expression swell" is used [CITATION].', '1904.02990-1-7-2': 'However, the difference is not really explained and stays in the dark.', '1904.02990-1-7-3': 'On the other hand it is sometimes hard to put one tool into its right category, i.e., either automatic differentiation or symbolic differentiation.', '1904.02990-1-7-4': 'For instance, the Theano framework [CITATION] that has been widely used in the machine learning community is sometimes said to use automatic differentiation and sometimes said to use symbolic differentiation [CITATION].', '1904.02990-1-7-5': 'Here, we explain why it is hard to tell both approaches apart by showing that they are in fact equivalent.', '1904.02990-1-7-6': 'They both perform the same operations when computing derivatives.', '1904.02990-1-7-7': 'The only difference they have is the underlying data structure.', '1904.02990-1-7-8': 'Automatic differentiation operates on directed acyclic graphs (DAGs) whereas symbolic differentiation operates on expression trees or expression forests if one allows common subexpressions.', '1904.02990-1-7-9': 'However, when we allow common subexpressions in symbolic differentiation, then forward mode automatic differentiation and symbolic differentiation compute the same result.', '1904.02990-1-7-10': 'We will also show that the phenomenon of "expression swell" does not originate from the differentiation process but from transforming a DAG into a tree when disallowing common subexpressions.', '1904.02990-1-8-0': '# Expression Representation', '1904.02990-1-9-0': 'By expressions we understand mathematical expressions like [MATH].', '1904.02990-1-9-1': 'They can be represented by expression DAGs (also known as computational graphs or execution trace), expression trees, or expression forests with common subexpressions.', '1904.02990-1-9-2': 'Figure [REF] illustrates the difference for the function [MATH].', '1904.02990-1-9-3': 'Obviously, an expression tree is also an expression DAG.', '1904.02990-1-9-4': 'On the other hand, an expression DAG can be converted into a tree simply by unfolding.', '1904.02990-1-9-5': 'The execution of a computer program results in an expression DAG.', '1904.02990-1-9-6': 'For instance, the following Python code results in the above expression DAG.', '1904.02990-1-10-0': '0.2', '1904.02990-1-11-0': '0.78', '1904.02990-1-12-0': 'def f(x1, x2): t1 = x1 + x2 f = sin(t1) * cos(t1) return f', '1904.02990-1-13-0': 'When unfolding an expression DAG into an expression tree without common subexpressions it can become exponentially large.', '1904.02990-1-13-1': 'An example is given in Figure [REF].', '1904.02990-1-13-2': 'However, when allowing common subexpressions it can be converted one-to-one where the resulting forest has the same size as the DAG.', '1904.02990-1-13-3': 'The following Python code corresponds to this example.', '1904.02990-1-14-0': '0.2', '1904.02990-1-15-0': '0.78', '1904.02990-1-16-0': 'def f(x1): t1 = x1 * x1 t2 = t1 * t1 f = t2 * t2 return f', '1904.02990-1-17-0': 'Now, we see an expression swell when converting a DAG into a tree.', '1904.02990-1-17-1': 'However, this is totally independent of computing derivatives.', '1904.02990-1-17-2': 'When allowing common subexpressions, we do not see an expression swell.', '1904.02990-1-17-3': 'This expression swell is often put forward as an argument for distinguishing between automatic and symbolic differentiation.', '1904.02990-1-17-4': 'Here, we see that it is not connected to computing derivatives.', '1904.02990-1-17-5': 'Rather, it is a matter of the expression representation.', '1904.02990-1-17-6': 'Note, that when allowing common subexpressions such an expression swell does not occur.', '1904.02990-1-18-0': '# Equivalence of Forward Mode Automatic Differentiation and Symbolic Differentiation', '1904.02990-1-19-0': 'In this section we will first review forward mode automatic differentiation and symbolic differentiation, and then we will show their equivalence.', '1904.02990-1-20-0': '## Forward Mode Automatic Differentiation', '1904.02990-1-21-0': 'Given an expression DAG [MATH], forward mode automatic differentiation proceeds as follows when computing the derivative with respect to the input variable [MATH].', '1904.02990-1-21-1': 'Each internal node [MATH] will eventually store the derivative [MATH] that is commonly denoted as [MATH].', '1904.02990-1-21-2': 'Forward mode proceeds from input to output nodes.', '1904.02990-1-21-3': 'At the nodes representing the input variable [MATH], the derivative [MATH] is stored.', '1904.02990-1-21-4': 'Then, the derivatives that are stored at the remaining nodes, here called [MATH], are iteratively computed by summing over all their incoming edges using the following equation: [EQUATION] where the [MATH] are the partial derivatives that have been computed before and are stored at the nodes [MATH].', '1904.02990-1-21-5': 'Finally, the derivative of the function [MATH] that is represented by the expression DAG is stored at the output node.', '1904.02990-1-22-0': '## Symbolic Differentiation', '1904.02990-1-23-0': 'Symbolic differentiation applies the following two rules iteratively to a given function in order to compute its derivative.', '1904.02990-1-24-0': 'If the function is unary, e.g., sine, cosine, etc., it applies the following rule [EQUATION]', '1904.02990-1-24-1': 'In case of binary functions, e.g, addition, multiplication, etc., symbolic differentiation applies [EQUATION]', '1904.02990-1-24-2': 'All rules from Calculus 101 can be reduced to these two rules.', '1904.02990-1-24-3': 'For instance, the multiplication rule is commonly known as [MATH].', '1904.02990-1-24-4': 'The following sequence shows that this follows from Equation [REF].', '1904.02990-1-24-5': 'We have [EQUATION] where the binary function [MATH] is the multiplication operation, [MATH], and [MATH].', '1904.02990-1-25-0': 'Please note, that the input expression is usually stored in an expression tree or expression forest with common subexpressions.', '1904.02990-1-25-1': 'Now there are two important questions that remain to be answered:', '1904.02990-1-26-0': 'Question 1: How does symbolic differentiation proceed when encountering common subexpressions?', '1904.02990-1-27-0': 'We illustrate this problem by the following example: Suppose we are differentiating a multiplication node [MATH].', '1904.02990-1-27-1': 'The derivative would be [MATH].', '1904.02990-1-27-2': 'Note, that [MATH] (and also [MATH]) appear in the original function as well as in the derivative.', '1904.02990-1-27-3': 'There are three possibilities to store the result.', '1904.02990-1-28-0': 'Copy the whole subtree representing [MATH].', '1904.02990-1-28-1': 'Store a pointer to the subtree representing [MATH].', '1904.02990-1-28-2': 'Introduce [MATH] as a common subexpression.', '1904.02990-1-29-0': 'All three options are valid possibilities.', '1904.02990-1-29-1': 'Option 1 will result in a tree as the output, Option 2 will result in a DAG, and Option 3 in a forest where each tree represents a common subexpression.', '1904.02990-1-29-2': 'Conceptually, Option 2 and Option 3 are identical.', '1904.02990-1-29-3': 'And this is also how one would implement symbolic differentiation.', '1904.02990-1-29-4': 'It is much easier to just store a pointer to the subtree [MATH] instead of copying the whole subtree [MATH].', '1904.02990-1-29-5': 'And since common subexpressions are allowed, turning this into a forest with common subexpressions is also trivial.', '1904.02990-1-30-0': 'Question 2: Should intermediate results be stored?', '1904.02990-1-31-0': 'Consider for instance the expression [MATH].', '1904.02990-1-31-1': 'Blindly computing the derivative would require to compute the derivative of [MATH] twice.', '1904.02990-1-31-2': 'Whenever there is the need for computing the derivative of a subtree [MATH], then one can simply check, if it has been done before.', '1904.02990-1-31-3': 'In this case, one does not compute it again but can reuse the old result.', '1904.02990-1-32-0': '## Equivalence', '1904.02990-1-33-0': 'Each internal node in an expression DAG can have either one or two incoming edges.', '1904.02990-1-33-1': 'In case a node has one incoming edge, the differentiation Rule [REF] for forward mode automatic differentiation is equivalent to the differentiation Rule [REF] for symbolic differentiation.', '1904.02990-1-33-2': 'In case of two incoming edges, it is equivalent to rule [REF] in symbolic differentiation.', '1904.02990-1-33-3': 'So obviously, both approaches perform the same operations.', '1904.02990-1-34-0': 'In symbolic differentiation, if one stores only a pointer to common subtrees, or introduces a common subexpression (Option 2 and Option 3 in Question 1) then this is exactly how automatic differentiation proceeds.', '1904.02990-1-35-0': 'Furthermore, if one stores intermediate results and reuses them (Question 2) then symbolic differentiation and forward mode automatic differentiation are identical.', '1904.02990-1-36-0': '# Demystifying the Common Myths', '1904.02990-1-37-0': 'It should be obvious now, that the myths that are commonly encountered when talking about automatic and symbolic differentiation are not true.', '1904.02990-1-37-1': 'The distinction between them is rather artificial.', '1904.02990-1-37-2': 'Especially, the claim of the "expression swell", i.e., careless symbolic differentiation can easily produce exponentially large expressions is not true.', '1904.02990-1-37-3': 'Instead, it will lead to the same result as forward mode automatic differentiation when intermediate results are stored as common subexpressions.', '1904.02990-1-37-4': 'This can be done easily.', '1904.02990-1-37-5': 'Hence, also the claim that common subexpressions need to be carefully determined after the differentiation process is not true.', '1904.02990-1-37-6': 'They can be simply collected during the differentiation process.', '1904.02990-1-38-0': 'When the input is a tree of size [MATH], then the resulting tree of the derivative will be of size at most [MATH] when storing only pointers to common subtrees and at most [MATH] when copying every subtree during the symbolic differentiation process.', '1904.02990-1-38-1': 'There is no exponential growth here.', '1904.02990-1-38-2': 'This misconception can be explained maybe by the following fact.', '1904.02990-1-38-3': 'When turning a DAG into a tree, one might see an exponential growth.', '1904.02990-1-38-4': 'But this is unrelated to differentiation.', '1904.02990-1-38-5': 'And if one turns a DAG into a forest with common subexpression then the size of the output stays the same (Myth 1).', '1904.02990-1-39-0': "## Speelpenning's Example", '1904.02990-1-40-0': "Often, Speelpenning's example [CITATION] is brought forward to illustrate that symbolic differentiation is inefficient.", '1904.02990-1-40-1': "Speelpenning's example is the following.", '1904.02990-1-40-2': 'Consider the function [MATH].', '1904.02990-1-40-3': 'When computing the gradient with respect to [MATH], symbolic differentiation would output [EQUATION]', '1904.02990-1-40-4': 'It is argued that the output is unnecessary large and there are quite a number of common subexpressions that now need to be identified.', '1904.02990-1-40-5': 'This is not true.', '1904.02990-1-40-6': 'What is displayed is the final output.', '1904.02990-1-40-7': 'However, this is the final output after the removal of common subexpressions.', '1904.02990-1-40-8': 'In between, they have already been computed in the same way as in forward mode automatic differentiation.', '1904.02990-1-40-9': 'Hence, symbolic differentiation is as efficient as forward mode automatic differentiation.', '1904.02990-1-41-0': "Summing up, Speelpenning's example rather serves as an example why reverse mode automatic differentiation is more efficient in case of many input variables and one output function (Myth 4).", '1904.02990-1-42-0': '## Control Structures', '1904.02990-1-43-0': 'When using operator overloading in automatic differentiation, control structures are easily circumvented since they do not appear in the computational graph, also known as the execution trace.', '1904.02990-1-43-1': 'That is, even when the code contains control structures, the computational graph/expression DAG will not.', '1904.02990-1-43-2': 'The same reasoning also applies to symbolic differentiation (Myth 2 and 3).', '1904.02990-1-44-0': '# Conclusion', '1904.02990-1-45-0': 'We have shown in this note that forward mode automatic differentiation and symbolic differentiation are algorithmically equivalent, i.e., they both perform the same set of operations when computing derivatives.', '1904.02990-1-45-1': 'The big contribution of automatic differentiation is rather of conceptual nature.', '1904.02990-1-45-2': 'It introduced the surprising idea that one can compute the derivative of any computer program.', '1904.02990-1-45-3': 'While this is a natural, yet very smart idea in hindsight, coming up with it for sure was not trivial.', '1904.02990-1-45-4': 'It has also led to many insights and algorithmic approaches like reverse mode, edge elimination, etc., and to automatic differentiation tools that compute derivatives of even very complex computer programs.'}

{'1904.02990-2-0-0': 'We show that forward mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives.', '1904.02990-2-0-1': 'This is in stark contrast to the common claim that they are substantially different.', '1904.02990-2-0-2': 'The difference is often illustrated by claiming that symbolic differentiation suffers from "expression swell" whereas automatic differentiation does not.', '1904.02990-2-0-3': 'Here, we show that this statement is not true.', '1904.02990-2-0-4': '"Expression swell" refers to the phenomenon of a much larger representation of the derivative as opposed to the representation of the original function.', '1904.02990-2-1-0': '# Common Myths', '1904.02990-2-2-0': 'When distinguishing automatic differentiation from symbolic differentiation one frequently encounters the following myths:', '1904.02990-2-3-0': 'Symbolic differentiation leads to "expression swell", i.e., careless symbolic differentiation can easily produce exponentially large symbolic expressions.', '1904.02990-2-3-1': 'Hence, automatic differentiation is more efficient than symbolic differentiation.', '1904.02990-2-3-2': 'Symbolic and automatic differentiation are very different, especially in the presence of control flow statements like if or while loops.', '1904.02990-2-3-3': 'Symbolic differentiation faces the problem of converting a computer program into a single expression.', '1904.02990-2-3-4': 'After computing a derivative using symbolic differentiation, which might lead to an exponential growth, one can reduce the size of the resulting expressions by using common subexpressions.', '1904.02990-2-3-5': 'However, this is has to be done very carefully and is not straightforward and the intermediate representation will still see an exponential growth.', '1904.02990-2-4-0': 'The purpose of this paper is to shed some light onto these issues and show, that they are not true.', '1904.02990-2-4-1': 'In fact, we will show that forward mode automatic differentiation and symbolic differentiation are equivalent.', '1904.02990-2-5-0': '# Introduction', '1904.02990-2-6-0': 'Computing derivatives is a fundamental task in computer science, especially in optimization and machine learning.', '1904.02990-2-6-1': 'Most optimization schemes rely on derivative information when minimizing a function.', '1904.02990-2-6-2': 'However, computing derivatives by hand is error prone and can be a time consuming task, especially when the function to be differentiated is more complex.', '1904.02990-2-6-3': 'Hence, methods for automatically computing derivatives have been designed.', '1904.02990-2-6-4': 'The two major approaches that can be found in the literature are automatic differentiation and symbolic differentiation.', '1904.02990-2-6-5': 'Symbolic differentiation is basically what one knows from high school.', '1904.02990-2-6-6': 'Automatic differentiation refers to the fact, that one can compute derivatives even for computer programs that compute functions.', '1904.02990-2-6-7': 'Automatic differentiation is often also referred to as algorithmic differentiation.', '1904.02990-2-6-8': 'The literature and tools on automatic differentiation is extensive, see, e.g., [CITATION].', '1904.02990-2-6-9': 'Its popularity increased significantly over the last few years, especially in the area of machine learning/deep learning [CITATION] where it is necessary to compute gradients of loss functions of deep nets.', '1904.02990-2-7-0': 'It is often claimed in the literature that automatic differentiation is not symbolic differentiation.', '1904.02990-2-7-1': 'To tell them apart, the phenomenon of "expression swell" is used [CITATION].', '1904.02990-2-7-2': 'However, the difference is not really explained and stays in the dark.', '1904.02990-2-7-3': 'On the other hand it is sometimes hard to put one tool into its right category, i.e., either automatic differentiation or symbolic differentiation.', '1904.02990-2-7-4': 'For instance, the Theano framework [CITATION] that has been widely used in the machine learning community is sometimes said to use automatic differentiation and sometimes said to use symbolic differentiation [CITATION].', '1904.02990-2-7-5': 'Here, we explain why it is hard to tell both approaches apart by showing that they are in fact equivalent.', '1904.02990-2-7-6': 'They both perform the same operations when computing derivatives.', '1904.02990-2-7-7': 'The only difference they have is the underlying data structure.', '1904.02990-2-7-8': 'Automatic differentiation operates on directed acyclic graphs (DAGs) whereas symbolic differentiation operates on expression trees or expression forests if one allows common subexpressions.', '1904.02990-2-7-9': 'However, when we allow common subexpressions in symbolic differentiation, then forward mode automatic differentiation and symbolic differentiation compute the same result.', '1904.02990-2-7-10': 'We will also show that the phenomenon of "expression swell" does not originate from the differentiation process but from transforming a DAG into a tree when disallowing common subexpressions.', '1904.02990-2-8-0': '# Expression Representation', '1904.02990-2-9-0': 'By expressions we understand mathematical expressions like [MATH].', '1904.02990-2-9-1': 'They can be represented by expression DAGs (also known as computational graphs or execution trace), expression trees, or expression forests with common subexpressions.', '1904.02990-2-9-2': 'Figure [REF] illustrates the difference for the function [MATH].', '1904.02990-2-9-3': 'Obviously, an expression tree is also an expression DAG.', '1904.02990-2-9-4': 'On the other hand, an expression DAG can be converted into a tree simply by unfolding.', '1904.02990-2-9-5': 'The execution of a computer program results in an expression DAG.', '1904.02990-2-9-6': 'For instance, the following Python code results in the above expression DAG.', '1904.02990-2-10-0': '0.2', '1904.02990-2-11-0': '0.78', '1904.02990-2-12-0': 'def f(x1, x2): t1 = x1 + x2 f = sin(t1) * cos(t1) return f', '1904.02990-2-13-0': 'When unfolding an expression DAG into an expression tree without common subexpressions it can become exponentially large.', '1904.02990-2-13-1': 'An example is given in Figure [REF].', '1904.02990-2-13-2': 'However, when allowing common subexpressions it can be converted one-to-one where the resulting forest has the same size as the DAG.', '1904.02990-2-13-3': 'The following Python code corresponds to this example.', '1904.02990-2-14-0': '0.2', '1904.02990-2-15-0': '0.78', '1904.02990-2-16-0': 'def f(x1): t1 = x1 * x1 t2 = t1 * t1 f = t2 * t2 return f', '1904.02990-2-17-0': 'Now, we see an expression swell when converting a DAG into a tree.', '1904.02990-2-17-1': 'However, this is totally independent of computing derivatives.', '1904.02990-2-17-2': 'When allowing common subexpressions, we do not see an expression swell.', '1904.02990-2-17-3': 'This expression swell is often put forward as an argument for distinguishing between automatic and symbolic differentiation.', '1904.02990-2-17-4': 'Here, we see that it is not connected to computing derivatives.', '1904.02990-2-17-5': 'Rather, it is a matter of the expression representation.', '1904.02990-2-17-6': 'Note, that when allowing common subexpressions such an expression swell does not occur.', '1904.02990-2-18-0': '# Equivalence of Forward Mode Automatic Differentiation and Symbolic Differentiation', '1904.02990-2-19-0': 'In this section we will first review forward mode automatic differentiation and symbolic differentiation, and then we will show their equivalence.', '1904.02990-2-20-0': '## Forward Mode Automatic Differentiation', '1904.02990-2-21-0': 'Given an expression DAG [MATH], forward mode automatic differentiation proceeds as follows when computing the derivative with respect to the input variable [MATH].', '1904.02990-2-21-1': 'Each internal node [MATH] will eventually store the derivative [MATH] that is commonly denoted as [MATH].', '1904.02990-2-21-2': 'Forward mode proceeds from input to output nodes.', '1904.02990-2-21-3': 'At the nodes representing the input variable [MATH], the derivative [MATH] is stored.', '1904.02990-2-21-4': 'Then, the derivatives that are stored at the remaining nodes, here called [MATH], are iteratively computed by summing over all their incoming edges using the following equation: [EQUATION] where the [MATH] are the partial derivatives that have been computed before and are stored at the nodes [MATH].', '1904.02990-2-21-5': 'Finally, the derivative of the function [MATH] that is represented by the expression DAG is stored at the output node.', '1904.02990-2-22-0': '## Symbolic Differentiation', '1904.02990-2-23-0': 'Symbolic differentiation applies the following two rules iteratively to a given function in order to compute its derivative.', '1904.02990-2-24-0': 'If the function is unary, e.g., sine, cosine, etc., it applies the following rule [EQUATION]', '1904.02990-2-24-1': 'In case of binary functions, e.g, addition, multiplication, etc., symbolic differentiation applies [EQUATION]', '1904.02990-2-24-2': 'All rules from Calculus 101 can be reduced to these two rules.', '1904.02990-2-24-3': 'For instance, the multiplication rule is commonly known as [MATH].', '1904.02990-2-24-4': 'The following sequence shows that this follows from Equation [REF].', '1904.02990-2-24-5': 'We have [EQUATION] where the binary function [MATH] is the multiplication operation, [MATH], and [MATH].', '1904.02990-2-25-0': 'Please note, that the input expression is usually stored in an expression tree or expression forest with common subexpressions.', '1904.02990-2-25-1': 'Now there are two important questions that remain to be answered:', '1904.02990-2-26-0': 'Question 1: How does symbolic differentiation proceed when encountering common subexpressions?', '1904.02990-2-27-0': 'We illustrate this problem by the following example: Suppose we are differentiating a multiplication node [MATH].', '1904.02990-2-27-1': 'The derivative would be [MATH].', '1904.02990-2-27-2': 'Note, that [MATH] (and also [MATH]) appear in the original function as well as in the derivative.', '1904.02990-2-27-3': 'There are three possibilities to store the result.', '1904.02990-2-28-0': 'Copy the whole subtree representing [MATH].', '1904.02990-2-28-1': 'Store a pointer to the subtree representing [MATH].', '1904.02990-2-28-2': 'Introduce [MATH] as a common subexpression.', '1904.02990-2-29-0': 'All three options are valid possibilities.', '1904.02990-2-29-1': 'Option 1 will result in a tree as the output, Option 2 will result in a DAG, and Option 3 in a forest where each tree represents a common subexpression.', '1904.02990-2-29-2': 'Conceptually, Option 2 and Option 3 are identical.', '1904.02990-2-29-3': 'And this is also how one would implement symbolic differentiation.', '1904.02990-2-29-4': 'It is much easier to just store a pointer to the subtree [MATH] instead of copying the whole subtree [MATH].', '1904.02990-2-29-5': 'And since common subexpressions are allowed, turning this into a forest with common subexpressions is also trivial.', '1904.02990-2-30-0': 'Question 2: Should intermediate results be stored?', '1904.02990-2-31-0': 'Consider for instance the expression [MATH].', '1904.02990-2-31-1': 'Blindly computing the derivative would require to compute the derivative of [MATH] twice.', '1904.02990-2-31-2': 'Whenever there is the need for computing the derivative of a subtree [MATH], then one can simply check, if it has been done before.', '1904.02990-2-31-3': 'In this case, one does not compute it again but can reuse the old result.', '1904.02990-2-32-0': '## Equivalence', '1904.02990-2-33-0': 'Each internal node in an expression DAG can have either one or two incoming edges.', '1904.02990-2-33-1': 'In case a node has one incoming edge, the differentiation Rule [REF] for forward mode automatic differentiation is equivalent to the differentiation Rule [REF] for symbolic differentiation.', '1904.02990-2-33-2': 'In case of two incoming edges, it is equivalent to rule [REF] in symbolic differentiation.', '1904.02990-2-33-3': 'So obviously, both approaches perform the same operations.', '1904.02990-2-34-0': 'In symbolic differentiation, if one stores only a pointer to common subtrees, or introduces a common subexpression (Option 2 and Option 3 in Question 1) then this is exactly how automatic differentiation proceeds.', '1904.02990-2-35-0': 'Furthermore, if one stores intermediate results and reuses them (Question 2) then symbolic differentiation and forward mode automatic differentiation are identical.', '1904.02990-2-36-0': '# Demystifying the Common Myths', '1904.02990-2-37-0': 'It should be obvious now, that the myths that are commonly encountered when talking about automatic and symbolic differentiation are not true.', '1904.02990-2-37-1': 'The distinction between them is rather artificial.', '1904.02990-2-37-2': 'Especially, the claim of the "expression swell", i.e., careless symbolic differentiation can easily produce exponentially large expressions is not true.', '1904.02990-2-37-3': 'Instead, it will lead to the same result as forward mode automatic differentiation when intermediate results are stored as common subexpressions.', '1904.02990-2-37-4': 'This can be done easily.', '1904.02990-2-37-5': 'Hence, also the claim that common subexpressions need to be carefully determined after the differentiation process is not true.', '1904.02990-2-37-6': 'They can be simply collected during the differentiation process.', '1904.02990-2-38-0': 'When the input is a tree of size [MATH], then the resulting tree of the derivative will be of size at most [MATH] when storing only pointers to common subtrees and at most [MATH] when copying every subtree during the symbolic differentiation process.', '1904.02990-2-38-1': 'There is no exponential growth here.', '1904.02990-2-38-2': 'This misconception can be explained maybe by the following fact.', '1904.02990-2-38-3': 'When turning a DAG into a tree, one might see an exponential growth.', '1904.02990-2-38-4': 'But this is unrelated to differentiation.', '1904.02990-2-38-5': 'And if one turns a DAG into a forest with common subexpression then the size of the output stays the same (Myth 1).', '1904.02990-2-39-0': "## Speelpenning's Example", '1904.02990-2-40-0': "Often, Speelpenning's example [CITATION] is brought forward to illustrate that symbolic differentiation is inefficient.", '1904.02990-2-40-1': "Speelpenning's example is the following.", '1904.02990-2-40-2': 'Consider the function [MATH].', '1904.02990-2-40-3': 'When computing the gradient with respect to [MATH], symbolic differentiation would output [EQUATION]', '1904.02990-2-40-4': 'It is argued that the output is unnecessary large and there are quite a number of common subexpressions that now need to be identified.', '1904.02990-2-40-5': 'This is not true.', '1904.02990-2-40-6': 'What is displayed is the final output.', '1904.02990-2-40-7': 'However, this is the final output after the removal of common subexpressions.', '1904.02990-2-40-8': 'In between, they have already been computed in the same way as in forward mode automatic differentiation.', '1904.02990-2-40-9': 'Hence, symbolic differentiation is as efficient as forward mode automatic differentiation (Myth 4).', '1904.02990-2-41-0': "Summing up, Speelpenning's example rather serves as an example why reverse mode automatic differentiation is more efficient in case of many input variables and one output function.", '1904.02990-2-42-0': '## Control Structures', '1904.02990-2-43-0': 'When using operator overloading in automatic differentiation, control structures are easily circumvented since they do not appear in the computational graph, also known as the execution trace.', '1904.02990-2-43-1': 'That is, even when the code contains control structures, the computational graph/expression DAG will not.', '1904.02990-2-43-2': 'The same reasoning also applies to symbolic differentiation (Myth 2 and 3).', '1904.02990-2-44-0': '# Conclusion', '1904.02990-2-45-0': 'We have shown in this note that forward mode automatic differentiation and symbolic differentiation are algorithmically equivalent, i.e., they both perform the same set of operations when computing derivatives.', '1904.02990-2-45-1': 'The big contribution of automatic differentiation is rather of conceptual nature.', '1904.02990-2-45-2': 'It introduced the surprising idea that one can compute the derivative of any computer program.', '1904.02990-2-45-3': 'While this is a natural, yet very smart idea in hindsight, coming up with it for sure was not trivial.', '1904.02990-2-45-4': 'It has also led to many insights and algorithmic approaches like reverse mode, edge elimination, etc., and to automatic differentiation tools that compute derivatives of even very complex computer programs.'}

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[]

[]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1904.02990

1311.0882

{'1311.0882-1-0-0': 'One dimensional tight binding models such as Aubry-Andre-Harper (AAH) model (with onsite cosine potential) and the integrable Maryland model (with onsite tangent potential) have been the subject of extensive theoretical research in localization studies.', '1311.0882-1-0-1': 'AAH can be directly mapped onto the two dimensional Hofstadter model which manifests the integer quantum Hall topology on a lattice.', '1311.0882-1-0-2': 'However, no such connection has been made for the Maryland model (MM).', '1311.0882-1-0-3': 'In this work, we describe a generalized model that contains AAH and MM as the limiting cases with the MM lying precisely at a topological quantum phase transition (TQPT) point.', '1311.0882-1-0-4': 'A remarkable feature of this critical point is that the 1D MM retains well defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the TQPT.', '1311.0882-1-1-0': 'Introduction:- Integer Quantum Hall Effect (IQH) is a canonical example of a gapped bulk topological phase with no generic symmetry protection.', '1311.0882-1-1-1': 'IQH can be captured by the 2D Hofstadter model [CITATION], a 2D lattice tight binding model with non-zero flux per unit cell.', '1311.0882-1-1-2': 'Hofstadter model can be mapped onto the 1D Aubry-Andre-Harper [CITATION] (AAH) model, a 1D tight binding chain with onsite cosine potential.', '1311.0882-1-1-3': 'The momentum corresponding to the second dimension of the equivalent 2D Hofstadter model appears as the phase of the onsite cosine potential in the 1D AAH model.', '1311.0882-1-1-4': 'The cosine modulation in the AAH model can be commensurate or incommensurate with respect to the lattice spacing depending on the rational or irrational value of the flux per plaquette.', '1311.0882-1-1-5': 'Aubry and Andre [CITATION] identified a localization transition for the incommensurate AAH model.', '1311.0882-1-1-6': 'This result led to an extensive theoretical investigation of the AAH model in the context of localization studies [CITATION].', '1311.0882-1-1-7': 'Recent experimental developments in photonic crystals [CITATION] and ultracold atoms [CITATION] have realized this localization phenomena in 1D quasiperiodic AAH lattices.', '1311.0882-1-2-0': "A completely different example of a 1D tight binding model with an onsite tangent modulation is presented by the 1D 'Maryland model'.", '1311.0882-1-2-1': "The 'Maryland Model' (MM) was proposed and solved exactly by Grempel et.", '1311.0882-1-2-2': 'al. [CITATION].', '1311.0882-1-2-3': 'MM has one-to-one correspondence with the quantum kicked rotor problem which has been experimentally realized in ultra cold atoms [CITATION], and has been extensively studied [CITATION].', '1311.0882-1-2-4': 'We discover in the current work a completely unexpected deep mathematical connection between MM and IQH, which has remained unappreciated in the literature.', '1311.0882-1-2-5': 'In addition, we show that MM presents an intriguing example of a topological quantum phase transition (TQPT).', '1311.0882-1-3-0': "The incommensurate Maryland model is a 1D quasicrystal (QC) with a special 'quasiperiodic' symmetry [CITATION].", '1311.0882-1-3-1': 'A family of 1D QCs taken together (generalized AAH, Fibonacci [CITATION]) [CITATION] has been topologically classified with an equivalent IQH topology in 2D.', '1311.0882-1-3-2': 'This classification was identified by connecting different models of QCs with the same topological invariant corresponding to the real space 2D lattice with a flux [CITATION].', '1311.0882-1-3-3': 'An argument was made [CITATION] and subsequently debated [CITATION] that this quasiperiodic symmetry allows one to associate 2D IQH invariants to each 1D member of the family of QCs [CITATION].', '1311.0882-1-3-4': 'The fact that the MM belongs to this 1D quasicrystal symmetry class and was not associated with the IQH topology calls for an investigation of this model from a fresh perspective.', '1311.0882-1-3-5': 'We base our arguments only on the well established connection between families of 1D tight binding models with periodic modulation and 2D IQH topology [CITATION], and comment on implications for the classification of families of 1D quasicrystals, noting that a single 1D quasiperiodic model is topologically trivial.', '1311.0882-1-4-0': 'The motivation of this work is two-fold: We want to understand the existence/nonexistence of the IQH topology in the Maryland model.', '1311.0882-1-4-1': 'Once we answer this question, we can understand its impact on the general classification of 1D QC families.', '1311.0882-1-4-2': 'In addition, MM being integrable with a relatively simple form of an exact solution allows us to gain deeper analytical insight into both of these questions [CITATION].', '1311.0882-1-5-0': 'In this letter, we take a fresh approach in understanding the relationship between the IQH topology and the Maryland model.', '1311.0882-1-5-1': 'We construct a family of 1D tight binding models paramterized by a phase with a general onsite modulation potential that contains AAH and MM as limiting cases.', '1311.0882-1-5-2': 'This enables us to interpolate the general potential between the two models.', '1311.0882-1-5-3': 'We construct the equivalent real space 2D lattice model by taking an inverse Fourier transform with respect to this phase parameter.', '1311.0882-1-5-4': 'We analyze the energy spectrum of the general 1D model as a function of the phase parameter.', '1311.0882-1-5-5': 'We identify the topological invariants for this general model using the theory of electric polarization [CITATION] which provides a natural framework to identify IQH invariants.', '1311.0882-1-5-6': 'Based on this analysis we explicitly show that the Maryland model sits at the critical point of a quantum phase transition to the topologically trivial state.', '1311.0882-1-5-7': 'The criticality of the Maryland model allows us to associate topological invariants to it in a purely mathematical sense using the limiting procedure along the deformation path in the parameter space.', '1311.0882-1-5-8': 'We show both numerically and analytically that even though the 1D gaps are preserved throughout the deformation from AAH to MM, the energy gaps in the equivalent 2D model close at the TQPT, as required by general considerations.', '1311.0882-1-5-9': 'We discuss the consequences of this result for the topological classification of 1D QC families [CITATION].', '1311.0882-1-6-0': 'Model:- We consider a 1D tight binding chain of size [MATH] with an onsite potential modulation [MATH] described by the following Hamiltonian,', '1311.0882-1-7-0': 'H(,)=-_n=1^N-1 t (c^_n+1c_n+c^_n c_n+1)-_n=1^N V_n(,)c^_nc_n,', '1311.0882-1-8-0': 'V_n(,)=2[MATH]((2n b+2)1+ (2n b+)), where [MATH] and [MATH] are fermionic creation and annihilation operators on the site [MATH], [MATH], [MATH], [MATH], and [MATH] is the amplitude of the nearest neighbour hopping.', '1311.0882-1-8-1': 'We emphasize, however, that our work and all conclusions are equally valid for the corresponding bosonic case since we are considering a noninteracting 1D quantum system.', '1311.0882-1-8-2': 'The onsite potential, [MATH], is characterized by the strength [MATH], period [MATH] and the phase parameter [MATH].', '1311.0882-1-8-3': 'The parameter [MATH] interpolates between the limiting cases AAH ([MATH]) and MM ([MATH]), [EQUATION]', '1311.0882-1-8-4': 'This general onsite potential is a smooth function of [MATH] in the open interval [MATH].', '1311.0882-1-8-5': '[MATH] has singularities at [MATH] corresponding to the integrable MM which we approach asymptotically in a limiting sense and we define TQPT in terms of these singularities.', '1311.0882-1-8-6': 'Note that one can define a general expansion [MATH] that contain the pole structure of MM asymptotically.', '1311.0882-1-8-7': '[MATH] corresponds to a simple example in this general family.', '1311.0882-1-9-0': '2D ancestor:- The phase parameter [MATH] allows one to associate the 1D AAH model with a dual 2D lattice by taking an inverse Fourier transform with respect to [MATH].', '1311.0882-1-9-1': 'The resulting real space lattice is the 2D Hofstdater model with a flux [MATH] per unit cell, which characterizes the IQH topology.', '1311.0882-1-9-2': 'The same idea applies to the Hamiltonian in Eq. ([REF]),', '1311.0882-1-10-0': 'H_2D()=_-^dH(,)e^im,', '1311.0882-1-11-0': 'c_nc_n()=12_me^imc_n,m.', '1311.0882-1-12-0': 'In this way we associate a 2D lattice (n,m) in the real space, [MATH], to the family of 1D Hamiltonians parameterized by [MATH].', '1311.0882-1-12-1': 'We now ask what happens to the IQH topology as we approach the Maryland model ([MATH]) limit in Eq. ([REF]).', '1311.0882-1-13-0': 'The resulting real space 2D Hamiltonian equivalent to [MATH] (up to a constant energy shift) reads, [EQUATION]', '1311.0882-1-13-1': 'The hopping strength in the [MATH]-direction [MATH] is written in terms of a contour integral, [EQUATION] where we have defined [MATH] on a unit circle contour.', '1311.0882-1-13-2': '[MATH] describes the hopping amplitude from site [MATH] to [MATH], i.e. a hopping of range [MATH] term is the constant shift in the onsite energy).', '1311.0882-1-13-3': 'The contour integration can be performed exactly, [EQUATION]', '1311.0882-1-13-4': 'Note that [MATH] in Eq. ([REF]) is defined in a limiting sense at the special points [MATH] (AAH and MM).', '1311.0882-1-13-5': 'Fig. [REF] plots the absolute value of the hopping amplitude [MATH] in the [MATH]-direction for different values of the hopping range [MATH] as a function of [MATH].', '1311.0882-1-13-6': 'In the limiting case of [MATH] (AAH) only the [MATH] term survives with [MATH] which corresponds to the nearest neighbor hopping of the Hofstadter model.', '1311.0882-1-13-7': 'As [MATH] increases, hopping terms of longer range [MATH] in the [MATH]-direction acquire non-vanishing amplitudes.', '1311.0882-1-13-8': 'In the limiting case of [MATH] (MM) the integral in Eq. ([REF]) diverges, which can be interpreted as the corresponding dual 2D lattice acquiring long range hopping terms of arbitrarily large [MATH] in the [MATH]-direction (all of equal unit amplitude).', '1311.0882-1-13-9': 'This arbitrarily long range hopping singularity is indicative of a quantum phase transition occurring at the critical points [MATH].', '1311.0882-1-13-10': 'To elucidate the physical nature of these [MATH] critical points further, we analyze the spectral band structure and the topological invariants of Eq. ([REF]) as a function of [MATH].', '1311.0882-1-14-0': 'Band structure:- We impose open boundary conditions on the 1D tight binding Hamiltonian, [MATH], in Eq. ([REF]) and numerically diagonalize it for the system size of [MATH] sites.', '1311.0882-1-14-1': 'It is instructive to plot the resulting energy bands as a function of the phase parameter [MATH].', '1311.0882-1-14-2': 'Note that tuning [MATH] captures the 2D band structure in the hybrid space [MATH].', '1311.0882-1-14-3': 'We start with the case of a commensurate modulation by setting [MATH] and [MATH].', '1311.0882-1-14-4': 'Fig. [REF] shows the resulting band structure as a function of the phase parameter [MATH] for four different values of [MATH] between 0 and 1.', '1311.0882-1-14-5': 'The case of AAH ([MATH]), top left panel of Figure [REF], demonstrates a clear and well defined set of gaps reflecting the robust integer quantum Hall topology of the 2D Hofstadter model.', '1311.0882-1-14-6': 'For [MATH] and [MATH], the band gaps gradually decrease.', '1311.0882-1-14-7': 'All spectral band gaps close (scale to zero with the system size) precisely at the critical point [MATH], see the bottom right panel of Fig. [REF], which corresponds to the MM.', '1311.0882-1-14-8': 'The gapless nature of the 2D spectrum for the MM case ([MATH]) in the hybrid space [MATH] is explicitly confirmed using the exact analytical expression (in an infinite system) for the MM spectrum with commensurate modulation [CITATION].', '1311.0882-1-15-0': 'Closing of the spectral gaps coincides with the hopping divergence in the 2D lattice and indicates a TQPT in the system as [MATH] (i.e. at the MM point).', '1311.0882-1-15-1': 'The interesting aspect of MM is that the 2D spectrum is gapless in the reciprocal space [MATH] whereas the 1D spectrum has well defined gaps for each value of [MATH].', '1311.0882-1-15-2': 'Here [MATH] is the Fourier image of the site index [MATH].', '1311.0882-1-15-3': 'The fact that the 1D MM spectrum has gaps whereas the corresponding dual 2D spectrum is gapless makes perfect sense since the non-trivial TQPT can only exist in the 2D space.', '1311.0882-1-16-0': 'Chern number from polarization theory:- In the following, we change [MATH] from 0 to 1 and track the change in the IQH topological invariant associated with the 2D system (Eq. ([REF])) in the hybrid space [MATH].', '1311.0882-1-16-1': 'An ideal tool for this task is the polarization of the 1D chain defined in the hybrid space.', '1311.0882-1-16-2': 'Modern theory of electric polarization of crystalline insulators [CITATION] is formulated in terms of the geometrical Berry phase of spectral energy bands.', '1311.0882-1-16-3': 'The polarization of a finite 1D insulator is given simply by the average charge center of the hybrid wannier function (HWF) ([MATH]) of the system [CITATION],', '1311.0882-1-17-0': 'n()=_nn (n,)_n(n,),', '1311.0882-1-18-0': '(n,)=_occupied states n,n,,', '1311.0882-1-19-0': 'where [MATH] is the real space site index and [MATH] is the hybrid eigenstate of the system.', '1311.0882-1-19-1': 'The polarization thus obtained is a gauge dependent function of the phase parameter [MATH].', '1311.0882-1-19-2': 'When the chemical potential is fixed in a spectral gap, adiabatic sweeping of this phase parameter by a period [MATH] results in the transfer of an amount of electric charge proportional to the Chern number of the gap.', '1311.0882-1-19-3': 'This transfer of charge is reflected in the discontinuity of the 1D polarization with the adiabatic change in the phase [MATH].', '1311.0882-1-19-4': 'This demonstrates the topological nature of the charge pumping with respect to the adiabatic sweep of [MATH].', '1311.0882-1-19-5': "This is analogous to (and equivalent to) Laughlin's IQH gauge argument [CITATION].", '1311.0882-1-19-6': 'The discontinuity of the polarization is a robust feature of the IQH and was recently proposed [CITATION] as a tool to measure topological invariants directly in the Hofstadter model realized in cold atomic systems [CITATION].', '1311.0882-1-19-7': 'Note that the generalized 1D chain (Eq. ([REF])) has well defined gaps in the spectrum for any fixed [MATH] and [MATH] (even in the case of the critical Maryland model) which allows us to define the 1D polarization in terms of HWF centers in the whole parameter space.', '1311.0882-1-20-0': 'In Fig [REF], we plot the shift in the HWF centers as a function of the phase [MATH] for four different values of [MATH] (for [MATH]) for which we have plotted the spectrum in Fig. [REF].', '1311.0882-1-20-1': 'In these plots we set the chemical potential in the gap above the top of the lowest band in the AAH limit ([MATH]).', '1311.0882-1-20-2': 'We keep the chemical potential fixed at this point as we deform AAH to MM and monitor the shift in the HWF centers.', '1311.0882-1-20-3': 'In the limit of AAH, the HWF center as a function of [MATH] shows a one unit cell jump corresponding to the Chern number [MATH], or, equivalently a transfer of charge [MATH] by a distance of one unit cell.', '1311.0882-1-20-4': 'We monitor this jump (invariant) as we deform AAH ([MATH]) to MM ([MATH]) keeping the chemical potential fixed.', '1311.0882-1-20-5': 'Note that the polarization jump survives in the MM limit [MATH], see Fig. [REF] bottom right.', '1311.0882-1-20-6': 'In the MM limit, these topological invariants are not robust with respect to small perturbations, e.g. parabolic confinement, disorder, that mix the eigenstates with energies above and below the chemical potential and in this way destroy the topological quantization.', '1311.0882-1-20-7': 'This demonstrates the change in the topological class of the system precisely at [MATH] which shows that the Maryland model is precisely at the TQPT point.', '1311.0882-1-20-8': 'The remarkable fact is that there are topological features preserved by the wave functions of the Maryland model that can be defined in a purely mathematical sense relying on the infinitesimal spectral gap at the filling fractions that correspond to the non-trivial topological state of the AAH family.', '1311.0882-1-21-0': 'Topological classification of 1D quasicrystals:- We have shown that the Maryland model [CITATION] represents a critical point for a quantum phase transition to a topologically trivial state.', '1311.0882-1-21-1': 'This result has important implications for the topological classification of 1D quasicrystals [CITATION].', '1311.0882-1-21-2': 'Families of 1D incommensurate tight binding models manifest a special \'quasiperiodic\' translational invariance arising from the "freedom" associated with the phase parameter of the periodic term.', '1311.0882-1-21-3': 'An infinite chain described by the Hamiltonian Eq. ([REF]) with an irrational value [MATH] is an example of such a family parameterized by the phase [MATH].', '1311.0882-1-21-4': 'An arbitrary shift in [MATH] can be compensated by a shift along the chain [MATH].', '1311.0882-1-21-5': 'Note that this is true only at irrational values of [MATH] since only in this case [MATH] forms a dense set mod [MATH].', '1311.0882-1-21-6': 'It has been argued [CITATION] that this quasiperiodic translational invariance allows one to assign the same Chern number to each member of the family of QCs, i.e. for each value of the phase parameter [MATH].', '1311.0882-1-21-7': 'This interpretation has been challenged in Ref. [CITATION].', '1311.0882-1-21-8': 'The quasiperiodic translation symmetry is preserved in the case of the Maryland model ([MATH]) which sits exactly at the critical point of a 2D TQPT.', '1311.0882-1-22-0': 'In Fig. [REF] we plot the band structure and the change in the polarization as a function of the phase [MATH] for the incommensurate AAH and MM.', '1311.0882-1-22-1': 'We choose the flux fraction to be a truncated Liouville constant (Liouville numbers are irrational numbers infinitely close to rational numbers).', '1311.0882-1-22-2': 'Note that the finite size Maryland model still demonstrates the presence of a non-zero Chern number in the same restricted sense as we found for the commensurate case.', '1311.0882-1-22-3': 'The topological invariants truly vanish only in the limit of the infinite system size.', '1311.0882-1-22-4': 'This can be clearly shown by using the exact analytic expression for the continuous MM spectrum for the infinite system size,', '1311.0882-1-23-0': 'E_(k_n,)=2(b+)^2+^2(b + ),', '1311.0882-1-24-0': 'where [MATH] is the Fourier image of [MATH] and [MATH] labels different bands.', '1311.0882-1-24-1': 'Not the that bands are degenerate with respect to [MATH].', '1311.0882-1-24-2': 'Remarkably, in the same way as we found in the commensurate case the band gaps in the incommensurate 1D model (Eq. ([REF])) for fixed [MATH] remain open for all values of [MATH] whereas the equivalent 2D model becomes gapless as we approach critical points [MATH].', '1311.0882-1-24-3': 'Within the class of 1D models with quasiperiodic symmetry, Maryland model manifests a 2D topological phase transition as a function of the deformation parameter [MATH] which can only be realized by sweeping the phase [MATH] (see Fig. [REF]).', '1311.0882-1-25-0': 'Conclusion:- We have identified a previously unknown topological feature of the Maryland model introduced in Refs. [CITATION] in the context of Anderson localization and kicked quantum rotor studies.', '1311.0882-1-25-1': 'We show that this model represents a topological quantum phase transition point in a class of corresponding 2D lattice models with IQH topology.', '1311.0882-1-25-2': 'The criticality allows us to associate topological invariants with the Maryland model in a restricted mathematical sense at the special filling factors that are adiabatically connected to the spectral gaps in the 1D Aubry-Andre-Harper model.', '1311.0882-1-25-3': 'Our theory presented in this work establishes deep and unexpected mathematical connections between 2D topological models and a family of 1D incommensurate localization models.'}

{'1311.0882-2-0-0': 'One dimensional tight binding models such as Aubry-Andre-Harper (AAH) model (with onsite cosine potential) and the integrable Maryland model (with onsite tangent potential) have been the subject of extensive theoretical research in localization studies.', '1311.0882-2-0-1': 'AAH can be directly mapped onto the two dimensional Hofstadter model which manifests the integer quantum Hall topology on a lattice.', '1311.0882-2-0-2': 'However, no such connection has been made for the Maryland model (MM).', '1311.0882-2-0-3': 'In this work, we describe a generalized model that contains AAH and MM as the limiting cases with the MM lying precisely at a topological quantum phase transition (TQPT) point.', '1311.0882-2-0-4': 'A remarkable feature of this critical point is that the 1D MM retains well defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the TQPT.', '1311.0882-2-1-0': 'Integer Quantum Hall Effect (IQH) is a canonical example of a gapped bulk topological phase with no generic symmetry protection.', '1311.0882-2-1-1': 'IQH can be captured by the 2D Hofstadter model [CITATION], a 2D lattice tight binding model with non-zero flux per unit cell.', '1311.0882-2-1-2': 'Hofstadter model can be mapped onto the 1D Aubry-Andre-Harper [CITATION] (AAH) model, a 1D tight binding chain with onsite cosine potential.', '1311.0882-2-1-3': 'Aubry and Andre [CITATION] identified a localization transition in the AAH model with modulation incommensurate with the lattice (corresponding to an irrational value of flux).', '1311.0882-2-1-4': 'This result led to an extensive theoretical investigation of the AAH model in the context of localization studies [CITATION].', '1311.0882-2-1-5': 'Recent experimental developments in photonic crystals [CITATION] and ultracold atoms [CITATION] have realized this localization phenomena in 1D quasiperiodic AAH lattices.', '1311.0882-2-2-0': "A completely different example of a 1D tight binding model with an onsite tangent modulation is presented by the 1D 'Maryland model'.", '1311.0882-2-2-1': "The 'Maryland Model' (MM) was proposed and solved exactly by Grempel et.", '1311.0882-2-2-2': 'al. [CITATION].', '1311.0882-2-2-3': 'MM has one-to-one correspondence with the quantum kicked rotor problem which has been experimentally realized in ultra cold atoms [CITATION], and has been extensively studied [CITATION].', '1311.0882-2-2-4': 'We discover in the current work a completely unexpected deep mathematical connection between MM and IQH, which has remained unappreciated in the literature.', '1311.0882-2-2-5': 'In addition, we show that MM presents an intriguing example of a topological quantum phase transition (TQPT).', '1311.0882-2-3-0': "Maryland model with period of onsite potential incommensurate with lattice spacing presents an example of a 1D quasicrystal (QC) for which a special 'quasiperiodic' transatlion symmetry was recently identified [CITATION].", '1311.0882-2-3-1': 'A family of 1D QCs taken together (generalized AAH, Fibonacci [CITATION]) [CITATION] has been topologically classified with an equivalent IQH topology in 2D.', '1311.0882-2-3-2': 'This classification was identified by connecting different models of QCs with the same topological invariant corresponding to the real space 2D lattice with a flux [CITATION].', '1311.0882-2-3-3': 'An argument was made [CITATION] and subsequently debated [CITATION] that this quasiperiodic translation symmetry allows one to associate 2D IQH invariants to each 1D member of the family of QCs [CITATION].', '1311.0882-2-3-4': 'The fact that the MM belongs to this 1D quasicrystal symmetry class and was not associated with the IQH topology calls for an investigation of this model from a fresh perspective.', '1311.0882-2-3-5': 'We base our arguments only on the well established connection between families of 1D tight binding models with periodic modulation and 2D IQH topology [CITATION].', '1311.0882-2-4-0': 'In this letter, we take a fresh approach in understanding the relationship between the IQH topology and the Maryland model.', '1311.0882-2-4-1': 'We construct a family of 1D tight binding models parameterized by a phase with a general onsite modulation potential that contains AAH and MM as limiting cases.', '1311.0882-2-4-2': 'We construct the equivalent real space 2D lattice model by taking an inverse Fourier transform with respect to this phase parameter.', '1311.0882-2-4-3': 'We analyze the energy spectrum of the general 1D model as a function of the phase parameter.', '1311.0882-2-4-4': 'We identify the topological invariants for this general model using the theory of electric polarization [CITATION] which provides a natural framework to study IQH invariants.', '1311.0882-2-4-5': 'Based on this analysis we explicitly show that the Maryland model sits at the critical point of a quantum phase transition to the topologically trivial state.', '1311.0882-2-4-6': 'The criticality of the Maryland model allows us to associate topological invariants to it in a purely mathematical sense using the limiting procedure along the deformation path in the parameter space.', '1311.0882-2-4-7': 'We show that even though the 1D gaps are preserved throughout the deformation from AAH to MM, the energy gaps in the equivalent 2D model close at the TQPT, as required by general considerations.', '1311.0882-2-4-8': 'We discuss the consequences of this result for the topological classification of 1D QC families [CITATION].', '1311.0882-2-5-0': 'We consider a 1D tight binding chain of size [MATH] with an onsite potential modulation [MATH],', '1311.0882-2-6-0': 'H(,)=-_n=1^N-1 t (c^_n+1c_n+c^_n c_n+1)-_n=1^N V_n(,)c^_nc_n,', '1311.0882-2-7-0': 'V_n(,)=2[MATH]((2n b+2)1+ (2n b+)), where [MATH] and [MATH] are creation and annihilation operators on the site [MATH], [MATH], [MATH], [MATH], and [MATH] is the amplitude of the nearest neighbour hopping.', '1311.0882-2-7-1': 'The onsite potential, [MATH], is characterized by the strength [MATH], period [MATH] and the phase parameter [MATH].', '1311.0882-2-7-2': 'The parameter [MATH] interpolates between the limiting cases AAH ([MATH]) and MM ([MATH]), [EQUATION]', '1311.0882-2-7-3': 'This general onsite potential is a smooth function of [MATH] in the open interval [MATH].', '1311.0882-2-7-4': '[MATH] has singularities at [MATH] corresponding to the integrable MM which we approach asymptotically in a limiting sense and we define TQPT in terms of these singularities.', '1311.0882-2-7-5': '[MATH] is a specific example of a generic [MATH] periodic onsite potential [MATH], where [MATH] is an analytic function everywhere except in the limit of singular MM where it acquires isolated poles.', '1311.0882-2-8-0': '2D ancestor:- Taking an inverse Fourier transform with respect to [MATH] results in a real space lattice which is the 2D Hofstadter model with a flux [MATH] per unit cell.', '1311.0882-2-8-1': 'The same idea applies to the Hamiltonian in Eq. ([REF]),', '1311.0882-2-9-0': 'H_2D()=_-^dH(,)e^im,', '1311.0882-2-10-0': 'c_nc_n()=12_me^i mc_n,m,', '1311.0882-2-11-0': 'where [MATH].', '1311.0882-2-11-1': 'The resulting real space 2D Hamiltonian equivalent to [MATH] (up to a constant energy shift) reads, [EQUATION] where [EQUATION] describes the hopping amplitude from site [MATH] to [MATH], i.e. a hopping of range [MATH] term is the constant shift in the onsite energy).', '1311.0882-2-11-2': 'Note that [MATH] in Eq. ([REF]) is defined in a limiting sense at the special points [MATH] (AAH and MM) [CITATION].', '1311.0882-2-11-3': 'Fig. [REF] plots the absolute value of the hopping amplitude [MATH] for different values of the hopping range [MATH] as a function of [MATH].', '1311.0882-2-11-4': 'In the limiting case of [MATH] (AAH) only the [MATH] term survives, [MATH], which corresponds to the nearest neighbor hopping of the Hofstadter model.', '1311.0882-2-11-5': 'As [MATH] increases, hopping terms of longer range [MATH] in the [MATH]-direction acquire non-vanishing amplitudes.', '1311.0882-2-11-6': 'In the limiting case of [MATH] (MM) the dual 2D lattice acquires long range hopping terms of arbitrarily large [MATH] in the [MATH]-direction all of equal unit amplitude, see Eq. ([REF]).', '1311.0882-2-11-7': 'This arbitrarily long range hopping singularity is indicative of a quantum phase transition occurring at the critical points [MATH].', '1311.0882-2-11-8': 'To elucidate the physical nature of these [MATH] critical points further, we analyze the band structure and the topological invariants of Eq. ([REF]) as a function of [MATH].', '1311.0882-2-12-0': 'Band structure:- We impose open boundary conditions on the 1D tight binding Hamiltonian, [MATH], in Eq. ([REF]) and numerically diagonalize it for the system size of [MATH] sites.', '1311.0882-2-12-1': 'It is instructive to plot the resulting energy bands as a function of the phase parameter [MATH], which captures the 2D band structure in the hybrid space [MATH].', '1311.0882-2-12-2': 'We start with the case of a commensurate modulation by setting [MATH] and [MATH].', '1311.0882-2-12-3': 'Fig. [REF] shows the resulting band structure as a function of the phase parameter [MATH] for four different values of [MATH].', '1311.0882-2-12-4': 'The case of AAH ([MATH]), top left panel of Figure [REF], demonstrates a well defined set of gaps reflecting the robust integer quantum Hall topology of the 2D Hofstadter model.', '1311.0882-2-12-5': 'For [MATH] and [MATH], the band gaps gradually decrease.', '1311.0882-2-12-6': 'All band gaps close (scale to zero with the system size) precisely at the critical point [MATH] (as explicitly shown using the exact spectrum [CITATION]).', '1311.0882-2-12-7': 'The gapless nature of the 2D spectrum for the MM case ([MATH]) in the hybrid space [MATH] is explicitly confirmed using the exact analytical expression for the MM spectrum with commensurate modulation [CITATION].', '1311.0882-2-13-0': 'Closing of the spectral gaps coincides with the hopping range divergence in the 2D lattice and indicates a TQPT in the system as [MATH] (i.e. at the MM point).', '1311.0882-2-13-1': 'The interesting aspect of MM is that the 2D spectrum is gapless in the reciprocal space [MATH] whereas the 1D spectrum has well defined gaps for each value of [MATH].', '1311.0882-2-13-2': 'Here [MATH] is the Fourier image of the site index [MATH].', '1311.0882-2-13-3': 'The fact that the 1D MM spectrum has gaps whereas the corresponding dual 2D spectrum is gapless makes perfect sense since the non-trivial TQPT can only exist in the 2D space.', '1311.0882-2-13-4': 'The scale invariance of the system at the transition point can also be explicityly demonstrated [CITATION].', '1311.0882-2-14-0': 'Chern number from polarization theory:- In the following, we change [MATH] from 0 to 1 and track the change in the IQH topological invariant associated with the 2D system (Eq. ([REF])) in the hybrid space [MATH].', '1311.0882-2-14-1': 'An ideal tool for this task is the polarization of the 1D chain defined in the hybrid space [CITATION].', '1311.0882-2-14-2': 'The polarization of a finite 1D insulator is given by the average charge center of the hybrid Wannier function (HWF) ([MATH]) of the system [CITATION],', '1311.0882-2-15-0': 'n()=_n n (n,)_n(n,),', '1311.0882-2-16-0': '(n,)=_occupied states n,n,, where [MATH] is the real space site index and [MATH] is the hybrid eigenstate of the system, and the angular brackets [MATH] stand for the ground state expectation value given a fixed filling factor.', '1311.0882-2-17-0': 'Non-zero Chern number is reflected in a discontinuity of [MATH] as a function of the phase (or gauge) parameter [MATH].', '1311.0882-2-17-1': 'This discontinuity is a robust feature of the IQH and was recently proposed [CITATION] as a tool to measure topological invariants directly in 2D cold atomic systems [CITATION].', '1311.0882-2-17-2': 'Note that the generalized 1D chain (Eq. ([REF])) has well defined gaps in the spectrum for any fixed [MATH] and [MATH] (including the Maryland model) which allows us to define the 1D polarization in terms of HWF centers in the whole parameter space.', '1311.0882-2-18-0': 'In Fig [REF], we plot the shift in the HWF centers for the same values of [MATH] (for [MATH]) as in Fig. [REF].', '1311.0882-2-18-1': 'We fix the filling factor (particle number per site) such that the chemical potential is in the gap above the top of the lowest band in the AAH limit ([MATH]).', '1311.0882-2-18-2': 'In the limit of AAH, the HWF center as a function of [MATH] shows a one unit cell jump corresponding to the Chern number [MATH], or, equivalently, a transfer of charge [MATH] by a distance of one unit cell as [MATH] changes by a period, reflecting topological charge pumping [CITATION].', '1311.0882-2-18-3': 'We monitor this jump (invariant) as we deform AAH ([MATH]) to MM ([MATH]) keeping the filling factor fixed.', '1311.0882-2-18-4': 'Note that the polarization jump corresponding to the topological charge transfer survives in the MM limit [MATH], see Fig. [REF] bottom right.', '1311.0882-2-18-5': 'It may seem paradoxical at first that we can associate a Chern number with a gapless system.', '1311.0882-2-18-6': 'The limiting procedure [MATH] allows to project on to the states that are connected to the a topological band defined for [MATH].', '1311.0882-2-18-7': 'Note that the topology is not robust as any infinitesimal perturbation may mix the states thereby violating the quantization of the topological response.', '1311.0882-2-18-8': 'Such behavior is expected of a critical phase at [MATH] on general grounds.', '1311.0882-2-18-9': 'Note the additional discontinuities appearing in HWF shift [MATH] in the case of MM, Fig. [REF] bottom right arise due to the divergent onsite potential effectively breaking the system up into smaller subsystems coupled by tunneling.', '1311.0882-2-19-0': "Topological classification of 1D quasicrystals:- Families of 1D incommensurate tight binding models manifest a special 'quasiperiodic' translational invariance: an arbitrary shift in the phase [MATH] can be compensated by a shift along the chain [MATH].", '1311.0882-2-19-1': 'Note that this is true only at irrational values of [MATH] since only in this case [MATH] forms a dense set mod [MATH].', '1311.0882-2-19-2': 'It has been argued [CITATION] that this quasiperiodic translational invariance allows one to assign the same Chern number to each member of the family of QCs, i.e. for each value of the phase parameter [MATH].', '1311.0882-2-19-3': 'This interpretation has been challenged in Ref. [CITATION].', '1311.0882-2-19-4': 'The quasiperiodic translation symmetry is preserved in the case of the Maryland model ([MATH]) which sits exactly at the critical point of a 2D TQPT.', '1311.0882-2-20-0': 'In Fig. [REF] we plot the band structure and the change in the polarization as a function of the phase [MATH] for the incommensurate AAH and MM.', '1311.0882-2-20-1': 'We choose the flux fraction to be a truncated Liouville constant (Liouville numbers are irrational numbers infinitely close to rational numbers).', '1311.0882-2-20-2': 'Note that the finite size Maryland model still demonstrates the presence of a non-zero Chern number in the same restricted sense as we found for the commensurate case.', '1311.0882-2-20-3': 'The constant slope of [MATH] in Fig. ([REF]) manifests the constant Berry curvature (as a function of [MATH]), see [CITATION] for details.', '1311.0882-2-20-4': "The latter being a signature of the 'quasiperiodic' translation invariance as noted by Kraus et.", '1311.0882-2-20-5': 'al in Ref. [CITATION].', '1311.0882-2-20-6': 'Remarkably, the spectrum is gapped in the incommensurate (Liouville) 1D model Eq. ([REF]) (for fixed [MATH]) for [MATH] and forms a dense set for [MATH] (rather than a continuous set), whereas the equivalent 2D model becomes gapless as we approach critical points [MATH].', '1311.0882-2-20-7': 'The details of the 1D spectrum depend on the type of the irrational number [MATH] however at no value the spectrum becomes absolutely continuous [CITATION].', '1311.0882-2-20-8': 'Within the class of 1D models with quasiperiodic symmetry, Maryland model manifests a 2D topological phase transition as a function of the deformation parameter [MATH] which can only be realized by sweeping the phase [MATH] (see Fig. [REF]).', '1311.0882-2-21-0': 'Conclusion:- We have identified a previously unknown topological feature of the Maryland model introduced in Refs. [CITATION] in the context of Anderson localization and kicked quantum rotor studies.', '1311.0882-2-21-1': 'We show that this model represents a topological quantum phase transition point in a class of corresponding 2D lattice models with IQH topology.', '1311.0882-2-21-2': 'The criticality allows us to associate topological invariants with the Maryland model in a restricted mathematical sense at the special filling factors that are adiabatically connected to the spectral gaps in the 1D Aubry-Andre-Harper model.', '1311.0882-2-21-3': 'Our theory presented in this work establishes deep and unexpected mathematical connections between 2D topological models and a family of 1D incommensurate localization models.'}

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[]

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[]

['1311.0882-1-2-2', '1311.0882-1-6-0', '1311.0882-1-7-0', '1311.0882-1-9-2', '1311.0882-1-10-0', '1311.0882-1-11-0', '1311.0882-1-16-3', '1311.0882-1-17-0', '1311.0882-1-18-0', '1311.0882-1-22-4', '1311.0882-1-23-0', '1311.0882-2-2-2', '1311.0882-2-5-0', '1311.0882-2-6-0', '1311.0882-2-8-1', '1311.0882-2-9-0', '1311.0882-2-10-0', '1311.0882-2-11-0', '1311.0882-2-14-2', '1311.0882-2-15-0', '1311.0882-2-20-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1311.0882

1901.08956

{'1901.08956-1-0-0': 'It is now widely recognized that the Shannon measure of information is a fundamental tool that can be employed across many fields.', '1901.08956-1-0-1': 'The quantum mechanical notion of entropy created by von Neumann is a measure of the purity of a quantum state described by a density matrix.', '1901.08956-1-0-2': 'Jaynes applied information theoretic entropy to both statistical mechanics and quantum theory using the principle of maximum entropy.', '1901.08956-1-0-3': 'We apply the Shannon measure, as developed by Ben-Naim, to the general problem of characterizing the information about a particular observable that is missing from a quantum state.', '1901.08956-1-0-4': 'The quantum operator entropy of a pure or mixed quantum state is the Shannon measure on the set of probabilities for the possible results when measuring the associated observable.', '1901.08956-1-0-5': 'The von Neumann entropy is then the special case where the quantum operator is the density matrix itself.', '1901.08956-1-0-6': 'We examine how these distinct entropies vary in three example systems: unitary expansion into unoccupied states, a model thermal system in equilibrium, and transients in unitary quasi-adiabatic switching between two states.', '1901.08956-1-0-7': 'The entropies corresponding to each operator quantify missing information about the corresponding observable.', '1901.08956-1-1-0': '# Introduction', '1901.08956-1-2-0': 'Entropy as a concept was defined in historical order by Clausius, Boltzmann, Gibbs, von Neumann, and Shannon.', '1901.08956-1-2-1': 'Conceptually, it would have been clearer if the order were reversed.', '1901.08956-1-2-2': 'The Shannon definition is the most fundamental, which could then be applied to physical quantum systems, with classical statistical mechanics following as an approximation to the quantum case.', '1901.08956-1-3-0': 'Shannon chose to use the word "entropy" from the field of statistical mechanics for a quantity he variously described as measuring "choice," "information", "uncertainty", or "surprise" [CITATION].', '1901.08956-1-3-1': 'As the mathematical theory of communication he invented became the field of information theory, and in due course was turned back onto analyzing statistical mechanics, the layering of these various concepts often became confusing.', '1901.08956-1-3-2': 'Information and uncertainty, for example, seem to be opposite one another.', '1901.08956-1-3-3': 'The more information one has, the less uncertainty.', '1901.08956-1-3-4': 'Whose choice is involved in the entropy of a physical system?', '1901.08956-1-4-0': 'Ben-Naim has done the world a great favor in relentlessly clarifying the quantity defined by Shannon as a measure of the missing information associated with a probability distribution [CITATION].', '1901.08956-1-4-1': 'If all one knows is the probability distribution for a finite set of discrete possible events, the Shannon measure quantifies the amount of information, measured in bits, that one is missing.', '1901.08956-1-4-2': 'In that sense, it is the difference between the incomplete knowledge captured in a probability distribution, and certainty about which event will occur.', '1901.08956-1-4-3': 'What Ben-Naim prefers to call the Shannon Measure of Information (SMI) (or Shannon Missing Information), represents, as it were, the part of the graduated cylinder that is empty of fluid.', '1901.08956-1-4-4': 'The SMI is the more general concept; the thermodynamic entropy [MATH] is a special case of the SMI applied to a particular class of physical problems.', '1901.08956-1-4-5': 'The Shannon measure quantifies how much remains to be specified, given a partial specification by probabilities.', '1901.08956-1-4-6': 'Ben-Naim also rightly inveighs against interpreting physical entropy as a measure of "disorder," a concept too vague to be scientifically quantifiable.', '1901.08956-1-4-7': 'What counts as order is entirely subjective.', '1901.08956-1-5-0': 'The work of Jaynes was a major breakthrough in reversing the historical sequence to make more conceptual sense of physical entropy [CITATION].', '1901.08956-1-5-1': 'He showed how both classical and quantum statistical mechanics could be grounded in the Shannon concept of entropy, using his principle of maximum entropy to find the optimal probability distribution, given some limited information.', '1901.08956-1-5-2': 'Here we apply the Shannon measure to quantify the missing information about an observable in a system described by a quantum state.', '1901.08956-1-5-3': "This is aided enormously by returning again and again to Ben-Naim's insistence that what entropy measures is in fact missing information.", '1901.08956-1-5-4': 'The approach taken here is somewhat pedagogical in reviewing the Shannon and Jaynes formulation, using the Ben-Naim lens.', '1901.08956-1-5-5': 'The expert reader is asked to be tolerant of this, because the interpretation of the quantum operator entropy is aided by carefully stating these results in this language.', '1901.08956-1-6-0': 'The main result of the current work is to expand the definition of entropy by associating with any quantum operator [MATH] that corresponds to an observable a related entropy [MATH].', '1901.08956-1-6-1': 'This entropy quantifies the amount of missing information regarding the property [MATH] in the pure or mixed quantum state.', '1901.08956-1-6-2': 'For example, the position operator [MATH] generates an associated entropy [MATH] which captures how much information about position is missing.', '1901.08956-1-6-3': 'The familiar von Neumann entropy is then seen to be the special case when the observable is the density matrix [MATH].', '1901.08956-1-6-4': 'Though the von Neumann entropy is constant under unitary time evolution, other operator entropies need not be.', '1901.08956-1-6-5': 'We examine the interpretation of these results in three simple model systems: two involving isolated pure-state unitary time evolution, and one examining a steady state system in thermal equilibrium.', '1901.08956-1-7-0': 'The next section defines the Shannon measure of information in terms familiar to readers of both Jaynes and Ben-Naim.', '1901.08956-1-7-1': 'Section [REF] explains succinctly how the maximum entropy principle leads to a unique specification for the optimal probability distribution given known constraints.', '1901.08956-1-7-2': 'In Section [REF] we define quantum operator entropies for observables in both pure and mixed quantum systems.', '1901.08956-1-7-3': 'In Section [REF] we examine the different entropies for the case of unitary expansion into unoccupied states in a highly-connected system that models much more complex systems with many degrees of freedom.', '1901.08956-1-7-4': 'In this case, the quantum operator entropy for position seems to capture features in free expansion that connect to both the classical entropy and the Second Law better than does the von Neumann entropy.', '1901.08956-1-7-5': 'Section [REF] develops the Jaynes approach to equilibrium quantum statistical mechanics using the matrix calculus formalism.', '1901.08956-1-7-6': 'Those results are then employed in Section [REF] to explore how the different quantum entropies illuminate the simple problem of a quantum dot chain at thermal equilibrium .', '1901.08956-1-7-7': 'The final example, considered in Section [REF] is the time-dependent quantum problem of quasi-adiabatic two-state switching.', '1901.08956-1-7-8': 'The goal is to examine the role of each operator entropy in characterizing information missing from the quantum state in a dynamic and transient, rather than equilibrium, situation.', '1901.08956-1-8-0': '# Shannon measure of information', '1901.08956-1-9-0': "We follow Ben-Naim in defining the SMI, (measured in bits) as Shannon's measure on a probability distribution [MATH] over [MATH] possible outcomes.", '1901.08956-1-9-1': '[EQUATION]', '1901.08956-1-9-2': 'As an example, suppose the outcome of an event yields one of [MATH] numbers [EQUATION]', '1901.08956-1-9-3': 'The selection of a labeled ball out of a large urn containing very many such balls is a classic example of this sort of event.', '1901.08956-1-9-4': 'If the probability of each event is the same, we would assign probabilities [MATH], as shown in Figure [REF]a.', '1901.08956-1-9-5': 'We take probability to simply mean the quantification of incomplete knowledge about what will occur.', '1901.08956-1-9-6': 'The uniform probability distribution over the possibilities represents no knowledge of the outcome.', '1901.08956-1-9-7': 'The uniform probability shown in Figure [REF]a has an SMI of 3 bits.', '1901.08956-1-9-8': 'If all probabilities were 0 except for that corresponding to the third outcome, [MATH], we would have certainty that the outcome would be [MATH] and the SMI would be 0-no missing information.', '1901.08956-1-9-9': 'The probability distribution shown in Figure [REF]b [MATH] has an SMI of 2.67 bits.', '1901.08956-1-9-10': 'Figure [REF]c shows the case when [MATH] for which the SMI is 2.0 bits.', '1901.08956-1-9-11': 'Note that the horizontal axis correspond to the value of the outcome [MATH], not the index of the outcome [MATH].', '1901.08956-1-10-0': 'We focus here exclusively on the case with a finite number of possible outcomes-quantum mechanically a Hilbert space of finite dimension.', '1901.08956-1-10-1': 'We will not address the case of continuous degrees of freedom, nor the non-trivial question of how large to construct the underlying space; both are discussed in [CITATION].', '1901.08956-1-11-0': 'The SMI of a probability distribution can be related to the number of answers to yes/no questions (bits) it would take to extract a specific result using an optimal questioning scheme.', '1901.08956-1-11-1': 'We imagine a chooser who has selected one outcome (perhaps by drawing a ball from the urn or by making a physical measurement).', '1901.08956-1-11-2': 'The questioner knows only the probability distribution [MATH] and can ask a series of binary questions about the outcome.', '1901.08956-1-11-3': 'Consider one of the probability distributions over [MATH] possible outcomes shown in Figure [REF].', '1901.08956-1-11-4': 'A non-optimal questioning scheme would be: Is [MATH] 0?', '1901.08956-1-11-5': 'Is [MATH] 1.2?', '1901.08956-1-11-6': 'Is [MATH] 3.2, etc.', '1901.08956-1-11-7': 'On average this would take [MATH] questions to yield the result.', '1901.08956-1-11-8': 'For the uniform distribution the optimal strategy would be a binary search, making a sequence of bipartitions of the set of possibilities.', '1901.08956-1-12-0': 'Is [MATH] in [MATH]?', '1901.08956-1-12-1': '(Assume the answer is "yes.")', '1901.08956-1-12-2': 'Is [MATH] in [MATH]?', '1901.08956-1-12-3': '(Assume the answer is "no.")', '1901.08956-1-12-4': 'Is [MATH]?', '1901.08956-1-12-5': '(If it is not, then [MATH] is [MATH].)', '1901.08956-1-13-0': 'Thus for the uniform probability case, 3 questions will always suffice.', '1901.08956-1-13-1': 'Three bits of information are missing-the SMI is 3.', '1901.08956-1-14-0': 'For the probabilities shown in Figure [REF]c, there is an 80% chance that the result for [MATH] is either 4.75 or 6.1.', '1901.08956-1-14-1': 'The logical first question is then: Is the result one of these two?', '1901.08956-1-14-2': 'Most of the time it will be, and the questioner will only have to ask one more question to get the result.', '1901.08956-1-14-3': 'So the number of questions needed is 2.', '1901.08956-1-14-4': 'But, of course, sometimes the questioner will be less lucky and have to ask more questions to deduce which of the remaining 6 options the chooser has picked.', '1901.08956-1-14-5': 'The optimal strategy at each stage is to create a bipartition of the remaining possibilities that has the largest SMI.', '1901.08956-1-14-6': 'This yields the maximal information "payoff" from the answer to each question.', '1901.08956-1-14-7': '(A more detailed description of the optimal algorithm is given in reference [CITATION].)', '1901.08956-1-14-8': 'Over 10,000 simulations, with with the same probability distribution, the average number of questions required was 2.30.', '1901.08956-1-14-9': 'The general result is that the average number of yes/no questions will be between the SMI and the SMI+1.', '1901.08956-1-14-10': 'For large sets, this will become essentially equal to the SMI.', '1901.08956-1-15-0': 'The Shannon measure is then quantifying the information that is missing from the probability distribution, which could be extracted given the answers to a number of questions equal to the SMI (or more precisely in the interval [SMI, SMI+1]).', '1901.08956-1-15-1': 'This is the number of missing bits of information.', '1901.08956-1-16-0': '# The Jaynes maximum entropy principle', '1901.08956-1-17-0': 'If we know nothing about an outcome, it makes sense to assign equal probabilities to all the possibilities.', '1901.08956-1-17-1': 'But what if we know [MATH], just not enough to narrow down the possibilities completely.', '1901.08956-1-17-2': 'Jaynes argued that the only unbiased way to assign the probabilities was to find the distribution that maximizes the SMI, subject to the constraints that captured what one does know.', '1901.08956-1-17-3': 'To do otherwise is to implicitly assume knowledge about the events one does not really have.', '1901.08956-1-18-0': 'Suppose we want to choose the set of probabilities [MATH] for outcomes [MATH] which maximize [MATH] subject to the constraints (a) that the probabilities sum to one, and (b) that the average value of [MATH] is known to be [MATH].', '1901.08956-1-18-1': '[EQUATION] [EQUATION]', '1901.08956-1-18-2': 'We use the method of Lagrange multipliers and form the Lagrangian: [EQUATION]', '1901.08956-1-18-3': 'For convenience (and with forethought) we are defining the Lagrange multipliers as [MATH] and [MATH], and use here the natural logarithm.', '1901.08956-1-18-4': 'We set the derivatives of the Lagrangian to zero to find the extrema.', '1901.08956-1-18-5': 'The derivatives with respect to [MATH] and [MATH] just reproduce the constraint equations.', '1901.08956-1-18-6': 'Setting the derivatives with respect to [MATH] to zero yields [EQUATION]', '1901.08956-1-18-7': 'Applying the normalization condition, we have [EQUATION] therefore [EQUATION]', '1901.08956-1-18-8': 'So we get a result with the familiar Boltzmann form.', '1901.08956-1-18-9': '[EQUATION]', '1901.08956-1-19-0': '[EQUATION]', '1901.08956-1-19-1': 'The value of [MATH] is related to the value of [MATH] through the relation [EQUATION]', '1901.08956-1-19-2': 'Figure [REF] shows the resulting exponential probability distribution for the 8 outcomes of [MATH] in ([REF]) corresponding to different values of the average [MATH].', '1901.08956-1-19-3': 'For [MATH], shown in Figure [REF]a, and [MATH], shown in Figure [REF]b, the probabilities have the familiar Boltzmann-type exponential behavior and [MATH] is positive.', '1901.08956-1-19-4': 'A uniform distribution corresponds to [MATH].', '1901.08956-1-19-5': 'To achieve higher [MATH] values, [MATH] becomes negative.', '1901.08956-1-19-6': 'The relationship between a given [MATH] and the correspond value of [MATH], from ([REF]), is shown graphically in Figure [REF].', '1901.08956-1-19-7': 'An important point here is that the Boltzmann profile results from just asking the question: What is the probability distribution which maximizes the SMI and has a specified [MATH].', '1901.08956-1-20-0': 'To derive all the basic results of classical statistical mechanics we simply consider the possible energies [MATH] of a physical system which is in equilibrium and thermal contact with a reservoir at temperature [MATH].', '1901.08956-1-20-1': 'We identify [MATH], and the constraint is that the average energy is the same as that of the reservoir, [MATH].', '1901.08956-1-20-2': 'The value of the equilibrium thermodynamic entropy [MATH] is a constant times the amount of missing information in the probability distribution that maximizes the SMI, given the constraint that the average energy is U .', '1901.08956-1-20-3': '[EQUATION]', '1901.08956-1-20-4': 'The free energy [MATH] is then defined to be [EQUATION] yielding the familiar [EQUATION]', '1901.08956-1-20-5': 'The development of standard classical thermodynamics from this information theoretic basis was the original breakthrough of Jaynes [CITATION] and is described in references [CITATION] and [CITATION].', '1901.08956-1-21-0': '# Entropies of Quantum Operators', '1901.08956-1-22-0': 'To apply the Shannon measure of information in the quantum mechanical case we consider a Hermitian operator representing the observable [MATH] written in its eigen-basis.', '1901.08956-1-22-1': '[EQUATION]', '1901.08956-1-22-2': 'We assume that the set of states [MATH] form an orthonormal basis.', '1901.08956-1-22-3': 'If a measurement of [MATH] is made, the result will be one of the eigenvalues of [MATH] with probabilities [MATH].', '1901.08956-1-22-4': 'It is natural then to define the Shannon measure on this set of probabilities as the entropy associated with [MATH].', '1901.08956-1-22-5': '[EQUATION]', '1901.08956-1-22-6': 'Pure states.', '1901.08956-1-22-7': 'If the system is in a pure quantum state [MATH], then the probability that a measurement of [MATH] yields [MATH] is given by the Born Rule.', '1901.08956-1-22-8': '[EQUATION]', '1901.08956-1-22-9': 'So that [EQUATION]', '1901.08956-1-22-10': 'What information is missing?', '1901.08956-1-22-11': 'For a pure state [MATH] measures the number of bits of information that are missing from the universe concerning what value of [MATH] will be measured.', '1901.08956-1-22-12': 'The quantum state of the system [MATH] contains everything there is to know about the system at time [MATH].', '1901.08956-1-22-13': 'Because of fundamental quantum indeterminism, that is not enough to pin down which eigenvalue of [MATH] will be measured (unless, of course, [MATH] happens to be an eigensate of [MATH]).', '1901.08956-1-22-14': 'We now know from recent Bell test experiments that this indeterminism is a fundamental feature of the reality [CITATION].', '1901.08956-1-22-15': 'It is not just a feature of quantum mechanics as we currently understand it, nor is it just an expression of the limited information of an observer.', '1901.08956-1-22-16': 'The result of a measurement is fundamentally underdetermined by the the state of the universe immediately prior to the measurement being made.', '1901.08956-1-22-17': 'The missing information about [MATH] in ([REF]) is really missing.', '1901.08956-1-23-0': 'Mixed states.', '1901.08956-1-23-1': 'For a pure quantum state the density operator corresponding to [MATH] is simply [EQUATION]', '1901.08956-1-23-2': 'We consider a system [MATH] which is either coupled dynamically to a reservoir system [MATH] or has been so in the past.', '1901.08956-1-23-3': 'The quantum state of the composite system is not in general simply a direct product of the state of each subsystem but rather an entangled state.', '1901.08956-1-23-4': 'If system [MATH] has a complete basis set [MATH] and the (reservoir) system [MATH] has a basis set [MATH], we can write the density operator for the pure composite system [EQUATION]', '1901.08956-1-23-5': 'The matrix elements of the reduced density matrix for system [MATH] alone is [EQUATION] or more compactly [EQUATION] where we now omit the superscript [MATH].', '1901.08956-1-23-6': 'A density operator that cannot be written in the form of ([REF]) is said to represent a "mixed state."', '1901.08956-1-24-0': 'The density operator can be written in the basis of its own eigenstates.', '1901.08956-1-24-1': '[EQUATION]', '1901.08956-1-24-2': "For a pure state only one of the [MATH]'s is nonzero.", '1901.08956-1-25-0': 'The probability [MATH] that a measurement of [MATH] for the system yields [MATH] can be calculated for the mixed state using the density operator and the projection operator onto the [MATH] eigenstate of [MATH].', '1901.08956-1-25-1': '[EQUATION]', '1901.08956-1-25-2': 'The quantum operator entropy for a mixed (or pure) state is then given by applying ([REF]) to ([REF]).', '1901.08956-1-25-3': '[EQUATION]', '1901.08956-1-25-4': 'The expression in ([REF]) includes the previous expression in ([REF]) as a special case when [MATH] represents a pure state.', '1901.08956-1-26-0': 'Again, [MATH] is providing a measure (in bits) of missing information.', '1901.08956-1-26-1': 'For a mixed state, the source of this missing information is two-fold.', '1901.08956-1-26-2': 'Quantum indeterminacy still limits the information about a future measurement that is present in the current state of the system.', '1901.08956-1-26-3': "But in addition there is also information missing about the reservoir's state and the mutual information characterizing the entanglement between system and reservoir.", '1901.08956-1-26-4': 'The system density matrix [MATH] is not a complete description of the quantum state of the system, but it is the best possible local description of what is known about the system.', '1901.08956-1-26-5': 'A system that is entangled with a reservoir, viewed in isolation, has no quantum state-the quantum entanglement by definition precludes an adequate local description.', '1901.08956-1-26-6': 'For the physical world, there is a fact-of-the-matter about the global quantum state that includes both system and reservoir described by ([REF]).', '1901.08956-1-26-7': 'But one who has only the local information about the system contained in [MATH] has less information.', '1901.08956-1-27-0': 'Examples of quantum operator entropies.', '1901.08956-1-27-1': 'If we use a basis set of discrete position eigenstates [MATH] (anticipating the quantum dot chain shown in Figure [REF]) we can define the quantum operator entropy for [MATH], the position operator.', '1901.08956-1-27-2': '[EQUATION]', '1901.08956-1-27-3': 'The quantum operator entropy for the Hamiltonian [MATH] with allowed energies [MATH] and eigenstates [MATH] is [EQUATION]', '1901.08956-1-27-4': 'The quantum operator entropy for the density operator [MATH] itself, from ([REF]) is [EQUATION]', '1901.08956-1-27-5': 'The value of the trace is invariant under a unitary transformation so we can transform both [MATH] and the projection operator [MATH] into the eigenbasis of the density operator.', '1901.08956-1-27-6': 'In that basis, both are diagonal, and so it is clear that [EQUATION] and we can write [EQUATION]', '1901.08956-1-27-7': 'The quantum operator entropy for the density operator is the SMI of the diagonal elements of the density matrix.', '1901.08956-1-27-8': 'We recognize ([REF]) as the average value of [MATH], we can write [EQUATION]', '1901.08956-1-27-9': 'The entropy [MATH] is identical to the von Neumann entropy (in bits).', '1901.08956-1-27-10': 'For a pure state, all but one of the [MATH] is zero, so the entropy [MATH] is zero.', '1901.08956-1-28-0': 'In general each operator entropy [MATH] can take on different values because each quantifies something different.', '1901.08956-1-28-1': 'If what is known about the system is [MATH], [MATH] is the information that is missing about position, or more precisely, about the outcome of position measurements.', '1901.08956-1-28-2': 'It is the answer to the question: How many bits of information are not known about the outcomes of a position measurement if all one knows is [MATH]?', '1901.08956-1-28-3': '[MATH] is the information that is missing about energy.', '1901.08956-1-28-4': '[MATH] is the information that is missing about which quantum state the system will be found in.', '1901.08956-1-28-5': 'It measures the "mixedness" or purity of the state.', '1901.08956-1-28-6': '[MATH], the von Neumann entropy, is invariant under unitary transformations of the basis.', '1901.08956-1-28-7': 'Other operator entropies [MATH] are not invariant-they are tied to the eigen-basis of the particular operator [MATH].', '1901.08956-1-29-0': '# Example: Entropy in unitary free expansion', '1901.08956-1-30-0': 'We consider here unitary evolution of a closed system with a time-independent Hamiltonian.', '1901.08956-1-30-1': 'Our purpose is to examine the different roles played by the von Neumann entropy [MATH], the energy entropy [MATH], and the positional entropy [MATH].', '1901.08956-1-31-0': 'The system consists of a random array of 1024 fixed sites.', '1901.08956-1-31-1': 'Each site is labeled with an index [MATH] and is at a randomly chosen position [MATH].', '1901.08956-1-31-2': 'The site positions are shown graphically in Figure [REF]a as red dots.', '1901.08956-1-31-3': 'The basis states for the system are the set of all the states localized on each site [MATH].', '1901.08956-1-31-4': 'The on-site energy for each is [MATH].', '1901.08956-1-32-0': 'Off-diagonal elements of the Hamiltonian couple each site to several nearby sites with a fixed coupling matrix element [MATH].', '1901.08956-1-32-1': 'Each site is coupled to a randomly chosen subset of its 50 nearest sites.', '1901.08956-1-32-2': 'The coupling to neighbors for six representative sites is shown in Figure [REF]a as blue lines connecting red dots.', '1901.08956-1-32-3': 'Figure [REF]b shows the connectivity of the sites with a blue dot on row [MATH], column [MATH], if site [MATH] is coupled to site [MATH].', '1901.08956-1-33-0': 'The Hamiltonian for the system is [EQUATION]', '1901.08956-1-33-1': 'Here [MATH] if two sites are connected, and 0 if they are not.', '1901.08956-1-33-2': 'The eigenvalues of [MATH] are denoted [MATH].', '1901.08956-1-34-0': 'The purpose of the randomness in this model, in both the position and connectivity of the sites, is to minimize the artifacts of geometric regularity on the dynamics.', '1901.08956-1-34-1': 'We want to see how these different entropy measure change due to fundamental unitary dynamics without the regularities of constructive and destructive interference that dominate, for example, the evolution of a similar system on a regular lattice.', '1901.08956-1-35-0': 'To construct the connections between sites, nine passes through all the sites are made, adding a connection between each site and another site randomly chosen from among its 50 closest neighbors.', '1901.08956-1-35-1': 'The result is that the number of connections for each site varies between 10 and 32, with a mean of 18.', '1901.08956-1-35-2': '(The situation is complicated by the fact that site [MATH] might have site [MATH] as one of its 50 nearest neighbors, while site [MATH] does not have site [MATH] as one of its 50 nearest neighbors.)', '1901.08956-1-36-0': 'The dynamic problem we solve is the expansion of the state from an initially spatially confined state.', '1901.08956-1-36-1': 'The initial state has an equal probability distributed among the 64 sites that are closest to the origin.', '1901.08956-1-36-2': 'We solve for the time development of the state function using the time development operator.', '1901.08956-1-36-3': '[EQUATION]', '1901.08956-1-36-4': 'The expansion of the state into the surrounding state space is shown by the snapshots of the probability density in Figure [REF].', '1901.08956-1-36-5': 'The time scale is set by the characteristic tunneling time between sites [MATH].', '1901.08956-1-36-6': 'The figure shows how the probability expands much like a classical gas into the available states.', '1901.08956-1-36-7': 'Since the expectation value of the energy is constant during unitary evolution, the system cannot de-excite, and quantum interference fluctuations persist indefinitely.', '1901.08956-1-37-0': 'Figure [REF] shows the the calculated entropies [MATH], [MATH], and [MATH], the von Neumann entropy, during the expansion shown in Figure [REF].', '1901.08956-1-37-1': 'The von Neumann entropy is, of course, constant during the unitary evolution and is in fact 0 because the state is always pure.', '1901.08956-1-38-0': 'The energy entropy [MATH] is also constant during the expansion, but it is not [MATH].', '1901.08956-1-38-1': 'The energy eigenstate occupation probabilities cannot change during unitary time development so [MATH] is independent of time.', '1901.08956-1-38-2': '[MATH] is 4.7 bits (rather than 0) because the initial state is not a Hamiltonian eigenstate, so there are many energy eigenvalues that could be measured.', '1901.08956-1-38-3': '[MATH] characterizes the missing information in the probability distribution of those energy measurements.', '1901.08956-1-39-0': 'The positional entropy [MATH] characterizes the missing information about position.', '1901.08956-1-39-1': 'At [MATH], it is exactly 6 bits because the probability is uniformly distributed among [MATH] sites.', '1901.08956-1-39-2': 'As the expansion progresses in increase to above [MATH].', '1901.08956-1-39-3': 'If the distribution were distributed completely evenly among the [MATH] sites, [MATH] would be 10.', '1901.08956-1-39-4': 'The quantum oscillations prevent that.', '1901.08956-1-39-5': 'Nevertheless, it is clear that the increase in the quantum mechanical measure [MATH] more closely resembles the increase in the classical thermodynamic entropy of the ideal gas; expanding the volume by a factor of 16 would increase the classical entropy by [MATH] bits.', '1901.08956-1-40-0': 'We should not miss the import and surprise of this result.', '1901.08956-1-40-1': 'This is a closed system with no information moving to or from the environment and the time evolution is purely unitary via equation ([REF]).', '1901.08956-1-40-2': 'The von Neumann entropy, as well as other entropic measures derived from it (e.g., linear entropy, entropy of formation) quantify the missing information about the purity of the state.', '1901.08956-1-40-3': 'The purity is unchanged by unitary time evolution so one might expect to see any entropic measure constant in time in Figure [REF].', '1901.08956-1-40-4': 'But we have generalized the notion of entropic measure to capture the missing information concerning other observables-notably here the position.', '1901.08956-1-40-5': 'The entropy of position [MATH] grows in time because position information is decreasing as the state function spreads out.', '1901.08956-1-40-6': 'The intuitive connection to the classical case of Joule expansion of an ideal gas is clear.', '1901.08956-1-41-0': '# Quantum Statistical Mechanics', '1901.08956-1-42-0': 'We now consider canonical quantum statistical mechanics behavior.', '1901.08956-1-42-1': 'The system is assumed to be in thermal contact with the much larger reservoir which is at temperature [MATH].', '1901.08956-1-42-2': 'Thermal contact means energy can flow back and forth and the average energy of the system will be the same as that of the reservoir.', '1901.08956-1-42-3': 'At equilibrium the expectation values of all observables are constant in time; any previous transient behavior is over.', '1901.08956-1-42-4': 'The system can be in a number of energy eigenstates because of the energy fluctuations from the reservoir.', '1901.08956-1-42-5': 'What probabilities should we assign to these eigenstates probabilities?', '1901.08956-1-43-0': 'Following the Jaynes approach, we will find the probability density that maximizes [MATH] (which is the von Neumann entropy [MATH]), subject to the constraints that the density operator is normalized and that the expectation value of the energy is fixed.', '1901.08956-1-43-1': 'Why should we optimize [MATH] and not another entropy?', '1901.08956-1-43-2': 'The reason is that the reservoir affects the system in two ways.', '1901.08956-1-43-3': 'Firstly, it sets the average energy of the system due to the energy flowing back and forth through the thermal contact.', '1901.08956-1-43-4': 'But secondly, at equilibrium, the system is coupled to, and hence heavily entangled with, the large reservoir.', '1901.08956-1-43-5': 'Maximum entanglement means maximum "mixed-ness" of the system, and that is what is measured by [MATH].', '1901.08956-1-43-6': 'The mixing of many quantum states would be larger still, of course, if the energy were not constrained.', '1901.08956-1-43-7': 'The maximum "mixed-ness" would entail equal probability of all energy eigenstates.', '1901.08956-1-43-8': 'It is the competition between entanglement and the average energy constraint which are being balanced.', '1901.08956-1-43-9': 'The example in Section [REF] will illustrate this balance.', '1901.08956-1-44-0': 'We use the method of Lagrange multipliers, as in Section [REF], and construct the scalar Lagrangian as a function of the density matrix [MATH] in any basis.', '1901.08956-1-44-1': '[EQUATION]', '1901.08956-1-44-2': 'As before, finding the extremum with respect to the Lagrange multipliers reproduces the constraint equations.', '1901.08956-1-44-3': 'We want to minimize with respect to every element of the density matrix.', '1901.08956-1-44-4': 'To do that it is convenient to employ the techniques of matrix calculus [CITATION].', '1901.08956-1-44-5': 'Given a function [MATH], we define the matrix derivative of the scalar function [MATH] of a matrix [MATH].', '1901.08956-1-44-6': '[EQUATION]', '1901.08956-1-44-7': 'This is understood to be the matrix whose [MATH] element is the derivative of [MATH] with respect to the matrix element [MATH].', '1901.08956-1-44-8': '[EQUATION]', '1901.08956-1-44-9': 'We need only a few results from this formalism.', '1901.08956-1-44-10': 'One can show [EQUATION]', '1901.08956-1-44-11': 'The logarithm here is the matrix logarithm, not an element-by-element logarithm.', '1901.08956-1-44-12': 'Finding the extremum of [MATH] in ([REF]) with respect to all elements of [MATH] yields the following; [EQUATION]', '1901.08956-1-44-13': 'Applying the normalizing constraint on [MATH] we find [EQUATION] so [EQUATION] and therefore ([REF]) becomes [EQUATION]', '1901.08956-1-44-14': 'We identify [MATH] as the thermodynamic partition function and identify the Lagrange multiplier [MATH] with the inverse temperature [MATH].', '1901.08956-1-45-0': 'Equation ([REF]) is fundamental for equilibrium quantum statistical mechanics.', '1901.08956-1-45-1': 'The density matrix is diagonal in the energy basis, which is also then the eigen-basis of [MATH], so [MATH].', '1901.08956-1-45-2': 'The vanishing of all off-diagonal terms in the eigen-basis (i.e., coherences) does not, however, mean that quantum correlations are necessarily absent in other bases.', '1901.08956-1-46-0': '# Example: Thermal quantum dot chain', '1901.08956-1-47-0': 'We consider a chain of [MATH] tunnel-coupled quantum dots with coupling energy [MATH], as shown schematically in Figure [REF]a.', '1901.08956-1-47-1': 'The system is in thermal equilibrium with a reservoir at temperture [MATH].', '1901.08956-1-47-2': 'Our goal here is to examine the differences between the von Neumann entropy and the positional entropy [MATH] in a steady state system at equilbrium.', '1901.08956-1-47-3': 'We use the basis of localized states [MATH] with position [MATH] (in arbitrary units).', '1901.08956-1-47-4': 'The Hamiltonian for the system is written [EQUATION]', '1901.08956-1-47-5': 'The density matrix at non-zero temperature [MATH] is calculated directly using ([REF]).', '1901.08956-1-47-6': 'The positional entropy [MATH] is calculated using ([REF]) and the von Neumann entropy [MATH] is calculated from ([REF]).', '1901.08956-1-48-0': 'Figure [REF]b-e shows the probability for each dot at various temperatures.', '1901.08956-1-48-1': 'The thermal energy in units of [MATH] is shown for each sub-figure with [MATH].', '1901.08956-1-48-2': 'At low temperatures, the von Neumann entropy is near zero, because only one state-the ground state-has appreciable occupancy.', '1901.08956-1-48-3': 'The system is nearly in a pure state.', '1901.08956-1-48-4': 'But the positional entropy [MATH] is still substantial ([MATH] at [MATH]), because the quantum ground state maintains correlations between dot occupancy.', '1901.08956-1-48-5': 'The density matrix in the position basis has substantial off-diagonal elements.', '1901.08956-1-48-6': 'While the energy is certain, there are still almost 3 bits of missing information about the results of a position measurement.', '1901.08956-1-48-7': '(The probability distribution is, of course, reminiscent of the ground state of the infinite square well.)', '1901.08956-1-49-0': 'As the temperature increases, the occupancy probability for excited states increases, as shown in Figure [REF] for the same 8-dot system.', '1901.08956-1-49-1': 'At high temperature ([MATH]) the system is nearly completely delocalized in space with [MATH] and also spread out in energy with [MATH].', '1901.08956-1-49-2': 'The off-diagonal elements of the density matrix in the position basis are now much smaller.', '1901.08956-1-49-3': 'The competition between maximization of [MATH] and the constraint on the average energy expressed by maximizing [MATH] in ([REF]) is visible here.', '1901.08956-1-49-4': 'As the temperature goes from low to high, the energy constraint is eased, the von Neuman entropy increases, and the wavefunction spreads out in both space and energy.', '1901.08956-1-49-5': 'In the high temperature limit, the density matrix becomes diagonal in space as well as energy.', '1901.08956-1-50-0': '# Example: Quasi-adiabatic two state switching', '1901.08956-1-51-0': 'We consider a very simple two-state quantum system undergoing unitary time evolution, with a time-dependent Hamiltonian.', '1901.08956-1-51-1': 'Like the free-expansion example in Section [REF], this is a completely isolated system and thermal effects play no role-just unitary evolution of a pure state.', '1901.08956-1-51-2': 'Our goal is to see the difference between various quantum operator entropies when the Hamiltonian is explicitly time-dependent.', '1901.08956-1-52-0': 'The basis states for the system are [MATH] and [MATH], which we can consider neighboring quantum dots at positions [MATH] and [MATH] (in arbitrary units).', '1901.08956-1-52-1': 'The position operator is then simply [EQUATION]', '1901.08956-1-52-2': 'State [MATH], say the left dot, has site energy [MATH] but state [MATH] has an an additional bias energy which is a function of time [MATH].', '1901.08956-1-52-3': 'We will take this bias to be a linear function in time which switches the ground state of the system from [MATH] to [MATH] during a switching time [MATH].', '1901.08956-1-52-4': 'The Hamiltonian for the system is [EQUATION] where the potential bias is [EQUATION]', '1901.08956-1-52-5': 'Here [MATH] determines the magnitude of the energy swing over the time [MATH].', '1901.08956-1-53-0': 'We calculate the unitary evolution of the isolated system by solving the von Neumann equation for the density matrix.', '1901.08956-1-53-1': '[EQUATION]', '1901.08956-1-53-2': 'The initial state is taken to be the ground state at [MATH].', '1901.08956-1-53-3': 'Figure [REF]a shows the energies of the instantaneous ground state, [MATH] and excited state [MATH] as functions of [MATH], where the characteristic time [MATH], and total switching time [MATH].', '1901.08956-1-53-4': 'The bias potential [MATH] in arbitrary units.', '1901.08956-1-53-5': 'The anti-crossing of the two energy levels due to the coupling [MATH] is clear.', '1901.08956-1-53-6': 'The dotted line shows the expectation value of the energy, which rather closely tracks the instantaneous ground state energy, indicating that the switching is gradual enough to be considered quasi-adiabatic.', '1901.08956-1-53-7': 'The degree of adiabaticity of the switching can be characterized by the adiabatic parameter [MATH]: [EQUATION]', '1901.08956-1-53-8': 'As [MATH] becomes larger, the process becomes more and more adiabatic.', '1901.08956-1-53-9': 'The excess energy deposited in the system will be exponentially reduced with larger [MATH] [CITATION].', '1901.08956-1-53-10': 'Here we have chosen a slightly fast switching speed which for which [MATH], precisely so that some non-adiabaticity is apparent.', '1901.08956-1-54-0': 'Figure [REF]b shows the state occupation probabilities [MATH] and [MATH] as functions of time.', '1901.08956-1-54-1': 'The switching from state [MATH] to state [MATH] is fairly smooth, but some quantum oscillations are still visible.', '1901.08956-1-54-2': 'These oscillations would be smaller if [MATH] were longer, or [MATH] were larger, or [MATH] were smaller.', '1901.08956-1-54-3': '(Halving both [MATH] and [MATH] would give curves with exactly the same shape.)', '1901.08956-1-54-4': 'As it is, it is clear that the system is having some trouble keeping up with the ramping of [MATH].', '1901.08956-1-55-0': 'Figure [REF]c shows the calculated positional entropy [MATH], the energy entropy [MATH], and the von Neumann entropy, during the switching event.', '1901.08956-1-55-1': 'Because the time-evolution in ([REF]) is unitary, the initially pure state is always pure and the von Neumann entropy remains zero.', '1901.08956-1-56-0': 'Initially the system is in state [MATH], which is also an energy eigenstate so [MATH].', '1901.08956-1-56-1': 'As the crossing is approached, the dynamics mixes in some of the excited state.', '1901.08956-1-56-2': 'When the energy levels cross, the quantum state is predominantly (but not entirely) in the symmetric state [MATH].', '1901.08956-1-56-3': 'The small admixture of the excited, antisymmeteric state [MATH] accounts for the increase in [MATH].', '1901.08956-1-56-4': 'Though [MATH] remains small in this example, it is clearly nonzero and exhibits the characteristic quantum oscillations.', '1901.08956-1-57-0': 'The positional entropy increases more dramatically as the crossing point is approached and [MATH] bit when [MATH].', '1901.08956-1-57-1': 'At this time, the occupancy probability for each dot is [MATH], as seen in Figure [REF]b.', '1901.08956-1-57-2': 'The wavefunction has expanded to include both available states.', '1901.08956-1-57-3': 'This is reminiscent of the expansion, and corresponding increase in positional entropy we observed in Section [REF].', '1901.08956-1-57-4': 'But now as time moves forward that entropy increase is reversed and [MATH] drops down toward zero again.', '1901.08956-1-57-5': 'Why is the entropy not always increasing?', '1901.08956-1-57-6': 'Because the Hamiltonian is time-dependent and the ramping energy of state [MATH] "pushes" the system back into a more localized state.', '1901.08956-1-57-7': 'This is analogous to pistons in the classical examples re-compressing an expanded gas.', '1901.08956-1-57-8': 'Indeed one could easily do the comparable process with the model in Section [REF]; but this simple model is adequate to see the behavior.', '1901.08956-1-58-0': '# Discussion', '1901.08956-1-59-0': 'The application of Shannon information theory to statistical mechanics achieved by Jaynes was a landmark step.', '1901.08956-1-59-1': 'The maximum entropy principle, when applied to the density matrix with the constraint of a given energy average, produces the Boltzmann expression ([REF]) for the density operator that is the starting point for quantum statistical mechanics.', '1901.08956-1-59-2': 'Thermal contact with a large reservoir sets the average energy and destroys coherences between energy (or density matrix) eigenstates.', '1901.08956-1-60-0': 'But it is helpful to understand that the von Neumann entropy is special case of the more general process of using the Shannon information theoretic measure to determine how much information about a specific observable is missing from the a state specificated by [MATH] (pure or mixed).', '1901.08956-1-60-1': 'The quantum operator entropy [MATH] quantifies how many bits of information about [MATH] are missing from [MATH].', '1901.08956-1-60-2': 'The von Neumann entropy is special in that it is the case when the observable [MATH] is the density matrix itself, [MATH].', '1901.08956-1-61-0': 'This is not the same as simply taking the Shannon measure of the diagonal values of the density matrix in an arbitrary representation.', '1901.08956-1-61-1': 'Jaynes dismissed proposals by others to define the quantity [EQUATION] as the entropy because it depended on the representation chosen.', '1901.08956-1-61-2': '[CITATION].', '1901.08956-1-61-3': 'This is quite right.', '1901.08956-1-61-4': 'Not just any basis set will do.', '1901.08956-1-61-5': 'To be meaningful as a measure of (missing) information about an actual observable [MATH], the states must be an orthonormal set of eigenstates of [MATH].', '1901.08956-1-61-6': 'Moreover, there are different entropies associated with different quantities and we do not expect that in general the entropy associated with different observables will be equal.', '1901.08956-1-62-0': 'Having different entropy measures allows us to capture different aspects of the dynamics.', '1901.08956-1-62-1': 'We have seen in Section [REF] that the positional entropy [MATH] exhibits the expansion and entropy increase that is a feature of large systems, yet without resort to a course-graining procedure.', '1901.08956-1-62-2': 'The very simple model exhibits a basic feature of highly-coupled systems under unitary time evolution-namely, the probability spreads out.', '1901.08956-1-62-3': 'It is present in this simple unitary pure-state evolution but it is not captured by the von Neumann measure of entropy.', '1901.08956-1-62-4': 'It is captured by the positional entropy [MATH].', '1901.08956-1-62-5': 'The reason is simply that they measure different things- overall state purity vs. spread among positional eigenstates.', '1901.08956-1-62-6': 'This is surely an example of how dynamics stands behind the increase in entropy predicted by the Second Law.', '1901.08956-1-62-7': 'The Second Law must ultimately be a feature of physical dynamics-there is no law of nature that says things move from less probable states to more probable states.', '1901.08956-1-62-8': 'Rather, there is the Schrodinger equation applied to the relevant (perhaps very large) state space.', '1901.08956-1-62-9': 'If work is not being done on the system (as it is in Section [REF]), then unitary Schrodinger dynamics causes the probability density to spread out to available states and the information about what state the system is in decreases.', '1901.08956-1-62-10': 'The missing information increases.', '1901.08956-1-62-11': 'The amount of the increase is basis-dependent because the observable in question defines a specific basis.', '1901.08956-1-62-12': 'This is a feature of the Schrodinger equation (as well as most master equations).', '1901.08956-1-63-0': 'Focusing on entropy as missing information forces consideration of (a) what is the information about and, (b) for whom (or what) is the information missing.', '1901.08956-1-63-1': 'Does the entropy represent an objective property of the physical system, or a property of the knowledge state of an observer?', '1901.08956-1-64-0': 'For a dilute classical ideal gas, the thermodynamic entropy (in bits) is given by the Sakur-Tetrode equation [CITATION].', '1901.08956-1-64-1': '[EQUATION]', '1901.08956-1-64-2': 'Ben Naim has shown how to derive this result using Shannon information theory and the Jaynes maximum entropy principle [CITATION].', '1901.08956-1-64-3': 'What is the missing information in [MATH] about, and for whom is it missing?', '1901.08956-1-64-4': 'The missing information is, of course, about the exact microstate of all the particles in the gas.', '1901.08956-1-64-5': 'It is missing for anyone who knows only the values of the number of particles, the volume, and the temperature of the gas, [MATH], [MATH], and [MATH], but nothing else about the instantaneous microstate.', '1901.08956-1-64-6': 'If one had more information about the microstate of this particular gas at this moment in time, one would compute a different entropy.', '1901.08956-1-64-7': 'In this sense perhaps, Shannon\'s notion of entropy as "uncertainly" applies-it is the uncertainty of the observer about the microstate.', '1901.08956-1-64-8': 'In what sense is this entropy a property of the gas itself?', '1901.08956-1-64-9': 'Only in this very limited sense: [MATH] does quantify how many microstates of the gas are compatible with the macroscopic variables, and one could (arguably) consider that number a feature of the gas.', '1901.08956-1-64-10': 'That number is roughly [MATH], which for a liter of argon at S.T.P. is about [MATH], far more microstates than will visited by a liter of gas in the lifetime of the universe.', '1901.08956-1-64-11': 'The entropy quantifies a vast amount of missing information.', '1901.08956-1-64-12': 'But that information is not missing from the classical gas itself, which is in a specific microstate at a particular time.', '1901.08956-1-65-0': 'We now know as an experimental fact that for a pure quantum state, the situation is fundamentally different.', '1901.08956-1-65-1': 'Because of the recent suite of Bell test experiments, [CITATION], we know that the information about the result of a measurement of an observable [MATH] is not contained in the physical world at the instant just prior to the measurement being made.', '1901.08956-1-65-2': 'Quantum measurements do not (in general) disclose a pre-existing value.', '1901.08956-1-65-3': 'The quantum operator entropy [MATH] therefore reflects information that is about the observable Q, and is information missing from the physical world.', '1901.08956-1-65-4': "It is in that strange sense, an objective property of the physical system and certainly not a property of the observer's knowledge.", '1901.08956-1-65-5': 'Perhaps one can speak of Shannon\'s notion of "choice" applying here.', '1901.08956-1-65-6': 'When confronted with a measurement of [MATH], the physical world has [MATH] bits of choice that are unconstrained by the physical law which has prescribed the present [MATH].', '1901.08956-1-66-0': 'For mixed states, the information missing that is quantified by [MATH] is certainly information about potential measurements of the observable [MATH].', '1901.08956-1-66-1': 'But the missing information is a subtle combination of fundamental quantum information missing from the physical system and information the observer is missing.', '1901.08956-1-66-2': 'The observer does not have detailed information which is physically present in the reservoir, to which the system is (or has been) coupled, nor that information which is contained in the quantum correlations that remain between reservoir and system.', '1901.08956-1-66-3': 'The density matrix could be said to represent the best possible strictly local account of the state of the system.', '1901.08956-1-66-4': 'But of course, this account is one constructed by the observer.', '1901.08956-1-66-5': 'The system does not really have a local state-it is entangled with the reservoir.', '1901.08956-1-66-6': 'The eigenvalues of the density matrix [MATH] give probabilities for the outcomes of projective measurements of the system onto the eigenstates [MATH].', '1901.08956-1-66-7': 'But such a measurement should not be interpreted as simply disclosing which one of those states the system was already in.', '1901.08956-1-67-0': 'Quantifying missing information, of various sorts, with the Shannon entropy measure can bring clarity to both classical and quantum notions of entropy.'}

{'1901.08956-2-0-0': 'For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant.', '1901.08956-2-0-1': 'Yet the second law of thermodynamics, and our experience, show that entropy increases with time.', '1901.08956-2-0-2': 'Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator.', '1901.08956-2-0-3': 'These entropies characterize the missing information about a particular observable inherent in the quantum state itself.', '1901.08956-2-0-4': 'The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix.', '1901.08956-2-0-5': 'We examine pure state unitary evolution in a simple model system comprised of a set of highly interconnected states and a constant Hamiltonian.', '1901.08956-2-0-6': 'The time development is completely reversible with no loss of quantum information and no course graining is applied.', '1901.08956-2-0-7': 'The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.', '1901.08956-2-1-0': 'Entropy as a concept was defined in historical order by Clausius, Boltzmann, Gibbs, von Neumann, and Shannon.', '1901.08956-2-1-1': 'Conceptually, it would have been clearer if the order were reversed.', '1901.08956-2-1-2': 'The Shannon information-theoretic definition is the most fundamental, which can then be applied to physical quantum systems, with classical statistical mechanics following as the classical limit of the quantum case.', '1901.08956-2-2-0': 'Shannon chose to use the word "entropy" from the field of statistical mechanics for a quantity he variously described as measuring "choice," "information", "uncertainty", or "surprise" [CITATION].', '1901.08956-2-2-1': 'As the mathematical theory of communication he invented became the field of information theory, and in due course was turned back onto analyzing statistical mechanics, the layering of these various concepts often became confusing.', '1901.08956-2-2-2': 'Information and uncertainty, for example, seem to be opposite one another.', '1901.08956-2-2-3': 'The more information one has, the less uncertainty.', '1901.08956-2-2-4': 'Whose choice is involved in the entropy of a physical system?', '1901.08956-2-3-0': 'Ben-Naim has done the world a great favor by relentlessly clarifying the quantity defined by Shannon as a measure of the missing information associated with a probability distribution [CITATION].', '1901.08956-2-3-1': 'If all one knows is the probability distribution for a finite set of discrete possible events, the Shannon measure quantifies the amount of information, measured in bits, that one is missing.', '1901.08956-2-3-2': 'It is the difference between the incomplete knowledge captured in a probability distribution, and certainty about which event will occur.', '1901.08956-2-3-3': 'What Ben-Naim prefers to call the Shannon Measure of Information (SMI) (or Shannon Missing Information), represents, as it were, the part of the graduated cylinder that is empty of fluid.', '1901.08956-2-3-4': 'The SMI is the more general concept; the thermodynamic entropy [MATH] is a special case of the SMI applied to a particular class of physical problems.', '1901.08956-2-3-5': 'Ben-Naim also rightly inveighs against interpreting physical entropy as a measure of "disorder," a concept too vague to be scientifically quantifiable.', '1901.08956-2-3-6': 'What counts as order is entirely subjective.', '1901.08956-2-3-7': "Ben-Naim builds on the work of Jaynes who, reversing the historical sequence, showed specifically how statistical mechanical entropy was a particular application of Shannon's information theoretic entropy [CITATION].", '1901.08956-2-4-0': 'In this letter we focus on the quantum mechanical operator entropy [MATH], associated with a Hermitian operator [MATH], as formulated by Ingarden [CITATION].', '1901.08956-2-4-1': 'This operator entropy quantifies the amount of information about the property [MATH] that is missing in the (pure or mixed) quantum state.', '1901.08956-2-4-2': 'For example, the position operator [MATH] generates an associated entropy [MATH] which captures how much information about position is missing.', '1901.08956-2-4-3': 'The familiar von Neumann entropy is then seen to be the special case of quantum operator entropy when the operator is the density matrix [MATH].', '1901.08956-2-4-4': 'Though the von Neumann entropy is zero for a pure state and constant under unitary time evolution, other operator entropies need not be.', '1901.08956-2-4-5': 'We examine the interpretation of these results in a model system and find that entropy-increasing second law behavior is apparent even in purely unitary time evolution with no coarse-graining [CITATION] or loss of quantum information.', '1901.08956-2-5-0': "Defining quantum operator entropy.-We follow Ben-Naim in defining the SMI (Shannon Measure of Information), in bits, as Shannon's measure on a probability distribution [MATH] over [MATH] possible outcomes.", '1901.08956-2-5-1': '[EQUATION]', '1901.08956-2-5-2': 'The notation is useful here to distinguish this quantity from the physical or thermodynamic entropy [MATH]; it is simply a measure on a probability distribution.', '1901.08956-2-6-0': 'To apply the Shannon measure of information in the quantum mechanical case we consider a Hermitian operator [MATH] written in its eigen-basis.', '1901.08956-2-6-1': '[EQUATION]', '1901.08956-2-6-2': 'We assume that the set of states [MATH] are chosen to form an orthonormal basis.', '1901.08956-2-6-3': 'If a measurement of [MATH] is made, the result will be one of the eigenvalues of [MATH] with probabilities [MATH].', '1901.08956-2-6-4': 'It is natural then to define the Shannon measure on this set of probabilities as the entropy associated with [MATH].', '1901.08956-2-6-5': '[EQUATION]', '1901.08956-2-6-6': 'This quantity was introduced by Ingarden [CITATION] and has been studied by Anza and Vedral in the context of thermalization [CITATION], which is not our purpose here.', '1901.08956-2-7-0': 'If the system is in a pure quantum state [MATH], then the probability that a measurement of [MATH] yields [MATH] is given by the Born Rule.', '1901.08956-2-7-1': '[EQUATION]', '1901.08956-2-7-2': 'So that [EQUATION]', '1901.08956-2-7-3': 'Entropy quantifies missing information, so what information is missing?', '1901.08956-2-7-4': 'For an observable [MATH] and a pure state [MATH], [MATH] measures the number of bits of information that are missing from the universe concerning what value of [MATH] will be obtained if a measurement of [MATH] is made.', '1901.08956-2-7-5': 'The quantum state of the system [MATH] contains everything there is to know about the system at time [MATH].', '1901.08956-2-7-6': 'Because of fundamental quantum indeterminism, that is not enough to pin down which eigenvalue of [MATH] will be measured (unless, of course, [MATH] happens to be an eigensate of [MATH]).', '1901.08956-2-7-7': 'We now know from recent Bell test experiments that this indeterminism is a fundamental feature of reality [CITATION].', '1901.08956-2-7-8': 'It is not just a feature of quantum mechanics as we currently understand it, nor is it just an expression of the limited information of an observer.', '1901.08956-2-7-9': 'The quantum operator entropy [MATH] therefore reflects information about the observable Q that is missing from the physical world.', '1901.08956-2-7-10': "It is, in that somewhat strange sense, an objective property of the physical system and not a subjective property of the observer's knowledge.", '1901.08956-2-7-11': 'Perhaps one can speak of Shannon\'s notion of "choice" applying here.', '1901.08956-2-7-12': 'When confronted with a measurement of [MATH], the physical world has [MATH] bits of choice in the outcome that are unconstrained by the physical law, a law which has determined the present [MATH].', '1901.08956-2-7-13': 'The missing information about [MATH] in ([REF]) is really missing.', '1901.08956-2-8-0': 'A mixed state describes a system [MATH] which is either coupled dynamically to a reservoir system [MATH] or has been so in the past.', '1901.08956-2-8-1': 'The quantum state of the composite system is not in general simply a direct product of the state of each subsystem but rather an entangled state [MATH].', '1901.08956-2-8-2': 'The density operator for the composite system is defined by [EQUATION]', '1901.08956-2-8-3': 'The density operator for the system alone is defined using the partial trace over the reservoir degrees of freedom.', '1901.08956-2-8-4': '[EQUATION]', '1901.08956-2-8-5': 'We can write the density operator in the basis of its own eigenstates.', '1901.08956-2-8-6': '[EQUATION]', '1901.08956-2-8-7': "For a pure state only one of the [MATH]'s is nonzero.", '1901.08956-2-9-0': 'The probability [MATH] that a measurement of [MATH] for the system yields [MATH] can be calculated for the mixed state using the density operator and the projection operator onto the [MATH] eigenstate of [MATH].', '1901.08956-2-9-1': '[EQUATION]', '1901.08956-2-9-2': 'The quantum operator entropy for a mixed (or pure) state is then given by applying ([REF]) to ([REF]).', '1901.08956-2-10-0': 'The expression in ([REF]) includes the previous expression in ([REF]) as a special case when [MATH] represents a pure state.', '1901.08956-2-11-0': 'Again, [MATH] is providing a measure (in bits) of missing information.', '1901.08956-2-11-1': 'For a mixed state, the source of this missing information is two-fold.', '1901.08956-2-11-2': 'Quantum indeterminacy still limits the information about a future measurement that is present in the current state of the system.', '1901.08956-2-11-3': "But in addition there is also information missing about the reservoir's state and the mutual information characterizing the entanglement between system and reservoir.", '1901.08956-2-11-4': "The system's reduced density matrix [MATH] is not a complete description of the quantum state of the system, but it is the best possible local description.", '1901.08956-2-11-5': 'For the physical world, there is a fact-of-the-matter about the global quantum state that includes both system and reservoir [MATH].', '1901.08956-2-11-6': 'But the local description of the system alone represented by [MATH] has less information.', '1901.08956-2-12-0': 'Examples of quantum operator entropies.-If we use a basis set of discrete position eigenstates [MATH] we can define the quantum operator entropy for [MATH], the position operator.', '1901.08956-2-12-1': '[EQUATION]', '1901.08956-2-12-2': 'The quantum operator entropy for the Hamiltonian [MATH] with allowed energies [MATH] and eigenstates [MATH] is [EQUATION]', '1901.08956-2-12-3': 'The quantum operator entropy for the density operator [MATH] itself, from ([REF]) is [EQUATION]', '1901.08956-2-12-4': 'Because [EQUATION] we can write [EQUATION]', '1901.08956-2-12-5': 'The quantum operator entropy for the density operator is the SMI of the diagonal elements of the density matrix.', '1901.08956-2-12-6': 'We recognize ([REF]) as the expectation value of [MATH], and so write [EQUATION]', '1901.08956-2-12-7': 'The entropy [MATH] is identical to the von Neumann entropy [MATH] (in bits).', '1901.08956-2-12-8': 'For a pure state, the entropy [MATH] is zero.', '1901.08956-2-13-0': 'In general each operator entropy [MATH] can take on different values because each quantifies something different.', '1901.08956-2-13-1': 'If what is known about the system is [MATH], [MATH] is the amount of information that is missing about position, or more precisely, about the outcome of position measurements.', '1901.08956-2-13-2': 'It is the answer to the question: How many bits of information are not known about the outcome of a position measurement if all one knows is [MATH]?', '1901.08956-2-13-3': '[MATH] is the amount of information about energy that is missing.', '1901.08956-2-13-4': '[MATH] is the amount of information that is missing about which quantum state the system will be found in.', '1901.08956-2-13-5': 'It measures the "mixedness" or purity of the state.', '1901.08956-2-13-6': '[MATH], the von Neumann entropy, is invariant under unitary transformations of the basis.', '1901.08956-2-13-7': 'Other operator entropies [MATH] are not invariant-they are tied to the eigen-basis of the particular operator [MATH].', '1901.08956-2-14-0': 'Entropy change of a pure state under unitary evolution.-We consider here a model problem of unitary evolution of a pure state in a closed system with a time-independent Hamiltonian.', '1901.08956-2-14-1': 'The purpose is to examine the different roles played by the von Neumann entropy [MATH], the energy entropy [MATH], and the positional entropy [MATH].', '1901.08956-2-15-0': 'The system consists of a random array of [MATH] fixed sites.', '1901.08956-2-15-1': 'Each site is labeled with an index [MATH] and is at a randomly chosen position [MATH].', '1901.08956-2-15-2': 'The site positions are shown graphically in Figure [REF]a as red dots.', '1901.08956-2-15-3': 'We use as basis states for the system the set of all the position eigenstates localized on each site, [MATH].', '1901.08956-2-15-4': 'The on-site energy for each is [MATH].', '1901.08956-2-16-0': 'Off-diagonal elements of the Hamiltonian couple each site to several nearby sites with a fixed coupling matrix element [MATH].', '1901.08956-2-16-1': 'Each site is coupled to a randomly chosen subset of its 50 nearest sites.', '1901.08956-2-16-2': 'The coupling to neighbors for six representative sites is shown in Figure [REF]a as blue lines connecting red dots.', '1901.08956-2-16-3': 'Figure [REF]b shows the connectivity of the sites with a blue dot on row [MATH], column [MATH], if site [MATH] is coupled to site [MATH].', '1901.08956-2-17-0': 'The Hamiltonian for the system is [EQUATION]', '1901.08956-2-17-1': 'Here [MATH] if two sites are connected, and 0 if they are not.', '1901.08956-2-18-0': 'The purpose of the randomness in this model is to minimize the artifacts of geometric regularity on the dynamics.', '1901.08956-2-18-1': 'We want to see how these different operator entropies change due to fundamental unitary dynamics without the patterns of constructive and destructive interference that dominate, for example, the evolution of a similar system on a regular lattice.', '1901.08956-2-19-0': 'To construct the connections between sites, nine passes through all the sites are made, adding a connection between each site and another site randomly chosen from among its 50 closest neighbors.', '1901.08956-2-19-1': 'The result is that the number of connections for each site varies between 10 and 32, with a mean of 18.', '1901.08956-2-19-2': '(The situation is complicated by the fact that site [MATH] might have site [MATH] as one of its 50 nearest neighbors, while site [MATH] does not have site [MATH] as one of its 50 nearest neighbors.)', '1901.08956-2-20-0': 'The dynamic problem we solve is the expansion of the state from an initially spatially confined state.', '1901.08956-2-20-1': 'The initial state has an equal probability distributed among the 64 sites that are closest to the origin.', '1901.08956-2-20-2': 'We solve for the time development of the state function using the unitary time development operator.', '1901.08956-2-20-3': '[EQUATION]', '1901.08956-2-20-4': 'The expansion of the state into the surrounding state space is shown by the snapshots of the probability density in Figure [REF].', '1901.08956-2-20-5': 'The time scale is set by the characteristic tunneling time between connected sites [MATH].', '1901.08956-2-20-6': 'The figure shows how the probability expands much like a classical gas into the available states.', '1901.08956-2-20-7': 'Since the expectation value of the energy is constant during unitary evolution, the system cannot de-excite, and quantum interference fluctuations persist indefinitely.', '1901.08956-2-21-0': 'Figure [REF] shows the the calculated entropies [MATH], [MATH], and [MATH], the von Neumann entropy, during the expansion shown in Figure [REF].', '1901.08956-2-21-1': 'The von Neumann entropy is, of course, constant during the unitary evolution and is in fact 0 because the state is always pure.', '1901.08956-2-22-0': 'The energy entropy [MATH] is also constant during the expansion, but it is not [MATH].', '1901.08956-2-22-1': 'The energy eigenstate occupation probabilities cannot change during unitary time development so [MATH] is independent of time.', '1901.08956-2-22-2': '[MATH] is 4.7 bits (rather than 0) because the initial state is not a Hamiltonian eigenstate, so there are many energy eigenvalues that could be measured.', '1901.08956-2-22-3': '[MATH] characterizes the missing information in the probability distribution of those energy measurements.', '1901.08956-2-23-0': 'The positional entropy [MATH] characterizes the missing information about position.', '1901.08956-2-23-1': 'At [MATH], it is exactly 6 bits because the probability is uniformly distributed among [MATH] sites.', '1901.08956-2-23-2': 'As the expansion progresses it increase to a value between [MATH] and [MATH].', '1901.08956-2-23-3': 'If the distribution were distributed completely evenly among the [MATH] sites, [MATH] would be 10.', '1901.08956-2-23-4': 'The initial quantum confinement means that the isolated system is excited and it has no way of de-exciting.', '1901.08956-2-23-5': 'If it did, [MATH] would approach a value of 10 bits.', '1901.08956-2-23-6': 'Nevertheless, it is clear that the increase in the quantum mechanical measure [MATH] resembles the increase in the classical thermodynamic entropy of an ideal gas for which [MATH] bits.', '1901.08956-2-23-7': 'Expanding the volume by a factor of 16 would increase the classical statistical mechanical entropy by [MATH] bits.', '1901.08956-2-24-0': 'We emphasise that despite the increase in position entropy [MATH] shown in Figure [REF], the system is evolving in a completely reversible way.', '1901.08956-2-24-1': 'No information about the quantum state is being lost-there is no coarse graining in this calculation.', '1901.08956-2-24-2': 'Equation ([REF]) can be inverted so we could use the state [MATH] at any time [MATH] to reconstruct precisely the initial state [MATH].', '1901.08956-2-24-3': 'The constant purity of the state is precisely reflected in the unchanging value of the von Neumann entropy [MATH].', '1901.08956-2-24-4': 'What is changing is the amount of missing information in the quantum state about position, and precisely that is quantified by [MATH].', '1901.08956-2-25-0': 'Discussion -It is clear that the actual positions of the sites [MATH], the eigenvalues of the position operator, play a limited role in the Hamiltonian ([REF]).', '1901.08956-2-25-1': 'The positions yield a simple algorithm for determining the connectivity between the basis states-random choices from among the 50 nearest neighbors-and a way of visualizing the evolution as in Figure [REF].', '1901.08956-2-25-2': 'What matters is really just the connectivity that is shown in Figure [REF]b and the coupling strengths.', '1901.08956-2-25-3': 'For example, we could interpret each basis state [MATH] as representing a particular nuclear and electronic configuration of a set of molecules.', '1901.08956-2-25-4': 'For each configuration there are presumably a set of accessible other configurations that are dynamically coupled by the Hamiltonian, giving a sense of "nearby states."', '1901.08956-2-25-5': 'The essential point is that the increase in [MATH] is capturing quantitatively the very familiar feature of unitary evolution that the wavefunction tends to spread out into accessible states.', '1901.08956-2-25-6': 'Nevertheless, in many cases position eigenstates are especially selected for survival by decoherence through environmental entanglement [CITATION], so it is not a mistake to focus on them here.', '1901.08956-2-26-0': 'Different operator entropy measures allows us to capture different aspects of the dynamics.', '1901.08956-2-26-1': 'The von Neumann entropy [MATH] captures the "mixedness" of a state, which is constant under unitary evolution.', '1901.08956-2-26-2': 'We have seen that by contrast the positional entropy [MATH] increases in a way consistent with the second law of thermodynamics, without recourse to a course-graining procedure.', '1901.08956-2-26-3': 'The second law must ultimately be a feature of physical dynamics.', '1901.08956-2-26-4': 'There is no law of nature that says systems move from less probable states to more probable states.', '1901.08956-2-26-5': 'There is rather just the dynamics of the Schrodinger equation applied to the relevant state space.'}

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1901.08956

{'1901.08956-3-0-0': 'For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant.', '1901.08956-3-0-1': 'Yet the second law of thermodynamics, and our experience, show that entropy increases with time.', '1901.08956-3-0-2': 'Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator.', '1901.08956-3-0-3': 'These entropies characterize the missing information about a particular observable inherent in the quantum state itself.', '1901.08956-3-0-4': 'The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix.', '1901.08956-3-0-5': 'We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian.', '1901.08956-3-0-6': 'An initially confined state is subject to free expansion into available states.', '1901.08956-3-0-7': 'The time development is completely reversible with no loss of quantum information and no course graining is applied.', '1901.08956-3-0-8': 'The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.', '1901.08956-3-1-0': '# Introduction', '1901.08956-3-2-0': 'Entropy as a concept was defined in historical order by Clausius, Boltzmann, Gibbs, von Neumann, and Shannon.', '1901.08956-3-2-1': 'Conceptually, it would have been clearer if the order were reversed.', '1901.08956-3-2-2': 'The Shannon information-theoretic definition is the most fundamental, which can then be applied to physical quantum systems, with classical statistical mechanics following as the classical limit of the quantum case.', '1901.08956-3-3-0': 'Shannon chose to use the word "entropy" from the field of statistical mechanics for a quantity he variously described as measuring "choice," "information", "uncertainty", or "surprise" [CITATION].', '1901.08956-3-3-1': 'As the mathematical theory of communication he invented became the field of information theory, and in due course was turned back onto analyzing statistical mechanics, the layering of these various concepts often became confusing.', '1901.08956-3-3-2': 'Information and uncertainty, for example, seem to be opposite one another.', '1901.08956-3-3-3': 'The more information one has, the less uncertainty.', '1901.08956-3-3-4': 'Whose choice is involved in the entropy of a physical system?', '1901.08956-3-4-0': 'Ben-Naim has done the world a great favor by relentlessly clarifying the quantity defined by Shannon as a measure of the missing information associated with a probability distribution [CITATION].', '1901.08956-3-4-1': 'If all one knows is the probability distribution for a finite set of discrete possible events, the Shannon measure quantifies the amount of information, measured in bits, that one is missing.', '1901.08956-3-4-2': 'It is the difference between the incomplete knowledge captured in a probability distribution, and certainty about which event will occur.', '1901.08956-3-4-3': 'What Ben-Naim prefers to call the Shannon Measure of Information (SMI) (or Shannon Missing Information), represents, as it were, the part of the graduated cylinder that is empty of fluid.', '1901.08956-3-4-4': 'The SMI is the more general concept; the thermodynamic entropy [MATH] is a special case of the SMI applied to a particular class of physical problems.', '1901.08956-3-4-5': 'Ben-Naim also rightly inveighs against interpreting physical entropy as a measure of "disorder," a concept too vague to be scientifically quantifiable.', '1901.08956-3-4-6': 'What counts as order is entirely subjective.', '1901.08956-3-4-7': "Ben-Naim builds on the work of Jaynes who, reversing the historical sequence, showed specifically how statistical mechanical entropy was a particular application of Shannon's information theoretic entropy [CITATION].", '1901.08956-3-5-0': 'Here we focus on the quantum mechanical operator entropy [MATH], associated with a Hermitian operator [MATH], as formulated by Ingarden [CITATION] and described in Section II.', '1901.08956-3-5-1': 'This operator entropy quantifies the amount of information about the property [MATH] that is missing in the (pure or mixed) quantum state.', '1901.08956-3-5-2': 'For example, the position operator [MATH] generates an associated entropy [MATH] which captures how much information about position is missing.', '1901.08956-3-5-3': 'The familiar von Neumann entropy is then seen to be the special case of quantum operator entropy when the operator is the density matrix [MATH].', '1901.08956-3-5-4': 'Though the von Neumann entropy is zero for a pure state and constant under unitary time evolution, other operator entropies need not be.', '1901.08956-3-6-0': 'We examine the interpretation of these quantities in a model system with topological disorder.', '1901.08956-3-6-1': 'Section III examines the free expansion of an initially localized system, tracking several operator entropies.', '1901.08956-3-6-2': 'We see the increase in the position entropy in time which parallels classical second law behavior even in purely unitary time evolution with no coarse-graining [CITATION] or loss of quantum information.', '1901.08956-3-6-3': 'The position entropy saturates at levels than can be predicted by sampling random superpositions of energy eigenstates.', '1901.08956-3-6-4': 'In this way the behavior is connected to the notions of typicality that are proving so helpful in quantum statistical mechanics [CITATION].', '1901.08956-3-6-5': 'In Section IV we discuss the second law of thermodynamics and time-reversibility for this system.', '1901.08956-3-6-6': 'Results for thermal equilibrium in the model system are briefly discussed in Section V.', '1901.08956-3-7-0': '# Quantum operator entropy', '1901.08956-3-8-0': '## Definition', '1901.08956-3-9-0': "We follow Ben-Naim in defining the SMI (Shannon Measure of Information), in bits, as Shannon's measure on a probability distribution [MATH] over [MATH] possible outcomes.", '1901.08956-3-9-1': '[EQUATION]', '1901.08956-3-9-2': 'The notation is useful here to distinguish this quantity from the physical or thermodynamic entropy [MATH]; it is simply a measure on a probability distribution.', '1901.08956-3-10-0': 'To apply the Shannon measure of information in the quantum mechanical case we consider a Hermitian operator [MATH] written in its eigen-basis.', '1901.08956-3-10-1': '[EQUATION]', '1901.08956-3-10-2': 'We assume that the set of states [MATH] are chosen to form an orthonormal basis.', '1901.08956-3-10-3': 'If a measurement of [MATH] is made, the result will be one of the eigenvalues of [MATH] with probabilities [MATH].', '1901.08956-3-10-4': 'It is natural then to define the Shannon measure on this set of probabilities as the entropy associated with [MATH].', '1901.08956-3-10-5': '[EQUATION]', '1901.08956-3-10-6': 'This quantity was introduced by Ingarden [CITATION] and has been studied by Anza and Vedral, [CITATION], Hu et al. [CITATION], and others.', '1901.08956-3-11-0': 'If the system is in a pure quantum state [MATH], then the probability that a measurement of [MATH] yields [MATH] is given by the Born Rule.', '1901.08956-3-11-1': '[EQUATION]', '1901.08956-3-11-2': 'So that [EQUATION]', '1901.08956-3-11-3': 'Entropy quantifies missing information, so what information is missing?', '1901.08956-3-11-4': 'For an observable [MATH] and a pure state [MATH], [MATH] measures the number of bits of information that are missing from the universe concerning what value of [MATH] will be obtained if a measurement of [MATH] is made.', '1901.08956-3-11-5': 'The quantum state of the system [MATH] contains everything there is to know about the system at time [MATH].', '1901.08956-3-11-6': 'Because of fundamental quantum indeterminism, that is not enough to pin down which eigenvalue of [MATH] will be measured (unless, of course, [MATH] happens to be an eigensate of [MATH]).', '1901.08956-3-11-7': 'We now know from recent Bell test experiments that this indeterminism is a fundamental feature of reality [CITATION].', '1901.08956-3-11-8': 'It is not just a feature of quantum mechanics as we currently understand it, nor is it just an expression of the limited information of an observer.', '1901.08956-3-11-9': 'The quantum operator entropy [MATH] therefore reflects information about the observable Q that is missing from the physical world.', '1901.08956-3-11-10': "It is, in that somewhat strange sense, an objective property of the physical system and not a subjective property of the observer's knowledge.", '1901.08956-3-11-11': 'Perhaps one can speak of Shannon\'s notion of "choice" applying here.', '1901.08956-3-11-12': 'When confronted with a measurement of [MATH], the physical world has [MATH] bits of choice in the outcome that are unconstrained by the physical law, a law which has determined the present [MATH].', '1901.08956-3-11-13': 'The missing information about [MATH] in ([REF]) is really missing.', '1901.08956-3-12-0': 'A mixed state describes a system [MATH] which is either coupled dynamically to a reservoir system [MATH] or has been so in the past.', '1901.08956-3-12-1': 'The quantum state of the composite system is not in general simply a direct product of the state of each subsystem but rather an entangled state [MATH].', '1901.08956-3-12-2': 'The density operator for the composite system is defined by [EQUATION]', '1901.08956-3-12-3': 'The density operator for the system alone is defined using the partial trace over the reservoir degrees of freedom.', '1901.08956-3-12-4': '[EQUATION]', '1901.08956-3-12-5': 'We can write the density operator in the basis of its own eigenstates.', '1901.08956-3-12-6': '[EQUATION]', '1901.08956-3-12-7': "For a pure state only one of the [MATH]'s is nonzero.", '1901.08956-3-13-0': 'The probability [MATH] that a measurement of [MATH] for the system yields [MATH] can be calculated for the mixed state using the density operator and the projection operator onto the [MATH] eigenstate of [MATH].', '1901.08956-3-13-1': '[EQUATION]', '1901.08956-3-13-2': 'The quantum operator entropy for a mixed (or pure) state is then given by applying ([REF]) to ([REF]).', '1901.08956-3-14-0': 'The expression in ([REF]) includes the previous expression in ([REF]) as a special case when [MATH] represents a pure state.', '1901.08956-3-15-0': 'Again, [MATH] is providing a measure (in bits) of missing information.', '1901.08956-3-15-1': 'For a mixed state, the source of this missing information is two-fold.', '1901.08956-3-15-2': 'Quantum indeterminacy still limits the information about a future measurement that is present in the current state of the system.', '1901.08956-3-15-3': "But in addition there is also information missing about the reservoir's state and the mutual information characterizing the entanglement between system and reservoir.", '1901.08956-3-15-4': "The system's reduced density matrix [MATH] is not a complete description of the quantum state of the system, but it is the best possible local description.", '1901.08956-3-15-5': 'For the physical world, there is a fact-of-the-matter about the global quantum state that includes both system and reservoir [MATH].', '1901.08956-3-15-6': 'But the local description of the system alone represented by [MATH] has less information.', '1901.08956-3-16-0': '## Examples of quantum operator entropies', '1901.08956-3-17-0': 'If we use a basis set of discrete position eigenstates [MATH] we can define the quantum operator entropy for [MATH], the position operator.', '1901.08956-3-17-1': '[EQUATION]', '1901.08956-3-17-2': 'The quantum operator entropy for the Hamiltonian [MATH] with allowed energies [MATH] and eigenstates [MATH] is [EQUATION]', '1901.08956-3-17-3': 'The quantum operator entropy for the density operator [MATH] itself, from ([REF]) is [EQUATION]', '1901.08956-3-17-4': 'Because [EQUATION] we can write [EQUATION]', '1901.08956-3-17-5': 'The quantum operator entropy for the density operator is the SMI of the diagonal elements of the density matrix.', '1901.08956-3-17-6': 'We recognize ([REF]) as the expectation value of [MATH], and so write [EQUATION]', '1901.08956-3-17-7': 'The entropy [MATH] is identical to the von Neumann entropy [MATH] (in bits).', '1901.08956-3-17-8': 'For a pure state, the entropy [MATH] is zero.', '1901.08956-3-18-0': 'In general each operator entropy [MATH] can take on different values because each quantifies something different.', '1901.08956-3-18-1': 'If what is known about the system is [MATH], [MATH] is the amount of information that is missing about position, or more precisely, about the outcome of position measurements.', '1901.08956-3-18-2': 'It is the answer to the question: How many bits of information are not known about the outcome of a position measurement if all one knows is [MATH]?', '1901.08956-3-18-3': '[MATH] is the amount of information about energy that is missing.', '1901.08956-3-18-4': '[MATH] is the amount of information that is missing about which quantum state the system will be found in.', '1901.08956-3-18-5': 'It measures the "mixedness" or purity of the state.', '1901.08956-3-18-6': '[MATH], the von Neumann entropy, is invariant under unitary transformations of the basis.', '1901.08956-3-18-7': 'Other operator entropies [MATH] are not invariant-they are tied to the eigen-basis of the particular operator [MATH].', '1901.08956-3-19-0': '# Entropy change of a pure state under unitary evolution', '1901.08956-3-20-0': '## Model system', '1901.08956-3-21-0': 'We consider here a model problem of unitary evolution of a pure state in a closed system with a time-independent Hamiltonian.', '1901.08956-3-21-1': 'The purpose is to examine the different roles played by the von Neumann entropy [MATH], the energy entropy [MATH], and the positional entropy [MATH].', '1901.08956-3-22-0': 'The system consists of a random array of [MATH] fixed sites.', '1901.08956-3-22-1': 'Each site is labeled with an index [MATH] and is at a randomly chosen position [MATH] a unit square.', '1901.08956-3-22-2': 'The site positions are shown graphically in Figure [REF]a as red dots.', '1901.08956-3-22-3': 'We use as basis states for the system the set of all the position eigenstates localized on each site, [MATH].', '1901.08956-3-22-4': 'The on-site energy for each is [MATH].', '1901.08956-3-23-0': 'Off-diagonal elements of the Hamiltonian couple each site to several nearby sites with a fixed coupling matrix element [MATH].', '1901.08956-3-23-1': 'Each site is coupled to a randomly chosen subset of its 50 nearest sites.', '1901.08956-3-23-2': 'The coupling to neighbors for six representative sites is shown in Figure [REF]a as blue lines connecting red dots.', '1901.08956-3-23-3': 'Figure [REF]b shows the connectivity of the sites with a blue dot on row [MATH], column [MATH], if site [MATH] is coupled to site [MATH].', '1901.08956-3-24-0': 'The Hamiltonian for the system is [EQUATION]', '1901.08956-3-24-1': 'Here [MATH] if two sites are connected, and 0 if they are not.', '1901.08956-3-25-0': 'The randomness in this model minimizes the artifacts of geometric regularity on the dynamics.', '1901.08956-3-25-1': 'We want to see how these different operator entropies change due to fundamental unitary dynamics without the patterns of constructive and destructive interference that dominate, for example, the evolution of a similar system on a regular lattice.', '1901.08956-3-25-2': 'This topologically disordered model is similar to, but distinct from, the Lifshitz model for disordered semiconductors [CITATION] for which [MATH] is simply a function of the distance between sites.', '1901.08956-3-25-3': 'The high multiple connectivity also lets us generalize the interpretation of the model as will be discussed below.', '1901.08956-3-26-0': 'To construct the connections between sites, nine passes through all the sites are made, adding a connection between each site and another site randomly chosen from among its 50 closest neighbors.', '1901.08956-3-26-1': 'The result is that the number of connections for each site varies between 10 and 32, with a mean of 18.', '1901.08956-3-26-2': '(The situation is complicated by the fact that site [MATH] might have site [MATH] as one of its 50 nearest neighbors, while site [MATH] does not have site [MATH] as one of its 50 nearest neighbors.)', '1901.08956-3-27-0': '## Unitary free expansion', '1901.08956-3-28-0': 'The dynamic problem we solve is the expansion of the state from an initially spatially confined state.', '1901.08956-3-28-1': 'The initial state has an equal probability distributed among the 64 sites that are closest to the origin.', '1901.08956-3-28-2': 'We solve for the time development of the state function using the unitary time development operator.', '1901.08956-3-28-3': '[EQUATION]', '1901.08956-3-28-4': 'The expansion of the state into the surrounding state space is shown by the snapshots of the probability density in Figure [REF].', '1901.08956-3-28-5': 'The time scale is set by the characteristic tunneling time between connected sites [EQUATION]', '1901.08956-3-28-6': 'The figure shows how the probability expands much like a classical gas into the available states.', '1901.08956-3-28-7': 'Since the expectation value of the energy is constant during unitary evolution, the system cannot de-excite, and quantum interference fluctuations persist indefinitely.', '1901.08956-3-29-0': 'Figure [REF] shows the the calculated entropies [MATH], [MATH], and [MATH], the von Neumann entropy, during the expansion shown in Figure [REF].', '1901.08956-3-29-1': 'The von Neumann entropy is, of course, constant during the unitary evolution and is in fact 0 because the state is always pure.', '1901.08956-3-30-0': 'The energy entropy [MATH] is also constant during the expansion, but it is not [MATH].', '1901.08956-3-30-1': 'The energy eigenstate occupation probabilities cannot change during unitary time development so [MATH] is independent of time.', '1901.08956-3-30-2': '[MATH] is 4.86 bits (rather than 0) because the initial state is not a Hamiltonian eigenstate, so there are many energy eigenvalues that could be measured.', '1901.08956-3-30-3': '[MATH] characterizes the missing information in the probability distribution of those energy measurements, shown in Figure [REF].', '1901.08956-3-31-0': 'The positional entropy [MATH] characterizes the missing information about position.', '1901.08956-3-31-1': 'At [MATH], it is exactly 6 bits because the probability is uniformly distributed among [MATH] sites.', '1901.08956-3-31-2': 'As the expansion progresses it increase to a value between [MATH] and [MATH].', '1901.08956-3-31-3': 'If the distribution were distributed completely evenly among the [MATH] sites, [MATH] would be 10.', '1901.08956-3-31-4': 'The initial quantum confinement means that the isolated system is excited and it has no way of de-exciting.', '1901.08956-3-31-5': 'If it did, [MATH] would approach a value of 10 bits.', '1901.08956-3-31-6': 'Nevertheless, it is clear that the increase in the quantum mechanical measure [MATH] resembles the increase in the classical thermodynamic entropy of an ideal gas for which [MATH] bits.', '1901.08956-3-31-7': 'Expanding the volume by a factor of 16 would increase the classical statistical mechanical entropy by [MATH] bits.', '1901.08956-3-31-8': 'The dashed line in Figure [REF] is the position entropy of a random superpostition of energy eigenstates, as discussed further in Section [REF].', '1901.08956-3-32-0': 'We emphasise that despite the increase in position entropy [MATH] shown in Figure [REF], the system is evolving in a completely reversible way.', '1901.08956-3-32-1': 'No information about the quantum state is being lost.', '1901.08956-3-32-2': 'Equation ([REF]) can be inverted so we could use the state [MATH] at any time [MATH] to reconstruct precisely the initial state [MATH].', '1901.08956-3-32-3': 'The constant purity of the state is precisely reflected in the unchanging value of the von Neumann entropy [MATH].', '1901.08956-3-32-4': 'What is changing is the amount of missing information in the quantum state about position, and precisely that is quantified by [MATH].', '1901.08956-3-33-0': 'The Hamiltonian for the system given by ([REF]) has eigenvalues [MATH].', '1901.08956-3-33-1': 'It is helpful to scale energies relative to the ground state in units of [MATH].', '1901.08956-3-33-2': '[EQUATION]', '1901.08956-3-33-3': 'The scaled energy eigenvalues for the particular Hamiltonian (i.e., random interconnections) shown in Figures [REF], [REF], and [REF] extend from [MATH] to [MATH].', '1901.08956-3-33-4': 'The expectation value of the scaled energy is [MATH], and is independent of time.', '1901.08956-3-33-5': 'This energy reflects the kinetic energy of confining the system into the initial [MATH] states.', '1901.08956-3-34-0': 'One might describe the apparent saturation of [MATH] as the system "thermalization," but it is important to note that it is not in a thermal equilibrium state.', '1901.08956-3-34-1': 'Figure [REF] shows the probability distribution for the 50 lowest eigenenergies.', '1901.08956-3-34-2': 'The red line is a thermal Boltzmann distribution.', '1901.08956-3-34-3': 'Obviously the state is far from being thermal in energy.', '1901.08956-3-34-4': 'Because the time evolution is unitary, this probability distribution is constant in time.', '1901.08956-3-35-0': '## Differing random configurations', '1901.08956-3-36-0': 'The specific configuration of the random local connectivities of the Hamiltonian ([REF]) affects the details of [MATH], but not the overall shape of the saturation to a typical value.', '1901.08956-3-36-1': 'Figure [REF] shows [MATH] and [MATH] for 10 different random configuration, chosen in the same way as describe above, with the initial state localized uniformly across the 64 sites nearest to the origin (hence [MATH]).', '1901.08956-3-36-2': 'Because of the differences in ([REF]), the scaled energy expectation values [MATH] vary between 3.38 and 5.88 across this set of configurations.', '1901.08956-3-36-3': 'The energy operator entropy [MATH] varies similarly between 4.28 and 5.46 bits and are, of course, constant in time.', '1901.08956-3-36-4': 'The position entropy [MATH] in each case increases and saturates around the same values, despite the difference in the configurations.', '1901.08956-3-36-5': 'The variations in time due to quantum interference fluctuations are of the same magnitude as the differences between different configurations.', '1901.08956-3-36-6': 'The dashed line in Figure [REF] shows the position entropy for the average value of [MATH] over 300 random superpositions of the energy eigenstates (RaSEE) for the first of the ten Hamiltonians, [MATH] bits.', '1901.08956-3-36-7': 'The exact value so obtained varies slightly with the specifics of each configuration because the eigenvalue spectrum of each is different in detail.', '1901.08956-3-36-8': 'But again, the variation across configurations is comparable in magnitude to that of the quantum fluctuations in time.', '1901.08956-3-37-0': '## Differing initial confinements', '1901.08956-3-38-0': 'Figure [REF] shows the position entropy [MATH] under unitary time evolution for different initial states.', '1901.08956-3-38-1': 'The Hamiltonian (and random interconnectivity) is identical to that which produced the time development shown in Figures [REF] and [REF].', '1901.08956-3-38-2': 'The initial states are in each case chosen to be a uniform distribution of the [MATH] sites nearest the origin where [EQUATION]', '1901.08956-3-38-3': 'The corresponding values of [MATH] are 2, 3, 4, 5, 6, and 7 bits.', '1901.08956-3-38-4': 'In contrast to the situation of Figure [REF], the Hamiltonian here is exactly the same for all cases; only the initial condition is varied.', '1901.08956-3-38-5': 'The tightly confined states have a higher energy expectation value.', '1901.08956-3-38-6': 'For these initial states with [MATH] from 4 to 128 we have [EQUATION]', '1901.08956-3-38-7': 'The behavior of [MATH] is basically the same in each case, increasing to a value of about 9.4 bits, with persistent quantum oscillations, as the probability expands to fill the available state space.', '1901.08956-3-38-8': 'It is notable that the systems saturate to roughly the same value, though the energy expectation values are very different.', '1901.08956-3-39-0': 'The dashed line in Figure [REF] shows the RaSEE position entropy, averaged over 300 random superpositions of energy eigenstates, with a resulting value of [MATH].', '1901.08956-3-39-1': 'For the ensemble of random superpositions, the expectation values of the energy are high compared with ([REF]), averaging [MATH].', '1901.08956-3-39-2': 'Nevertheless the RaSEE value is remarkably consistent with the saturation value of [MATH] regardless of the initial confinement.', '1901.08956-3-40-0': '## Random superpositions of energy eigenstates (RaSEE)', '1901.08956-3-41-0': 'We have seen in our model system that for different initial confinements and for different random connectivities, unitary time-evolution ([REF]) drives the the position entropy [MATH] toward the same average value, about 9.39 bits, though with small and persistent quantum oscillations.', '1901.08956-3-41-1': 'How could this value be predicted prior to solving the detailed dynamics of ([REF])?', '1901.08956-3-42-0': 'For a given N-dimensional Hamiltonian [MATH] with eigenvalues [MATH] and eigenstates [MATH], we can construct a random superposition of energy eigenstates (RaSEE).', '1901.08956-3-42-1': 'This involves choosing points randomly and uniformly distributed on the surface of the unit sphere in a Hilbert space of [MATH] dimensions ([MATH]).', '1901.08956-3-42-2': 'To do this we use the method of Marsaglia [CITATION].', '1901.08956-3-42-3': 'We first construct a vector [MATH] of length [MATH] whose components [MATH] are random normal deviates with unit variance.', '1901.08956-3-42-4': 'The vector is then normalized to unit length, [MATH], and random phase factors [MATH] are chosen from a uniform distribution over the interval [MATH].', '1901.08956-3-42-5': 'We construct an RaSEE (we pronounce this "racy") state as a weighted sum of the lowest [MATH] energy eigenvectors.', '1901.08956-3-42-6': '[EQUATION]', '1901.08956-3-42-7': 'Figure [REF] shows the values of [MATH], the expectation value of position, and the expectation value of the scaled energy for 900 states RaSEE states created using ([REF]).', '1901.08956-3-42-8': 'We use the same Hamiltonian (with the same connectivity) as that used to calculate the results of Figures [REF] and [REF].', '1901.08956-3-42-9': 'The figure shows the results of limiting the summation in ([REF]) to the lowest [MATH]= 256, 512, or 1024 energy eigenstates, the latter being the full spectrum.', '1901.08956-3-42-10': 'The mean value of [MATH] for all the samples is 9.39 [MATH] 0.02 bits.', '1901.08956-3-42-11': 'Neither the mean value of [MATH] nor the variance is affected by changing [MATH].', '1901.08956-3-42-12': 'The mean value of [MATH] is [MATH] (centered in the unit square), and as [MATH] is raised the variance decreases.', '1901.08956-3-42-13': 'The mean value of [MATH] over the RaSEE samples does increase as larger swaths of the energy spectrum are sampled: [MATH]=13.2 for [MATH]=256, 15.3 for [MATH]=512, and 18.7 for [MATH]=1024.', '1901.08956-3-43-0': 'The RaSEE states are not, of course, stationary states.', '1901.08956-3-43-1': 'Figure [REF] shows how [MATH] and position expectation values vary in time under unitary evolution for ten different RaSEE initial states.', '1901.08956-3-43-2': 'Position expectation values [MATH] and [MATH] fluctuate but stay close to the mean of 1/2.', '1901.08956-3-43-3': 'The [MATH] values also vary in time but remain close to the mean value of 9.39 bits.', '1901.08956-3-43-4': 'If the system size is doubled to N=2048 (not shown), the mean RaSEE value of [MATH] becomes 10.39 [MATH] 0.01 bits- one more bit of missing position information and an even tighter variance.', '1901.08956-3-44-0': 'What we see in Figures [REF] and [REF] is that initially localized states, regardless of how localized they are, expand and [MATH] increases and saturates around a "typical" value of 9.39 bits.', '1901.08956-3-44-1': 'This value comes from the detailed unitary dynamics for different initial states, but is independent of the details.', '1901.08956-3-44-2': 'In fact, we can get a very good estimate of the saturation value of [MATH], without having to solve the full dynamics, by simply picking a single RaSEE state using ([REF]) and calculating the corresponding [MATH].', '1901.08956-3-44-3': 'If we construct a population of such states we get an even better estimate of the [MATH] saturation value.', '1901.08956-3-44-4': 'Even if the disorder is varied, so that the energy eigenvalues and eigenstates are slightly different, as in Figure [REF], we see the saturation value is well-approximated by the RaSEE result for any one of the configurations.', '1901.08956-3-45-0': 'The vast majority of RaSEE states have "typical" values of [MATH], the same values to which unitary dynamics drives the system regardless of its initial confinement, as in Figure [REF].', '1901.08956-3-45-1': 'But RaSEE states are not typical in terms of energy expectation values.', '1901.08956-3-45-2': 'For [MATH]=1024 (full spectrum) energy expectation values for RaSEE states are significantly higher than all but the most confined states.', '1901.08956-3-46-0': 'One might expect that the same typical value of [MATH] could be obtained by any unbiased sampling of the surface of the unit sphere in the [MATH]-dimensional Hilbert space of the the problem, using the eigenstates of any observable.', '1901.08956-3-46-1': 'They are connected by a unitary transformation, so all would seem to be equivalent.', '1901.08956-3-46-2': 'That would be true if we were calculating the expectation value of an observable, which cannot depend on the basis used.', '1901.08956-3-46-3': 'But the operator entropy ([REF]) is not an expectation value and is in fact a very non-linear function of the state.', '1901.08956-3-46-4': 'If we construct, for example a random superposition of position eigenstates rather than energy eigenstates, we get a typical value of [MATH]= 8.95 bits, which does not match the temporal saturation value.', '1901.08956-3-46-5': 'The fact that the saturation value of [MATH] is accurately generated by sampling the RaSEE states is presumably related to the special role of energy in the time propagation of ([REF]).', '1901.08956-3-47-0': '# Second law of thermodynamics', '1901.08956-3-48-0': 'The increase in entropy associated with the free expansion of the system we have seen in Figures [REF] and [REF] suggests a connection to the second law of thermodynamics.', '1901.08956-3-48-1': 'We should note that the von Neuman entropy is constant in these cases.', '1901.08956-3-48-2': 'Also, many limit entropy as a thermodynamic concept to systems in thermal equilibrium with the environment.', '1901.08956-3-48-3': 'This is not the case here (though see Section [REF])-any of the quantum operator entropies are simply properties of a quantum state determined by ([REF]) or ([REF]) and vary in time as the state varies in time.', '1901.08956-3-48-4': 'Each entropy characterizes a quantum state at a particular time (how much information of a particular kind is missing), but an entropy is not an expectation value of an observable.', '1901.08956-3-49-0': 'As Lesovik has pointed out [CITATION], the microscopic origin of the increase of entropy can be attributed to the dynamics of the Schrodinger equation itself.', '1901.08956-3-49-1': 'The spreading of the wave packet, of which the current model is an elaboration, is a feature of the basic structure of the Hamiltonian and the time development operator.', '1901.08956-3-50-0': 'One then has to deal with the apparent paradox of the time-reversibility of ([REF]) in light of the second law.', '1901.08956-3-50-1': 'It is helpful to illustrate this in the current model with its attendant complexity.', '1901.08956-3-50-2': 'Let us start with a confined (localized) state with uniform probability over the 64 sites closes to the origin, such as is shown in Figure [REF]a, and which we will denote [MATH].', '1901.08956-3-50-3': 'Now define a state [MATH] constructed by using the time-reversed version of ([REF]), moving backward in time [MATH] with our model Hamiltonian.', '1901.08956-3-50-4': '[EQUATION]', '1901.08956-3-50-5': 'We now use this state as an initial state, [MATH] and solve ([REF]) forward in time.', '1901.08956-3-50-6': 'The resultant [MATH] is shown in Figure [REF].', '1901.08956-3-50-7': 'We do indeed see second-law-violating behavior for [MATH] inasmuch as there is an abrupt drop in the entropy from its steady state value to a value of 6 bits as the precise combination of magnitudes, phases, and interference combine to reconstruct the localized state at [MATH].', '1901.08956-3-50-8': 'Moreover, the quantum fluctuations prior to and after that point cause [MATH] to move both up and down around the typical level of 9.39 bits (which is a feature of almost any RaSEE state).', '1901.08956-3-50-9': 'So the change in the position entropy moment-by-moment is not monotonically non-negative even away from the sudden recovery of localization.', '1901.08956-3-51-0': 'If one could control all the phases and amplitudes that define the quantum state, nothing in the physical law prevents one from constructing a state like [MATH] that behaves in this second-law violating way.', '1901.08956-3-51-1': 'Indeed, for a small number of bits Lesovik and coworkers have done just that on the IBM quantum computer [CITATION].', '1901.08956-3-51-2': 'The time-symmetry is broken, not by ([REF]) or by a cosmological condition, but by the difficulty of constructing, artificially or naturally, an initial state sufficiently well tuned in phases and magnitudes to produce even the brief and fleeting reduction in entropy we see here.', '1901.08956-3-51-3': 'This point was made by Lesovik in [CITATION] and is underscored by the present model behavior.', '1901.08956-3-52-0': 'One might suspect that the complex interference that results in the reduction of [MATH] from about 9.39 to 6 is fragile, in the sense that any small perturbation of the initial state [MATH] would destroy it.', '1901.08956-3-52-1': 'To test the resilience of the drop in entropy, we alter the initial state [MATH] by adding to it a fraction of a RaSEE state and renormalizing.', '1901.08956-3-52-2': '[EQUATION]', '1901.08956-3-52-3': 'Here the real scalar [MATH] determines the amount of of the RaSEE state in the initial state.', '1901.08956-3-52-4': 'The inset of Figure [REF] shows [MATH], the value of the position entropy at the moment of localization recovery, as a function of [MATH].', '1901.08956-3-52-5': 'For small values of [MATH], [MATH] at this minimum increases linearly, recovering to the steady-state (and RaSEE) value of 9.39 bits by about [MATH].', '1901.08956-3-52-6': 'Different disordered connectivities produce essentially identical results.', '1901.08956-3-52-7': 'Thus, achieving the sudden moment of localization does not in fact require an exquisite tuning of the initial state.', '1901.08956-3-52-8': 'Getting the initial state slightly wrong will not wipe out the later localization, just diminish it.', '1901.08956-3-53-0': 'Nevertheless, it is worth observing that the momentary localization of the state is very brief, lasting only about [MATH].', '1901.08956-3-53-1': 'Thus in the long history of this carefully prepared system, there is a fleeting blip in time when it spontaneously localizes for a moment.', '1901.08956-3-53-2': 'That moment, if achieved, will almost certainly not be repeated in many lifetimes of the universe.', '1901.08956-3-54-0': '# Thermal ground state', '1901.08956-3-55-0': 'We have considered the time evolution of a pure quantum state during free expansion.', '1901.08956-3-55-1': 'We now briefly describe the application of quantum operator entropies to a state in thermal equilibrium with a reservoir at temperature [MATH].', '1901.08956-3-55-2': 'In that case, the system degrees of freedom interact with the reservoir degrees of freedom so that energy flows between system and reservoir and their quantum states become entangled.', '1901.08956-3-55-3': 'The best local quantum description one can give for the system is then a reduced density matrix ([REF]) where the unknown reservoir degrees of freedom have been traced out.', '1901.08956-3-55-4': 'As Jaynes showed, the optimal reduced density matrix in this case is the one which maximizes the von Neumann entropy over variations of each element of the density matrix [CITATION].', '1901.08956-3-55-5': 'This is the right entropy to maximize because the effect of the reservoir is precisely to mix system states through entanglement with the reservoir and it is this "mixedness" that is quantified by the von Neumann entropy.', '1901.08956-3-55-6': 'The result of the maximum entropy procedure is the canonical expression for the density operator.', '1901.08956-3-55-7': '[EQUATION]', '1901.08956-3-55-8': 'The density matix is diagonal in basis of Hamiltonian eigenstates for the system.', '1901.08956-3-55-9': 'The von Neumannn entropy is the Shannon entropy of the diagonal elements of [MATH], and is therefore equal to the energy entropy, [MATH].', '1901.08956-3-55-10': 'The off-diagonal elements of the density operator in the position basis do not vanish, and the position entropy [MATH] remains as a measure of the missing information about position for the state defined by ([REF]).', '1901.08956-3-56-0': 'Figure [REF] shows the position entropy, von Neumann entropy, and the expectation value of the scaled energy [MATH] as a function of temperature for the model system described by Figures [REF], [REF], and [REF].', '1901.08956-3-56-1': 'In the low temperature limit [MATH] and [MATH] as the system cools to the ground state.', '1901.08956-3-56-2': 'The position entropy of the ground state is in this case [MATH] bits (notably larger than the RaSEE value).', '1901.08956-3-56-3': 'In the high temperature limit both [MATH] and [MATH] go to 10 bits, the completely delocalized state over 1024 sites.', '1901.08956-3-57-0': '# Discussion', '1901.08956-3-58-0': 'The topologically disordered model described here is of more general applicability than it might at first seem.', '1901.08956-3-58-1': 'It is clear that the actual positions of the sites [MATH], the eigenvalues of the position operator, play a limited role in the Hamiltonian ([REF]).', '1901.08956-3-58-2': 'These positions yield a simple algorithm for determining the connectivity between the basis states-random choices from among the 50 nearest neighbors-and a way of visualizing the evolution as in Figure [REF].', '1901.08956-3-58-3': 'But what really matters is just the connectivity that is shown in Figure [REF]b and the coupling strengths.', '1901.08956-3-58-4': 'We could, for example, re-interpret each basis state [MATH] as representing a particular many-body nuclear and electronic configuration for a molecule.', '1901.08956-3-58-5': 'For each configuration there are a set of other accessible configurations that are dynamically coupled by the Hamiltonian, giving a sense of "nearby states," but without a regular pattern.', '1901.08956-3-58-6': 'The essential point is that the increase in [MATH] is capturing quantitatively the very familiar feature of unitary evolution that a localized wavefunction tends to spread out into accessible states, however the states are defined.', '1901.08956-3-58-7': 'Nevertheless, in many cases position eigenstates are especially selected for survival by decoherence through environmental entanglement [CITATION], so it is not a mistake to focus on them here.', '1901.08956-3-59-0': 'Different operator entropy measures allows us to capture different aspects of the dynamics.', '1901.08956-3-59-1': 'The von Neumann entropy [MATH] captures the "mixedness" of a state, which is constant under unitary evolution.', '1901.08956-3-59-2': 'We have seen that by contrast the positional entropy [MATH] increases in a way consistent with the second law of thermodynamics, without recourse to a course-graining procedure.', '1901.08956-3-59-3': 'The second law must ultimately be a feature of physical dynamics.', '1901.08956-3-59-4': 'There is no law of nature that says systems move from less probable states to more probable states.', '1901.08956-3-59-5': 'There is rather just the dynamics of the Schrodinger equation applied to the relevant state space.', '1901.08956-3-60-0': 'The author gratefully acknowledges helpful conversatins with Harvey Brown.'}

{'1901.08956-4-0-0': 'For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant.', '1901.08956-4-0-1': 'Yet the second law of thermodynamics, and our experience, show that entropy increases with time.', '1901.08956-4-0-2': 'Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator.', '1901.08956-4-0-3': 'These entropies characterize the missing information about a particular observable inherent in the quantum state itself.', '1901.08956-4-0-4': 'The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix.', '1901.08956-4-0-5': 'We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian.', '1901.08956-4-0-6': 'An initially confined state is subject to free expansion into available states.', '1901.08956-4-0-7': 'The time development is completely reversible with no loss of quantum information and no course graining is applied.', '1901.08956-4-0-8': 'The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.', '1901.08956-4-1-0': '# Introduction', '1901.08956-4-2-0': 'Entropy as a concept was defined in historical order by Clausius, Boltzmann, Gibbs, von Neumann, and Shannon.', '1901.08956-4-2-1': 'Conceptually, it would have been clearer if the order were reversed.', '1901.08956-4-2-2': 'The Shannon information-theoretic definition is the most fundamental, which can then be applied to physical quantum systems, with classical statistical mechanics following as the classical limit of the quantum case.', '1901.08956-4-3-0': 'Shannon chose to use the word "entropy" from the field of statistical mechanics for a quantity he variously described as measuring "choice," "information", "uncertainty", or "surprise" [CITATION].', '1901.08956-4-3-1': 'As the mathematical theory of communication he invented became the field of information theory, and in due course was turned back onto analyzing statistical mechanics, the layering of these various concepts often became confusing.', '1901.08956-4-3-2': 'Information and uncertainty, for example, seem to be opposite one another.', '1901.08956-4-3-3': 'The more information one has, the less uncertainty.', '1901.08956-4-3-4': 'Whose choice is involved in the entropy of a physical system?', '1901.08956-4-4-0': 'Ben-Naim has done the world a great favor by relentlessly clarifying the quantity defined by Shannon as a measure of the missing information associated with a probability distribution [CITATION].', '1901.08956-4-4-1': 'If all one knows is the probability distribution for a finite set of discrete possible events, the Shannon measure quantifies the amount of information, measured in bits, that one is missing.', '1901.08956-4-4-2': 'It is the difference between the incomplete knowledge captured in a probability distribution, and certainty about which event will occur.', '1901.08956-4-4-3': 'What Ben-Naim prefers to call the Shannon Measure of Information (SMI) (or Shannon Missing Information), represents, as it were, the part of the graduated cylinder that is empty of fluid.', '1901.08956-4-4-4': 'The SMI is the more general concept; the thermodynamic entropy [MATH] is a special case of the SMI applied to a particular class of physical problems.', '1901.08956-4-4-5': 'Ben-Naim also rightly inveighs against interpreting physical entropy as a measure of "disorder," a concept too vague to be scientifically quantifiable.', '1901.08956-4-4-6': 'What counts as order is entirely subjective.', '1901.08956-4-4-7': "Ben-Naim builds on the work of Jaynes who, reversing the historical sequence, showed specifically how statistical mechanical entropy was a particular application of Shannon's information theoretic entropy [CITATION].", '1901.08956-4-5-0': 'Here we focus on the quantum mechanical operator entropy [MATH], associated with a Hermitian operator [MATH], as formulated by Ingarden [CITATION] and described in Section II.', '1901.08956-4-5-1': 'This operator entropy quantifies the amount of information about the property [MATH] that is missing in the (pure or mixed) quantum state.', '1901.08956-4-5-2': 'For example, the position operator [MATH] generates an associated entropy [MATH] which captures how much information about position is missing.', '1901.08956-4-5-3': 'The familiar von Neumann entropy is then seen to be the special case of quantum operator entropy when the operator is the density matrix [MATH].', '1901.08956-4-5-4': 'Though the von Neumann entropy is zero for a pure state and constant under unitary time evolution, other operator entropies need not be.', '1901.08956-4-6-0': 'We examine the interpretation of these quantities in a model system with topological disorder.', '1901.08956-4-6-1': 'Section III examines the free expansion of an initially localized system, tracking several operator entropies.', '1901.08956-4-6-2': 'We see the increase in the position entropy in time which parallels classical second law behavior even in purely unitary time evolution with no coarse-graining [CITATION] or loss of quantum information.', '1901.08956-4-6-3': 'The position entropy saturates at levels than can be predicted by sampling random superpositions of energy eigenstates.', '1901.08956-4-6-4': 'In this way the behavior is connected to the notions of typicality that are proving so helpful in quantum statistical mechanics [CITATION].', '1901.08956-4-6-5': 'In Section IV we discuss the second law of thermodynamics and time-reversibility for this system.', '1901.08956-4-6-6': 'Results for thermal equilibrium in the model system are briefly discussed in Section V.', '1901.08956-4-7-0': '# Quantum operator entropy', '1901.08956-4-8-0': '## Definition', '1901.08956-4-9-0': "We follow Ben-Naim in defining the SMI (Shannon Measure of Information), in bits, as Shannon's measure on a probability distribution [MATH] over [MATH] possible outcomes.", '1901.08956-4-9-1': '[EQUATION]', '1901.08956-4-9-2': 'The notation is useful here to distinguish this quantity from the physical or thermodynamic entropy [MATH]; it is simply a measure on a probability distribution.', '1901.08956-4-10-0': 'To apply the Shannon measure of information in the quantum mechanical case we consider a Hermitian operator [MATH] written in its eigen-basis.', '1901.08956-4-10-1': '[EQUATION]', '1901.08956-4-10-2': 'We assume that the set of states [MATH] are chosen to form an orthonormal basis.', '1901.08956-4-10-3': 'If a measurement of [MATH] is made, the result will be one of the eigenvalues of [MATH] with probabilities [MATH].', '1901.08956-4-10-4': 'It is natural then to define the Shannon measure on this set of probabilities as the entropy associated with [MATH].', '1901.08956-4-10-5': '[EQUATION]', '1901.08956-4-10-6': 'This quantity was introduced by Ingarden [CITATION] and has been studied by Anza and Vedral, [CITATION], Hu et al. [CITATION], and others.', '1901.08956-4-11-0': 'If the system is in a pure quantum state [MATH], then the probability that a measurement of [MATH] yields [MATH] is given by the Born Rule.', '1901.08956-4-11-1': '[EQUATION]', '1901.08956-4-11-2': 'So that [EQUATION]', '1901.08956-4-11-3': 'Entropy quantifies missing information, so what information is missing?', '1901.08956-4-11-4': 'For an observable [MATH] and a pure state [MATH], [MATH] measures the number of bits of information that are missing from the universe concerning what value of [MATH] will be obtained if a measurement of [MATH] is made.', '1901.08956-4-11-5': 'The quantum state of the system [MATH] contains everything there is to know about the system at time [MATH].', '1901.08956-4-11-6': 'Because of fundamental quantum indeterminism, that is not enough to pin down which eigenvalue of [MATH] will be measured (unless, of course, [MATH] happens to be an eigensate of [MATH]).', '1901.08956-4-11-7': 'We now know from recent Bell test experiments that this indeterminism is a fundamental feature of reality [CITATION].', '1901.08956-4-11-8': 'It is not just a feature of quantum mechanics as we currently understand it, nor is it just an expression of the limited information of an observer.', '1901.08956-4-11-9': 'The quantum operator entropy [MATH] therefore reflects information about the observable Q that is missing from the physical world.', '1901.08956-4-11-10': "It is, in that somewhat strange sense, an objective property of the physical system and not a subjective property of the observer's knowledge.", '1901.08956-4-11-11': 'Perhaps one can speak of Shannon\'s notion of "choice" applying here.', '1901.08956-4-11-12': 'When confronted with a measurement of [MATH], the physical world has [MATH] bits of choice in the outcome that are unconstrained by the physical law, a law which has determined the present [MATH].', '1901.08956-4-11-13': 'The missing information about [MATH] in ([REF]) is really missing.', '1901.08956-4-12-0': 'A mixed state describes a system [MATH] which is either coupled dynamically to a reservoir system [MATH] or has been so in the past.', '1901.08956-4-12-1': 'The quantum state of the composite system is not in general simply a direct product of the state of each subsystem but rather an entangled state [MATH].', '1901.08956-4-12-2': 'The density operator for the composite system is defined by [EQUATION]', '1901.08956-4-12-3': 'The density operator for the system alone is defined using the partial trace over the reservoir degrees of freedom.', '1901.08956-4-12-4': '[EQUATION]', '1901.08956-4-12-5': 'We can write the density operator in the basis of its own eigenstates.', '1901.08956-4-12-6': '[EQUATION]', '1901.08956-4-12-7': "For a pure state only one of the [MATH]'s is nonzero.", '1901.08956-4-13-0': 'The probability [MATH] that a measurement of [MATH] for the system yields [MATH] can be calculated for the mixed state using the density operator and the projection operator onto the [MATH] eigenstate of [MATH].', '1901.08956-4-13-1': '[EQUATION]', '1901.08956-4-13-2': 'The quantum operator entropy for a mixed (or pure) state is then given by applying ([REF]) to ([REF]).', '1901.08956-4-14-0': 'The expression in ([REF]) includes the previous expression in ([REF]) as a special case when [MATH] represents a pure state.', '1901.08956-4-15-0': 'Again, [MATH] is providing a measure (in bits) of missing information.', '1901.08956-4-15-1': 'For a mixed state, the source of this missing information is two-fold.', '1901.08956-4-15-2': 'Quantum indeterminacy still limits the information about a future measurement that is present in the current state of the system.', '1901.08956-4-15-3': "But in addition there is also information missing about the reservoir's state and the mutual information characterizing the entanglement between system and reservoir.", '1901.08956-4-15-4': "The system's reduced density matrix [MATH] is not a complete description of the quantum state of the system, but it is the best possible local description.", '1901.08956-4-15-5': 'For the physical world, there is a fact-of-the-matter about the global quantum state that includes both system and reservoir [MATH].', '1901.08956-4-15-6': 'But the local description of the system alone represented by [MATH] has less information.', '1901.08956-4-16-0': '## Examples of quantum operator entropies', '1901.08956-4-17-0': 'If we use a basis set of discrete position eigenstates [MATH] we can define the quantum operator entropy for [MATH], the position operator.', '1901.08956-4-17-1': '[EQUATION]', '1901.08956-4-17-2': 'The quantum operator entropy for the Hamiltonian [MATH] with allowed energies [MATH] and eigenstates [MATH] is [EQUATION]', '1901.08956-4-17-3': 'The quantum operator entropy for the density operator [MATH] itself, from ([REF]) is [EQUATION]', '1901.08956-4-17-4': 'Because [EQUATION] we can write [EQUATION]', '1901.08956-4-17-5': 'The quantum operator entropy for the density operator is the SMI of the diagonal elements of the density matrix.', '1901.08956-4-17-6': 'We recognize ([REF]) as the expectation value of [MATH], and so write [EQUATION]', '1901.08956-4-17-7': 'The entropy [MATH] is identical to the von Neumann entropy [MATH] (in bits).', '1901.08956-4-17-8': 'For a pure state, the entropy [MATH] is zero.', '1901.08956-4-18-0': 'In general each operator entropy [MATH] can take on different values because each quantifies something different.', '1901.08956-4-18-1': 'If what is known about the system is [MATH], [MATH] is the amount of information that is missing about position, or more precisely, about the outcome of position measurements.', '1901.08956-4-18-2': 'It is the answer to the question: How many bits of information are not known about the outcome of a position measurement if all one knows is [MATH]?', '1901.08956-4-18-3': '[MATH] is the amount of information about energy that is missing.', '1901.08956-4-18-4': '[MATH] is the amount of information that is missing about which quantum state the system will be found in.', '1901.08956-4-18-5': 'It measures the "mixedness" or purity of the state.', '1901.08956-4-18-6': '[MATH], the von Neumann entropy, is invariant under unitary transformations of the basis.', '1901.08956-4-18-7': 'Other operator entropies [MATH] are not invariant-they are tied to the eigen-basis of the particular operator [MATH].', '1901.08956-4-19-0': '# Entropy change of a pure state under unitary evolution', '1901.08956-4-20-0': '## Model system', '1901.08956-4-21-0': 'We consider here a model problem of unitary evolution of a pure state in a closed system with a time-independent Hamiltonian.', '1901.08956-4-21-1': 'The purpose is to examine the different roles played by the von Neumann entropy [MATH], the energy entropy [MATH], and the positional entropy [MATH].', '1901.08956-4-22-0': 'The system consists of a random array of [MATH] fixed sites.', '1901.08956-4-22-1': 'Each site is labeled with an index [MATH] and is at a randomly chosen position [MATH] a unit square.', '1901.08956-4-22-2': 'The site positions are shown graphically in Figure [REF]a as dots.', '1901.08956-4-22-3': 'We use as basis states for the system the set of all the position eigenstates localized on each site, [MATH].', '1901.08956-4-22-4': 'The on-site energy for each is [MATH].', '1901.08956-4-23-0': 'Off-diagonal elements of the Hamiltonian couple each site to several nearby sites with a fixed coupling matrix element [MATH].', '1901.08956-4-23-1': 'Each site is coupled to a randomly chosen subset of its 50 nearest sites.', '1901.08956-4-23-2': 'The coupling to neighbors for six representative sites is shown in Figure [REF]a as lines connecting the dots.', '1901.08956-4-23-3': 'Figure [REF]b shows the connectivity of the sites with a dot on row [MATH], column [MATH], if site [MATH] is coupled to site [MATH].', '1901.08956-4-24-0': 'The Hamiltonian for the system is [EQUATION]', '1901.08956-4-24-1': 'Here [MATH] if two sites are connected, and 0 if they are not.', '1901.08956-4-25-0': 'The randomness in this model minimizes the artifacts of geometric regularity on the dynamics.', '1901.08956-4-25-1': 'We want to see how these different operator entropies change due to fundamental unitary dynamics without the patterns of constructive and destructive interference that dominate, for example, the evolution of a similar system on a regular lattice.', '1901.08956-4-25-2': 'This topologically disordered model is similar to, but distinct from, the Lifshitz model for disordered semiconductors [CITATION] for which [MATH] is simply a function of the distance between sites.', '1901.08956-4-25-3': 'The high multiple connectivity also lets us generalize the interpretation of the model as will be discussed below.', '1901.08956-4-26-0': 'To construct the connections between sites, nine passes through all the sites are made, adding a connection between each site and another site randomly chosen from among its 50 closest neighbors.', '1901.08956-4-26-1': 'The result is that the number of connections for each site varies between 10 and 32, with a mean of 18.', '1901.08956-4-26-2': '(The situation is complicated by the fact that site [MATH] might have site [MATH] as one of its 50 nearest neighbors, while site [MATH] does not have site [MATH] as one of its 50 nearest neighbors.)', '1901.08956-4-27-0': '## Unitary free expansion', '1901.08956-4-28-0': 'The dynamic problem we solve is the expansion of the state from an initially spatially confined state.', '1901.08956-4-28-1': 'The initial state has an equal probability distributed among the 64 sites that are closest to the origin.', '1901.08956-4-28-2': 'We solve for the time development of the state function using the unitary time development operator.', '1901.08956-4-28-3': '[EQUATION]', '1901.08956-4-28-4': 'The expansion of the state into the surrounding state space is shown by the snapshots of the probability density in Figure [REF].', '1901.08956-4-28-5': 'The time scale is set by the characteristic tunneling time between connected sites [EQUATION]', '1901.08956-4-28-6': 'The figure shows how the probability expands much like a classical gas into the available states.', '1901.08956-4-28-7': 'Since the expectation value of the energy is constant during unitary evolution, the system cannot de-excite, and quantum interference fluctuations persist indefinitely.', '1901.08956-4-29-0': 'Figure [REF] shows the the calculated entropies [MATH], [MATH], and [MATH], the von Neumann entropy, during the expansion shown in Figure [REF].', '1901.08956-4-29-1': 'The von Neumann entropy is, of course, constant during the unitary evolution and is in fact 0 because the state is always pure.', '1901.08956-4-30-0': 'The energy entropy [MATH] is also constant during the expansion, but it is not [MATH].', '1901.08956-4-30-1': 'The energy eigenstate occupation probabilities cannot change during unitary time development so [MATH] is independent of time.', '1901.08956-4-30-2': '[MATH] is 4.86 bits (rather than 0) because the initial state is not a Hamiltonian eigenstate, so there are many energy eigenvalues that could be measured.', '1901.08956-4-30-3': '[MATH] characterizes the missing information in the probability distribution of those energy measurements, shown in Figure [REF].', '1901.08956-4-31-0': 'The positional entropy [MATH] characterizes the missing information about position.', '1901.08956-4-31-1': 'At [MATH], it is exactly 6 bits because the probability is uniformly distributed among [MATH] sites.', '1901.08956-4-31-2': 'As the expansion progresses it increase to a value between [MATH] and [MATH].', '1901.08956-4-31-3': 'If the distribution were distributed completely evenly among the [MATH] sites, [MATH] would be 10.', '1901.08956-4-31-4': 'The initial quantum confinement means that the isolated system is excited and it has no way of de-exciting.', '1901.08956-4-31-5': 'If it did, [MATH] would approach a value of 10 bits.', '1901.08956-4-31-6': 'Nevertheless, it is clear that the increase in the quantum mechanical measure [MATH] resembles the increase in the classical thermodynamic entropy of an ideal gas for which [MATH] bits.', '1901.08956-4-31-7': 'Expanding the volume by a factor of 16 would increase the classical statistical mechanical entropy by [MATH] bits.', '1901.08956-4-31-8': 'The dashed line in Figure [REF] is the position entropy of a random superpostition of energy eigenstates, as discussed further in Section [REF].', '1901.08956-4-32-0': 'We emphasise that despite the increase in position entropy [MATH] shown in Figure [REF], the system is evolving in a completely reversible way.', '1901.08956-4-32-1': 'No information about the quantum state is being lost.', '1901.08956-4-32-2': 'Equation ([REF]) can be inverted so we could use the state [MATH] at any time [MATH] to reconstruct precisely the initial state [MATH].', '1901.08956-4-32-3': 'The constant purity of the state is precisely reflected in the unchanging value of the von Neumann entropy [MATH].', '1901.08956-4-32-4': 'What is changing is the amount of missing information in the quantum state about position, and precisely that is quantified by [MATH].', '1901.08956-4-33-0': 'The Hamiltonian for the system given by ([REF]) has eigenvalues [MATH].', '1901.08956-4-33-1': 'It is helpful to scale energies relative to the ground state in units of [MATH].', '1901.08956-4-33-2': '[EQUATION]', '1901.08956-4-33-3': 'The scaled energy eigenvalues for the particular Hamiltonian (i.e., random interconnections) shown in Figures [REF], [REF], and [REF] extend from [MATH] to [MATH].', '1901.08956-4-33-4': 'The expectation value of the scaled energy is [MATH], and is independent of time.', '1901.08956-4-33-5': 'This energy reflects the kinetic energy of confining the system into the initial [MATH] states.', '1901.08956-4-34-0': 'One might describe the apparent saturation of [MATH] as the system "thermalization," but it is important to note that it is not in a thermal equilibrium state.', '1901.08956-4-34-1': 'Figure [REF] shows the probability distribution for the 50 lowest eigenenergies.', '1901.08956-4-34-2': 'The line is a thermal Boltzmann distribution.', '1901.08956-4-34-3': 'Obviously the state is far from being thermal in energy.', '1901.08956-4-34-4': 'Because the time evolution is unitary, this probability distribution is constant in time.', '1901.08956-4-35-0': '## Differing random configurations', '1901.08956-4-36-0': 'The specific configuration of the random local connectivities of the Hamiltonian ([REF]) affects the details of [MATH], but not the overall shape of the saturation to a typical value.', '1901.08956-4-36-1': 'Figure [REF] shows [MATH] and [MATH] for 10 different random configuration, chosen in the same way as describe above, with the initial state localized uniformly across the 64 sites nearest to the origin (hence [MATH]).', '1901.08956-4-36-2': 'Because of the differences in ([REF]), the scaled energy expectation values [MATH] vary between 3.38 and 5.88 across this set of configurations.', '1901.08956-4-36-3': 'The energy operator entropy [MATH] varies similarly between 4.28 and 5.46 bits and are, of course, constant in time.', '1901.08956-4-36-4': 'The position entropy [MATH] in each case increases and saturates around the same values, despite the difference in the configurations.', '1901.08956-4-36-5': 'The variations in time due to quantum interference fluctuations are of the same magnitude as the differences between different configurations.', '1901.08956-4-36-6': 'The dashed line in Figure [REF] shows the position entropy for the average value of [MATH] over 300 random superpositions of the energy eigenstates (RaSEE) for the first of the ten Hamiltonians, [MATH] bits.', '1901.08956-4-36-7': 'The exact value so obtained varies slightly with the specifics of each configuration because the eigenvalue spectrum of each is different in detail.', '1901.08956-4-36-8': 'But again, the variation across configurations is comparable in magnitude to that of the quantum fluctuations in time.', '1901.08956-4-37-0': '## Differing initial confinements', '1901.08956-4-38-0': 'Figure [REF] shows the position entropy [MATH] under unitary time evolution for different initial states.', '1901.08956-4-38-1': 'The Hamiltonian (and random interconnectivity) is identical to that which produced the time development shown in Figures [REF] and [REF].', '1901.08956-4-38-2': 'The initial states are in each case chosen to be a uniform distribution of the [MATH] sites nearest the origin where [EQUATION]', '1901.08956-4-38-3': 'The corresponding values of [MATH] are 2, 3, 4, 5, 6, and 7 bits.', '1901.08956-4-38-4': 'In contrast to the situation of Figure [REF], the Hamiltonian here is exactly the same for all cases; only the initial condition is varied.', '1901.08956-4-38-5': 'The tightly confined states have a higher energy expectation value.', '1901.08956-4-38-6': 'For these initial states with [MATH] from 4 to 128 we have [EQUATION]', '1901.08956-4-38-7': 'The behavior of [MATH] is basically the same in each case, increasing to a value of about 9.4 bits, with persistent quantum oscillations, as the probability expands to fill the available state space.', '1901.08956-4-38-8': 'It is notable that the systems saturate to roughly the same value, though the energy expectation values are very different.', '1901.08956-4-39-0': 'The dashed line in Figure [REF] shows the RaSEE position entropy, averaged over 300 random superpositions of energy eigenstates, with a resulting value of [MATH].', '1901.08956-4-39-1': 'For the ensemble of random superpositions, the expectation values of the energy are high compared with ([REF]), averaging [MATH].', '1901.08956-4-39-2': 'Nevertheless the RaSEE value is remarkably consistent with the saturation value of [MATH] regardless of the initial confinement.', '1901.08956-4-40-0': '## Random superpositions of energy eigenstates (RaSEE)', '1901.08956-4-41-0': 'We have seen in our model system that for different initial confinements and for different random connectivities, unitary time-evolution ([REF]) drives the the position entropy [MATH] toward the same average value, about 9.39 bits, though with small and persistent quantum oscillations.', '1901.08956-4-41-1': 'How could this value be predicted prior to solving the detailed dynamics of ([REF])?', '1901.08956-4-42-0': 'For a given N-dimensional Hamiltonian [MATH] with eigenvalues [MATH] and eigenstates [MATH], we can construct a random superposition of energy eigenstates (RaSEE).', '1901.08956-4-42-1': 'This involves choosing points randomly and uniformly distributed on the surface of the unit sphere in a Hilbert space of [MATH] dimensions ([MATH]).', '1901.08956-4-42-2': 'To do this we use the method of Marsaglia [CITATION].', '1901.08956-4-42-3': 'We first construct a vector [MATH] of length [MATH] whose components [MATH] are random normal deviates with unit variance.', '1901.08956-4-42-4': 'The vector is then normalized to unit length, [MATH], and random phase factors [MATH] are chosen from a uniform distribution over the interval [MATH].', '1901.08956-4-42-5': 'We construct an RaSEE (we pronounce this "racy") state as a weighted sum of the lowest [MATH] energy eigenvectors.', '1901.08956-4-42-6': '[EQUATION]', '1901.08956-4-42-7': 'Figure [REF] shows the values of [MATH], the expectation value of position, and the expectation value of the scaled energy for 900 states RaSEE states created using ([REF]).', '1901.08956-4-42-8': 'We use the same Hamiltonian (with the same connectivity) as that used to calculate the results of Figures [REF] and [REF].', '1901.08956-4-42-9': 'The figure shows the results of limiting the summation in ([REF]) to the lowest [MATH]= 256, 512, or 1024 energy eigenstates, the latter being the full spectrum.', '1901.08956-4-42-10': 'The mean value of [MATH] for all the samples is 9.39 [MATH] 0.02 bits.', '1901.08956-4-42-11': 'Neither the mean value of [MATH] nor the variance is affected by changing [MATH].', '1901.08956-4-42-12': 'The mean value of [MATH] is [MATH] (centered in the unit square), and as [MATH] is raised the variance decreases.', '1901.08956-4-42-13': 'The mean value of [MATH] over the RaSEE samples does increase as larger swaths of the energy spectrum are sampled: [MATH]=13.2 for [MATH]=256, 15.3 for [MATH]=512, and 18.7 for [MATH]=1024.', '1901.08956-4-43-0': 'The RaSEE states are not, of course, stationary states.', '1901.08956-4-43-1': 'Figure [REF] shows how [MATH] and position expectation values vary in time under unitary evolution for ten different RaSEE initial states.', '1901.08956-4-43-2': 'Position expectation values [MATH] and [MATH] fluctuate but stay close to the mean of 1/2.', '1901.08956-4-43-3': 'The [MATH] values also vary in time but remain close to the mean value of 9.39 bits.', '1901.08956-4-43-4': 'If the system size is doubled to N=2048 (not shown), the mean RaSEE value of [MATH] becomes 10.39 [MATH] 0.01 bits- one more bit of missing position information and an even tighter variance.', '1901.08956-4-44-0': 'What we see in Figures [REF] and [REF] is that initially localized states, regardless of how localized they are, expand and [MATH] increases and saturates around a "typical" value of 9.39 bits.', '1901.08956-4-44-1': 'This value comes from the detailed unitary dynamics for different initial states, but is independent of the details.', '1901.08956-4-44-2': 'In fact, we can get a very good estimate of the saturation value of [MATH], without having to solve the full dynamics, by simply picking a single RaSEE state using ([REF]) and calculating the corresponding [MATH].', '1901.08956-4-44-3': 'If we construct a population of such states we get an even better estimate of the [MATH] saturation value.', '1901.08956-4-44-4': 'Even if the disorder is varied, so that the energy eigenvalues and eigenstates are slightly different, as in Figure [REF], we see the saturation value is well-approximated by the RaSEE result for any one of the configurations.', '1901.08956-4-45-0': 'The vast majority of RaSEE states have "typical" values of [MATH], the same values to which unitary dynamics drives the system regardless of its initial confinement, as in Figure [REF].', '1901.08956-4-45-1': 'But RaSEE states are not typical in terms of energy expectation values.', '1901.08956-4-45-2': 'For [MATH]=1024 (full spectrum) energy expectation values for RaSEE states are significantly higher than all but the most confined states.', '1901.08956-4-46-0': 'One might expect that the same typical value of [MATH] could be obtained by any unbiased sampling of the surface of the unit sphere in the [MATH]-dimensional Hilbert space of the the problem, using the eigenstates of any observable.', '1901.08956-4-46-1': 'They are connected by a unitary transformation, so all would seem to be equivalent.', '1901.08956-4-46-2': 'That would be true if we were calculating the expectation value of an observable, which cannot depend on the basis used.', '1901.08956-4-46-3': 'But the operator entropy ([REF]) is not an expectation value and is in fact a very non-linear function of the state.', '1901.08956-4-46-4': 'If we construct, for example a random superposition of position eigenstates rather than energy eigenstates, we get a typical value of [MATH]= 8.95 bits, which does not match the temporal saturation value.', '1901.08956-4-46-5': 'The fact that the saturation value of [MATH] is accurately generated by sampling the RaSEE states is presumably related to the special role of energy in the time propagation of ([REF]).', '1901.08956-4-47-0': '# Second law of thermodynamics', '1901.08956-4-48-0': 'The increase in entropy associated with the free expansion of the system we have seen in Figures [REF] and [REF] suggests a connection to the second law of thermodynamics.', '1901.08956-4-48-1': 'We should note that the von Neuman entropy is constant in these cases.', '1901.08956-4-48-2': 'Also, many limit entropy as a thermodynamic concept to systems in thermal equilibrium with the environment.', '1901.08956-4-48-3': 'This is not the case here (though see Section [REF])-any of the quantum operator entropies are simply properties of a quantum state determined by ([REF]) or ([REF]) and vary in time as the state varies in time.', '1901.08956-4-48-4': 'Each entropy characterizes a quantum state at a particular time (how much information of a particular kind is missing), but an entropy is not an expectation value of an observable.', '1901.08956-4-49-0': 'As Lesovik has pointed out [CITATION], the microscopic origin of the increase of entropy can be attributed to the dynamics of the Schrodinger equation itself.', '1901.08956-4-49-1': 'The spreading of the wave packet, of which the current model is an elaboration, is a feature of the basic structure of the Hamiltonian and the time development operator.', '1901.08956-4-50-0': 'One then has to deal with the apparent paradox of the time-reversibility of ([REF]) in light of the second law.', '1901.08956-4-50-1': 'It is helpful to illustrate this in the current model with its attendant complexity.', '1901.08956-4-50-2': 'Let us start with a confined (localized) state with uniform probability over the 64 sites closes to the origin, such as is shown in Figure [REF]a, and which we will denote [MATH].', '1901.08956-4-50-3': 'Now define a state [MATH] constructed by using the time-reversed version of ([REF]), moving backward in time [MATH] with our model Hamiltonian.', '1901.08956-4-50-4': '[EQUATION]', '1901.08956-4-50-5': 'We now use this state as an initial state, [MATH] and solve ([REF]) forward in time.', '1901.08956-4-50-6': 'The resultant [MATH] is shown in Figure [REF].', '1901.08956-4-50-7': 'We do indeed see second-law-violating behavior for [MATH] inasmuch as there is an abrupt drop in the entropy from its steady state value to a value of 6 bits as the precise combination of magnitudes, phases, and interference combine to reconstruct the localized state at [MATH].', '1901.08956-4-50-8': 'Moreover, the quantum fluctuations prior to and after that point cause [MATH] to move both up and down around the typical level of 9.39 bits (which is a feature of almost any RaSEE state).', '1901.08956-4-50-9': 'So the change in the position entropy moment-by-moment is not monotonically non-negative even away from the sudden recovery of localization.', '1901.08956-4-51-0': 'If one could control all the phases and amplitudes that define the quantum state, nothing in the physical law prevents one from constructing a state like [MATH] that behaves in this second-law-violating way.', '1901.08956-4-51-1': 'Indeed, for a small number of bits Lesovik and coworkers have done just that on the IBM quantum computer [CITATION].', '1901.08956-4-51-2': 'The time-symmetry is broken, not by ([REF]) or by a cosmological condition, but by the difficulty of constructing, artificially or naturally, an initial state sufficiently well tuned in phases and magnitudes to produce even the brief and fleeting reduction in entropy we see here.', '1901.08956-4-51-3': 'This point was made by Lesovik in [CITATION] and is underscored by the present model behavior.', '1901.08956-4-52-0': 'One might suspect that the complex interference that results in the reduction of [MATH] from about 9.39 to 6 is fragile, in the sense that any small perturbation of the initial state [MATH] would destroy it.', '1901.08956-4-52-1': 'To test the resilience of the drop in entropy, we alter the initial state [MATH] by adding to it a fraction of a RaSEE state and renormalizing.', '1901.08956-4-52-2': '[EQUATION]', '1901.08956-4-52-3': 'Here the real scalar [MATH] determines the amount of of the RaSEE state in the initial state.', '1901.08956-4-52-4': 'The inset of Figure [REF] shows [MATH], the value of the position entropy at the moment of localization recovery, as a function of [MATH].', '1901.08956-4-52-5': 'For small values of [MATH], [MATH] at this minimum increases linearly, recovering to the steady-state (and RaSEE) value of 9.39 bits by about [MATH].', '1901.08956-4-52-6': 'Different disordered connectivities produce essentially identical results.', '1901.08956-4-52-7': 'Thus, achieving the sudden moment of localization does not in fact require an exquisite tuning of the initial state.', '1901.08956-4-52-8': 'Getting the initial state slightly wrong will not wipe out the later localization, just diminish it.', '1901.08956-4-53-0': 'Nevertheless, it is worth observing that the momentary localization of the state is very brief, lasting only about [MATH].', '1901.08956-4-53-1': 'Thus in the long history of this carefully prepared system, there is a fleeting blip in time when it spontaneously localizes for a moment.', '1901.08956-4-53-2': 'That moment, if achieved, will almost certainly not be repeated in many lifetimes of the universe.', '1901.08956-4-54-0': '# Thermal ground state', '1901.08956-4-55-0': 'We have considered the time evolution of a pure quantum state during free expansion.', '1901.08956-4-55-1': 'We now briefly describe the application of quantum operator entropies to a state in thermal equilibrium with a reservoir at temperature [MATH].', '1901.08956-4-55-2': 'In that case, the system degrees of freedom interact with the reservoir degrees of freedom so that energy flows between system and reservoir and their quantum states become entangled.', '1901.08956-4-55-3': 'The best local quantum description one can give for the system is then a reduced density matrix ([REF]) where the unknown reservoir degrees of freedom have been traced out.', '1901.08956-4-55-4': 'As Jaynes showed, the optimal reduced density matrix in this case is the one which maximizes the von Neumann entropy over variations of each element of the density matrix [CITATION].', '1901.08956-4-55-5': 'This is the right entropy to maximize because the effect of the reservoir is precisely to mix system states through entanglement with the reservoir and it is this "mixedness" that is quantified by the von Neumann entropy.', '1901.08956-4-55-6': 'The result of the maximum entropy procedure is the canonical expression for the density operator.', '1901.08956-4-55-7': '[EQUATION]', '1901.08956-4-55-8': 'The density matix is diagonal in basis of Hamiltonian eigenstates for the system.', '1901.08956-4-55-9': 'The von Neumannn entropy is the Shannon entropy of the diagonal elements of [MATH], and is therefore equal to the energy entropy, [MATH].', '1901.08956-4-55-10': 'The off-diagonal elements of the density operator in the position basis do not vanish, and the position entropy [MATH] remains as a measure of the missing information about position for the state defined by ([REF]).', '1901.08956-4-56-0': 'Figure [REF] shows the position entropy, von Neumann entropy, and the expectation value of the scaled energy [MATH] as a function of temperature for the model system described by Figures [REF], [REF], and [REF].', '1901.08956-4-56-1': 'In the low temperature limit [MATH] and [MATH] as the system cools to the ground state.', '1901.08956-4-56-2': 'The position entropy of the ground state is in this case [MATH] bits (notably larger than the RaSEE value).', '1901.08956-4-56-3': 'In the high temperature limit both [MATH] and [MATH] go to 10 bits, the completely delocalized state over 1024 sites.', '1901.08956-4-57-0': '# Discussion', '1901.08956-4-58-0': 'The topologically disordered model described here is of more general applicability than it might at first seem.', '1901.08956-4-58-1': 'It is clear that the actual positions of the sites [MATH], the eigenvalues of the position operator, play a limited role in the Hamiltonian ([REF]).', '1901.08956-4-58-2': 'These positions yield a simple algorithm for determining the connectivity between the basis states-random choices from among the 50 nearest neighbors-and a way of visualizing the evolution as in Figure [REF].', '1901.08956-4-58-3': 'But what really matters is just the connectivity that is shown in Figure [REF]b and the coupling strengths.', '1901.08956-4-58-4': 'We could, for example, re-interpret each basis state [MATH] as representing a particular many-body nuclear and electronic configuration for a molecule.', '1901.08956-4-58-5': 'For each configuration there are a set of other accessible configurations that are dynamically coupled by the Hamiltonian, giving a sense of "nearby states," but without a regular pattern.', '1901.08956-4-58-6': 'The essential point is that the increase in [MATH] is capturing quantitatively the very familiar feature of unitary evolution that a localized wavefunction tends to spread out into accessible states, however the states are defined.', '1901.08956-4-58-7': 'Nevertheless, in many cases position eigenstates are especially selected for survival by decoherence through environmental entanglement [CITATION], so it is not a mistake to focus on them here.', '1901.08956-4-59-0': 'Different operator entropy measures allows us to capture different aspects of the dynamics.', '1901.08956-4-59-1': 'The von Neumann entropy [MATH] captures the "mixedness" of a state, which is constant under unitary evolution.', '1901.08956-4-59-2': 'We have seen that by contrast the positional entropy [MATH] increases in a way consistent with the second law of thermodynamics, without recourse to a course-graining procedure.', '1901.08956-4-59-3': 'The second law must ultimately be a feature of physical dynamics.', '1901.08956-4-59-4': 'There is no law of nature that says systems move from less probable states to more probable states.', '1901.08956-4-59-5': 'There is rather just the dynamics of the Schrodinger equation applied to the relevant state space.'}

physics-0606257

{'physics-0606257-1-0-0': 'Larmor precession of laser-polarized atoms contained in anti-relaxation-coated cells, detected via nonlinear magneto-optical rotation (NMOR) is a promising technique for a new generation of ultra-sensitive atomic magnetometers.', 'physics-0606257-1-0-1': 'For magnetic fields directed along the light propagation direction, resonances in NMOR appear when linearly polarized light is frequency- or amplitude-modulated at twice the Larmor frequency.', 'physics-0606257-1-0-2': 'Because the frequency of these resonances depends on the magnitude but not the direction of the field, they are useful for scalar magnetometry.', 'physics-0606257-1-0-3': 'New NMOR resonances at the Larmor frequency appear when the magnetic field is tilted away from the light propagation direction in the plane defined by the light propagation and polarization vectors.', 'physics-0606257-1-0-4': 'These new resonances, studied both experimentally and with a density matrix calculation in the present work, offer a convenient method for NMOR-based vector magnetometry.', 'physics-0606257-1-1-0': 'When linearly polarized light, frequency- or amplitude-modulated at [MATH], resonantly interacts with an atomic vapor in the presence of a magnetic field, the polarization of the light can be observed to rotate synchronously with the modulation.', 'physics-0606257-1-1-1': 'This effect is known as nonlinear magneto-optical rotation with frequency-modulated light (FM NMOR) [CITATION] or amplitude-modulated light (AMOR) [CITATION].', 'physics-0606257-1-1-2': 'It occurs when [MATH] is a subharmonic of the quantum beat frequency; the quantum beat frequency is at the first or second harmonic of the Larmor frequency [MATH] for the lowest-order effect discussed here.', 'physics-0606257-1-1-3': 'Higher-order effects involve quantum beat frequencies at other multiples of [MATH] [CITATION].', 'physics-0606257-1-1-4': 'The width of the resonance between [MATH] and [MATH] is given by the relaxation rate of the atomic ground-state coherences.', 'physics-0606257-1-1-5': 'When the atomic vapor is contained in a paraffin-coated cell, in which ground-state atomic coherences can survive on the order a second, the widths of the resonances can be as small as 0.6 Hz [CITATION].', 'physics-0606257-1-1-6': 'These resonances allow extremely precise measurements of the magnetic field over a wide field range, with magnetometric sensitivities exceeding [MATH] for low fields (competitive with the best atomic magnetometers, see for example Ref. [CITATION], as well as SQUID magnetometers [CITATION]) and reaching [MATH] for higher fields (up to 1 G) [CITATION].', 'physics-0606257-1-1-7': 'The FM NMOR method used in the present work, as well as AMOR, can be applied to studies of nuclear magnetic resonance and magnetic-resonance imaging [CITATION], measurements of geophysical fields [CITATION], and tests of fundamental symmetries [CITATION].', 'physics-0606257-1-1-8': 'The same approach can also be used in construction of chip-scale atomic magnetometers [CITATION].', 'physics-0606257-1-2-0': 'Although the magnetometric method based on FM NMOR enables sensitive measurements of the magnetic field, the measurements are scalar, i.e., the position of a given resonance depends only on the magnitude, and not the direction, of the magnetic field.', 'physics-0606257-1-2-1': 'However, the relative magnitudes of the FM NMOR resonances can depend on the magnetic field direction.', 'physics-0606257-1-2-2': 'Thus, a detailed analysis of the FM NMOR signal could give some information about the direction of the magnetic field.', 'physics-0606257-1-2-3': 'Towards this end, we study here the dependence of the FM NMOR signal on the magnetic field direction.', 'physics-0606257-1-3-0': 'In the Faraday geometry, in which the magnetic field is along the light propagation direction, the main resonance occurs at [MATH], because of the symmetry of the optically pumped state, as discussed below.', 'physics-0606257-1-3-1': 'We find that when the magnetic field direction is tilted in the plane perpendicular to the light polarization axis, only this resonance at [MATH] is observed, with its amplitude depending on the tilt angle.', 'physics-0606257-1-3-2': 'However, when the magnetic field is tilted toward the light polarization axis, a new resonance appears at [MATH]; the relative magnitudes of the resonances at [MATH] and [MATH] depend on the tilt angle.', 'physics-0606257-1-4-0': 'The scheme of the experiment is shown in Fig. [REF].', 'physics-0606257-1-5-0': 'An anti-relaxation-coated buffer-gas-free vapor cell, containing isotopically enriched [MATH]Rb, was placed within a four-layer magnetic shield.', 'physics-0606257-1-5-1': 'The magnetic shield provided passive attenuation of dc magnetic fields by a factor of 10[MATH] [CITATION].', 'physics-0606257-1-5-2': 'A set of three mutually orthogonal magnetic-field coils placed inside the innermost layer enabled compensation of the residual average magnetic field and first-order magnetic field gradients inside the shield.', 'physics-0606257-1-5-3': 'The coils were also used for generation of an arbitrarily oriented magnetic field inside the shield.', 'physics-0606257-1-5-4': 'The rubidium atoms interacted with an [MATH]-directed, 2 mm diameter laser light beam, linearly polarized along the [MATH]-axis.', 'physics-0606257-1-5-5': 'An external cavity diode laser was tuned to the rubidium D[MATH] line (795 nm) and its central frequency was stabilized with a dichroic atomic vapor laser lock [CITATION] at the low-frequency wing of the [MATH] transition.', 'physics-0606257-1-5-6': 'The laser-modulation frequency ranged from 100 Hz to 400 Hz with 300 MHz (peak to peak) modulation depth and light power was [MATH]W. Upon passing through the cell, the light was analyzed using a balanced polarimeter.', 'physics-0606257-1-5-7': 'The polarimetric signal was detected with a lock-in amplifier at the first harmonic of [MATH] and stored on a computer.', 'physics-0606257-1-6-0': 'The FM NMOR signals were studied as the magnetic field direction was tilted in both the [MATH]-plane (perpendicular to the light-polarization axis) and the [MATH]-plane (containing the light-propagation vector and light-polarization axis).', 'physics-0606257-1-6-1': 'Figure [REF] shows the signals with the magnetic field [MATH] in the [MATH]-plane at various angles to the light propagation direction.', 'physics-0606257-1-7-0': 'The strength of the magnetic field was equal for all the measurements ([MATH] Hz).', 'physics-0606257-1-7-1': 'The main resonance occurs at [MATH].', 'physics-0606257-1-7-2': 'The small resonance at [MATH] is a result of using the modulated pump beam as a probe.', 'physics-0606257-1-7-3': 'For the frequency-modulated probe the time dependence of the light-polarization plane is observed, even if the atomic polarization in the medium is static.', 'physics-0606257-1-7-4': 'This effect has been analyzed in detail in Ref. [CITATION] Sec. IV.', 'physics-0606257-1-7-5': 'The amplitudes of the resonances decrease with increasing angle in the [MATH]-plane (Fig. [REF]).', 'physics-0606257-1-8-0': 'The experimental signals are in qualitative agreement with the predictions of a density-matrix calculation of FM NMOR for a model [MATH] transition (Fig. [REF]).', 'physics-0606257-1-9-0': 'The observed dependence on the magnetic-field direction can be understood by considering the time evolution of the atomic polarization.', 'physics-0606257-1-9-1': 'In the absence of a magnetic field, the linearly polarized light creates atomic alignment (polarization with a preferred axis, but no preferred direction) along the [MATH]-axis.', 'physics-0606257-1-9-2': 'This alignment can be visualized with a method described in, for example, Ref. [CITATION]: a surface is plotted whose radius in a given direction is equal to the probability of finding the maximum projection of angular momentum along that direction.', 'physics-0606257-1-9-3': 'When a magnetic field is applied, the alignment precesses around the magnetic-field direction at the Larmor frequency [Fig. [REF](a)].', 'physics-0606257-1-10-0': 'Since in this case [MATH] is perpendicular to the alignment axis, the polarization returns to its original state after a [MATH] rotation, i.e., in half a Larmor period .', 'physics-0606257-1-10-1': 'Thus optical rotation, induced by the rotating linear dichroism, is periodic at twice the Larmor frequency [Fig. [REF](b)].', 'physics-0606257-1-10-2': 'Because only linear dichroism transverse to the light propagation direction (i.e., in the [MATH] plane) can produce rotation, the amplitude of the rotation decreases as the cosine of the angle between [MATH] and the light propagation direction, as seen experimentally in Fig. [REF].', 'physics-0606257-1-11-0': 'Figure [REF] shows the FM NMOR signals for the magnetic field tilted in the [MATH] plane at various angles to the light propagation direction; corresponding theoretical results for a [MATH] transition are shown in Fig. [REF].', 'physics-0606257-1-12-0': 'When the magnetic field is parallel to the light propagation direction a strong FM NMOR resonance is observed only at [MATH].', 'physics-0606257-1-12-1': 'However, when the magnetic field is tilted toward [MATH]-axis an additional resonance appears at [MATH].', 'physics-0606257-1-12-2': 'The amplitude of this resonance becomes more pronounced with increasing the angle, while the amplitude of the resonance at 2[MATH] decreases (Fig. [REF]).', 'physics-0606257-1-13-0': 'However, when the magnetic field is tilted by more than 60[MATH] the amplitude of the resonance recorded at [MATH] also starts to decrease and reaches zero when the magnetic field is directed along the [MATH]-axis.', 'physics-0606257-1-14-0': 'Once again, these signals can be explained by considering the time-evolution of atomic polarization [Fig. [REF](a)].', 'physics-0606257-1-15-0': 'Because the magnetic field is no longer perpendicular to the alignment axis, the polarization takes a full Larmor period to return to its original state.', 'physics-0606257-1-15-1': 'The anisotropy of the medium in the [MATH]-plane is modulated at two frequencies: [MATH] and [MATH], as reflected in the plot of time-dependent optical rotation [Fig. [REF](b)].', 'physics-0606257-1-15-2': 'The larger the tilt angle between [MATH] and the light propagation direction, the bigger the difference between the polarization states at the beginning and middle of the Larmor periods, increasing the signal at [MATH].', 'physics-0606257-1-15-3': 'However, for large angles, the magnetic field direction is nearly along the alignment axis, reducing the effect of the field on the polarization and thus reducing the optical rotation.', 'physics-0606257-1-16-0': 'The data presented in Figs. [REF] and [REF] can be also understood in a language of atomic coherences.', 'physics-0606257-1-16-1': 'When the magnetic field is in the [MATH]-plane, the light polarization axis is perpendicular to the magnetic field.', 'physics-0606257-1-16-2': 'Thus, with the quantization axis along the magnetic field, the light can only create coherence between the [MATH] and [MATH] Zeeman sublevels.', 'physics-0606257-1-16-3': 'The frequency splitting between these sublevels is 2[MATH], so the resonance is observed at this frequency.', 'physics-0606257-1-16-4': 'However, when the magnetic field is tilted in the [MATH]-plane, the light is a linear superposition of polarizations parallel and perpendicular to the magnetic field.', 'physics-0606257-1-16-5': 'In this case, the light can create coherences between sublevels with [MATH] and [MATH], so resonances are observed at both [MATH] and 2[MATH].', 'physics-0606257-1-17-0': 'In conclusion, we have studied resonances in nonlinear magneto-optical rotation with frequency-modulated light (FM NMOR) when the magnetic field is tilted at angles away from the direction of light propagation.', 'physics-0606257-1-17-1': 'When the magnetic field is tilted in the plane defined by the light polarization and light propagation vectors, a new FM NMOR resonance appears at the Larmor frequency.', 'physics-0606257-1-17-2': 'The amplitude of the resonance depends on the tilt angle and thus can be useful for vector magnetometry or for aligning a scalar magnetometer to reduce heading errors.', 'physics-0606257-1-17-3': 'When the magnetic field is tilted in the orthogonal plane, no new resonance appears at the Larmor frequency.', 'physics-0606257-1-17-4': 'For tilt angles in both planes, the amplitude of the FM NMOR resonance at twice the Larmor frequency decreases with increasing tilt angle.', 'physics-0606257-1-17-5': 'The effects have been modeled with density matrix calculations, yielding good agreement with the experimental observations, and can be understood through visualization [CITATION] of the time-dependence of the atomic polarization moments.', 'physics-0606257-1-18-0': 'The authors would like to acknowledge M. Auzinsh, M. P. Ledbetter, A. Cingoz, and J. Higbie for fruitful discussions.', 'physics-0606257-1-18-1': 'This work is supported by DOD MURI grant N-00014-05-1-0406, KBN grant 1 P03B 102 30, and a NSF US-Poland collaboration grant.', 'physics-0606257-1-18-2': 'One of the author (S.P.) is a scholar of the project co-financed from the European Social Fund.'}

{'physics-0606257-2-0-0': 'Larmor precession of laser-polarized atoms contained in anti-relaxation-coated cells, detected via nonlinear magneto-optical rotation (NMOR) is a promising technique for a new generation of ultra-sensitive atomic magnetometers.', 'physics-0606257-2-0-1': 'For magnetic fields directed along the light propagation direction, resonances in NMOR appear when linearly polarized light is frequency- or amplitude-modulated at twice the Larmor frequency.', 'physics-0606257-2-0-2': 'Because the frequency of these resonances depends on the magnitude but not the direction of the field, they are useful for scalar magnetometry.', 'physics-0606257-2-0-3': 'New NMOR resonances at the Larmor frequency appear when the magnetic field is tilted away from the light propagation direction in the plane defined by the light propagation and polarization vectors.', 'physics-0606257-2-0-4': 'These new resonances, studied both experimentally and with a density matrix calculation in the present work, offer a convenient method for NMOR-based vector magnetometry.', 'physics-0606257-2-1-0': 'When linearly polarized light, frequency- or amplitude-modulated at [MATH], resonantly interacts with an atomic vapor in the presence of a magnetic field, the polarization of the light can be observed to rotate synchronously with the modulation.', 'physics-0606257-2-1-1': 'This effect is known as nonlinear magneto-optical rotation with frequency-modulated light (FM NMOR) [CITATION] or amplitude-modulated light (AMOR) [CITATION].', 'physics-0606257-2-1-2': 'It occurs when [MATH] is a subharmonic of the quantum beat frequency; the quantum beat frequency is at the first or second harmonic of the Larmor frequency [MATH] for the lowest-order effect discussed here.', 'physics-0606257-2-1-3': 'Higher-order effects involve quantum beat frequencies at other multiples of [MATH] [CITATION].', 'physics-0606257-2-1-4': 'The width of the resonance between [MATH] and [MATH] is given by the relaxation rate of the atomic ground-state coherences.', 'physics-0606257-2-1-5': 'When the atomic vapor is contained in a paraffin-coated cell, in which ground-state atomic coherences can survive on the order a second, the widths of the resonances can be as small as 0.6 Hz [CITATION].', 'physics-0606257-2-1-6': 'These resonances allow extremely precise measurements of the magnetic field over a wide field range, with magnetometric sensitivities exceeding [MATH] for low fields (competitive with the best atomic magnetometers, see for example Ref. [CITATION], as well as SQUID magnetometers [CITATION]) and reaching [MATH] for higher fields (up to 1 G) [CITATION].', 'physics-0606257-2-1-7': 'The FM NMOR method used in the present work, as well as AMOR, can be applied to studies of nuclear magnetic resonance and magnetic-resonance imaging [CITATION], measurements of geophysical fields [CITATION], and tests of fundamental symmetries [CITATION].', 'physics-0606257-2-1-8': 'The same approach can also be used in construction of chip-scale atomic magnetometers [CITATION].', 'physics-0606257-2-2-0': 'Although the magnetometric method based on FM NMOR enables sensitive measurements of the magnetic field, the measurements are scalar, i.e., the position of a given resonance depends only on the magnitude, and not the direction, of the magnetic field.', 'physics-0606257-2-2-1': 'However, the relative magnitudes of the FM NMOR resonances can depend on the magnetic field direction.', 'physics-0606257-2-2-2': 'Thus, a detailed analysis of the FM NMOR signal could give some information about the direction of the magnetic field.', 'physics-0606257-2-2-3': 'Towards this end, we study here the dependence of the FM NMOR signal on the magnetic field direction.', 'physics-0606257-2-3-0': 'In the Faraday geometry, in which the magnetic field is along the light propagation direction, the main resonance occurs at [MATH], because of the symmetry of the optically pumped state, as discussed below.', 'physics-0606257-2-3-1': 'We find that when the magnetic field direction is tilted in the plane perpendicular to the light polarization axis, only this resonance at [MATH] is observed, with its amplitude depending on the tilt angle.', 'physics-0606257-2-3-2': 'However, when the magnetic field is tilted toward the light polarization axis, a new resonance appears at [MATH]; the relative magnitudes of the resonances at [MATH] and [MATH] depend on the tilt angle.', 'physics-0606257-2-4-0': 'The scheme of the experiment is shown in Fig. [REF].', 'physics-0606257-2-5-0': 'An anti-relaxation-coated buffer-gas-free vapor cell, containing isotopically enriched [MATH]Rb, was placed within a four-layer magnetic shield.', 'physics-0606257-2-5-1': 'The magnetic shield provided passive attenuation of dc magnetic fields by a factor of 10[MATH] [CITATION].', 'physics-0606257-2-5-2': 'A set of three mutually orthogonal magnetic-field coils placed inside the innermost layer enabled compensation of the residual average magnetic field and first-order magnetic field gradients inside the shield.', 'physics-0606257-2-5-3': 'The coils were also used for generation of an arbitrarily oriented magnetic field inside the shield.', 'physics-0606257-2-5-4': 'The rubidium atoms interacted with an [MATH]-directed, 2 mm diameter laser light beam, linearly polarized along the [MATH]-axis.', 'physics-0606257-2-5-5': 'An external cavity diode laser was tuned to the rubidium D[MATH] line (795 nm) and its central frequency was stabilized with a dichroic atomic vapor laser lock [CITATION] at the low-frequency wing of the [MATH] transition.', 'physics-0606257-2-5-6': 'The laser-modulation frequency ranged from 100 Hz to 400 Hz with 300 MHz (peak to peak) modulation depth and light power was [MATH]W. Upon passing through the cell, the light was analyzed using a balanced polarimeter.', 'physics-0606257-2-5-7': 'The polarimetric signal was detected with a lock-in amplifier at the first harmonic of [MATH] and stored on a computer.', 'physics-0606257-2-6-0': 'The FM NMOR signals were studied as the magnetic field direction was tilted in both the [MATH]-plane (perpendicular to the light-polarization axis) and the [MATH]-plane (containing the light-propagation vector and light-polarization axis).', 'physics-0606257-2-6-1': 'Figure [REF] shows the signals with the magnetic field [MATH] in the [MATH]-plane at various angles to the light propagation direction.', 'physics-0606257-2-7-0': 'The strength of the magnetic field was equal for all the measurements ([MATH] Hz).', 'physics-0606257-2-7-1': 'The main resonance occurs at [MATH].', 'physics-0606257-2-7-2': 'The small resonance at [MATH] is a result of using the modulated pump beam as a probe.', 'physics-0606257-2-7-3': 'For the frequency-modulated probe the time dependence of the light-polarization plane is observed, even if the atomic polarization in the medium is static.', 'physics-0606257-2-7-4': 'This effect has been analyzed in detail in Ref. [CITATION] Sec. IV.', 'physics-0606257-2-7-5': 'The amplitudes of the resonances decrease with increasing angle in the [MATH]-plane (Fig. [REF]).', 'physics-0606257-2-8-0': 'The experimental signals are in qualitative agreement with the predictions of a density-matrix calculation of FM NMOR for a model [MATH] transition (Fig. [REF]).', 'physics-0606257-2-9-0': 'The observed dependence on the magnetic-field direction can be understood by considering the time evolution of the atomic polarization.', 'physics-0606257-2-9-1': 'In the absence of a magnetic field, the linearly polarized light creates atomic alignment (polarization with a preferred axis, but no preferred direction) along the [MATH]-axis.', 'physics-0606257-2-9-2': 'This alignment can be visualized with a method described in, for example, Ref. [CITATION]: a surface is plotted whose radius in a given direction is equal to the probability of finding the maximum projection of angular momentum along that direction.', 'physics-0606257-2-9-3': 'When a magnetic field is applied, the alignment precesses around the magnetic-field direction at the Larmor frequency [Fig. [REF](a)].', 'physics-0606257-2-10-0': 'Since in this case [MATH] is perpendicular to the alignment axis, the polarization returns to its original state after a [MATH] rotation, i.e., in half a Larmor period .', 'physics-0606257-2-10-1': 'Thus optical rotation, induced by the rotating linear dichroism, is periodic at twice the Larmor frequency [Fig. [REF](b)].', 'physics-0606257-2-10-2': 'Because only linear dichroism transverse to the light propagation direction (i.e., in the [MATH] plane) can produce rotation, the amplitude of the rotation decreases as the cosine of the angle between [MATH] and the light propagation direction, as seen experimentally in Fig. [REF].', 'physics-0606257-2-11-0': 'Figure [REF] shows the FM NMOR signals for the magnetic field tilted in the [MATH] plane at various angles to the light propagation direction; corresponding theoretical results for a [MATH] transition are shown in Fig. [REF].', 'physics-0606257-2-12-0': 'When the magnetic field is parallel to the light propagation direction a strong FM NMOR resonance is observed only at [MATH].', 'physics-0606257-2-12-1': 'However, when the magnetic field is tilted toward [MATH]-axis an additional resonance appears at [MATH].', 'physics-0606257-2-12-2': 'The amplitude of this resonance becomes more pronounced with increasing the angle, while the amplitude of the resonance at 2[MATH] decreases (Fig. [REF]).', 'physics-0606257-2-13-0': 'However, when the magnetic field is tilted by more than 60[MATH] the amplitude of the resonance recorded at [MATH] also starts to decrease and reaches zero when the magnetic field is directed along the [MATH]-axis.', 'physics-0606257-2-14-0': 'Once again, these signals can be explained by considering the time-evolution of atomic polarization [Fig. [REF](a)].', 'physics-0606257-2-15-0': 'Because the magnetic field is no longer perpendicular to the alignment axis, the polarization takes a full Larmor period to return to its original state.', 'physics-0606257-2-15-1': 'The anisotropy of the medium in the [MATH]-plane is modulated at two frequencies: [MATH] and [MATH], as reflected in the plot of time-dependent optical rotation [Fig. [REF](b)].', 'physics-0606257-2-15-2': 'The larger the tilt angle between [MATH] and the light propagation direction, the bigger the difference between the polarization states at the beginning and middle of the Larmor periods, increasing the signal at [MATH].', 'physics-0606257-2-15-3': 'However, for large angles, the magnetic field direction is nearly along the alignment axis, reducing the effect of the field on the polarization and thus reducing the optical rotation.', 'physics-0606257-2-16-0': 'The data presented in Figs. [REF] and [REF] can be also understood in a language of atomic coherences.', 'physics-0606257-2-16-1': 'When the magnetic field is in the [MATH]-plane, the light polarization axis is perpendicular to the magnetic field.', 'physics-0606257-2-16-2': 'Thus, with the quantization axis along the magnetic field, the light can only create coherence between the [MATH] and [MATH] Zeeman sublevels.', 'physics-0606257-2-16-3': 'The frequency splitting between these sublevels is 2[MATH], so the resonance is observed at this frequency.', 'physics-0606257-2-16-4': 'However, when the magnetic field is tilted in the [MATH]-plane, the light is a linear superposition of polarizations parallel and perpendicular to the magnetic field.', 'physics-0606257-2-16-5': 'In this case, the light can create coherences between sublevels with [MATH] and [MATH], so resonances are observed at both [MATH] and 2[MATH].', 'physics-0606257-2-17-0': 'In conclusion, we have studied resonances in nonlinear magneto-optical rotation with frequency-modulated light (FM NMOR) when the magnetic field is tilted at angles away from the direction of light propagation.', 'physics-0606257-2-17-1': 'When the magnetic field is tilted in the plane defined by the light polarization and light propagation vectors, a new FM NMOR resonance appears at the Larmor frequency.', 'physics-0606257-2-17-2': 'The amplitude of the resonance depends on the tilt angle and thus can be useful for vector magnetometry or for aligning a scalar magnetometer to reduce heading errors.', 'physics-0606257-2-17-3': 'When the magnetic field is tilted in the orthogonal plane, no new resonance appears at the Larmor frequency.', 'physics-0606257-2-17-4': 'For tilt angles in both planes, the amplitude of the FM NMOR resonance at twice the Larmor frequency decreases with increasing tilt angle.', 'physics-0606257-2-17-5': 'The effects have been modeled with density matrix calculations, yielding good agreement with the experimental observations, and can be understood through visualization [CITATION] of the time-dependence of the atomic polarization moments.', 'physics-0606257-2-18-0': 'The authors would like to acknowledge M. Auzinsh, M. P. Ledbetter, A. Cingoz, and J. Higbie for fruitful discussions.', 'physics-0606257-2-18-1': 'This work is supported by DOD MURI grant N-00014-05-1-0406, KBN grant 1 P03B 102 30, and a NSF US-Poland collaboration grant.', 'physics-0606257-2-18-2': 'One of the author (S.P.) is a scholar of the project co-financed from the European Social Fund.'}

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[]

[]

[]

[]

['physics-0606257-1-4-0', 'physics-0606257-2-4-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/physics/0606257

1803.05578

{'1803.05578-1-0-0': 'A2BCD: An Asynchronous Accelerated Block Coordinate Descent Algorithm With Optimal Complexity', '1803.05578-1-1-0': 'Department of Mathematics, University of California, Los Angeles, USA', '1803.05578-1-2-0': 'synchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm ([MATH]), the first asynchronous Nesterov-accelerated algorithm that achieves optimal complexity.', '1803.05578-1-2-1': 'This parallel algorithm solves the unconstrained convex minimization problem, using [MATH] computing nodes which compute updates to shared solution vectors, in an asynchronous fashion with no central coordination.', '1803.05578-1-2-2': 'Nodes in asynchronous algorithms do not wait for updates from other nodes before starting a new iteration, but simply compute updates using the most recent solution information available.', '1803.05578-1-2-3': 'This allows them to complete iterations much faster than traditional ones, especially at scale, by eliminating the costly synchronization penalty of traditional algorithms.', '1803.05578-1-3-0': 'We first prove that [MATH] converges linearly to a solution with a fast accelerated rate that matches the recently proposed [MATH], so long as the maximum delay is not too large.', '1803.05578-1-3-1': 'Somewhat surprisingly, [MATH] pays no complexity penalty for using outdated information.', '1803.05578-1-3-2': 'We then prove lower complexity bounds for randomized coordinate descent methods, which show that [MATH] (and hence [MATH]) has optimal complexity to within a constant factor.', '1803.05578-1-3-3': 'We confirm with numerical experiments that [MATH] outperforms [MATH], which is the current fastest coordinate descent algorithm, even at small scale.', '1803.05578-1-3-4': 'We also derive and analyze a second-order ordinary differential equation, which is the continuous-time limit of our algorithm, and prove it converges linearly to a solution with a similar accelerated rate.', '1803.05578-1-4-0': '2', '1803.05578-1-5-0': '# Introduction', '1803.05578-1-6-0': 'In this paper, we propose and prove the convergence of the Asynchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm ([MATH]), the first asynchronous Nesterov-accelerated algorithm that achieves the state-of-the-art complexity.', '1803.05578-1-6-1': 'We aim to find the minimizer [MATH] of the unconstrained minimization problem: [EQUATION] where [MATH] is [MATH]-strongly convex for [MATH] with [MATH]-Lipschitz gradient [MATH].', '1803.05578-1-6-2': '[MATH] is composed of coordinate blocks [MATH].', '1803.05578-1-6-3': 'The coordinate blocks of the gradient [MATH] are assumed [MATH]-Lipschitz with respect to the [MATH]th block.', '1803.05578-1-6-4': 'That is, [MATH]: [EQUATION] where [MATH] is the projection onto the [MATH]th block of [MATH].', '1803.05578-1-6-5': 'Let [MATH] be the average block Lipschitz constant.', '1803.05578-1-6-6': 'Our algorithm can be applied to non-strongly convex objectives using the black box reduction techniques proposed in [CITATION].', '1803.05578-1-7-0': 'Coordinate descent methods, in which a chosen coordinate block [MATH] is updated at every iteration, are a popular way to solve [REF].', '1803.05578-1-7-1': 'Randomized block coordinate descent (RBCD, [CITATION]) updates a uniformly randomly chosen coordinate block [MATH] with a gradient-descent-like step: [MATH].', '1803.05578-1-7-2': 'This algorithm decreases the error [MATH] to [MATH] in [MATH] iterations.', '1803.05578-1-8-0': "Using a series of averaging and extrapolation steps, accelerated RBCD improves RBCD's iteration complexity [MATH] to [MATH], which leads to much faster convergence when [MATH] is large.", '1803.05578-1-8-1': 'Finally, using a special probability distribution for the random block index [MATH], non-uniform accelerated coordinate descent method [CITATION] (NUACDM) can further decrease the complexity to [MATH], which can be up to [MATH] times faster than accelerated RBCD, since some [MATH] can be significantly smaller than [MATH].', '1803.05578-1-8-2': 'NUACDM is the current state-of-the-art coordinate descent algorithm for solving [REF].', '1803.05578-1-9-0': 'Our [MATH] algorithm generalizes NUACDM to the asynchronous-parallel case.', '1803.05578-1-9-1': 'We solve [REF] with a collection of [MATH] computing nodes that continually read a shared-access solution vector [MATH] into local memory then compute a block gradient [MATH], which is used to update shared solution vectors [MATH].', '1803.05578-1-10-0': 'A traditional synchronous-parallel implementation is organized into rounds of computation: Every computing node must complete an update in order for the next iteration to begin.', '1803.05578-1-10-1': 'However, this synchronization process can be extremely costly, since the lateness of a single node can halt the entire system.', '1803.05578-1-10-2': 'This becomes increasingly problematic with scale, as differences in node computing speeds, load balancing, random network delays, and bandwidth constraints mean that a synchronous-parallel solver may spend more time waiting than computing a solution.', '1803.05578-1-11-0': 'Computing nodes in an asynchronous solver do not wait for others to complete and share their updates before starting the next iteration, but simply continue to update the solution vectors with the most recent information available, without any central coordination.', '1803.05578-1-11-1': 'This eliminates costly idle time, meaning that asynchronous algorithms can be much faster than traditional ones, since they have much faster iterations.', '1803.05578-1-11-2': 'However, since many iterations may occur between the time that a node reads the solution vector, and the time that its computed update is applied, effectively the solution vector is being updated with outdated information.', '1803.05578-1-11-3': 'At iteration [MATH], the block gradient [MATH] is computed at a delayed iterate [MATH] defined as: [EQUATION] for delay parameters [MATH].', '1803.05578-1-11-4': 'Here [MATH] denotes how many iterations out of date coordinate block [MATH] is at iteration [MATH].', '1803.05578-1-11-5': 'Different block may be out of date by different amounts, which is known as an inconsistent read.', '1803.05578-1-11-6': 'We assume that [MATH] for some constant [MATH].', '1803.05578-1-12-0': "Our results: In this paper, we prove that [MATH] attains NUACDM's state-of-the-art iteration complexity to highest order for solving [REF], so long as delays are not too large.", '1803.05578-1-12-1': 'Since asynchronous algorithms have much faster iterations, and [MATH] needs essentially the same number of epochs as NUACDM to compute a solution of a target accuracy, we expect [MATH] to be faster than all existing coordinate descent algorithms.', '1803.05578-1-12-2': 'We confirm this with computational experiments, comparing A2BCD to NUACDM, which is the current fastest block coordinate descent algorithm.', '1803.05578-1-13-0': 'We also prove that A2BCD (and hence NUACDM) has optimal complexity to within a constant factor over a fairly general class of randomized block coordinate descent algorithms.', '1803.05578-1-13-1': 'We do this by proving that any algorithm [MATH] in this class must in general complete at least [MATH] random gradient evaluations to decrease the error by a factor of [MATH].', '1803.05578-1-14-0': 'These results are significant, because it was an open question whether Nesterov-type acceleration was compatible with asynchronicity.', '1803.05578-1-14-1': 'Not only is this possible, but asynchronous algorithms may even attain optimal complexity in this setting.', '1803.05578-1-14-2': 'In light of the above, it also seems plausible that accelerated incremental algorithms for finite sum problems [MATH] can be made asynchronous-parallel, too.', '1803.05578-1-14-3': 'Since asynchronous algorithms have faster iterations, and can attain optimal complexity, we argue that asynchronous algorithms may drastically outperform traditional ones in a wide variety of circumstances.', '1803.05578-1-15-0': 'We also derive a second-order ordinary differential equation (ODE), which is the continuous-time limit of [MATH].', '1803.05578-1-15-1': 'This extends the ODE found in SuBoydCandes2014_differential to an asynchronous accelerated algorithm minimizing a strongly convex function.', '1803.05578-1-15-2': "We prove this ODE linearly converges to a solution with rate similar to [MATH]'s, without needing to resort to the restarting technique employed in SuBoydCandes2014_differential.", '1803.05578-1-15-3': 'We prove this result using techniques that motivate and clarify the proof of the main results, as well as the strategies that we employ (for instance the use of Lyapunov functions).', '1803.05578-1-16-0': '# Main results', '1803.05578-1-17-0': 'We define the condition number [MATH], and let [MATH] be the smallest block Lipschitz constant.', '1803.05578-1-17-1': 'We should consider functions [MATH] where it is efficient to calculate blocks of the gradient, so that coordinate-wise parallelization is efficient.', '1803.05578-1-17-2': 'That is, the function should be "coordinate friendly" [CITATION].', '1803.05578-1-17-3': 'This turns out to be a rather wide class of algorithms.', '1803.05578-1-18-0': 'Even if the objective in not coordinate friendly, it may sometimes be transformed into one that is.', '1803.05578-1-18-1': 'For instance, consider the regularized ERM problem: [EQUATION] for feature vector [MATH] and [MATH] convex.', '1803.05578-1-18-2': 'It is possible to transform this to an equivalent dual problem: [EQUATION] (see [CITATION]).', '1803.05578-1-18-3': 'This dual problem is coordinate friendly, and hence can be solved efficiently by coordinate methods such as [MATH], even though the primal is not.', '1803.05578-1-19-0': "To calculate the [MATH]'th iteration of the algorithm from iteration [MATH], we use a block of the gradient [MATH].", '1803.05578-1-19-1': 'Finally, we assume that the delays [MATH] are independent of the block sequence [MATH], but otherwise arbitrary (though this assumption can be relaxed.', '1803.05578-1-19-2': 'See [CITATION]).', '1803.05578-1-20-0': '[Asynchronous Accelerated Randomized Block Coordinate Descent ([MATH])] Let [MATH] be [MATH]-strongly convex, and let its gradient [MATH] be [MATH]-Lipschitz with block coordinate Lipschitz parameters [MATH] [REF].', '1803.05578-1-20-1': 'Using these parameters, we sample [MATH] in an independent and identically distributed (IID) fashion according to [EQUATION]', '1803.05578-1-20-2': 'Let [MATH] be the maximum asynchronous delay in the parallel system (or an overestimate of this).', '1803.05578-1-20-3': 'We define the asynchronicity parameter [MATH], which is proportional to the maximum delay [MATH], and quantifies how strongly asynchronicity will affect convergence: [EQUATION]', '1803.05578-1-20-4': 'We use the above system parameters and [MATH] to define the coefficients: [EQUATION]', '1803.05578-1-20-5': 'The [MATH] algorithm is hence defined via the iteration: [EQUATION]', '1803.05578-1-20-6': 'Here [MATH] may be underestimated, and [MATH] may be overestimated if exact values are unavailable.', '1803.05578-1-20-7': 'Notice that [MATH] can be eliminated from the above iteration, and the block gradient [MATH] only needs to be calculated once per iteration.', '1803.05578-1-20-8': 'A larger (or overestimated) maximum delay [MATH] will cause a larger asynchronicity parameter [MATH], which leads to more conservative step sizes to compensate.', '1803.05578-1-21-0': 'The convergence of this algorithm can be stated in terms of a Lyapunov function that we will define shortly.', '1803.05578-1-21-1': 'We first introduce the asynchronicity error, a powerful tool for analyzing asynchronous algorithms used in several recent works [CITATION].', '1803.05578-1-21-2': 'This error is a weighted sum of the history of the algorithm, with the weights [MATH] decreasing as one goes further back in time.', '1803.05578-1-21-3': 'This error appears naturally in the analysis.', '1803.05578-1-21-4': 'Much like a well-chosen basis in linear algorithm, it appears to be a natural quantity to consider when analyzing convergence of asynchronous algorithms.', '1803.05578-1-22-0': '[Asynchronicity error] Using the above parameters, we define: [EQUATION] for coefficients [EQUATION]', '1803.05578-1-22-1': 'Here we define [MATH] for all [MATH].', '1803.05578-1-23-0': 'The determination of the coefficients [MATH] is in general a very involved process of trial and error, intuition, and balancing competing requirements.', '1803.05578-1-23-1': 'Obtaining a convergence proof and optimal rates relies on the skillful choice of these coefficients [MATH].', '1803.05578-1-23-2': "The algorithm doesn't depend on the coefficients, however; they are only used in the analysis.", '1803.05578-1-24-0': 'We define [MATH] as the expectation of [MATH] conditional on conditioned on [MATH], [MATH], [MATH], and [MATH].', '1803.05578-1-24-1': 'To simplify notation, we assume that the minimizer [MATH], and that [MATH] with no loss in generality.', '1803.05578-1-24-2': 'To present our convergence results, we define the Lyapunov function: [EQUATION] for [EQUATION]', '1803.05578-1-24-3': 'We will show this Lyapunov function converges linearly to zero in expectation.', '1803.05578-1-24-4': 'We define the iteration complexity [MATH] with respect to some error [MATH] as the number of iterations [MATH] such that the expected error [MATH] decreases to less than [MATH].', '1803.05578-1-25-0': "We now present this paper's first main contribution.", '1803.05578-1-25-1': '[EQUATION]', '1803.05578-1-25-2': 'This result is proven in Section [REF].', '1803.05578-1-25-3': 'In practice, asynchronous algorithms are far more resilient to delays than the theory predicts.', '1803.05578-1-25-4': '[MATH] can be much larger without negatively affecting the convergence rate and complexity.', '1803.05578-1-25-5': 'This is perhaps because we are limited to a worst-case analysis, which is not representative of the average-case performance.', '1803.05578-1-25-6': 'Hence in our numerical experiments, we generously underestimate [MATH], and the algorithm still converges without any apparent penalty.', '1803.05578-1-26-0': 'Authors in [CITATION] (Theorem 5.1) obtain a linear convergence rate of [MATH] for NUACDM which leads to the corresponding iteration complexity of [MATH] as [MATH].', '1803.05578-1-26-1': 'And hence, we have: [EQUATION]', '1803.05578-1-26-2': 'Hence when [MATH], or equivalently, when [EQUATION] the complexity of [MATH] asymptotically matches that of NUACDM.', '1803.05578-1-26-3': 'Hence [MATH] combines state-of-the-art complexity with the faster iterations and superior scaling that asynchronous iterations allow.', '1803.05578-1-27-0': 'We now present some special cases of the conditions on the maximum delay [MATH] required for good complexity.', '1803.05578-1-28-0': 'Let the conditions of Theorem [REF] hold.', '1803.05578-1-28-1': 'Additionally, let all coordinate-wise Lipschitz constants [MATH] be equal (i.e. [MATH]).', '1803.05578-1-28-2': 'Then we have [MATH] when [EQUATION]', '1803.05578-1-28-3': 'Further, let us assume all coordinate-wise Lipschitz constants [MATH] equal [MATH].', '1803.05578-1-28-4': 'Then [MATH], when [EQUATION]', '1803.05578-1-29-0': '[Reduction to synchronous case] Notice that when [MATH], we have [MATH], [MATH] and hence [MATH].', '1803.05578-1-29-1': 'Thus [MATH] becomes equivalent to NUACDM, the Lyapunov function [MATH] becomes equivalent to one found in [CITATION](pg.', '1803.05578-1-29-2': '9), and Theorem [REF] yields the same complexity to highest order.', '1803.05578-1-30-0': '## Optimality', '1803.05578-1-31-0': '[MATH] and hence [MATH] are in fact optimal among a wide class of randomized block gradient algorithms.', '1803.05578-1-31-1': 'We consider a class of algorithms slightly wider than that considered in [CITATION] to encompass asynchronous algorithms.', '1803.05578-1-31-2': 'The authors also only considered functions [MATH]-strongly convex functions [MATH] with [MATH]-Lipschitz gradient [MATH], and did not consider function with coordinate Lipschitz constants [MATH].', '1803.05578-1-31-3': 'For a subset [MATH], we let [MATH] (inconsistent read) denote the set of vectors [MATH] such that [EQUATION] for some vectors [MATH].', '1803.05578-1-31-4': 'Here [MATH] denotes the [MATH]th component of vector [MATH].', '1803.05578-1-31-5': 'That is, the coordinates of [MATH] are some combination of coordinates of vectors in [MATH].', '1803.05578-1-31-6': 'Let [MATH].', '1803.05578-1-32-0': '[Asynchronous Randomized Incremental Algorithms Class] Consider the unconstrained minimization problem [REF] for [MATH] that is [MATH]-strongly convex with [MATH]-Lipschitz gradient, with block-coordinate-wise Lipschitz constants [MATH].', '1803.05578-1-32-1': 'We define the class [MATH] as algorithms [MATH] on this problem such that:', '1803.05578-1-33-0': '1.', '1803.05578-1-33-1': 'For each parameter set [MATH], [MATH] has an associated IID random variable [MATH] with some fixed distribution [MATH] for [MATH].', '1803.05578-1-34-0': 'This is a rather general class: [MATH] can be constructed from any inconsistent reading of past iterates [MATH], and any past gradient of an inconsistent read [MATH].', '1803.05578-1-35-0': 'For any algorithm [MATH], and parameter set [MATH] for the unconstrained minimization problem, and [MATH], there is a dimension [MATH], a corresponding function [MATH] on [MATH], and a starting point [MATH], such that [EQUATION]', '1803.05578-1-35-1': 'Hence the complexity [MATH] for all algorithms in [MATH] satisfies [EQUATION]', '1803.05578-1-35-2': 'Our proof in Section [REF] follows very similar lines to LanZhou2015_optimal, which is inspired by Nesterov (see for instance Nesterov2013_introductory).', '1803.05578-1-36-0': '# ODE Analysis', '1803.05578-1-37-0': 'In this section we present and analyze an ODE which is the continuous-time limit of [MATH].', '1803.05578-1-37-1': 'This ODE is a strongly convex, and asynchronous version of the ODE found in [CITATION].', '1803.05578-1-37-2': 'For simplicity, assume [MATH].', '1803.05578-1-37-3': 'We rescale [MATH] so that the strong convexity modulus [MATH], and hence [MATH].', '1803.05578-1-38-0': 'Taking the discrete limit of synchronous [MATH] (i.e. accelerated RBCD), we can derive the following ODE (Section [REF]): [EQUATION]', '1803.05578-1-38-1': 'For compactness, we have omitted the [MATH] from time-varying functions [MATH], [MATH], [MATH], etc.', '1803.05578-1-38-2': 'We define the parameter [MATH], and the energy: [EQUATION]', '1803.05578-1-38-3': 'This is very similar to the Lyapunov function discussed in Section [REF], with [MATH] fulfilling the role of [MATH], and [MATH] (since there is no delay yet).', '1803.05578-1-38-4': 'Much like the traditional analysis in the proof of Theorem [REF], we can derive a linear convergence result that has a similar rate.', '1803.05578-1-39-0': 'If [MATH] satisfies [REF], then the energy [MATH] satisfies [MATH].', '1803.05578-1-39-1': 'This implies [MATH], which is equivalent to: [EQUATION]', '1803.05578-1-39-2': 'See Section [REF].', '1803.05578-1-40-0': 'We may also analyze an asynchronous version of [REF] to motivate our proof of the main theorem in Section [REF]: [EQUATION] [MATH] is a delayed version of [MATH] defined similarly to [REF], with the delay bounded by [MATH].', '1803.05578-1-41-0': 'Unfortunately, the energy satisfies (see Section [REF], [REF]): [EQUATION]', '1803.05578-1-41-1': 'Since the right-hand side can be positive, the energy [MATH] is not decreasing in general.', '1803.05578-1-41-2': 'But, we may add a continuous-time asynchronicity error (see [CITATION]), much like in Definition [REF], to create a decreasing energy.', '1803.05578-1-41-3': 'Let [MATH] and [MATH] be arbitrary constants that will be set later.', '1803.05578-1-41-4': 'Define: [EQUATION]', '1803.05578-1-41-5': 'When [MATH], the asynchronicity error [MATH] satisfies: [EQUATION]', '1803.05578-1-41-6': 'See Section [REF].', '1803.05578-1-42-0': 'Adding this error to the Lyapunov function serves a similar purpose in the continuous-time case as in the proof of Theorem [REF] (see Lemma [REF]).', '1803.05578-1-42-1': 'It allows us to negate [MATH] units of [MATH] for the cost of creating [MATH] units of [MATH].', '1803.05578-1-43-0': 'Notice how this convergence condition is similar to Corollary [REF], but a little looser.', '1803.05578-1-43-1': '[EQUATION]', '1803.05578-1-43-2': 'The last inequality followed from the condition on [MATH].', '1803.05578-1-44-0': 'The preceding should act as a guide to understanding the convergence of [MATH].', '1803.05578-1-44-1': 'It may make clearer the use of the Lyapunov function to establish convergence in the synchronous case, the function of the asynchronicity error in the proof of the asynchronous case, and may hopefully elucidate the logic and general strategy of the proof in Section [REF].', '1803.05578-1-45-0': '# Parallel implementation', '1803.05578-1-46-0': 'We now describe a parallel shared-memory implementation of [MATH].', '1803.05578-1-46-1': 'We have a collection of [MATH] computing nodes whose role is to pick block indices [MATH] and compute the corresponding block gradients [MATH], and then use these gradient to update the shared solution vectors.', '1803.05578-1-46-2': 'We have eliminated [MATH] from the iteration for simplicity.', '1803.05578-1-46-3': '[H] Shared-memory implementation of A2BCD.', '1803.05578-1-47-0': '[1] Inputs: Function parameters [MATH], [MATH], [MATH].', '1803.05578-1-47-1': 'Implementation parameters [MATH], [MATH].', '1803.05578-1-47-2': 'Shared data: Solution vectors [MATH] and [MATH].', '1803.05578-1-47-3': 'Iteration counter [MATH].', '1803.05578-1-47-4': 'Node local data: Block index [MATH], working gradient blocks [MATH].', '1803.05578-1-47-5': 'Algorithm parameters [MATH].', '1803.05578-1-48-0': 'Calculate parameters [MATH] via [REF].', '1803.05578-1-48-1': 'Set [MATH].', '1803.05578-1-48-2': 'not converged, nodes [MATH] Choose [MATH] in an IID fashion from [MATH] according to distribution [REF].', '1803.05578-1-48-3': 'Potentially read [MATH] from shared memory into [MATH], depending on a reading-frequency policy.', '1803.05578-1-48-4': 'Calculate the gradient [MATH] Output [MATH].', '1803.05578-1-49-0': 'Synchronous algorithms must read the shared solution vectors at every iteration.', '1803.05578-1-49-1': 'However nodes in [MATH] can use far less network bandwidth by reading these vectors only occasionally, at the expense of increasing the maximum delay [MATH].', '1803.05578-1-49-2': "Hence there will be a continuous trade-off between conserving bandwidth (by reading less frequently) and keeping iteration complexity low (by making sure [MATH] and hence [MATH] doesn't get too large) that can be tuned for added performance.", '1803.05578-1-50-0': '# Related work', '1803.05578-1-51-0': '## Acceleration', '1803.05578-1-52-0': "Continuing our discussion from Section [REF], Nesterov acceleration is a method for improving an algorithm's iteration complexity's dependence the condition number [MATH].", '1803.05578-1-52-1': 'Nesterov-accelerated methods have been proposed and discovered in many settings [CITATION], including for coordinate descent algorithms (algorithms that use 1 gradient block [MATH] or minimize with respect to [MATH] coordinate block per iteration), and incremental algorithms (algorithms for finite sum problems [MATH] that use [MATH] function gradient [MATH] per iteration).', '1803.05578-1-52-2': 'Such algorithms can often be augmented to solve composite minimization problems (minimization for objective of the form [MATH], especially for nonsomooth [MATH]), or include constraints.', '1803.05578-1-53-0': 'When all block coordinate Lipschitz constants [MATH] are equal, accelerated RBCD has a complexity of [MATH], which is optimal among a certain class of randomized block coordinate descent algorithms to within a constant factor (see Section [REF]).', '1803.05578-1-53-1': 'However Allen-ZhuQuRichtarikYuan2015_even assuming coordinate-wise Lipschitz constants, and using a non-uniform distribution over block indices [MATH] allows them to improve the complexity to [MATH], which can be up to [MATH] faster than accelerated RBCD.', '1803.05578-1-53-2': 'Our [MATH] has the same complexity as the NUACDM algorithms proposed in Allen-ZhuQuRichtarikYuan2015_even to highest order.', '1803.05578-1-53-3': 'This complexity is optimal to within a constant factor, which is shown in this paper.', '1803.05578-1-54-0': '## Asynchronous algorithms', '1803.05578-1-55-0': 'Asynchronous algorithms were proposed in ChazanMiranker1969_chaotic to solve linear systems.', '1803.05578-1-55-1': 'General convergence results and theory were developed later in Bertsekas1983_distributed,BertsekasTsitsiklis1997_parallel,TsengBertsekasTsitsiklis1990_partially,LuoTseng1992_convergence,LuoTseng1993_convergence,Tseng1991_rate for partially and totally asynchronous systems, with essentially-cyclic block sequence [MATH].', '1803.05578-1-55-2': 'More recently, there has been renewed interest in asynchronous algorithms with random block coordinate updates.', '1803.05578-1-55-3': 'Linear and sublinear convergence results were proven for asynchronous RBCD LiuWright2015_asynchronous,AvronDruinskyGupta2014, and similar was proven for asynchronous SGD RechtReWrightNiu2011_hogwild, and variance reduction algorithms ReddiHefnySraPoczosSmola2015_variance,LeblondPedregosaLacoste-Julien2017_asaga.', '1803.05578-1-56-0': 'In PengXuYanYin2016_arock, authors proposed and analyzed an asynchronous fixed-point fixed-point algorithm called ARock, that takes proximal algorithms, forward-backward, ADMM, etc. as special cases.', '1803.05578-1-56-1': 'HannahYin2017_unboundeda,SunHannahYin2017_asynchronousb,PengXuYanYin2016_convergence showed that many of the assumptions used in prior work (such as bounded delay [MATH]) were unrealistic and unnecessary in general.', '1803.05578-1-56-2': 'In HannahYin2017_more the authors showed that asynchronous iterations will complete far more iterations per second, and that a wide class of asynchronous algorithms, including asynchronous [MATH], have the same iteration complexity as their traditional counterparts.', '1803.05578-1-56-3': 'Hence certain asynchronous algorithms can be expected to significantly outperform traditional ones.', '1803.05578-1-57-0': 'In [CITATION] authors propose a novel asynchronous catalyst-accelerated [CITATION] primal-dual algorithmic framework to solve regularized ERM problems.', '1803.05578-1-57-1': 'Instead of using outdated information, they structure the parallel updates so that the data that an update depends on is up to date (though the rest of the data may not be).', '1803.05578-1-57-2': 'However catalyst acceleration incurs a logarithmic penalty over Nesterov acceleration in general.', '1803.05578-1-58-0': '# Numerical experiments', '1803.05578-1-59-0': 'To investigate the performance of [MATH], we performed numerical tests on the ridge regression problem: [EQUATION] where [MATH] contains [MATH] samples and [MATH] features, and [MATH] is the label vector.', '1803.05578-1-59-1': 'The dual formulation of [REF] is (see for instance [CITATION]) [EQUATION] for an objective [MATH] which is a strongly convex with Lipschitz gradient equal to [EQUATION]', '1803.05578-1-59-2': 'We compare three algorithms: A2BCD (which is asynchronous accelerated), synchronous NUACDM (which is synchronous accelerated), and asynchronous RBCD (which is asynchronous and non-accelerated).', '1803.05578-1-59-3': 'At each iteration, each thread selects one block [MATH] based on the probability distribution [REF], reads the shared solution vector, and computes [MATH], (see Algorithm [REF]).', '1803.05578-1-59-4': 'However the threads in synchronous NUACDM must wait until the updates from all other threads are received and applied to the solution vector before they can start the next iteration.', '1803.05578-1-59-5': 'The nodes of asynchronous algorithms do not wait, but simply compute with the most recent updates from the other nodes.', '1803.05578-1-60-0': 'We use the dataset w8a.t (14951 samples and 300 features) from LIBSVM [CITATION].', '1803.05578-1-60-1': 'The algorithm is implemented in a multi-threaded fashion using C++11 and GNU Scientific Library.', '1803.05578-1-60-2': 'We used 40 threads on two 2.5GHz 10-core Intel Xeon E5-2670v2 processors.', '1803.05578-1-60-3': 'During our tests, we set [MATH] (since underestimation of this parameter leads to better performance) and used a uniform Lipshitz constant for all blocks, and [MATH].', '1803.05578-1-60-4': 'For simplicity, we first compute and cache [MATH] in a dense format.', '1803.05578-1-61-0': 'To compare solution times, we first ensure perfect load balancing.', '1803.05578-1-61-1': 'We see that A2BCD slightly outperforms NUACDM (see figure [REF]).', '1803.05578-1-61-2': 'This is because both have similar complexity and complete a similar number of iterations per second.', '1803.05578-1-61-3': 'However A2BCD will significantly outperform NUACDM when the synchronization penalty becomes larger (see next paragraph).', '1803.05578-1-61-4': 'This will be the case when the computation of updates [MATH] is of heterogeneous difficulty, when there are large differences in the computational power of the nodes, or when there is significant network delay.', '1803.05578-1-61-5': 'These factors are the hallmarks of large-scale applications.', '1803.05578-1-62-0': 'To better simulate real-world applications, we measure performance when the the subproblem difficulty is heterogeneous (i.e. imperfect load balancing).', '1803.05578-1-62-1': 'We do this with an uneven coordinate block distribution, using [MATH] large blocks (200 coordinates per such block), and [MATH] small blocks (20 coordinates per such block).', '1803.05578-1-62-2': 'The threads continue to pick blocks in the same fashion as the last test.', '1803.05578-1-62-3': 'This imbalance means that A2BCD will complete epochs significantly faster than synchronous NUACDM (see Figure [REF]).', '1803.05578-1-62-4': 'Therefore, since A2BCD still has the same complexity as NUACDM (see Figure [REF]), and completes epochs much faster, we expect it to compute a solution of a given accuracy much faster.', '1803.05578-1-62-5': 'Our experiments confirm this (see Figure [REF]).', '1803.05578-1-63-0': '[1]ReferencesReferences We first recall a couple of inequalities for convex functions.', '1803.05578-1-64-0': 'Let [MATH] be [MATH]-strongly convex with [MATH]-Lipschitz gradient.', '1803.05578-1-64-1': 'Then we have: [EQUATION]', '1803.05578-1-65-0': 'We also find it convenient to define the norm: [EQUATION]', '1803.05578-1-66-0': '## Starting point', '1803.05578-1-67-0': 'First notice that using the definition [REF] of [MATH] we have: [EQUATION]', '1803.05578-1-67-1': 'We have the following general identity: [EQUATION]', '1803.05578-1-67-2': 'It can also easily be verified from [REF] that we have: [EQUATION]', '1803.05578-1-67-3': 'Using [REF] on term [MATH], [REF] on term [MATH], and recalling the definition [REF] on term [MATH], we have from [REF]: [EQUATION]', '1803.05578-1-67-4': 'This inequality is our starting point.', '1803.05578-1-67-5': 'We analyze the terms on the second line in the next section.', '1803.05578-1-68-0': '## The Cross Term', '1803.05578-1-69-0': 'To analyze these terms, we need a small lemma.', '1803.05578-1-69-1': 'This lemma is fundamental in allowing us to deal with asynchronicity.', '1803.05578-1-70-0': 'Let [MATH].', '1803.05578-1-70-1': 'Let the delay be bounded by [MATH].', '1803.05578-1-70-2': 'Then: [EQUATION]', '1803.05578-1-70-3': 'See [CITATION].', '1803.05578-1-71-0': 'We have: [EQUATION]', '1803.05578-1-71-1': 'The terms in bold in [REF] and [REF] are a result of the asynchronicity, and are identically [MATH] in its absence.', '1803.05578-1-72-0': 'Our strategy is to separately analyze terms that appear in the traditional analysis of [CITATION], and the terms that result from asynchronicity.', '1803.05578-1-72-1': 'We first prove [REF]: [EQUATION] [REF] follows from strong convexity ([REF] with [MATH] and [MATH]), and the fact that [MATH] is [MATH]-Lipschitz.', '1803.05578-1-72-2': 'The term due to asynchronicity becomes: [EQUATION] using Lemma [REF] with [MATH].', '1803.05578-1-72-3': 'Combining this with [REF] completes the proof of [REF].', '1803.05578-1-73-0': 'We now prove [REF]: [EQUATION]', '1803.05578-1-73-1': 'Here the last line follows from Lemma [REF] with [MATH], [MATH].', '1803.05578-1-73-2': 'We can complete the proof using the following identity that can be easily obtained from [REF]: [EQUATION]', '1803.05578-1-74-0': '## Function-value term', '1803.05578-1-75-0': 'Much like [CITATION], we need a [MATH] term in the Lyapunov function (see the middle of page [MATH]).', '1803.05578-1-75-1': 'However we additionally need to consider asynchronicity when analyzing the growth of this term.', '1803.05578-1-75-2': 'Again terms due to asynchronicity are emboldened.', '1803.05578-1-76-0': 'We have: [EQUATION]', '1803.05578-1-76-1': 'From the definition [REF] of [MATH], we can see that [MATH] is supported on block [MATH].', '1803.05578-1-76-2': 'Since each gradient block [MATH] is [MATH] Lipschitz with respect to changes to block [MATH], we can use [REF] to obtain: [EQUATION]', '1803.05578-1-76-3': 'Here the last line followed from the definition [REF] of the norm [MATH].', '1803.05578-1-76-4': 'We now analyze the middle term: [EQUATION]', '1803.05578-1-76-5': 'We then apply Lemma [REF] to this with [MATH] to yield: [EQUATION]', '1803.05578-1-76-6': 'Finally to complete the proof, we combine [REF], with [REF].', '1803.05578-1-77-0': '## Asynchronicity error', '1803.05578-1-78-0': 'The previous inequalities produced difference terms of the form [MATH].', '1803.05578-1-78-1': 'The following lemma shows how these errors can be incorporated into a Lyapunov function.', '1803.05578-1-79-0': 'Let [MATH] and consider the asynchronicity error and corresponding coefficients: [EQUATION]', '1803.05578-1-79-1': 'This sum satisfies: [EQUATION] [Interpretation] This result means that an asynchronicity error term [MATH] can negate a series of difference terms [MATH] at the cost of producing an additional error [MATH], while maintaining a convergence rate of [MATH].', '1803.05578-1-79-2': 'This essentially converts difference terms, which are hard to deal with, into a [MATH] term which can be negated by other terms in the Lyapunov function.', '1803.05578-1-79-3': 'The proof is straightforward.', '1803.05578-1-79-4': '[EQUATION]', '1803.05578-1-79-5': 'Noting the following completes the proof: [EQUATION]', '1803.05578-1-79-6': 'Given that [MATH] allows us to negate difference terms, we now analyze the cost [MATH] of this negation.', '1803.05578-1-80-0': 'We have: [EQUATION]', '1803.05578-1-80-1': 'Here [REF] following from [REF], the definition of [MATH].', '1803.05578-1-80-2': '[REF] follows from the inequality [MATH].', '1803.05578-1-80-3': 'The rest is simple algebraic manipulation.', '1803.05578-1-80-4': '[EQUATION]', '1803.05578-1-80-5': 'Finally, to complete the proof, we prove [MATH].', '1803.05578-1-80-6': '[EQUATION]', '1803.05578-1-80-7': 'Rearranging the definition of [MATH], we have: [EQUATION]', '1803.05578-1-80-8': 'Using this on [REF], we have: [EQUATION]', '1803.05578-1-80-9': 'This completes the proof.', '1803.05578-1-81-0': '## Master inequality', '1803.05578-1-82-0': 'We are finally in a position to bring together all the all the previous results together into a master inequality for the Lyapunov function [MATH] (defined in [REF]).', '1803.05578-1-82-1': 'After this lemma is proven, we will prove that the right hand size is negative, which will imply that [MATH] linearly converges to [MATH] with rate [MATH].', '1803.05578-1-83-0': '[Master inequality] We have: [EQUATION]', '1803.05578-1-83-1': 'We now collect and organize the similar terms of this inequality.', '1803.05578-1-83-2': '[EQUATION]', '1803.05578-1-83-3': 'Now finally, we add the function-value and asynchronicity terms to our analysis.', '1803.05578-1-83-4': 'We use Lemma [REF] is with [MATH], and [EQUATION]', '1803.05578-1-83-5': 'Notice that this choice of [MATH] will recover the coefficient formula given in [REF].', '1803.05578-1-83-6': 'Hence we have: [EQUATION]', '1803.05578-1-83-7': 'Notice [MATH].', '1803.05578-1-83-8': 'Finally, combining [REF] and [REF] completes the proof.', '1803.05578-1-84-0': 'In the next section, we will prove that every coefficient on the right hand side of [REF] is [MATH] or less, which will complete the proof of Theorem [REF].', '1803.05578-1-85-0': '## Proof of main theorem', '1803.05578-1-86-0': 'The coefficients of [MATH], [MATH], and [MATH] in Lemma [REF] are non-positive.', '1803.05578-1-87-0': 'The coefficient [MATH] of [MATH] is identically [MATH] via the definition [REF] of [MATH].', '1803.05578-1-87-1': 'The coefficient [MATH] of [MATH] is identically [MATH] via the definition [REF] of [MATH].', '1803.05578-1-88-0': 'First notice from the definition [REF] of [MATH]: [EQUATION]', '1803.05578-1-88-1': 'Here the last line followed since [MATH] and [MATH].', '1803.05578-1-88-2': 'We now analyze the coefficient of [MATH].', '1803.05578-1-88-3': '[EQUATION]', '1803.05578-1-88-4': 'The coefficient [MATH] of [MATH] in Lemma [REF] is non-positive.', '1803.05578-1-88-5': '[EQUATION]', '1803.05578-1-88-6': 'The coefficient [MATH] of [MATH] in Lemma [REF] is non-positive.', '1803.05578-1-89-0': 'We first need to bound [MATH].', '1803.05578-1-89-1': '[EQUATION]', '1803.05578-1-89-2': 'It can be easily verified that if [MATH] and [MATH], then [MATH].', '1803.05578-1-89-3': 'Using this fact with [MATH] and [MATH], we have: [EQUATION]', '1803.05578-1-89-4': 'We now analyze the coefficient of [MATH] [EQUATION]', '1803.05578-1-89-5': 'The coefficient [MATH] of [MATH] in [REF] is non-positive.', '1803.05578-1-89-6': '[EQUATION]', '1803.05578-1-89-7': 'Here the last inequality follows since [MATH] and [MATH].', '1803.05578-1-89-8': 'We now rearrange the definition of [MATH] to yield the identity: [EQUATION]', '1803.05578-1-89-9': 'Using this, we have: [EQUATION]', '1803.05578-1-89-10': 'Here the last line followed since [MATH], [MATH], and [MATH].', '1803.05578-1-89-11': 'Hence the proof is complete.', '1803.05578-1-90-0': '[Proof of Theorem [REF]] Using the master inequality [REF] in combination with the previous Lemmas [REF], [REF], [REF], and [REF], we have: [EQUATION]', '1803.05578-1-90-1': 'When we have: [EQUATION] then the Lyapunov function [MATH] has decreased below [MATH] in expectation.', '1803.05578-1-90-2': 'Hence the complexity [MATH] satisfies: [EQUATION]', '1803.05578-1-90-3': 'Now it can be shown that for [MATH], we have: [EQUATION]', '1803.05578-1-90-4': 'Since [MATH], we have [MATH].', '1803.05578-1-90-5': 'Hence: [EQUATION]', '1803.05578-1-90-6': 'An expression for [MATH], the complexity of [MATH] follows by similar reasoning.', '1803.05578-1-90-7': '[EQUATION]', '1803.05578-1-90-8': 'Finally we have: [EQUATION] which completes the proof.', '1803.05578-1-91-0': '# Ordinary Differential Equation Analysis', '1803.05578-1-92-0': '## Derivation of ODE for synchronous A2BCD', '1803.05578-1-93-0': 'If we take expectations with respect to [MATH], then synchronous (no delay) [MATH] becomes: [EQUATION]', '1803.05578-1-93-1': 'We find it convenient to define [MATH].', '1803.05578-1-93-2': 'Inspired by this, we consider the following iteration: [EQUATION] for coefficients: [EQUATION] [MATH] is a discretization scale parameter that will be sent to [MATH] to obtain an ODE analogue of synchronous [MATH].', '1803.05578-1-93-3': 'We first use [REF] to eliminate [MATH] from from [REF].', '1803.05578-1-93-4': '[EQUATION]', '1803.05578-1-93-5': 'We now eliminate [MATH] using [REF]: [EQUATION]', '1803.05578-1-93-6': 'Now to derive an ODE, we let [MATH].', '1803.05578-1-93-7': 'Then [MATH].', '1803.05578-1-93-8': 'Hence the above becomes: [EQUATION]', '1803.05578-1-93-9': 'We now look at some of the terms in this equation to find the highest-order dependence on [MATH].', '1803.05578-1-93-10': '[EQUATION]', '1803.05578-1-93-11': 'We also have: [EQUATION]', '1803.05578-1-93-12': 'Hence using these facts on [REF], we have: [EQUATION]', '1803.05578-1-93-13': 'Taking the limit as [MATH], we obtain the ODE: [EQUATION]', '1803.05578-1-94-0': '## Convergence proof for synchronous ODE', '1803.05578-1-95-0': '[EQUATION]', '1803.05578-1-95-1': 'Hence we have [MATH].', '1803.05578-1-95-2': 'Therefore [MATH].', '1803.05578-1-95-3': 'That is: [EQUATION] which implies: [EQUATION]', '1803.05578-1-96-0': '## Asynchronicity error lemma', '1803.05578-1-97-0': 'This result is the continuous-time analogue of Lemma [REF].', '1803.05578-1-97-1': 'First notice that [MATH] and [MATH].', '1803.05578-1-97-2': 'We also have: [EQUATION]', '1803.05578-1-97-3': 'Hence using [MATH]: [EQUATION]', '1803.05578-1-97-4': 'Now when [MATH], we have [MATH].', '1803.05578-1-97-5': 'Hence when [MATH], we have: [EQUATION] and the result easily follows.', '1803.05578-1-98-0': '## Convergence analysis for the asynchronous ODE', '1803.05578-1-99-0': 'We consider the same energy as in the synchronous case (that is, the ODE in [REF]).', '1803.05578-1-99-1': 'Similar to before, we have: [EQUATION] where the final equality follows from the proof in Section [REF].', '1803.05578-1-99-2': 'Hence [EQUATION]', '1803.05578-1-99-3': 'Now we present an inequality that is similar to [REF].', '1803.05578-1-100-0': 'Let [MATH].', '1803.05578-1-100-1': 'Then: [EQUATION]', '1803.05578-1-100-2': 'Since [MATH] is a delayed version of [MATH], we have: [MATH] for some function [MATH] (though this can be easily generalized to an inconsistent read).', '1803.05578-1-100-3': 'Recall that for [MATH], we have [MATH].', '1803.05578-1-100-4': 'Hence [EQUATION]', '1803.05578-1-100-5': 'We use this lemma twice on [MATH] and [MATH] in [REF] with [MATH] and [MATH] respectively, to yield: [EQUATION]', '1803.05578-1-100-6': 'The proof of convergence is completed in Section [REF].', '1803.05578-1-101-0': '# Optimality proof', '1803.05578-1-102-0': 'For parameter set [MATH], we construct a block-separable function [MATH] on the space [MATH] (separated into [MATH] blocks of size [MATH]), which will imply this lower bound.', '1803.05578-1-102-1': 'Define [MATH].', '1803.05578-1-102-2': 'We define the matrix [MATH] via: [EQUATION]', '1803.05578-1-102-3': 'Hence we define [MATH] on [MATH] via: [EQUATION] which is clearly [MATH]-strongly convex and [MATH]-Lipschitz on [MATH].', '1803.05578-1-102-4': 'From Lemma 8 of [CITATION], we know that this function has unique minimizer [EQUATION]', '1803.05578-1-102-5': 'Finally, we define [MATH] via: [EQUATION]', '1803.05578-1-102-6': 'Now let [MATH] be the [MATH]th unit vector of the [MATH]th block of size [MATH] in [MATH].', '1803.05578-1-102-7': 'For [MATH], we define the subspaces [EQUATION] [MATH] is the subspace with the first [MATH] components of block [MATH] nonzero, the first [MATH] components of block [MATH] nonzero, etc.', '1803.05578-1-102-8': 'First notice that [MATH].', '1803.05578-1-102-9': 'Also, clearly, we have: [EQUATION] [MATH] is supported on the [MATH]th block, hence why all the other indices are [MATH].', '1803.05578-1-102-10': 'The patten of nonzeros in [MATH] means that the gradient will have at most [MATH] more nonzero on the [MATH]th block (see [CITATION]).', '1803.05578-1-103-0': 'Let the initial point [MATH] belong to [MATH].', '1803.05578-1-103-1': 'Let [MATH] be the number of times we have had [MATH] for [MATH].', '1803.05578-1-103-2': 'By induction on [REF] using [REF], we have: [EQUATION]', '1803.05578-1-103-3': 'Hence if [MATH] and [MATH], then [EQUATION]', '1803.05578-1-103-4': 'Therefore for all [MATH], we have: [EQUATION]', '1803.05578-1-103-5': 'To evaluate this expectation, we note: [EQUATION]', '1803.05578-1-103-6': 'Hence [EQUATION]', '1803.05578-1-103-7': 'For any [MATH], we may select the starting iterate [MATH] by defining its block [MATH] via: [EQUATION]', '1803.05578-1-103-8': 'For such a choice of [MATH], we have [EQUATION]', '1803.05578-1-103-9': 'Hence for this choice of [MATH]: [EQUATION]', '1803.05578-1-103-10': 'Now notice: [EQUATION]', '1803.05578-1-103-11': 'Hence [EQUATION]', '1803.05578-1-103-12': 'Now if we let [MATH], then we have: [EQUATION]', '1803.05578-1-103-13': 'Now let us take the minimum of the right-hand side over the parameters [MATH], subject to [MATH].', '1803.05578-1-103-14': 'The solution to this minimization is clearly: [EQUATION]', '1803.05578-1-103-15': 'Hence [EQUATION]', '1803.05578-1-103-16': 'Hence the complexity [MATH] satisfies: [EQUATION]'}

{'1803.05578-2-0-0': 'A2BCD: An Asynchronous Accelerated Block Coordinate Descent Algorithm With Optimal Complexity', '1803.05578-2-1-0': 'Department of Mathematics, University of California, Los Angeles, USA', '1803.05578-2-2-0': 'synchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm ([MATH]), the first asynchronous Nesterov-accelerated algorithm that achieves optimal complexity.', '1803.05578-2-2-1': 'This parallel algorithm solves the unconstrained convex minimization problem, using [MATH] computing nodes which compute updates to shared solution vectors, in an asynchronous fashion with no central coordination.', '1803.05578-2-2-2': 'Nodes in asynchronous algorithms do not wait for updates from other nodes before starting a new iteration, but simply compute updates using the most recent solution information available.', '1803.05578-2-2-3': 'This allows them to complete iterations much faster than traditional ones, especially at scale, by eliminating the costly synchronization penalty of traditional algorithms.', '1803.05578-2-3-0': 'We first prove that [MATH] converges linearly to a solution with a fast accelerated rate that matches the recently proposed [MATH], so long as the maximum delay is not too large.', '1803.05578-2-3-1': 'Somewhat surprisingly, [MATH] pays no complexity penalty for using outdated information.', '1803.05578-2-3-2': 'We then prove lower complexity bounds for randomized coordinate descent methods, which show that [MATH] (and hence [MATH]) has optimal complexity to within a constant factor.', '1803.05578-2-3-3': 'We confirm with numerical experiments that [MATH] outperforms [MATH], which is the current fastest coordinate descent algorithm, even at small scale.', '1803.05578-2-3-4': 'We also derive and analyze a second-order ordinary differential equation, which is the continuous-time limit of our algorithm, and prove it converges linearly to a solution with a similar accelerated rate.', '1803.05578-2-4-0': '# Introduction', '1803.05578-2-5-0': 'In this paper, we propose and prove the convergence of the Asynchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm ([MATH]), the first asynchronous Nesterov-accelerated algorithm that achieves optimal complexity.', '1803.05578-2-5-1': 'No previous attempts have been able to prove a speedup for asynchronous Nesterov acceleration.', '1803.05578-2-5-2': 'We aim to find the minimizer [MATH] of the unconstrained minimization problem: [EQUATION] where [MATH] is [MATH]-strongly convex for [MATH] with [MATH]-Lipschitz gradient [MATH].', '1803.05578-2-5-3': '[MATH] is composed of coordinate blocks [MATH].', '1803.05578-2-5-4': 'The coordinate blocks of the gradient [MATH] are assumed [MATH]-Lipschitz with respect to the [MATH]th block.', '1803.05578-2-5-5': 'That is, [MATH]: [EQUATION] where [MATH] is the projection onto the [MATH]th block of [MATH].', '1803.05578-2-5-6': 'Let [MATH] be the average block Lipschitz constant.', '1803.05578-2-5-7': 'These conditions on [MATH] are assumed throughout this whole paper.', '1803.05578-2-5-8': 'Our algorithm can be applied to non-strongly convex objectives ([MATH]) or non-smooth objectives using the black box reduction techniques proposed in [CITATION].', '1803.05578-2-6-0': 'Coordinate descent methods, in which a chosen coordinate block [MATH] is updated at every iteration, are a popular way to solve [REF].', '1803.05578-2-6-1': 'Randomized block coordinate descent (RBCD, [CITATION]) updates a uniformly randomly chosen coordinate block [MATH] with a gradient-descent-like step: [MATH].', '1803.05578-2-6-2': 'This algorithm decreases the error [MATH] to [MATH] in [MATH] iterations.', '1803.05578-2-7-0': "Using a series of averaging and extrapolation steps, accelerated RBCD Nesterov2012_efficiency improves RBCD's iteration complexity [MATH] to [MATH], which leads to much faster convergence when [MATH] is large.", '1803.05578-2-7-1': 'This rate is optimal when all [MATH] are equal LanZhou2017_optimal.', '1803.05578-2-7-2': 'Finally, using a special probability distribution for the random block index [MATH], non-uniform accelerated coordinate descent method Allen-ZhuQuRichtarikYuan2016_even (NUACDM) can further decrease the complexity to [MATH], which can be up to [MATH] times faster than accelerated RBCD, since some [MATH] can be significantly smaller than [MATH].', '1803.05578-2-7-3': 'NUACDM is the current state-of-the-art coordinate descent algorithm for solving [REF].', '1803.05578-2-8-0': 'Our [MATH] algorithm generalizes NUACDM to the asynchronous-parallel case.', '1803.05578-2-8-1': 'We solve [REF] with a collection of [MATH] computing nodes that continually read a shared-access solution vector [MATH] into local memory then compute a block gradient [MATH], which is used to update shared solution vectors [MATH].', '1803.05578-2-8-2': 'Proving convergence in the asynchronous case requires extensive new technical machinery.', '1803.05578-2-9-0': 'A traditional synchronous-parallel implementation is organized into rounds of computation: Every computing node must complete an update in order for the next iteration to begin.', '1803.05578-2-9-1': 'However, this synchronization process can be extremely costly, since the lateness of a single node can halt the entire system.', '1803.05578-2-9-2': 'This becomes increasingly problematic with scale, as differences in node computing speeds, load balancing, random network delays, and bandwidth constraints mean that a synchronous-parallel solver may spend more time waiting than computing a solution.', '1803.05578-2-10-0': 'Computing nodes in an asynchronous solver do not wait for others to complete and share their updates before starting the next iteration, but simply continue to update the solution vectors with the most recent information available, without any central coordination.', '1803.05578-2-10-1': 'This eliminates costly idle time, meaning that asynchronous algorithms can be much faster than traditional ones, since they have much faster iterations.', '1803.05578-2-10-2': 'For instance, random network delays cause asynchronous algorithms to complete iterations [MATH] time faster than synchronous algorithms at scale.', '1803.05578-2-10-3': 'This and other factors that influence the speed of iterations are discussed in [CITATION].', '1803.05578-2-10-4': 'However, since many iterations may occur between the time that a node reads the solution vector, and the time that its computed update is applied, effectively the solution vector is being updated with outdated information.', '1803.05578-2-10-5': 'At iteration [MATH], the block gradient [MATH] is computed at a delayed iterate [MATH] defined as: [EQUATION] for delay parameters [MATH].', '1803.05578-2-10-6': 'Here [MATH] denotes how many iterations out of date coordinate block [MATH] is at iteration [MATH].', '1803.05578-2-10-7': 'Different block may be out of date by different amounts, which is known as an inconsistent read .', '1803.05578-2-10-8': 'We assume that [MATH] for some constant [MATH].', '1803.05578-2-11-0': "Our results: In this paper, we prove that [MATH] attains NUACDM's state-of-the-art iteration complexity to highest order for solving [REF], so long as delays are not too large.", '1803.05578-2-11-1': 'Hence we prove that there is no significant complexity penalty, despite the use of outdated information.', '1803.05578-2-11-2': 'The proof is very different from that of [CITATION], and involves significant technical innovations and formidable complexity related to the analysis of asynchronicity.', '1803.05578-2-12-0': 'Since asynchronous algorithms have much faster iterations, and [MATH] needs essentially the same number of epochs as NUACDM to compute a solution of a target accuracy, we expect [MATH] to be faster than all existing coordinate descent algorithms.', '1803.05578-2-12-1': 'We confirm this with computational experiments, comparing A2BCD to NUACDM, which is the current fastest block coordinate descent algorithm.', '1803.05578-2-13-0': 'We also prove that A2BCD (and hence NUACDM) has optimal complexity to within a constant factor over a fairly general class of randomized block coordinate descent algorithms.', '1803.05578-2-13-1': 'We do this by proving that any algorithm [MATH] in this class must in general complete at least [MATH] random gradient evaluations to decrease the error by a factor of [MATH].', '1803.05578-2-13-2': 'This extends results in [CITATION] to the case where [MATH] are not all equal, and the algorithm in question can be asynchronous.', '1803.05578-2-14-0': 'These results are significant, because it was an open question whether Nesterov-type acceleration was compatible with asynchronicity.', '1803.05578-2-14-1': 'Not only is this possible, but asynchronous algorithms may even attain optimal complexity in this setting.', '1803.05578-2-14-2': 'In light of the above, it also seems plausible that accelerated incremental algorithms for finite sum problems [MATH] can be made asynchronous-parallel, too.', '1803.05578-2-15-0': 'We also derive a second-order ordinary differential equation (ODE), which is the continuous-time limit of [MATH].', '1803.05578-2-15-1': 'This extends the ODE found in SuBoydCandes2014_differential to an asynchronous accelerated algorithm minimizing a strongly convex function.', '1803.05578-2-15-2': "We prove this ODE linearly converges to a solution with the same rate as [MATH]'s, without needing to resort to the restarting technique employed in SuBoydCandes2014_differential.", '1803.05578-2-15-3': 'We prove this result using techniques that motivate and clarify the our proof strategy of the main result.', '1803.05578-2-16-0': '# Main results', '1803.05578-2-17-0': 'We define the condition number [MATH], and let [MATH] be the smallest block Lipschitz constant.', '1803.05578-2-17-1': 'We should consider functions [MATH] where it is efficient to calculate blocks of the gradient, so that coordinate-wise parallelization is efficient.', '1803.05578-2-17-2': 'That is, the function should be "coordinate friendly" [CITATION].', '1803.05578-2-17-3': 'This turns out to be a rather wide class of algorithms.', '1803.05578-2-17-4': 'So for instance, while the [MATH]-regularized empirical risk minimization problem is not coordinate friendly in general, the equivalent dual problem is, and hence can be solved efficiently by [MATH] (see [CITATION], and Section [REF]).', '1803.05578-2-18-0': "To calculate the [MATH]'th iteration of the algorithm from iteration [MATH], we use a block of the gradient [MATH].", '1803.05578-2-18-1': 'Finally, we assume that the delays [MATH] are independent of the block sequence [MATH], but otherwise arbitrary (This assumption can be relaxed.', '1803.05578-2-18-2': 'See [CITATION]).', '1803.05578-2-19-0': '[Asynchronous Accelerated Randomized Block Coordinate Descent ([MATH])] Let [MATH] be [MATH]-strongly convex, and let its gradient [MATH] be [MATH]-Lipschitz with block coordinate Lipschitz parameters [MATH] [REF].', '1803.05578-2-19-1': 'Using these parameters, we sample [MATH] in an independent and identically distributed (IID) fashion according to [EQUATION]', '1803.05578-2-19-2': 'Let [MATH] be the maximum asynchronous delay in the parallel system (or an overestimate of this).', '1803.05578-2-19-3': 'We define the dimensionless asynchronicity parameter [MATH], which is proportional to the maximum delay [MATH], and quantifies how strongly asynchronicity will affect convergence: [EQUATION]', '1803.05578-2-19-4': 'We use the above system parameters and [MATH] to define the coefficients [MATH], and [MATH] via [REF],[REF], and [REF].', '1803.05578-2-19-5': '[EQUATION]', '1803.05578-2-19-6': 'Hence [MATH] algorithm is hence defined via the iterations: [REF],[REF], and [REF]: [EQUATION]', '1803.05578-2-19-7': 'Here [MATH] may be underestimated, and [MATH] may be overestimated if exact values are unavailable.', '1803.05578-2-19-8': 'Notice that [MATH] can be eliminated from the above iteration, and the block gradient [MATH] only needs to be calculated once per iteration.', '1803.05578-2-19-9': 'A larger (or overestimated) maximum delay [MATH] will cause a larger asynchronicity parameter [MATH], which leads to more conservative step sizes to compensate.', '1803.05578-2-20-0': 'The convergence of this algorithm can be stated in terms of a Lyapunov function that we will define shortly.', '1803.05578-2-20-1': 'We first introduce the asynchronicity error, a powerful tool for analyzing asynchronous algorithms used in several recent works [CITATION].', '1803.05578-2-20-2': 'This error is a weighted sum of the history of the algorithm, with the weights [MATH] decreasing as one goes further back in time.', '1803.05578-2-20-3': 'This error appears naturally in the analysis.', '1803.05578-2-20-4': 'Much like a well-chosen basis in linear algorithm, it appears to be a natural quantity to consider when analyzing convergence of asynchronous algorithms.', '1803.05578-2-21-0': '[Asynchronicity error] Using the above parameters, we define: [EQUATION]', '1803.05578-2-21-1': 'Here we define [MATH] for all [MATH].', '1803.05578-2-22-0': 'The determination of the coefficients [MATH] is in general a very involved process of trial and error, intuition, and balancing competing requirements.', '1803.05578-2-22-1': 'Obtaining a convergence proof and optimal rates relies on the skillful choice of these coefficients [MATH].', '1803.05578-2-22-2': "The algorithm doesn't depend on the coefficients, however; they are only used in the analysis.", '1803.05578-2-23-0': 'We define [MATH] as the expectation of [MATH] conditional on conditioned on [MATH], [MATH], [MATH], and [MATH].', '1803.05578-2-23-1': 'To simplify notation, we assume that the minimizer [MATH], and that [MATH] with no loss in generality.', '1803.05578-2-23-2': 'We define the Lyapunov function: [EQUATION]', '1803.05578-2-23-3': 'We define the iteration complexity [MATH] with respect to some error [MATH] as the number of iterations [MATH] such that the expected error [MATH] decreases to less than [MATH].', '1803.05578-2-24-0': "We now present this paper's first main contribution.", '1803.05578-2-24-1': '[EQUATION]', '1803.05578-2-24-2': 'This result is proven in Section [REF].', '1803.05578-2-24-3': 'A stronger result can be proven, however this adds too much to the complexity of the proof.', '1803.05578-2-24-4': 'See Section [REF] for a discussion.', '1803.05578-2-24-5': 'In practice, asynchronous algorithms are far more resilient to delays than the theory predicts.', '1803.05578-2-24-6': '[MATH] can be much larger without negatively affecting the convergence rate and complexity.', '1803.05578-2-24-7': 'This is perhaps because we are limited to a worst-case analysis, which is not representative of the average-case performance.', '1803.05578-2-25-0': 'Authors in [CITATION] (Theorem 5.1) obtain a linear convergence rate of [MATH] for NUACDM which leads to the corresponding iteration complexity of [MATH] as [MATH].', '1803.05578-2-25-1': 'And hence, we have: [EQUATION]', '1803.05578-2-25-2': 'Hence when [MATH], or equivalently, when [MATH] the complexity of [MATH] asymptotically matches that of NUACDM.', '1803.05578-2-25-3': 'Hence [MATH] combines state-of-the-art complexity with the faster iterations and superior scaling that asynchronous iterations allow.', '1803.05578-2-26-0': 'We now present some special cases of the conditions on the maximum delay [MATH] required for good complexity.', '1803.05578-2-27-0': 'Let the conditions of Theorem [REF] hold.', '1803.05578-2-27-1': 'Additionally, let all coordinate-wise Lipschitz constants [MATH] be equal (i.e. [MATH]).', '1803.05578-2-27-2': 'Then we have [MATH] when [MATH].', '1803.05578-2-27-3': 'Further, let us assume all coordinate-wise Lipschitz constants [MATH] equal [MATH].', '1803.05578-2-27-4': 'Then [MATH], when [MATH]', '1803.05578-2-28-0': '[Reduction to synchronous case] Notice that when [MATH], we have [MATH], [MATH] and hence [MATH].', '1803.05578-2-28-1': 'Thus [MATH] becomes equivalent to NUACDM, the Lyapunov function [MATH] becomes equivalent to one found in [CITATION](pg.', '1803.05578-2-28-2': '9), and Theorem [REF] yields the same complexity to highest order.', '1803.05578-2-29-0': '## Optimality', '1803.05578-2-30-0': '[MATH] and hence [MATH] are in fact optimal among a wide class of randomized block gradient algorithms.', '1803.05578-2-30-1': 'We consider a class of algorithms slightly wider than that considered in [CITATION] to encompass asynchronous algorithms.', '1803.05578-2-30-2': 'Their result only apply to the case where all [MATH] are equal in this setting.', '1803.05578-2-30-3': 'Our result applies for unequal [MATH], and potentially asynchronous algorithms.', '1803.05578-2-30-4': 'For a subset [MATH], we let [MATH] (inconsistent read) denote the set of vectors [MATH] such that [MATH] for some vectors [MATH].', '1803.05578-2-30-5': 'Here [MATH] denotes the [MATH]th component of vector [MATH].', '1803.05578-2-30-6': 'That is, the coordinates of [MATH] are some combination of coordinates of vectors in [MATH].', '1803.05578-2-30-7': 'Let [MATH].', '1803.05578-2-31-0': '[Asynchronous Randomized Incremental Algorithms Class] Consider the unconstrained minimization problem [REF] for [MATH] that is [MATH]-strongly convex with [MATH]-Lipschitz gradient, with block-coordinate-wise Lipschitz constants [MATH].', '1803.05578-2-31-1': 'We define the class [MATH] as algorithms [MATH] on this problem such that:', '1803.05578-2-32-0': '1.', '1803.05578-2-32-1': 'For each parameter set [MATH], [MATH] has an associated IID random variable [MATH] with some fixed distribution [MATH] for [MATH].', '1803.05578-2-33-0': '2.', '1803.05578-2-33-1': 'For [MATH], the iterates of [MATH] satisfy: [EQUATION]', '1803.05578-2-33-2': 'This is a rather general class: [MATH] can be constructed from any inconsistent reading of past iterates [MATH], and any past gradient of an inconsistent read [MATH].', '1803.05578-2-34-0': 'For any algorithm [MATH], and parameter set [MATH] for the unconstrained minimization problem, and [MATH], there is a dimension [MATH], a corresponding function [MATH] on [MATH], and a starting point [MATH], such that [EQUATION]', '1803.05578-2-34-1': 'Hence the complexity [MATH] for all algorithms in [MATH] satisfies [EQUATION]', '1803.05578-2-34-2': 'Our proof in Section [REF] follows very similar lines to LanZhou2017_optimal, which is inspired by Nesterov Nesterov2013_introductory.', '1803.05578-2-35-0': '# ODE Analysis', '1803.05578-2-36-0': 'In this section we present and analyze an ODE which is the continuous-time limit of [MATH].', '1803.05578-2-36-1': 'This ODE is a strongly convex, and asynchronous version of the ODE found in [CITATION].', '1803.05578-2-36-2': 'For simplicity, assume [MATH].', '1803.05578-2-36-3': 'We rescale [MATH] so that the strong convexity modulus [MATH], and hence [MATH].', '1803.05578-2-36-4': 'Taking the discrete limit of synchronous [MATH] (i.e. accelerated RBCD), we can derive the following ODE (see Section [REF]): [EQUATION]', '1803.05578-2-36-5': 'We define the parameter [MATH], and the energy: [EQUATION]', '1803.05578-2-36-6': 'This is very similar to the Lyapunov function discussed in [REF], with [MATH] fulfilling the role of [MATH], and [MATH] (since there is no delay yet).', '1803.05578-2-36-7': 'Much like the traditional analysis in the proof of Theorem [REF], we can derive a linear convergence result that has a similar rate.', '1803.05578-2-36-8': 'See Section [REF] for the proof.', '1803.05578-2-37-0': 'If [MATH] satisfies [REF], then the energy [MATH] satisfies [MATH].', '1803.05578-2-37-1': 'This implies [MATH], which is equivalent to: [EQUATION]', '1803.05578-2-37-2': 'We may also analyze an asynchronous version of [REF] to motivate our proof of the main theorem in Section [REF]: [EQUATION] [MATH] is a delayed version of [MATH] defined similarly to [REF], with the delay bounded by [MATH].', '1803.05578-2-38-0': 'Unfortunately, the energy satisfies (see Section [REF], [REF]): [EQUATION]', '1803.05578-2-38-1': 'Since the right-hand side can be positive, the energy [MATH] may not be decreasing in general.', '1803.05578-2-38-2': 'But, we may add a continuous-time version of the asynchronicity error (see [CITATION]), much like in Definition [REF], to create a decreasing energy.', '1803.05578-2-38-3': 'Let [MATH] and [MATH] be arbitrary constants that will be set later.', '1803.05578-2-38-4': 'Define: [EQUATION]', '1803.05578-2-38-5': 'When [MATH], the asynchronicity error [MATH] satisfies: [EQUATION]', '1803.05578-2-38-6': 'See Section [REF] for the proof.', '1803.05578-2-38-7': 'Adding this error to the Lyapunov function serves a similar purpose in the continuous-time case as in the proof of Theorem [REF] (see Lemma [REF]).', '1803.05578-2-38-8': 'It allows us to negate [MATH] units of [MATH] for the cost of creating [MATH] units of [MATH].', '1803.05578-2-39-0': 'Notice how this convergence condition is similar to Corollary [REF], but a little looser.', '1803.05578-2-40-0': '0.9 [EQUATION]', '1803.05578-2-41-0': 'The preceding should act as a guide to understanding the convergence of [MATH].', '1803.05578-2-41-1': 'It may make clearer the use of the Lyapunov function to establish convergence in the synchronous case, the function of the asynchronicity error in the proof of the asynchronous case, and may hopefully elucidate the logic and general strategy of the proof in Section [REF].', '1803.05578-2-42-0': '# Related work', '1803.05578-2-43-0': 'We now discuss related work that was not addressed in Section [REF].', '1803.05578-2-43-1': "Nesterov acceleration is a method for improving an algorithm's iteration complexity's dependence the condition number [MATH].", '1803.05578-2-43-2': 'Nesterov-accelerated methods have been proposed and discovered in many settings [CITATION], including for coordinate descent algorithms (algorithms that use 1 gradient block [MATH] or minimize with respect to [MATH] coordinate block per iteration), and incremental algorithms (algorithms for finite sum problems [MATH] that use [MATH] function gradient [MATH] per iteration).', '1803.05578-2-43-3': 'Such algorithms can often be augmented to solve composite minimization problems (minimization for objective of the form [MATH], especially for nonsomooth [MATH]), or include constraints.', '1803.05578-2-44-0': 'Asynchronous algorithms were proposed in ChazanMiranker1969_chaotic to solve linear systems.', '1803.05578-2-44-1': 'General convergence results and theory were developed later in Bertsekas1983_distributed,BertsekasTsitsiklis1997_parallel,TsengBertsekasTsitsiklis1990_partially,LuoTseng1992_convergence,LuoTseng1993_convergence,Tseng1991_rate for partially and totally asynchronous systems, with essentially-cyclic block sequence [MATH].', '1803.05578-2-44-2': 'More recently, there has been renewed interest in asynchronous algorithms with random block coordinate updates.', '1803.05578-2-44-3': 'Linear and sublinear convergence results were proven for asynchronous RBCD LiuWright2015_asynchronous,AvronDruinskyGupta2014, and similar was proven for asynchronous SGD RechtReWrightNiu2011_hogwild, and variance reduction algorithms J.ReddiHefnySraPoczosSmola2015_variance,LeblondPedregosaLacoste-Julien2017_asaga.', '1803.05578-2-45-0': 'In PengXuYanYin2016_arock, authors proposed and analyzed an asynchronous fixed-point algorithm called ARock, that takes proximal algorithms, forward-backward, ADMM, etc. as special cases.', '1803.05578-2-45-1': 'HannahYin2017_unboundeda,SunHannahYin2017_asynchronousb,PengXuYanYin2017_convergence showed that many of the assumptions used in prior work (such as bounded delay [MATH]) were unrealistic and unnecessary in general.', '1803.05578-2-45-2': 'In HannahYin2017_more the authors showed that asynchronous iterations will complete far more iterations per second, and that a wide class of asynchronous algorithms, including asynchronous [MATH], have the same iteration complexity as their traditional counterparts.', '1803.05578-2-45-3': 'Hence certain asynchronous algorithms can be expected to significantly outperform traditional ones.', '1803.05578-2-46-0': 'In [CITATION] authors propose a novel asynchronous catalyst-accelerated [CITATION] primal-dual algorithmic framework to solve regularized ERM problems.', '1803.05578-2-46-1': 'Instead of using outdated information, they structure the parallel updates so that the data that an update depends on is up to date (though the rest of the data may not be).', '1803.05578-2-46-2': 'However catalyst acceleration incurs a logarithmic penalty over Nesterov acceleration in general.', '1803.05578-2-46-3': 'Authors in [CITATION] skillfully devised accelerated schemes for asynchronous coordinate descent and SVRG using momentum compensation techniques.', '1803.05578-2-46-4': 'Although their complexity results have the improved [MATH] dependence on the condition number, they do not prove an asynchronous speedup.', '1803.05578-2-46-5': 'Their complexity is [MATH] times larger than our complexity.', '1803.05578-2-46-6': 'Since [MATH] is necessarily greater than [MATH], their results imply that adding more computing nodes will increase running time.', '1803.05578-2-47-0': '# Numerical experiments', '1803.05578-2-48-0': 'To investigate the performance of [MATH], we solve the ridge regression problem.', '1803.05578-2-48-1': 'Consider the following primal and corresponding dual objective (see for instance [CITATION]): [EQUATION] where [MATH] is a matrix of [MATH] samples and [MATH] features, and [MATH] is a label vector.', '1803.05578-2-48-2': 'We let [MATH] where [MATH] are the column blocks of [MATH].', '1803.05578-2-48-3': 'We compare [MATH] (which is asynchronous accelerated), synchronous [MATH] (which is synchronous accelerated), and asynchronous [MATH] (which is asynchronous non-accelerated).', '1803.05578-2-48-4': 'At each iteration, each node randomly selects a coordinate block according to [REF], calculates the corresponding block gradient, and uses it to apply an update to the shared solution vectors.', '1803.05578-2-48-5': 'Nodes in synchronous [MATH] implementation must wait until all nodes apply an update before they can start the next iteration, but the asynchronous algorithms simply compute with the most up-to-date information available.', '1803.05578-2-49-0': 'We use the datasets [MATH] (47272 samples, 300 features) and [MATH] (20242 samples, 47236 features) from LIBSVM [CITATION].', '1803.05578-2-49-1': 'The algorithm is implemented in a multi-threaded fashion using C++11 and GNU Scientific Library with a shared memory architecture.', '1803.05578-2-49-2': 'We use 40 threads on two 2.5GHz 10-core Intel Xeon E5-2670v2 processors.', '1803.05578-2-50-0': 'To estimate [MATH], one can first performed a dry run with all coefficient set to [MATH] to estimate [MATH].', '1803.05578-2-50-1': 'All function parameters can be calculated exactly for this problem in terms of the data matrix and [MATH].', '1803.05578-2-50-2': 'We can then use these parameters and this tau to calculate [MATH].', '1803.05578-2-50-3': '[MATH] and [MATH] merely change the parameters, and do not change execution patterns of the processors.', '1803.05578-2-50-4': "Hence their parameter specification doesn't affect the observed delay.", '1803.05578-2-50-5': 'Through simple tuning though, we found that [MATH] resulted in good performance.', '1803.05578-2-51-0': 'In tuning for general problems, there are theoretical reasons why it is difficult to attain acceleration without some prior knowledge of [MATH], the strong convexity modulus [CITATION].', '1803.05578-2-51-1': 'Ideally [MATH] is pre-specified for instance in a regularization term.', '1803.05578-2-51-2': 'If the Lipschitz constants [MATH] cannot be calculated directly (which is rarely the case for the classic dual problem of empirical risk minimization objectives), the line-search method discussed in RouxSchmidtBach2012_stochastic Section 4 can be used.', '1803.05578-2-52-0': 'A critical ingredient in the efficient implementation of [MATH] and [MATH] for this problem is the efficient update scheme discussed in LeeSidford2013_efficient.', '1803.05578-2-52-1': 'In linear regression applications such as this, it is essential to be able to efficiently maintain or recover [MATH].', '1803.05578-2-52-2': 'This is because calculating block gradients requires the vector [MATH], and without an efficient way to recover [MATH], block gradient evaluations are essentially [MATH] as expensive as full-gradient calculations.', '1803.05578-2-52-3': 'Unfortunately, every accelerated iteration results in dense updates to [MATH] because of the averaging step in [REF].', '1803.05578-2-52-4': 'Hence [MATH] must be recalculated from scratch.', '1803.05578-2-53-0': 'However [CITATION] introduces a linear transformation that allows for an equivalent iteration that results in sparse updates to new iteration variables [MATH] and [MATH].', '1803.05578-2-53-1': 'The original purpose of this transformation was to ensure that the averaging steps (e.g. [REF]) do not dominate the computational cost for sparse problems.', '1803.05578-2-53-2': 'However we find a more important secondary use which applies to both sparse and dense problems.', '1803.05578-2-53-3': 'Since the updates to [MATH] and [MATH] are sparse coordinate-block updates, the vectors [MATH], and [MATH] can be efficiently maintained, and therefore block gradients can be efficiently calculated.', '1803.05578-2-53-4': 'Implementation details are discussed in more detail in Appendix [REF].', '1803.05578-2-54-0': 'In Table [REF], we plot the sub-optimality vs. time for decreasing values of [MATH], which corresponds to increasingly large condition numbers [MATH].', '1803.05578-2-54-1': "When [MATH] is small, acceleration doesn't result in a significantly better convergence rate, and hence [MATH] and async-RBCD both outperform sync-[MATH] since they complete faster iterations at similar complexity.", '1803.05578-2-54-2': 'Acceleration for low [MATH] has unnecessary overhead, which means async-RBCD can be quite competitive.', '1803.05578-2-54-3': 'When [MATH] becomes large, async-RBCD is no longer competitive, since it has a poor convergence rate.', '1803.05578-2-54-4': 'We observe that [MATH] and sync-[MATH] have essentially the same convergence rate, but [MATH] is up to [MATH] faster than sync-[MATH] because it completes much faster iterations.', '1803.05578-2-54-5': 'We observe this advantage despite the fact that we are in an ideal environment for synchronous computation: A small, homogeneous, high-bandwidth, low-latency cluster.', '1803.05578-2-54-6': "In large-scale heterogeneous systems with greater synchronization overhead, bandwidth constraints, and latency, we expect [MATH]'s advantage to be much larger.", '1803.05578-2-55-0': '[1]ReferencesReferences We first recall a couple of inequalities for convex functions.', '1803.05578-2-56-0': 'Let [MATH] be [MATH]-strongly convex with [MATH]-Lipschitz gradient.', '1803.05578-2-56-1': 'Then we have: [EQUATION]', '1803.05578-2-57-0': 'We also find it convenient to define the norm: [EQUATION]', '1803.05578-2-58-0': '## Starting point', '1803.05578-2-59-0': 'First notice that using the definition [REF] of [MATH] we have: [EQUATION]', '1803.05578-2-59-1': 'We have the following general identity: [EQUATION]', '1803.05578-2-59-2': 'It can also easily be verified from [REF] that we have: [EQUATION]', '1803.05578-2-59-3': 'Using [REF] on term [MATH], [REF] on term [MATH], and recalling the definition [REF] on term [MATH], we have from [REF]: [EQUATION]', '1803.05578-2-59-4': 'This inequality is our starting point.', '1803.05578-2-59-5': 'We analyze the terms on the second line in the next section.', '1803.05578-2-60-0': '## The Cross Term', '1803.05578-2-61-0': 'To analyze these terms, we need a small lemma.', '1803.05578-2-61-1': 'This lemma is fundamental in allowing us to deal with asynchronicity.', '1803.05578-2-62-0': 'Let [MATH].', '1803.05578-2-62-1': 'Let the delay be bounded by [MATH].', '1803.05578-2-62-2': 'Then: [EQUATION]', '1803.05578-2-62-3': 'See [CITATION].', '1803.05578-2-63-0': 'We have: [EQUATION]', '1803.05578-2-63-1': 'The terms in bold in [REF] and [REF] are a result of the asynchronicity, and are identically [MATH] in its absence.', '1803.05578-2-64-0': 'Our strategy is to separately analyze terms that appear in the traditional analysis of [CITATION], and the terms that result from asynchronicity.', '1803.05578-2-64-1': 'We first prove [REF]: [EQUATION] [REF] follows from strong convexity ([REF] with [MATH] and [MATH]), and the fact that [MATH] is [MATH]-Lipschitz.', '1803.05578-2-64-2': 'The term due to asynchronicity becomes: [EQUATION] using Lemma [REF] with [MATH].', '1803.05578-2-64-3': 'Combining this with [REF] completes the proof of [REF].', '1803.05578-2-65-0': 'We now prove [REF]: [EQUATION]', '1803.05578-2-65-1': 'Here the last line follows from Lemma [REF] with [MATH], [MATH].', '1803.05578-2-65-2': 'We can complete the proof using the following identity that can be easily obtained from [REF]: [EQUATION]', '1803.05578-2-66-0': '## Function-value term', '1803.05578-2-67-0': 'Much like [CITATION], we need a [MATH] term in the Lyapunov function (see the middle of page [MATH]).', '1803.05578-2-67-1': 'However we additionally need to consider asynchronicity when analyzing the growth of this term.', '1803.05578-2-67-2': 'Again terms due to asynchronicity are emboldened.', '1803.05578-2-68-0': 'We have: [EQUATION]', '1803.05578-2-68-1': 'From the definition [REF] of [MATH], we can see that [MATH] is supported on block [MATH].', '1803.05578-2-68-2': 'Since each gradient block [MATH] is [MATH] Lipschitz with respect to changes to block [MATH], we can use [REF] to obtain: [EQUATION]', '1803.05578-2-68-3': 'Here the last line followed from the definition [REF] of the norm [MATH].', '1803.05578-2-68-4': 'We now analyze the middle term: [EQUATION]', '1803.05578-2-68-5': 'We then apply Lemma [REF] to this with [MATH] to yield: [EQUATION]', '1803.05578-2-68-6': 'Finally to complete the proof, we combine [REF], with [REF].', '1803.05578-2-69-0': '## Asynchronicity error', '1803.05578-2-70-0': 'The previous inequalities produced difference terms of the form [MATH].', '1803.05578-2-70-1': 'The following lemma shows how these errors can be incorporated into a Lyapunov function.', '1803.05578-2-71-0': 'Let [MATH] and consider the asynchronicity error and corresponding coefficients: [EQUATION]', '1803.05578-2-71-1': 'This sum satisfies: [EQUATION] [Interpretation] This result means that an asynchronicity error term [MATH] can negate a series of difference terms [MATH] at the cost of producing an additional error [MATH], while maintaining a convergence rate of [MATH].', '1803.05578-2-71-2': 'This essentially converts difference terms, which are hard to deal with, into a [MATH] term which can be negated by other terms in the Lyapunov function.', '1803.05578-2-71-3': 'The proof is straightforward.', '1803.05578-2-71-4': '[EQUATION]', '1803.05578-2-71-5': 'Noting the following completes the proof: [EQUATION]', '1803.05578-2-71-6': 'Given that [MATH] allows us to negate difference terms, we now analyze the cost [MATH] of this negation.', '1803.05578-2-72-0': 'We have: [EQUATION]', '1803.05578-2-72-1': 'Here [REF] following from [REF], the definition of [MATH].', '1803.05578-2-72-2': '[REF] follows from the inequality [MATH].', '1803.05578-2-72-3': 'The rest is simple algebraic manipulation.', '1803.05578-2-72-4': '[EQUATION]', '1803.05578-2-72-5': 'Finally, to complete the proof, we prove [MATH].', '1803.05578-2-72-6': '[EQUATION]', '1803.05578-2-72-7': 'Rearranging the definition of [MATH], we have: [EQUATION]', '1803.05578-2-72-8': 'Using this on [REF], we have: [EQUATION]', '1803.05578-2-72-9': 'This completes the proof.', '1803.05578-2-73-0': '## Master inequality', '1803.05578-2-74-0': 'We are finally in a position to bring together all the all the previous results together into a master inequality for the Lyapunov function [MATH] (defined in [REF]).', '1803.05578-2-74-1': 'After this lemma is proven, we will prove that the right hand size is negative, which will imply that [MATH] linearly converges to [MATH] with rate [MATH].', '1803.05578-2-75-0': '[Master inequality] We have: [EQUATION]', '1803.05578-2-75-1': 'We now collect and organize the similar terms of this inequality.', '1803.05578-2-75-2': '[EQUATION]', '1803.05578-2-75-3': 'Now finally, we add the function-value and asynchronicity terms to our analysis.', '1803.05578-2-75-4': 'We use Lemma [REF] is with [MATH], and [EQUATION]', '1803.05578-2-75-5': 'Notice that this choice of [MATH] will recover the coefficient formula given in [REF].', '1803.05578-2-75-6': 'Hence we have: [EQUATION]', '1803.05578-2-75-7': 'Notice [MATH].', '1803.05578-2-75-8': 'Finally, combining [REF] and [REF] completes the proof.', '1803.05578-2-76-0': 'In the next section, we will prove that every coefficient on the right hand side of [REF] is [MATH] or less, which will complete the proof of Theorem [REF].', '1803.05578-2-77-0': '## Proof of main theorem', '1803.05578-2-78-0': 'The coefficients of [MATH], [MATH], and [MATH] in Lemma [REF] are non-positive.', '1803.05578-2-79-0': 'The coefficient [MATH] of [MATH] is identically [MATH] via the definition [REF] of [MATH].', '1803.05578-2-79-1': 'The coefficient [MATH] of [MATH] is identically [MATH] via the definition [REF] of [MATH].', '1803.05578-2-80-0': 'First notice from the definition [REF] of [MATH]: [EQUATION]', '1803.05578-2-80-1': 'Here the last line followed since [MATH] and [MATH].', '1803.05578-2-80-2': 'We now analyze the coefficient of [MATH].', '1803.05578-2-80-3': '[EQUATION]', '1803.05578-2-80-4': 'The coefficient [MATH] of [MATH] in Lemma [REF] is non-positive.', '1803.05578-2-80-5': '[EQUATION]', '1803.05578-2-80-6': 'The coefficient [MATH] of [MATH] in Lemma [REF] is non-positive.', '1803.05578-2-81-0': 'We first need to bound [MATH].', '1803.05578-2-81-1': '[EQUATION]', '1803.05578-2-81-2': 'It can be easily verified that if [MATH] and [MATH], then [MATH].', '1803.05578-2-81-3': 'Using this fact with [MATH] and [MATH], we have: [EQUATION]', '1803.05578-2-81-4': 'We now analyze the coefficient of [MATH] [EQUATION]', '1803.05578-2-81-5': 'The coefficient [MATH] of [MATH] in [REF] is non-positive.', '1803.05578-2-81-6': '[EQUATION]', '1803.05578-2-81-7': 'Here the last inequality follows since [MATH] and [MATH].', '1803.05578-2-81-8': 'We now rearrange the definition of [MATH] to yield the identity: [EQUATION]', '1803.05578-2-81-9': 'Using this, we have: [EQUATION]', '1803.05578-2-81-10': 'Here the last line followed since [MATH], [MATH], and [MATH].', '1803.05578-2-81-11': 'Hence the proof is complete.', '1803.05578-2-82-0': '[Proof of Theorem [REF]] Using the master inequality [REF] in combination with the previous Lemmas [REF], [REF], [REF], and [REF], we have: [EQUATION]', '1803.05578-2-82-1': 'When we have: [EQUATION] then the Lyapunov function [MATH] has decreased below [MATH] in expectation.', '1803.05578-2-82-2': 'Hence the complexity [MATH] satisfies: [EQUATION]', '1803.05578-2-82-3': 'Now it can be shown that for [MATH], we have: [EQUATION]', '1803.05578-2-82-4': 'Since [MATH], we have [MATH].', '1803.05578-2-82-5': 'Hence: [EQUATION]', '1803.05578-2-82-6': 'An expression for [MATH], the complexity of [MATH] follows by similar reasoning.', '1803.05578-2-82-7': '[EQUATION]', '1803.05578-2-82-8': 'Finally we have: [EQUATION] which completes the proof.', '1803.05578-2-83-0': '# Ordinary Differential Equation Analysis', '1803.05578-2-84-0': '## Derivation of ODE for synchronous A2BCD', '1803.05578-2-85-0': 'If we take expectations with respect to [MATH], then synchronous (no delay) [MATH] becomes: [EQUATION]', '1803.05578-2-85-1': 'We find it convenient to define [MATH].', '1803.05578-2-85-2': 'Inspired by this, we consider the following iteration: [EQUATION] for coefficients: [EQUATION] [MATH] is a discretization scale parameter that will be sent to [MATH] to obtain an ODE analogue of synchronous [MATH].', '1803.05578-2-85-3': 'We first use [REF] to eliminate [MATH] from from [REF].', '1803.05578-2-85-4': '[EQUATION]', '1803.05578-2-85-5': 'We now eliminate [MATH] using [REF]: [EQUATION]', '1803.05578-2-85-6': 'Now to derive an ODE, we let [MATH].', '1803.05578-2-85-7': 'Then [MATH].', '1803.05578-2-85-8': 'Hence the above becomes: [EQUATION]', '1803.05578-2-85-9': 'We now look at some of the terms in this equation to find the highest-order dependence on [MATH].', '1803.05578-2-85-10': '[EQUATION]', '1803.05578-2-85-11': 'We also have: [EQUATION]', '1803.05578-2-85-12': 'Hence using these facts on [REF], we have: [EQUATION]', '1803.05578-2-85-13': 'Taking the limit as [MATH], we obtain the ODE: [EQUATION]', '1803.05578-2-86-0': '## Convergence proof for synchronous ODE', '1803.05578-2-87-0': '[EQUATION]', '1803.05578-2-87-1': 'Hence we have [MATH].', '1803.05578-2-87-2': 'Therefore [MATH].', '1803.05578-2-87-3': 'That is: [EQUATION] which implies: [EQUATION]', '1803.05578-2-88-0': '## Asynchronicity error lemma', '1803.05578-2-89-0': 'This result is the continuous-time analogue of Lemma [REF].', '1803.05578-2-89-1': 'First notice that [MATH] and [MATH].', '1803.05578-2-89-2': 'We also have: [EQUATION]', '1803.05578-2-89-3': 'Hence using [MATH]: [EQUATION]', '1803.05578-2-89-4': 'Now when [MATH], we have [MATH].', '1803.05578-2-89-5': 'Hence when [MATH], we have: [EQUATION] and the result easily follows.', '1803.05578-2-90-0': '## Convergence analysis for the asynchronous ODE', '1803.05578-2-91-0': 'We consider the same energy as in the synchronous case (that is, the ODE in [REF]).', '1803.05578-2-91-1': 'Similar to before, we have: [EQUATION] where the final equality follows from the proof in Section [REF].', '1803.05578-2-91-2': 'Hence [EQUATION]', '1803.05578-2-91-3': 'Now we present an inequality that is similar to [REF].', '1803.05578-2-92-0': 'Let [MATH].', '1803.05578-2-92-1': 'Then: [EQUATION]', '1803.05578-2-92-2': 'Since [MATH] is a delayed version of [MATH], we have: [MATH] for some function [MATH] (though this can be easily generalized to an inconsistent read).', '1803.05578-2-92-3': 'Recall that for [MATH], we have [MATH].', '1803.05578-2-92-4': 'Hence [EQUATION]', '1803.05578-2-92-5': 'We use this lemma twice on [MATH] and [MATH] in [REF] with [MATH] and [MATH] respectively, to yield: [EQUATION]', '1803.05578-2-92-6': 'The proof of convergence is completed in Section [REF].', '1803.05578-2-93-0': '# Optimality proof', '1803.05578-2-94-0': 'For parameter set [MATH], we construct a block-separable function [MATH] on the space [MATH] (separated into [MATH] blocks of size [MATH]), which will imply this lower bound.', '1803.05578-2-94-1': 'Define [MATH].', '1803.05578-2-94-2': 'We define the matrix [MATH] via: [EQUATION]', '1803.05578-2-94-3': 'Hence we define [MATH] on [MATH] via: [EQUATION] which is clearly [MATH]-strongly convex and [MATH]-Lipschitz on [MATH].', '1803.05578-2-94-4': 'From Lemma 8 of [CITATION], we know that this function has unique minimizer [EQUATION]', '1803.05578-2-94-5': 'Finally, we define [MATH] via: [EQUATION]', '1803.05578-2-94-6': 'Now let [MATH] be the [MATH]th unit vector of the [MATH]th block of size [MATH] in [MATH].', '1803.05578-2-94-7': 'For [MATH], we define the subspaces [EQUATION] [MATH] is the subspace with the first [MATH] components of block [MATH] nonzero, the first [MATH] components of block [MATH] nonzero, etc.', '1803.05578-2-94-8': 'First notice that [MATH].', '1803.05578-2-94-9': 'Also, clearly, we have: [EQUATION] [MATH] is supported on the [MATH]th block, hence why all the other indices are [MATH].', '1803.05578-2-94-10': 'The patten of nonzeros in [MATH] means that the gradient will have at most [MATH] more nonzero on the [MATH]th block (see [CITATION]).', '1803.05578-2-95-0': 'Let the initial point [MATH] belong to [MATH].', '1803.05578-2-95-1': 'Let [MATH] be the number of times we have had [MATH] for [MATH].', '1803.05578-2-95-2': 'By induction on [REF] using [REF], we have: [EQUATION]', '1803.05578-2-95-3': 'Hence if [MATH] and [MATH], then [EQUATION]', '1803.05578-2-95-4': 'Therefore for all [MATH], we have: [EQUATION]', '1803.05578-2-95-5': 'To evaluate this expectation, we note: [EQUATION]', '1803.05578-2-95-6': 'Hence [EQUATION]', '1803.05578-2-95-7': 'For any [MATH], we may select the starting iterate [MATH] by defining its block [MATH] via: [EQUATION]', '1803.05578-2-95-8': 'For such a choice of [MATH], we have [EQUATION]', '1803.05578-2-95-9': 'Hence for this choice of [MATH]: [EQUATION]', '1803.05578-2-95-10': 'Now notice: [EQUATION]', '1803.05578-2-95-11': 'Hence [EQUATION]', '1803.05578-2-95-12': 'Now if we let [MATH], then we have: [EQUATION]', '1803.05578-2-95-13': 'Now let us take the minimum of the right-hand side over the parameters [MATH], subject to [MATH].', '1803.05578-2-95-14': 'The solution to this minimization is clearly: [EQUATION]', '1803.05578-2-95-15': 'Hence [EQUATION]', '1803.05578-2-95-16': 'Hence the complexity [MATH] satisfies: [EQUATION]', '1803.05578-2-96-0': '# Extensions', '1803.05578-2-97-0': 'As mentioned, a stronger result than Theorem [REF] is possible.', '1803.05578-2-97-1': 'In the case when [MATH] for all [MATH], we can consider a slight modification of the coefficients: [EQUATION] for the asynchronicity parameter: [EQUATION]', '1803.05578-2-97-2': 'This leads to complexity: [EQUATION]', '1803.05578-2-97-3': 'Here there is no restriction on [MATH] as in Theorem [REF], and hence there is no restriction on [MATH].', '1803.05578-2-97-4': 'Assuming [MATH] gives optimal complexity to within a constant factor.', '1803.05578-2-97-5': 'Notice then that the resulting condition of [MATH] [EQUATION] now essentially matches the one in Theorem [REF] in Section [REF].', '1803.05578-2-97-6': 'While this result is stronger, it increases the complexity of the proof substantially.', '1803.05578-2-97-7': 'So in the interests of space and simplicity, we do not prove this stronger result.', '1803.05578-2-98-0': '# Efficient Implementation', '1803.05578-2-99-0': 'As mentioned in Section [REF], authors in [CITATION] proposed a linear transformation of an accelerated RBCD scheme that results in sparse coordinate updates.', '1803.05578-2-99-1': 'Our proposed algorithm can be given a similar efficient implementation.', '1803.05578-2-99-2': 'We may eliminate [MATH] from [MATH], and derive the equivalent iteration below: [EQUATION] where [MATH] and [MATH] are defined in the obvious way.', '1803.05578-2-99-3': 'Hence we define auxiliary variables [MATH] defined via: [EQUATION]', '1803.05578-2-99-4': 'These clearly follow the iteration: [EQUATION]', '1803.05578-2-99-5': 'Since the vector [MATH] is sparse, we can evolve variables [MATH], and [MATH] in a sparse manner, and recover the original iteration variables at the end of the algorithm via [REF].', '1803.05578-2-100-0': 'The gradient of the dual function is given by: [EQUATION]', '1803.05578-2-100-1': 'As mentioned before, it is necessary to maintain or recover [MATH] to calculate block gradients.', '1803.05578-2-100-2': 'Since [MATH] can be recovered via the linear relation in [REF], and the gradient is an affine function, we maintain the auxiliary vectors [MATH] and [MATH] instead.', '1803.05578-2-101-0': 'Hence we propose the following efficient implementation in Algorithm [REF].', '1803.05578-2-101-1': 'We used this to generate the results in Table [REF].', '1803.05578-2-101-2': 'We also note also that it can improve performance to periodically recover [MATH] and [MATH], reset the values of [MATH], [MATH], and [MATH] to [MATH], [MATH], and [MATH] respectively, and restarting the scheme (which can be done cheaply in time [MATH]).', '1803.05578-2-102-0': 'We let [MATH] represent [MATH], and [MATH] represent [MATH].', '1803.05578-2-102-1': '[MATH] is the Kronecker product.', '1803.05578-2-102-2': 'Each computing node has local outdated versions of [MATH] which we denote [MATH] respectively.', '1803.05578-2-102-3': 'We also find it convenient to define: [EQUATION] [H] Shared-memory implementation of A2BCD [1] Inputs: Function parameters [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '1803.05578-2-102-4': 'Delay [MATH] (obtained in dry run).', '1803.05578-2-102-5': 'Starting vectors [MATH], [MATH].', '1803.05578-2-102-6': 'Shared data: Solution vectors [MATH], [MATH]; auxiliary vectors [MATH], [MATH]; sparsifying matrix [MATH] Node local data: Solution vectors [MATH], [MATH], auxiliary vectors [MATH], [MATH], sparsifying matrix [MATH].', '1803.05578-2-102-7': 'Calculate parameters [MATH] via [REF].', '1803.05578-2-102-8': 'Set [MATH].', '1803.05578-2-102-9': 'Initializations: [MATH], [MATH], [MATH], [MATH], [MATH].', '1803.05578-2-102-10': 'not converged, each computing node asynchronous Randomly select block [MATH] via [REF].', '1803.05578-2-102-11': 'Read shared data into local memory: [MATH], [MATH], [MATH], [MATH], [MATH].', '1803.05578-2-102-12': 'Compute block gradient: [MATH] Compute quantity [MATH] Shared memory updates: Update [MATH], calculate inverse [MATH].', '1803.05578-2-102-13': '[MATH][MATH]kk+1[MATH] y v B p q [MATH]y[MATH]'}

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['1803.05578-1-83-8', '1803.05578-2-75-8'], ['1803.05578-1-89-0', '1803.05578-2-81-0'], ['1803.05578-1-89-2', '1803.05578-2-81-2'], ['1803.05578-1-89-3', '1803.05578-2-81-3'], ['1803.05578-1-89-4', '1803.05578-2-81-4'], ['1803.05578-1-89-5', '1803.05578-2-81-5'], ['1803.05578-1-89-7', '1803.05578-2-81-7'], ['1803.05578-1-89-8', '1803.05578-2-81-8'], ['1803.05578-1-89-9', '1803.05578-2-81-9'], ['1803.05578-1-89-10', '1803.05578-2-81-10'], ['1803.05578-1-89-11', '1803.05578-2-81-11'], ['1803.05578-1-15-0', '1803.05578-2-15-0'], ['1803.05578-1-15-1', '1803.05578-2-15-1'], ['1803.05578-1-100-2', '1803.05578-2-92-2'], ['1803.05578-1-100-3', '1803.05578-2-92-3'], ['1803.05578-1-100-5', '1803.05578-2-92-5'], ['1803.05578-1-100-6', '1803.05578-2-92-6'], ['1803.05578-1-22-1', '1803.05578-2-21-1'], ['1803.05578-1-3-0', '1803.05578-2-3-0'], ['1803.05578-1-3-1', '1803.05578-2-3-1'], ['1803.05578-1-3-2', '1803.05578-2-3-2'], ['1803.05578-1-3-3', '1803.05578-2-3-3'], ['1803.05578-1-3-4', '1803.05578-2-3-4'], ['1803.05578-1-76-1', '1803.05578-2-68-1'], ['1803.05578-1-76-2', '1803.05578-2-68-2'], ['1803.05578-1-76-3', '1803.05578-2-68-3'], ['1803.05578-1-76-4', '1803.05578-2-68-4'], ['1803.05578-1-76-5', '1803.05578-2-68-5'], ['1803.05578-1-76-6', '1803.05578-2-68-6'], ['1803.05578-1-99-0', '1803.05578-2-91-0'], ['1803.05578-1-99-1', '1803.05578-2-91-1'], ['1803.05578-1-99-3', '1803.05578-2-91-3'], ['1803.05578-1-8-2', '1803.05578-2-7-3'], ['1803.05578-1-102-0', '1803.05578-2-94-0'], ['1803.05578-1-102-2', '1803.05578-2-94-2'], ['1803.05578-1-102-3', '1803.05578-2-94-3'], ['1803.05578-1-102-5', '1803.05578-2-94-5'], ['1803.05578-1-102-6', '1803.05578-2-94-6'], ['1803.05578-1-102-7', '1803.05578-2-94-7'], ['1803.05578-1-102-8', '1803.05578-2-94-8'], ['1803.05578-1-102-9', '1803.05578-2-94-9'], ['1803.05578-1-102-10', '1803.05578-2-94-10'], ['1803.05578-1-33-1', '1803.05578-2-32-1'], ['1803.05578-1-78-0', '1803.05578-2-70-0'], ['1803.05578-1-78-1', '1803.05578-2-70-1'], ['1803.05578-1-31-0', '1803.05578-2-30-0'], ['1803.05578-1-31-4', '1803.05578-2-30-5'], ['1803.05578-1-31-5', '1803.05578-2-30-6'], ['1803.05578-1-10-0', '1803.05578-2-9-0'], ['1803.05578-1-10-1', '1803.05578-2-9-1'], ['1803.05578-1-10-2', '1803.05578-2-9-2'], ['1803.05578-1-44-0', '1803.05578-2-41-0'], ['1803.05578-1-44-1', '1803.05578-2-41-1'], ['1803.05578-1-13-0', '1803.05578-2-13-0'], ['1803.05578-1-13-1', '1803.05578-2-13-1'], ['1803.05578-1-84-0', '1803.05578-2-76-0'], ['1803.05578-1-52-2', '1803.05578-2-43-3'], ['1803.05578-1-88-0', '1803.05578-2-80-0'], ['1803.05578-1-88-1', '1803.05578-2-80-1'], ['1803.05578-1-88-2', '1803.05578-2-80-2'], ['1803.05578-1-88-4', '1803.05578-2-80-4'], ['1803.05578-1-88-6', '1803.05578-2-80-6'], ['1803.05578-1-35-0', '1803.05578-2-34-0'], ['1803.05578-1-35-1', '1803.05578-2-34-1'], ['1803.05578-1-17-0', '1803.05578-2-17-0'], ['1803.05578-1-17-1', '1803.05578-2-17-1'], ['1803.05578-1-17-2', '1803.05578-2-17-2'], ['1803.05578-1-17-3', '1803.05578-2-17-3'], ['1803.05578-1-72-0', '1803.05578-2-64-0'], ['1803.05578-1-72-1', '1803.05578-2-64-1'], ['1803.05578-1-72-2', '1803.05578-2-64-2'], ['1803.05578-1-72-3', '1803.05578-2-64-3'], ['1803.05578-1-21-0', '1803.05578-2-20-0'], ['1803.05578-1-21-1', '1803.05578-2-20-1'], ['1803.05578-1-21-2', '1803.05578-2-20-2'], ['1803.05578-1-21-3', '1803.05578-2-20-3'], ['1803.05578-1-21-4', '1803.05578-2-20-4'], ['1803.05578-1-97-0', '1803.05578-2-89-0'], ['1803.05578-1-97-1', '1803.05578-2-89-1'], ['1803.05578-1-97-2', '1803.05578-2-89-2'], ['1803.05578-1-97-3', '1803.05578-2-89-3'], ['1803.05578-1-97-4', '1803.05578-2-89-4'], ['1803.05578-1-97-5', '1803.05578-2-89-5'], ['1803.05578-1-14-0', '1803.05578-2-14-0'], ['1803.05578-1-14-1', '1803.05578-2-14-1'], ['1803.05578-1-14-2', '1803.05578-2-14-2'], ['1803.05578-1-19-0', '1803.05578-2-18-0'], ['1803.05578-1-9-0', '1803.05578-2-8-0'], ['1803.05578-1-9-1', '1803.05578-2-8-1'], ['1803.05578-1-69-0', '1803.05578-2-61-0'], 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['1803.05578-1-69-1', '1803.05578-2-61-1'], ['1803.05578-1-56-2', '1803.05578-2-45-2'], ['1803.05578-1-56-3', '1803.05578-2-45-3'], ['1803.05578-1-40-0', '1803.05578-2-37-2'], ['1803.05578-1-93-0', '1803.05578-2-85-0'], ['1803.05578-1-93-1', '1803.05578-2-85-1'], ['1803.05578-1-93-2', '1803.05578-2-85-2'], ['1803.05578-1-93-3', '1803.05578-2-85-3'], ['1803.05578-1-93-5', '1803.05578-2-85-5'], ['1803.05578-1-93-6', '1803.05578-2-85-6'], ['1803.05578-1-93-8', '1803.05578-2-85-8'], ['1803.05578-1-93-9', '1803.05578-2-85-9'], ['1803.05578-1-93-11', '1803.05578-2-85-11'], ['1803.05578-1-93-12', '1803.05578-2-85-12'], ['1803.05578-1-93-13', '1803.05578-2-85-13'], ['1803.05578-1-67-0', '1803.05578-2-59-0'], ['1803.05578-1-67-1', '1803.05578-2-59-1'], ['1803.05578-1-67-2', '1803.05578-2-59-2'], ['1803.05578-1-67-3', '1803.05578-2-59-3'], ['1803.05578-1-67-4', '1803.05578-2-59-4'], ['1803.05578-1-67-5', '1803.05578-2-59-5'], ['1803.05578-1-79-0', '1803.05578-2-71-0'], ['1803.05578-1-79-1', '1803.05578-2-71-1'], ['1803.05578-1-79-2', '1803.05578-2-71-2'], ['1803.05578-1-79-3', '1803.05578-2-71-3'], ['1803.05578-1-79-5', '1803.05578-2-71-5'], ['1803.05578-1-79-6', '1803.05578-2-71-6'], ['1803.05578-1-24-0', '1803.05578-2-23-0'], ['1803.05578-1-24-1', '1803.05578-2-23-1'], ['1803.05578-1-24-4', '1803.05578-2-23-3'], ['1803.05578-1-28-0', '1803.05578-2-27-0'], ['1803.05578-1-28-1', '1803.05578-2-27-1'], ['1803.05578-1-28-3', '1803.05578-2-27-3'], ['1803.05578-1-90-0', '1803.05578-2-82-0'], ['1803.05578-1-90-1', '1803.05578-2-82-1'], ['1803.05578-1-90-2', '1803.05578-2-82-2'], ['1803.05578-1-90-3', '1803.05578-2-82-3'], ['1803.05578-1-90-4', '1803.05578-2-82-4'], ['1803.05578-1-90-6', '1803.05578-2-82-6'], ['1803.05578-1-90-8', '1803.05578-2-82-8'], ['1803.05578-1-20-0', '1803.05578-2-19-0'], ['1803.05578-1-20-1', '1803.05578-2-19-1'], ['1803.05578-1-20-2', '1803.05578-2-19-2'], ['1803.05578-1-20-6', '1803.05578-2-19-7'], ['1803.05578-1-20-7', '1803.05578-2-19-8'], ['1803.05578-1-20-8', '1803.05578-2-19-9'], ['1803.05578-1-63-0', '1803.05578-2-55-0'], ['1803.05578-1-11-0', '1803.05578-2-10-0'], ['1803.05578-1-11-1', '1803.05578-2-10-1'], ['1803.05578-1-11-2', '1803.05578-2-10-4'], ['1803.05578-1-11-3', '1803.05578-2-10-5'], ['1803.05578-1-11-4', '1803.05578-2-10-6'], ['1803.05578-1-11-6', '1803.05578-2-10-8'], ['1803.05578-1-82-0', '1803.05578-2-74-0'], ['1803.05578-1-82-1', '1803.05578-2-74-1'], ['1803.05578-1-2-0', '1803.05578-2-2-0'], ['1803.05578-1-2-1', '1803.05578-2-2-1'], ['1803.05578-1-2-2', '1803.05578-2-2-2'], ['1803.05578-1-2-3', '1803.05578-2-2-3'], ['1803.05578-1-0-0', '1803.05578-2-0-0'], ['1803.05578-1-103-0', '1803.05578-2-95-0'], ['1803.05578-1-103-1', '1803.05578-2-95-1'], ['1803.05578-1-103-2', '1803.05578-2-95-2'], ['1803.05578-1-103-3', '1803.05578-2-95-3'], ['1803.05578-1-103-4', '1803.05578-2-95-4'], ['1803.05578-1-103-5', '1803.05578-2-95-5'], ['1803.05578-1-103-7', '1803.05578-2-95-7'], ['1803.05578-1-103-8', '1803.05578-2-95-8'], ['1803.05578-1-103-9', '1803.05578-2-95-9'], ['1803.05578-1-103-12', '1803.05578-2-95-12'], ['1803.05578-1-103-13', '1803.05578-2-95-13'], ['1803.05578-1-103-14', '1803.05578-2-95-14'], ['1803.05578-1-103-16', '1803.05578-2-95-16'], ['1803.05578-1-55-0', '1803.05578-2-44-0'], ['1803.05578-1-55-1', '1803.05578-2-44-1'], ['1803.05578-1-55-2', '1803.05578-2-44-2'], ['1803.05578-1-57-1', '1803.05578-2-46-1'], ['1803.05578-1-57-2', '1803.05578-2-46-2'], ['1803.05578-1-71-1', '1803.05578-2-63-1'], ['1803.05578-1-80-1', '1803.05578-2-72-1'], ['1803.05578-1-80-2', '1803.05578-2-72-2'], ['1803.05578-1-80-3', '1803.05578-2-72-3'], ['1803.05578-1-80-5', '1803.05578-2-72-5'], ['1803.05578-1-80-7', '1803.05578-2-72-7'], ['1803.05578-1-80-8', '1803.05578-2-72-8'], ['1803.05578-1-80-9', '1803.05578-2-72-9'], ['1803.05578-1-27-0', '1803.05578-2-26-0'], ['1803.05578-1-29-0', '1803.05578-2-28-0'], ['1803.05578-1-29-2', '1803.05578-2-28-2'], ['1803.05578-1-73-0', '1803.05578-2-65-0'], ['1803.05578-1-73-1', '1803.05578-2-65-1'], ['1803.05578-1-73-2', '1803.05578-2-65-2'], ['1803.05578-1-26-1', '1803.05578-2-25-1'], ['1803.05578-1-26-3', '1803.05578-2-25-3'], ['1803.05578-1-34-0', '1803.05578-2-33-2'], ['1803.05578-1-75-0', '1803.05578-2-67-0'], ['1803.05578-1-75-1', '1803.05578-2-67-1'], ['1803.05578-1-75-2', '1803.05578-2-67-2'], ['1803.05578-1-12-0', '1803.05578-2-11-0'], ['1803.05578-1-12-1', '1803.05578-2-12-0'], ['1803.05578-1-12-2', '1803.05578-2-12-1'], ['1803.05578-1-41-0', '1803.05578-2-38-0'], ['1803.05578-1-41-3', '1803.05578-2-38-3'], ['1803.05578-1-41-5', '1803.05578-2-38-5'], ['1803.05578-1-42-0', '1803.05578-2-38-7'], ['1803.05578-1-42-1', '1803.05578-2-38-8'], ['1803.05578-1-37-0', '1803.05578-2-36-0'], ['1803.05578-1-37-1', '1803.05578-2-36-1'], ['1803.05578-1-37-2', '1803.05578-2-36-2'], ['1803.05578-1-37-3', '1803.05578-2-36-3'], ['1803.05578-1-38-2', '1803.05578-2-36-5'], ['1803.05578-1-38-4', '1803.05578-2-36-7'], ['1803.05578-1-102-4', '1803.05578-2-94-4'], ['1803.05578-1-31-1', '1803.05578-2-30-1'], ['1803.05578-1-52-1', '1803.05578-2-43-2'], ['1803.05578-1-57-0', '1803.05578-2-46-0'], ['1803.05578-1-29-1', '1803.05578-2-28-1'], ['1803.05578-1-26-0', '1803.05578-2-25-0']]

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[]

[['1803.05578-1-15-3', '1803.05578-2-15-3'], ['1803.05578-1-35-2', '1803.05578-2-34-2'], ['1803.05578-1-59-0', '1803.05578-2-48-0'], ['1803.05578-1-59-2', '1803.05578-2-48-3'], ['1803.05578-1-59-3', '1803.05578-2-48-4'], ['1803.05578-1-59-5', '1803.05578-2-48-5'], ['1803.05578-1-60-1', '1803.05578-2-49-1'], ['1803.05578-1-24-2', '1803.05578-2-23-2'], ['1803.05578-1-28-4', '1803.05578-2-27-4'], ['1803.05578-1-20-4', '1803.05578-2-19-4'], ['1803.05578-1-20-5', '1803.05578-2-19-6']]

[]

['1803.05578-1-1-0', '1803.05578-1-4-0', '1803.05578-1-19-2', '1803.05578-1-25-0', '1803.05578-1-25-1', '1803.05578-1-31-6', '1803.05578-1-32-1', '1803.05578-1-33-0', '1803.05578-1-39-0', '1803.05578-1-39-1', '1803.05578-1-39-2', '1803.05578-1-41-4', '1803.05578-1-41-6', '1803.05578-1-43-1', '1803.05578-1-47-0', '1803.05578-1-47-1', '1803.05578-1-47-2', '1803.05578-1-47-3', '1803.05578-1-47-4', '1803.05578-1-47-5', '1803.05578-1-48-1', '1803.05578-1-64-0', '1803.05578-1-64-1', '1803.05578-1-65-0', '1803.05578-1-70-0', '1803.05578-1-70-1', '1803.05578-1-70-2', '1803.05578-1-70-3', '1803.05578-1-71-0', '1803.05578-1-76-0', '1803.05578-1-79-4', '1803.05578-1-80-0', '1803.05578-1-80-4', '1803.05578-1-80-6', '1803.05578-1-83-2', '1803.05578-1-83-7', '1803.05578-1-86-0', '1803.05578-1-87-0', '1803.05578-1-87-1', '1803.05578-1-88-3', '1803.05578-1-88-5', '1803.05578-1-89-1', '1803.05578-1-89-6', '1803.05578-1-90-5', '1803.05578-1-90-7', '1803.05578-1-93-4', '1803.05578-1-93-7', '1803.05578-1-93-10', '1803.05578-1-95-0', '1803.05578-1-95-1', '1803.05578-1-95-2', '1803.05578-1-95-3', '1803.05578-1-99-2', '1803.05578-1-100-0', '1803.05578-1-100-1', '1803.05578-1-100-4', '1803.05578-1-102-1', '1803.05578-1-103-6', '1803.05578-1-103-10', '1803.05578-1-103-11', '1803.05578-1-103-15', '1803.05578-2-1-0', '1803.05578-2-18-2', '1803.05578-2-19-5', '1803.05578-2-24-0', '1803.05578-2-24-1', '1803.05578-2-30-7', '1803.05578-2-31-1', '1803.05578-2-32-0', '1803.05578-2-33-0', '1803.05578-2-38-4', '1803.05578-2-40-0', '1803.05578-2-49-0', '1803.05578-2-56-0', '1803.05578-2-56-1', '1803.05578-2-57-0', '1803.05578-2-62-0', '1803.05578-2-62-1', '1803.05578-2-62-2', '1803.05578-2-62-3', '1803.05578-2-63-0', '1803.05578-2-68-0', '1803.05578-2-71-4', '1803.05578-2-72-0', '1803.05578-2-72-4', '1803.05578-2-72-6', '1803.05578-2-75-2', '1803.05578-2-75-7', '1803.05578-2-78-0', '1803.05578-2-79-0', '1803.05578-2-79-1', '1803.05578-2-80-3', '1803.05578-2-80-5', '1803.05578-2-81-1', '1803.05578-2-81-6', '1803.05578-2-82-5', '1803.05578-2-82-7', '1803.05578-2-85-4', '1803.05578-2-85-7', '1803.05578-2-85-10', '1803.05578-2-87-0', '1803.05578-2-87-1', '1803.05578-2-87-2', '1803.05578-2-87-3', '1803.05578-2-91-2', '1803.05578-2-92-0', '1803.05578-2-92-1', '1803.05578-2-92-4', '1803.05578-2-94-1', '1803.05578-2-95-6', '1803.05578-2-95-10', '1803.05578-2-95-11', '1803.05578-2-95-15', '1803.05578-2-102-0', '1803.05578-2-102-1', '1803.05578-2-102-2', '1803.05578-2-102-3', '1803.05578-2-102-4', '1803.05578-2-102-5', '1803.05578-2-102-6', '1803.05578-2-102-7', '1803.05578-2-102-8', '1803.05578-2-102-9', '1803.05578-2-102-10', '1803.05578-2-102-11', '1803.05578-2-102-12', '1803.05578-2-102-13']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1803.05578

astro-ph-0303624

{'astro-ph-0303624-1-0-0': 'We study the effect of random extra-galactic magnetic fields on the propagation of protons of energy larger than [MATH] eV.', 'astro-ph-0303624-1-0-1': 'We show that for reasonable field values (in the 0.1 [MATH]G range) the transition between diffusive and ballistic regimes occurs in the same energy range as the GZK cutoff (a few [MATH] eV).', 'astro-ph-0303624-1-0-2': 'The usual interpretation of the flux reduction above the GZK energy in terms of a sudden reduction of the visible horizon is modified.', 'astro-ph-0303624-1-0-3': 'Moreover, since the size of the diffusion sphere of a continuous source of cosmic rays is of the order of 10 Mpc, the local structure of the Universe and therefore of potential local astrophysical sources plays a dominant role in the expected spectrum.', 'astro-ph-0303624-1-0-4': 'Under reasonable assumptions on the sources configurations the expected GZK cutoff is not visible.', 'astro-ph-0303624-1-1-0': '# Introduction', 'astro-ph-0303624-1-2-0': 'The cosmic microwave background is expected to limit the travel distance of nucleons and nuclei above 30 EeV due to photo-disintegration or photo-production of pions.', 'astro-ph-0303624-1-2-1': 'Those interactions impose a black body cut-off in the spectrum of high energy cosmic rays known as the GZK cutoff [CITATION] strongly limiting their visible flux.', 'astro-ph-0303624-1-3-0': 'The usual interpretation of the expected reduction of the flux above 100 EeV lies in the sudden reduction of the visible Universe from a few Gpc below 10 EeV to less than 20 Mpc above a few 100 EeV.', 'astro-ph-0303624-1-3-1': 'This simple view may however be strongly modified in the presence of extra-galactic magnetic fields of the order of 0.1 [MATH]G.', 'astro-ph-0303624-1-3-2': 'Such strong fields are compatible with current observations and upper limits from Faraday rotation measurements which indicate field strengths at the [MATH]G level within the central Mpc region of galaxy clusters [CITATION].', 'astro-ph-0303624-1-3-3': 'Relatively strong magnetic fields may exist in intergalactic space with coherence lengths of the order of a Mpc.', 'astro-ph-0303624-1-4-0': 'In such a scenario a charged particle of 10 EeV has a Larmor radius of 100 kpc and propagates diffusively in the Universe while at higher energies (above 100 EeV, say) the trajectories are essentially ballistic.', 'astro-ph-0303624-1-4-1': 'Therefore the clear picture of the GZK cutoff imposed by particle dynamics gets blurred by another effect taking place at the same energy: the transition between ballistic and diffusive transport regimes.', 'astro-ph-0303624-1-5-0': 'Another point to note is that in such a field and for attenuation times of the order of a few Gyr (attenuation of pre-GZK protons by pair production), the radius of the diffusion sphere of a continuous source is about 10 Mpc.', 'astro-ph-0303624-1-5-1': 'On this scale, the Universe cannot be taken as isotropic and uniform.', 'astro-ph-0303624-1-5-2': 'This indicates that the local distribution of matter will play a dominant role in the observed energy spectra.', 'astro-ph-0303624-1-6-0': 'Given these ingredients, it is tempting to imagine a scenario where a few sources in our local Universe would dominantly contribute to the observed UHECR fluxes at all energies.', 'astro-ph-0303624-1-6-1': 'These sources would be sufficiently far away so that we lie outside their low-energy (1-10 EeV) diffusion spheres but close enough so that their high energy component does not get significantly attenuated.', 'astro-ph-0303624-1-7-0': 'In the following we explore this scenario, the next section describes our magnetic field model and our particle propagation Monte Carlo.', 'astro-ph-0303624-1-7-1': 'Section 3 discuss the magnetic horizon for charged particles and section 4 presents our result.', 'astro-ph-0303624-1-8-0': '# A simplistic magnetized Universe', 'astro-ph-0303624-1-9-0': '## Field modelling', 'astro-ph-0303624-1-10-0': 'We simulated a magnetized Universe following the method of [CITATION].', 'astro-ph-0303624-1-10-1': 'Our turbulent field has a zero mean value and fluctuates following a Gaussian distribution.', 'astro-ph-0303624-1-10-2': 'The spectrum of fluctuation is a power law [EQUATION].', 'astro-ph-0303624-1-10-3': 'The normalization [MATH] is obtained through the Wiener-Kintchine theorem by imposing the average fluctuation strength [MATH]: [EQUATION] leading to [EQUATION]', 'astro-ph-0303624-1-10-4': 'The maximum ([MATH]) and minimum ([MATH]) modes are given respectively by the coherence length of our field model [MATH] and the size of our field grid [MATH].', 'astro-ph-0303624-1-10-5': 'We used a Kolmogorov fluctuation spectrum corresponding to [MATH].', 'astro-ph-0303624-1-10-6': 'In Fourier space the field verifies [MATH] to satisfy [MATH] and we pick up a random phase in the transverse plane to complete the field description.', 'astro-ph-0303624-1-10-7': 'In real space the field is computed on a 15 kpc grid over a volume of 1 Mpc[MATH].', 'astro-ph-0303624-1-10-8': 'This volume is then translated periodically to fill our model Universe.', 'astro-ph-0303624-1-10-9': 'The value of the field at any point in space is obtained through a linear interpolation of the 8 closest vertices of the grid.', 'astro-ph-0303624-1-11-0': '## particle trajectories and sources spectra', 'astro-ph-0303624-1-12-0': 'For each Monte-Carlo simulation, we compute a field according to the previous paragraph and we follow a number of protons as long as their energy is above 10 EeV or their propagation time is below 3 Gyr.', 'astro-ph-0303624-1-12-1': 'At each step along their trajectory we solve the equation of motion in the local field and compute the energy loss due to pion photo-production, simulating each interaction.', 'astro-ph-0303624-1-12-2': 'The losses due to pair production are treated continuously.', 'astro-ph-0303624-1-12-3': 'Neutrons, if produced, are also followed until they transform again into a proton via the photo-production of pions or until they decay.', 'astro-ph-0303624-1-13-0': 'To compute the spectrum of a source at a given distance [MATH] from the observer, we generate a set of protons at the origin and record the times (and the corresponding energies) of each crossing of a sphere of radius [MATH], centered on the origin.', 'astro-ph-0303624-1-13-1': 'By construction our spectrum does not depend on the crossing position on the sphere, therefore we record them all.', 'astro-ph-0303624-1-13-2': 'Moreover the effective detection probability being very small (the surface of a detector is negligible compared to the surface of the sphere), we compute the spectrum at various radii following the same set of particles.', 'astro-ph-0303624-1-14-0': 'Fig. [REF] shows the distribution of particle energies as a function of trajectory length (equivalent to time) for an observer located on a sphere 12 Mpc away from the source, and for an initial energy of 800 EeV.', 'astro-ph-0303624-1-15-0': 'For radii comparable to or smaller than [MATH] the orientation of the field on the surface of the sphere may not average properly an may lead to strong distortion in the time distribution of the particle crossings (a particle trapped along a field line parallel to the sphere surface may cross it many times over a small distance).', 'astro-ph-0303624-1-15-1': 'To avoid this effect we regenerate a field configuration every 100 generated protons.', 'astro-ph-0303624-1-16-0': 'From the recorded information we are able to compute the probability [MATH] for a particle produced at time [MATH] with energy [MATH] to be detected at a given distance [MATH] at time [MATH] with an energy [MATH].', 'astro-ph-0303624-1-16-1': 'From these probability tables, and assuming an isotropic distribution of sources [MATH], the spectrum observed today (time t) from sources located between [MATH] and [MATH] is given by: [EQUATION] where [MATH] is the source injection spectrum and where we have neglected all cosmological effects.', 'astro-ph-0303624-1-17-0': '# Magnetic horizon', 'astro-ph-0303624-1-18-0': 'As we mentioned in the introduction, for field strength of the order of 0.1 [MATH]G with Mpc coherence length, particle trajectories below a few tens of EeV are well described by the diffusion equation.', 'astro-ph-0303624-1-18-1': 'Several diffusive regime may be distinguished within a given field configuration, depending on the particle energy.', 'astro-ph-0303624-1-18-2': 'For the pure random field we are considering, three regimes have been identified and can be parameterized with the following diffusion coefficients (in Mpc[MATH]/Myr) [CITATION]: [EQUATION]', 'astro-ph-0303624-1-18-3': 'The transition energy [MATH] is evaluated from the Larmor radius, and the field coherence length [MATH], giving: [EQUATION]', 'astro-ph-0303624-1-18-4': 'Below the GZK cutoff energy, where losses do not play a crucial role, the Green function for the diffusion is given by: [EQUATION]', 'astro-ph-0303624-1-18-5': 'From this solution, one can estimate the radius of the diffusion sphere as a function of time and energy: [EQUATION]', 'astro-ph-0303624-1-18-6': 'For a given source distance, [MATH], the energy scale separating the ballistic and diffusive regimes can be estimated by comparing the corresponding propagation times: [MATH] and [MATH].', 'astro-ph-0303624-1-18-7': 'Fig. [REF] shows this energy as a function of source distance for a random field of [MATH]G.', 'astro-ph-0303624-1-18-8': 'It is remarkable that for sources within the GZK sphere (10-100 Mpc), the transition occurs around the GZK energy (i.e. a few tens of EeV).', 'astro-ph-0303624-1-19-0': 'The existence of a diffusion sphere whose radius grows much slower than [MATH] gives rise to a magnetic horizon which limits the distance up to which a given particle can escape from its source before losing most of its energy.', 'astro-ph-0303624-1-19-1': 'This horizon is given by: [EQUATION] where [MATH] is the energy loss time.', 'astro-ph-0303624-1-20-0': 'Using the Monte Carlo described in the previous section, we computed this distance as the maximum radius reached by 68% of the particles over a propagation time [MATH].', 'astro-ph-0303624-1-20-1': 'This horizon is shown on Fig. [REF] at lower energy (below 10 EeV) it is given by Eqn. [REF] and the parameterization of the diffusion coefficients given in Eqn. [REF].', 'astro-ph-0303624-1-21-0': '# Results', 'astro-ph-0303624-1-22-0': 'All the following results have been obtained for continuous sources with power-law injection spectra of index 2.3 and maximum energy of 1000 EeV.', 'astro-ph-0303624-1-23-0': 'We calculated the observed spectra for various source positions (45 values distributed uniformly in log scale from 1 Mpc to 1 Gpc) and turbulent field values ranging from 3 to 300 nG.', 'astro-ph-0303624-1-23-1': 'Fig. [REF] shows a comparison of these spectra for sources at 1, 10, 50 and 100 Mpc obtained with a 100 nG random field.', 'astro-ph-0303624-1-23-2': 'The solid angle effect has been taken away so that the normalization and spectra at all distances would be identical in the absence of magnetic field and energy losses.', 'astro-ph-0303624-1-23-3': 'As expected, the spectrum of a very close source (1 Mpc) shows a softening (index changing form [MATH] to about [MATH]) at low energies due to the accumulation of low energy particles, while at high energy the original spectrum is restored.', 'astro-ph-0303624-1-23-4': 'For such a nearby source there is of course no GZK cutoff, but this configuration would lead to anisotropy in the sky incompatible with the already available data.', 'astro-ph-0303624-1-23-5': 'At intermediate distance (10 Mpc) the flux is only slightly reduced at high energy and the contribution (corrected by E[MATH]) at 10 EeV and 1000 EeV are roughly the same, leading to a nearly flat spectrum in the 10-1000 EeV range.', 'astro-ph-0303624-1-24-0': 'Further away (above 50 Mpc) the flux is strongly attenuated at high energy because of the GZK cutoff but also at low energy because of the diffusive regime.', 'astro-ph-0303624-1-24-1': 'One should also note that increasing the distance between the source and the observer leads to a stronger GZK cutoff because the mean travel length increase faster than the source distance.', 'astro-ph-0303624-1-24-2': 'Moreover, for greater source distances, the diffusive regime extends to higher energy, as shown by Fig. [REF].', 'astro-ph-0303624-1-25-0': 'The main conclusion is therefore that far away sources have a negligible contribution to the total flux at all energies (and not only above GZK energy).', 'astro-ph-0303624-1-25-1': 'In other words, the presence of random magnetic fields leads to a non trivial configuration of sources contributing to the spectrum, even in the case of a uniform distribution.', 'astro-ph-0303624-1-26-0': 'In Fig. [REF] we compare the observed flux for a uniform distribution of sources between 10 and 1000 Mpc with and without a 300 nG random magnetic field.', 'astro-ph-0303624-1-26-1': 'In the later case the reduction of the low energy flux is clearly visible as the observer lies mostly outside the diffusion spheres.', 'astro-ph-0303624-1-26-2': 'The exact form of the spectrum strongly depends on the source configuration on a scale (a few tens of Mpc) where the Universe cannot be taken as uniform.', 'astro-ph-0303624-1-26-3': 'Therefore the local configuration of sources may lead to a spectrum that lies anywhere between the shapes of those obtained from individual sources at 10 or 50 Mpc as shown on Fig. [REF].', 'astro-ph-0303624-1-27-0': 'This comparison also shows that for a field of 300 nG, propagation becomes purely ballistic only above 500 EeV and the transition between the diffusive and ballistic regimes spreads over half a decade above 100 EeV.', 'astro-ph-0303624-1-27-1': 'This transition implies a flattening of the spectrum at energies higher than [MATH].', 'astro-ph-0303624-1-27-2': 'The position of this flattening strongly depends on the position of the most contributing sources, and on the value of random fields.', 'astro-ph-0303624-1-27-3': 'Increasing the value of the random fields ([MATH]G) leads to particle diffusion on small scales (a few tens of Mpc) even at energies higher than [MATH], and therefore moves the transition to the ballistic regime to energies so high ([MATH] eV) that this transition becomes invisible.', 'astro-ph-0303624-1-27-4': 'In such a situation a strong GZK cutoff is restored as the signature of the much shorter energy loss time of post GZK particles.', 'astro-ph-0303624-1-28-0': '# Conclusion', 'astro-ph-0303624-1-29-0': 'We have modeled a fully magnetized Universe to study the effect of random magnetic fields on the spectra of UHECR sources.', 'astro-ph-0303624-1-29-1': 'We showed that for magnetic fields in the 0.1 [MATH]G range, distant sources (above a few tens of Mpc) no longer contribute to the observed spectra at all energies.', 'astro-ph-0303624-1-29-2': 'In such a model the GZK cutoff is therefore no longer visible.', 'astro-ph-0303624-1-29-3': 'We are aware that a fully magnetized Universe is not a realistic model of our real environment.', 'astro-ph-0303624-1-29-4': 'Voids are known to exist, and, in those regions random fields would rather be in the nano Gauss range.', 'astro-ph-0303624-1-29-5': 'However, our main argument "far away sources do not contribute to the visible spectrum of UHECR at all energies" would be maintained if all sources are surrounded by a random field of order 0.1[MATH]G over distances of several Mpc, as expected for sources that lie within galaxies or clusters.', 'astro-ph-0303624-1-30-0': 'We did not model any particular configuration of sources to study the anisotropy effect, but given the slowness of the transition between the two propagation regimes and the fact that the transport becomes only purely ballistic above 500 EeV, we expect that anisotropy would only be visible above 100 EeV where the data is currently to scarce to make any definitive statement.', 'astro-ph-0303624-1-30-1': 'A very recent paper [CITATION] effectively shows that even with more localized magnetic fields the expected anisotropy from local sources is not ruled out by the current data.', 'astro-ph-0303624-1-30-2': 'One should also note that a few sources localized in the magnetic sheet around the local supercluster would suppress the bump before [MATH], visible on Fig. [REF], because low-energies particles leak through the walls [CITATION].', 'astro-ph-0303624-1-31-0': 'Finally we want to stress that this work was done to emphasize a mechanism and is therefore based on a simplified model of a magnetized Universe.', 'astro-ph-0303624-1-31-1': 'While doing this study we went in detail through the numerous literature on the subject [CITATION] (and references therein).', 'astro-ph-0303624-1-31-2': 'We realize that some of our arguments and conclusions are spread in these papers, however not brought together nor summarized in a simple model as we did here.'}

{'astro-ph-0303624-2-0-0': 'We study the effect of random extra-galactic magnetic fields on the propagation of protons of energy larger than [MATH] eV.', 'astro-ph-0303624-2-0-1': 'We show that for reasonable field values (in the 100 nG range) the transition between diffusive and ballistic regimes occurs in the same energy range as the GZK cutoff (a few [MATH] eV).', 'astro-ph-0303624-2-0-2': 'The usual interpretation of the flux reduction above the GZK energy in terms of a sudden reduction of the visible horizon is modified.', 'astro-ph-0303624-2-0-3': 'Moreover, since the size of the diffusion sphere of a continuous source of cosmic rays is of the order of 10 Mpc, the local structure of the Universe and therefore of potential local astrophysical sources plays a dominant role in the expected spectrum.', 'astro-ph-0303624-2-0-4': 'Under reasonable assumptions on the sources configurations the expected GZK cutoff is reduced.', 'astro-ph-0303624-2-1-0': '# Introduction', 'astro-ph-0303624-2-2-0': 'The cosmic microwave background is expected to limit the travel distance of nucleons and nuclei above 30 EeV, due to photo-disintegration or photo-production of pions.', 'astro-ph-0303624-2-2-1': 'These interactions impose a cut-off in the spectrum of high-energy cosmic rays known as the GZK cutoff [CITATION], strongly limiting their visible flux.', 'astro-ph-0303624-2-3-0': 'The usual interpretation of the expected reduction of the flux above 100 EeV lies in the sudden reduction of the visible Universe from a few Gpc below 10 EeV to less than 20 Mpc above a few 100 EeV.', 'astro-ph-0303624-2-3-1': 'This simple view may however be strongly modified in the presence of extra-galactic magnetic fields of the order of 100 nG.', 'astro-ph-0303624-2-3-2': 'Such strong fields are compatible with current observations and upper limits from Faraday rotation measurements which indicate field strengths at the [MATH]G level within the central Mpc region of galaxy clusters [CITATION].', 'astro-ph-0303624-2-3-3': 'Relatively strong magnetic fields may exist in intergalactic space with coherence lengths of the order of a Mpc.', 'astro-ph-0303624-2-4-0': 'In realistic models of the Universe, however, such strong fields cannot fill the entire intergalactic volume, and should rather be concentrated along the high density sheets where matter also concentrates, as suggested by large-scale structure formation models [CITATION].', 'astro-ph-0303624-2-4-1': 'The transport of UHECRs in this kind of environments, and their resulting spectrum observed on Earth, depend on the particular geometry of the fields, combined with that of the sources, and on the intensity of the magnetic fields.', 'astro-ph-0303624-2-4-2': 'This has been investigated by a number of authors using various assumptions for the distance of one or a few local sources, and working out UHECR propagation in a model of the local supercluster [CITATION].', 'astro-ph-0303624-2-4-3': 'It has been found that under some circumstances, the UHECR spectrum could show an attenuated GZK suppression.', 'astro-ph-0303624-2-5-0': 'In this paper, we focus on the effect of strong magnetic fields around our Galaxy, assuming that the local group lies within a supercluster with higher than average magnetic field.', 'astro-ph-0303624-2-5-1': 'In order to obtain a clear physical understanding of this effect alone, we use a spherically symmetric configuration and a homogeneous field in the Universe.', 'astro-ph-0303624-2-5-2': 'This is clearly not representative of the observed walls-and-voids structure of our Universe on all scales, but it allows us to emphasize a basic mechanism using a complete Monte-Carlo propagation model, and disentangling the configuration specific effects from the influence of strong fields.', 'astro-ph-0303624-2-6-0': 'As a matter of fact, in magnetic fields of a 100 nG, a charged particle of 10 EeV has a Larmor radius of 100 kpc and propagates diffusively while at higher energies (above 100 EeV, say) the trajectories are essentially ballistic.', 'astro-ph-0303624-2-6-1': 'Therefore the clear picture of the GZK cutoff imposed by particle dynamics gets blurred by another effect taking place at the same energy: the transition between ballistic and diffusive transport regimes.', 'astro-ph-0303624-2-7-0': 'Another point to note is that in such a field and for attenuation times of the order of a few Gyr (attenuation of pre-GZK protons by pair production), the radius of the diffusion sphere of a continuous source is only 10 Mpc.', 'astro-ph-0303624-2-7-1': 'Therefore, far away sources ([MATH] Mpc, say), would not be visible at any energy.', 'astro-ph-0303624-2-8-0': 'Given the above ingredients, nearby sources ([MATH] Mpc) appear to be responsible for the observed UHECR spectrum at all energies.', 'astro-ph-0303624-2-8-1': 'These sources would be sufficiently far away so that we lie outside their low-energy (1-10 EeV) diffusion spheres, therefore reducing the corresponding flux, but close enough so that their high-energy component does not get significantly attenuated by energy losses.', 'astro-ph-0303624-2-8-2': 'This can reduce the usually inferred flux difference between energies below and above the GZK energy.', 'astro-ph-0303624-2-8-3': 'In addition, on a 10 Mpc scale the Universe cannot be taken as isotropic and uniform, which indicates that the local distribution of matter will play a dominant role in the actually observed energy spectra.', 'astro-ph-0303624-2-9-0': 'In the following we explore this scenario.', 'astro-ph-0303624-2-9-1': 'The next section describes our magnetic field model and our particle propagation Monte-Carlo.', 'astro-ph-0303624-2-9-2': 'Section 3 discusses the magnetic horizon for charged particles and section 4 presents our results.', 'astro-ph-0303624-2-10-0': '# A simplistic magnetized Universe', 'astro-ph-0303624-2-11-0': '## Field modelling', 'astro-ph-0303624-2-12-0': 'We simulated a magnetized Universe following the method of [CITATION].', 'astro-ph-0303624-2-12-1': 'Our turbulent field has a zero mean value and fluctuates following a Gaussian distribution.', 'astro-ph-0303624-2-12-2': 'The spectrum of fluctuations is a power law [EQUATION]', 'astro-ph-0303624-2-12-3': 'The normalization [MATH] is obtained through the Wiener-Kintchine theorem by imposing the average fluctuation strength [MATH]: [EQUATION] leading to [EQUATION]', 'astro-ph-0303624-2-12-4': 'The maximum ([MATH]) and minimum ([MATH]) modes are set respectively by the size of the simulation box [MATH]1 Mpc (related to the coherence length of our field model) and the step size of the grid [MATH]15 kpc.', 'astro-ph-0303624-2-12-5': 'We used a Kolmogorov fluctuation spectrum corresponding to [MATH].', 'astro-ph-0303624-2-12-6': 'In Fourier space, the field verifies [MATH] to satisfy div[MATH] and we pick up a random phase in the transverse plane to complete the field description.', 'astro-ph-0303624-2-12-7': 'In real space the field is computed on a 15 kpc grid over a volume of 1 Mpc[MATH].', 'astro-ph-0303624-2-12-8': 'This volume is then translated periodically to fill our modelled Universe.', 'astro-ph-0303624-2-12-9': 'The value of the field at any point in space is obtained through a linear interpolation of the 8 closest vertices of the grid.', 'astro-ph-0303624-2-13-0': '## Particle trajectories and sources spectra', 'astro-ph-0303624-2-14-0': 'For each Monte-Carlo simulation, we compute a field configuration according to the above method and we follow a number of protons as long as their energy is above 10 EeV or their propagation time is below 3 Gyr.', 'astro-ph-0303624-2-14-1': 'At each step along their trajectory, we solve the equation of motion in the local field and compute the energy loss due to pion photo-production, simulating each interaction.', 'astro-ph-0303624-2-14-2': 'The losses due to pair production are treated continuously.', 'astro-ph-0303624-2-14-3': 'Neutrons, if produced, are also followed until they transform again into a proton via the photo-production of pions or until they decay.', 'astro-ph-0303624-2-15-0': 'To compute the spectrum of a source at a given distance [MATH] from the observer, we generate a set of protons at the origin and record the times (and the corresponding energies) of each crossing of a sphere of radius [MATH], centered on the origin.', 'astro-ph-0303624-2-15-1': 'By construction, our spectrum does not depend on the crossing position on the sphere, therefore we record them all.', 'astro-ph-0303624-2-15-2': 'Moreover the effective detection probability being very small (the surface of a detector is negligible compared to the surface of the sphere), we compute the spectrum at various radii following the same set of particles.', 'astro-ph-0303624-2-16-0': 'Fig. [REF] shows the distribution of particle energies as a function of trajectory length (equivalent to time) for an observer located on a sphere 12 Mpc away from the source, and for an initial energy of 800 EeV.', 'astro-ph-0303624-2-17-0': 'For radii comparable to or smaller than [MATH] the orientation of the field on the surface of the sphere may not average properly and may lead to strong distortion in the time distribution of the particle crossings (a particle trapped along a field line parallel to the sphere surface may cross it many times over a small distance).', 'astro-ph-0303624-2-17-1': 'To avoid this effect we regenerate a field configuration every 100 generated protons.', 'astro-ph-0303624-2-18-0': 'From the recorded information we are able to compute the probability [MATH] for a particle produced at time [MATH] with energy [MATH] to be detected at a given distance [MATH] at time [MATH] with an energy [MATH].', 'astro-ph-0303624-2-18-1': 'From these probability tables, and assuming an isotropic distribution of sources [MATH], the spectrum observed today (time t) from sources located between [MATH] and [MATH] is given by: [EQUATION] where [MATH] is the source injection spectrum and where we have neglected all cosmological effects.', 'astro-ph-0303624-2-19-0': '# Magnetic horizon', 'astro-ph-0303624-2-20-0': 'As we mentioned in the introduction, for field strengths of the order of 100 nG with Mpc coherence length, particle trajectories below a few tens of EeV are well described by the diffusion equation.', 'astro-ph-0303624-2-20-1': 'Several diffusive regime may be distinguished within a given field configuration, depending on the particle energy.', 'astro-ph-0303624-2-20-2': 'For the pure random field we are considering, three regimes have been identified and can be parameterized with the following diffusion coefficients (in Mpc[MATH]/Myr) [CITATION]: [EQUATION]', 'astro-ph-0303624-2-20-3': 'The transition energy [MATH] can be evaluated from the Larmor radius and the field coherence length [MATH], giving: [EQUATION]', 'astro-ph-0303624-2-20-4': 'Of course, the transition actually takes place over a range of energies around [MATH].', 'astro-ph-0303624-2-21-0': 'Below the GZK cutoff energy, where losses do not play a crucial role, the Green function for the diffusion is given by: [EQUATION]', 'astro-ph-0303624-2-21-1': 'From this solution, one can estimate the radius of the diffusion sphere as a function of time and energy: [EQUATION]', 'astro-ph-0303624-2-21-2': 'For a given source distance [MATH], the energy scale separating the ballistic and diffusive regimes can be estimated by comparing the corresponding propagation times: [MATH] and [MATH].', 'astro-ph-0303624-2-21-3': 'Fig. [REF] shows this energy as a function of source distance for a random field of 100 nG.', 'astro-ph-0303624-2-21-4': 'It is remarkable that for sources within the GZK sphere (10-100 Mpc), the transition occurs around the GZK energy (i.e. a few tens of EeV).', 'astro-ph-0303624-2-22-0': 'The existence of a diffusion sphere whose radius grows much slower than [MATH] gives rise to a magnetic horizon which limits the distance up to which a given particle can escape from its source before losing most of its energy.', 'astro-ph-0303624-2-22-1': 'This horizon is given by: [EQUATION] where [MATH] is the energy loss time.', 'astro-ph-0303624-2-23-0': 'Using the Monte Carlo described in the previous section, we computed this distance as the maximum radius reached by 68% of the particles over a propagation time [MATH].', 'astro-ph-0303624-2-23-1': 'This horizon is shown on Fig. [REF].', 'astro-ph-0303624-2-23-2': 'At lower energy (below 10 EeV), it is given by Eqn. [REF] and the parameterization of the diffusion coefficients given in Eqn. [REF].', 'astro-ph-0303624-2-23-3': 'Different values of turbulent magnetic fields have been used.', 'astro-ph-0303624-2-23-4': 'For relatively strong fields (300 nG), the horizon is the same before and after the GZK energy ; implying that the sudden reduction of the visible Universe at the origin of the expected suppression of the flux above 100 EeV does not occur in that case.', 'astro-ph-0303624-2-24-0': '# Results', 'astro-ph-0303624-2-25-0': 'All the following results have been obtained for continuous sources with power-law injection spectra of index 2.3 and maximum energy of 1000 EeV.', 'astro-ph-0303624-2-25-1': 'This slope is typical of particle acceleration at relativistic shockwave [CITATION].', 'astro-ph-0303624-2-26-0': 'We calculated the observed spectra for various source positions (45 values distributed uniformly in log scale from 1 Mpc to 1 Gpc) and turbulent field values ranging from 3 to 300 nG.', 'astro-ph-0303624-2-26-1': 'Fig. [REF] shows a comparison of these spectra for sources at 1, 10, 50 and 100 Mpc obtained with a 100 nG random field.', 'astro-ph-0303624-2-26-2': 'The solid angle effect has been taken away so that the normalization and spectra at all distances would be identical in the absence of magnetic field and energy losses.', 'astro-ph-0303624-2-26-3': 'As expected, the spectrum of a very close source (1 Mpc) shows a softening (index changing form [MATH] to about [MATH]) at low energies due to the accumulation of low energy particles, while at high energy the original spectrum is restored.', 'astro-ph-0303624-2-26-4': 'For such a nearby source there is of course no GZK cutoff, but this configuration would lead to an anisotropy in the sky incompatible with the already available data.', 'astro-ph-0303624-2-26-5': 'At intermediate distance (10 Mpc) the flux is only slightly reduced at high energy and the contribution (corrected by E[MATH]) at 10 EeV and 1000 EeV are roughly the same, leading to a nearly flat spectrum in the 10-1000 EeV range.', 'astro-ph-0303624-2-27-0': 'Further away (above 100 Mpc) the flux is strongly attenuated at high energy because of the GZK cutoff but also at low energy because of the diffusive regime.', 'astro-ph-0303624-2-27-1': 'One should also note that increasing the distance between the source and the observer leads to a stronger GZK cutoff because the mean travel length increases faster than the source distance.', 'astro-ph-0303624-2-27-2': 'Moreover, for greater source distances, the diffusive regime extends to higher energy, as shown by Fig. [REF].', 'astro-ph-0303624-2-28-0': 'The main conclusion is therefore that far away sources have a negligible contribution to the total flux at all energies (and not only above GZK energy).', 'astro-ph-0303624-2-28-1': 'In other words, the presence of random magnetic fields leads to a non trivial configuration of sources contributing to the spectrum, even in the case of a uniform distribution.', 'astro-ph-0303624-2-29-0': 'In Fig. [REF] we compare the observed flux for a uniform distribution of sources between 10 and 1000 Mpc with and without a 300 nG random magnetic field.', 'astro-ph-0303624-2-29-1': 'In the later case the reduction of the low energy flux is clearly visible as the observer lies mostly outside the diffusion spheres.', 'astro-ph-0303624-2-29-2': 'The exact form of the spectrum strongly depends on the source configuration on a scale (a few tens of Mpc) where the Universe cannot be taken as uniform.', 'astro-ph-0303624-2-29-3': 'Therefore the local configuration of sources may lead to a spectrum that lies anywhere between the shapes of those obtained from individual sources at 10 or 50 Mpc as shown on Fig. [REF].', 'astro-ph-0303624-2-30-0': 'This comparison also shows that for a field of 300 nG, propagation becomes purely ballistic only above 500 EeV and the transition between the diffusive and ballistic regimes spreads over half a decade above 100 EeV.', 'astro-ph-0303624-2-30-1': 'This transition implies a flattening of the spectrum at energies higher than [MATH].', 'astro-ph-0303624-2-30-2': 'The position of this flattening strongly depends on the position of the most contributing sources, and on the value of random fields.', 'astro-ph-0303624-2-30-3': 'Increasing the value of the random fields ([MATH]G) leads to particle diffusion on small scales (a few tens of Mpc) even at energies higher than [MATH], and therefore moves the transition to the ballistic regime to energies so high ([MATH] eV) that this transition becomes invisible.', 'astro-ph-0303624-2-30-4': 'In such a situation a strong GZK cutoff is restored as the signature of the much shorter energy loss time of post GZK particles.', 'astro-ph-0303624-2-31-0': '# Conclusion', 'astro-ph-0303624-2-32-0': 'We have modelled a fully magnetized Universe to study the effect of random magnetic fields on the spectra of UHECR sources, in a spherically symmetric Universe, in order to get rid of any additional effect related to the geometry of magnetic fields and source distributions.', 'astro-ph-0303624-2-32-1': 'We showed that for magnetic fields in the 100 nG range, sources more distant than about 100 Mpc do not contribute to the observed fluxes at all energies.', 'astro-ph-0303624-2-32-2': 'In such a model, the argument of the GZK cutoff in its original form in terms of a sudden reduction of the horizon is modified.', 'astro-ph-0303624-2-33-0': 'We are aware that a fully magnetized Universe is not a realistic model of our environment.', 'astro-ph-0303624-2-33-1': 'Voids are known to exist, where random fields would rather be around a few nG, or more less.', 'astro-ph-0303624-2-33-2': 'In such regions, UHECRs propagate ballistically even at low energies where GZK losses are unimportant, and therefore the particular effect which we discussed above does not appear.', 'astro-ph-0303624-2-33-3': 'However, our main argument that "far away sources do not contribute to the visible spectrum of UHECRs at all energies" would be maintained if all sources are surrounded by a random field of order 100 nG over distances of several Mpc, as expected for sources that lie within galaxies or clusters.', 'astro-ph-0303624-2-34-0': 'By using a locally homogeneous Universe, we implicitely restricted ourselves to one sheet of magnetic fields and neglected the contribution of other high-density sheets that may lie in our neighbourhood.', 'astro-ph-0303624-2-34-1': "While this is a direct consequence of our choice to isolate the 'magnetic cutoff' effect from any other effect influencing the shape of the spectrum, it may also be qualitatively justified by the fact that the highest energy particles would not be able to travel the distances between two neighbouring sheets without losing some energy, and the lowest energy particles would be largely reflected as they approach the strongly enhanced magnetic fields in our own sheet, resulting in a reduced flux contribution.", 'astro-ph-0303624-2-34-2': 'Such an (imperfect) confinement of UHECRs between two magnetic walls contributes to isolate regions of high magnetic fields from one another.', 'astro-ph-0303624-2-35-0': 'To illustrate the effect of non uniform fields, with alternating voids and sheets, we also investigated models in which the fields are concentrated in gaussian concentric shells of width [MATH] (as in the description of planar magnetic sheets in Ref. [CITATION]) or [MATH], and radii increasing by steps of 30 Mpc, respectively 15 Mpc.', 'astro-ph-0303624-2-35-1': 'As shown in Figure [REF], the reduction of the low energy part of the spectrum (between [MATH] and [MATH] eV) is still visible in this field configuration, and the general effect of reducing the GZK suppression remains.', 'astro-ph-0303624-2-35-2': 'More realistic field configurations thus do not seem to affect the general arguments emphasized here.', 'astro-ph-0303624-2-36-0': 'We did not model any particular configuration of sources to study the anisotropy effect, but given the slowness of the transition between the two propagation regimes and the fact that the transport becomes only purely ballistic above 500 EeV, we expect that anisotropies would be visible only above 100 EeV where the data is currently too scarce for one to make any definitive statement.', 'astro-ph-0303624-2-36-1': 'It has been shown recently that even with more localized magnetic fields the expected anisotropy from local sources is not ruled out by the current data [CITATION].', 'astro-ph-0303624-2-36-2': 'One should also note that a few sources localized in the magnetic sheet around the local supercluster would suppress the bump before [MATH], visible on Fig. [REF], because low-energy particles leak through the walls [CITATION].'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0303624

1109.4504

{'1109.4504-1-0-0': 'The relation between free carrier absorption and intersubband transitions in semiconductor heterostructures is resolved by comparing a sequence of structures.', '1109.4504-1-0-1': 'Our numerical and analytical results show that free carrier absorption essentially evolves from the intersubband transitions in the limit of infinite wells with vanishing barrier width.', '1109.4504-1-0-2': 'This shows that a proper treatment of intersubband transitions is fully sufficient to simulate the electronic absorption in heterostructure THz devices.', '1109.4504-1-1-0': 'The absorption of electromagnetic radiation due to the interaction with electrons in bulk crystals is essentially determined by two distinct effects: (i) The free carrier absorption (FCA), which is directly related to the electrical conductivity and drops with frequency on the scale of the inverse scattering time.', '1109.4504-1-1-1': '(ii) Interband transitions, which are typically described via the dipole moments induced by the coupling between states in different bands.', '1109.4504-1-1-2': 'For most crystals these transition energies are of the order of eV and thus this dominates the response around the optical spectrum.', '1109.4504-1-1-3': 'In addition to these electronic features, optical phonons provide absorption in the far infrared region, which is not addressed here.', '1109.4504-1-2-0': 'Semiconductor heterostructures provide an additional quantization of the electronic states in the growth direction (denoted by [MATH]), and commonly the absorption between these subbands is treated analogously to the interband transitions in bulk crystals [CITATION].', '1109.4504-1-2-1': 'The standard treatment relies on the envelope functions [MATH] for the subbands [MATH] with energies [MATH] and areal electron densities [MATH] using expressions for the absorption coefficient [MATH] like [EQUATION] see, e.g., [CITATION] where the matrix element [EQUATION] describes the coupling strength.', '1109.4504-1-2-2': 'Throughout this work we assume the polarization of the electric field to point in [MATH]-direction.', '1109.4504-1-2-3': 'This scheme is also routinely applied for the calculation of the gain spectrum of quantum cascade lasers (QCLs).', '1109.4504-1-2-4': '[CITATION] In this context the broadening [MATH] can be either added in a phenomenological way [CITATION] or by detailed calculations, see, e.g. [CITATION].', '1109.4504-1-2-5': 'It can also be seen as a limiting case of a full quantum kinetic calculation.[', '1109.4504-1-3-0': 'While the conventional treatment of intersubband transitions is well accepted for transitions in the infrared, this approach is less obvious for THz systems, which have become of high interest.[', '1109.4504-1-3-1': '[CITATION] Here, FCA-related features might turn up as a strong competing mechanism to the intersubband gain transition in analogy to the bulk case where both FCA and interband transitions occur as separate processes.', '1109.4504-1-3-2': 'In order to demonstrate the potential relevance, we consider the standard expression for FCA in bulk systems[CITATION] [EQUATION] where [MATH] is the elementary charge, [MATH] the effective mass, and [MATH] the volume density of electrons in the conduction band.', '1109.4504-1-3-3': '[MATH] is the refractive index and [MATH] the vacuum permeability (SI units are used).', '1109.4504-1-3-4': 'Finally [MATH] is the scattering time.', '1109.4504-1-3-5': 'As an example for GaAs with a doping of [MATH] and [MATH] ps (corresponding to a mobility of 6000 cm[MATH]/Vs at 300K [CITATION]) one obtains [MATH] for a frequency [MATH] THz.', '1109.4504-1-3-6': 'This is larger than typical gain coefficients in THz quantum cascade lasers [CITATION].', '1109.4504-1-3-7': 'Thus, bulk FCA would provide a strong obstacle in achieving lasing in such structures and its proper treatment in heterostructures is of crucial importance for the description of QCLs or other THz heterostructure devices.', '1109.4504-1-3-8': '(For a typical infrared laser, in contrast, it was shown that FCA in the cascade structure does not play a role [CITATION].)', '1109.4504-1-3-9': 'In Ref. [CITATION] FCA was only considered in the waveguide layers but not the QCL structure itself, where the absorption was determined by intersubband transitions.', '1109.4504-1-3-10': 'Furthermore, in [CITATION] it was shown that processes as described by Eq. ([REF]) dominate the absorption of light (with [MATH]-polarized electric field) for quantum wells.', '1109.4504-1-4-0': 'In this context the question arises how such a treatment based on intersubband transitions is related to the FCA in the bulk.', '1109.4504-1-4-1': 'Is FCA related to the seemingly dominating intersubband processes or does it stem from further processes not identified yet?', '1109.4504-1-4-2': 'In the latter case, such processes could strongly alter the THz performance of heterostructure devices.', '1109.4504-1-4-3': 'In order to shed light on this important issue we present a detailed study on the unfolding of FCA starting from different types of heterostructure.', '1109.4504-1-4-4': 'Our main conclusion is that the absorption due to intersubband transitions evolves into the bulk FCA for superlattice (SL) structures with small barriers.', '1109.4504-1-4-5': 'This shows that a proper treatment of intersubband transitions provides a complete description of gain and absorption processes in heterostructure devices.', '1109.4504-1-5-0': '# From superlattice to bulk', '1109.4504-1-6-0': 'We consider four GaAs-Al[MATH]Ga[MATH]As SLs with constant period [MATH] nm.', '1109.4504-1-6-1': 'The barrier width is set equal to 0.5 nm, 1.5 nm, 2.5 nm, and 3.5 nm, respectively, and a homogeneous doping with [MATH] is used.', '1109.4504-1-6-2': 'The sample with the 2.5 nm barrier has been investigated in [CITATION], which motivates our choice.', '1109.4504-1-6-3': 'Fig. [REF](a) shows the calculated band structures assuming effective masses of [MATH] and [MATH] for GaAs and Al[MATH]Ga[MATH]As, respectively, where [MATH] is the free electron mass, as well as a conduction band offset of 276 meV.', '1109.4504-1-6-4': '[CITATION] Further information on the structures is given in table [REF].', '1109.4504-1-7-0': "Here the zero-field conductivity [MATH] is evaluated from the nonequilibrium Green's function (NEGF) model following [CITATION], which includes scattering processes from phonons, impurities, interface roughness (with an average height of one monolayer and a length correlation of 10 nm) and alloy disorder in an approximate way.", '1109.4504-1-7-1': 'This program also calculates the absorption in linear response to the optical field [CITATION] as given in Fig. [REF](b).', '1109.4504-1-7-2': 'Using [MATH] the conductivity for the 0.5 nm barrier structure provides a scattering time of [MATH] fs.', '1109.4504-1-7-3': 'This value agrees roughly with the momentum scattering rate [MATH]ps in a SL miniband (which is the sum of the elastic and inelastic scattering rate [CITATION]) extracted from several highly doped, wide band GaAs/AlAs SLs at room temperature.', '1109.4504-1-7-4': '[CITATION] The value is much smaller than the bulk scattering time of 0.2 ps, as scattering is enhanced due to the presence of rough interfaces in all SLs (which are particular strong scatterers for small barrier widths, when the wave functions highly penetrate through the barriers).', '1109.4504-1-7-5': 'In addition, the assumption of a constant scattering time is only expected to be of semi-quantitative nature, the same holds for the approximations in matrix elements used.', '1109.4504-1-7-6': '(For a more detailed treatment of roughness scattering in thin barriers, see [CITATION].)', '1109.4504-1-7-7': 'Using [MATH] fs the Drude expression ([REF]) fits the absorption quite well, demonstrating, that these small barriers actually provide almost the bulk free carrier absorption behavior.', '1109.4504-1-8-0': 'With increasing barrier thickness the conductivity becomes smaller due to the reduced coupling between the quantum wells.', '1109.4504-1-8-1': 'Accordingly, there is a decrease in the low frequency absorption [EQUATION] as it follows from electrodynamics.', '1109.4504-1-8-2': '[CITATION] Here our numerical calculations are in full agreement, as we do not employ the rotating wave approximation and include broadening in a fully consistent way.', '1109.4504-1-8-3': 'Furthermore, for thicker barriers, the absorption between the minibands becomes more prominent and thus the absorption increases close to the photon energy required to overcome the gap between the first and the second miniband, as indicated by the arrows in Fig. [REF](b).', '1109.4504-1-8-4': 'The shift of the peak positions with respect to the minigaps can be related to scattering induced level shifts.', '1109.4504-1-8-5': 'For the 2.5 nm barrier the results are in good agreement with the measurements reported in [CITATION].', '1109.4504-1-8-6': 'The onset of absorption around 100 meV is slightly sharper in the experiment, which may be attributed to less rough interfaces or to the limited accuracy of the various approximations used for the scattering potentials.', '1109.4504-1-9-0': 'For SLs the absorption can be understood within the common miniband picture.', '1109.4504-1-9-1': 'For low frequencies intra-miniband processes dominate, which are easily treated in semiclassical transport models providing for zero electric field [CITATION] [EQUATION]', '1109.4504-1-9-2': 'This behavior was experimentally observed in [CITATION].', '1109.4504-1-9-3': 'Here, [MATH] for large miniband widths.', '1109.4504-1-9-4': 'With decreasing miniband width, the increase of [MATH] reduces [MATH].', '1109.4504-1-9-5': 'An even stronger reduction arises, if the miniband width drops below either [MATH] or the Fermi energy, see [CITATION] for details.', '1109.4504-1-9-6': 'For all superlattice structures studied by our NEGF model, we found good agreement with ([REF]) for low frequencies.', '1109.4504-1-9-7': 'Some examples are shown in the inset of Fig. [REF](b).', '1109.4504-1-9-8': 'As a further example, the calculated absorption spectrum at 65 K for the structure of [CITATION] can be fitted by [MATH]ps (not shown here).', '1109.4504-1-9-9': 'This is in good agreement with the experimental value of 0.18 ps, which demonstrates the quality of the NEGF approach.', '1109.4504-1-10-0': 'For higher frequencies, intersubband transitions between the minibands can describe the absorption between 60 and 200 meV very well.', '1109.4504-1-10-1': 'See, e.g., the results of the calculations in [CITATION], which fully agree with our more sophisticated Greens function approach.', '1109.4504-1-11-0': 'We conclude that the absorption of SLs at zero bias can be well described by the Drude-like miniband conduction result ([REF]) for low frequencies and by common inter-miniband transitions for higher frequencies.', '1109.4504-1-11-1': 'As shown in Fig. [REF](b), the combination of both features evolves into the bulk FCA ([REF]) if the barrier width becomes small.', '1109.4504-1-12-0': '# From multiple well to superlattice', '1109.4504-1-13-0': 'Now we want to study, how the SL absorption arises from the behavior of systems containg few wells, which show distinct absorption peaks between discrete levels.', '1109.4504-1-13-1': 'Fig. [REF] shows the absorption for multi-quantum-well structures, as presented in Fig. [REF](a) for the case of two wells.', '1109.4504-1-13-2': 'Here, all parameters correspond to the SL with a 1.5 nm barrier discussed above.', '1109.4504-1-13-3': 'For the double well structure, essentially the two lowest subbands are occupied in thermal equilibrium, and one observes clear absorption peaks corresponding to the separations between the subbands, see Fig. [REF](b).', '1109.4504-1-13-4': 'As the dipole matrix element ([REF]) vanishes for equal parity of the states, not all possible transitions are visible.', '1109.4504-1-13-5': 'The observed peak structure can be directly described by the standard intersubband expressions ([REF]).', '1109.4504-1-13-6': 'Furthermore, there is zero absorption in the limit of zero frequency as no dc current along the structure is possible, compare Eq. ([REF]).', '1109.4504-1-14-0': 'With increasing well numbers, the peaks III and IV of the double well split up and form the continuous absorption between 60 and 200 meV due to the transitions between the first and the second SL miniband, see Fig. [REF](c).', '1109.4504-1-14-1': 'While this is quite expected, peak I does not show any clear splitting, but shifts to lower frequencies, approaching the intra-miniband absorption.', '1109.4504-1-14-2': 'This behavior can be understood by a detailed study of the multi-quantum-well eigenstates.', '1109.4504-1-14-3': 'Here, a tight binding model for [MATH] wells with next neighbor coupling [MATH] shows (see the appendix for details): (i) There are [MATH] eigenstates, labeled by an index [MATH] according to their energy [MATH].', '1109.4504-1-14-4': 'Here [MATH] is of the order of [MATH].', '1109.4504-1-14-5': '(ii) The matrix element [MATH] from Eq. ([REF]) is small unless for neighboring states, i.e. [MATH].', '1109.4504-1-14-6': 'Thus, the transitions between neighboring states dominate, explaining the strong absorption around [MATH] visible in Fig. [REF](c), where [MATH] essentially corresponds to the miniband width of the infinite structure.', '1109.4504-1-14-7': 'Together with a tail at higher frequencies due to broadening of these transitions this explains the appearance of the Drude-like miniband absorption for the SL in the limit of large [MATH].', '1109.4504-1-14-8': 'For [MATH] the evolution is not smooth as any finite sequence of quantum wells has a zero dc conductivity in contrast to an infinite SL and thus the absorption must vanish according to Eq. ([REF]).', '1109.4504-1-15-0': '# Conclusion We demonstrated how the common bulk free carrier absorption evolves from standard intersubband absorption in heterostructures for electromagnetic waves with an electric field pointing in growth direction.', '1109.4504-1-15-1': 'Here the well-studied SL absorption constitutes an intermediate case, which can be entirely understood on the basis of common intersubband absorption processes in the limit of a growing number of quantum wells.', '1109.4504-1-15-2': 'For decreasing SL barrier width the combination of inter and intra-miniband absorption evolves into the standard FCA of the bulk crystal.', '1109.4504-1-15-3': 'The most relevant consequence is that there is no need to bother about any additional FCA-related absorption processes, provided all intersubband transitions are properly taken into account.', '1109.4504-1-15-4': 'A consistency check for the calculated gain/absorption spectrum is whether Eq. ([REF]) is satisfied in the low frequency limit.', '1109.4504-1-16-0': 'Financial support from the Swedish Research Council (VR) and the French ANR agency (ROOTS project)is gratefully acknowledged.', '1109.4504-1-17-0': '# Appendix', '1109.4504-1-18-0': 'We consider a multi-quantum-well structure with [MATH] wells centered at [MATH], where [MATH].', '1109.4504-1-18-1': 'The ground state of the isolated well [MATH] has the wavefunction [MATH] and the energy [MATH].', '1109.4504-1-18-2': 'Restricting to a nearest neighbor coupling [MATH] (which is negative for the lowest subband), the eigenenergies are [EQUATION] and the eigenstates read [MATH] with [EQUATION]', '1109.4504-1-18-3': 'If the overlap between the states in different wells is negligible, i.e. [MATH], we find [MATH], which can be directly evaluated.', '1109.4504-1-18-4': 'If [MATH] is even we find [MATH] as both states have the same parity with respect to [MATH].', '1109.4504-1-18-5': 'For odd [MATH], some algebra yields [EQUATION]', '1109.4504-1-18-6': 'For [MATH] we thus have:', '1109.4504-1-19-0': 'As the square of [MATH] enters the absorption ([REF]), it becomes clear that the transitions with [MATH] highly dominate the absorption spectrum.', '1109.4504-1-19-1': 'The energy difference of the corresponding states ([REF]) for these transitions is less than [MATH] with an average of approximately [MATH].'}

{'1109.4504-2-0-0': 'The relation between free carrier absorption and intersubband transitions in semiconductor heterostructures is resolved by comparing a sequence of structures.', '1109.4504-2-0-1': 'Our numerical and analytical results show how free carrier absorption evolves from the intersubband transitions in the limit of an infinite number of wells with vanishing barrier width.', '1109.4504-2-0-2': 'It is explicitly shown that the integral of the absorption over frequency matches the value obtained by the f-sum rule.', '1109.4504-2-0-3': 'This shows that a proper treatment of intersubband transitions is fully sufficient to simulate the entire electronic absorption in heterostructure THz devices.', '1109.4504-2-1-0': '# Introduction', '1109.4504-2-2-0': 'The absorption of electromagnetic radiation due to the interaction with electrons in bulk crystals is essentially determined by two distinct effects: (i) The free carrier absorption (FCA), which is directly related to the electrical conductivity and drops with frequency on the scale of the inverse scattering time.', '1109.4504-2-2-1': '(ii) Interband transitions, which are typically described via the dipole moments induced by the coupling between states in different bands.', '1109.4504-2-2-2': 'For most crystals these transition energies are of the order of eV and thus this dominates the response around the optical spectrum.', '1109.4504-2-2-3': 'In addition to these electronic features, optical phonons provide absorption in the far infrared region, which is not addressed here.', '1109.4504-2-3-0': 'Semiconductor heterostructures provide an additional effective potential for the electron in the conduction band causing a further quantization of the electronic states in the growth direction (denoted by [MATH]).', '1109.4504-2-3-1': 'Taking into account the degrees of freedom for motion in the [MATH] plane, this establishes subbands within the conduction band.', '1109.4504-2-3-2': 'Commonly, the absorption between these subbands is treated analogously to the interband transitions in bulk crystals.', '1109.4504-2-3-3': 'The standard treatment relies on the envelope functions [MATH] for the subbands [MATH] with energies [MATH] and areal electron densities [MATH] using expressions for the absorption coefficient [MATH] as [CITATION] [EQUATION] where [MATH] and the counter-rotating terms are neglected.', '1109.4504-2-3-4': 'Here [MATH] is the elementary charge, [MATH] is the refractive index, and [MATH] is the vacuum permeability (SI units are used).', '1109.4504-2-3-5': 'The matrix element [EQUATION] describes the coupling strength.', '1109.4504-2-3-6': 'Throughout this work we assume the polarization of the electric field to point in [MATH]-direction and that the wave propagates in a waveguide of effective thickness [MATH] which is filled by the (layered) semiconductor material.', '1109.4504-2-3-7': 'This scheme is also routinely applied for the calculation of the gain spectrum of quantum cascade lasers (QCLs).[', '1109.4504-2-3-8': '[CITATION] In this context the broadening [MATH] can be either added in a phenomenological way [CITATION] or by detailed calculations, see, e.g., Ref. UnumaJAP2003.', '1109.4504-2-3-9': 'It can also be seen as a limiting case of a full quantum kinetic calculation.[', '1109.4504-2-4-0': 'While the conventional treatment of intersubband transitions is well accepted for transitions in the infrared, this approach is less obvious for THz systems, which have become of high interest.[', '1109.4504-2-4-1': '[CITATION] Here, FCA-related features might turn up as a strong competing mechanism to the intersubband gain transition in analogy to the bulk case where both FCA and interband transitions occur as separate processes.', '1109.4504-2-4-2': 'In order to demonstrate the potential relevance, we consider the standard expression for FCA in bulk systems[CITATION] [EQUATION] where [MATH] is the effective mass, [MATH] the volume density of electrons in the conduction band, and [MATH] is the scattering time.', '1109.4504-2-4-3': 'As an example for GaAs with a doping of [MATH] and [MATH] ps (corresponding to a mobility of 6000 cm[MATH]/Vs at 300K [CITATION]) one obtains [MATH] for a frequency [MATH] THz.', '1109.4504-2-4-4': 'This is larger than typical gain coefficients in THz quantum cascade lasers.', '1109.4504-2-4-5': '[CITATION] Thus, bulk FCA would provide a strong obstacle in achieving lasing in such structures and its proper treatment in heterostructures is of crucial importance for the description of QCLs or other THz heterostructure devices.', '1109.4504-2-4-6': '(For a typical infrared laser, in contrast, it was shown that FCA in the cascade structure does not play a role.[', '1109.4504-2-4-7': '[CITATION]) In Ref. AjiliJAP2006 FCA was only considered in the waveguide layers but not the QCL structure itself, where the absorption was determined by intersubband transitions.', '1109.4504-2-4-8': 'Furthermore, in Ref. VurgaftmanPRB1999 it was shown that processes as described by Eq. ([REF]) dominate the absorption of light (with [MATH]-polarized electric field) for quantum wells.', '1109.4504-2-5-0': 'In this context the question arises how such a treatment based on intersubband transitions is related to the FCA in the bulk.', '1109.4504-2-5-1': 'Is FCA related to the seemingly dominating intersubband processes or does it stem from further processes not identified yet?', '1109.4504-2-5-2': 'In the latter case, such processes could strongly alter the THz performance of heterostructure devices.', '1109.4504-2-5-3': 'In order to shed light on this important issue we present a detailed study on the unfolding of FCA starting from different types of heterostructure.', '1109.4504-2-5-4': 'Our main conclusion is that the absorption due to intersubband transitions evolves into the bulk FCA for vanishing barrier widths.', '1109.4504-2-5-5': 'This shows that a proper treatment of intersubband transitions provides a complete description of gain and absorption processes in heterostructure devices.', '1109.4504-2-6-0': '# From superlattice to bulk', '1109.4504-2-7-0': 'We consider four GaAs-Al[MATH]Ga[MATH]As superlattices[CITATION] (SLs) with constant period [MATH] nm.', '1109.4504-2-7-1': 'The barrier width is set equal to 0.5 nm, 1.5 nm, 2.5 nm, and 3.5 nm, respectively, and a homogeneous doping with [MATH] is used.', '1109.4504-2-7-2': 'The sample with the 2.5 nm barrier has been investigated in Ref. HelmPRB1993, which motivates our choice.', '1109.4504-2-7-3': 'Fig. [REF](a) shows the calculated minibands assuming effective masses of [MATH] and [MATH] for GaAs and Al[MATH]Ga[MATH]As, respectively, where [MATH] is the free electron mass, as well as a conduction band offset of 276 meV.[', '1109.4504-2-7-4': '[CITATION] Further information on the structures is given in table [REF].', '1109.4504-2-8-0': "Here the zero-field conductivity [MATH] is evaluated from the nonequilibrium Green's function (NEGF) model following Ref. [CITATION], which includes scattering processes from phonons, impurities, interface roughness (with an average height of one monolayer and a length correlation of 10 nm), and alloy disorder in an approximate way.", '1109.4504-2-8-1': 'This program also calculates the absorption in linear response to the optical field [CITATION] as given in Fig. [REF](b).', '1109.4504-2-8-2': 'Using [MATH] the conductivity for the 0.5 nm barrier structure provides a scattering time of [MATH] fs.', '1109.4504-2-8-3': 'This value agrees roughly with the momentum scattering rate [MATH]ps (which is the sum of the elastic and inelastic scattering rate[CITATION]) extracted from several highly doped GaAs/AlAs SLs with narrow barriers at room temperature.', '1109.4504-2-8-4': '[CITATION] This value is much smaller than the bulk scattering time of 0.2 ps, as scattering is enhanced due to the presence of rough interfaces in all SLs (which are particular strong scatterers for small barrier widths, when the wave functions highly penetrate through the barriers).', '1109.4504-2-8-5': 'In addition, the assumption of a constant scattering time is only expected to be of semi-quantitative nature, the same holds for the approximations in matrix elements used.', '1109.4504-2-8-6': '(For a more detailed treatment of roughness scattering in thin barriers, see Ref. [CITATION].)', '1109.4504-2-8-7': 'Using [MATH] fs, the Drude expression ([REF]) fits the absorption quite well, demonstrating, that these small barriers actually provide almost the bulk free carrier absorption behavior.', '1109.4504-2-9-0': 'With increasing barrier thickness the conductivity becomes smaller due to the reduced coupling between the quantum wells.', '1109.4504-2-9-1': 'Accordingly, there is a decrease in the low frequency absorption [EQUATION] as it follows from electrodynamics.', '1109.4504-2-9-2': '[CITATION] Here our numerical calculations are in full agreement, as we do not employ the rotating wave approximation and include broadening in a fully consistent way.', '1109.4504-2-9-3': 'Furthermore, for thicker barriers, the absorption between the minibands becomes more prominent and thus the absorption increases close to the photon energy required to overcome the gap between the first and the second miniband, as indicated by the arrows in Fig. [REF](b).', '1109.4504-2-9-4': 'The shift of the peak positions with respect to the minigaps can be related to scattering induced level shifts.', '1109.4504-2-9-5': 'For the 2.5 nm barrier the results are in good agreement with the measurements reported in Ref. [CITATION].', '1109.4504-2-9-6': 'The onset of absorption around 100 meV is slightly sharper in the experiment, which may be attributed to less rough interfaces or to the limited accuracy of the various approximations used for the scattering potentials.', '1109.4504-2-10-0': 'For SLs the absorption can be understood within the common miniband picture.', '1109.4504-2-10-1': 'For low frequencies intra-miniband processes dominate, which are easily treated in semiclassical transport models providing for zero electric field[CITATION] [EQUATION]', '1109.4504-2-10-2': 'This behavior was experimentally observed in Refs. [CITATION].', '1109.4504-2-10-3': 'Here, [MATH] for large miniband widths.', '1109.4504-2-10-4': 'With decreasing miniband width, the increase of [MATH] reduces [MATH].', '1109.4504-2-10-5': 'An even stronger reduction arises, if the miniband width drops below either [MATH] or the Fermi energy, see Ref. [CITATION] for details.', '1109.4504-2-10-6': 'For all superlattice structures studied by our NEGF model, we found good agreement with ([REF]) for low frequencies.', '1109.4504-2-10-7': 'Some examples are shown in the inset of Fig. [REF](b).', '1109.4504-2-10-8': 'As a further example, the calculated absorption spectrum at 65 K for the structure of Ref. [CITATION] can be fitted by [MATH]ps (not shown here).', '1109.4504-2-10-9': 'This is in good agreement with the experimental value of 0.18 ps, which demonstrates the quality of the NEGF approach.', '1109.4504-2-11-0': 'For higher frequencies, transitions between the minibands can describe the absorption between 60 and 200 meV very well.', '1109.4504-2-11-1': 'See, e.g., the results of the calculations in Ref. [CITATION], which fully agree with our more sophisticated NEGF approach.', '1109.4504-2-12-0': 'We conclude that the absorption of SLs at zero bias can be well described by the Drude-like miniband conduction result ([REF]) for low frequencies and by common inter-miniband transitions for higher frequencies.', '1109.4504-2-12-1': 'As shown in Fig. [REF](b), the combination of both features evolves into the bulk FCA ([REF]) if the barrier width becomes small.', '1109.4504-2-13-0': '# From multiple well to superlattice', '1109.4504-2-14-0': 'Now we want to study, how the SL absorption arises from the behavior of systems containing few wells, which show distinct absorption peaks between discrete levels.', '1109.4504-2-14-1': 'Fig. [REF] shows the absorption for multi-quantum-well structures, as presented in Fig. [REF](a) for the case of two wells.', '1109.4504-2-14-2': 'Here, all parameters correspond to the SL with a 1.5 nm barrier discussed above.', '1109.4504-2-14-3': 'For the double well structure, essentially the two lowest subbands are occupied in thermal equilibrium, and one observes clear absorption peaks corresponding to the separations between the subbands, see Fig. [REF](b).', '1109.4504-2-14-4': 'As the dipole matrix element ([REF]) vanishes for equal parity of the states, not all possible transitions are visible.', '1109.4504-2-14-5': 'The observed peak structure can be directly described by the standard intersubband expressions ([REF]).', '1109.4504-2-14-6': 'Furthermore, there is zero absorption in the limit of zero frequency as no dc current along the structure is possible, compare Eq. ([REF]).', '1109.4504-2-15-0': 'With increasing well numbers, the peaks III and IV of the double well split up and form the continuous absorption between 60 and 200 meV due to the transitions between the first and the second SL miniband, see Fig. [REF](c).', '1109.4504-2-15-1': 'While this is quite expected, peak I does not show any clear splitting, but shifts to lower frequencies, approaching the intra-miniband absorption.', '1109.4504-2-15-2': 'This behavior can be understood by a detailed study of the multi-quantum-well eigenstates.', '1109.4504-2-15-3': 'Here, a tight binding model for [MATH] wells with next neighbor coupling [MATH] shows (see the appendix [REF] for details): (i) There are [MATH] eigenstates, labeled by an index [MATH] according to their energy [MATH].', '1109.4504-2-15-4': 'Here [MATH] is of the order of [MATH].', '1109.4504-2-15-5': '(ii) The matrix element [MATH] from Eq. ([REF]) is small unless for neighboring states, i.e. [MATH].', '1109.4504-2-15-6': 'Thus, the transitions between neighboring states dominate, explaining the strong absorption around [MATH] visible in Fig. [REF](c), where [MATH] essentially corresponds to the miniband width of the infinite structure.', '1109.4504-2-15-7': 'Together with a tail at higher frequencies due to broadening of these transitions this explains the appearance of the Drude-like miniband absorption for the SL in the limit of large [MATH].', '1109.4504-2-15-8': 'For [MATH] the evolution is not smooth as any finite sequence of quantum wells has a zero dc conductivity in contrast to an infinite SL and thus the absorption must vanish according to Eq. ([REF]).', '1109.4504-2-16-0': '# The integrated absorption', '1109.4504-2-17-0': 'Summing over all possible intersubband transitions ([REF]), we obtain the total absorption [MATH].', '1109.4504-2-17-1': 'Here the discrete index [MATH] runs over all (infinitely many) eigenstates of the heterostructure of finite length, including states which correspond to unbounded states with energies far above the barrier potential.', '1109.4504-2-17-2': 'Integrating over all frequencies provides [EQUATION] under the assumption [MATH] - otherwise the counter-rotating terms become of relevance, which had been neglected here.', '1109.4504-2-17-3': 'In appendix [REF] we show that the same integral relation is more generally obtained for arbitrary level spacings [MATH] within our NEGF model, which also covers dispersive gain.[', '1109.4504-2-18-0': 'Following Ref. PeetersPRB1993, Eq. ([REF]) can be simplified by the Thomas-Reiche-Kuhn sum rule[CITATION] (also called f-sum rule[CITATION]) which reads for a parabolic band with effective mass [MATH] [EQUATION] and provides the integrated absorption [EQUATION] where [MATH] is the average three-dimensional carrier density in the waveguide.', '1109.4504-2-19-0': 'For a bulk semiconductor, the free carrier absorption ([REF]) provides after integration over energy [EQUATION] which fully agrees with the intersubband result ([REF]) for equal total densities [MATH].', '1109.4504-2-19-1': 'Thus the total FCA in a bulk semiconductor equals the total intersubband absorption within the conduction band for a finite heterostructure of finite length, which shows the direct relation between these.', '1109.4504-2-19-2': 'More generally, Eqs. ([REF],[REF]) establish a general rule for the integrated absorption within the conduction band of a semiconductor under conditions, where the approximation of a constant effective mass is justified.', '1109.4504-2-19-3': 'In this context superlattices appear as an intermediate case, where the inter-miniband absorption and the Drude-like intra-miniband absorption add up to the full result.', '1109.4504-2-19-4': '[CITATION]', '1109.4504-2-20-0': 'Our numerical data in Fig. [REF](b) exhibit the integrated absorption [MATH] for all curves.', '1109.4504-2-20-1': 'The data from Fig. [REF] provide [MATH], where the additional factor takes into account the undoped region of 6 nm between adjacent multiple quantum wells ([MATH] is the number of well/barrier combinations with a length of 10 nm each).', '1109.4504-2-20-2': 'These values are slightly below the value of [MATH] given by Eq. ([REF]) using the GaAs effective mass.', '1109.4504-2-20-3': 'This minor discrepancy of less than 7% can be easily attributed to some absorption at higher frequencies and the impact of the barrier material with a larger mass.', '1109.4504-2-20-4': 'We conclude, that the absorption obtained by our NEGF code is in excellent agreement with the rule ([REF]).', '1109.4504-2-21-0': 'More generally the effect of the semiconductor heterostructure can be understood as shifting the absorption strength within the frequency space, as explicitly demonstrated by our calculations.', '1109.4504-2-21-1': 'This perception has actually been used in the design of QCL structures, where the unavoidable free carrier absorption is deflected from the frequency region of operation by a proper choice of heterostructures [CITATION], see, e.g., Ref. [CITATION].', '1109.4504-2-22-0': '# Conclusion We demonstrated how the common bulk free carrier absorption evolves from standard intersubband absorption in heterostructures for electromagnetic waves with an electric field pointing in growth direction.', '1109.4504-2-22-1': 'Here the well-studied SL absorption constitutes an intermediate case, which can be entirely understood on the basis of common intersubband absorption processes in the limit of a growing number of quantum wells.', '1109.4504-2-22-2': 'For decreasing SL barrier width the combination of inter- and intra-miniband absorption evolves into the standard FCA of the bulk crystal.', '1109.4504-2-22-3': 'This behavior reflects a redistribution of absorption strength, while the integrated absorption is constant.', '1109.4504-2-22-4': 'The most relevant consequence is that there is no need to bother about any additional FCA-related absorption processes, provided all intersubband transitions are properly taken into account.', '1109.4504-2-22-5': 'A consistency check for the calculated gain/absorption spectrum is whether Eq. ([REF]) is satisfied in the low frequency limit and the integrated absorption matches Eqs. ([REF],[REF]).', '1109.4504-2-23-0': 'We thank J. Faist for helpful discussions.', '1109.4504-2-23-1': 'Financial support from the Swedish Research Council (VR) and the French ANR agency (ROOTS project) is gratefully acknowledged.', '1109.4504-2-24-0': '# Analytical calculation for coupled wells', '1109.4504-2-25-0': 'We consider a multi-quantum-well structure with [MATH] wells centered at [MATH], where [MATH].', '1109.4504-2-25-1': 'The ground state of the isolated well [MATH] has the wavefunction [MATH] and the energy [MATH].', '1109.4504-2-25-2': 'Restricting to a nearest neighbor coupling [MATH] (which is negative for the lowest subband), the eigenenergies are [EQUATION] and the eigenstates read [MATH] with [EQUATION]', '1109.4504-2-25-3': 'If the overlap between the states in different wells is negligible, i.e. [MATH], we find [MATH], which can be directly evaluated.', '1109.4504-2-25-4': 'If [MATH] is even we find [MATH] as both states have the same parity with respect to [MATH].', '1109.4504-2-25-5': 'For odd [MATH], some algebra yields [EQUATION]', '1109.4504-2-25-6': 'For [MATH] we thus have:', '1109.4504-2-26-0': 'As the square of [MATH] enters the absorption ([REF]), it becomes clear that the transitions with [MATH] highly dominate the absorption spectrum.', '1109.4504-2-26-1': 'The energy difference of the corresponding states ([REF]) for these transitions is less than [MATH] with an average of approximately [MATH].', '1109.4504-2-27-0': "# Total absorption with the Green's function model", '1109.4504-2-28-0': 'Here we refer to the formulation of our NEGF model as outlined in Ref. [CITATION].', '1109.4504-2-28-1': "Here gain is evaluated within linear response around the stationary state characterized by the Green's functions [MATH].", '1109.4504-2-28-2': 'In order to simplify the analysis, nondiagonal [MATH] are neglected here - they are, however, fully included in our numerical implementation.', '1109.4504-2-28-3': 'Then the absorption resulting from the pair of states [MATH] can be written as [EQUATION] which is essentially the last equation of the appendix in Ref. [CITATION] with the counter-rotating term added.', '1109.4504-2-28-4': 'Inserting the spectral function[CITATION] [MATH] and its occupied part [MATH], which is assumed to be real, we find [EQUATION]', '1109.4504-2-28-5': 'The terms [MATH] provide the physical origin of dispersive gain as sketched in Refs. [CITATION].', '1109.4504-2-28-6': 'The signs of the counter-rotating terms [MATH] seem to contradict our intuition, as the first one appears to relate to emission and the second to absorption.', '1109.4504-2-28-7': 'However, in this formulation the sign is defined via the difference in energy between the initial and the final state, where only one a specific combination is used in the prefactor [MATH].', '1109.4504-2-29-0': 'Using the general relations [EQUATION] integration of the terms from Eq. ([REF]) over frequency provides [EQUATION] so that the sum over all different pairs [MATH] equals the second line of Eq. ([REF]).', '1109.4504-2-29-1': 'Thus the integrated absorption ([REF]) also holds for the more involved absorption terms ([REF]) of the NEGF model which include the dispersive gain.'}

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[]

['1109.4504-1-18-6', '1109.4504-2-19-4', '1109.4504-2-25-6']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1109.4504

cond-mat-0302389

{'cond-mat-0302389-1-0-0': 'We study ground state and finite temperature properties of disordered heavy fermion metals by using a generalization of dynamical mean field theory which incorporates Anderson localization effects.', 'cond-mat-0302389-1-0-1': 'The emergence of a non-Fermi liquid metallic behavior even at moderate disorder is shown to be a universal phenomenon resulting from local density of states fluctuations.', 'cond-mat-0302389-1-0-2': 'This behavior is found to have a character of an electronic Griffiths phase, and can be thought of as a precursor of Anderson localization in a strongly correlated host.', 'cond-mat-0302389-1-0-3': 'The temperature dependence of the conducting properties of the system reveal a non-trivial interplay between disorder and inelastic processes, which are reminiscent of the Mooij correlations observed in many disordered metals.', 'cond-mat-0302389-1-1-0': '# Introduction', 'cond-mat-0302389-1-2-0': 'The interplay of disorder and strong correlations remains one of the least understood topics of contemporary condensed matter physics.', 'cond-mat-0302389-1-2-1': 'These effects are believed to bear relevance to many problems that have attracted recent attention, such as the metal-insulator transition (MIT) in two-dimensional electron systems.', 'cond-mat-0302389-1-2-2': '[CITATION] Disorder effects are also likely to be important for the understanding of the puzzling non-Fermi liquid (NFL) behavior of several heavy fermion compounds.', 'cond-mat-0302389-1-2-3': '[CITATION] In some of these systems, impurities seem to play only a subsidiary role: the explanation for the anomalous behavior is more likely to be found in the physics of quantum criticality,[CITATION] even though a complete description is still lacking.[', 'cond-mat-0302389-1-2-4': '[CITATION] However, in other heavy fermion systems, disorder seems to play a more essential role and seems to be at the origin of the NFL behavior.[', 'cond-mat-0302389-1-3-0': 'Several attempts have been made to address theoretically the role of disorder in heavy fermion compounds (see an overview below).', 'cond-mat-0302389-1-3-1': 'Many experimental results can be described within the so-called Kondo disorder model (KDM)[CITATION] or, equivalently, the dynamical mean field theory (DMFT) of disordered Kondo/Anderson lattices,[CITATION] at least above the lowest temperatures.', 'cond-mat-0302389-1-3-2': 'Essential to this description is the consideration of the full distribution of local Kondo temperatures [MATH].', 'cond-mat-0302389-1-3-3': 'For sufficient disorder, it has a large weight as [MATH] describing the presence of dilute low-[MATH] spins that dominate the thermodynamic and transport properties.', 'cond-mat-0302389-1-3-4': 'However, a deficiency of the initial KDM/DMFT scheme was its neglect of fluctuations in the conduction electron local density of states (DOS), a quantity that is crucial for the determination of [MATH].', 'cond-mat-0302389-1-3-5': 'This was remedied by two of us through a generalization of the DMFT that incorporates such Anderson localization effects while keeping its local treatment of correlations.', 'cond-mat-0302389-1-3-6': '[CITATION] One important result of this study is the emergence of a power-law distribution of Kondo temperatures [MATH], where [MATH] depends continuously on the strength of disorder [MATH] and decreases as [MATH] is increased.', 'cond-mat-0302389-1-3-7': 'As the distribution becomes more singular, several thermodynamic quantities become divergent, in a manner characteristic of Griffiths phases.', 'cond-mat-0302389-1-3-8': '[CITATION] The system eventually localizes at a critical disorder strength [MATH].', 'cond-mat-0302389-1-3-9': 'Since there is no magnetic phase transition this has been dubbed an electronic Griffiths phase.', 'cond-mat-0302389-1-4-0': 'This initial work[CITATION] employed the slave boson large-N theory[CITATION] to solve the auxiliary single-impurity problems posed by the method.', 'cond-mat-0302389-1-4-1': 'While versatile, powerful and yet computationally cheap, this approach presents some disadvantages, the main one being the difficulty of working at finite temperatures.', 'cond-mat-0302389-1-4-2': 'It should be reminded that, since we deal with wide distributions of [MATH], we need to be able to describe well the full crossover from [MATH] to [MATH], which is not possible with the slave boson large-N treatment.', 'cond-mat-0302389-1-4-3': 'In particular, conspicuously missing are inelastic scattering processes.', 'cond-mat-0302389-1-4-4': 'Besides, though giving a good description of the low-energy Fermi-liquid regime of the single-impurity problem, this treatment does not incorporate high- and intermediate-energy incoherent processes.', 'cond-mat-0302389-1-4-5': 'Another impurity solver is therefore needed to assess the importance of these intrinsically finite-[MATH] and finite-energy features.', 'cond-mat-0302389-1-4-6': 'A particularly useful method, able to fill this gap at a reasonable computational cost, is second order perturbation theory in [MATH].', 'cond-mat-0302389-1-4-7': 'We have used this method to both complement and cross-check the slave boson results.', 'cond-mat-0302389-1-5-0': 'Besides the results of the initial work,[CITATION] which have been confirmed by both methods, some of our main conclusions are: (i) there is a subtle interplay between conduction and f-electron site disorder that leads to a surprising non-monotonic dependence of the conducting properties on disorder, a feature that is likely unique to Kondo/Anderson as opposed to Hubbard models; (ii) localization effects are essential for the determination of the distribution of Kondo temperatures, especially if one starts from an experimentally measured discrete distribution; (iii) the interplay between disorder and inelastic processes can lead to a temperature dependence of the conducting properties that is reminiscent of the ones found by Mooij and others in several strongly correlated disordered metals.', 'cond-mat-0302389-1-5-1': '[CITATION]', 'cond-mat-0302389-1-6-0': 'This paper is organized as follows.', 'cond-mat-0302389-1-6-1': 'We review the disorder-based mechanisms of non-Fermi liquid behavior in the next subsection.', 'cond-mat-0302389-1-6-2': 'Section [REF] describes the model of disordered Anderson lattices we studied and the methods we employed to solve it.', 'cond-mat-0302389-1-6-3': 'Section [REF] focuses on the detailed results obtained within the slave boson large-N method.', 'cond-mat-0302389-1-6-4': 'This expands considerably on the previously published results.', 'cond-mat-0302389-1-6-5': '[CITATION] In Section [REF], we show the results obtained with perturbation theory.', 'cond-mat-0302389-1-6-6': 'Finally, we wrap up with a general discussion of the strengths and limitations of this study and point out possible future directions in Section [REF].', 'cond-mat-0302389-1-6-7': 'Some details of the computational procedures are given in an Appendix.', 'cond-mat-0302389-1-7-0': '## Brief overview of disorder-based mechanisms of non-Fermi liquid behavior', 'cond-mat-0302389-1-8-0': '### Kondo-disorder models and the electronic Griffiths phase', 'cond-mat-0302389-1-9-0': 'The KDM was proposed early on to account for the temperature dependence of the Cu nuclear magnetic resonance (NMR) line widths in UCu[MATH]Pd[MATH]).', 'cond-mat-0302389-1-9-1': '[CITATION] It assumed that disorder in a heavy fermion system generates random spatial fluctuations of the exchange coupling constant [MATH] between local moments and conduction electrons (the Kondo coupling).', 'cond-mat-0302389-1-9-2': 'Each local moment was assumed to undergo the Kondo effect in a manner that is completely uncorrelated with the others and each with a characteristic energy scale, its Kondo temperature [MATH].', 'cond-mat-0302389-1-9-3': "Even narrow Kondo coupling distributions lead to a wide distribution of Kondo temperatures due to the latter's exponential dependence on the former.", 'cond-mat-0302389-1-9-4': 'As a result, at low temperatures, many spins are quenched while a few percent remain unquenched and dominate, giving rise to singular, NFL thermodynamic properties (specific heat and magnetic susceptibility).', 'cond-mat-0302389-1-9-5': 'The NMR results in UCu[MATH]Pd[MATH]) are well described within this picture if the distribution function [MATH] is such that [MATH][CITATION] The KDM gained a natural theoretical setting within the DMFT[CITATION] of a disordered Anderson/Kondo lattice.', 'cond-mat-0302389-1-9-6': '[CITATION] In this approach, each conduction electron site exchanges single particle excitations with an average "cavity" bath, which is in turn self-consistently determined.', 'cond-mat-0302389-1-9-7': 'This treatment becomes exact in the limit of infinite dimensionality and is the natural generalization of the Curie-Weiss mean field theory of magnets to a fermionic system.', 'cond-mat-0302389-1-9-8': 'Its treatment of disorder is equivalent to the well-known coherent potential approximation (CPA).', 'cond-mat-0302389-1-9-9': '[CITATION] In an Anderson lattice description, the localized f-electron is hybridized with its adjacent conduction electron orbital and spatial fluctuations are preserved through the random distribution of hybridization strengths.', 'cond-mat-0302389-1-9-10': 'The local moments are no longer independent since their distribution self-consistently determines the cavity bath.', 'cond-mat-0302389-1-9-11': 'Besides showing that the KDM corresponds to a rigorous limit of a microscopic Hamiltonian, the DMFT enabled the calculation of other properties such as the resistivity[CITATION], dynamic magnetic susceptibility,[CITATION] optical conductivity[CITATION] and magneto-resistance,[CITATION] with good agreement with experiments.', 'cond-mat-0302389-1-9-12': 'The non-Fermi liquid behavior of these quantities hinged on the condition that [MATH] Subsequent experiments of muon spin rotation[CITATION] and NMR in high fields[CITATION] showed some inconsistencies with the KDM/DMFT, suggesting that inter-site correlations, which are absent from that approach, may play a crucial role at the lowest temperatures.', 'cond-mat-0302389-1-9-13': 'However, annealing studies have further emphasized that the consideration of disorder effects is indispensable.', 'cond-mat-0302389-1-9-14': '[CITATION]', 'cond-mat-0302389-1-10-0': 'More recently, two of us have gone beyond the assumptions of the KDM/DMFT by incorporating localization effects.', 'cond-mat-0302389-1-10-1': 'The importance of this modification is unquestionable.', 'cond-mat-0302389-1-10-2': 'Disorder scatters the conduction electrons giving rise to spatial fluctuations in their wave function amplitude.', 'cond-mat-0302389-1-10-3': 'These Anderson localization precursor effects in turn give rise to fluctuations of the conduction electron local DOS.', 'cond-mat-0302389-1-10-4': 'The Kondo temperatures are exponential functions of the local DOS and will show a wide distribution for mild disorder strengths, even in the absence of fluctuations in [MATH].[', 'cond-mat-0302389-1-10-5': '[CITATION] Besides, direct experimental determination of the distribution of [MATH] from x-ray absorption fine-structure (XAFS) experiments in UCu[MATH]Pd[MATH] have shown that additional conduction electron disorder is necessary for the interpretation of the results within the KDM/DMFT.[', 'cond-mat-0302389-1-10-6': '[CITATION] The average cavity bath of the DMFT, however, completely neglects DOS fluctuations and a more general treatment is necessary.', 'cond-mat-0302389-1-10-7': 'Progress could be made by means of the statistical dynamical mean field theory (statDMFT), which incorporates the full distribution of the conduction electron local DOS, while keeping the treatment of local correlations already present in the DMFT.', 'cond-mat-0302389-1-10-8': "[CITATION] The treatment involves solving a fully self-consistent loop: the f-electron fluid gives rise to an effective disorder potential for the conduction electrons, while the latter's DOS fluctuations determine the distribution of Kondo temperatures.", 'cond-mat-0302389-1-10-9': 'We enumerate the main conclusions of our analysis:[CITATION]', 'cond-mat-0302389-1-11-0': '### Magnetic Griffiths phase scenario', 'cond-mat-0302389-1-12-0': 'An alternative theoretical scenario for disorder-induced non-Fermi liquid behavior is the magnetic Griffiths phase.', 'cond-mat-0302389-1-12-1': '[CITATION] In the vicinity of magnetic phase transitions disorder fluctuations induce rare regions with an enhanced local critical temperature.', 'cond-mat-0302389-1-12-2': 'These large clusters are ordered on the scale of the correlation length and act as effective spins.', 'cond-mat-0302389-1-12-3': 'Though rare in occurrence they carry a considerable amount of magnetic entropy and the overall effect is the appearance of singular, thermodynamic responses.', 'cond-mat-0302389-1-12-4': 'This magnetic Griffiths phase picture has been advocated as a source of NFL behavior in disordered heavy fermion systems.', 'cond-mat-0302389-1-12-5': '[CITATION] However, very recent results seem to point to several difficulties encountered when this scenario is applied to experimental systems, as follows.', 'cond-mat-0302389-1-13-0': 'The entropy problem.', 'cond-mat-0302389-1-13-1': 'The amount of magnetic entropy observed experimentally in most disordered heavy fermion systems seems much too high to be compatible with the magnetic Griffiths phase picture.', 'cond-mat-0302389-1-13-2': 'Taking the measured specific heat of, say, UCu[MATH]Pd[MATH], we estimate that about 5% of the sample would have to participate in the spin-[MATH] clusters.', 'cond-mat-0302389-1-13-3': 'This implies an average cluster separation of [MATH] lattice constants, ruling out cluster sizes exceeding this distance.', 'cond-mat-0302389-1-13-4': 'These small clusters suggest instead that the "unquenched" localized moments of the KDM/DMFT or the electronic Griffiths phase offer a much more natural explanation.', 'cond-mat-0302389-1-14-0': 'Effects of dissipation.', 'cond-mat-0302389-1-14-1': 'Other important limitations of the magnetic Griffiths phase scheme have also been emphasized in recent work by Millis, Morr, and Schmalian,[CITATION] who have carefully examined the effects of dissipation caused by the metallic bath.', 'cond-mat-0302389-1-14-2': 'This work suggests that the dissipation caused by itinerant electrons is so pronounced that quantum tunneling of even moderately sized magnetic clusters will be suppressed.', 'cond-mat-0302389-1-14-3': 'Although the emergence of a magnetic Griffiths phase is a well established phenomenon in disordered insulating magnets, this result seems to bring into question its relevance to itinerant systems.', 'cond-mat-0302389-1-15-0': '### Spin glass precursors', 'cond-mat-0302389-1-16-0': 'Finally, another possibility is to invoke the proximity to a spin glass quantum phase transition.', 'cond-mat-0302389-1-16-1': 'Several theoretical schemes predicting non-Fermi liquid behavior in the vicinity of a spin-glass quantum critical point have been proposed.[', 'cond-mat-0302389-1-16-2': '[CITATION] Spin glass phases have been identified in the phase diagram of some heavy fermion alloys (UCu[MATH]Pd[MATH], for [MATH])[CITATION] and structurally disordered compounds (URh[MATH]Ge[MATH]).[', 'cond-mat-0302389-1-16-3': '[CITATION] More interestingly, evidence of glassy dynamics in the absence of freezing at very low temperatures has been seen in UCu[MATH]Pd[MATH], [MATH])[CITATION] and Ce(Ru[MATH]Rh[MATH]Si[MATH],[CITATION] with conflicting results pointing to a very low freezing temperature in UCu[MATH]Pd[MATH].[', 'cond-mat-0302389-1-16-4': '[CITATION] These experiments seem to suggest that, if there is a true spin glass transition, freezing temperatures are strongly suppressed in a wide portion of the phase diagram.', 'cond-mat-0302389-1-17-0': '# The model and its solution', 'cond-mat-0302389-1-18-0': '## The statistical dynamical mean field theory', 'cond-mat-0302389-1-19-0': 'A simplified Hamiltonian capable of capturing the essential physics of disordered metals with localized moments is provided by a disordered Anderson lattice[EQUATION] where[EQUATION]', 'cond-mat-0302389-1-19-1': 'In Eqs. ([REF]-[REF]), [MATH]) annihilates a conduction (f-) electron on site [MATH] with spin projection [MATH], [MATH] is the nearest neighbor hopping amplitude, and we introduce random conduction electron on-site energies ([MATH]) and hybridization matrix elements [MATH].', 'cond-mat-0302389-1-19-2': 'These are chosen from given distributions [MATH] and [MATH] taken to be either square or Gaussian, with width and standard deviation [MATH], respectively.', 'cond-mat-0302389-1-19-3': 'We have also studied discrete cases of [MATH].', 'cond-mat-0302389-1-19-4': 'There is a large degree of uncertainty as to a realistic model of disorder for heavy fermion alloys.', 'cond-mat-0302389-1-19-5': 'A rather thorough study of the local f-site environment in the alloys UCu[MATH]Pd[MATH] and [MATH]) was carried out in Refs. [CITATION] through XAFS experiments.', 'cond-mat-0302389-1-19-6': 'These authors were able to determine the amount of Pd/Cu site interchange as well as the U-Cu bond length distributions.', 'cond-mat-0302389-1-19-7': 'In order to accommodate both types of fluctuations one must, in principle, allow for a distribution of both hybridization strengths and on-site conduction electron energies.', 'cond-mat-0302389-1-19-8': 'On the other hand, when the local moments are randomly replaced by non-magnetic elements (the so-called "Kondo holes"), spatial fluctuations of [MATH] should also be included.', 'cond-mat-0302389-1-19-9': '[CITATION] Throughout the paper, we use [MATH] as energy unit, except where indicated otherwise.', 'cond-mat-0302389-1-20-0': 'We worked within the framework of the statDMFT.', 'cond-mat-0302389-1-20-1': '[CITATION] This treatment is able to incorporate both strong local correlations and Anderson localization effects in a fully self-consistent fashion.', 'cond-mat-0302389-1-20-2': 'Although the method has been described before in the context of the disordered Hubbard model,[CITATION] we will briefly review it with the dual goal of setting the notation and extending it to the disordered Anderson lattice.', 'cond-mat-0302389-1-20-3': 'It starts by focusing on a generic unit cell [MATH] of the lattice, containing an f-site and its adjoining conduction electron Wannier state, and writing its effective action in imaginary time as [EQUATION] where [MATH].', 'cond-mat-0302389-1-20-4': 'In writing Eqs. ([REF]-[REF]), a simplification has been made of retaining only quadratic contributions in fermionic fields after integrating out the other sites (besides the instantaneous Hubbard term), much like the usual dynamical mean field theory.', 'cond-mat-0302389-1-20-5': '[CITATION] The bath (or "cavity") function [MATH] in Eq. ([REF]) is given by[EQUATION] where the sum extends over the [MATH] nearest neighbors and [EQUATION] is the Green\'s function for propagation from nearest neighbor site [MATH] to nearest neighbor site [MATH], calculated with the site [MATH] removed.', 'cond-mat-0302389-1-20-6': 'Integrating out the remaining conduction electron [MATH], we get the effective action of an auxiliary single-impurity Anderson model at each site [MATH][EQUATION] where the hybridization function for the f-site is, in Matsubara frequency space, [EQUATION]', 'cond-mat-0302389-1-20-7': 'The solution of this impurity problem is the major difficulty in this treatment.', 'cond-mat-0302389-1-20-8': 'We implemented two different methods of solution, which will be expanded upon in the next subsections.', 'cond-mat-0302389-1-20-9': "The aim is to calculate the local f-electron Green's function[EQUATION] under the dynamics dictated by ([REF]).", 'cond-mat-0302389-1-20-10': 'It is conveniently parameterized by its self-energy[EQUATION]', 'cond-mat-0302389-1-20-11': "It is also convenient to define a local conduction electron Green's function[EQUATION] such that[EQUATION] where[EQUATION]", 'cond-mat-0302389-1-20-12': 'We note that the [MATH] function describes the local scattering of the conduction electrons off the f-shell at site [MATH], incorporating information about both elastic and inelastic processes.', 'cond-mat-0302389-1-21-0': "All the information a generic site [MATH] has about the rest of the lattice is encoded in the bath function ([REF]), which should be viewed as a functional of the conduction electron lattice Green's function.", 'cond-mat-0302389-1-21-1': 'Since a fully analytical treatment is impossible, we have to solve the equations numerically.', 'cond-mat-0302389-1-21-2': 'For this purpose, we formulated the problem on a Bethe lattice, where things are considerably simplified as explained in Ref. [CITATION].', 'cond-mat-0302389-1-21-3': "In this case, nearest neighbors [MATH] and [MATH] become disconnected once [MATH] is removed and only local Green's functions survive [EQUATION].", 'cond-mat-0302389-1-21-4': 'Finally, these last objects can be computed from an action at site [MATH] in almost all aspects identical to ([REF]-[REF]), the only difference now being that the bath function sum runs over the [MATH] nearest neighbors (here labeled by [MATH]) only[EQUATION]', 'cond-mat-0302389-1-21-5': 'Note that, on the right-hand side, we do not need to specify that site [MATH] has been excluded as the removal of [MATH] completely disconnects sites labeled by [MATH] from site [MATH] (this property is specific to the Bethe lattice).', 'cond-mat-0302389-1-21-6': 'The reappearance of [MATH], whose distribution is identical to that of [MATH] since all sites are equivalent, closes the loop and establishes a recursive set of stochastic equations.', 'cond-mat-0302389-1-21-7': 'When the interaction is turned off ([MATH]), this treatment reduces to the well-known self-consistent theory of localization,[CITATION] here generalized to a two-band lattice.', 'cond-mat-0302389-1-21-8': 'When we take the coordination to infinity [MATH] keeping [MATH], our treatment reduces to the DMFT of correlations and disorder.', 'cond-mat-0302389-1-21-9': '[CITATION] In the latter case, the disorder treatment is equivalent to the CPA,[CITATION] which has no localization transition.', 'cond-mat-0302389-1-22-0': 'A full solution of Eqs. ([REF]-[REF]) for given distributions [MATH] and/or [MATH] involves solving an ensemble of impurity problems self-consistently.', 'cond-mat-0302389-1-22-1': 'Physically, the conduction electrons propagate through a disordered lattice and scatter off conduction site potential fluctuations as well as f-site resonances.', 'cond-mat-0302389-1-22-2': 'These resonances, in turn, describe the formation of localized moments, whose local Kondo temperatures fluctuate as well, reflecting a disordered conduction sea environment.', 'cond-mat-0302389-1-22-3': 'Complete statistical information can in principle be obtained from the distributions of the various renormalized local quantities.', 'cond-mat-0302389-1-22-4': 'We stress that a random distribution of any bare parameter causes all renormalized quantities to fluctuate as a result of the self-consistent nature of our treatment.', 'cond-mat-0302389-1-22-5': 'Therefore, even if we include only fluctuations in the conduction sea through [MATH], a distribution of Kondo temperatures ensues.', 'cond-mat-0302389-1-22-6': '[CITATION] This is easily seen from the approximate formula for the Kondo temperature in the Kondo limit ([MATH])[CITATION][EQUATION] where [MATH] is the local density of states (DOS) seen by the f-site[EQUATION] and [MATH] is the Kondo coupling constant, given in the Kondo limit by[EQUATION]', 'cond-mat-0302389-1-22-7': 'Even if [MATH] is not random, the local DOS is, because of the denominator in Eq. ([REF]).', 'cond-mat-0302389-1-22-8': 'As a result of the strong exponential dependence in ([REF]), even mild localization effects can be strongly enhanced and should be seriously considered, specially in disordered heavy fermion systems.', 'cond-mat-0302389-1-23-0': '## The impurity solvers', 'cond-mat-0302389-1-24-0': 'An important part of the method we employed is the solution of the impurity problems posed by the ensemble of effective actions given by Eq. ([REF]) and its counterpart for a site with one nearest neighbor removed.', 'cond-mat-0302389-1-24-1': 'We concentrated mostly on two methods of solution, which we now briefly describe: the slave boson, large-N based, mean field theory and second order perturbation theory in [MATH].', 'cond-mat-0302389-1-24-2': 'In order to unclutter the notation, we drop in the next subsections the site index [MATH] and the superscript [MATH].', 'cond-mat-0302389-1-24-3': 'Details of the numerical treatment are given in the Appendix.', 'cond-mat-0302389-1-25-0': '### Slave boson mean field theory', 'cond-mat-0302389-1-26-0': 'This method gives a good description of the low temperature, Fermi liquid regime of the Anderson impurity problem in the limit [MATH] and is extensively covered in the literature.', 'cond-mat-0302389-1-26-1': '[CITATION] Its main advantage is the ability to capture the zero temperature fixed point correctly as well as the exponential nature of the low energy scale.', 'cond-mat-0302389-1-26-2': 'Its treatment of the self-energy, however, does not incorporate inelastic processes to leading order.', 'cond-mat-0302389-1-26-3': 'Besides, it has a spurious phase transition at a finite temperature, where in reality there should be only a smooth crossover.', 'cond-mat-0302389-1-26-4': 'For these reasons, we confine it to the zero temperature limit, where it is a useful guide.', 'cond-mat-0302389-1-26-5': 'As applied to our problem, the method has been described in Appendix D of Ref. [CITATION] and we will merely state the results, generalized to the Matsubara frequency axis and at [MATH].', 'cond-mat-0302389-1-26-6': "The local f-electron Green's function is given by[EQUATION] where the last equality defines the local f-electron quasi-particle Green's function and the variational parameters [MATH] (renormalized f-energy) and [MATH] (quasi-particle residue) are determined from the solution of the set of equations [EQUATION]", 'cond-mat-0302389-1-26-7': 'Using [EQUATION]', 'cond-mat-0302389-1-26-8': 'Eqs. ([REF]-[REF]) simplify to[EQUATION]', 'cond-mat-0302389-1-26-9': 'Eq. ([REF]) becomes in this approximation[EQUATION]', 'cond-mat-0302389-1-27-0': '### Second order perturbation theory', 'cond-mat-0302389-1-28-0': 'The perturbative solution of the single-impurity Anderson model with particle-hole symmetry was thoroughly analyzed by Yamada and Yosida.[', 'cond-mat-0302389-1-28-1': '[CITATION] The series expansion in [MATH] for physical quantities such as the specific heat and the spin susceptibility converges very fast and even second order results can be useful.[', 'cond-mat-0302389-1-28-2': '[CITATION] Extension of the perturbative treatment to the case without particle-hole symmetry poses considerable difficulties.', 'cond-mat-0302389-1-28-3': 'A particularly useful proposal is the use of an interpolative self-energy which recovers the atomic ([MATH]) and high frequency limits.[', 'cond-mat-0302389-1-28-4': '[CITATION] Further improvements of the method were later suggested.[', 'cond-mat-0302389-1-29-0': "The procedure consists in defining an unperturbed f-electron Green's function[EQUATION] with a new parameter [MATH] to be determined later, which vanishes at particle-hole symmetry.", 'cond-mat-0302389-1-29-1': "The interacting Green's function is given in ([REF]).", 'cond-mat-0302389-1-29-2': 'The interpolative self-energy is[CITATION][EQUATION] where[EQUATION] and[EQUATION]', 'cond-mat-0302389-1-29-3': "The last equation ([REF]) is the usual second order diagram using the unperturbed Green's function ([REF]) for the internal lines.", 'cond-mat-0302389-1-29-4': 'The parameters [MATH] and [MATH] are determined by imposing the high frequency and atomic limits, respectively, and are given by[CITATION][EQUATION] where [EQUATION]', 'cond-mat-0302389-1-29-5': 'Different schemes have been proposed in order to fix the free parameter [MATH].[', 'cond-mat-0302389-1-29-6': '[CITATION] At zero temperature, one can ensure that the low energy Fermi liquid behavior is obtained by imposing the Friedel sum rule.[', 'cond-mat-0302389-1-29-7': '[CITATION] This procedure cannot be easily generalized to finite temperatures, however.', 'cond-mat-0302389-1-29-8': 'One option is to fix [MATH] at its zero temperature value even at finite temperatures.', 'cond-mat-0302389-1-29-9': 'Alternatively, one can require at any temperature[CITATION][EQUATION] which makes [MATH].', 'cond-mat-0302389-1-29-10': 'Finally, a third possibility is imposing [MATH].[', 'cond-mat-0302389-1-29-11': '[CITATION] These three alternatives have been rather carefully compared in Ref. potthoffetal at [MATH] and checked against exact diagonalization.', 'cond-mat-0302389-1-29-12': 'The first two methods were shown to be almost equivalent whereas the third one is inferior.', 'cond-mat-0302389-1-29-13': 'Moreover, comparisons at finite temperatures with Quantum Monte Carlo results confirmed the adequacy of imposing Eq. ([REF]).[', 'cond-mat-0302389-1-29-14': '[CITATION] Specific applications to a clean Anderson lattice model further corroborated this conclusion.[', 'cond-mat-0302389-1-29-15': '[CITATION] Thus, our results were based on imposing condition ([REF]).', 'cond-mat-0302389-1-29-16': 'It should be remembered, however, that the perturbative solution predicts a characteristic energy scale that is quantitatively incorrect at large values of [MATH], since it is unable to capture the correct exponential dependence.', 'cond-mat-0302389-1-29-17': 'Nevertheless, for moderate interactions, it still gives reasonable results.', 'cond-mat-0302389-1-29-18': 'Within its limitations, this perturbative scheme is a relatively flexible low-cost tool to tackle the impurity problem with the great advantage of being able to naturally account for inelastic processes.', 'cond-mat-0302389-1-30-0': 'We note that a direct comparison between the slave boson mean field theory results and second order perturbation theory is not possible because the former is limited to the [MATH] limit, which is obviously outside the region of validity of the latter.', 'cond-mat-0302389-1-30-1': 'The main interest of an analysis of both methods, however, resides in the exploration of the importance of inelastic processes, which are absent in the slave boson mean field treatment.', 'cond-mat-0302389-1-31-0': '# Slave boson mean field theory results', 'cond-mat-0302389-1-32-0': 'We now present the results obtained at [MATH] using the slave boson mean field theory as an impurity solver.', 'cond-mat-0302389-1-32-1': 'Most of our results were obtained for a uniform distribution of on-site conduction electron energies [EQUATION].', 'cond-mat-0302389-1-32-2': 'In Section [REF] we also show results for a discrete distribution of hybridization strengths [MATH].', 'cond-mat-0302389-1-33-0': '## Conduction electron typical density of states', 'cond-mat-0302389-1-34-0': 'To understand the overall behavior as a function of disorder, it is instructive to consider the transport properties of the conduction electrons.', 'cond-mat-0302389-1-34-1': 'Since there are no interactions among them in our model, their behavior is that of a disordered non-interacting electron system.', 'cond-mat-0302389-1-34-2': 'There are two sources of disorder, as can be seen in Eq. ([REF]): fluctuations of the local on-site energies [MATH] and of the f-shell resonances described by [MATH].', 'cond-mat-0302389-1-34-3': 'They are not independent, however, since they are inextricably tied by self-consistency.', 'cond-mat-0302389-1-34-4': 'Their combined effect acts to decrease the conduction electron mobility.', 'cond-mat-0302389-1-35-0': '### Typical density of states:', 'cond-mat-0302389-1-36-0': 'an order parameter for localization', 'cond-mat-0302389-1-37-0': 'A useful measure of this mobility is given by the typical value of the local escape rate.', 'cond-mat-0302389-1-37-1': "This is encoded in the imaginary part of the local conduction electron Green's function (the local DOS) at zero frequency, [MATH].", 'cond-mat-0302389-1-37-2': 'We will, from now on, drop the superscript denoting the removal of a nearest neighbor so as to lighten the notation.', 'cond-mat-0302389-1-37-3': 'As shown originally by Anderson, the typical value of the local DOS vanishes when the electrons are localized and can be viewed as an order parameter for the localization transition.', 'cond-mat-0302389-1-37-4': '[CITATION] A convenient way of accessing the typical value is furnished by the geometric average[EQUATION] where the overbar denotes a disorder average.', 'cond-mat-0302389-1-37-5': 'By contrast, the arithmetic average[EQUATION] is finite at the transition.', 'cond-mat-0302389-1-37-6': 'A thorough analysis of the critical behavior of the local DOS distribution in the non-interacting Bethe lattice localization problem was carried out in Ref. [CITATION].', 'cond-mat-0302389-1-38-0': 'In Fig. [REF] we show the typical conduction electron DOS as a function of disorder for several values of the chemical potential.', 'cond-mat-0302389-1-39-0': '### Proximity to the Kondo insulator', 'cond-mat-0302389-1-40-0': 'We can identify three qualitatively distinct behaviors.', 'cond-mat-0302389-1-40-1': '[CITATION] For [MATH] and for [MATH], [MATH] is a monotonically decreasing function of disorder.', 'cond-mat-0302389-1-40-2': 'For [MATH] the clean system is a Kondo insulator[CITATION] and [MATH].', 'cond-mat-0302389-1-40-3': 'As disorder is introduced in the Kondo insulator, [MATH] initially increases, reaching a maximum at about [MATH], after which it decreases monotonically.', 'cond-mat-0302389-1-40-4': 'Finally, for [MATH] and [MATH] initially decreases, passes through a minimum around [MATH], then increases up to a maximum at about [MATH] and eventually becomes monotonically decreasing.', 'cond-mat-0302389-1-40-5': 'For all values of [MATH], the typical DOS vanishes at a disorder-induced metal insulator transition (MIT) at [MATH].', 'cond-mat-0302389-1-41-0': 'These distinct behaviors can be traced back to how close the clean system is to the Kondo insulator point[CITATION] at [MATH].', 'cond-mat-0302389-1-41-1': '[CITATION] If we start from the clean insulator, the introduction of disorder acts to create states inside the gap, thus increasing the DOS at the chemical potential.', 'cond-mat-0302389-1-41-2': 'This increase continues until the gap is essentially washed out and the system becomes a bad metal.', 'cond-mat-0302389-1-41-3': 'After that, localization intervenes and [MATH] starts to decrease towards the MIT.', 'cond-mat-0302389-1-42-0': 'For fillings close to but not at the Kondo insulator point, the clean system is a heavy fermion metal at [MATH].', 'cond-mat-0302389-1-42-1': 'The f-resonances ([MATH]) coherently scatter the conduction electrons creating a strongly renormalized Fermi liquid.', 'cond-mat-0302389-1-42-2': 'In the slave boson treatment, [MATH] diverges at [MATH], see Eq. ([REF]), corresponding to the limit of unitary scattering, with a maximally allowed phase shift [MATH].', 'cond-mat-0302389-1-42-3': 'We can view its value at the Fermi level [MATH] (which is real at [MATH] as an effective potential coming from the f-electrons.', 'cond-mat-0302389-1-42-4': 'The closer the system is to the Kondo insulator, the larger the value of [MATH], the insulator being signalled by the divergence of this quantity (or equivalently by [MATH]).', 'cond-mat-0302389-1-42-5': 'The effect of disorder is to immediately start generating spatial fluctuations of the f-resonances, with different phase shift values at the chemical potential.', 'cond-mat-0302389-1-42-6': 'Proximity to the insulator implies large, random, almost unitary scattering potentials.', 'cond-mat-0302389-1-42-7': 'As a result, metallic coherence is efficiently destroyed and the typical conduction electron DOS is strongly suppressed.[', 'cond-mat-0302389-1-42-8': '[CITATION] The important role played by the unitary scatterers was emphasized in Refs. mirandavlad2 and mirandavlad1, where the distribution [MATH] was directly computed and its weight at [MATH] was shown to correlate with the destruction of coherence.', 'cond-mat-0302389-1-43-0': 'There is another equivalent way of understanding these effects.', 'cond-mat-0302389-1-43-1': 'For small dopings away from the Kondo insulator, carriers are introduced at the edges of the valence or conduction bands defined by the Kondo insulator gap, which have a small DOS (in the Bethe lattice, band edges have a square root shape as in three dimensions).', 'cond-mat-0302389-1-43-2': 'As has been known for a long time, a region of small DOS is particularly sensitive to localization effects introduced by disorder.', 'cond-mat-0302389-1-44-0': 'As in the previous case, further increase of disorder acts to wash out the nearby Kondo pseudo-gap and the behavior then becomes very similar to the disordered Kondo insulator.', 'cond-mat-0302389-1-44-1': 'We thus have a region with a rather non-intuitive increasing [MATH], which can be ascribed to the proximity to the Kondo insulator fixed point.', 'cond-mat-0302389-1-44-2': 'The behavior at fillings well away from the Kondo insulator is much less influenced by the pseudo-gap, see Fig. [REF] for [MATH].', 'cond-mat-0302389-1-44-3': 'Although there is a rapid initial decrease of [MATH], followed by a much slower dependence, the typical DOS does not exhibit the unconventional increase with disorder observed at other fillings.', 'cond-mat-0302389-1-45-0': '### Role of the hybridization strength', 'cond-mat-0302389-1-46-0': 'It is interesting to note that the critical value of disorder for the MIT [MATH] depends on the hybridization strength.', 'cond-mat-0302389-1-46-1': 'In Fig. [REF], we show the disorder dependence of the typical conduction electron DOS for different values of [MATH].', 'cond-mat-0302389-1-46-2': 'There is hardly any change in [MATH] as we go from [MATH] to [MATH] (cf. also Fig. [REF]).', 'cond-mat-0302389-1-46-3': 'However, for [MATH], the critical disorder strength is clearly enhanced.', 'cond-mat-0302389-1-46-4': 'This figure also illustrates the non-trivial nature of the self-consistency.', 'cond-mat-0302389-1-46-5': 'Indeed, the two types of disorder coming from fluctuations in [MATH] and [MATH] are clearly not independent, since the addition of f-site disorder as we turn on [MATH] from [MATH] to [MATH] acts to increase the mobility for [MATH].', 'cond-mat-0302389-1-46-6': 'The self-consistently determined solutions of the impurity problems effectively help screen the conduction electron disorder.', 'cond-mat-0302389-1-46-7': 'Note also how an increased value of [MATH] pushes the "dip-hump" structure to higher values of disorder.', 'cond-mat-0302389-1-46-8': 'Since the Kondo insulating gap increases with the hybridization strength, this is consistent with our explanation for the nature of this non-monotonic behavior.', 'cond-mat-0302389-1-47-0': '## Distribution of the conduction electron local density of states', 'cond-mat-0302389-1-48-0': 'One of the great advantages of the present approach is the possibility of monitoring complete distribution functions.', 'cond-mat-0302389-1-48-1': 'Many of the features exhibited in Figs. [REF] and [REF] can be directly read off the distribution of [MATH].', 'cond-mat-0302389-1-48-2': 'We show this by plotting [MATH] for several disorder strengths and different chemical potential values in Fig. [REF] (we use "log" for the base 10 logarithm).', 'cond-mat-0302389-1-48-3': 'It follows from the definition of [MATH], Eq. ([REF]), that it is obtained by raising 10 to the power of the average of this distribution (we use powers of 10 for ease of computation).', 'cond-mat-0302389-1-48-4': 'For weak disorder in the metallic cases (Figs. [REF]a and b), [MATH] is approximately Gaussian, even though the bare disorder is uniform, a feature shared by several physical quantities.', 'cond-mat-0302389-1-48-5': '[CITATION] This is due to the presence of correlations between many distant lattice sites mediated by the extended conduction electron wave function, which introduces a sort of averaging effect.', 'cond-mat-0302389-1-48-6': 'In the Kondo insulator case (Fig. [REF]c), however, the distribution is not Gaussian at weak disorder.', 'cond-mat-0302389-1-48-7': 'Keeping in mind that [MATH]) in the clean Kondo insulator, it is clear that the introduction of weak disorder has to generate weight at very small [MATH].', 'cond-mat-0302389-1-48-8': 'Indeed, [MATH] shows a divergence at [MATH] for [MATH].', 'cond-mat-0302389-1-49-0': 'For large values of [MATH] the distribution becomes extremely broad, spanning many orders of magnitude.', 'cond-mat-0302389-1-49-1': 'In the case [MATH] (Fig. [REF]a), corresponding to a system well away from the Kondo insulator filling, the distribution broadens and its maximum steadily shifts towards lower values as disorder is increased.', 'cond-mat-0302389-1-49-2': 'This is to be expected from the monotonic behavior of [MATH].', 'cond-mat-0302389-1-49-3': 'Likewise, at [MATH], the non-monotonic behavior of the typical value is also clearly reflected in [MATH] (see Fig. [REF]b and compare it to Fig. [REF]a).', 'cond-mat-0302389-1-50-0': 'As we saw, at the Kondo insulating chemical potential [MATH] and for [MATH], the distribution shows a divergence at [MATH].', 'cond-mat-0302389-1-50-1': 'A similar diverging tendency is observed at [MATH] (Fig. [REF]b) and [MATH].', 'cond-mat-0302389-1-50-2': 'This is precisely the disorder value where the minimum of [MATH] occurs (cf. Fig. [REF]a) and which we have been ascribing to the presence of many unitary scatterers due to the nearby Kondo insulator.', 'cond-mat-0302389-1-50-3': 'The similarity between the two distributions strengthens further our case for the importance of the proximity to the Kondo insulator.', 'cond-mat-0302389-1-50-4': 'Additionally and consistent with this, the divergence is totally absent at [MATH], where the role played by the Kondo insulator fixed point is much less important.', 'cond-mat-0302389-1-51-0': 'It is also interesting to observe in Fig. [REF]c, how the Kondo gap is washed out by disorder: at [MATH], where [MATH] peaks (Fig. [REF]a), most of the weight of the distribution is already at sizeable values of the DOS and its shape is very similar to the metallic cases.', 'cond-mat-0302389-1-52-0': '## Distribution of Kondo temperatures', 'cond-mat-0302389-1-53-0': 'We now proceed to the analysis of the physical properties related to the ensemble of impurity problems.', 'cond-mat-0302389-1-53-1': 'As shown before[CITATION] the distribution of Kondo temperatures of the various f-sites is log-normal for weak disorder, but broadens and acquires a power-law shape at intermediate values of [MATH].', 'cond-mat-0302389-1-53-2': 'Once this power-law becomes singular enough, a Griffiths phase is entered with diverging thermodynamic responses.', 'cond-mat-0302389-1-53-3': '[CITATION]', 'cond-mat-0302389-1-54-0': '### Universality at weak disorder', 'cond-mat-0302389-1-55-0': 'We have noticed that for weak disorder, the shape of the distribution of various quantities, including the Kondo temperature, is universal, irrespective of the shape of the bare distribution of disorder.', 'cond-mat-0302389-1-56-0': 'A nice illustration of this effect is given by the case where the bare disorder is a discrete distribution.', 'cond-mat-0302389-1-56-1': 'As an example, we take the discrete distribution of hybridization strengths, [MATH], determined in Ref. [CITATION] from XAFS measurements in UCu[MATH]Pd[MATH].', 'cond-mat-0302389-1-56-2': 'The resulting distribution of [MATH] is shown in Fig. [REF]a.', 'cond-mat-0302389-1-56-3': 'It is continuous and has a log-normal shape.', 'cond-mat-0302389-1-56-4': 'In Fig. [REF]b we show the distribution of [MATH] (smooth dot-dashed line) and compare it to the discrete distribution obtained in the DMFT (vertical solid lines), which corresponds to the limit of infinite coordination.', 'cond-mat-0302389-1-56-5': 'We also include in the figure the results of the Kondo disorder model[CITATION] (KDM) (vertical dashed lines), which is very similar to the DMFT.', 'cond-mat-0302389-1-56-6': 'The only differences between the KDM and the DMFT are that, in the former, no self-consistency is imposed and a Kondo instead of an Anderson lattice model is used.', 'cond-mat-0302389-1-56-7': 'The difference between the results of the DMFT/KDM and the statDMFT is striking.', 'cond-mat-0302389-1-56-8': 'The fluctuations of the conduction electron wave functions incorporated in the statDMFT smooth out the discrete results of the DMFT into a universal continuous form.', 'cond-mat-0302389-1-56-9': 'This comparison also shows that hybridization disorder alone is not able to generate non-Fermi liquid behavior in this case, since the distribution of Kondo temperatures goes to zero as [MATH], a point that was stressed in Refs. [CITATION] However, if disorder in the conduction electron sites, [MATH], is also included a singular behavior can be obtained (not shown).', 'cond-mat-0302389-1-56-10': 'The inclusion of conduction electron disorder is reasonable in UCu[MATH]Pd[MATH], since the Cu-Pd interchange affects both [MATH] and [MATH].', 'cond-mat-0302389-1-57-0': '### Emergence of the Electronic Griffiths phase', 'cond-mat-0302389-1-58-0': 'In order to identify the emergence of the Griffiths phase, we next study the evolution of the distribution of Kondo temperatures as the width [MATH] of [MATH] is varied (with no disorder in [MATH]).', 'cond-mat-0302389-1-58-1': 'Typical results are shown in Fig. [REF], corresponding to [MATH].', 'cond-mat-0302389-1-58-2': 'As the disorder increases, we find that the overall width, but most significantly, the size of the low-[MATH] tail rapidly grows.', 'cond-mat-0302389-1-58-3': 'These tails assume a power-law form [MATH], with the power [MATH] being a monotonically decreasing function of disorder.', 'cond-mat-0302389-1-58-4': 'The thermodynamic response of the system assumes a singular, non-Fermi liquid form as soon as [MATH], which happens for sufficiently strong disorder [MATH].', 'cond-mat-0302389-1-58-5': 'Since this behavior does not reflect any thermodynamic phase transition, it assumes the character of an electronic Griffiths phase.', 'cond-mat-0302389-1-58-6': 'Here, singular behavior emerges due to the presence of exponentially rare events (in our case Kondo spins) which nevertheless provide an exponentially large contribution to thermodynamic and transport properties and thus dominate the macroscopic behavior of the system.', 'cond-mat-0302389-1-58-7': 'The [MATH]-dependence of the exponent [MATH] can be easily obtained by fitting the tails of these distributions; a representative behavior for several values of the chemical potential is shown in Fig. [REF].', 'cond-mat-0302389-1-59-0': '### NFL and the proximity to the Kondo insulator', 'cond-mat-0302389-1-60-0': 'In the previous section we have seen how the proximity of the Kondo insulator plays a crucial role in determining the disorder dependence of the localization and transport properties, and leads to the surprising "bad metal" behavior for a wide parameter range.', 'cond-mat-0302389-1-60-1': 'This was argued to reflect the enhanced sensitivity to disorder of those electronic states that are very close to the Kondo insulator, and which are most easily affected by localization effects.', 'cond-mat-0302389-1-60-2': 'We have also established that NFL behavior also emerges as a result of disorder-induced density of states fluctuations.', 'cond-mat-0302389-1-60-3': 'It is then natural to ask how sensitive this emergence of NFL behavior is to the proximity to the Kondo insulator, which in the clean limit emerges only in a narrow parameter range, close (in our case) to [MATH].', 'cond-mat-0302389-1-61-0': 'To address this question, we have systematically investigated the evolution of [MATH] as a function of the distance to the Kondo insulator, i.e. as a function of the chemical potential [MATH].', 'cond-mat-0302389-1-61-1': 'The behavior for weak disorder ([MATH]) is shown in Fig. [REF].', 'cond-mat-0302389-1-61-2': 'Despite what one would naively expect, these result clearly demonstrate that the distributions are the broadest far from the Kondo insulator.', 'cond-mat-0302389-1-61-3': 'As we can see on this figure, the distributions narrow down as the Kondo insulator is approached from either side.', 'cond-mat-0302389-1-61-4': 'As a result, we may expect that the critical disorder strength [MATH] necessary for the emergence of NFL behavior should increase closer to the Kondo insulator.', 'cond-mat-0302389-1-61-5': 'This surprising result is confirmed by examining the [MATH]-dependence of the exponent [MATH] as shown in Fig. [REF].', 'cond-mat-0302389-1-61-6': 'As we can see there, for a given [MATH], the exponent [MATH] is indeed smaller, and [MATH] decreases for larger [MATH] (far from the Kondo insulator).', 'cond-mat-0302389-1-62-0': 'At first sight, these findings seem in contradiction to what one may expect, since we have found that the typical density of states decreases close to the Kondo insulator.', 'cond-mat-0302389-1-62-1': 'Naively, one could then expect the Kondo temperatures to be depressed as well, leading to broader distributions and enhanced NFL behavior.', 'cond-mat-0302389-1-62-2': 'On the other hand, we know that the Kondo temperature remains finite within the Kondo insulator, despite the fact that the density of states at the Fermi energy vanishes there.', 'cond-mat-0302389-1-62-3': 'Although surprising at first sight, this curious feature of Kondo insulators is at present well understood.', 'cond-mat-0302389-1-62-4': 'It is called the "strong coupling Kondo effect".', 'cond-mat-0302389-1-62-5': '[CITATION] It reflects the fact that the Kondo screening is not determined only by electronic states precisely at the Fermi energy, but also by all the states in an energy interval of order [MATH] around the Fermi energy.', 'cond-mat-0302389-1-62-6': 'Indeed, the average value of the Kondo temperature (see Fig. [REF]) is the highest precisely near the Kondo insulator.', 'cond-mat-0302389-1-62-7': 'In the presence of disorder, the value of the Kondo temperature is determined by a certain weighted average of the density of states over this extended energy interval.', 'cond-mat-0302389-1-62-8': 'When localization is present, only the states closest to the Kondo gap band edge will be appreciably affected, but since not only those states determine [MATH], the net effect is washed away.', 'cond-mat-0302389-1-62-9': 'We thus conclude that the proximity to the Kondo insulator, in contrast to transport, does not have appreciable effect on the emergence of the NFL behavior.', 'cond-mat-0302389-1-62-10': 'Indeed, the critical value of disorder [MATH] required for the emergence of NFL behavior is found to have a remarkably weak [MATH] dependence.', 'cond-mat-0302389-1-63-0': '### Universality at strong disorder', 'cond-mat-0302389-1-64-0': 'At strong disorder we expect the density of states fluctuations to completely wash out any trace of the Kondo gap, and in addition to broaden the conduction band, making it very flat and featureless.', 'cond-mat-0302389-1-64-1': 'As a result, all quantities are expected to have an extremely weak [MATH] dependence, leading to a more universal behavior of all quantities.', 'cond-mat-0302389-1-64-2': 'Such behavior is indeed seen at sufficiently strong disorder, where the typical DOS curves (Fig. [REF]) are seen to merge around [MATH].', 'cond-mat-0302389-1-64-3': "Similar behavior is seen in Fig. [REF], which shows [MATH] for different [MATH]'s at [MATH].", 'cond-mat-0302389-1-64-4': 'This universal behavior is even more striking if this distribution is plotted on a log-log scale (Fig. [REF]), where an almost perfect power-law tail ([MATH]) is seen to span several decades for all the values of [MATH].', 'cond-mat-0302389-1-65-0': '## Distribution of the hybridization function', 'cond-mat-0302389-1-66-0': 'A key input to the determination of the Kondo temperature is the hybridization function [MATH] of Eq. ([REF]).', 'cond-mat-0302389-1-66-1': 'We show in Fig. [REF]a the distribution of its imaginary part calculated at the chemical potential ([MATH]).', 'cond-mat-0302389-1-66-2': 'Note that, for a featureless bath, it appears in the exponential of the Kondo temperature formula, Eq. ([REF]), which is thus very sensitive to it.', 'cond-mat-0302389-1-66-3': 'It can be seen in Fig. [REF]a that its distribution is very regular and retains a bell-shaped structure for any [MATH].', 'cond-mat-0302389-1-66-4': 'It also inherits the non-monotonic behavior observed in the conduction electron local DOS (cf. Fig. [REF]b).', 'cond-mat-0302389-1-67-0': 'It is tempting to try to calculate the distribution of Kondo temperatures from the distribution of [MATH], by naively applying the Kondo temperature formula, Eq. ([REF]).', 'cond-mat-0302389-1-67-1': 'This procedure fails, however: the actual distribution of [MATH] has much lower weight at small [MATH] values than is predicted by the Kondo temperature formula.', 'cond-mat-0302389-1-67-2': 'The explanation for this failure lies in the fact, already alluded to before, that [MATH] is determined by a weighted average of [MATH] over a region around the Fermi level.', 'cond-mat-0302389-1-67-3': 'This can be glimpsed from the strong frequency dependence of the typical (geometric average) hybridization function close to the chemical potential, as shown, for frequencies on the real axis, in Fig. [REF]b.', 'cond-mat-0302389-1-67-4': 'It shows a robust and well-defined pseudo-gap, inherited from the nearby Kondo insulator, and a tiny narrow peak at the chemical potential.', 'cond-mat-0302389-1-67-5': 'This narrow peak is easy to understand: spatial fluctuations due to disorder give rise to narrow peaks within the pseudo-gap, most typically at the chemical potential.', 'cond-mat-0302389-1-67-6': 'However, as we have remarked before, the Kondo temperature can be finite even if the density of states is zero or almost zero at the chemical potential.', 'cond-mat-0302389-1-67-7': '[CITATION] In this case, the spectral weight right at the chemical potential is unimportant for the determination of the Kondo temperature.', 'cond-mat-0302389-1-67-8': 'It is dominated by a whole range of spectral density away from the Fermi level.', 'cond-mat-0302389-1-67-9': 'Since far from the pseudo-gap region the density of states is much larger and hence much less affected by the spatial fluctuations, the distribution of Kondo temperatures is narrower than one might guess based on the distribution of [MATH] and the Kondo temperature formula.', 'cond-mat-0302389-1-68-0': 'When considered together, the results of Section [REF] show the importance of a self-consistent solution of the problem, with a non-trivial interplay between spatial fluctuations due to localization and strong correlation effects.', 'cond-mat-0302389-1-68-1': 'However, an important feature that is missed in the slave boson treatment of the impurity problems is the presence of inelastic scattering.', 'cond-mat-0302389-1-68-2': 'This will be considered in the next section, where we show the results obtained with perturbation theory in the interaction.', 'cond-mat-0302389-1-69-0': '# Perturbation theory results', 'cond-mat-0302389-1-70-0': 'We consider now the results obtained at finite [MATH] using second order perturbation theory as the impurity solver.', 'cond-mat-0302389-1-71-0': '## Results for a fixed temperature', 'cond-mat-0302389-1-72-0': 'Fig. [REF] shows the results for the conduction electron typical DOS near the Fermi level as a function of the disorder parameter [MATH] for different values of the interaction energy [MATH] and the hybridization [MATH] at [MATH].', 'cond-mat-0302389-1-72-1': 'In order to understand them, we looked at scatter plots of realizations of the local effective f-shell potential (real and imaginary parts of [MATH]) and the corresponding bare potential [MATH] at the same site, both seen by the conduction electrons.', 'cond-mat-0302389-1-72-2': 'These results are presented for two different values of the hybridization, [MATH] and [MATH], and two different values of the disorder parameter, [MATH] and [MATH], in Figs. [REF] to [REF].', 'cond-mat-0302389-1-72-3': 'The other parameters used in the calculation were [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0302389-1-72-4': 'The figures present the results obtained in the statDMFT calculation (represented by dots) as well as the results of DMFT (full line), in which the disorder treatment reduces to CPA.', 'cond-mat-0302389-1-72-5': '[CITATION] It is important to notice that we have [MATH] (as [MATH] at finite [MATH]), implying the presence of inelastic scattering, a feature absent in the slave boson treatment of the last section.', 'cond-mat-0302389-1-72-6': 'Besides, the imaginary part of the self-energy gets folded into the real part of [MATH] as well.', 'cond-mat-0302389-1-72-7': 'This is a peculiar feature of a two-band model, where the effective conduction electron self-energy represented by [MATH] has a real part for which inelastic processes contribute.', 'cond-mat-0302389-1-72-8': 'Thus, we should keep in mind that, even though in an effective description of conduction electron processes [MATH] and [MATH] are associated with elastic scattering while [MATH] is related to inelastic processes, both [MATH] and [MATH] contain information on f-electron collisions.', 'cond-mat-0302389-1-73-0': 'Let us compare the results for [MATH] for both [MATH] and [MATH], which are in Figs. [REF] and [REF].', 'cond-mat-0302389-1-73-1': 'The first observation we make is that for [MATH] the values of [MATH] are mainly concentrated around zero, while for [MATH] they are distributed in a wider range of values.', 'cond-mat-0302389-1-73-2': 'On the other hand, the results for [MATH] show that the inelastic scattering is stronger for [MATH] than for [MATH].', 'cond-mat-0302389-1-73-3': 'Concerning the results for [MATH], the great concentration of sites with large [MATH] explains the great decrease in the typical DOS for low disorder shown in Fig. [REF].', 'cond-mat-0302389-1-73-4': 'These sites act as almost unitary scatterers, which give rise to a maximally allowed scattering phase shift ([MATH]) for the conduction electrons, and represent droplets of Kondo insulator within the metal, in close parallel to the slave boson results discussed of Section [REF].', 'cond-mat-0302389-1-73-5': '[CITATION] For [MATH], although the inelastic scattering is stronger, it cannot compensate for the much narrower distribution of [MATH].', 'cond-mat-0302389-1-73-6': 'Thus, the DOS does not decrease as fast as for [MATH].', 'cond-mat-0302389-1-74-0': 'As the disorder increases to intermediate values ([MATH]), the distribution of [MATH] becomes larger, causing a steady decrease in the typical DOS for [MATH].', 'cond-mat-0302389-1-74-1': 'However, for [MATH], the typical DOS presents a non-monotonic behavior similar to the slave boson results for [MATH] and [MATH].', 'cond-mat-0302389-1-74-2': 'As we explained in Section [REF], this is a result of the fact that, as the disorder increases, the concentration of unitary scatterers first increases, then saturates and the bare disorder dominates over the f-related one ([MATH]).', 'cond-mat-0302389-1-74-3': '[CITATION] Indeed, even for [MATH] we notice the presence of a slight "shoulder" in the typical DOS around [MATH], whose origin is the same as that of the non-monotonic behavior at [MATH].', 'cond-mat-0302389-1-74-4': 'We call attention to the slave boson results of Fig. [REF], which similarly show that a decrease of the hybridization strength moves the "dip-hump" feature to smaller disorder values.', 'cond-mat-0302389-1-74-5': 'Moreover, as the current calculation was done at finite temperature, inelastic scattering also plays some role in smoothing out this feature.', 'cond-mat-0302389-1-75-0': 'As the disorder continues to increase and the distribution of [MATH] becomes broader, the typical DOS for both [MATH] and [MATH] decreases.', 'cond-mat-0302389-1-75-1': 'These results tend to the non-interacting one, as is expected if only the bare disorder plays a role.', 'cond-mat-0302389-1-75-2': 'Indeed, comparing the results for [MATH] (Figs. [REF] and [REF]) we notice that the realizations for which the bare disorder [MATH] is large have [MATH] around zero, meaning that the real part of the f-shell disorder is not important.', 'cond-mat-0302389-1-75-3': 'Besides, the values of [MATH] are small for these realizations.', 'cond-mat-0302389-1-75-4': 'In the case of the realizations for which [MATH] is small, [MATH] attains values that are larger than the results for [MATH] (cf. Figs. [REF] and [REF]).', 'cond-mat-0302389-1-76-0': 'The above discussion has focused on the conduction electron viewpoint.', 'cond-mat-0302389-1-76-1': 'Let us now consider how the presence of disorder in [MATH] is seen by the f-electrons.', 'cond-mat-0302389-1-76-2': 'For the auxiliary one-impurity problem, the important scale is the Kondo temperature [MATH], which measures the coupling between the impurity and the conduction electron bath.', 'cond-mat-0302389-1-76-3': 'The presence of disorder in [MATH] generates a distribution of [MATH], as we have seen before.', 'cond-mat-0302389-1-76-4': 'Thus at finite temperature some of the sites have [MATH], forming a singlet state with the bath, while the sites for which [MATH] represent an almost free spin, scattering conduction electrons incoherently.', 'cond-mat-0302389-1-76-5': 'This incoherence is characterized by a large amount of inelastic scattering, signaled by a significant imaginary part of the self-energy.', 'cond-mat-0302389-1-76-6': 'Going back to the results for [MATH], the fact that the inelastic scattering is stronger for [MATH] than for [MATH] reflects the larger number of incoherent sites (with [MATH]) in the former, since the smaller the hybridization, the smaller the Kondo temperature (see Eqs. [REF] and [REF]).', 'cond-mat-0302389-1-76-7': 'On the other hand, the sites with large [MATH] for [MATH], which are responsible for the great decrease in the typical DOS, represent sites with [MATH].', 'cond-mat-0302389-1-77-0': 'Fig. [REF] also shows the results for the typical DOS for [MATH], and [MATH].', 'cond-mat-0302389-1-77-1': 'In this case, the system has particle-hole symmetry in the clean limit, presenting a gap in its DOS (the Kondo insulator).', 'cond-mat-0302389-1-77-2': 'This explains the fact that for small disorder the typical DOS decreases as [MATH] decreases.', 'cond-mat-0302389-1-77-3': 'The overall behavior here bears strong similarity with the slave boson results of Fig. [REF]a (circles) and the explanation for it has been given in Section [REF].', 'cond-mat-0302389-1-78-0': '## Temperature dependence', 'cond-mat-0302389-1-79-0': 'Fig. [REF] presents the results for the typical DOS as a function of [MATH] for different temperatures.', 'cond-mat-0302389-1-79-1': 'The other parameters used were [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0302389-1-79-2': 'Here is where the interplay between inelastic and elastic processes proves to be fairly non-trivial and the use of a technique that incorporates both, such as perturbation theory, is crucial.', 'cond-mat-0302389-1-79-3': 'First we note that for the lowest disorder value ([MATH]), the typical DOS decreases with increasing temperature.', 'cond-mat-0302389-1-79-4': 'This is made more clear in the inset.', 'cond-mat-0302389-1-79-5': 'On the other hand, for [MATH], this tendency is reversed.', 'cond-mat-0302389-1-79-6': 'Finally, in between these two extremes, the temperature dependence can be non-monotonic.', 'cond-mat-0302389-1-79-7': 'A better sense of the overall behavior can be grasped by plotting the inverse of the typical DOS as a function of [MATH] for different values of [MATH], as shown in Fig. [REF].', 'cond-mat-0302389-1-79-8': 'It is clear that for [MATH], the inverse typical DOS shows a peak as a function of [MATH], which gradually moves to zero temperature as disorder is increased.', 'cond-mat-0302389-1-80-0': 'The temperature dependence of the typical DOS can be rationalized by looking at the corresponding changes in the distribution of [MATH].', 'cond-mat-0302389-1-80-1': 'For this purpose, we focus on the disorder value of [MATH], for which [MATH] has a clear maximum at [MATH].', 'cond-mat-0302389-1-80-2': 'We show in Figs. [REF] and [REF] the scatter plots of [MATH] at [MATH] and [MATH], respectively, which should be compared to the higher temperature results ([MATH]) of Fig. [REF].', 'cond-mat-0302389-1-80-3': 'From Fig. [REF] we see that the inverse typical DOS increases as we go from [MATH] to [MATH] and then decreases as the temperature is varied up to [MATH].', 'cond-mat-0302389-1-80-4': 'As can be seen from the figures, this non-monotonic behavior is governed by the effective f-disorder encoded in the distribution of [MATH].', 'cond-mat-0302389-1-80-5': 'Indeed, its variance increases from [MATH] to [MATH] but decreases from [MATH] to [MATH].', 'cond-mat-0302389-1-80-6': 'Note that the imaginary part of [MATH] always increases with increasing temperature, reflecting the enhancement of inelastic processes.', 'cond-mat-0302389-1-80-7': 'In terms of [MATH], this is the same as saying that as the temperature increases the number of sites with [MATH], which have a stronger inelastic scattering, becomes larger.', 'cond-mat-0302389-1-80-8': 'However, in the interval from about [MATH] up to [MATH], the increase of [MATH] is outweighed by the narrowing of the distribution of [MATH], which is the dominant contribution.', 'cond-mat-0302389-1-80-9': 'The above analysis can be similarly extended to the other values of disorder shown in Fig. [REF].', 'cond-mat-0302389-1-81-0': 'All along we have been using the typical DOS as a measure of the conducting properties of the system.', 'cond-mat-0302389-1-81-1': 'This is justified by its interpretation as a escape rate from a lattice site and the fact that it vanishes at the localization transition.', 'cond-mat-0302389-1-81-2': '[CITATION] Ideally, one would like to calculate the conductivity instead.', 'cond-mat-0302389-1-81-3': 'This is a difficult task in the present scheme, however, although an approximate calculation can be performed, which becomes accurate close to the localization transition.', 'cond-mat-0302389-1-81-4': "[CITATION] It requires the calculation of the propagator between two different sites, which goes beyond our current method, whose focus is on local Green's functions only.", 'cond-mat-0302389-1-81-5': 'Even in view of all these caveats, however, it is tempting to use the inverse typical DOS as an approximate measure of the resistivity, specially far from the weakly disordered region ([MATH]).', 'cond-mat-0302389-1-81-6': 'If this is done, then the fanning out of the "resistivity" curves of Fig. [REF] as [MATH] is reminiscent of the Mooij correlations,[CITATION] originally observed in disordered transition metal alloys, but which are also seen in heavy fermion alloys,[CITATION] doped Kondo insulators[CITATION] and even in two-dimensional systems as in the metal-oxide-semiconductor field-effect transistors.[', 'cond-mat-0302389-1-81-7': '[CITATION] Our results and the discussion above show that the interplay between localization effects and electron-electron interactions can give rise to the Mooij correlations, without the need to invoke other sources, such as electron-phonon interactions.[', 'cond-mat-0302389-1-81-8': '[CITATION] In addition, we have pointed out how the rapid drop in the typical DOS is a consequence of the proximity to the Kondo insulator, a region where localization effects are particularly large.', 'cond-mat-0302389-1-81-9': 'Taken together, these observations point to a close connection between Mooij correlations and localization effects in the vicinity of a Kondo or a Mott insulator.', 'cond-mat-0302389-1-81-10': 'It would be interesting to test these ideas by a direct calculation of the resistivity within a scheme capable of incorporating both localization and strong correlations.', 'cond-mat-0302389-1-82-0': 'Finally, regarding the DMFT calculation presented in all the figures discussed in this Section, we note that the results obtained for [MATH] fall approximately in the region where the concentration of realizations in the statDMFT calculation is largest.', 'cond-mat-0302389-1-82-1': 'There is no reason to expect DMFT and the statDMFT to give similar results, except at weak disorder.', 'cond-mat-0302389-1-82-2': 'Nevertheless, our results show that, surprisingly, DMFT can serve as a rough guide for the most probable values of [MATH].', 'cond-mat-0302389-1-82-3': 'There is a sizeable discrepancy, however, between the DMFT and the statDMFT results for [MATH] at the highest temperature ([MATH]) and low disorder.', 'cond-mat-0302389-1-82-4': 'The DMFT line in this case is an underestimate of the realizations obtained within the statDMFT.', 'cond-mat-0302389-1-82-5': 'As the temperature is lowered a better agreement is obtained.', 'cond-mat-0302389-1-82-6': 'Thus, besides its inherent neglect of localization effects, DMFT should be used only as a lower bound when gauging the importance of inelastic processes in disordered Anderson lattices.', 'cond-mat-0302389-1-83-0': '# Discussion and conclusions', 'cond-mat-0302389-1-84-0': 'We have in this paper extensively characterized the physics of disordered Anderson lattices within the statDMFT scheme, which is able to incorporate both localization effects and the local correlations coming from electron-electron interactions.', 'cond-mat-0302389-1-84-1': 'This was done using both large-N methods and perturbation theory for the auxiliary single impurity problems.', 'cond-mat-0302389-1-84-2': 'These are in a sense complementary approaches.', 'cond-mat-0302389-1-84-3': 'On the one hand, large-N theory is ideal for ground state properties, where inelastic effects are absent.', 'cond-mat-0302389-1-84-4': 'In particular, it affords a quick and reliable way of calculating Kondo temperatures (with the correct exponential dependence) and scattering phase shifts at [MATH].', 'cond-mat-0302389-1-84-5': 'Without [MATH] corrections, however, it is unsuitable for a finite temperature calculation.', 'cond-mat-0302389-1-84-6': 'On the other hand, second order perturbation theory has the advantage of being equally flexible with the added bonus of incorporating inelastic processes and the temperature dependence of the scattering phase shifts.', 'cond-mat-0302389-1-84-7': 'Nevertheless, it fails to capture the exponential nature of the low temperature scale of the single impurity problem.', 'cond-mat-0302389-1-84-8': 'Taken together, the two methods have enabled us to put on firm grounds the conclusions laid out in previous work,[CITATION] namely, the emergence of an electronic Griffiths phase in Anderson lattices governed by the proximity to the disorder-induced localization transition.', 'cond-mat-0302389-1-84-9': 'Furthermore, the perturbation theory treatment has also suggested a mechanism behind the ubiquitous observation of Mooij correlations in the resistivity of disordered materials.', 'cond-mat-0302389-1-85-0': 'Even within the confines of the approximations of the statDMFT scheme, there are still outstanding issues that we would like to resolve.', 'cond-mat-0302389-1-85-1': 'Even though a fully analytical treatment is probably impossible, it might be feasible to devise an approximate parameterization of the statDMFT on the Bethe lattice, specially at weak disorder.', 'cond-mat-0302389-1-85-2': 'In this respect, the universality of the distributions of "dressed" quantities, such as the various local Green\'s functions and the Kondo temperatures, which are either Gaussian or log-normal is a useful guide.', 'cond-mat-0302389-1-85-3': 'A "toy" model that assumes a simple form for the distribution of [MATH] can be written down, which recovers the Griffiths singularities obtained in the numerical treatment.', 'cond-mat-0302389-1-85-4': '[CITATION] This "toy" model may prove useful for a calculation of the resistivity and for generalizations of the statDMFT treatment.', 'cond-mat-0302389-1-86-0': 'On the other hand, a specially important effect neglected in the statDMFT is two-particle inter-site correlations, particularly in the spin channel.', 'cond-mat-0302389-1-86-1': 'Indeed, the proliferation of poorly quenched low-[MATH] spins in our treatment generates a large amount of entropy that must be relieved at low temperatures through inter-site correlations.', 'cond-mat-0302389-1-86-2': 'Indeed, experimental evidence in favor of spin-glass dynamics at low temperatures in UCu[MATH]Pd[MATH][CITATION] and URh[MATH]Ge[MATH][CITATION] makes the inclusion of inter-site correlations more pressing.', 'cond-mat-0302389-1-86-3': 'A promising avenue of attack would be to remain true to the spirit of the DMFT and use its extended version.', 'cond-mat-0302389-1-86-4': 'Several treatments along these lines have been attempted.', 'cond-mat-0302389-1-86-5': '[CITATION] The challenge in our case is to incorporate both the dynamical inter-site correlations of the latter treatments and the spatial fluctuations of Kondo temperatures of the electronic Griffiths phase we find in our approach.', 'cond-mat-0302389-1-86-6': 'We defer the discussion of this problem to a future publication.', 'cond-mat-0302389-1-87-0': 'We thank M. J. Rozenberg for useful discussions.', 'cond-mat-0302389-1-87-1': 'We also thank C. H. Booth for providing us his XAFS data.', 'cond-mat-0302389-1-87-2': 'This work was supported by FAPESP through grants 99/00895-9 (MCOA), 98/12741-3 (EM), 01/00719-8 (EM), by CNPq through grant 301222/97-5 (EM), and by the NSF through grants DMR-9974311 and NSF-0234215 (VD).', 'cond-mat-0302389-1-88-0': '# Brief description of the numerical method', 'cond-mat-0302389-1-89-0': 'The set of stochastic equations defined in ([REF]-[REF]) must be solved in two steps.', 'cond-mat-0302389-1-89-1': 'First, action ([REF]-[REF]) is solved with the bath function with [MATH] nearest neighbors defined in ([REF]).', 'cond-mat-0302389-1-89-2': "This determines self-consistently the distribution of local conduction electron Green's functions with one nearest neighbor removed [MATH].", 'cond-mat-0302389-1-89-3': 'Next, the same action ([REF]-[REF]) is solved, this time with the bath function in ([REF]), constructed from the previously determined [MATH].', 'cond-mat-0302389-1-89-4': 'This step involves no self-consistency and yields the distribution of [MATH] as output.', 'cond-mat-0302389-1-89-5': 'Since the latter bath function is a sum over [MATH] nearest neighbors its statistical fluctuations are reduced compared to the former one.', 'cond-mat-0302389-1-89-6': 'Thus, [MATH] is more narrowly distributed than [MATH].', 'cond-mat-0302389-1-89-7': 'Yet, we expect the qualitative behavior to be the same.', 'cond-mat-0302389-1-89-8': 'We have therefore focused on the first step of the procedure only.', 'cond-mat-0302389-1-90-0': 'For a given impurity solver, this disordered Bethe lattice problem was solved for [MATH], by sampling the distribution of [MATH] from an ensemble of [MATH] sites, as proposed originally in Ref. [CITATION].', 'cond-mat-0302389-1-90-1': 'We have generally used [MATH] since we have checked that the results do not change by taking [MATH].', 'cond-mat-0302389-1-90-2': 'We have thus determined the distribution of various local properties.', 'cond-mat-0302389-1-91-0': 'The equations were solved on a discrete mesh along the Matsubara axis.', 'cond-mat-0302389-1-91-1': 'The mesh is set by the Matsubara frequencies in the perturbative treatment at finite temperatures and by an arbitrary finite discrete mesh in the slave boson mean field theory at [MATH] (up to 32000 points).', 'cond-mat-0302389-1-91-2': 'The choice of the imaginary frequency axis is due to a greater numerical stability.', 'cond-mat-0302389-1-91-3': "When the disorder is strong, the various Green's functions show large fluctuations.", 'cond-mat-0302389-1-91-4': 'However, these are much more pronounced on the real frequency axis, where they give rise to several peaks and gaps.', 'cond-mat-0302389-1-92-0': 'The slave boson treatment consists in finding the two mean field parameters [MATH] and [MATH] by solving the set of two non-linear Eqs. ([REF]-[REF]).', 'cond-mat-0302389-1-92-1': 'For that, we used the Powell hybrid method.', 'cond-mat-0302389-1-92-2': 'The integrals were calculated with standard adaptive quadrature routines.', 'cond-mat-0302389-1-92-3': 'Since we used a finite frequency mesh, it was important to extrapolate the value of [MATH] with the asymptotic form [MATH] for values of [MATH] greater than the largest mesh value.', 'cond-mat-0302389-1-92-4': 'In this fashion, a wide range of Kondo temperatures is covered, going down to almost machine precision in the clean metallic case.', 'cond-mat-0302389-1-92-5': 'When disorder is present a given impurity problem may not have a solution even at [MATH], because the strong spatial fluctuations may cause the local DOS to vanish at the Fermi level.', 'cond-mat-0302389-1-92-6': 'In this case, there are two possible regimes for the impurity, which have been carefully analyzed.', 'cond-mat-0302389-1-92-7': '[CITATION] The analysis shows that there is a critical coupling constant [MATH] such that the ground state is a singlet for [MATH] (the so-called "strong coupling Kondo effect"), whereas the local moment remains unquenched if [MATH].', 'cond-mat-0302389-1-92-8': 'When a solution could not be found, this corresponded to either a free spin ([MATH]) or a Kondo temperature which is smaller than the smallest value we can reach with our numerical code.', 'cond-mat-0302389-1-92-9': 'In either case, we set [MATH], effectively decoupling the free moment from the rest of the lattice.', 'cond-mat-0302389-1-92-10': 'Yet, we were still able to span several decades of energy scales.', 'cond-mat-0302389-1-93-0': 'In the perturbative treatment, the solution of each impurity problem is found by solving a set of two non-linear equations for [MATH] and [MATH], which is defined by Eqs. ([REF]) and ([REF]).', 'cond-mat-0302389-1-93-1': 'As in the slave boson treatment we used the Powell hybrid method.', 'cond-mat-0302389-1-93-2': 'The calculation of the second order correction for the self-energy involves Fourier transforms as, according to Eq. ([REF]), it has a simpler form in imaginary time rather than in frequency space.', 'cond-mat-0302389-1-93-3': 'For this, we used the Fast Fourier Transform algorithm.', 'cond-mat-0302389-1-93-4': '[CITATION]'}

{'cond-mat-0302389-2-0-0': 'We study ground state and finite temperature properties of disordered heavy fermion metals by using a generalization of dynamical mean field theory which incorporates Anderson localization effects.', 'cond-mat-0302389-2-0-1': 'The emergence of a non-Fermi liquid metallic behavior even at moderate disorder is shown to be a universal phenomenon resulting from local density of states fluctuations.', 'cond-mat-0302389-2-0-2': 'This behavior is found to have a character of an electronic Griffiths phase, and can be thought of as a precursor of Anderson localization in a strongly correlated host.', 'cond-mat-0302389-2-0-3': 'The temperature dependence of the conducting properties of the system reveal a non-trivial interplay between disorder and inelastic processes, which are reminiscent of the Mooij correlations observed in many disordered metals.', 'cond-mat-0302389-2-1-0': '# Introduction', 'cond-mat-0302389-2-2-0': 'The interplay of disorder and strong correlations remains one of the least understood topics of contemporary condensed matter physics.', 'cond-mat-0302389-2-2-1': 'These effects are believed to bear relevance to many problems that have attracted recent attention, such as the metal-insulator transition (MIT) in two-dimensional electron systems.', 'cond-mat-0302389-2-2-2': '[CITATION] Disorder effects are also likely to be important for the understanding of the puzzling non-Fermi liquid (NFL) behavior of several heavy fermion compounds.', 'cond-mat-0302389-2-2-3': '[CITATION] In some of these systems, impurities seem to play only a subsidiary role: the explanation for the anomalous behavior is more likely to be found in the physics of quantum criticality,[CITATION] even though a complete description is still lacking.[', 'cond-mat-0302389-2-2-4': '[CITATION] However, in other heavy fermion systems, disorder seems to play a more essential role and seems to be at the origin of the NFL behavior.[', 'cond-mat-0302389-2-3-0': 'Several attempts have been made to address theoretically the role of disorder in heavy fermion compounds (see an overview below).', 'cond-mat-0302389-2-3-1': 'Many experimental results can be described within the so-called Kondo disorder model (KDM)[CITATION] or, equivalently, the dynamical mean field theory (DMFT) of disordered Kondo/Anderson lattices,[CITATION] at least above the lowest temperatures.', 'cond-mat-0302389-2-3-2': 'Essential to this description is the consideration of the full distribution of local Kondo temperatures [MATH].', 'cond-mat-0302389-2-3-3': 'For sufficient disorder, it has a large weight as [MATH] describing the presence of dilute low-[MATH] spins that dominate the thermodynamic and transport properties.', 'cond-mat-0302389-2-3-4': 'However, the KDM/DMFT predictions relied on a fine tuning of the bare disorder that suggested that a more accurate microscopic foundation was necessary.', 'cond-mat-0302389-2-3-5': 'Conspicuously missing in this scheme were fluctuations in the conduction electron local density of states (DOS), a quantity that is crucial for the determination of [MATH].', 'cond-mat-0302389-2-3-6': 'This was remedied by two of us through a generalization of the DMFT that incorporates such Anderson localization effects while keeping its local treatment of correlations.', 'cond-mat-0302389-2-3-7': '[CITATION] As a result, the question of the extreme sensitivity to the bare disorder was solved.', 'cond-mat-0302389-2-3-8': 'One important result of this study is the emergence of a power-law distribution of Kondo temperatures [MATH], where [MATH] depends continuously on the strength of disorder [MATH] and decreases as [MATH] is increased.', 'cond-mat-0302389-2-3-9': 'As the distribution becomes more singular, several thermodynamic quantities become divergent, in a manner characteristic of Griffiths phases.', 'cond-mat-0302389-2-3-10': '[CITATION] The system eventually localizes at a critical disorder strength [MATH].', 'cond-mat-0302389-2-3-11': 'Since there is no magnetic phase transition this has been dubbed an electronic Griffiths phase.', 'cond-mat-0302389-2-4-0': 'This initial work[CITATION] employed the slave boson large-N theory[CITATION] to solve the auxiliary single-impurity problems posed by the method.', 'cond-mat-0302389-2-4-1': 'While versatile, powerful and yet computationally cheap, this approach presents some disadvantages, the main one being the difficulty of working at finite temperatures.', 'cond-mat-0302389-2-4-2': 'It should be reminded that, since we deal with wide distributions of [MATH], we need to be able to describe well the full crossover from [MATH] to [MATH], which is not possible with the slave boson large-N treatment.', 'cond-mat-0302389-2-4-3': 'In particular, conspicuously missing are inelastic scattering processes.', 'cond-mat-0302389-2-4-4': 'Besides, though giving a good description of the low-energy Fermi-liquid regime of the single-impurity problem, this treatment does not incorporate high- and intermediate-energy incoherent processes.', 'cond-mat-0302389-2-4-5': 'Another impurity solver is therefore needed to assess the importance of these intrinsically finite-[MATH] and finite-energy features.', 'cond-mat-0302389-2-4-6': 'A particularly useful method, able to fill this gap at a reasonable computational cost, is second order perturbation theory in [MATH].', 'cond-mat-0302389-2-4-7': 'We have used this method to both complement and cross-check the slave boson results.', 'cond-mat-0302389-2-5-0': 'Besides the results of the initial work,[CITATION] which have been confirmed by both methods, some of our main conclusions are: (i) there is a subtle interplay between conduction and f-electron site disorder that leads to a surprising non-monotonic dependence of the conducting properties on disorder, a feature that is likely unique to Kondo/Anderson as opposed to Hubbard models; (ii) localization effects are essential for the determination of the distribution of Kondo temperatures and a KDM/DMFT description is clearly insufficient, especially if one starts from an experimentally measured discrete distribution; (iii) the interplay between disorder and inelastic processes can lead to a temperature dependence of the conducting properties that is reminiscent of the ones found by Mooij and others in several strongly correlated disordered metals.', 'cond-mat-0302389-2-5-1': '[CITATION]', 'cond-mat-0302389-2-6-0': 'This paper is organized as follows.', 'cond-mat-0302389-2-6-1': 'We review the disorder-based mechanisms of non-Fermi liquid behavior in the next subsection.', 'cond-mat-0302389-2-6-2': 'Section [REF] describes the model of disordered Anderson lattices we studied and the methods we employed to solve it.', 'cond-mat-0302389-2-6-3': 'Section [REF] focuses on the detailed results obtained within the slave boson large-N method.', 'cond-mat-0302389-2-6-4': 'This expands considerably on the previously published results.', 'cond-mat-0302389-2-6-5': '[CITATION] In Section [REF], we show the results obtained with perturbation theory.', 'cond-mat-0302389-2-6-6': 'Finally, we wrap up with a general discussion of the strengths and limitations of this study and point out possible future directions in Section [REF].', 'cond-mat-0302389-2-6-7': 'Some details of the computational procedures are given in an Appendix.', 'cond-mat-0302389-2-7-0': '## Brief overview of disorder-based mechanisms of non-Fermi liquid behavior', 'cond-mat-0302389-2-8-0': '### Kondo-disorder models and the electronic Griffiths phase', 'cond-mat-0302389-2-9-0': 'The KDM was proposed early on to account for the temperature dependence of the Cu nuclear magnetic resonance (NMR) line widths in UCu[MATH]Pd[MATH]).', 'cond-mat-0302389-2-9-1': '[CITATION] It assumed that disorder in a heavy fermion system generates random spatial fluctuations of the exchange coupling constant [MATH] between local moments and conduction electrons (the Kondo coupling).', 'cond-mat-0302389-2-9-2': 'Each local moment was assumed to undergo the Kondo effect in a manner that is completely uncorrelated with the others and each with a characteristic energy scale, its Kondo temperature [MATH].', 'cond-mat-0302389-2-9-3': "Even narrow Kondo coupling distributions lead to a wide distribution of Kondo temperatures due to the latter's exponential dependence on the former.", 'cond-mat-0302389-2-9-4': 'As a result, at low temperatures, many spins are quenched while a few percent remain unquenched and dominate, giving rise to singular, NFL thermodynamic properties (specific heat and magnetic susceptibility).', 'cond-mat-0302389-2-9-5': 'The NMR results in UCu[MATH]Pd[MATH]) are well described within this picture if the distribution function [MATH] is such that [MATH][CITATION] The KDM gained a natural theoretical setting within the DMFT[CITATION] of a disordered Anderson/Kondo lattice.', 'cond-mat-0302389-2-9-6': '[CITATION] In this approach, each conduction electron site exchanges single particle excitations with an average "cavity" bath, which is in turn self-consistently determined.', 'cond-mat-0302389-2-9-7': 'This treatment becomes exact in the limit of infinite dimensionality and is the natural generalization of the Curie-Weiss mean field theory of magnets to a fermionic system.', 'cond-mat-0302389-2-9-8': 'Its treatment of disorder is equivalent to the well-known coherent potential approximation (CPA).', 'cond-mat-0302389-2-9-9': '[CITATION] In an Anderson lattice description, the localized f-electron is hybridized with its adjacent conduction electron orbital and spatial fluctuations are preserved through the random distribution of hybridization strengths.', 'cond-mat-0302389-2-9-10': 'The local moments are no longer independent since their distribution self-consistently determines the cavity bath.', 'cond-mat-0302389-2-9-11': 'Besides showing that the KDM corresponds to a rigorous limit of a microscopic Hamiltonian, the DMFT enabled the calculation of other properties such as the resistivity[CITATION], dynamic magnetic susceptibility,[CITATION] optical conductivity[CITATION] and magneto-resistance,[CITATION] with good agreement with experiments.', 'cond-mat-0302389-2-9-12': 'The non-Fermi liquid behavior of these quantities hinged on the condition that [MATH] Subsequent experiments of muon spin rotation[CITATION] and NMR in high fields[CITATION] showed some inconsistencies with the KDM/DMFT, suggesting that inter-site correlations, which are absent from that approach, may play a crucial role at the lowest temperatures.', 'cond-mat-0302389-2-9-13': 'However, annealing studies have further emphasized that the consideration of disorder effects is indispensable.', 'cond-mat-0302389-2-9-14': '[CITATION]', 'cond-mat-0302389-2-10-0': 'Despite its success, the KDM/DMFT description suffered from a basic deficiency, which can be ascribed to its extreme sensitivity to the bare disorder distribution.', 'cond-mat-0302389-2-10-1': 'Indeed, the connection between the distribution of [MATH] and the distribution of Kondo temperatures is too rigid and a proper description always relies on fine tuning.', 'cond-mat-0302389-2-10-2': 'In particular, discrete distributions of bare parameters can never generate a [MATH] such that [MATH].', 'cond-mat-0302389-2-10-3': 'Likewise, power law distributions of [MATH] are often necessary and it is not clear how they can be obtained within the KDM/DMFT.', 'cond-mat-0302389-2-11-0': 'More recently, two of us have pointed out that this can be solved through the inclusion of localization effects.', 'cond-mat-0302389-2-11-1': 'From a basic theoretical point of view, the importance of this modification is unquestionable.', 'cond-mat-0302389-2-11-2': 'Disorder scatters the conduction electrons giving rise to spatial fluctuations in their wave function amplitude.', 'cond-mat-0302389-2-11-3': 'These Anderson localization precursor effects in turn give rise to fluctuations of the conduction electron local DOS.', 'cond-mat-0302389-2-11-4': 'The Kondo temperatures are exponential functions of the local DOS and will show a wide distribution for mild disorder strengths, even in the absence of fluctuations in [MATH].[', 'cond-mat-0302389-2-11-5': '[CITATION] Besides, direct experimental determination of the distribution of [MATH] from x-ray absorption fine-structure (XAFS) experiments in UCu[MATH]Pd[MATH] have shown that additional conduction electron disorder is necessary for the interpretation of the results within the KDM/DMFT.[', 'cond-mat-0302389-2-11-6': '[CITATION] Finally, the addition of localization effects has proved to be just the necessary ingredient for the elimination of the extreme sensitivity to the bare disorder and for a much more universal description.', 'cond-mat-0302389-2-12-0': 'The average cavity bath of the DMFT, however, completely neglects DOS fluctuations and a more general treatment is necessary.', 'cond-mat-0302389-2-12-1': 'Progress could be made by means of the statistical dynamical mean field theory (statDMFT), which incorporates the full distribution of the conduction electron local DOS, while keeping the treatment of local correlations already present in the DMFT.', 'cond-mat-0302389-2-12-2': "[CITATION] The treatment involves solving a fully self-consistent loop: the f-electron fluid gives rise to an effective disorder potential for the conduction electrons, while the latter's DOS fluctuations determine the distribution of Kondo temperatures.", 'cond-mat-0302389-2-12-3': 'We enumerate the main conclusions of our analysis:[CITATION]', 'cond-mat-0302389-2-13-0': '### Magnetic Griffiths phase scenario', 'cond-mat-0302389-2-14-0': 'An alternative theoretical scenario for disorder-induced non-Fermi liquid behavior is the magnetic Griffiths phase.', 'cond-mat-0302389-2-14-1': '[CITATION] In the vicinity of magnetic phase transitions disorder fluctuations induce rare regions with an enhanced local critical temperature.', 'cond-mat-0302389-2-14-2': 'These large clusters are ordered on the scale of the correlation length and act as effective spins.', 'cond-mat-0302389-2-14-3': 'Though rare in occurrence they carry a considerable amount of magnetic entropy and the overall effect is the appearance of singular, thermodynamic responses.', 'cond-mat-0302389-2-14-4': 'This magnetic Griffiths phase picture has been advocated as a source of NFL behavior in disordered heavy fermion systems.', 'cond-mat-0302389-2-14-5': '[CITATION] However, very recent results seem to point to several difficulties encountered when this scenario is applied to experimental systems, as follows.', 'cond-mat-0302389-2-15-0': 'The entropy problem.', 'cond-mat-0302389-2-15-1': 'The amount of magnetic entropy observed experimentally in most disordered heavy fermion systems seems much too high to be compatible with the magnetic Griffiths phase picture.', 'cond-mat-0302389-2-15-2': 'Taking the measured specific heat of, say, UCu[MATH]Pd[MATH], we estimate that about 5% of the sample would have to participate in the spin-[MATH] clusters.', 'cond-mat-0302389-2-15-3': 'This implies an average cluster separation of [MATH] lattice constants, ruling out cluster sizes exceeding this distance.', 'cond-mat-0302389-2-15-4': 'These small clusters suggest instead that the "unquenched" localized moments of the KDM/DMFT or the electronic Griffiths phase offer a much more natural explanation.', 'cond-mat-0302389-2-16-0': 'Effects of dissipation.', 'cond-mat-0302389-2-16-1': 'Other important limitations of the magnetic Griffiths phase scheme have also been emphasized in recent work by Millis, Morr, and Schmalian,[CITATION] who have carefully examined the effects of dissipation caused by the metallic bath.', 'cond-mat-0302389-2-16-2': 'This work suggests that the dissipation caused by itinerant electrons is so pronounced that quantum tunneling of even moderately sized magnetic clusters will be suppressed.', 'cond-mat-0302389-2-16-3': 'Although the emergence of a magnetic Griffiths phase is a well established phenomenon in disordered insulating magnets, this result seems to bring into question its relevance to itinerant systems.', 'cond-mat-0302389-2-17-0': '### Spin glass precursors', 'cond-mat-0302389-2-18-0': 'Finally, another possibility is to invoke the proximity to a spin glass quantum phase transition.', 'cond-mat-0302389-2-18-1': 'Several theoretical schemes predicting non-Fermi liquid behavior in the vicinity of a spin-glass quantum critical point have been proposed.[', 'cond-mat-0302389-2-18-2': '[CITATION] Spin glass phases have been identified in the phase diagram of some heavy fermion alloys (UCu[MATH]Pd[MATH], for [MATH])[CITATION] and structurally disordered compounds (URh[MATH]Ge[MATH]).[', 'cond-mat-0302389-2-18-3': '[CITATION] More interestingly, evidence of glassy dynamics in the absence of freezing at very low temperatures has been seen in UCu[MATH]Pd[MATH], [MATH])[CITATION] and Ce(Ru[MATH]Rh[MATH]Si[MATH],[CITATION] with conflicting results pointing to a very low freezing temperature in UCu[MATH]Pd[MATH].[', 'cond-mat-0302389-2-18-4': '[CITATION] These experiments seem to suggest that, if there is a true spin glass transition, freezing temperatures are strongly suppressed in a wide portion of the phase diagram.', 'cond-mat-0302389-2-19-0': '# The model and its solution', 'cond-mat-0302389-2-20-0': '## The statistical dynamical mean field theory', 'cond-mat-0302389-2-21-0': 'A simplified Hamiltonian capable of capturing the essential physics of disordered metals with localized moments is provided by a disordered Anderson lattice[EQUATION] where[EQUATION]', 'cond-mat-0302389-2-21-1': 'In Eqs. ([REF]-[REF]), [MATH]) annihilates a conduction (f-) electron on site [MATH] with spin projection [MATH], [MATH] is the nearest neighbor hopping amplitude, [MATH] is the chemical potential, [MATH] is the f-site Coulomb repulsion, [MATH] is the f-energy level, and we introduce random conduction electron on-site energies ([MATH]) and hybridization matrix elements [MATH].', 'cond-mat-0302389-2-21-2': 'These are chosen from given distributions [MATH] and [MATH] taken to be either square or Gaussian, with width and standard deviation [MATH], respectively.', 'cond-mat-0302389-2-21-3': 'We have also studied discrete cases of [MATH].', 'cond-mat-0302389-2-21-4': 'There is a large degree of uncertainty as to a realistic model of disorder for heavy fermion alloys.', 'cond-mat-0302389-2-21-5': 'A rather thorough study of the local f-site environment in the alloys UCu[MATH]Pd[MATH] and [MATH]) was carried out in Refs. [CITATION] through XAFS experiments.', 'cond-mat-0302389-2-21-6': 'These authors were able to determine the amount of Pd/Cu site interchange as well as the U-Cu bond length distributions.', 'cond-mat-0302389-2-21-7': 'In order to accommodate both types of fluctuations one must, in principle, allow for a distribution of both hybridization strengths and on-site conduction electron energies.', 'cond-mat-0302389-2-21-8': 'On the other hand, when the local moments are randomly replaced by non-magnetic elements (the so-called "Kondo holes"), spatial fluctuations of [MATH] should also be included.', 'cond-mat-0302389-2-21-9': '[CITATION] Throughout the paper, we use the half bandwidth [MATH] as energy unit.', 'cond-mat-0302389-2-21-10': 'For the Bethe lattice (to be introduced later), [MATH] if the coordination number is 3.', 'cond-mat-0302389-2-21-11': 'This should be contrasted to Ref. [CITATION], where [MATH] is measured in units of [MATH].', 'cond-mat-0302389-2-21-12': 'Note also that the f-level energy [MATH] is always measured relative to the chemical potential.', 'cond-mat-0302389-2-21-13': 'This is to ensure that, by making it negative and large enough in absolute value, we always work in the local moment/Kondo regime.', 'cond-mat-0302389-2-22-0': 'We worked within the framework of the statDMFT.', 'cond-mat-0302389-2-22-1': '[CITATION] This treatment is able to incorporate both strong local correlations and Anderson localization effects in a fully self-consistent fashion.', 'cond-mat-0302389-2-22-2': 'Although the method has been described before in the context of the disordered Hubbard model,[CITATION] we will briefly review it with the dual goal of setting the notation and extending it to the disordered Anderson lattice.', 'cond-mat-0302389-2-22-3': 'It starts by focusing on a generic unit cell [MATH] of the lattice, containing an f-site and its adjoining conduction electron Wannier state, and writing its effective action in imaginary time as [EQUATION] where [MATH].', 'cond-mat-0302389-2-22-4': 'In writing Eqs. ([REF]-[REF]), a simplification has been made of retaining only quadratic contributions in fermionic fields after integrating out the other sites (besides the instantaneous Hubbard term), much like the usual dynamical mean field theory.', 'cond-mat-0302389-2-22-5': '[CITATION] The bath (or "cavity") function [MATH] in Eq. ([REF]) is given by[EQUATION] where the sum extends over the [MATH] nearest neighbors and [EQUATION] is the Green\'s function for propagation from nearest neighbor site [MATH] to nearest neighbor site [MATH], calculated with the site [MATH] removed.', 'cond-mat-0302389-2-22-6': 'Integrating out the remaining conduction electron [MATH], we get the effective action of an auxiliary single-impurity Anderson model at each site [MATH][EQUATION] where the hybridization function for the f-site is, in Matsubara frequency space, [EQUATION]', 'cond-mat-0302389-2-22-7': 'The solution of this impurity problem is the major difficulty in this treatment.', 'cond-mat-0302389-2-22-8': 'We implemented two different methods of solution, which will be expanded upon in the next subsections.', 'cond-mat-0302389-2-22-9': "The aim is to calculate the local f-electron Green's function[EQUATION] under the dynamics dictated by ([REF]).", 'cond-mat-0302389-2-22-10': 'It is conveniently parameterized by its self-energy[EQUATION]', 'cond-mat-0302389-2-22-11': "It is also convenient to define a local conduction electron Green's function[EQUATION] such that[EQUATION] where[EQUATION]", 'cond-mat-0302389-2-22-12': 'We note that the [MATH] function describes the local scattering of the conduction electrons off the f-shell at site [MATH], incorporating information about both elastic and inelastic processes.', 'cond-mat-0302389-2-23-0': "All the information a generic site [MATH] has about the rest of the lattice is encoded in the bath function ([REF]), which should be viewed as a functional of the conduction electron lattice Green's function.", 'cond-mat-0302389-2-23-1': 'Since a fully analytical treatment is impossible, we have to solve the equations numerically.', 'cond-mat-0302389-2-23-2': 'For this purpose, we formulated the problem on a Bethe lattice, where things are considerably simplified as explained in Ref. [CITATION].', 'cond-mat-0302389-2-23-3': "In this case, nearest neighbors [MATH] and [MATH] become disconnected once [MATH] is removed and only local Green's functions survive [EQUATION].", 'cond-mat-0302389-2-23-4': 'Finally, these last objects can be computed from an action at site [MATH] in almost all aspects identical to ([REF]-[REF]), the only difference now being that the bath function sum runs over the [MATH] nearest neighbors (here labeled by [MATH]) only[EQUATION]', 'cond-mat-0302389-2-23-5': 'Note that, on the right-hand side, we do not need to specify that site [MATH] has been excluded as the removal of [MATH] completely disconnects sites labeled by [MATH] from site [MATH] (this property is specific to the Bethe lattice).', 'cond-mat-0302389-2-23-6': 'The reappearance of [MATH], whose distribution is identical to that of [MATH] since all sites are equivalent, closes the loop and establishes a recursive set of stochastic equations.', 'cond-mat-0302389-2-23-7': 'When the interaction is turned off ([MATH]), this treatment reduces to the well-known self-consistent theory of localization,[CITATION] here generalized to a two-band lattice.', 'cond-mat-0302389-2-23-8': 'When we take the coordination to infinity [MATH] keeping [MATH], our treatment reduces to the DMFT of correlations and disorder.', 'cond-mat-0302389-2-23-9': '[CITATION] In the latter case, the disorder treatment is equivalent to the CPA,[CITATION] which has no localization transition.', 'cond-mat-0302389-2-24-0': 'A full solution of Eqs. ([REF]-[REF]) for given distributions [MATH] and/or [MATH] involves solving an ensemble of impurity problems self-consistently.', 'cond-mat-0302389-2-24-1': 'Physically, the conduction electrons propagate through a disordered lattice and scatter off conduction site potential fluctuations as well as f-site resonances.', 'cond-mat-0302389-2-24-2': 'These resonances, in turn, describe the formation of localized moments, whose local Kondo temperatures fluctuate as well, reflecting a disordered conduction sea environment.', 'cond-mat-0302389-2-24-3': 'Complete statistical information can in principle be obtained from the distributions of the various renormalized local quantities.', 'cond-mat-0302389-2-24-4': 'We stress that a random distribution of any bare parameter causes all renormalized quantities to fluctuate as a result of the self-consistent nature of our treatment.', 'cond-mat-0302389-2-24-5': 'Therefore, even if we include only fluctuations in the conduction sea through [MATH], a distribution of Kondo temperatures ensues.', 'cond-mat-0302389-2-24-6': '[CITATION] This is easily seen from the approximate formula for the Kondo temperature in the Kondo limit ([MATH])[CITATION][EQUATION] where [MATH] is the local density of states (DOS) seen by the f-site[EQUATION] and [MATH] is the Kondo coupling constant, given in the Kondo limit by[EQUATION]', 'cond-mat-0302389-2-24-7': 'Even if [MATH] is not random, the local DOS is, because of the denominator in Eq. ([REF]).', 'cond-mat-0302389-2-24-8': 'As a result of the strong exponential dependence in ([REF]), even mild localization effects can be strongly enhanced and should be seriously considered, specially in disordered heavy fermion systems.', 'cond-mat-0302389-2-25-0': '## The impurity solvers', 'cond-mat-0302389-2-26-0': 'An important part of the method we employed is the solution of the impurity problems posed by the ensemble of effective actions given by Eq. ([REF]) and its counterpart for a site with one nearest neighbor removed.', 'cond-mat-0302389-2-26-1': 'We concentrated mostly on two methods of solution, which we now briefly describe: the slave boson, large-N based, mean field theory and second order perturbation theory in [MATH].', 'cond-mat-0302389-2-26-2': 'In order to unclutter the notation, we drop in the next subsections the site index [MATH] and the superscript [MATH].', 'cond-mat-0302389-2-26-3': 'Details of the numerical treatment are given in the Appendix.', 'cond-mat-0302389-2-27-0': '### Slave boson mean field theory', 'cond-mat-0302389-2-28-0': 'This method gives a good description of the low temperature, Fermi liquid regime of the Anderson impurity problem in the limit [MATH] and is extensively covered in the literature.', 'cond-mat-0302389-2-28-1': '[CITATION] Its main advantage is the ability to capture the zero temperature fixed point correctly as well as the exponential nature of the low energy scale.', 'cond-mat-0302389-2-28-2': 'Its treatment of the self-energy, however, does not incorporate inelastic processes to leading order.', 'cond-mat-0302389-2-28-3': 'Besides, it has a spurious phase transition at a finite temperature, where in reality there should be only a smooth crossover.', 'cond-mat-0302389-2-28-4': 'For these reasons, we confine it to the zero temperature limit, where it is a useful guide.', 'cond-mat-0302389-2-28-5': 'As applied to our problem, the method has been described in Appendix D of Ref. [CITATION] and we will merely state the results, generalized to the Matsubara frequency axis and at [MATH].', 'cond-mat-0302389-2-28-6': "The local f-electron Green's function is given by[EQUATION] where the last equality defines the local f-electron quasi-particle Green's function and the variational parameters [MATH] (renormalized f-energy) and [MATH] (quasi-particle residue) are determined from the solution of the set of equations [EQUATION]", 'cond-mat-0302389-2-28-7': 'Using [EQUATION]', 'cond-mat-0302389-2-28-8': 'Eqs. ([REF]-[REF]) simplify to[EQUATION]', 'cond-mat-0302389-2-28-9': 'Eq. ([REF]) becomes in this approximation[EQUATION]', 'cond-mat-0302389-2-29-0': '### Second order perturbation theory', 'cond-mat-0302389-2-30-0': 'The perturbative solution of the single-impurity Anderson model with particle-hole symmetry was thoroughly analyzed by Yamada and Yosida.[', 'cond-mat-0302389-2-30-1': '[CITATION] The series expansion in [MATH] for physical quantities such as the specific heat and the spin susceptibility converges very fast and even second order results can be useful.[', 'cond-mat-0302389-2-30-2': '[CITATION] Extension of the perturbative treatment to the case without particle-hole symmetry poses considerable difficulties.', 'cond-mat-0302389-2-30-3': 'A particularly useful proposal is the use of an interpolative self-energy which recovers the atomic ([MATH]) and high frequency limits.[', 'cond-mat-0302389-2-30-4': '[CITATION] Further improvements of the method were later suggested.[', 'cond-mat-0302389-2-31-0': "The procedure consists in defining an unperturbed f-electron Green's function[EQUATION] with a new parameter [MATH] to be determined later, which vanishes at particle-hole symmetry.", 'cond-mat-0302389-2-31-1': "The interacting Green's function is given in ([REF]).", 'cond-mat-0302389-2-31-2': 'The interpolative self-energy is[CITATION][EQUATION] where[EQUATION] and[EQUATION]', 'cond-mat-0302389-2-31-3': "The last equation ([REF]) is the usual second order diagram using the unperturbed Green's function ([REF]) for the internal lines.", 'cond-mat-0302389-2-31-4': 'The parameters [MATH] and [MATH] are determined by imposing the high frequency and atomic limits, respectively, and are given by[CITATION][EQUATION] where [EQUATION]', 'cond-mat-0302389-2-31-5': 'Different schemes have been proposed in order to fix the free parameter [MATH].[', 'cond-mat-0302389-2-31-6': '[CITATION] At zero temperature, one can ensure that the low energy Fermi liquid behavior is obtained by imposing the Friedel sum rule.[', 'cond-mat-0302389-2-31-7': '[CITATION] This procedure cannot be easily generalized to finite temperatures, however.', 'cond-mat-0302389-2-31-8': 'One option is to fix [MATH] at its zero temperature value even at finite temperatures.', 'cond-mat-0302389-2-31-9': 'Alternatively, one can require at any temperature[CITATION][EQUATION] which makes [MATH].', 'cond-mat-0302389-2-31-10': 'Finally, a third possibility is imposing [MATH].[', 'cond-mat-0302389-2-31-11': '[CITATION] These three alternatives have been rather carefully compared in Ref. potthoffetal at [MATH] and checked against exact diagonalization.', 'cond-mat-0302389-2-31-12': 'The first two methods were shown to be almost equivalent whereas the third one is inferior.', 'cond-mat-0302389-2-31-13': 'Moreover, comparisons at finite temperatures with Quantum Monte Carlo results confirmed the adequacy of imposing Eq. ([REF]).[', 'cond-mat-0302389-2-31-14': '[CITATION] Specific applications to a clean Anderson lattice model further corroborated this conclusion.[', 'cond-mat-0302389-2-31-15': '[CITATION] Thus, our results were based on imposing condition ([REF]).', 'cond-mat-0302389-2-31-16': 'It should be remembered, however, that the perturbative solution predicts a characteristic energy scale that is quantitatively incorrect at large values of [MATH], since it is unable to capture the correct exponential dependence.', 'cond-mat-0302389-2-31-17': 'Nevertheless, for moderate interactions, it still gives reasonable results.', 'cond-mat-0302389-2-31-18': 'Within its limitations, this perturbative scheme is a relatively flexible low-cost tool to tackle the impurity problem with the great advantage of being able to naturally account for inelastic processes.', 'cond-mat-0302389-2-32-0': 'We note that a direct comparison between the slave boson mean field theory results and second order perturbation theory is not possible because the former is limited to the [MATH] limit, which is obviously outside the region of validity of the latter.', 'cond-mat-0302389-2-32-1': 'The main interest of an analysis of both methods, however, resides in the exploration of the importance of inelastic processes, which are absent in the slave boson mean field treatment.', 'cond-mat-0302389-2-33-0': '# Slave boson mean field theory results', 'cond-mat-0302389-2-34-0': 'We now present the results obtained at [MATH] using the slave boson mean field theory as an impurity solver.', 'cond-mat-0302389-2-34-1': 'Most of our results were obtained for a uniform distribution of on-site conduction electron energies [EQUATION].', 'cond-mat-0302389-2-34-2': 'In Section [REF] we also show results for a discrete distribution of hybridization strengths [MATH].', 'cond-mat-0302389-2-35-0': '## Conduction electron typical density of states', 'cond-mat-0302389-2-36-0': 'To understand the overall behavior as a function of disorder, it is instructive to consider the transport properties of the conduction electrons.', 'cond-mat-0302389-2-36-1': 'Since there are no interactions among them in our model, their behavior is that of a disordered non-interacting electron system.', 'cond-mat-0302389-2-36-2': 'There are two sources of disorder, as can be seen in Eq. ([REF]): fluctuations of the local on-site energies [MATH] and of the f-shell resonances described by [MATH].', 'cond-mat-0302389-2-36-3': 'They are not independent, however, since they are inextricably tied by self-consistency.', 'cond-mat-0302389-2-36-4': 'Their combined effect acts to decrease the conduction electron mobility.', 'cond-mat-0302389-2-37-0': '### Typical density of states:', 'cond-mat-0302389-2-38-0': 'an order parameter for localization', 'cond-mat-0302389-2-39-0': 'A useful measure of this mobility is given by the typical value of the local escape rate.', 'cond-mat-0302389-2-39-1': "This is encoded in the imaginary part of the local conduction electron Green's function (the local DOS) at zero frequency, [MATH].", 'cond-mat-0302389-2-39-2': 'We will, from now on, drop the superscript denoting the removal of a nearest neighbor so as to lighten the notation.', 'cond-mat-0302389-2-39-3': 'As shown originally by Anderson, the typical value of the local DOS vanishes when the electrons are localized and can be viewed as an order parameter for the localization transition.', 'cond-mat-0302389-2-39-4': '[CITATION] A convenient way of accessing the typical value is furnished by the geometric average[EQUATION] where the overbar denotes a disorder average.', 'cond-mat-0302389-2-39-5': 'By contrast, the arithmetic average[EQUATION] is finite at the transition.', 'cond-mat-0302389-2-39-6': 'A thorough analysis of the critical behavior of the local DOS distribution in the non-interacting Bethe lattice localization problem was carried out in Ref. [CITATION].', 'cond-mat-0302389-2-40-0': 'In Fig. [REF] we show the typical conduction electron DOS as a function of disorder for several values of the chemical potential.', 'cond-mat-0302389-2-41-0': '### Proximity to the Kondo insulator', 'cond-mat-0302389-2-42-0': 'We can identify three qualitatively distinct behaviors.', 'cond-mat-0302389-2-42-1': '[CITATION] For [MATH] and for [MATH], [MATH] is a monotonically decreasing function of disorder.', 'cond-mat-0302389-2-42-2': 'For [MATH] the clean system is a Kondo insulator[CITATION] and [MATH].', 'cond-mat-0302389-2-42-3': 'As disorder is introduced in the Kondo insulator, [MATH] initially increases, reaching a maximum at about [MATH], after which it decreases monotonically.', 'cond-mat-0302389-2-42-4': 'Finally, for [MATH] and [MATH] initially decreases, passes through a minimum around [MATH], then increases up to a maximum at about [MATH] and eventually becomes monotonically decreasing.', 'cond-mat-0302389-2-42-5': 'For all values of [MATH], the typical DOS vanishes at a disorder-induced metal insulator transition (MIT) at [MATH].', 'cond-mat-0302389-2-43-0': 'These distinct behaviors can be traced back to how close the clean system is to the Kondo insulator point[CITATION] at [MATH].', 'cond-mat-0302389-2-43-1': '[CITATION] If we start from the clean insulator, the introduction of disorder acts to create states inside the gap, thus increasing the DOS at the chemical potential.', 'cond-mat-0302389-2-43-2': 'This increase continues until the gap is essentially washed out and the system becomes a bad metal.', 'cond-mat-0302389-2-43-3': 'After that, localization intervenes and [MATH] starts to decrease towards the MIT.', 'cond-mat-0302389-2-44-0': 'For fillings close to but not at the Kondo insulator point, the clean system is a heavy fermion metal at [MATH].', 'cond-mat-0302389-2-44-1': 'The f-resonances ([MATH]) coherently scatter the conduction electrons creating a strongly renormalized Fermi liquid.', 'cond-mat-0302389-2-44-2': 'In the slave boson treatment, [MATH] diverges at [MATH], see Eq. ([REF]), corresponding to the limit of unitary scattering, with a maximally allowed phase shift [MATH].', 'cond-mat-0302389-2-44-3': 'We can view its value at the Fermi level [MATH] (which is real at [MATH] as an effective potential coming from the f-electrons.', 'cond-mat-0302389-2-44-4': 'The closer the system is to the Kondo insulator, the larger the value of [MATH], the insulator being signaled by the divergence of this quantity (or equivalently by [MATH]).', 'cond-mat-0302389-2-44-5': 'The effect of disorder is to immediately start generating spatial fluctuations of the f-resonances, with different phase shift values at the chemical potential.', 'cond-mat-0302389-2-44-6': 'Proximity to the insulator implies large, random, almost unitary scattering potentials.', 'cond-mat-0302389-2-44-7': 'As a result, metallic coherence is efficiently destroyed and the typical conduction electron DOS is strongly suppressed.[', 'cond-mat-0302389-2-44-8': '[CITATION] The important role played by the unitary scatterers was emphasized in Refs. mirandavlad2 and mirandavlad1, where the distribution [MATH] was directly computed and its weight at [MATH] was shown to correlate with the destruction of coherence.', 'cond-mat-0302389-2-45-0': 'There is another equivalent way of understanding these effects.', 'cond-mat-0302389-2-45-1': 'For small dopings away from the Kondo insulator, carriers are introduced at the edges of the valence or conduction bands defined by the Kondo insulator gap, which have a small DOS (in the Bethe lattice, band edges have a square root shape as in three dimensions).', 'cond-mat-0302389-2-45-2': 'As has been known for a long time, a region of small DOS is particularly sensitive to localization effects introduced by disorder.', 'cond-mat-0302389-2-46-0': 'As in the previous case, further increase of disorder acts to wash out the nearby Kondo pseudo-gap and the behavior then becomes very similar to the disordered Kondo insulator.', 'cond-mat-0302389-2-46-1': 'We thus have a region with a rather non-intuitive increasing [MATH], which can be ascribed to the proximity to the Kondo insulator fixed point.', 'cond-mat-0302389-2-46-2': 'The behavior at fillings well away from the Kondo insulator is much less influenced by the pseudo-gap, see Fig. [REF] for [MATH].', 'cond-mat-0302389-2-46-3': 'Although there is a rapid initial decrease of [MATH], followed by a much slower dependence, the typical DOS does not exhibit the unconventional increase with disorder observed at other fillings.', 'cond-mat-0302389-2-47-0': '### Role of the hybridization strength', 'cond-mat-0302389-2-48-0': 'It is interesting to note that the critical value of disorder for the MIT [MATH] depends on the hybridization strength.', 'cond-mat-0302389-2-48-1': 'In Fig. [REF], we show the disorder dependence of the typical conduction electron DOS for different values of [MATH].', 'cond-mat-0302389-2-48-2': 'There is hardly any change in [MATH] as we go from [MATH] to [MATH] (cf. also Fig. [REF]).', 'cond-mat-0302389-2-48-3': 'However, for [MATH], the critical disorder strength is clearly enhanced.', 'cond-mat-0302389-2-48-4': 'This figure also illustrates the non-trivial nature of the self-consistency.', 'cond-mat-0302389-2-48-5': 'Indeed, the two types of disorder coming from fluctuations in [MATH] and [MATH] are clearly not independent, since the addition of f-site disorder as we turn on [MATH] from [MATH] to [MATH] acts to increase the mobility for [MATH].', 'cond-mat-0302389-2-48-6': 'The self-consistently determined solutions of the impurity problems effectively help screen the conduction electron disorder.', 'cond-mat-0302389-2-48-7': 'Note also how an increased value of [MATH] pushes the "dip-hump" structure to higher values of disorder.', 'cond-mat-0302389-2-48-8': 'Since the Kondo insulating gap increases with the hybridization strength, this is consistent with our explanation for the nature of this non-monotonic behavior.', 'cond-mat-0302389-2-49-0': '## Distribution of the conduction electron local density of states', 'cond-mat-0302389-2-50-0': 'One of the great advantages of the present approach is the possibility of monitoring complete distribution functions.', 'cond-mat-0302389-2-50-1': 'Many of the features exhibited in Figs. [REF] and [REF] can be directly read off the distribution of [MATH].', 'cond-mat-0302389-2-50-2': 'We show this by plotting [MATH] for several disorder strengths and different chemical potential values in Fig. [REF] (we use "log" for the base 10 logarithm).', 'cond-mat-0302389-2-50-3': 'It follows from the definition of [MATH], Eq. ([REF]), that it is obtained by raising 10 to the power of the average of this distribution (we use powers of 10 for ease of computation).', 'cond-mat-0302389-2-50-4': 'For weak disorder in the metallic cases (Figs. [REF]a and b), [MATH] is approximately Gaussian, even though the bare disorder is uniform, a feature shared by several physical quantities.', 'cond-mat-0302389-2-50-5': '[CITATION] This is due to the presence of correlations between many distant lattice sites mediated by the extended conduction electron wave function, which introduces a sort of averaging effect.', 'cond-mat-0302389-2-50-6': 'In the Kondo insulator case (Fig. [REF]c), however, the distribution is not Gaussian at weak disorder.', 'cond-mat-0302389-2-50-7': 'Keeping in mind that [MATH]) in the clean Kondo insulator, it is clear that the introduction of weak disorder has to generate weight at very small [MATH].', 'cond-mat-0302389-2-50-8': 'Indeed, [MATH] shows a divergence at [MATH] for [MATH].', 'cond-mat-0302389-2-51-0': 'For large values of [MATH] the distribution becomes extremely broad, spanning many orders of magnitude.', 'cond-mat-0302389-2-51-1': 'In the case [MATH] (Fig. [REF]a), corresponding to a system well away from the Kondo insulator filling, the distribution broadens and its maximum steadily shifts towards lower values as disorder is increased.', 'cond-mat-0302389-2-51-2': 'This is to be expected from the monotonic behavior of [MATH].', 'cond-mat-0302389-2-51-3': 'Likewise, at [MATH], the non-monotonic behavior of the typical value is also clearly reflected in [MATH] (see Fig. [REF]b and compare it to Fig. [REF]a).', 'cond-mat-0302389-2-52-0': 'As we saw, at the Kondo insulating chemical potential [MATH] and for [MATH], the distribution shows a divergence at [MATH].', 'cond-mat-0302389-2-52-1': 'A similar diverging tendency is observed at [MATH] (Fig. [REF]b) and [MATH].', 'cond-mat-0302389-2-52-2': 'This is precisely the disorder value where the minimum of [MATH] occurs (cf. Fig. [REF]a) and which we have been ascribing to the presence of many unitary scatterers due to the nearby Kondo insulator.', 'cond-mat-0302389-2-52-3': 'The similarity between the two distributions strengthens further our case for the importance of the proximity to the Kondo insulator.', 'cond-mat-0302389-2-52-4': 'Additionally and consistent with this, the divergence is totally absent at [MATH], where the role played by the Kondo insulator fixed point is much less important.', 'cond-mat-0302389-2-53-0': 'It is also interesting to observe in Fig. [REF]c, how the Kondo gap is washed out by disorder: at [MATH], where [MATH] peaks (Fig. [REF]a), most of the weight of the distribution is already at sizeable values of the DOS and its shape is very similar to the metallic cases.', 'cond-mat-0302389-2-54-0': '## Distribution of Kondo temperatures', 'cond-mat-0302389-2-55-0': 'We now proceed to the analysis of the physical properties related to the ensemble of impurity problems.', 'cond-mat-0302389-2-55-1': 'As shown before[CITATION] the distribution of Kondo temperatures of the various f-sites is log-normal for weak disorder, but broadens and acquires a power-law shape at intermediate values of [MATH].', 'cond-mat-0302389-2-55-2': 'Once this power-law becomes singular enough, a Griffiths phase is entered with diverging thermodynamic responses.', 'cond-mat-0302389-2-55-3': '[CITATION]', 'cond-mat-0302389-2-56-0': '### Universality at weak disorder', 'cond-mat-0302389-2-57-0': 'We have noticed that for weak disorder, the shape of the distribution of various quantities, including the Kondo temperature, is universal, irrespective of the shape of the bare distribution of disorder.', 'cond-mat-0302389-2-58-0': 'A nice illustration of this effect is given by the case where the bare disorder is a discrete distribution.', 'cond-mat-0302389-2-58-1': 'As an example, we take the discrete distribution of hybridization strengths, [MATH], determined in Ref. [CITATION] from XAFS measurements in UCu[MATH]Pd[MATH].', 'cond-mat-0302389-2-58-2': 'The resulting distribution of [MATH] is shown in Fig. [REF]a.', 'cond-mat-0302389-2-58-3': 'It is continuous and has a log-normal shape.', 'cond-mat-0302389-2-58-4': 'In Fig. [REF]b we show the distribution of [MATH] (smooth dot-dashed line) and compare it to the discrete distribution obtained in the DMFT (vertical solid lines), which corresponds to the limit of infinite coordination.', 'cond-mat-0302389-2-58-5': 'We also include in the figure the results of the Kondo disorder model[CITATION] (KDM) (vertical dashed lines), which is very similar to the DMFT.', 'cond-mat-0302389-2-58-6': 'The only differences between the KDM and the DMFT are that, in the former, no self-consistency is imposed and a Kondo instead of an Anderson lattice model is used.', 'cond-mat-0302389-2-58-7': 'The difference between the results of the KDM/DMFT and the statDMFT is striking.', 'cond-mat-0302389-2-58-8': 'The fluctuations of the conduction electron wave functions incorporated in the statDMFT smooth out the discrete results of the DMFT into a universal continuous form.', 'cond-mat-0302389-2-58-9': 'A description of the NFL behavior within the KDM/DMFT theory would be clearly impossible.', 'cond-mat-0302389-2-58-10': 'This comparison also shows that this level of hybridization disorder alone is not able to generate non-Fermi liquid behavior even in statDMFT, since the distribution of Kondo temperatures goes to zero as [MATH], a point that was stressed in Refs. [CITATION] However, if disorder in the conduction electron sites, [MATH], is also included a singular behavior can be obtained (not shown).', 'cond-mat-0302389-2-58-11': 'The inclusion of conduction electron disorder is reasonable in UCu[MATH]Pd[MATH], since the Cu-Pd interchange affects both [MATH] and [MATH].', 'cond-mat-0302389-2-59-0': '### Emergence of the Electronic Griffiths phase', 'cond-mat-0302389-2-60-0': 'In order to identify the emergence of the Griffiths phase, we next study the evolution of the distribution of Kondo temperatures as the width [MATH] of [MATH] is varied (with no disorder in [MATH]).', 'cond-mat-0302389-2-60-1': 'Typical results are shown in Fig. [REF], corresponding to [MATH].', 'cond-mat-0302389-2-60-2': 'As the disorder increases, we find that the overall width, but most significantly, the size of the low-[MATH] tail rapidly grows.', 'cond-mat-0302389-2-60-3': 'These tails assume a power-law form [MATH], with the power [MATH] being a monotonically decreasing function of disorder.', 'cond-mat-0302389-2-60-4': 'Once again, this behavior cannot be obtained in the rigid scheme of the KDM/DMFT without unjustified fine tuning.', 'cond-mat-0302389-2-60-5': 'The thermodynamic response of the system assumes a singular, non-Fermi liquid form as soon as [MATH], which happens for sufficiently strong disorder [MATH].', 'cond-mat-0302389-2-60-6': 'Since this behavior does not reflect any thermodynamic phase transition, it assumes the character of an electronic Griffiths phase.', 'cond-mat-0302389-2-60-7': 'Here, singular behavior emerges due to the presence of exponentially rare events (in our case Kondo spins) which nevertheless provide an exponentially large contribution to thermodynamic and transport properties and thus dominate the macroscopic behavior of the system.', 'cond-mat-0302389-2-60-8': 'The [MATH]-dependence of the exponent [MATH] can be easily obtained by fitting the tails of these distributions; a representative behavior for several values of the chemical potential is shown in Fig. [REF].', 'cond-mat-0302389-2-61-0': '### NFL and the proximity to the Kondo insulator', 'cond-mat-0302389-2-62-0': 'In the previous section we have seen how the proximity of the Kondo insulator plays a crucial role in determining the disorder dependence of the localization and transport properties, and leads to the surprising "bad metal" behavior for a wide parameter range.', 'cond-mat-0302389-2-62-1': 'This was argued to reflect the enhanced sensitivity to disorder of those electronic states that are very close to the Kondo insulator, and which are most easily affected by localization effects.', 'cond-mat-0302389-2-62-2': 'We have also established that NFL behavior also emerges as a result of disorder-induced density of states fluctuations.', 'cond-mat-0302389-2-62-3': 'It is then natural to ask how sensitive this emergence of NFL behavior is to the proximity to the Kondo insulator, which in the clean limit emerges only in a narrow parameter range, close (in our case) to [MATH].', 'cond-mat-0302389-2-63-0': 'To address this question, we have systematically investigated the evolution of [MATH] as a function of the distance to the Kondo insulator, i.e. as a function of the chemical potential [MATH].', 'cond-mat-0302389-2-63-1': 'The behavior for weak disorder ([MATH]) is shown in Fig. [REF].', 'cond-mat-0302389-2-63-2': 'Despite what one would naively expect, these result clearly demonstrate that the distributions are the broadest far from the Kondo insulator.', 'cond-mat-0302389-2-63-3': 'As we can see on this figure, the distributions narrow down as the Kondo insulator is approached from either side.', 'cond-mat-0302389-2-63-4': 'As a result, we may expect that the critical disorder strength [MATH] necessary for the emergence of NFL behavior should increase closer to the Kondo insulator.', 'cond-mat-0302389-2-63-5': 'This surprising result is confirmed by examining the [MATH]-dependence of the exponent [MATH] as shown in Fig. [REF].', 'cond-mat-0302389-2-63-6': 'As we can see there, for a given [MATH], the exponent [MATH] is indeed smaller, and [MATH] decreases for larger [MATH] (far from the Kondo insulator).', 'cond-mat-0302389-2-64-0': 'At first sight, these findings seem in contradiction to what one may expect, since we have found that the typical density of states decreases close to the Kondo insulator.', 'cond-mat-0302389-2-64-1': 'Naively, one could then expect the Kondo temperatures to be depressed as well, leading to broader distributions and enhanced NFL behavior.', 'cond-mat-0302389-2-64-2': 'On the other hand, we know that the Kondo temperature remains finite within the Kondo insulator, despite the fact that the density of states at the Fermi energy vanishes there.', 'cond-mat-0302389-2-64-3': 'Although surprising at first sight, this curious feature of Kondo insulators is at present well understood.', 'cond-mat-0302389-2-64-4': 'It is called the "strong coupling Kondo effect".', 'cond-mat-0302389-2-64-5': '[CITATION] It reflects the fact that the Kondo screening is not determined only by electronic states precisely at the Fermi energy, but also by all the states in an energy interval of order [MATH] around the Fermi energy.', 'cond-mat-0302389-2-64-6': 'Indeed, the average value of the Kondo temperature (see Fig. [REF]) is the highest precisely near the Kondo insulator.', 'cond-mat-0302389-2-64-7': 'In the presence of disorder, the value of the Kondo temperature is determined by a certain weighted average of the density of states over this extended energy interval.', 'cond-mat-0302389-2-64-8': 'When localization is present, only the states closest to the Kondo gap band edge will be appreciably affected, but since not only those states determine [MATH], the net effect is washed away.', 'cond-mat-0302389-2-64-9': 'We thus conclude that the proximity to the Kondo insulator, in contrast to transport, does not have appreciable effect on the emergence of the NFL behavior.', 'cond-mat-0302389-2-64-10': 'Indeed, the critical value of disorder [MATH] required for the emergence of NFL behavior is found to have a remarkably weak [MATH] dependence.', 'cond-mat-0302389-2-65-0': '### Universality at strong disorder', 'cond-mat-0302389-2-66-0': 'At strong disorder we expect the density of states fluctuations to completely wash out any trace of the Kondo gap, and in addition to broaden the conduction band, making it very flat and featureless.', 'cond-mat-0302389-2-66-1': 'As a result, all quantities are expected to have an extremely weak [MATH] dependence, leading to a more universal behavior of all quantities.', 'cond-mat-0302389-2-66-2': 'Such behavior is indeed seen at sufficiently strong disorder, where the typical DOS curves (Fig. [REF]) are seen to merge around [MATH].', 'cond-mat-0302389-2-66-3': "Similar behavior is seen in Fig. [REF], which shows [MATH] for different [MATH]'s at [MATH].", 'cond-mat-0302389-2-66-4': 'This universal behavior is even more striking if this distribution is plotted on a log-log scale (Fig. [REF]), where an almost perfect power-law tail ([MATH]) is seen to span several decades for all the values of [MATH].', 'cond-mat-0302389-2-66-5': 'This is a remarkable example of universality generated by DOS fluctuations, completely absent in the KDM/DMFT treatment.', 'cond-mat-0302389-2-67-0': '## Distribution of the hybridization function', 'cond-mat-0302389-2-68-0': 'A key input to the determination of the Kondo temperature is the hybridization function [MATH] of Eq. ([REF]).', 'cond-mat-0302389-2-68-1': 'We show in Fig. [REF]a the distribution of its imaginary part calculated at the chemical potential ([MATH]).', 'cond-mat-0302389-2-68-2': 'Note that, for a featureless bath, it appears in the exponential of the Kondo temperature formula, Eq. ([REF]), which is thus very sensitive to it.', 'cond-mat-0302389-2-68-3': 'It can be seen in Fig. [REF]a that its distribution is very regular and retains a bell-shaped structure for any [MATH].', 'cond-mat-0302389-2-68-4': 'It also inherits the non-monotonic behavior observed in the conduction electron local DOS (cf. Fig. [REF]b).', 'cond-mat-0302389-2-69-0': 'It is tempting to try to calculate the distribution of Kondo temperatures from the distribution of [MATH], by naively applying the Kondo temperature formula, Eq. ([REF]).', 'cond-mat-0302389-2-69-1': 'This procedure fails, however: the actual distribution of [MATH] has much lower weight at small [MATH] values than is predicted by the Kondo temperature formula.', 'cond-mat-0302389-2-69-2': 'The explanation for this failure lies in the fact, already alluded to before, that [MATH] is determined by a weighted average of [MATH] over a region around the Fermi level.', 'cond-mat-0302389-2-69-3': 'This can be glimpsed from the strong frequency dependence of the typical (geometric average) hybridization function close to the chemical potential, as shown, for frequencies on the real axis, in Fig. [REF]b.', 'cond-mat-0302389-2-69-4': 'It shows a robust and well-defined pseudo-gap, inherited from the nearby Kondo insulator, and a tiny narrow peak at the chemical potential.', 'cond-mat-0302389-2-69-5': 'This narrow peak is easy to understand: spatial fluctuations due to disorder give rise to narrow peaks within the pseudo-gap, most typically at the chemical potential.', 'cond-mat-0302389-2-69-6': 'However, as we have remarked before, the Kondo temperature can be finite even if the density of states is zero or almost zero at the chemical potential.', 'cond-mat-0302389-2-69-7': '[CITATION] In this case, the spectral weight right at the chemical potential is unimportant for the determination of the Kondo temperature.', 'cond-mat-0302389-2-69-8': 'It is dominated by a whole range of spectral density away from the Fermi level.', 'cond-mat-0302389-2-69-9': 'Since far from the pseudo-gap region the density of states is much larger and hence much less affected by the spatial fluctuations, the distribution of Kondo temperatures is narrower than one might guess based on the distribution of [MATH] and the Kondo temperature formula.', 'cond-mat-0302389-2-70-0': 'When considered together, the results of Section [REF] show the importance of a self-consistent solution of the problem, with a non-trivial interplay between spatial fluctuations due to localization and strong correlation effects.', 'cond-mat-0302389-2-70-1': 'However, an important feature that is missed in the slave boson treatment of the impurity problems is the presence of inelastic scattering.', 'cond-mat-0302389-2-70-2': 'This will be considered in the next section, where we show the results obtained with perturbation theory in the interaction.', 'cond-mat-0302389-2-71-0': '# Perturbation theory results', 'cond-mat-0302389-2-72-0': 'We consider now the results obtained at finite [MATH] using second order perturbation theory as the impurity solver.', 'cond-mat-0302389-2-73-0': '## Results for a fixed temperature', 'cond-mat-0302389-2-74-0': 'Fig. [REF] shows the results for the conduction electron typical DOS near the Fermi level as a function of the disorder parameter [MATH] for different values of the interaction energy [MATH] and the hybridization [MATH] at [MATH].', 'cond-mat-0302389-2-74-1': 'In order to understand them, we looked at scatter plots of realizations of the local effective f-shell potential (real and imaginary parts of [MATH]) and the corresponding bare potential [MATH] at the same site, both seen by the conduction electrons.', 'cond-mat-0302389-2-74-2': 'These results are presented for two different values of the hybridization, [MATH] and [MATH], and two different values of the disorder parameter, [MATH] and [MATH], in Figs. [REF] to [REF].', 'cond-mat-0302389-2-74-3': 'The other parameters used in the calculation were [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0302389-2-74-4': 'The figures present the results obtained in the statDMFT calculation (represented by dots) as well as the results of DMFT (full line), in which the disorder treatment reduces to CPA.', 'cond-mat-0302389-2-74-5': '[CITATION] It is important to notice that we have [MATH] (as [MATH] at finite [MATH]), implying the presence of inelastic scattering, a feature absent in the slave boson treatment of the last section.', 'cond-mat-0302389-2-74-6': 'Besides, the imaginary part of the self-energy gets folded into the real part of [MATH] as well.', 'cond-mat-0302389-2-74-7': 'This is a peculiar feature of a two-band model, where the effective conduction electron self-energy represented by [MATH] has a real part for which inelastic processes contribute.', 'cond-mat-0302389-2-74-8': 'Thus, we should keep in mind that, even though in an effective description of conduction electron processes [MATH] and [MATH] are associated with elastic scattering while [MATH] is related to inelastic processes, both [MATH] and [MATH] contain information on f-electron collisions.', 'cond-mat-0302389-2-75-0': 'Let us compare the results for [MATH] for both [MATH] and [MATH], which are in Figs. [REF] and [REF].', 'cond-mat-0302389-2-75-1': 'The first observation we make is that for [MATH] the values of [MATH] are mainly concentrated around zero, while for [MATH] they are distributed in a wider range of values.', 'cond-mat-0302389-2-75-2': 'On the other hand, the results for [MATH] show that the inelastic scattering is stronger for [MATH] than for [MATH].', 'cond-mat-0302389-2-75-3': 'Concerning the results for [MATH], the great concentration of sites with large [MATH] explains the great decrease in the typical DOS for low disorder shown in Fig. [REF].', 'cond-mat-0302389-2-75-4': 'These sites act as almost unitary scatterers, which give rise to a maximally allowed scattering phase shift ([MATH]) for the conduction electrons, and represent droplets of Kondo insulator within the metal, in close parallel to the slave boson results discussed of Section [REF].', 'cond-mat-0302389-2-75-5': '[CITATION] For [MATH], although the inelastic scattering is stronger, it cannot compensate for the much narrower distribution of [MATH].', 'cond-mat-0302389-2-75-6': 'Thus, the DOS does not decrease as fast as for [MATH].', 'cond-mat-0302389-2-76-0': 'As the disorder increases to intermediate values ([MATH]), the distribution of [MATH] becomes larger, causing a steady decrease in the typical DOS for [MATH].', 'cond-mat-0302389-2-76-1': 'However, for [MATH], the typical DOS presents a non-monotonic behavior similar to the slave boson results for [MATH] and [MATH].', 'cond-mat-0302389-2-76-2': 'As we explained in Section [REF], this is a result of the fact that, as the disorder increases, the concentration of unitary scatterers first increases, then saturates and the bare disorder dominates over the f-related one ([MATH]).', 'cond-mat-0302389-2-76-3': '[CITATION] Indeed, even for [MATH] we notice the presence of a slight "shoulder" in the typical DOS around [MATH], whose origin is the same as that of the non-monotonic behavior at [MATH].', 'cond-mat-0302389-2-76-4': 'We call attention to the slave boson results of Fig. [REF], which similarly show that a decrease of the hybridization strength moves the "dip-hump" feature to smaller disorder values.', 'cond-mat-0302389-2-76-5': 'Moreover, as the current calculation was done at finite temperature, inelastic scattering also plays some role in smoothing out this feature.', 'cond-mat-0302389-2-77-0': 'As the disorder continues to increase and the distribution of [MATH] becomes broader, the typical DOS for both [MATH] and [MATH] decreases.', 'cond-mat-0302389-2-77-1': 'These results tend to the non-interacting one, as is expected if only the bare disorder plays a role.', 'cond-mat-0302389-2-77-2': 'Indeed, comparing the results for [MATH] (Figs. [REF] and [REF]) we notice that the realizations for which the bare disorder [MATH] is large have [MATH] around zero, meaning that the real part of the f-shell disorder is not important.', 'cond-mat-0302389-2-77-3': 'Besides, the values of [MATH] are small for these realizations.', 'cond-mat-0302389-2-77-4': 'In the case of the realizations for which [MATH] is small, [MATH] attains values that are larger than the results for [MATH] (cf. Figs. [REF] and [REF]).', 'cond-mat-0302389-2-78-0': 'The above discussion has focused on the conduction electron viewpoint.', 'cond-mat-0302389-2-78-1': 'Let us now consider how the presence of disorder in [MATH] is seen by the f-electrons.', 'cond-mat-0302389-2-78-2': 'For the auxiliary one-impurity problem, the important scale is the Kondo temperature [MATH], which measures the coupling between the impurity and the conduction electron bath.', 'cond-mat-0302389-2-78-3': 'The presence of disorder in [MATH] generates a distribution of [MATH], as we have seen before.', 'cond-mat-0302389-2-78-4': 'Thus at finite temperature some of the sites have [MATH], forming a singlet state with the bath, while the sites for which [MATH] represent an almost free spin, scattering conduction electrons incoherently.', 'cond-mat-0302389-2-78-5': 'This incoherence is characterized by a large amount of inelastic scattering, signaled by a significant imaginary part of the self-energy.', 'cond-mat-0302389-2-78-6': 'Going back to the results for [MATH], the fact that the inelastic scattering is stronger for [MATH] than for [MATH] reflects the larger number of incoherent sites (with [MATH]) in the former, since the smaller the hybridization, the smaller the Kondo temperature (see Eqs. [REF] and [REF]).', 'cond-mat-0302389-2-78-7': 'On the other hand, the sites with large [MATH] for [MATH], which are responsible for the great decrease in the typical DOS, represent sites with [MATH].', 'cond-mat-0302389-2-79-0': 'Fig. [REF] also shows the results for the typical DOS for [MATH], and [MATH].', 'cond-mat-0302389-2-79-1': 'In this case, the system has particle-hole symmetry in the clean limit, presenting a gap in its DOS (the Kondo insulator).', 'cond-mat-0302389-2-79-2': 'This explains the fact that for small disorder the typical DOS decreases as [MATH] decreases.', 'cond-mat-0302389-2-79-3': 'The overall behavior here bears strong similarity with the slave boson results of Fig. [REF]a (circles) and the explanation for it has been given in Section [REF].', 'cond-mat-0302389-2-80-0': '## Temperature dependence', 'cond-mat-0302389-2-81-0': 'Fig. [REF] presents the results for the typical DOS as a function of [MATH] for different temperatures.', 'cond-mat-0302389-2-81-1': 'The other parameters used were [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0302389-2-81-2': 'Here is where the interplay between inelastic and elastic processes proves to be fairly non-trivial and the use of a technique that incorporates both, such as perturbation theory, is crucial.', 'cond-mat-0302389-2-81-3': 'First we note that for the lowest disorder value ([MATH]), the typical DOS decreases with increasing temperature.', 'cond-mat-0302389-2-81-4': 'This is made more clear in the inset.', 'cond-mat-0302389-2-81-5': 'On the other hand, for [MATH], this tendency is reversed.', 'cond-mat-0302389-2-81-6': 'Finally, in between these two extremes, the temperature dependence can be non-monotonic.', 'cond-mat-0302389-2-81-7': 'A better sense of the overall behavior can be grasped by plotting the inverse of the typical DOS as a function of [MATH] for different values of [MATH], as shown in Fig. [REF].', 'cond-mat-0302389-2-81-8': 'It is clear that for [MATH], the inverse typical DOS shows a peak as a function of [MATH], which gradually moves to zero temperature as disorder is increased.', 'cond-mat-0302389-2-82-0': 'The temperature dependence of the typical DOS can be rationalized by looking at the corresponding changes in the distribution of [MATH].', 'cond-mat-0302389-2-82-1': 'For this purpose, we focus on the disorder value of [MATH], for which [MATH] has a clear maximum at [MATH].', 'cond-mat-0302389-2-82-2': 'We show in Figs. [REF] and [REF] the scatter plots of [MATH] at [MATH] and [MATH], respectively, which should be compared to the higher temperature results ([MATH]) of Fig. [REF].', 'cond-mat-0302389-2-82-3': 'From Fig. [REF] we see that the inverse typical DOS increases as we go from [MATH] to [MATH] and then decreases as the temperature is varied up to [MATH].', 'cond-mat-0302389-2-82-4': 'As can be seen from the figures, this non-monotonic behavior is governed by the effective f-disorder encoded in the distribution of [MATH].', 'cond-mat-0302389-2-82-5': 'Indeed, its variance increases from [MATH] to [MATH] but decreases from [MATH] to [MATH].', 'cond-mat-0302389-2-82-6': 'Note that the imaginary part of [MATH] always increases with increasing temperature, reflecting the enhancement of inelastic processes.', 'cond-mat-0302389-2-82-7': 'In terms of [MATH], this is the same as saying that as the temperature increases the number of sites with [MATH], which have a stronger inelastic scattering, becomes larger.', 'cond-mat-0302389-2-82-8': 'However, in the interval from about [MATH] up to [MATH], the increase of [MATH] is outweighed by the narrowing of the distribution of [MATH], which is the dominant contribution.', 'cond-mat-0302389-2-82-9': 'The above analysis can be similarly extended to the other values of disorder shown in Fig. [REF].', 'cond-mat-0302389-2-83-0': 'All along we have been using the typical DOS as a measure of the conducting properties of the system.', 'cond-mat-0302389-2-83-1': 'This is justified by its interpretation as a escape rate from a lattice site and the fact that it vanishes at the localization transition.', 'cond-mat-0302389-2-83-2': '[CITATION] Ideally, one would like to calculate the conductivity instead.', 'cond-mat-0302389-2-83-3': 'This is a difficult task in the present scheme, however, although an approximate calculation can be performed, which becomes accurate close to the localization transition.', 'cond-mat-0302389-2-83-4': "[CITATION] It requires the calculation of the propagator between two different sites, which goes beyond our current method, whose focus is on local Green's functions only.", 'cond-mat-0302389-2-83-5': 'Even in view of all these caveats, however, it is tempting to use the inverse typical DOS as an approximate measure of the resistivity, specially far from the weakly disordered region ([MATH]).', 'cond-mat-0302389-2-83-6': 'If this is done, then the fanning out of the "resistivity" curves of Fig. [REF] as [MATH] is reminiscent of the Mooij correlations,[CITATION] originally observed in disordered transition metal alloys, but which are also seen in heavy fermion alloys,[CITATION] doped Kondo insulators[CITATION] and even in two-dimensional systems as in the metal-oxide-semiconductor field-effect transistors.[', 'cond-mat-0302389-2-83-7': '[CITATION] Our results and the discussion above show that the interplay between localization effects and electron-electron interactions can give rise to the Mooij correlations, without the need to invoke other sources, such as electron-phonon interactions.[', 'cond-mat-0302389-2-83-8': '[CITATION] In addition, we have pointed out how the rapid drop in the typical DOS is a consequence of the proximity to the Kondo insulator, a region where localization effects are particularly large.', 'cond-mat-0302389-2-83-9': 'Taken together, these observations point to a close connection between Mooij correlations and localization effects in the vicinity of a Kondo or a Mott insulator.', 'cond-mat-0302389-2-83-10': 'It would be interesting to test these ideas by a direct calculation of the resistivity within a scheme capable of incorporating both localization and strong correlations.', 'cond-mat-0302389-2-84-0': 'Finally, regarding the DMFT calculation presented in all the figures discussed in this Section, we note that the results obtained for [MATH] fall approximately in the region where the concentration of realizations in the statDMFT calculation is largest.', 'cond-mat-0302389-2-84-1': 'There is no reason to expect DMFT and the statDMFT to give similar results, except at weak disorder.', 'cond-mat-0302389-2-84-2': 'Nevertheless, our results show that, surprisingly, DMFT can serve as a rough guide for the most probable values of [MATH].', 'cond-mat-0302389-2-84-3': 'There is a sizeable discrepancy, however, between the DMFT and the statDMFT results for [MATH] at the highest temperature ([MATH]) and low disorder.', 'cond-mat-0302389-2-84-4': 'The DMFT line in this case is an underestimate of the realizations obtained within the statDMFT.', 'cond-mat-0302389-2-84-5': 'As the temperature is lowered a better agreement is obtained.', 'cond-mat-0302389-2-84-6': 'Thus, besides its inherent neglect of localization effects, DMFT should be used only as a lower bound when gauging the importance of inelastic processes in disordered Anderson lattices.', 'cond-mat-0302389-2-85-0': '# Discussion and conclusions', 'cond-mat-0302389-2-86-0': 'We have in this paper extensively characterized the physics of disordered Anderson lattices within the statDMFT scheme, which is able to incorporate both localization effects and the local correlations coming from electron-electron interactions.', 'cond-mat-0302389-2-86-1': 'This was done using both large-N methods and perturbation theory for the auxiliary single impurity problems.', 'cond-mat-0302389-2-86-2': 'These are in a sense complementary approaches.', 'cond-mat-0302389-2-86-3': 'On the one hand, large-N theory is ideal for ground state properties, where inelastic effects are absent.', 'cond-mat-0302389-2-86-4': 'In particular, it affords a quick and reliable way of calculating Kondo temperatures (with the correct exponential dependence) and scattering phase shifts at [MATH].', 'cond-mat-0302389-2-86-5': 'Without [MATH] corrections, however, it is unsuitable for a finite temperature calculation.', 'cond-mat-0302389-2-86-6': 'On the other hand, second order perturbation theory has the advantage of being equally flexible with the added bonus of incorporating inelastic processes and the temperature dependence of the scattering phase shifts.', 'cond-mat-0302389-2-86-7': 'Nevertheless, it fails to capture the exponential nature of the low temperature scale of the single impurity problem.', 'cond-mat-0302389-2-86-8': 'Taken together, the two methods have enabled us to put on firm grounds the conclusions laid out in previous work,[CITATION] namely, the emergence of an electronic Griffiths phase in Anderson lattices governed by the proximity to the disorder-induced localization transition.', 'cond-mat-0302389-2-86-9': 'In particular, several inadequacies of the early Kondo Disorder Model (and its formulation as a Dynamical Mean Field Theory) have been given a better theoretical basis once localization effects were included.', 'cond-mat-0302389-2-86-10': 'The self-averaging effect introduced by the spatial fluctuations of the conduction electron wave functions induce a much higher degree of universality than is possible in the rigid KDM description.', 'cond-mat-0302389-2-86-11': 'Furthermore, the perturbation theory treatment has also suggested a mechanism behind the ubiquitous observation of Mooij correlations in the resistivity of disordered materials.', 'cond-mat-0302389-2-87-0': 'Even within the confines of the approximations of the statDMFT scheme, there are still outstanding issues that we would like to resolve.', 'cond-mat-0302389-2-87-1': 'Even though a fully analytical treatment is probably impossible, it might be feasible to devise an approximate parameterization of the statDMFT on the Bethe lattice, specially at weak disorder.', 'cond-mat-0302389-2-87-2': 'In this respect, the universality of the distributions of "dressed" quantities, such as the various local Green\'s functions and the Kondo temperatures, which are either Gaussian or log-normal is a useful guide.', 'cond-mat-0302389-2-87-3': 'A "toy" model that assumes a simple form for the distribution of [MATH] can be written down, which recovers the Griffiths singularities obtained in the numerical treatment.', 'cond-mat-0302389-2-87-4': '[CITATION] This "toy" model may prove useful for a calculation of the resistivity and for generalizations of the statDMFT treatment.', 'cond-mat-0302389-2-88-0': 'On the other hand, a specially important effect neglected in the statDMFT is two-particle inter-site correlations, particularly in the spin channel.', 'cond-mat-0302389-2-88-1': 'Indeed, the proliferation of poorly quenched low-[MATH] spins in our treatment generates a large amount of entropy that must be relieved at low temperatures through inter-site correlations.', 'cond-mat-0302389-2-88-2': 'Indeed, experimental evidence in favor of spin-glass dynamics at low temperatures in UCu[MATH]Pd[MATH][CITATION] and URh[MATH]Ge[MATH][CITATION] makes the inclusion of inter-site correlations more pressing.', 'cond-mat-0302389-2-88-3': 'A promising avenue of attack would be to remain true to the spirit of the DMFT and use its extended version.', 'cond-mat-0302389-2-88-4': 'Several treatments along these lines have been attempted.', 'cond-mat-0302389-2-88-5': '[CITATION] The challenge in our case is to incorporate both the dynamical inter-site correlations of the latter treatments and the spatial fluctuations of Kondo temperatures of the electronic Griffiths phase we find in our approach.', 'cond-mat-0302389-2-88-6': 'We defer the discussion of this problem to a future publication.', 'cond-mat-0302389-2-89-0': 'We thank M. J. Rozenberg for useful discussions.', 'cond-mat-0302389-2-89-1': 'We also thank C. H. Booth for providing us his XAFS data.', 'cond-mat-0302389-2-89-2': 'This work was supported by FAPESP through grants 99/00895-9 (MCOA), 98/12741-3 (EM), 01/00719-8 (EM), by CNPq through grant 301222/97-5 (EM), and by the NSF through grants DMR-9974311 and NSF-0234215 (VD).', 'cond-mat-0302389-2-90-0': '# Brief description of the numerical method', 'cond-mat-0302389-2-91-0': 'The set of stochastic equations defined in ([REF]-[REF]) must be solved in two steps.', 'cond-mat-0302389-2-91-1': 'First, action ([REF]-[REF]) is solved with the bath function with [MATH] nearest neighbors defined in ([REF]).', 'cond-mat-0302389-2-91-2': "This determines self-consistently the distribution of local conduction electron Green's functions with one nearest neighbor removed [MATH].", 'cond-mat-0302389-2-91-3': 'Next, the same action ([REF]-[REF]) is solved, this time with the bath function in ([REF]), constructed from the previously determined [MATH].', 'cond-mat-0302389-2-91-4': 'This step involves no self-consistency and yields the distribution of [MATH] as output.', 'cond-mat-0302389-2-91-5': 'Since the latter bath function is a sum over [MATH] nearest neighbors its statistical fluctuations are reduced compared to the former one.', 'cond-mat-0302389-2-91-6': 'Thus, [MATH] is more narrowly distributed than [MATH].', 'cond-mat-0302389-2-91-7': 'Yet, we expect the qualitative behavior to be the same.', 'cond-mat-0302389-2-91-8': 'We have therefore focused on the first step of the procedure only.', 'cond-mat-0302389-2-92-0': 'For a given impurity solver, this disordered Bethe lattice problem was solved for [MATH], by sampling the distribution of [MATH] from an ensemble of [MATH] sites, as proposed originally in Ref. [CITATION].', 'cond-mat-0302389-2-92-1': 'We have generally used [MATH] since we have checked that the results do not change by taking [MATH].', 'cond-mat-0302389-2-92-2': 'We have thus determined the distribution of various local properties.', 'cond-mat-0302389-2-93-0': 'The equations were solved on a discrete mesh along the Matsubara axis.', 'cond-mat-0302389-2-93-1': 'The mesh is set by the Matsubara frequencies in the perturbative treatment at finite temperatures and by an arbitrary finite discrete mesh in the slave boson mean field theory at [MATH] (up to 32000 points).', 'cond-mat-0302389-2-93-2': 'The choice of the imaginary frequency axis is due to a greater numerical stability.', 'cond-mat-0302389-2-93-3': "When the disorder is strong, the various Green's functions show large fluctuations.", 'cond-mat-0302389-2-93-4': 'However, these are much more pronounced on the real frequency axis, where they give rise to several peaks and gaps.', 'cond-mat-0302389-2-94-0': 'The slave boson treatment consists in finding the two mean field parameters [MATH] and [MATH] by solving the set of two non-linear Eqs. ([REF]-[REF]).', 'cond-mat-0302389-2-94-1': 'For that, we used the Powell hybrid method.', 'cond-mat-0302389-2-94-2': 'The integrals were calculated with standard adaptive quadrature routines.', 'cond-mat-0302389-2-94-3': 'Since we used a finite frequency mesh, it was important to extrapolate the value of [MATH] with the asymptotic form [MATH] for values of [MATH] greater than the largest mesh value.', 'cond-mat-0302389-2-94-4': 'In this fashion, a wide range of Kondo temperatures is covered, going down to almost machine precision in the clean metallic case.', 'cond-mat-0302389-2-94-5': 'When disorder is present a given impurity problem may not have a solution even at [MATH], because the strong spatial fluctuations may cause the local DOS to vanish at the Fermi level.', 'cond-mat-0302389-2-94-6': 'In this case, there are two possible regimes for the impurity, which have been carefully analyzed.', 'cond-mat-0302389-2-94-7': '[CITATION] The analysis shows that there is a critical coupling constant [MATH] such that the ground state is a singlet for [MATH] (the so-called "strong coupling Kondo effect"), whereas the local moment remains unquenched if [MATH].', 'cond-mat-0302389-2-94-8': 'When a solution could not be found, this corresponded to either a free spin ([MATH]) or a Kondo temperature which is smaller than the smallest value we can reach with our numerical code.', 'cond-mat-0302389-2-94-9': 'In either case, we set [MATH], effectively decoupling the free moment from the rest of the lattice.', 'cond-mat-0302389-2-94-10': 'Yet, we were still able to span several decades of energy scales.', 'cond-mat-0302389-2-95-0': 'In the perturbative treatment, the solution of each impurity problem is found by solving a set of two non-linear equations for [MATH] and [MATH], which is defined by Eqs. ([REF]) and ([REF]).', 'cond-mat-0302389-2-95-1': 'As in the slave boson treatment we used the Powell hybrid method.', 'cond-mat-0302389-2-95-2': 'The calculation of the second order correction for the self-energy involves Fourier transforms as, according to Eq. ([REF]), it has a simpler form in imaginary time rather than in frequency space.', 'cond-mat-0302389-2-95-3': 'For this, we used the Fast Fourier Transform algorithm.', 'cond-mat-0302389-2-95-4': '[CITATION]'}

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0302389

cond-mat-0503257

{'cond-mat-0503257-1-0-0': 'We review the theoretical aspects of pseudospin quantum computation using vertically coupled quantum dots in the quantum Hall regime.', 'cond-mat-0503257-1-0-1': 'We discuss the robustness and addressability of these collective, charge-based qubits.', 'cond-mat-0503257-1-0-2': 'The low energy Hilbert space of a coupled set of qubits yields an effective quantum Ising model tunable through external gates.', 'cond-mat-0503257-1-0-3': 'An experimental prediction of an even-odd effect in the Coulomb blockade spectra of the coupled quantum dot system probes the parameter regime necessary for realization of these qubits.', 'cond-mat-0503257-1-1-0': '# Introduction', 'cond-mat-0503257-1-2-0': 'Implementation of useful quantum algorithms requires large scale quantum information processing.', 'cond-mat-0503257-1-2-1': 'One perceived advantage of solid state quantum computing proposals has been the possibility of scaling up the system to produce nearly homogeneous arrays of qubits with tunable interactions.', 'cond-mat-0503257-1-2-2': 'Some rather promising proposals [CITATION] make use of real spin in semiconductor nanostructures as a natural two-level system with tunable couplings.', 'cond-mat-0503257-1-2-3': 'A potential advantage of real spin quantum computation, over charge-based proposals, is the long decoherence times for spin states in solids, [MATH]s or longer.', 'cond-mat-0503257-1-2-4': 'One particular demerit of these proposals is the difficulty in addressing a single spin.', 'cond-mat-0503257-1-2-5': 'Experimental techniques required to perform local manipulation of a single spin through applied magnetic fields push the limits of current technology [CITATION].', 'cond-mat-0503257-1-2-6': 'Similarly, single spin detection has proven difficult.', 'cond-mat-0503257-1-2-7': 'Recent experiments [CITATION] involving single spin detection have shown some success.', 'cond-mat-0503257-1-2-8': 'These measurements are an important first step in quantum computing with real spins but remain far from the goal of measuring several individual spins at specific locations.', 'cond-mat-0503257-1-3-0': 'An interesting solid state quantum computing implementation [CITATION] makes use of the coherent properties of bilayer quantum Hall states confined to nanostructures offering the possibility of charge-based quantum computing with long decoherence times.', 'cond-mat-0503257-1-3-1': 'Charge-based proposals yield the advantage of addressability.', 'cond-mat-0503257-1-3-2': 'Detection and manipulation of individual charges in quantum dots with single electron transistors is standard practice [CITATION].', 'cond-mat-0503257-1-3-3': 'Single charges in the solid state, however, are particularly sensitive to electric field fluctuations, phonons, and other very strong and common noise sources in solids.', 'cond-mat-0503257-1-3-4': 'As we will discuss below, the charge degree of freedom in quantum Hall states comprise many-body states which map onto a pseudospin and remain robust against external perturbations.', 'cond-mat-0503257-1-3-5': 'The quantum Hall liquid is incompressible and, by definition, possesses a gap to excited states in the charge sector.', 'cond-mat-0503257-1-3-6': 'Quantum Hall states confined to nanostructures [CITATION] therefore offer a robust charge-based qubit.', 'cond-mat-0503257-1-3-7': 'These systems are in direct analogy to the Cooper pair box system [CITATION] (another solid state, charge-based qubit) as a mesoscopic, scaled-down version of a coherent, bulk condensate in the solid state.', 'cond-mat-0503257-1-4-0': 'In what follows we review the microscopic theory and fundamental aspects of pseudospin quantum computing using bilayer quantum Hall systems confined to nanostructures[CITATION].', 'cond-mat-0503257-1-4-1': 'In Section [REF] we discuss the coherent properties of bilayer quantum Hall systems.', 'cond-mat-0503257-1-4-2': 'We draw an analogy between exciton condensation in these systems and low temperature superconductivity.', 'cond-mat-0503257-1-4-3': 'In Section [REF] we discuss the microscopic theory used to model vertically coupled quantum dots in the quantum Hall regime.', 'cond-mat-0503257-1-4-4': 'We discuss the parameters necessary to define a two-level system.', 'cond-mat-0503257-1-4-5': 'In Section [REF] we construct an effective, single pseudospin model for the qubit.', 'cond-mat-0503257-1-4-6': 'We outline several theoretical results which demonstrate the robust nature of pseudospin qubits.', 'cond-mat-0503257-1-4-7': 'In Section [REF] we review a derivation of a low-energy, pseudospin model describing Coulomb coupled pseudospin qubits.', 'cond-mat-0503257-1-4-8': 'The resulting quantum Ising model is sufficient for carrying out a universal set of quantum gates on a set of pseudospin qubits.', 'cond-mat-0503257-1-4-9': 'We conclude in Section [REF].', 'cond-mat-0503257-1-5-0': '# Exciton Condensation in Bilayer-Quantum Hall Systems', 'cond-mat-0503257-1-6-0': 'In this section we review the physics of exciton condensation in "bulk" quantum Hall bilayers.', 'cond-mat-0503257-1-6-1': 'We discuss the geometry, formalism, and phenomena in the bulk that will be relevant for our discussion of pseudospin quantum computation using a scaled down version of the bulk system: vertically coupled quantum dots in the quantum Hall regime.', 'cond-mat-0503257-1-6-2': 'The bulk system consists of two parallel two-dimensional electron gases separated by a barrier of thickness [MATH], for example.', 'cond-mat-0503257-1-6-3': 'A magnetic field oriented perpendicular to the plane quantizes the planar motion of electrons into Landau levels (LLs).', 'cond-mat-0503257-1-6-4': 'Along the direction perpendicular to the two dimensional plane the electrons lie in the lowest sub-band of the confinement potential.', 'cond-mat-0503257-1-6-5': 'The finite extent of the wavefunctions in the perpendicular direction allows for a small amount of single particle tunneling, [MATH], between the two two-dimensional electron gases.', 'cond-mat-0503257-1-6-6': '(The tunneling is equal to the symmetric-antisymmetric gap established by the double quantum well confining the electrons perpendicular to the plane.)', 'cond-mat-0503257-1-6-7': 'At low tunneling the interlayer physics is dominated by the Coulomb interaction.', 'cond-mat-0503257-1-6-8': 'Even without single particle tunneling the two layers correlate through the Coulomb interaction.', 'cond-mat-0503257-1-6-9': 'In fact the large Coulomb interaction, along with the large magnetic fields in these systems, polarizes the real electron spin in parameter regimes where the ground state is essentially ferromagnetic.', 'cond-mat-0503257-1-6-10': 'In what follows we take the system to be fully real-spin polarized.', 'cond-mat-0503257-1-7-0': 'Transport experiments on quantum Hall bilayers display a variety of spectacular phenomena [CITATION].', 'cond-mat-0503257-1-7-1': 'We discuss results associated with magnetic fields large enough to produce one flux quanta per electron, i.e. total filling [MATH].', 'cond-mat-0503257-1-7-2': 'Interlayer tunneling conductance measurements at this filling [CITATION] reveal a dramatic increase in tunneling conductance.', 'cond-mat-0503257-1-7-3': 'The dramatic increase has been associated with a spontaneously interlayer coherent phase supported by an equal number of electrons and correlation holes residing in each of the two dimensional layers[CITATION].', 'cond-mat-0503257-1-7-4': 'The resulting exciton condensate has been the subject of intense theoretical and experimental study [CITATION].', 'cond-mat-0503257-1-8-0': 'The fundamental properties of this neutral superfluid have been well established [CITATION].', 'cond-mat-0503257-1-8-1': 'We begin with a Hamiltonian describing electrons confined to bilayer systems: [EQUATION] where the interaction is given by: [EQUATION]', 'cond-mat-0503257-1-8-2': 'Here [MATH] is the electron effective mass, [MATH] is the dielectric constant of the host material and [MATH] is the center-to-center interlayer separation.', 'cond-mat-0503257-1-8-3': 'The indices [MATH] denote layer index (up or down) while the prime on the sum indicates [MATH] when [MATH].', 'cond-mat-0503257-1-8-4': 'We work in the symmetric gauge at magnetic field [MATH]: [MATH], in Eq. [REF], denotes the interlayer tunneling Hamiltonian which we take to be small.', 'cond-mat-0503257-1-9-0': 'The single particle energy spectrum is split into LLs.', 'cond-mat-0503257-1-9-1': 'The splitting is given by [MATH], where [MATH].', 'cond-mat-0503257-1-9-2': 'At large fields the kinetic energy is quenched to the lowest LL.', 'cond-mat-0503257-1-9-3': 'The basis states are given by: [EQUATION] where the planar coordinates [MATH] scale with the magnetic length [MATH].', 'cond-mat-0503257-1-9-4': 'The quantum number [MATH] represents the angular momentum.', 'cond-mat-0503257-1-9-5': 'It is the eigenvalue of the angular momentum operator: [MATH].', 'cond-mat-0503257-1-9-6': 'Recast in the basis of the lowest LL the problem simplifies.', 'cond-mat-0503257-1-9-7': 'Estimates [CITATION] of the effects of LL mixing (along with finite thickness perpendicular to the plane) find, at most, a 15% correction to energy differences.', 'cond-mat-0503257-1-9-8': 'In what follows we ignore finite thickness and LL mixing.', 'cond-mat-0503257-1-10-0': 'At [MATH] the Hartree-Fock solution of Eq. [REF] provides a surprisingly accurate solution over a large range of parameters.', 'cond-mat-0503257-1-10-1': 'The layer index enlarges the Hilbert space.', 'cond-mat-0503257-1-10-2': "A Hund's rule, applicable to layer index, picks out a single low energy state, the ground state in the Hartree-Fock approximation, to minimize the Coulomb energy cost.", 'cond-mat-0503257-1-10-3': 'The total ground state wavefunction, including the orbital and layer degrees of freedom, is generally given by: [MATH].', 'cond-mat-0503257-1-10-4': 'Here [MATH] is the antisymmetrization operator and [MATH] is a spinor dependent on layer index [MATH].', 'cond-mat-0503257-1-10-5': 'The orbital part takes a simple form [CITATION]: [EQUATION]', 'cond-mat-0503257-1-10-6': 'This solution represents the exact orbital part of the ground state in the limit [MATH] and [MATH].', 'cond-mat-0503257-1-10-7': 'This limit ensures our lowest LL projection while effectively lowering the interlayer separation in units of [MATH].', 'cond-mat-0503257-1-10-8': 'From this form of the wavefunction we see that each electron lies opposite a correlation hole in the neighboring layer.', 'cond-mat-0503257-1-10-9': 'The electron and its opposing zero form a neutral electron-hole pair.', 'cond-mat-0503257-1-10-10': 'These excitons condense to from an exciton condensate associated with a spontaneously broken symmetry, discussed below.', 'cond-mat-0503257-1-11-0': 'The spinors in the total wavefunction suggest a pseudospin interpretation of the layer index.', 'cond-mat-0503257-1-11-1': 'We formally define the pseudospin to be: [EQUATION] where [MATH] creates an electron in layer [MATH] with orbital angular momentum [MATH] and [MATH] are the usual Pauli matrices.', 'cond-mat-0503257-1-11-2': 'With this definition the eigenvalue of the pseudospin operator along the pseudospin [MATH] direction, [MATH], denotes the relative number difference between layers.', 'cond-mat-0503257-1-11-3': 'Along the pseudospin [MATH] direction the pseudospin operator is equivalent to the interlayer tunneling operator.', 'cond-mat-0503257-1-11-4': '[MATH] creates a bonding state between layers.', 'cond-mat-0503257-1-11-5': 'The pseudospin operator along the pseudospin [MATH] direction, [MATH], is equivalent to an interlayer current operator.', 'cond-mat-0503257-1-12-0': 'Consider a bilayer system with an interlayer bias adjusted to ensure an equal number of electrons in each layer, on average.', 'cond-mat-0503257-1-12-1': 'The many-body ground state will consist of a coherent superposition of orbital states [MATH] centered around [MATH].', 'cond-mat-0503257-1-12-2': 'The electrons and zeroes swap places in a coherent fashion, enhancing the interlayer tunneling in the process.', 'cond-mat-0503257-1-12-3': 'The coherent properties of the ground state can be seen explicitly in it second quantized form: [EQUATION]', 'cond-mat-0503257-1-12-4': 'This ground state has a form similar to the well known Bardeen-Cooper-Schrieffer (BCS) ground state of a superconductor: [EQUATION] where [MATH] denotes wave-vector, the arrows indicate real spin (in Eq. [REF]), and the coefficients [MATH] and [MATH] are fixed to ensure proper relative phases.', 'cond-mat-0503257-1-12-5': 'By comparison we see that, with a redefined vacuum ([MATH]), the excitons condense to form a neutral superfluid analogous to a condensation of cooper pairs in a BCS superconductor.', 'cond-mat-0503257-1-12-6': 'Eq. [REF] describes a coherent superposition of eigenstates of pseudospin centered around [MATH] while the BCS state captures a coherent superposition of number eigenstates.', 'cond-mat-0503257-1-13-0': 'Exciton condensation is associated with a spontaneous breaking of pseudospin symmetry.', 'cond-mat-0503257-1-13-1': 'The electrons must have some (albeit arbitrarily small) interlayer, single particle tunneling to enable passage between layers.', 'cond-mat-0503257-1-13-2': 'But even with an infinitesimally small amount of tunneling, [MATH], the system, in the coherent ground state, exhibits a large tunneling renormalized by the interaction: [EQUATION] where [MATH] is the total number of particles.', 'cond-mat-0503257-1-13-3': 'An arbitrarily small amount of tunneling spontaneously reorients the total pseudospin along the pseudospin [MATH] direction.', 'cond-mat-0503257-1-13-4': 'The observation [CITATION] of enhanced interlayer conductance, at [MATH], suggests the formation of an exciton condensate with an order parameter defined by Eq. [REF].', 'cond-mat-0503257-1-14-0': 'By analogy with superconductivity one may consider the mesoscopic version of the exciton condensate.', 'cond-mat-0503257-1-14-1': 'The mesoscopic version of a BCS superconductor is a superconducting grain which exhibits some of the coherent properties of superconductors but in a smaller system.', 'cond-mat-0503257-1-14-2': 'Vertically separated, lateral quantum dots in the quantum Hall regime offer the "mesoscopic" version of the bulk exciton condensate discussed above.', 'cond-mat-0503257-1-14-3': 'Figure [REF] summarizes the analogous properties of exciton condensates and superconductors.', 'cond-mat-0503257-1-15-0': 'In what follows we discuss the possibility of encoding quantum information in the layer degree of freedom of a mesoscopic exciton condensate.', 'cond-mat-0503257-1-15-1': 'We now discuss the properties of an individual qubit and the experimental prediction of an even-odd effect in the Coulomb blockade spectra caused by spontaneous interlayer phase coherence inherent in our qubit.', 'cond-mat-0503257-1-16-0': '# Bilayer Quantum Hall Droplets', 'cond-mat-0503257-1-17-0': 'We now define the model and parameter regime necessary to establish a robust two-level system in a mesoscopic version of the exciton condensate discussed above.', 'cond-mat-0503257-1-17-1': 'We review a proposal [CITATION] designed to characterize the two-level system and quantitatively test the interlayer coherence present in the system.', 'cond-mat-0503257-1-17-2': 'An even-odd effect in the Coulomb blockade spectra of such a device yields an accurate test of interlayer coherence and, as we will see, provides important information related to the two-level systems encoded in the pseudospin degree of freedom.', 'cond-mat-0503257-1-18-0': 'Consider a bilayer quantum Hall system at [MATH] with vanishingly small interlayer tunneling, as described in the previous section.', 'cond-mat-0503257-1-18-1': 'The application of an external confinement fixes the total number of particles [MATH].', 'cond-mat-0503257-1-18-2': 'It is sufficient to consider parabolic confinement: [EQUATION] where the confinement parameter, [MATH]meV in GaAs samples, can be adjusted with external gates.', 'cond-mat-0503257-1-18-3': 'In the presence of a parabolic confinement the single particle eigenstates of the non-interacting Hamiltonian are the Fock-Darwin states [CITATION].', 'cond-mat-0503257-1-18-4': 'In the large magnetic field limit, [MATH], the eigenstates reduce to Eq. [REF] but with the replacement: [MATH], where we define the modified magnetic length, [MATH].', 'cond-mat-0503257-1-18-5': 'These states provide a basis for our calculations.', 'cond-mat-0503257-1-19-0': 'With a small number of particles [MATH] the system forms a maximum density liquid with an equal number of particles and flux quanta piercing the system.', 'cond-mat-0503257-1-19-1': 'This few-body state, often called the maximum density droplet (MDD), is separated from excited states by a gap.', 'cond-mat-0503257-1-19-2': 'It is the mesoscopic realization of the bulk [MATH] quantum Hall state.', 'cond-mat-0503257-1-19-3': 'In the special case of an odd number of particles the two, degenerate low energy states have total pseudospin [MATH].', 'cond-mat-0503257-1-19-4': 'The [MATH]) indicates one excess charge in the top (bottom) layer.', 'cond-mat-0503257-1-19-5': 'These two states, denoted [MATH], define a two-level system arising from a competition between the Coulomb interaction and confinement.', 'cond-mat-0503257-1-19-6': 'An even number of particles distributed between the two layers yields one state with [MATH].', 'cond-mat-0503257-1-20-0': 'We study the low energy physics of a bilayer quantum Hall droplet (BQHD) with an odd number of particles using exact diagonalization.', 'cond-mat-0503257-1-20-1': 'In second quantization the lowest LL Hamiltonian describing the system, including confinement and finite tunneling now reads: [EQUATION] where the confinement redefines the basis length scale and adds the first term with coefficient [MATH].', 'cond-mat-0503257-1-20-2': 'Explicit expressions for the Coulomb matrix elements, [MATH], can be found in the literature [CITATION].', 'cond-mat-0503257-1-20-3': 'We diagonalize the system in the limit [MATH] and consider finite tunneling effects subsequently.', 'cond-mat-0503257-1-20-4': 'In this limit the total [MATH] component of pseudospin, [MATH], the total angular momentum, [MATH], and the total particle number are all good quantum numbers.', 'cond-mat-0503257-1-20-5': 'The first two terms in Eq. [REF] compete.', 'cond-mat-0503257-1-20-6': 'It is necessary to tune the magnetic field and confinement to the proper values ensuring that the MDD is the ground state of the system.', 'cond-mat-0503257-1-20-7': 'The angular momentum of the MDD (the orbital state defined by Eq. [REF]) is [MATH].', 'cond-mat-0503257-1-20-8': 'With [MATH], for example, we find that a range of confinements centered around [MATH] yield the MDD ground state.', 'cond-mat-0503257-1-20-9': 'Fig. [REF] plots the low energy Hilbert space of Eq. [REF] for [MATH] with [MATH] in the [MATH] sector as a function of total angular momentum.', 'cond-mat-0503257-1-20-10': 'This implies four (three) electrons in the top (bottom) layer.', 'cond-mat-0503257-1-20-11': 'The interlayer separation is chosen to be [MATH].', 'cond-mat-0503257-1-21-0': 'The arrows indicate the three lowest energy states.', 'cond-mat-0503257-1-21-1': 'The central arrow shows the [MATH] ground state.', 'cond-mat-0503257-1-21-2': 'The left and right-most arrows indicate the edge excitations.', 'cond-mat-0503257-1-21-3': 'It was found in Ref. [CITATION] that the gap to edge excitations remains finite for [MATH].', 'cond-mat-0503257-1-21-4': 'The many-body ground state found here is one of two degenerate states which form a two-level system, separated from excited states (with the same [MATH]) by [MATH] at [MATH].', 'cond-mat-0503257-1-22-0': 'For an odd number of particles, BQHD states with [MATH] have the lowest charging energy cost.', 'cond-mat-0503257-1-22-1': 'They are separated from states with higher [MATH] by the relative charging energy cost: [MATH], where we find [CITATION] [MATH] for [MATH] through an empirical fit to our numerical, exact diagonalization over several values of [MATH].', 'cond-mat-0503257-1-22-2': 'We find that for [MATH] the two-level system defined by [MATH] remains separated from excited states by an energy [MATH].', 'cond-mat-0503257-1-23-0': 'We now discuss the coherence properties of a BQHD in the MDD state.', 'cond-mat-0503257-1-23-1': 'As for the bulk system, the BQHD will develop a renormalized tunneling gap: [MATH], in the limit [MATH].', 'cond-mat-0503257-1-23-2': 'The Coulomb interaction enhances the single particle tunneling by a factor of order [MATH].', 'cond-mat-0503257-1-23-3': 'The definition of spontaneous interlayer phase coherence in the BQHD is then: [EQUATION]', 'cond-mat-0503257-1-23-4': 'In Fig. [REF] we plot [MATH] as a function of interlayer separation for several odd particle numbers in the MDD state.', 'cond-mat-0503257-1-24-0': 'We find [MATH] to be sizable for [MATH].', 'cond-mat-0503257-1-24-1': 'The quantity [MATH] provides a mesoscopic measure of spontaneous interlayer phase coherence with direct experimental relevance, as we now discuss.', 'cond-mat-0503257-1-25-0': 'The coherence order parameter, [MATH], may be related to the Coulomb-blockade peak spacing measured in experiments on BQHDs.', 'cond-mat-0503257-1-25-1': 'Coulomb-blockade peaks in the tunneling conductance of quantum dots arise when the gate voltage, [MATH], is tuned so that the total energy of the [MATH] electron system coincides with the energy of the [MATH] electron system.', 'cond-mat-0503257-1-25-2': 'The total energy of the bilayer quantum dot system includes the total charging energy cost, which is given by: [EQUATION] where [MATH] is the total capacitance of the double dot system and [MATH] is the lead capacitance to one of the two dots.', 'cond-mat-0503257-1-25-3': 'The enhanced tunneling gap leads to a splitting between the states [MATH] in systems with [MATH] odd, thereby modifying the total energy of the system.', 'cond-mat-0503257-1-25-4': 'For an even number of particles no such splitting exists.', 'cond-mat-0503257-1-25-5': 'There is only one state which minimizes the charging energy cost giving: [MATH].', 'cond-mat-0503257-1-25-6': 'Therefore the Coulomb-blockade peak spacing will alternate with an even or odd number of particles.', 'cond-mat-0503257-1-25-7': 'Fig. [REF] illustrates the even-odd effect in BQHDs.', 'cond-mat-0503257-1-26-0': 'The top panel, (a), shows the situation with no interlayer coherence: [MATH].', 'cond-mat-0503257-1-26-1': 'The bottom panel, (b), illustrates the even-odd effect arising from finite [MATH].', 'cond-mat-0503257-1-26-2': 'Observation of an even-odd effect in the Coulomb blockade peak spacing would provide evidence for conditions sufficient to establish a two-level system in a BQHD.', 'cond-mat-0503257-1-27-0': '# Pseudospin Qubits', 'cond-mat-0503257-1-28-0': 'The many-body two-level system defined by the states [MATH] in a BQHD offers a charge-based qubit with tunable effective magnetic fields.', 'cond-mat-0503257-1-28-1': 'In this section we discuss two important aspects of BQHD qubits: the tunability and robustness of the qubit.', 'cond-mat-0503257-1-28-2': 'The qubit proposal outlined here has several advantages and disadvantages in meeting the DiVincenzo criteria [CITATION], as opposed to other solid state quantum computing proposals.', 'cond-mat-0503257-1-28-3': 'The disadvantages include susceptibility to leakage (as compared to real spin [CITATION] which is inherently a two-level system) and charge noise.', 'cond-mat-0503257-1-28-4': 'The advantages include ease of addressability (detection and manipulation of a single electron charge, as opposed to a single electron spin), tunability thorough externally applied electric fields, and a certain rigidity against external perturbations (as compared to single electron, charge-based qubits).', 'cond-mat-0503257-1-29-0': 'In the reduced Hilbert space [MATH] (as defined for a BQHD with an odd number of particles in the MDD state) two parameters may rotate the pseudospin.', 'cond-mat-0503257-1-29-1': 'As noted previously: finite interlayer tunneling effectively rotates the pseudospin along the pseudospin [MATH] direction.', 'cond-mat-0503257-1-29-2': 'The parameter [MATH] can be tuned with external gates or by application of a real, in-plane magnetic field.', 'cond-mat-0503257-1-29-3': 'Both fields alter the spread of the electron density along the direction perpendicular to the two-dimensional plane thereby tuning the interlayer overlap.', 'cond-mat-0503257-1-29-4': 'These fields provide an effective magnetic field along the pseudospin [MATH] direction which is always negative.', 'cond-mat-0503257-1-30-0': 'An external bias, [MATH], applied perpendicular to the two dimensional plane of a BQHD energetically favors one pseudospin state over another.', 'cond-mat-0503257-1-30-1': '[MATH] acts as an effective magnetic field along the pseudospin [MATH] direction.', 'cond-mat-0503257-1-30-2': 'We may therefore rotate the pseudospin direction through any point on the Bloch sphere by pulsing [MATH] and [MATH].', 'cond-mat-0503257-1-30-3': 'In the reduced Hilbert space, [MATH], we have a reduced Hamiltonian [CITATION]: [MATH].', 'cond-mat-0503257-1-30-4': 'The collective state associated with an excess charge in either the top or bottom layer of a coherent BQHD in the MDD state therefore provides a fully tunable qubit.', 'cond-mat-0503257-1-31-0': 'Several potential noise sources relevant for a BQHD qubit have been quantitatively addressed in the literature [CITATION].', 'cond-mat-0503257-1-31-1': 'In Ref. [CITATION] it was shown that phase flip errors arising from an inhomogeneous, externally applied potential are strongly suppressed by increasing the total number of particles defining the qubit.', 'cond-mat-0503257-1-31-2': 'Our analysis [CITATION] of phase flip errors arising from fluctuations in homogeneous, external voltages in the leads (fluctuations in [MATH]) finds that, at low temperatures, the decoherence rate (the ratio of the dephasing rate to the elementary logic operation rate) can be made small enough to allow for fault-tolerant quantum computation.', 'cond-mat-0503257-1-31-3': 'More precisely, we find the ratio to be: [MATH] where [MATH] is the gate capacitance, [MATH] is the typical impedance of the voltage circuit and [MATH].', 'cond-mat-0503257-1-32-0': 'In addition to dephasing from voltage fluctuations there is also the possibility of leakage due to density perturbations such as phonons.', 'cond-mat-0503257-1-32-1': 'In Ref. [CITATION] it was shown that increasing the particle number suppresses the form factor associated with phonons coupling the states [MATH] to unwanted edge modes, [MATH].', 'cond-mat-0503257-1-32-2': 'The form factor: [EQUATION] was suppressed by more than an order of magnitude in increasing [MATH] from 1 to 9.', 'cond-mat-0503257-1-32-3': 'Here [MATH] is the wave-vector of a density perturbation in the plane caused, for example, by a phonon of wave-vector [MATH].', 'cond-mat-0503257-1-32-4': 'The above results suggest that the rigidity inherent in collective and incompressible BQHD states suppress decoherence mechanisms otherwise present in single-electron, charge-based qubits in quantum dots.', 'cond-mat-0503257-1-33-0': '# Coulomb Coupling', 'cond-mat-0503257-1-34-0': 'We now discuss the possibility of constructing a universal set of quantum gates by coupling BQHD qubits.', 'cond-mat-0503257-1-34-1': 'There are several types of inter-BQHD couplings possible.', 'cond-mat-0503257-1-34-2': 'Three examples are exchange, capacitive, and Coulomb coupling.', 'cond-mat-0503257-1-34-3': 'In what follows we discuss Coulomb coupled BQHDs.', 'cond-mat-0503257-1-35-0': 'Consider two neighboring BQHDs, as shown schematically in Fig. [REF].', 'cond-mat-0503257-1-35-1': 'The centers of the BQHDs are placed a distance [MATH] apart.', 'cond-mat-0503257-1-36-0': 'The Coulomb interaction between neighboring droplets will favor an anti-alignment of charge as depicted in Fig. [REF].', 'cond-mat-0503257-1-36-1': 'The charge distributes among two parallel disks with an excess of charge in the top (bottom) layer in the state [MATH].', 'cond-mat-0503257-1-36-2': 'The disk extends to roughly the largest orbital state, [MATH].', 'cond-mat-0503257-1-36-3': 'Fig. [REF] plots the density of two neighboring BQHDs placed with a center-to-center distance of [MATH].', 'cond-mat-0503257-1-36-4': 'We obtain the densities from exact diagonalization of Eq. [REF] for a single BQHD with [MATH], [MATH] and [MATH].', 'cond-mat-0503257-1-37-0': 'We consider the low energy Hilbert space of two, well separated BQHDs.', 'cond-mat-0503257-1-37-1': 'The large inter-BQHD separation ensures little overlap between electronic states in neighboring BQHDs.', 'cond-mat-0503257-1-37-2': 'We also require that [MATH] is large enough to ensure that a neighboring BQHD does not induce unwanted intra-BQHD excitations.', 'cond-mat-0503257-1-37-3': 'In this limit the Hilbert space of two neighboring BQHDs comprises a set of four product states [MATH].', 'cond-mat-0503257-1-37-4': 'In this basis the resulting inter-BQHD Coulomb interaction maps onto a pseudospin Ising interaction: [EQUATION] where we define the effective exchange splitting to be: [EQUATION] where: [EQUATION] [MATH] indicates the radial vector in the [MATH]-[MATH] plane in the left (right) BQHD and [MATH] is the inter-BQHD dielectric constant.', 'cond-mat-0503257-1-38-0': 'At large distances, [MATH], it was shown [CITATION] that the "exchange" coefficient exhibits dipolar behavior, [MATH].', 'cond-mat-0503257-1-38-1': 'However, at these large distances we found [MATH] to be negligibly small in GaAs devices, leaving only nearest neighbor interactions.', 'cond-mat-0503257-1-38-2': '[MATH] can be tuned with [MATH] or [MATH].', 'cond-mat-0503257-1-39-0': 'We generalize the double BQHD system to an arbitrary number of BQHDs.', 'cond-mat-0503257-1-39-1': 'Each BQHD can be thought of as a lattice site [MATH] containing a pseudospin interacting with its neighbor through an Ising interaction: [EQUATION]', 'cond-mat-0503257-1-39-2': 'The above model, with tunable coefficients, offers a universal set of quantum gates.', 'cond-mat-0503257-1-39-3': 'The effective magnetic fields are tunable through external electric fields.', 'cond-mat-0503257-1-39-4': '[MATH] is not so easily tuned.', 'cond-mat-0503257-1-39-5': 'But, an intermediate, idle BQHD with an odd or even number of electrons can turn the interaction between next nearest neighbors on or off.', 'cond-mat-0503257-1-39-6': 'Another possibility borrows techniques from nuclear magnetic resonance theory.', 'cond-mat-0503257-1-39-7': 'It is well known that similar models with fixed inter-spin interactions allow quantum computation with refocusing pulses [CITATION].', 'cond-mat-0503257-1-40-0': '# Conclusion', 'cond-mat-0503257-1-41-0': 'We have reviewed the theoretical aspects of pseudospin quantum computation using confined, bilayer quantum Hall states.', 'cond-mat-0503257-1-41-1': 'We argue that bilayer quantum Hall droplets offer the quantum Hall analogue of Cooper pair boxes.', 'cond-mat-0503257-1-41-2': 'The even-odd effect in Coulomb blockade spectra provides criteria sufficient in defining a two-level system in the low energy Hilbert space of a bilayer quantum Hall droplet.', 'cond-mat-0503257-1-41-3': 'The two-level system encoded in the which-layer degree of freedom remains robust against external perturbations.', 'cond-mat-0503257-1-41-4': 'Coulomb coupling offers the possibility of a universal set of quantum gates through a pseudospin, quantum Ising model.', 'cond-mat-0503257-1-42-0': 'We would like to thank J.K. Jain for valuable discussions.', 'cond-mat-0503257-1-42-1': 'We acknowledge support from ARDA and NSA-LPS.', 'cond-mat-0503257-1-43-0': '# References'}

{'cond-mat-0503257-2-0-0': 'We review the theoretical aspects of pseudospin quantum computation using vertically coupled quantum dots in the quantum Hall regime.', 'cond-mat-0503257-2-0-1': 'We discuss the robustness and addressability of these collective, charge-based qubits.', 'cond-mat-0503257-2-0-2': 'The low energy Hilbert space of a coupled set of qubits yields an effective quantum Ising model tunable through external gates.', 'cond-mat-0503257-2-0-3': 'An experimental prediction of an even-odd effect in the Coulomb blockade spectra of the coupled quantum dot system allows for a probe of the parameter regime necessary for realization of these qubits.', 'cond-mat-0503257-2-1-0': '# Introduction', 'cond-mat-0503257-2-2-0': 'Implementation of useful quantum algorithms requires large scale quantum information processing.', 'cond-mat-0503257-2-2-1': 'One perceived advantage of solid state quantum computing proposals has been the possibility of scaling up the system to produce nearly homogeneous arrays of qubits with tunable interactions.', 'cond-mat-0503257-2-2-2': 'Some rather promising proposals [CITATION] make use of real spin in semiconductor nanostructures as a natural two-level system with tunable couplings.', 'cond-mat-0503257-2-2-3': 'A potential advantage of real spin quantum computation, over charge-based proposals, is the long decoherence times for spin states in solids, [MATH]s or longer.', 'cond-mat-0503257-2-2-4': 'One particular demerit of these proposals is the difficulty in addressing a single spin.', 'cond-mat-0503257-2-2-5': 'Experimental techniques required to perform local manipulation of a single spin through applied magnetic fields push the limits of current technology [CITATION].', 'cond-mat-0503257-2-2-6': 'Similarly, single spin detection has proven difficult.', 'cond-mat-0503257-2-2-7': 'Recent experiments [CITATION] involving single spin detection have shown some success.', 'cond-mat-0503257-2-2-8': 'These measurements are an important first step in quantum computing with real spins but remain far from the goal of measuring several individual spins at specific locations.', 'cond-mat-0503257-2-3-0': 'An interesting solid state quantum computing implementation [CITATION] makes use of the coherent properties of bilayer quantum Hall states confined to nanostructures offering the possibility of charge-based quantum computing with long decoherence times.', 'cond-mat-0503257-2-3-1': 'Charge-based proposals yield the advantage of addressability.', 'cond-mat-0503257-2-3-2': 'Detection and manipulation of individual charges in quantum dots with single electron transistors is standard practice [CITATION].', 'cond-mat-0503257-2-3-3': 'Single charges in the solid state, however, are particularly sensitive to electric field fluctuations, phonons, and other very strong and common noise sources in solids.', 'cond-mat-0503257-2-3-4': 'As we will discuss below, the charge degree of freedom in quantum Hall states comprise many-body states which map onto a pseudospin and remain robust against external perturbations.', 'cond-mat-0503257-2-3-5': 'The quantum Hall liquid is incompressible and, by definition, possesses a gap to excited states in the charge sector.', 'cond-mat-0503257-2-3-6': 'Quantum Hall states confined to nanostructures [CITATION] therefore offer a many-body charge-based qubit which should be less susceptible to certain types of environmental noise than similar systems using single charges.', 'cond-mat-0503257-2-3-7': 'These systems are in direct analogy to the Cooper pair box system [CITATION] (another solid state, charge-based qubit) as a mesoscopic, scaled-down version of a coherent, bulk condensate in the solid state.', 'cond-mat-0503257-2-4-0': 'In what follows we review the microscopic theory and fundamental aspects of pseudospin quantum computing using bilayer quantum Hall systems confined to nanostructures[CITATION].', 'cond-mat-0503257-2-4-1': 'In Section [REF] we discuss the coherent properties of bilayer quantum Hall systems.', 'cond-mat-0503257-2-4-2': 'We draw an analogy between exciton condensation in these systems and low temperature superconductivity.', 'cond-mat-0503257-2-4-3': 'In Section [REF] we discuss the microscopic theory used to model vertically coupled quantum dots in the quantum Hall regime.', 'cond-mat-0503257-2-4-4': 'We discuss the parameters necessary to define a two-level system.', 'cond-mat-0503257-2-4-5': 'In Section [REF] we construct an effective, single pseudospin model for the qubit.', 'cond-mat-0503257-2-4-6': 'We outline several theoretical results which compare noise issues of the many-body pseudospin qubits discussed here with similar single-charge qubits.', 'cond-mat-0503257-2-4-7': 'In Section [REF] we review a derivation of a low-energy, pseudospin model describing Coulomb coupled pseudospin qubits.', 'cond-mat-0503257-2-4-8': 'The resulting quantum Ising model is sufficient for carrying out a universal set of quantum gates on a set of pseudospin qubits.', 'cond-mat-0503257-2-4-9': 'We conclude in Section [REF].', 'cond-mat-0503257-2-5-0': '# Exciton Condensation in Bilayer-Quantum Hall Systems', 'cond-mat-0503257-2-6-0': 'In this section we review the physics of exciton condensation in "bulk" quantum Hall bilayers.', 'cond-mat-0503257-2-6-1': 'We discuss the geometry, formalism, and phenomena in the bulk that will be relevant for our discussion of pseudospin quantum computation using a scaled down version of the bulk system: vertically coupled quantum dots in the quantum Hall regime.', 'cond-mat-0503257-2-6-2': 'The bulk system consists of two parallel two-dimensional electron gases separated by a barrier of thickness [MATH], for example.', 'cond-mat-0503257-2-6-3': 'A magnetic field oriented perpendicular to the plane quantizes the planar motion of electrons into Landau levels (LLs).', 'cond-mat-0503257-2-6-4': 'Along the direction perpendicular to the two dimensional plane the electrons lie in the lowest sub-band of the confinement potential.', 'cond-mat-0503257-2-6-5': 'The finite extent of the wavefunctions in the perpendicular direction allows for a small amount of single particle tunneling, [MATH], between the two two-dimensional electron gases.', 'cond-mat-0503257-2-6-6': '(The tunneling is equal to the symmetric-antisymmetric gap established by the double quantum well confining the electrons perpendicular to the plane.)', 'cond-mat-0503257-2-6-7': 'At low tunneling the interlayer physics is dominated by the Coulomb interaction.', 'cond-mat-0503257-2-6-8': 'Even without single particle tunneling the two layers correlate through the Coulomb interaction.', 'cond-mat-0503257-2-6-9': 'In fact the large Coulomb interaction, along with the large magnetic fields in these systems, polarizes the real electron spin in parameter regimes where the ground state is essentially ferromagnetic.', 'cond-mat-0503257-2-6-10': 'In what follows we take the system to be fully real-spin polarized.', 'cond-mat-0503257-2-7-0': 'Transport experiments on quantum Hall bilayers display a variety of spectacular phenomena [CITATION].', 'cond-mat-0503257-2-7-1': 'We discuss results associated with magnetic fields large enough to produce one flux quanta per electron, i.e. total filling [MATH].', 'cond-mat-0503257-2-7-2': 'Interlayer tunneling conductance measurements at this filling [CITATION] reveal a dramatic increase in tunneling conductance.', 'cond-mat-0503257-2-7-3': 'The dramatic increase has been associated with a spontaneously interlayer coherent phase supported by an equal number of electrons and correlation holes residing in each of the two dimensional layers[CITATION].', 'cond-mat-0503257-2-7-4': 'The resulting exciton condensate has been the subject of intense theoretical and experimental study [CITATION].', 'cond-mat-0503257-2-8-0': 'The fundamental properties of this neutral superfluid have been well established [CITATION].', 'cond-mat-0503257-2-8-1': 'We begin with a Hamiltonian describing electrons confined to bilayer systems: [EQUATION] where the interaction is given by: [EQUATION]', 'cond-mat-0503257-2-8-2': 'Here [MATH] is the electron effective mass, [MATH] is the dielectric constant of the host material and [MATH] is the center-to-center interlayer separation.', 'cond-mat-0503257-2-8-3': 'The indices [MATH] denote layer index (up or down) while the prime on the sum indicates [MATH] when [MATH].', 'cond-mat-0503257-2-8-4': 'We work in the symmetric gauge at magnetic field [MATH]: [MATH], in Eq. [REF], denotes the interlayer tunneling Hamiltonian which we take to be small.', 'cond-mat-0503257-2-9-0': 'The single particle energy spectrum is split into LLs.', 'cond-mat-0503257-2-9-1': 'The splitting is given by [MATH], where [MATH].', 'cond-mat-0503257-2-9-2': 'At large fields the kinetic energy is quenched to the lowest LL.', 'cond-mat-0503257-2-9-3': 'The basis states are given by: [EQUATION] where the planar coordinates [MATH] scale with the magnetic length [MATH].', 'cond-mat-0503257-2-9-4': 'The quantum number [MATH] represents the angular momentum.', 'cond-mat-0503257-2-9-5': 'It is the eigenvalue of the angular momentum operator: [MATH].', 'cond-mat-0503257-2-9-6': 'Recast in the basis of the lowest LL the problem simplifies.', 'cond-mat-0503257-2-9-7': 'Estimates [CITATION] of the effects of LL mixing (along with finite thickness perpendicular to the plane) find, at most, a 15% correction to energy differences.', 'cond-mat-0503257-2-9-8': 'In what follows we ignore finite thickness and LL mixing.', 'cond-mat-0503257-2-10-0': 'At [MATH] the Hartree-Fock solution of Eq. [REF] provides a surprisingly accurate solution over a large range of parameters, [MATH].', 'cond-mat-0503257-2-10-1': 'The layer index enlarges the Hilbert space.', 'cond-mat-0503257-2-10-2': "A Hund's rule, applicable to layer index, picks out a single low energy state, the ground state in the Hartree-Fock approximation, to minimize the Coulomb energy cost.", 'cond-mat-0503257-2-10-3': 'The total ground state wavefunction, including the orbital and layer degrees of freedom, is generally given by: [MATH].', 'cond-mat-0503257-2-10-4': 'Here [MATH] is the antisymmetrization operator and [MATH] is a spinor dependent on the set of all layer indices [MATH].', 'cond-mat-0503257-2-10-5': 'The orbital part takes a simple form [CITATION]: [EQUATION]', 'cond-mat-0503257-2-10-6': 'This solution represents the exact orbital part of the ground state in the limit [MATH] and [MATH].', 'cond-mat-0503257-2-10-7': 'This limit ensures our lowest LL projection while effectively lowering the interlayer separation in units of [MATH].', 'cond-mat-0503257-2-10-8': 'From this form of the wavefunction we see that each electron lies opposite a correlation hole in the neighboring layer.', 'cond-mat-0503257-2-10-9': 'The electron and its opposing zero form a neutral electron-hole pair.', 'cond-mat-0503257-2-10-10': 'These excitons condense to from an exciton condensate associated with a spontaneously broken symmetry, discussed below.', 'cond-mat-0503257-2-11-0': 'The spinors in the total wavefunction suggest a pseudospin interpretation of the layer index.', 'cond-mat-0503257-2-11-1': 'We formally define the pseudospin to be: [EQUATION] where [MATH] creates an electron in layer [MATH] with orbital angular momentum [MATH] and [MATH] are the usual Pauli matrices.', 'cond-mat-0503257-2-11-2': 'With this definition the eigenvalue of the pseudospin operator along the pseudospin [MATH] direction, [MATH], denotes the relative number difference between layers.', 'cond-mat-0503257-2-11-3': 'Along the pseudospin [MATH] direction the pseudospin operator is equivalent to the interlayer tunneling operator.', 'cond-mat-0503257-2-11-4': '[MATH] creates a bonding state between layers.', 'cond-mat-0503257-2-11-5': 'The pseudospin operator along the pseudospin [MATH] direction, [MATH], is equivalent to an interlayer current operator.', 'cond-mat-0503257-2-12-0': 'Consider a bilayer system with an interlayer bias adjusted to ensure an equal number of electrons in each layer, on average.', 'cond-mat-0503257-2-12-1': 'The many-body ground state will consist of a coherent superposition of orbital states [MATH] centered around [MATH].', 'cond-mat-0503257-2-12-2': 'The electrons and zeroes swap places in a coherent fashion, enhancing the interlayer tunneling in the process.', 'cond-mat-0503257-2-12-3': 'The coherent properties of the ground state can be seen explicitly in it second quantized form: [EQUATION]', 'cond-mat-0503257-2-12-4': 'This ground state has a form similar to the well known Bardeen-Cooper-Schrieffer (BCS) ground state of a superconductor: [EQUATION] where [MATH] denotes wave-vector, the arrows indicate real spin (in Eq. [REF]), and the coefficients [MATH] and [MATH] are fixed to ensure proper relative phases.', 'cond-mat-0503257-2-12-5': 'By comparison we see that, with a redefined vacuum ([MATH]), the excitons condense to form a neutral superfluid analogous to a condensation of cooper pairs in a BCS superconductor.', 'cond-mat-0503257-2-12-6': 'Eq. [REF] describes a coherent superposition of eigenstates of pseudospin centered around [MATH] while the BCS state captures a coherent superposition of number eigenstates.', 'cond-mat-0503257-2-13-0': 'Exciton condensation is associated with a spontaneous breaking of pseudospin symmetry.', 'cond-mat-0503257-2-13-1': 'The electrons must have some (albeit arbitrarily small) interlayer, single particle tunneling to enable passage between layers.', 'cond-mat-0503257-2-13-2': 'But even with an infinitesimally small amount of tunneling, [MATH], the system, in the coherent ground state, exhibits a large tunneling renormalized by the interaction: [EQUATION] where [MATH] is the total number of particles.', 'cond-mat-0503257-2-13-3': 'An arbitrarily small amount of tunneling spontaneously reorients the total pseudospin along the pseudospin [MATH] direction.', 'cond-mat-0503257-2-13-4': 'The observation [CITATION] of enhanced interlayer conductance, at [MATH], suggests the formation of an exciton condensate with an order parameter defined by Eq. [REF].', 'cond-mat-0503257-2-14-0': 'By analogy with superconductivity one may consider the mesoscopic version of the exciton condensate.', 'cond-mat-0503257-2-14-1': 'The mesoscopic version of a BCS superconductor is a superconducting grain which exhibits some of the coherent properties of superconductors but in a smaller system.', 'cond-mat-0503257-2-14-2': 'Vertically separated, lateral quantum dots in the quantum Hall regime offer the "mesoscopic" version of the bulk exciton condensate discussed above.', 'cond-mat-0503257-2-14-3': 'Figure [REF] summarizes the analogous properties of exciton condensates and superconductors.', 'cond-mat-0503257-2-15-0': 'In what follows we discuss the possibility of encoding quantum information in the layer degree of freedom of a mesoscopic exciton condensate.', 'cond-mat-0503257-2-15-1': 'In the next section we discuss the properties of an individual qubit and the experimental prediction of an even-odd effect in the Coulomb blockade spectra caused by spontaneous interlayer phase coherence inherent in our qubit.', 'cond-mat-0503257-2-16-0': '# Bilayer Quantum Hall Droplets', 'cond-mat-0503257-2-17-0': 'We now define the model and parameter regime necessary to establish a robust two-level system in a mesoscopic version of the exciton condensate discussed above.', 'cond-mat-0503257-2-17-1': 'We review a proposal [CITATION] designed to characterize the two-level system and quantitatively test the interlayer coherence present in the system.', 'cond-mat-0503257-2-17-2': 'An even-odd effect in the Coulomb blockade spectra of such a device yields an accurate test of interlayer coherence and, as we will see, provides important information related to the two-level systems encoded in the pseudospin degree of freedom.', 'cond-mat-0503257-2-18-0': 'Consider a bilayer quantum Hall system at [MATH] with vanishingly small interlayer tunneling, as described in the previous section.', 'cond-mat-0503257-2-18-1': 'The application of an external confinement fixes the total number of particles [MATH].', 'cond-mat-0503257-2-18-2': 'It is sufficient to consider parabolic confinement: [EQUATION] where the confinement parameter, [MATH]meV in GaAs samples, can be adjusted with external gates.', 'cond-mat-0503257-2-18-3': 'In the presence of a parabolic confinement the single particle eigenstates of the non-interacting Hamiltonian are the Fock-Darwin states [CITATION].', 'cond-mat-0503257-2-18-4': 'In the large magnetic field limit, [MATH], the eigenstates reduce to Eq. [REF] but with the replacement: [MATH], where we define the modified magnetic length, [MATH].', 'cond-mat-0503257-2-18-5': 'These states provide a basis for our calculations.', 'cond-mat-0503257-2-19-0': 'With a small number of particles [MATH] the system forms a maximum density liquid with an equal number of particles and flux quanta piercing the system.', 'cond-mat-0503257-2-19-1': 'This few-body state, often called the maximum density droplet (MDD), is separated from excited states by a gap.', 'cond-mat-0503257-2-19-2': 'It is the mesoscopic realization of the bulk [MATH] quantum Hall state.', 'cond-mat-0503257-2-19-3': 'In the special case of an odd number of particles the two, degenerate low energy states have total pseudospin [MATH].', 'cond-mat-0503257-2-19-4': 'The [MATH]) indicates one excess charge in the top (bottom) layer.', 'cond-mat-0503257-2-19-5': 'These two states, denoted [MATH], define a two-level system arising from a competition between the Coulomb interaction and confinement.', 'cond-mat-0503257-2-19-6': 'An even number of particles distributed between the two layers yields one state with [MATH].', 'cond-mat-0503257-2-20-0': 'We study the low energy physics of a bilayer quantum Hall droplet (BQHD) with an odd number of particles using exact diagonalization.', 'cond-mat-0503257-2-20-1': 'In second quantization the lowest LL Hamiltonian describing the system, including confinement and finite tunneling now reads: [EQUATION] where the confinement redefines the basis length scale and adds the first term with coefficient [MATH].', 'cond-mat-0503257-2-20-2': 'Explicit expressions for the Coulomb matrix elements, [MATH], can be found in the literature [CITATION].', 'cond-mat-0503257-2-20-3': 'We diagonalize the system in the limit [MATH] and consider finite tunneling effects subsequently.', 'cond-mat-0503257-2-20-4': 'In this limit the total [MATH] component of pseudospin, [MATH], the total angular momentum, [MATH], and the total particle number are all good quantum numbers.', 'cond-mat-0503257-2-20-5': 'The first two terms in Eq. [REF] compete.', 'cond-mat-0503257-2-20-6': 'It is necessary to tune the magnetic field and confinement to the proper values ensuring that the MDD is the ground state of the system.', 'cond-mat-0503257-2-20-7': 'The angular momentum of the MDD (the orbital state defined by Eq. [REF]) is [MATH].', 'cond-mat-0503257-2-20-8': 'With [MATH], for example, we find that a range of confinements centered around [MATH] yield the MDD ground state.', 'cond-mat-0503257-2-20-9': 'Fig. [REF] plots the low energy Hilbert space of Eq. [REF] for [MATH] with [MATH] in the [MATH] sector as a function of total angular momentum.', 'cond-mat-0503257-2-20-10': 'This implies four (three) electrons in the top (bottom) layer.', 'cond-mat-0503257-2-20-11': 'The interlayer separation is chosen to be [MATH].', 'cond-mat-0503257-2-21-0': 'The arrows indicate the three lowest energy states.', 'cond-mat-0503257-2-21-1': 'The central arrow shows the [MATH] ground state.', 'cond-mat-0503257-2-21-2': 'The left and right-most arrows indicate the edge excitations.', 'cond-mat-0503257-2-21-3': 'It was found in Ref. [CITATION] that the gap to edge excitations remains finite for [MATH].', 'cond-mat-0503257-2-21-4': 'The many-body ground state found here is one of two degenerate states which form a two-level system, separated from excited states (with the same [MATH]) by [MATH] at [MATH].', 'cond-mat-0503257-2-22-0': 'For an odd number of particles, BQHD states with [MATH] have the lowest charging energy cost.', 'cond-mat-0503257-2-22-1': 'They are separated from states with higher [MATH] by the relative charging energy cost: [MATH], where we find [CITATION] [MATH] for [MATH] through an empirical fit to our numerical, exact diagonalization over several values of [MATH].', 'cond-mat-0503257-2-22-2': 'We find that for [MATH] the two-level system defined by [MATH] remains separated from excited states by an energy [MATH].', 'cond-mat-0503257-2-23-0': 'We now discuss the coherence properties of a BQHD in the MDD state.', 'cond-mat-0503257-2-23-1': 'As for the bulk system, the BQHD will develop a renormalized tunneling gap: [MATH], in the limit [MATH].', 'cond-mat-0503257-2-23-2': 'The Coulomb interaction enhances the single particle tunneling by a factor of order [MATH].', 'cond-mat-0503257-2-23-3': 'The definition of spontaneous interlayer phase coherence in the BQHD is then: [EQUATION]', 'cond-mat-0503257-2-23-4': 'In Fig. [REF] we plot [MATH] as a function of interlayer separation for several odd particle numbers in the MDD state.', 'cond-mat-0503257-2-24-0': 'We find [MATH] to be sizable for [MATH].', 'cond-mat-0503257-2-24-1': 'The quantity [MATH] provides a mesoscopic measure of spontaneous interlayer phase coherence with direct experimental relevance.', 'cond-mat-0503257-2-25-0': 'The coherence order parameter, [MATH], may be related to the Coulomb-blockade peak spacing measured in experiments on BQHDs.', 'cond-mat-0503257-2-25-1': 'Coulomb-blockade peaks in the tunneling conductance of quantum dots arise when the gate voltage, [MATH], is tuned so that the total energy of the [MATH] electron system coincides with the energy of the [MATH] electron system.', 'cond-mat-0503257-2-25-2': 'The total energy of the bilayer quantum dot system includes the total charging energy cost: [EQUATION] where [MATH] is the total capacitance of the double dot system and [MATH] is the lead capacitance to one of the two dots.', 'cond-mat-0503257-2-25-3': 'The enhanced tunneling gap leads to a splitting between hybridized states formed from the states [MATH] in systems with [MATH] odd, thereby modifying the total energy of the system (see Fig. [REF]).', 'cond-mat-0503257-2-25-4': 'For an even number of particles no such splitting exists.', 'cond-mat-0503257-2-25-5': 'There is only one state which minimizes the charging energy cost giving: [MATH].', 'cond-mat-0503257-2-25-6': 'Therefore the Coulomb-blockade peak spacing will alternate with an even or odd number of particles.', 'cond-mat-0503257-2-25-7': 'Fig. [REF] illustrates the even-odd effect in BQHDs.', 'cond-mat-0503257-2-26-0': 'The top panel, (a), shows the situation with no interlayer coherence: [MATH].', 'cond-mat-0503257-2-26-1': 'The bottom panel, (b), illustrates the even-odd effect arising from finite [MATH].', 'cond-mat-0503257-2-26-2': 'Observation of an even-odd effect in the Coulomb blockade peak spacing would provide evidence for conditions sufficient to establish a two-level system in a BQHD.', 'cond-mat-0503257-2-27-0': '# Pseudospin Qubits', 'cond-mat-0503257-2-28-0': 'The many-body two-level system defined by the states [MATH] in a BQHD offers a charge-based qubit with tunable effective magnetic fields.', 'cond-mat-0503257-2-28-1': 'In this section we discuss two important aspects of BQHD qubits: the tunability and robustness of the qubit.', 'cond-mat-0503257-2-28-2': 'The qubit proposal outlined here has several advantages and disadvantages in meeting the DiVincenzo criteria [CITATION], as opposed to other solid state quantum computing proposals.', 'cond-mat-0503257-2-28-3': 'The disadvantages include susceptibility to leakage (as compared to real spin [CITATION] which is inherently a two-level system) and charge noise.', 'cond-mat-0503257-2-28-4': 'The advantages include ease of addressability (detection and manipulation of a single electron charge, as opposed to a single electron spin), tunability thorough externally applied electric fields, and a certain rigidity against external perturbations (as compared to single-electron, charge-based qubits).', 'cond-mat-0503257-2-29-0': 'In the reduced Hilbert space [MATH] (as defined for a BQHD with an odd number of particles in the MDD state) two parameters may rotate the pseudospin.', 'cond-mat-0503257-2-29-1': 'As noted previously: finite interlayer tunneling effectively rotates the pseudospin along the pseudospin [MATH] direction.', 'cond-mat-0503257-2-29-2': 'The parameter [MATH] can be tuned with external gates or by application of a real, in-plane magnetic field.', 'cond-mat-0503257-2-29-3': 'Both fields alter the spread of the electron density along the direction perpendicular to the two-dimensional plane thereby tuning the interlayer overlap.', 'cond-mat-0503257-2-29-4': 'These fields provide an effective magnetic field along the pseudospin [MATH] direction which is always negative.', 'cond-mat-0503257-2-30-0': 'An external bias, [MATH], applied perpendicular to the two dimensional plane of a BQHD energetically favors one pseudospin state over another.', 'cond-mat-0503257-2-30-1': '[MATH] acts as an effective magnetic field along the pseudospin [MATH] direction.', 'cond-mat-0503257-2-30-2': 'We may therefore rotate the pseudospin direction through any point on the Bloch sphere by pulsing [MATH] and [MATH].', 'cond-mat-0503257-2-30-3': 'In the reduced Hilbert space, [MATH], we have a reduced Hamiltonian [CITATION]: [MATH].', 'cond-mat-0503257-2-30-4': 'The collective state associated with an excess charge in either the top or bottom layer of a coherent BQHD in the MDD state therefore provides a fully tunable qubit.', 'cond-mat-0503257-2-31-0': 'Several potential error sources relevant for a BQHD qubit have been quantitatively addressed in the literature [CITATION].', 'cond-mat-0503257-2-31-1': 'Three types of error sources were studied in Refs. [CITATION]: I) spatially local decoherence II) spatially global decoherence III) leakage (as opposed to decoherence) due to density perturbations.', 'cond-mat-0503257-2-31-2': 'The first type was studied in Ref. [CITATION].', 'cond-mat-0503257-2-31-3': 'It was shown that phase flip errors arising from an inhomogeneous, externally applied potential are strongly suppressed by increasing the total number of particles defining the BQHD qubit.', 'cond-mat-0503257-2-31-4': 'The results shows that phase flip errors are suppressed by more than an order of magnitude by increasing [MATH] from 1 to 10.', 'cond-mat-0503257-2-32-0': 'The second type of error was studied in Ref. [CITATION].', 'cond-mat-0503257-2-32-1': 'The analysis of phase flip errors arising from fluctuations in spatially homogeneous, external voltages in the leads (fluctuations in [MATH]) finds that, at low temperatures, the decoherence rate (the ratio of the dephasing rate to the elementary logic operation rate) can be made small enough to allow for fault-tolerant quantum computation.', 'cond-mat-0503257-2-32-2': 'More precisely, we find the ratio to be: [MATH] where [MATH] is the typical impedance of the voltage circuit and [MATH].', 'cond-mat-0503257-2-32-3': 'The above analysis applies to voltage noise in leads attached to any charge based qubit, pseudospin or otherwise, modeled in the spin-boson formalism.', 'cond-mat-0503257-2-32-4': 'The above formula captures an implicit fact; isolating the system from the environment, [MATH], can reduce the decoherence rate.', 'cond-mat-0503257-2-33-0': 'In addition to dephasing from voltage fluctuations, there is also the possibility of leakage due to density perturbations such as phonons.', 'cond-mat-0503257-2-33-1': 'In Ref. [CITATION] it was shown that increasing the particle number suppresses the form factor associated with phonons coupling the states [MATH] to unwanted edge modes, [MATH].', 'cond-mat-0503257-2-33-2': 'The form factor: [EQUATION] was suppressed by more than an order of magnitude in increasing [MATH] from 1 to 9.', 'cond-mat-0503257-2-33-3': 'Here [MATH] is the wave-vector of a density perturbation in the plane caused, for example, by a phonon of wave-vector [MATH].', 'cond-mat-0503257-2-33-4': 'We emphasize that the diminished form factor suppresses leakage due to any density perturbation, including both acoustic and optical phonons.', 'cond-mat-0503257-2-33-5': 'The results of Refs. [CITATION] suggest that the rigidity inherent in collective and incompressible BQHD states suppress decoherence and leakage mechanisms otherwise present in single-electron, charge-based qubits in quantum dots.', 'cond-mat-0503257-2-34-0': '# Coulomb Coupling', 'cond-mat-0503257-2-35-0': 'We now discuss the possibility of constructing a universal set of quantum gates by coupling BQHD qubits.', 'cond-mat-0503257-2-35-1': 'There are several types of inter-BQHD couplings possible.', 'cond-mat-0503257-2-35-2': 'Three examples are exchange, capacitive, and Coulomb coupling.', 'cond-mat-0503257-2-35-3': 'In what follows we discuss Coulomb coupled BQHDs.', 'cond-mat-0503257-2-36-0': 'Consider two neighboring BQHDs, as shown schematically in Fig. [REF].', 'cond-mat-0503257-2-36-1': 'The centers of the BQHDs are placed a distance [MATH] apart.', 'cond-mat-0503257-2-37-0': 'The Coulomb interaction between neighboring droplets will favor an anti-alignment of charge as depicted in Fig. [REF].', 'cond-mat-0503257-2-37-1': 'The charge distributes among two parallel disks with an excess of charge in the top (bottom) layer in the state [MATH].', 'cond-mat-0503257-2-37-2': 'The disk extends to roughly the largest orbital state, [MATH].', 'cond-mat-0503257-2-37-3': 'Fig. [REF] plots the density of two neighboring BQHDs placed with a center-to-center distance of [MATH].', 'cond-mat-0503257-2-37-4': 'We obtain the densities from exact diagonalization of Eq. [REF] for a single BQHD with [MATH], [MATH] and [MATH].', 'cond-mat-0503257-2-38-0': 'We consider the low energy Hilbert space of two, well separated BQHDs.', 'cond-mat-0503257-2-38-1': 'The large inter-BQHD separation ensures little overlap between electronic states in neighboring BQHDs.', 'cond-mat-0503257-2-38-2': 'We also require that [MATH] is large enough so that a neighboring BQHD does not induce unwanted intra-BQHD excitations.', 'cond-mat-0503257-2-38-3': 'In this limit the Hilbert space of two neighboring BQHDs comprises a set of four product states [MATH].', 'cond-mat-0503257-2-38-4': 'In this basis the resulting inter-BQHD Coulomb interaction maps onto a pseudospin Ising interaction: [EQUATION] where we define the effective exchange splitting to be: [EQUATION] where: [EQUATION] [MATH] indicates the radial vector in the [MATH]-[MATH] plane in the left (right) BQHD and [MATH] is the inter-BQHD dielectric constant.', 'cond-mat-0503257-2-39-0': 'At large distances, [MATH], it was shown [CITATION] that the "exchange" coefficient exhibits dipolar behavior, [MATH].', 'cond-mat-0503257-2-39-1': 'However, at these large distances we find [MATH] to be negligibly small in GaAs devices, leaving only nearest neighbor interactions which can be tuned with [MATH] or [MATH].', 'cond-mat-0503257-2-39-2': 'Nonetheless, the long-range part of a dipolar coupling (albeit small) can give rise to errors in quantum gates constructed entirely from nearest neighbor interactions.', 'cond-mat-0503257-2-39-3': 'In Ref. [CITATION] it was shown that errors due to dipolar coupling can be treated with quantum error correction gating schemes.', 'cond-mat-0503257-2-40-0': 'We generalize the double BQHD system to an arbitrary number of BQHDs.', 'cond-mat-0503257-2-40-1': 'Each BQHD can be thought of as a lattice site [MATH] containing a pseudospin interacting with its neighbor through an Ising interaction: [EQUATION]', 'cond-mat-0503257-2-40-2': 'The above model, with tunable coefficients, offers a universal set of quantum gates [CITATION].', 'cond-mat-0503257-2-40-3': 'The effective magnetic fields are tunable through external electric fields.', 'cond-mat-0503257-2-40-4': '[MATH] is not so easily tuned.', 'cond-mat-0503257-2-40-5': 'But, an intermediate, idle BQHD with an odd or even number of electrons can turn the interaction between next nearest neighbors on or off.', 'cond-mat-0503257-2-40-6': 'Another possibility borrows techniques from nuclear magnetic resonance theory.', 'cond-mat-0503257-2-40-7': 'It is well known that similar models with fixed inter-spin interactions allow quantum computation with refocusing pulses [CITATION].', 'cond-mat-0503257-2-41-0': '# Conclusion', 'cond-mat-0503257-2-42-0': 'We have reviewed the theoretical aspects of pseudospin quantum computation using confined, bilayer quantum Hall states.', 'cond-mat-0503257-2-42-1': 'We argue that bilayer quantum Hall droplets offer the quantum Hall analogue of Cooper pair boxes.', 'cond-mat-0503257-2-42-2': 'The even-odd effect in Coulomb blockade spectra provides criteria sufficient in defining a two-level system in the low energy Hilbert space of a bilayer quantum Hall droplet.', 'cond-mat-0503257-2-42-3': 'The many-body, two-level system encoded in the which-layer degree of freedom provides advantages over single-charge based qubits when considering external perturbations due to the environment.', 'cond-mat-0503257-2-42-4': 'Coulomb coupling offers the possibility of a universal set of quantum gates through a pseudospin, quantum Ising model.', 'cond-mat-0503257-2-43-0': 'We would like to thank J.K. Jain for valuable discussions.', 'cond-mat-0503257-2-43-1': 'We acknowledge support from ARDA and NSA-LPS.', 'cond-mat-0503257-2-44-0': '# References'}

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[]

[['cond-mat-0503257-1-3-6', 'cond-mat-0503257-2-3-6'], ['cond-mat-0503257-1-41-3', 'cond-mat-0503257-2-42-3'], ['cond-mat-0503257-1-4-6', 'cond-mat-0503257-2-4-6']]

[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0503257

math-0402192

{'math-0402192-1-0-0': 'We prove here essentially sharp [MATH] linear and bilinear estimates for the wave equations on Minkowski space where we assume the initial data possesses additional regularity with respect to fractional powers of the angular momentum operators [MATH].', 'math-0402192-1-0-1': 'In this setting, the range of exponents [MATH] vastly improves over what is available for the wave equations based on translation invariant derivatives of the initial data, or uniform decay of the solution.', 'math-0402192-1-1-0': '# Introduction: Classical Strichartz estimates, improvements for spherically symmetric data, and Knapp counterexamples', 'math-0402192-1-2-0': 'The aim of this work is to prove mixed Lebesgue space estimates for solutions to the linear wave equation on Minkowski space in a setting where the initial data is assumed to possess extra regularity with respect to weighted derivatives in the angular variable.', 'math-0402192-1-2-1': 'These types of estimates arise naturally in applications to the scale invariant global existence theory of non-linear wave equations which do not possess certain "null" structures in their non-linearities, and where the initial data does not possess much decay at space-like infinity.', 'math-0402192-1-2-2': 'For example, in a companion to this article, we use the estimates proved here to show global existence and scattering for the (4+1) Yang-Mills equations in the Lorentz gauge with general charge.', 'math-0402192-1-3-0': 'All of the estimates we prove will be of "Strichartz type", i.e. [MATH] space-time estimates for solutions of [MATH].', 'math-0402192-1-3-1': 'Due to the presence of extra weighted angular regularity, we will get a significant gain over the usual estimates for the wave equations which are based solely on translation invariant derivatives of the initial data.', 'math-0402192-1-3-2': 'What separates our estimates from the "classical" Strichartz estimates, is that they are not solely based on the uniform decay of solutions to the wave equation.', 'math-0402192-1-3-3': 'Instead, it will be necessary for us to exploit a certain "wave packet" structure these solutions exhibit in radial coordinates.', 'math-0402192-1-3-4': 'This allows us to decompose our waves into a sum of pulses, each of which remains coherent for all time.', 'math-0402192-1-3-5': 'These pulses can then be treated on an essentially individual basis.', 'math-0402192-1-3-6': 'This works well to prove a certain weak endpoint estimate for the range we are considering.', 'math-0402192-1-3-7': 'We then interpolate our endpoint with the endpoint from [CITATION] to obtain the full sharp (up to an [MATH]) set of estimates.', 'math-0402192-1-4-0': 'We shall also consider multilinear type estimates which involve weighted angular regularity on one or any number of the factors.', 'math-0402192-1-4-1': 'For this set of estimates, we again obtain a vast improvement over the "classical" multilinear estimates for the wave equation (e.g. compared with [CITATION]).', 'math-0402192-1-4-2': "We shall state and prove these estimates in the context of T. Tao's dual scale machine for generating multilinear estimates (see [CITATION]).", 'math-0402192-1-4-3': 'All of the estimates we prove here are sharp, up to an arbitrarily small loss of angular regularity, when tested against Knapp counterexamples.', 'math-0402192-1-4-4': 'We will explain this in more detail in the sequel.', 'math-0402192-1-5-0': 'We now begin with a brief discussion of the usual Strichartz estimates for the wave equation.', 'math-0402192-1-5-1': 'The standard reference for this material at this point is the paper [CITATION].', 'math-0402192-1-5-2': 'Let [MATH] be a unit frequency solution to the wave equation on Minkowski space.', 'math-0402192-1-5-3': 'By this we mean a function [MATH] such that: [EQUATION]', 'math-0402192-1-5-4': 'Without loss of generality, we may assume that [MATH] is of the form [MATH], for some unit frequency function [MATH] of the spatial variable only.', 'math-0402192-1-5-5': 'Then two of the most basic quantities which determine the space-time behavior of [MATH] are the following estimates: [EQUATION]', 'math-0402192-1-5-6': 'By interpolating between [REF] and [REF], and using some standard duality arguments (specifically the [MATH] method), it is possible to show the following set of space-time estimates for [MATH]:', 'math-0402192-1-6-0': 'Let [MATH] be the number of spatial dimensions, and let [MATH], then the following estimate holds for [MATH]: [EQUATION] where [MATH], with the exception of the forbidden [MATH] endpoint on [MATH].', 'math-0402192-1-7-0': 'A key facet of the proof of [REF], is that it does not rely on any other property of the evolution operator [MATH] than the estimates [REF] and [REF].', 'math-0402192-1-7-1': 'In other words, the proof only uses the conservation of energy and the uniform decay estimate [REF].', 'math-0402192-1-7-2': 'Furthermore, the estimate [REF] is sharp in that one cannot improve the range of [MATH] indices stated above without replacing the [MATH] norm on the right hand side of [REF] with something else.', 'math-0402192-1-7-3': 'This can be seen as follows: Let us consider initial data sets [MATH], where [MATH] is the indicator function of a radially directed block of dimensions [MATH], lying along the [MATH] axis between [MATH].', 'math-0402192-1-7-4': 'Then a quick calculation using the integral formula: [EQUATION] shows that one has [MATH] on the space-time region [MATH]: [EQUATION]', 'math-0402192-1-7-5': 'Based on this one sees that: [EQUATION] where [MATH].', 'math-0402192-1-7-6': 'Therefore, the condition [MATH] of Theorem [REF] must be satisfied.', 'math-0402192-1-8-0': 'What is happening here is that the function [MATH] has space-time Fourier support in a block of dimensions [MATH] which is tangent to the light cone along the the [MATH]-axis.', 'math-0402192-1-8-1': 'Note that the block [MATH] in physical space is precisely the dual of this block.', 'math-0402192-1-8-2': 'In other words, for a larger and larger (in [MATH]) portion of the range of our integration, [MATH] acts as much like the Fourier transform of the indicator function of a block, as it does the Fourier transform of a measure supported on the light cone.', 'math-0402192-1-8-3': 'In effect the curvature of the light cone in Fourier space is becoming "immobilized" by this sequence of initial data sets, and it is precisely this curvature which is responsible for the decay rate [REF] (while [REF] would be true for any characteristic surface), and thus any generic dispersive behavior of solutions to the wave equation, including the "classical" Strichartz estimates.', 'math-0402192-1-8-4': 'Strangely enough, we shall see in a moment that the lack of curvature of the frequency light cone in the radial direction is ultimately responsible for a vast improvement to [REF].', 'math-0402192-1-8-5': 'We note here that initial data sets [MATH] are commonly referred to as Knapp counterexamples.', 'math-0402192-1-9-0': 'With the above consideration in mind, it is natural to wonder if that somehow the initial data [MATH] were forced to be more evenly spread out along the various radial directions in Fourier space, then one could gain some improvement on the range of indices in [REF].', 'math-0402192-1-9-1': 'However, any such improvement must somehow involve another mechanism than just the estimates [REF] and [REF].', 'math-0402192-1-9-2': 'This can be seen simply from the fact that both of these estimates are sharp even for spherically symmetric data (e.g. looking at waves which focus at lager and larger times).', 'math-0402192-1-9-3': 'Even so, it has been known for some time that, with the help of additional arguments based on the specific form of the integral representation for [MATH], one can obtain a significant improvement over [REF] for spherically symmetric initial data (see for example [CITATION] and [CITATION]).', 'math-0402192-1-9-4': 'To understand how this can happen, consider a unit frequency radially symmetric initial data set [MATH], such that [MATH] is a smooth bump function of the radial variable [MATH].', 'math-0402192-1-9-5': 'Then by using an integration by parts and stationary phase argument on the integral representation [REF], in conjunction with the phenomena of finite speed of propagation it is not difficult to see that one has the asymptotics (because we can let derivatives fall on [MATH], of course this works for smooth enough Fourier data even if it is not spherically symmetric): [EQUATION]', 'math-0402192-1-9-6': 'That is, at time t, [MATH] is essentially the indicator function of an [MATH] spherical shell of radius [MATH] multiplied by the amplitude [MATH].', 'math-0402192-1-9-7': 'Based on this, one can easily computes that: [EQUATION]', 'math-0402192-1-9-8': 'Therefore, in order for us to have that the [MATH] norm of [MATH] is finite, we need that [MATH], or equivalently, that [MATH].', 'math-0402192-1-9-9': 'This is a vast improvement over the requirement of [MATH] coming from the Knapp counterexamples.', 'math-0402192-1-9-10': 'The key to this improvement is that along with the uniform decay rate [REF], the waves [MATH] are highly localized in physical space along the radial variable.', 'math-0402192-1-10-0': 'Of course, an arbitrary spherically symmetric wave will not have the localization [REF].', 'math-0402192-1-10-1': 'However, it is possible to in a straightforward manner chop up a unit frequency spherically symmetric wave into a sum of pieces, each of which satisfy a time translated version of the asymptotic [REF].', 'math-0402192-1-10-2': 'This is accomplished via a suitable physical space localization of the radially symmetric Fourier transform as follows: We begin by rewriting the integral formula [REF] for radially symmetric initial data [MATH] as: [EQUATION] where [MATH] is the Bessel function of order [MATH].', 'math-0402192-1-10-3': 'We now us the well known asymptotics for Bessel functions of relatively small order (see [CITATION]): [EQUATION]', 'math-0402192-1-10-4': 'Here the function [MATH] is [MATH], and the remaining [MATH] have asymptotic expansions: [EQUATION] as [MATH].', 'math-0402192-1-10-5': 'In other words, the functions [MATH] and [MATH] are [MATH] with derivatives in [MATH] uniformly bounded for all [MATH] and [MATH].', 'math-0402192-1-10-6': 'Substituting the asymptotic [REF] into the integral formula, we may assume without loss of generality that we are trying to bound integrals of the form: [EQUATION] where [MATH] is a smooth function with derivatives in [MATH] uniformly bounded for all [MATH], and [MATH] is a smooth bump function on the interval [MATH].', 'math-0402192-1-10-7': 'It is now apparent that the integrals in [REF] are essentially time translated inverse Fourier transforms of a one dimensional unit frequency function.', 'math-0402192-1-10-8': 'Therefore, we can localize these integrals in physical space (i.e. the [MATH] variable) on an [MATH] scale.', 'math-0402192-1-10-9': 'This can be accomplished with the help of the so called [MATH]-transform (see [CITATION]), which is just a smoothed out redundant (over-sampled) version of the classical Shannon sampling.', 'math-0402192-1-10-10': 'Since the function [MATH] is compactly supported in the interval [MATH], we may take its Fourier series development: [EQUATION]', 'math-0402192-1-10-11': 'An important thing to notice here is that we can recover the [MATH] of [MATH] as a function on [MATH] in terms of the [MATH]: [EQUATION]', 'math-0402192-1-10-12': 'This can be seen from the Plancherel theorem and the fact that [MATH] is unit frequency, so the volume part of the integral in Fourier space which comes from integrating over spheres is [MATH].', 'math-0402192-1-10-13': 'Sticking the series [REF] into the the integrals [REF] yields: [EQUATION] where: [EQUATION]', 'math-0402192-1-10-14': 'Integrating by parts as many times as necessary in the above formula, we see that we have the asymptotic: [EQUATION]', 'math-0402192-1-10-15': 'Using the expansion [REF] and the asymptotic [REF] we can directly compute that for [MATH]: [EQUATION]', 'math-0402192-1-10-16': 'The manipulation to get the last line above follows from Holders inequality and the fact that [MATH].', 'math-0402192-1-10-17': 'By integrating each expression in this line term by term and using the inclusion [MATH] for [MATH] we arrive at the bound: [EQUATION]', 'math-0402192-1-10-18': 'Testing this last expression for [MATH] in time we see that: [EQUATION]', 'math-0402192-1-10-19': 'Now as long as [MATH], or equivalently [MATH], we have the result: [EQUATION]', 'math-0402192-1-10-20': 'Using the characterization [REF], we have shown that:', 'math-0402192-1-11-0': 'Interpolating [REF] with the energy estimate [REF], and by various rescalings and using Littlewood-Paley theory (as in the work [CITATION], we will discuss this in more detail in the sequel), the estimate [REF] can be extended to general spherical initial data and other [MATH] spaces in a straight forward manner.', 'math-0402192-1-11-1': 'We record this as:', 'math-0402192-1-12-0': 'Let [MATH] be a spherically symmetric function on [MATH] such that [MATH], and set [MATH], then the following estimates hold: [EQUATION] where [MATH], and [MATH].', 'math-0402192-1-13-0': 'In the language of [CITATION], Theorem [REF] says that the range of indices for Strichartz estimates for spherically symmetric initial data are non-sharp [MATH] admissible.', 'math-0402192-1-13-1': 'From the point of view of uniform decay and the general machinery of [CITATION], this is like saying that these solutions morally decay like [MATH].', 'math-0402192-1-13-2': 'Of course, solutions to wave equation in general (if you look everywhere in Minkowski space and take the supremum) only decay like [MATH].', 'math-0402192-1-13-3': 'The point here is that if they are spherically symmetric, they only do so on a relatively thin set.', 'math-0402192-1-13-4': 'ret', 'math-0402192-1-14-0': 'We would now like to prove a result like [REF] for non-spherical initial data.', 'math-0402192-1-14-1': 'By looking at the Knapp counterexample calculation [REF], we see that in order for us to avoid a contradiction, we need to replace the term [MATH] on the left hand side of that equation by something that is of the order [MATH].', 'math-0402192-1-14-2': 'Now, based on our experience with spherical data, it is natural to conjecture that we only need to replace the norm [MATH] with something that incorporates angular regularity.', 'math-0402192-1-14-3': 'What we will do then, is test for smoothness of [MATH] in the angular direction in Fourier space.', 'math-0402192-1-14-4': 'This is done by using the infinitesimal generators of the rotations on Euclidean space: [EQUATION]', 'math-0402192-1-14-5': 'One sees immediately that: [EQUATION]', 'math-0402192-1-14-6': 'Interpolating this with the identity, we see that: [EQUATION] where [MATH], and: [EQUATION] is the Laplacian on the sphere of radius [MATH].', 'math-0402192-1-14-7': 'Therefore, it is natural to expect that if we add [MATH] angular derivatives to the left hand side of [REF], we may get a true estimate in the range [MATH].', 'math-0402192-1-14-8': 'Furthermore, an estimate of this kind would be sharp.', 'math-0402192-1-14-9': 'This is precisely what we will prove in dimensions [MATH], with an [MATH] loss of angular regularity:', 'math-0402192-1-15-0': '[Strichartz estimates for angularly regular data] Let [MATH] be the number of spatial dimensions, and let [MATH] be a solution to the homogeneous wave equation [MATH].', 'math-0402192-1-15-1': 'Let [MATH] be the angular wave admissible Strichartz exponent, and let [MATH] be the classical wave admissible Strichartz exponent.', 'math-0402192-1-15-2': 'Then for every [MATH], there is a [MATH] depending only on [MATH] such that the following set of estimates hold: [EQUATION] where we have that [MATH], [MATH], [MATH] , [MATH] , and [MATH] .', 'math-0402192-1-15-3': 'All of the implicit constants in the above inequality depend on [MATH], [MATH], and [MATH].', 'math-0402192-1-15-4': 'Here: [EQUATION] with the analogous norm defined for [MATH].', 'math-0402192-1-16-0': 'A short calculation like the one done above shows that in fact (modulo [MATH] angular derivatives), all of the estimates [REF] are sharp when tested on Knapp counterexamples.', 'math-0402192-1-16-1': 'Therefore, in this sense, they are all endpoint estimates.', 'math-0402192-1-16-2': 'It would be interesting to try and remove the extra [MATH] in these.', 'math-0402192-1-16-3': 'We will not pursue this issue here, although we will do some extra work in the sequel to recover a sharp [MATH] dispersive estimate which we believe is a start in this direction (see Proposition [REF]).', 'math-0402192-1-17-0': 'For the case of [MATH] spatial dimensions, we will also prove an estimate of the type [MATH] which involves [MATH] an angular derivative.', 'math-0402192-1-17-1': 'However, to obtain the full range of [REF] in this dimension would require one to prove an [MATH] Strichartz estimate that involves [MATH] angular derivatives.', 'math-0402192-1-17-2': 'While it seems that this type of estimate is out of the reach of methods we use here, it should be attainable using the recent method wave packet of Wolff [CITATION].', 'math-0402192-1-17-3': 'In fact, all of the estimates [REF] should be able to be proved directly using that method, based on the fact that they correspond to estimates that should be true for angularly separated initial data with no extra angular regularity.', 'math-0402192-1-17-4': 'It would be interesting to pursue these issues further.', 'math-0402192-1-17-5': 'ret', 'math-0402192-1-18-0': 'The remainder of this paper is laid out as follows.', 'math-0402192-1-18-1': 'In the next section, we list briefly some of the basic notations we use here.', 'math-0402192-1-19-0': 'In the third section, we list some standard facts about analysis on the sphere that will be useful in the sequel, including formulas for Hankel transforms and Littlewood-Paley-Stein theory on the sphere.', 'math-0402192-1-19-1': 'We then use this machinery to reduce the proof of Theorem [REF] to a suitable set of "endpoint" estimates.', 'math-0402192-1-20-0': 'In the fourth section, we discuss the main tool to be used in this paper: a [MATH]-type transform for the Hankel transform.', 'math-0402192-1-20-1': 'This leads us to consider the localization properties in physical space of the bandwidth limited Hankel transform.', 'math-0402192-1-20-2': 'In particular, we provide detailed asymptotics for our Hankel-[MATH] transform which will form the backbone the Strichartz estimates to be proved here.', 'math-0402192-1-21-0': 'In the fifth, we prove an [MATH] dispersive estimate for the wave equation based on angular regularity with respect to the momentum operators [REF].', 'math-0402192-1-21-1': 'This can be interpolated with the energy estimate to prove our "endpoints" directly via integration in time, avoiding to use of any other analytic machinery such as usual [MATH] process or induction on scales.', 'math-0402192-1-22-0': 'In the sixth and final section, we give a brief description of how our dispersive estimate can be modified in a straight forward manner to accommodate multilinear phenomena.', 'math-0402192-1-22-1': 'This shows one strength of the method used here in that one gets the expected range of improved multilinear estimates virtually for free out of the machinery developed.', 'math-0402192-1-22-2': 'We will also discuss why these multilinear estimates are sharp, by testing them on a multilinear analog of the Knapp counterexamples introduced above.', 'math-0402192-1-23-0': '# Basic notation', 'math-0402192-1-24-0': 'We list here some of the basic notation used throughout this paper.', 'math-0402192-1-24-1': 'For quantities [MATH] and [MATH], we denote by [MATH] to mean that [MATH] for some large constant [MATH].', 'math-0402192-1-24-2': 'The constant [MATH] may change from line to line, but will always remain fixed for any given instance where this notation appears.', 'math-0402192-1-24-3': 'We will also use the notation [MATH] if there exists a constant [MATH] such that [MATH].', 'math-0402192-1-25-0': 'For a given function of tow variables, say [MATH], we denote the mixed Lebesgue spaces norms [MATH] of [MATH] via the formulas: [EQUATION]', 'math-0402192-1-25-1': 'For a given function of the spatial variable only, we denote its Fourier transform as: [EQUATION]', 'math-0402192-1-25-2': 'With this normalization of the Fourier variables, the Plancherel theorem becomes [MATH], and one has the Fourier inversion formula: [EQUATION]', 'math-0402192-1-25-3': 'Using the Fourier transform, we define the homogeneous Sobolev space of order [MATH] via the identity [MATH].', 'math-0402192-1-26-0': 'For every integer [MATH] we define the spatial Littlewood-Paley cutoff operator by the formula: [EQUATION] where [MATH] and [MATH] is a positive smooth bump function, [MATH] on the interval [MATH] and zero off the interval [MATH].', 'math-0402192-1-26-1': 'With this notation we have the following consequence of the Littlewood-Paley theorem for [MATH] and [MATH]: [EQUATION]', 'math-0402192-1-27-0': 'For a given function of the spatial variable only, we denote its forward and backward wave propagation via the formulas: [EQUATION]', 'math-0402192-1-27-1': 'If [MATH] is an arbitrary solution to the homogeneous wave equation, [MATH], with initial data [MATH] and [MATH], we may decompose it into forward and backward wave propagation in the following way: [EQUATION]', 'math-0402192-1-27-2': 'Notice that in the case where [MATH] is unit frequency, the functions: [EQUATION] have [MATH] norm comparable to the [MATH] norm of [MATH].', 'math-0402192-1-27-3': 'Therefore, in the sequel, we will always assume that our unit frequency waves are of the form [MATH], and we replace: [EQUATION]', 'math-0402192-1-28-0': '# Some results from analysis on the sphere.', 'math-0402192-1-29-0': 'We list here some basic results from Fourier analysis is spherical coordinates which will be used in our proof of Theorem [REF].', 'math-0402192-1-29-1': 'We have already introduced the two basic differential elements of analysis on the sphere, the rotation vector fields [MATH] and the spherical Laplacian [MATH].', 'math-0402192-1-29-2': 'For every integer [MATH], there exists a finite dimensional set of functions [MATH] on the sphere [MATH] with satisfy the equation: [EQUATION]', 'math-0402192-1-29-3': 'These sets of functions exhaust the set of eigenfunctions of [MATH] and can in fact be identified with the homogeneous polynomials [MATH] on [MATH] of degree [MATH] which satisfy: [EQUATION]', 'math-0402192-1-29-4': 'Setting [MATH], and introducing the natural inner product on [MATH]: [EQUATION] we have the following basic properties of the [MATH]:', 'math-0402192-1-30-0': '[Basic properties of spherical harmonics (see e.g. [CITATION]]', 'math-0402192-1-31-0': 'The dimension of the space [MATH] is [MATH].', 'math-0402192-1-31-1': 'The spaces [MATH] are mutually orthogonal.', 'math-0402192-1-31-2': 'That is [MATH] for [MATH].', 'math-0402192-1-31-3': 'Let [MATH] be any orthonormal basis of [MATH], then one has the following identity for all [MATH]: [MATH].', 'math-0402192-1-31-4': 'For each [MATH], we have the identity:', 'math-0402192-1-32-0': '[MATH]', 'math-0402192-1-33-0': 'From now on, we fix an orthonormal basis [MATH] for each [MATH].', 'math-0402192-1-33-1': 'For a given function [MATH], we may expand it in the [MATH] sense along this basis as follows: [EQUATION]', 'math-0402192-1-33-2': 'Using [REF], we can define the action of [MATH] on this [MATH] as follows: [EQUATION]', 'math-0402192-1-33-3': 'Then using item (2) and (4) of Lemma [REF], and the fact that [MATH] is a homogeneous harmonic polynomial of degree [MATH] on [MATH]), we see that we have the identity: [EQUATION]', 'math-0402192-1-33-4': 'Using this, we see that there is, for every test function [MATH] on [MATH] an equivalence of norms: [EQUATION] where [MATH] is the norm from line [REF].', 'math-0402192-1-33-5': 'We also make the definition: [EQUATION]', 'math-0402192-1-33-6': 'Now, using the fact that [MATH] is of the form [MATH], we have the following standard interpolation result (see [CITATION]):', 'math-0402192-1-34-0': 'We will also use here some Littlewood-Paley theory in the angular variable.', 'math-0402192-1-34-1': 'We proceed in analogy with [REF] and let [MATH] be a smooth bump function such that [MATH] on the interval [MATH] and vanishing off the interval [MATH].', 'math-0402192-1-34-2': 'For each [MATH] we denote its dyadic rescaling as [MATH].', 'math-0402192-1-34-3': 'Defining [MATH] and using the decomposition formula [REF], we define the angular frequency dyadic projections of a function [MATH] on the sphere as: [EQUATION]', 'math-0402192-1-34-4': 'We define [MATH] to be the constant [MATH], which is the average of [MATH] over the sphere.', 'math-0402192-1-34-5': 'A formula similar to [REF] can be used to define [MATH] for a function on the whole of [MATH].', 'math-0402192-1-35-0': "For a given function [MATH] on the sphere, we use line (1) from [REF], as well as the fact that [MATH] when [MATH] to prove Bernstein's inequality for the sphere: [EQUATION]", 'math-0402192-1-35-1': 'By rescaling [REF] to spheres of various radii, we have the following result on all of [MATH] (for [MATH], we of course replace the [MATH] on the right hand side by [MATH]): [EQUATION]', 'math-0402192-1-35-2': 'Here [MATH] denotes the sphere of radius [MATH] centered at the origin.', 'math-0402192-1-36-0': 'Next, we record the basic result which allows us to generate certain square function expressions in terms of the [MATH] for [MATH] spaces when [MATH]:', 'math-0402192-1-37-0': 'Using Proposition [REF] along with the decomposition [REF], we have the following estimates for functions on space-time, for [MATH] and [MATH]: [EQUATION]', 'math-0402192-1-38-0': '## Reduction of Theorem [REF] to a frequency localized endpoint', 'math-0402192-1-39-0': 'We now use the setup we have introduced above to reduce the proof of Theorem [REF] to the following frequency localized "endpoint" estimate:', 'math-0402192-1-40-0': 'Assuming now the validity of Proposition [REF], we prove Theorem [REF] as follows: Given exponents [MATH] such that [MATH] and [MATH], we first reduce things to the case where [MATH].', 'math-0402192-1-40-1': 'We define [MATH] and [MATH] via the formulas: [EQUATION]', 'math-0402192-1-40-2': 'Notice that we can find such a [MATH] and [MATH] due to the range of [MATH].', 'math-0402192-1-40-3': 'Therefore, interpolating with the energy estimate (using Proposition [REF]), it suffices to prove [REF] for indices [MATH] with [MATH].', 'math-0402192-1-41-0': 'Next, using the decomposition [REF] an rescaling the spatial frequency of each term in the resulting sum, it suffices to show that: [EQUATION]', 'math-0402192-1-41-1': 'To do this, we choose: [EQUATION] such that there exists an [MATH] with the property that: [EQUATION]', 'math-0402192-1-41-2': 'That such choices are possible follows from our assumptions on the range of [MATH] and the identity: [EQUATION] where we have set [MATH].', 'math-0402192-1-41-3': 'Because of the range [REF], we see that is is also possible to choose an [MATH] with the property that: [EQUATION]', 'math-0402192-1-41-4': 'Furthermore, using Proposition [REF] and possibly a Sobolev embedding, we see that we have the estimate: [EQUATION]', 'math-0402192-1-41-5': 'Interpolating this last line with the [MATH] endpoint of [REF] we have achieved [REF].', 'math-0402192-1-41-6': 'Therefore, in the sequel, we will concentrate on the proof of [REF].', 'math-0402192-1-42-0': '## The Hankel transform', 'math-0402192-1-43-0': 'Finally, to wrap things up for this section, we record here the following formula for the action of the inverse Fourier transform on the decomposition: [EQUATION]', 'math-0402192-1-43-1': 'As is well known, this is given by a series of Hankel transforms.', 'math-0402192-1-43-2': 'The formula is (see [CITATION]): [EQUATION]', 'math-0402192-1-43-3': 'Here [MATH] is the Bessel function of order [MATH].', 'math-0402192-1-43-4': 'For [MATH], this is given by the integral formula: [EQUATION]', 'math-0402192-1-44-0': '# The Hankel-[MATH] transform', 'math-0402192-1-45-0': 'As we see from the formula [REF] of the last subsection, it is possible to expand the expression [MATH] in terms of spherical harmonics as: [EQUATION] where the coefficients [MATH] are given by the Hankel transform formula: [EQUATION]', 'math-0402192-1-45-1': 'Here, as in the previous subsection, the [MATH] are the coefficients in the spherical harmonic expansion of [MATH].', 'math-0402192-1-45-2': 'Also, the coefficients [MATH] should not be confused with the inverse Fourier transform of the [MATH].', 'math-0402192-1-46-0': 'We would now like to be able to localize the expressions [REF] in a manner analogous to the localization of the integral [REF].', 'math-0402192-1-46-1': 'This would be a relatively simple matter, if we could show that the asymptotic [REF] held uniformly in [MATH].', 'math-0402192-1-46-2': 'That is, if there was an asymptotic of the form [REF] for [MATH] which held uniformly as [MATH].', 'math-0402192-1-46-3': 'Unfortunately, it is well known that this is only the case for the region [MATH] (see [CITATION]).', 'math-0402192-1-46-4': 'In the transition regions, that is when [MATH], the asymptotic for [MATH] becomes quite complicated.', 'math-0402192-1-46-5': 'Roughly speaking, it begins to loose oscillations in [MATH] while it gains decay in the parameter [MATH].', 'math-0402192-1-46-6': 'Because of this, it does not seem feasible to try and compute an approximate formula for [MATH] and then substitute it into the integrals [REF].', 'math-0402192-1-46-7': 'Instead we will use a more straight forward approach, by first localizing the [MATH] in frequency as a Fourier series, just as we had done for [MATH] in the integral [REF], and then computing the integral [REF] directly by using appropriate integral representations for the [MATH].', 'math-0402192-1-47-0': 'Since we are assuming that the initial data in Theorem [REF] is unit frequency, we will assume that all of the coefficient functions [MATH] in the integrals [REF] are supported on the interval [MATH].', 'math-0402192-1-47-1': 'We may take their Fourier series developments on the interval [MATH], and we record these as: [EQUATION]', 'math-0402192-1-47-2': 'Expanding the integral [REF] in terms of the above formula, we see that: [EQUATION] where: [EQUATION]', 'math-0402192-1-47-3': 'In the above formula [MATH] is a smooth bump function on the interval [MATH].', 'math-0402192-1-47-4': 'Notice that this is not necessarily equal to [MATH] on any interval because we have absorbed the volume element into our definition of [MATH].', 'math-0402192-1-47-5': 'We call the right hand side of [REF] the Hankel-[MATH] transform of the function [MATH].', 'math-0402192-1-47-6': 'We would now like to be able to give a precise bound on how well localized the functions [MATH] are in physical space for the various values of the half-integer parameter [MATH] and the real variable [MATH].', 'math-0402192-1-47-7': 'This brings us to the main result of this subsection:', 'math-0402192-1-48-0': 'The downside of the above asymptotic is of course the region governed by [REF].', 'math-0402192-1-48-1': 'When [MATH], one can see that the extra factor [REF] will allow this asymptotic to look like [REF].', 'math-0402192-1-48-2': 'Notice that this is consistent with the fact that one has the asymptotic [REF] for [MATH] in this region, and is what is responsible for the good localization [REF] for spherically symmetric waves.', 'math-0402192-1-48-3': 'Unfortunately, it does not seem like one can do much to improve [REF] in the region where [MATH] (except for the extra factor of [MATH]).', 'math-0402192-1-48-4': 'In fact, if one assumes there is an asymptotic for [MATH] in this region which is of the form [REF], and one sets the [MATH] in the sum [REF] for a fixed [MATH], by putting absolute values around the sum [REF] one would get an asymptotic that looks like: [MATH].', 'math-0402192-1-48-5': 'But in this case, [MATH] corresponds to a delta function along the radial variable in Fourier space, say supported at the point [MATH].', 'math-0402192-1-48-6': 'Therefore we would have shown a bound like [MATH] uniform in [MATH].', 'math-0402192-1-48-7': 'This violates the well known asymptotic for Bessel functions: [MATH] (see [CITATION]).', 'math-0402192-1-48-8': 'It would be interesting to know if there is a more coherent decomposition of the Hankel transform that could eliminate this problem.', 'math-0402192-1-49-0': '[proof of Proposition [REF]] The asymptotics [REF]-[REF] follow more or less directly from appropriate integral formulas for the [MATH].', 'math-0402192-1-49-1': 'We will need to split the proof into the two cases: [MATH] or [MATH] and [MATH].', 'math-0402192-1-50-0': 'Case 1: [MATH] or [MATH].', 'math-0402192-1-51-0': 'Here we use a standard integral representation for Bessel functions which differs from [REF].', 'math-0402192-1-51-1': 'For [MATH], one has the following formula: [EQUATION]', 'math-0402192-1-51-2': 'This can be proved by a simple recursive argument (see [CITATION], Chapter 4, Lemma 3.1).', 'math-0402192-1-51-3': 'All one has to is to show that for both the integral formulas [REF] and [REF], the function [MATH] satisfies the recursive relation: [EQUATION] for [MATH].', 'math-0402192-1-51-4': 'In light of [REF], the equality of [REF] and [REF] is reduced to showing that it is true when [MATH].', 'math-0402192-1-51-5': 'This can be achieved directly through a change of variables.', 'math-0402192-1-52-0': 'Now using periodicity, integrating over an adjacent interval of length [MATH], and averaging, we see that for [MATH] the following integral representation also holds: [EQUATION]', 'math-0402192-1-52-1': 'Moreover, by a direct calculation, its not hard to see that the recursive relation [REF] is satisfied by both the integrals [REF] and [REF] whenever [MATH].', 'math-0402192-1-52-2': 'Therefore, throughout the sequel, we may assume that [REF] is our definition of the Bessel function that appears in the integral formula [REF] for the function [MATH] in dimension [MATH].', 'math-0402192-1-52-3': 'Making this substitution yields: [EQUATION]', 'math-0402192-1-53-0': 'We begin by proving the asymptotic [REF].', 'math-0402192-1-53-1': 'In fact, we will prove a bit more.', 'math-0402192-1-53-2': 'We will show that the asymptotic [REF] holds for [MATH].', 'math-0402192-1-53-3': 'Our first step will be to pick up the decay in the [MATH] parameter by integrating by parts the expression [REF] [MATH] times.', 'math-0402192-1-53-4': "The resulting expression looks like: [EQUATION] where the [MATH] in the above formula denote some specific trigonometric polynomials of degree [MATH] who's exact form is not important for our analysis.", 'math-0402192-1-53-5': 'The next step is to gain the decay in [MATH] in conjunction with the damping in terms of inverse powers of [MATH].', 'math-0402192-1-53-6': 'To get this, we Taylor expand each [MATH], where [EQUATION] around the point [MATH].', 'math-0402192-1-53-7': 'Notice that this is consistent with the fact that we are investigating the region where [MATH] is bounded.', 'math-0402192-1-53-8': 'We now define the dimensional constant [MATH], we record this Taylor expansion as: [EQUATION] where [MATH] is some smooth function such that [MATH].', 'math-0402192-1-53-9': 'Substituting the Taylor expansion [REF] into the integral [REF], we see that we may write: [EQUATION] where: [EQUATION] and: [EQUATION]', 'math-0402192-1-53-10': 'We can further estimate the term [MATH] above by using the fact that [MATH] and all of its derivatives have rapid decay away from the origin.', 'math-0402192-1-53-11': 'This follows immediately from the fact that [MATH] is the inverse Fourier transform of a smooth [MATH] bump function.', 'math-0402192-1-53-12': 'We record this observation as: [EQUATION]', 'math-0402192-1-53-13': 'By adding things up and introducing a large enough constant, this allows us to write: [EQUATION]', 'math-0402192-1-53-14': 'Recalling now that we have set [MATH] and that we also have [MATH], we see that in order to achieve the bound [REF], all we need to do is to control the expression for [MATH].', 'math-0402192-1-53-15': 'This is easy to do because a moments inspection shows that in fact one has [MATH].', 'math-0402192-1-53-16': 'This can be readily seen for even dimensions, that is when [MATH] is even, because in this case the trigonometric polynomials under the integral sign in the expression for [MATH] are of degree strictly less than [MATH].', 'math-0402192-1-53-17': 'By orthogonality, the whole expression then integrates to zero.', 'math-0402192-1-53-18': 'In the case of odd dimension, each integral is still zero thanks to the fact that [MATH] is a half integer expression, where as the term: [EQUATION] is a trigonometric polynomial of integer degree.', 'math-0402192-1-53-19': 'Since we are integrating over the double torus, [MATH], a rescaling turns the expression under the integral sign in [MATH] into a product of even degree trigonometric polynomials and odd degree trigonometric polynomials.', 'math-0402192-1-53-20': 'Therefore one has the needed orthogonality.', 'math-0402192-1-53-21': 'This completes the proof of [REF].', 'math-0402192-1-54-0': 'We now turn our attention to proving the asymptotic [REF] for the regime where [MATH].', 'math-0402192-1-54-1': 'For the remainder of this section we will assume that [MATH] is positive, as the other case can be dealt with by a similar argument.', 'math-0402192-1-54-2': 'Furthermore, using the bound that we proved in the previous discussion, we can without loss of generality assume that [MATH].', 'math-0402192-1-55-0': 'Our first step is to split the integral on the right hand side of [REF] smoothly into the regions where [MATH] and otherwise.', 'math-0402192-1-55-1': 'To realize this split, we restrict the integral [REF] to the regions: [EQUATION]', 'math-0402192-1-55-2': 'Notice that by symmetry, we only need to consider the regions [MATH] and [MATH].', 'math-0402192-1-55-3': 'On these regions, a bit of explicit computation using Taylor expansions shows that: [EQUATION] where the function [MATH] in the first line above satisfies the bound: [EQUATION]', 'math-0402192-1-55-4': 'Notice that to get [REF] we have used the condition [MATH].', 'math-0402192-1-55-5': 'Next, using another simple calculation involving the Taylor series of trigonometric functions, as well as the estimates [REF], [REF] and [REF], we see that one may write for [MATH]: [EQUATION]', 'math-0402192-1-55-6': 'We are now in a position to bound the integral [REF] on the region [MATH].', 'math-0402192-1-55-7': 'To do this, we will employ a smooth cutoff function [MATH] such that: [EQUATION]', 'math-0402192-1-55-8': 'We are trying to prove the estimate [REF] for the expression: [EQUATION]', 'math-0402192-1-55-9': 'Integrating by parts [MATH] times in the above integral and using the bound [REF] (along with the fact that we are in dimension [MATH]) we compute: [EQUATION]', 'math-0402192-1-55-10': 'Thus, we have proved [REF] for this portion of things.', 'math-0402192-1-56-0': 'It remains to prove the estimate [REF] for the region [MATH].', 'math-0402192-1-56-1': 'We suggestively (c.f. [REF]) denote the cutoff here by [MATH].', 'math-0402192-1-56-2': 'The calculation is essentially the same as what was done above, except that here we can afford to be more careless about the powers of [MATH] which come up through integration by parts.', 'math-0402192-1-56-3': 'Integrating by parts [MATH] times and using the bound [REF] in conjunction with the estimate [REF], we see that: [EQUATION]', 'math-0402192-1-56-4': 'This completes the proof of [REF] and ends the demonstration of case 1.', 'math-0402192-1-57-0': 'Case 2: [MATH].', 'math-0402192-1-58-0': 'In this section, the main difficulty will be for us to incorporate the remainder term [REF] into our asymptotic.', 'math-0402192-1-58-1': 'In order to motivate the steps we will take here, we argue heuristically as follows: We first define the auxiliary function: [EQUATION]', 'math-0402192-1-58-2': 'It is clear that the main contribution to the integral [REF] integrating around points where the function [MATH] defined above vanishes.', 'math-0402192-1-58-3': 'By symmetry, we need only consider such points which are positive.', 'math-0402192-1-58-4': "We'll call the one closest to zero [MATH].", 'math-0402192-1-58-5': 'Now, expanding [MATH] around this zero gives: [EQUATION] where again [MATH].', 'math-0402192-1-58-6': 'Therefore, it is natural to expect that: [EQUATION] where the [MATH], [MATH], are the (unique) set of derivatives and derivatives of [MATH] which vanish at infinity.', 'math-0402192-1-58-7': 'These satisfy the bounds similar to [REF], even if [MATH] is positive: [EQUATION]', 'math-0402192-1-58-8': 'This, of course, comes from the fact that [MATH] is the inverse Fourier transform of a unit frequency function, and is therefore supported away from the origin in Fourier space.', 'math-0402192-1-58-9': 'Using [REF] we easily get the following bound for arbitrary non-negative [MATH]: [EQUATION]', 'math-0402192-1-59-0': 'In order to make the previous argument rigorous, we need to justify the two approximations used in the lines directly above [REF].', 'math-0402192-1-59-1': 'We will not be able to do this completely, which will be responsible for the extra term in the formula [REF] for [MATH].', 'math-0402192-1-59-2': 'As in the previous two subsections, we argue by isolating the interval of integration.', 'math-0402192-1-59-3': 'By symmetry, and the decay bound [REF], we can without loss of generality assume that we are integrating [REF] over the interval [MATH].', 'math-0402192-1-59-4': 'We will first need to go a little further and chop some more off of the left hand side of this interval.', 'math-0402192-1-59-5': "What we'll do is take [MATH] (we're still keeping [MATH]) to be such that: [EQUATION]", 'math-0402192-1-59-6': 'Taylor expanding [MATH] gives us the bound: [EQUATION]', 'math-0402192-1-59-7': 'Furthermore, we also have the bound: [EQUATION]', 'math-0402192-1-59-8': 'Now, using [REF] and [REF] above, in conjunction with the asymptotic [REF], we immediately see that: [EQUATION]', 'math-0402192-1-59-9': 'This is enough to give [REF] for this portion of things.', 'math-0402192-1-59-10': 'Therefore, it remains to compute the integral: [EQUATION]', 'math-0402192-1-59-11': 'Keeping the Taylor expansion [REF] in mind, we now make the following change of variable for [MATH]: [EQUATION]', 'math-0402192-1-59-12': 'Using this, we can write: [EQUATION]', 'math-0402192-1-59-13': 'Where [MATH] is the interval: [EQUATION]', 'math-0402192-1-59-14': 'The desired result will now follow by integrating by parts as many times as necessary the integral [REF].', 'math-0402192-1-59-15': 'However, some care needs to be taken in order to control terms involving [MATH], which can be as big as [MATH].', 'math-0402192-1-59-16': 'Also, one needs to know that the higher derivatives of [MATH] possess some decay in order to deal with terms where the derivatives fall on [MATH] instead of the exponential factor.', 'math-0402192-1-59-17': 'We address these issues now.', 'math-0402192-1-59-18': 'The first observation we make is that we may integrate the bound: [EQUATION] using the fact that [MATH] to get that: [EQUATION] over the range of integration.', 'math-0402192-1-59-19': 'However, this is not enough to address the issue of a product of the form [MATH].', 'math-0402192-1-59-20': 'To handle this, notice that one has the bound: [EQUATION] as can be seen from the fact that this bound is true for [MATH], and that upon differentiating both sides of this expression one is reduced to showing (the inequality for the derivatives is trivial for [MATH]): [EQUATION]', 'math-0402192-1-59-21': 'This then follows immediately from the increasing nature of [MATH] on this interval and from the identity [MATH].', 'math-0402192-1-59-22': 'Now, combining the bounds [REF], [REF] and [REF], we see that for any positive integers [MATH], one has the estimates: [EQUATION]', 'math-0402192-1-59-23': 'Finally, we record the fact that the higher derivatives of [MATH] satisfy the following simple bounds, which can be verified through direct calculation: [EQUATION]', 'math-0402192-1-60-0': 'We are now ready to bound [REF].', 'math-0402192-1-60-1': 'First, we integrate as many times as necessary, letting the derivatives fall on the term [MATH].', 'math-0402192-1-60-2': 'The resulting expression looks like: [EQUATION]', 'math-0402192-1-60-3': 'Expanding out the derivatives in the first term of the above expression, and using the bounds [REF] and [REF], we see that we can bound: [EQUATION]', 'math-0402192-1-60-4': 'Therefore, the integral in the first term above does not cause us any trouble.', 'math-0402192-1-60-5': 'As far as the boundary values are concerned, we can use the bounds [REF], [REF], [REF], and [REF] to show that: [EQUATION]', 'math-0402192-1-61-0': 'It remains to bound [REF] where we integrate by parts and let the derivatives fall on the exponential factor.', 'math-0402192-1-61-1': 'Doing this [MATH] times yields: [EQUATION]', 'math-0402192-1-61-2': 'To control the first term above, we bound: [EQUATION] where [MATH] is a constant, and the other [MATH] denote a homogeneous expression of weight [MATH] in the variables [MATH], where each [MATH] is given weight [MATH].', 'math-0402192-1-61-3': 'Therefore, using the bound [REF] (which in particular implies that [MATH]) as well as the bounds [REF], we see that we can estimate: [EQUATION]', 'math-0402192-1-61-4': 'Finally, to deal with the boundary terms on the right hand side of [REF] we use the bounds [REF] and [REF] and simply estimate: [EQUATION]', 'math-0402192-1-62-0': 'Combining the above estimates together, we have shown that: [EQUATION]', 'math-0402192-1-62-1': 'This completes the proof of the asymptotic [REF], and the demonstration of Proposition [REF].', 'math-0402192-1-63-0': '# Some [MATH] dispersive estimates for the wave equation; linear and bilinear estimates', 'math-0402192-1-64-0': 'We are now ready to directly proceed with the proof of estimate [REF].', 'math-0402192-1-64-1': 'As we have mentioned previously, we may assume that [MATH] is of the form [MATH], for some unit frequency and dyadic angular frequency function [MATH].', 'math-0402192-1-64-2': 'For this [MATH], we the formulas from lines [REF], [REF], [REF], [REF], and [REF] to expand its wave propagation into harmonics and Hankel-[MATH] transforms as follows: [EQUATION]', 'math-0402192-1-64-3': 'In the above expression, we have absorbed the constants [MATH] into each [MATH].', 'math-0402192-1-64-4': 'By orthogonality of everything in sight, we have that: [EQUATION]', 'math-0402192-1-64-5': 'We now state our main result as follows:', 'math-0402192-1-65-0': 'Before proceeding with the proof, let us first show briefly how the above estimate may be used to show [REF].', 'math-0402192-1-65-1': 'What we need to do is to show that for every [MATH], there exists an [MATH], such that the following estimate holds: [EQUATION] where the implicit constants depend on both [MATH] and [MATH].', 'math-0402192-1-65-2': 'We also need [MATH] to approach [MATH] as [MATH].', 'math-0402192-1-65-3': 'Now, interpolating [REF] with the energy estimate [REF] gives the following result for [MATH]: [EQUATION]', 'math-0402192-1-65-4': 'Choosing [MATH] according to the formula [MATH], we see that we have the following identity holds: [MATH].', 'math-0402192-1-65-5': 'Therefore, for this choice of [MATH], we may square [REF] and integrate directly in time to achieve [REF].', 'math-0402192-1-65-6': 'Also note that [MATH] as [MATH].', 'math-0402192-1-65-7': 'We are now reduced to proving the estimate [REF].', 'math-0402192-1-66-0': '[proof of Proposition [REF]] Writing everything out, and using a Cauchy-Schwartz in the sum over [MATH] for each fixed [MATH] we have that: [EQUATION]', 'math-0402192-1-66-1': 'Now, using the formula in item (3) of Lemma [REF], and the fact that [MATH] for [MATH], we have: [EQUATION] where the last line above follows from a Cauchy-Schwartz and the bound [MATH].', 'math-0402192-1-66-2': 'The proposition will now be shown if we can prove that: [EQUATION]', 'math-0402192-1-66-3': 'We now use the asymptotics of Proposition [REF] and a little computation to split: [EQUATION] where [MATH] is supported where [MATH], and: [EQUATION]', 'math-0402192-1-66-4': 'Here [MATH] is the function from line [REF].', 'math-0402192-1-66-5': 'Substituting [MATH] into the left hand side of the sum [REF], and using a Cauchy-Schwartz immediately yields the desired result for this half of things.', 'math-0402192-1-66-6': 'In order to finish up, then, we only need substitute to [MATH] into the right hand side.', 'math-0402192-1-66-7': 'Doing this and using a Cauchy-Schwartz and re-indexing, we see that it is enough to show: [EQUATION] uniform in [MATH] and [MATH].', 'math-0402192-1-66-8': 'This can be done by comparing things to the appropriate integrals and is left to the reader.', 'math-0402192-1-66-9': 'Notice that the convergence factor [MATH] avoids any logarithmic divergences in this sum.', 'math-0402192-1-66-10': 'This completes the proof of [REF] and therefore the proof of Theorem [REF].', 'math-0402192-1-67-0': '# Bilinear estimates for angularly regular data', 'math-0402192-1-68-0': 'As is well known, for applications in the lower dimensional setting, linear estimates of the form [REF] are not sufficient.', 'math-0402192-1-68-1': 'What is needed are multilinear versions.', 'math-0402192-1-68-2': 'An extremely versatile method for building these type of estimates is based on the "fine and coarse scale" machine of T. Tao (see [CITATION]).', 'math-0402192-1-68-3': 'The basic idea is to fix a scale, say [MATH] for [MATH], and then decompose the domain of spatial variable into cubes with side lengths of this scale.', 'math-0402192-1-68-4': 'Then, one replaces the usual [MATH] norm in the spatial variable with [MATH], where the [MATH] norm is taken on the "fine" scale of each individual cube, while the [MATH] norm represents the "coarse" scale which is summation over all cubes.', 'math-0402192-1-68-5': 'One reason this method is so important, is that it allows one to use the bilinear construction process directly in an iteration procedure where resorting to the canned estimates that this method ultimately provides may be unduly burdensome.', 'math-0402192-1-68-6': 'Also, the way these estimates are constructed will allow us to generate bilinear estimates where only one term in the product contains extra angular regularity.', 'math-0402192-1-68-7': 'These type of estimates are extremely important for applications when one proves inductive estimates via an iteration procedure where it is necessary to consider estimates for products with an angular derivative falling on one or the other term in the product.', 'math-0402192-1-68-8': 'Because of these considerations, we will content ourselves here with the dual scale estimates themselves, and not bother with listing out the various multilinear estimates which follow from them.', 'math-0402192-1-68-9': 'In the [MATH] and higher dimensions, the usual dual scale estimates read:', 'math-0402192-1-69-0': '["Improved" Strichartz estimates] Let [MATH] be the number of spatial dimensions.', 'math-0402192-1-69-1': 'Let [MATH] be given, and let [MATH] be a partition of [MATH] into cubes of side length [MATH].', 'math-0402192-1-69-2': 'Then if [MATH] is a unit frequency solution to the equation [MATH], the following estimates hold: [EQUATION] where [MATH].', 'math-0402192-1-70-0': 'We show here that both the range of [REF] and the power of [MATH] that appearers there can be significantly improved.', 'math-0402192-1-70-1': 'In what follows, we will only bother proving an estimate which is the analog of [REF].', 'math-0402192-1-70-2': 'More general estimates which involve various amounts of angular regularity can then be gained by interpolating this estimate with [REF].', 'math-0402192-1-70-3': 'Also, it is not so easy to work out the Littlewood-Paley theory in the angular variable for localized norms like those that appear on the left hand side of [REF].', 'math-0402192-1-70-4': 'However, since we are already loosing a small amount of angular regularity, one may simply replace the Littlewood-Paley sum [REF] by an [MATH] sum.', 'math-0402192-1-70-5': 'As we have noted before, this has no bearing on applications.', 'math-0402192-1-71-0': '["Improved" frequency localized Strichartz estimates for angularly regular data; endpoint case] Let [MATH] be the number of spatial dimensions, and let [MATH] be a unit frequency and angular frequency localized solution to the homogeneous wave equation [MATH].', 'math-0402192-1-71-1': 'Let [MATH] be given, and let [MATH] be a partition of [MATH] into cubes of side length [MATH].', 'math-0402192-1-71-2': 'Then for every [MATH], there is a [MATH] and [MATH] depending on [MATH], such that [MATH] as [MATH] such that the following estimate holds: [EQUATION]', 'math-0402192-1-72-0': 'Up to the small loss in [MATH] and [MATH], the estimate [REF] is sharp when tested against the bilinear analog of the Knapp counterexamples [REF].', 'math-0402192-1-72-1': 'We construct these briefly as follows.', 'math-0402192-1-72-2': 'We consider (frequency) initial data sets [MATH] which along with being highly localized in the angular variable, are also well localized in the radial variable.', 'math-0402192-1-72-3': 'That is, we are now assuming [MATH] is supported on a small square of dimensions [MATH], lying along the [MATH] axis between [MATH].', 'math-0402192-1-72-4': 'A quick calculation then shows that the integral: [EQUATION] behaves like [MATH] on the space-time region [MATH]: [EQUATION]', 'math-0402192-1-72-5': 'Choosing our cubes [MATH] with side lengths [MATH], we see that for any [MATH] (in fact for any [MATH]) the following is true: [EQUATION]', 'math-0402192-1-72-6': 'Therefore, using [REF], we see that [REF] is indeed sharp for this sequence of initial data.', 'math-0402192-1-72-7': 'Of course the condition [MATH] cannot be improved, even for spherically symmetric initial data.', 'math-0402192-1-72-8': 'In fact, one can also see for these type of waves (say with the asymptotic [REF]), [REF] is again sharp.', 'math-0402192-1-72-9': 'We leave this simple calculation to the interested reader.', 'math-0402192-1-72-10': 'ret', 'math-0402192-1-73-0': '[proof of estimate [REF]] The proof will be similar in spirit to that of [REF].', 'math-0402192-1-73-1': 'However, we will need to use orthogonality in a more fundamental way here.', 'math-0402192-1-73-2': 'This is because we will not be able to rely solely on [MATH] as we did in the proof of [REF].', 'math-0402192-1-73-3': 'Ultimately, this has to do with the fact that the wave packets [MATH] are not well localized in physical space when [MATH] and [MATH] (see the remark after Proposition [REF]).', 'math-0402192-1-74-0': 'In order to prove estimate [REF], we begin by writing the basic energy estimate [REF] in the following way (we are still using the notation from line [REF]): [EQUATION]', 'math-0402192-1-74-1': 'Our next step is to prove the following spatially localized fixed time estimate: [EQUATION]', 'math-0402192-1-75-0': 'As in the previous section, choosing [MATH] by the identity, [MATH], we have that [MATH].', 'math-0402192-1-75-1': 'Therefore, squaring [REF] and integrating directly in time, we will have achieved [REF].', 'math-0402192-1-75-2': 'Therefore, we now concentrate on proving [REF].', 'math-0402192-1-76-0': 'It suffices to show [REF] for a fixed [MATH].', 'math-0402192-1-76-1': 'Therefore, we will now assume that we are on a fixed cube [MATH].', 'math-0402192-1-76-2': 'Our first step is foliate [MATH] with the hypersurfaces [MATH], where [MATH] is the sphere of radius [MATH] centered at the origin.', 'math-0402192-1-76-3': 'Notice that for each [MATH], one has the area estimate [MATH].', 'math-0402192-1-76-4': 'Therefore, using a Cauchy-Schwartz, it suffices to prove the estimate: [EQUATION]', 'math-0402192-1-76-5': 'Next, we chop [MATH] into at most [MATH] dyadic pieces, which are of the form [MATH], where [MATH] is the radial dyadic region of size [MATH].', 'math-0402192-1-76-6': 'We only need to do this for [MATH], i.e we keep the ball of bounded radius (say radius [MATH]) as a single region [MATH].', 'math-0402192-1-76-7': 'Therefore, to show [REF], it suffices to estimate: [EQUATION]', 'math-0402192-1-76-8': 'Now fix [MATH].', 'math-0402192-1-76-9': 'To prove estimate [REF] here, we run the decomposition [REF] and decompose the resulting sum into the sum of pieces: [EQUATION] where: [EQUATION]', 'math-0402192-1-76-10': 'To estimate [REF] on the sum [MATH], we simply keep it as a whole object and use the Bernstein inequality [REF] to estimate: [EQUATION]', 'math-0402192-1-76-11': 'As was to be shown.', 'math-0402192-1-76-12': 'Therefore, we are reduced to bounding [MATH].', 'math-0402192-1-76-13': 'Notice that for each fixed [MATH], and for each term in this sum, we have [MATH].', 'math-0402192-1-76-14': 'Therefore, we can use the well localized asymptotic [REF] and a Cauchy-Schwartz in the [MATH] summation (in a way similar to the computation started on line [REF]) to bound this term in [MATH] by: [EQUATION]', 'math-0402192-1-76-15': 'Squaring this last expression, and integrating each term in the sum with respect to [MATH] (notice there is no extra volume element because we took the sup on [MATH]), we see that we have proved [REF] for this portion of things.', 'math-0402192-1-76-16': 'This then completes the proof of estimate [REF], and therefore the proof of [REF].', 'math-0402192-1-77-0': 'ret'}

{'math-0402192-2-0-0': 'We prove here essentially sharp [MATH] linear and bilinear estimates for the wave equations on Minkowski space where we assume the initial data possesses additional regularity with respect to fractional powers of the angular momentum operators [MATH].', 'math-0402192-2-0-1': 'In this setting, the range of exponents [MATH] vastly improves over what is available for the wave equations based on translation invariant derivatives of the initial data, or uniform decay of the solution.', 'math-0402192-2-1-0': '# Introduction: Classical Strichartz estimates, improvements for spherically symmetric data, and Knapp counterexamples', 'math-0402192-2-2-0': 'The aim of this work is to prove mixed Lebesgue space estimates for solutions to the linear wave equation on Minkowski space in a setting where the initial data is assumed to possess extra regularity with respect to weighted derivatives in the angular variable.', 'math-0402192-2-2-1': 'These types of estimates arise naturally in applications to the scale invariant global existence theory of non-linear wave equations which do not possess certain "null" structures in their non-linearities.', 'math-0402192-2-2-2': 'For example, in a companion to this article, we use the estimates proved here to show global existence and scattering for the (4+1) Yang-Mills equations in the Lorentz gauge for a certain class of small, scale invariant initial data.', 'math-0402192-2-3-0': 'All of the estimates we prove will be of "Strichartz type", i.e. [MATH] space-time estimates for solutions of [MATH].', 'math-0402192-2-3-1': 'Due to the presence of extra weighted angular regularity, we will get a significant gain over the usual estimates for the wave equations which are based solely on translation invariant derivatives of the initial data.', 'math-0402192-2-3-2': 'What separates our estimates from the "classical" Strichartz estimates, is that they are not solely based on the uniform decay of solutions to the wave equation.', 'math-0402192-2-3-3': 'Instead, we will exploit a certain "wave packet" structure these solutions exhibit in radial coordinates.', 'math-0402192-2-3-4': 'This allows us to decompose our waves into a sum of pulses, each of which remains coherent for all time.', 'math-0402192-2-3-5': 'These pulses can then be treated on an essentially individual basis.', 'math-0402192-2-3-6': 'This works well to prove a certain weak endpoint estimate for the range we are considering.', 'math-0402192-2-3-7': 'We then interpolate our endpoint with the endpoint from [CITATION] to obtain the full sharp (up to an [MATH]) set of estimates.', 'math-0402192-2-4-0': 'We shall also consider multilinear type estimates which involve weighted angular regularity on one or any number of the factors.', 'math-0402192-2-4-1': 'For this set of estimates, we again obtain a vast improvement over the "classical" multilinear estimates for the wave equation (e.g. compared with [CITATION]).', 'math-0402192-2-4-2': "We shall state and prove these estimates in the context of T. Tao's dual scale machine for generating multilinear estimates (see [CITATION]).", 'math-0402192-2-4-3': 'All of the estimates we prove here are sharp, up to an arbitrarily small loss of angular regularity, when tested against Knapp counterexamples.', 'math-0402192-2-4-4': 'We will explain this in more detail in the sequel.', 'math-0402192-2-5-0': 'After the proof of the estimates we describe here was discovered by the first author, a shorter proof was found by Igor Rodnianski for the case of [MATH] spatial dimensions.', 'math-0402192-2-5-1': 'We have included this in an appendix to the present work and have elected to retain the discussion of our original proof in the main body of the paper because it includes the development of machinery that is interesting in its own right and is perhaps more flexible.', 'math-0402192-2-5-2': 'Furthermore, our proof gives a lot of detailed and interesting information as to how wave propagation works on high spherical harmonics.', 'math-0402192-2-5-3': 'In particular, we provide what seems to be a sharp localization of band limited Hankel transforms, the type of which has been studied by previous authors (see [CITATION]).', 'math-0402192-2-5-4': 'It is likely that the general procedure we employ here which uses this type phase space localization is applicable to other dispersive phenomena (e.g. wave or Schrodinger equations) on spherically symmetric backgrounds.', 'math-0402192-2-6-0': 'We now begin with a brief discussion of the usual Strichartz estimates for the wave equation.', 'math-0402192-2-6-1': 'The standard reference for this material at this point is the paper [CITATION].', 'math-0402192-2-6-2': 'Let [MATH] be a unit frequency solution to the wave equation on Minkowski space.', 'math-0402192-2-6-3': 'By this we mean a function [MATH] such that: [EQUATION]', 'math-0402192-2-6-4': 'Without loss of generality, we may assume that [MATH] is of the form [MATH], for some unit frequency function [MATH] of the spatial variable only.', 'math-0402192-2-6-5': 'Then two of the most basic quantities which determine the space-time behavior of [MATH] are the following estimates: [EQUATION]', 'math-0402192-2-6-6': 'By interpolating between [REF] and [REF], and using some standard duality arguments (specifically the [MATH] method), it is possible to show the following set of space-time estimates for [MATH]:', 'math-0402192-2-7-0': 'Let [MATH] be the number of spatial dimensions, and let [MATH], then the following estimate holds for [MATH]: [EQUATION] where [MATH], with the exception of the forbidden [MATH] endpoint on [MATH].', 'math-0402192-2-8-0': 'A key facet of the proof of [REF], is that it does not rely on any other property of the evolution operator [MATH] than the estimates [REF] and [REF].', 'math-0402192-2-8-1': 'In other words, the proof only uses the conservation of energy and the uniform decay estimate [REF].', 'math-0402192-2-8-2': 'Furthermore, the estimate [REF] is sharp in that one cannot improve the range of [MATH] indices stated above without replacing the [MATH] norm on the right hand side of [REF] with something else.', 'math-0402192-2-8-3': 'This can be seen as follows: Let us consider initial data sets [MATH], where [MATH] is the indicator function of a radially directed block of dimensions [MATH], lying along the [MATH] axis between [MATH].', 'math-0402192-2-8-4': 'Then a quick calculation using the integral formula: [EQUATION] shows that one has [MATH] on the space-time region [MATH]: [EQUATION]', 'math-0402192-2-8-5': 'Based on this one sees that: [EQUATION] where [MATH].', 'math-0402192-2-8-6': 'Therefore, the condition [MATH] of Theorem [REF] must be satisfied.', 'math-0402192-2-8-7': 'We note here that initial data sets [MATH] are commonly referred to as Knapp counterexamples.', 'math-0402192-2-9-0': 'With the above consideration in mind, it is natural to wonder if that somehow the initial data [MATH] were forced to be more evenly spread out along the various radial directions in Fourier space, then one could gain some improvement on the range of indices in [REF].', 'math-0402192-2-9-1': 'However, any such improvement must somehow involve another mechanism than just the estimates [REF] and [REF].', 'math-0402192-2-9-2': 'This can be seen simply from the fact that both of these estimates are sharp even for spherically symmetric data (e.g. looking at waves which focus at lager and larger times).', 'math-0402192-2-9-3': 'Even so, it has been known for some time that, with the help of additional arguments based on the specific form of the integral representation for [MATH], one can obtain a significant improvement over [REF] for spherically symmetric initial data (see for example [CITATION] and [CITATION]).', 'math-0402192-2-9-4': 'To understand how this can happen, consider a unit frequency radially symmetric initial data set [MATH], such that [MATH] is a smooth bump function of the radial variable [MATH].', 'math-0402192-2-9-5': 'Then by using an integration by parts and stationary phase argument on the integral representation [REF], in conjunction with the phenomena of finite speed of propagation it is not difficult to see that one has the asymptotics (because we can let derivatives fall on [MATH], of course this works for smooth enough Fourier data even if it is not spherically symmetric): [EQUATION]', 'math-0402192-2-9-6': 'That is, at time t, [MATH] is essentially the indicator function of an [MATH] spherical shell of radius [MATH] multiplied by the amplitude [MATH].', 'math-0402192-2-9-7': 'Based on this, one can easily computes that: [EQUATION]', 'math-0402192-2-9-8': 'Therefore, in order for us to have that the [MATH] norm of [MATH] is finite, we need that [MATH], or equivalently, that [MATH].', 'math-0402192-2-9-9': 'This is a vast improvement over the requirement of [MATH] coming from the Knapp counterexamples.', 'math-0402192-2-9-10': 'The key to this improvement is that along with the uniform decay rate [REF], the waves [MATH] are highly localized in physical space along the radial variable.', 'math-0402192-2-10-0': 'Of course, an arbitrary spherically symmetric wave will not have the localization [REF].', 'math-0402192-2-10-1': 'However, it is possible to in a straightforward manner chop up a unit frequency spherically symmetric wave into a sum of pieces, each of which satisfy a time translated version of the asymptotic [REF].', 'math-0402192-2-10-2': 'This is accomplished via a suitable physical space localization of the radially symmetric Fourier transform as follows: We begin by rewriting the integral formula [REF] for radially symmetric initial data [MATH] as: [EQUATION] where [MATH] is the Bessel function of order [MATH].', 'math-0402192-2-10-3': 'We now us the well known asymptotics for Bessel functions of relatively small order (see [CITATION]): [EQUATION]', 'math-0402192-2-10-4': 'Here the function [MATH] is [MATH], and the remaining [MATH] have asymptotic expansions: [EQUATION] as [MATH].', 'math-0402192-2-10-5': 'In other words, the functions [MATH] and [MATH] are [MATH] with derivatives in [MATH] uniformly bounded for all [MATH] and [MATH].', 'math-0402192-2-10-6': 'Substituting the asymptotic [REF] into the integral formula, we may assume without loss of generality that we are trying to bound integrals of the form: [EQUATION] where [MATH] is a smooth function with derivatives in [MATH] uniformly bounded for all [MATH], and [MATH] is a smooth bump function on the interval [MATH].', 'math-0402192-2-10-7': 'It is now apparent that the integrals in [REF] are essentially time translated inverse Fourier transforms of a one dimensional unit frequency function.', 'math-0402192-2-10-8': 'Therefore, we can localize these integrals in physical space (i.e. the [MATH] variable) on an [MATH] scale.', 'math-0402192-2-10-9': 'This can be accomplished with the help of the so called [MATH]-transform (see [CITATION]), which is just a smoothed out redundant (over-sampled) version of the classical Shannon sampling.', 'math-0402192-2-10-10': 'Since the function [MATH] is compactly supported in the interval [MATH], we may take its Fourier series development: [EQUATION]', 'math-0402192-2-10-11': 'An important thing to notice here is that we can recover the [MATH] of [MATH] as a function on [MATH] in terms of the [MATH]: [EQUATION]', 'math-0402192-2-10-12': 'This can be seen from the Plancherel theorem and the fact that [MATH] is unit frequency, so the volume part of the integral in Fourier space which comes from integrating over spheres is [MATH].', 'math-0402192-2-10-13': 'Sticking the series [REF] into the the integrals [REF] yields: [EQUATION] where: [EQUATION]', 'math-0402192-2-10-14': 'Integrating by parts as many times as necessary in the above formula, we see that we have the asymptotic: [EQUATION]', 'math-0402192-2-10-15': 'Using the expansion [REF] and the asymptotic [REF] we can directly compute that for [MATH]: [EQUATION]', 'math-0402192-2-10-16': 'The manipulation to get the last line above follows from Holders inequality and the fact that [MATH].', 'math-0402192-2-10-17': 'By integrating each expression in this line term by term and using the inclusion [MATH] for [MATH] we arrive at the bound: [EQUATION]', 'math-0402192-2-10-18': 'Testing this last expression for [MATH] in time we see that: [EQUATION]', 'math-0402192-2-10-19': 'Now as long as [MATH], or equivalently [MATH], we have the result: [EQUATION]', 'math-0402192-2-10-20': 'Using the characterization [REF], we have shown that:', 'math-0402192-2-11-0': 'Interpolating [REF] with the energy estimate [REF], and by various rescalings and using Littlewood-Paley theory (as in the work [CITATION], we will discuss this in more detail in the sequel), the estimate [REF] can be extended to general spherical initial data and other [MATH] spaces in a straight forward manner.', 'math-0402192-2-11-1': 'We record this as:', 'math-0402192-2-12-0': 'Let [MATH] be a spherically symmetric function on [MATH] such that [MATH], and set [MATH], then the following estimates hold: [EQUATION] where [MATH], and [MATH].', 'math-0402192-2-13-0': 'In the language of [CITATION], Theorem [REF] says that the range of indices for Strichartz estimates for spherically symmetric initial data are non-sharp [MATH] admissible.', 'math-0402192-2-13-1': 'From the point of view of uniform decay and the general machinery of [CITATION], this is like saying that these solutions morally decay like [MATH].', 'math-0402192-2-13-2': 'Of course, solutions to wave equation in general (if you look everywhere in Minkowski space and take the supremum) only decay like [MATH].', 'math-0402192-2-13-3': 'The point here is that if they are spherically symmetric, they only do so on a relatively thin set.', 'math-0402192-2-13-4': 'ret', 'math-0402192-2-14-0': 'We would now like to prove a result like [REF] for non-spherical initial data.', 'math-0402192-2-14-1': 'By looking at the Knapp counterexample calculation [REF], we see that in order for us to avoid a contradiction, we need to replace the term [MATH] on the left hand side of that equation by something that is of the order [MATH].', 'math-0402192-2-14-2': 'Now, based on our experience with spherical data, it is natural to conjecture that we only need to replace the norm [MATH] with something that incorporates angular regularity.', 'math-0402192-2-14-3': 'What we will do then, is test for smoothness of [MATH] in the angular direction in Fourier space.', 'math-0402192-2-14-4': 'This is done by using the infinitesimal generators of the rotations on Euclidean space: [EQUATION]', 'math-0402192-2-14-5': 'One sees immediately that: [EQUATION]', 'math-0402192-2-14-6': 'Interpolating this with the identity, we see that: [EQUATION] where [MATH], and: [EQUATION] is the Laplacian on the sphere of radius [MATH].', 'math-0402192-2-14-7': 'Therefore, it is natural to expect that if we add [MATH] angular derivatives to the left hand side of [REF], we may get a true estimate in the range [MATH].', 'math-0402192-2-14-8': 'Furthermore, an estimate of this kind would be sharp.', 'math-0402192-2-14-9': 'This is precisely what we will prove in dimensions [MATH], with an [MATH] loss of angular regularity:', 'math-0402192-2-15-0': '[Strichartz estimates for angularly regular data] Let [MATH] be the number of spatial dimensions, and let [MATH] be a solution to the homogeneous wave equation [MATH].', 'math-0402192-2-15-1': 'Let [MATH] be the angular wave admissible Strichartz exponent, and let [MATH] be the classical wave admissible Strichartz exponent.', 'math-0402192-2-15-2': 'Then for every [MATH], there is a [MATH] depending only on [MATH] such that the following set of estimates hold: [EQUATION] where we have that [MATH], [MATH], [MATH] , [MATH] , and [MATH] .', 'math-0402192-2-15-3': 'All of the implicit constants in the above inequality depend on [MATH], [MATH], and [MATH].', 'math-0402192-2-15-4': 'Here: [EQUATION] with the analogous norm defined for [MATH].', 'math-0402192-2-16-0': 'A short calculation like the one done above shows that in fact (modulo [MATH] angular derivatives), all of the estimates [REF] are sharp when tested on Knapp counterexamples.', 'math-0402192-2-16-1': 'Therefore, in this sense, they are all endpoint estimates.', 'math-0402192-2-16-2': 'It would be interesting to try and remove the extra [MATH] in these.', 'math-0402192-2-16-3': 'We will not pursue this issue here, although we will do some extra work in the sequel to recover a sharp [MATH] dispersive estimate which could be a start in this direction (see Proposition [REF]).', 'math-0402192-2-17-0': 'For the case of [MATH] spatial dimensions, we will also prove an estimate of the type [MATH] and [MATH] respectively which involves [MATH] an angular derivative.', 'math-0402192-2-17-1': 'However, to obtain the full range of [REF] in the case of [MATH] spatial dimensions would require one to prove an [MATH] Strichartz estimate that involves [MATH] angular derivatives.', 'math-0402192-2-17-2': 'While it seems that this type of estimate is out of the reach of methods we use here, it should be attainable using the recent method wave packet of Wolff [CITATION].', 'math-0402192-2-17-3': 'In fact, all of the estimates [REF] should be able to be proved directly using that method, based on the fact that they correspond to estimates that should be true for angularly separated initial data with no extra angular regularity.', 'math-0402192-2-17-4': 'ret', 'math-0402192-2-18-0': 'The remainder of this paper is laid out as follows.', 'math-0402192-2-18-1': 'In the next section, we list briefly some of the basic notations we use here.', 'math-0402192-2-19-0': 'In the third section, we list some standard facts about analysis on the sphere that will be useful in the sequel, including formulas for Hankel transforms and Littlewood-Paley-Stein theory on the sphere.', 'math-0402192-2-19-1': 'We then use this machinery to reduce the proof of Theorem [REF] to a suitable set of "endpoint" estimates.', 'math-0402192-2-20-0': 'In the fourth section, we discuss the main tool to be used in this paper: a [MATH]-type transform for the Hankel transform.', 'math-0402192-2-20-1': 'This leads us to consider the localization properties in physical space of the bandwidth limited Hankel transform.', 'math-0402192-2-20-2': 'In particular, we provide detailed asymptotics for our Hankel-[MATH] transform which will form the backbone the Strichartz estimates to be proved here.', 'math-0402192-2-21-0': 'In the fifth, we prove an [MATH] dispersive estimate for the wave equation based on angular regularity with respect to the momentum operators [REF].', 'math-0402192-2-21-1': 'This can be interpolated with the energy estimate to prove our "endpoints" directly via integration in time, avoiding to use of any other analytic machinery such as usual [MATH] process or induction on scales.', 'math-0402192-2-22-0': 'In the sixth, we give a brief description of how our dispersive estimate can be modified in a straight forward manner to accommodate multilinear phenomena.', 'math-0402192-2-22-1': 'This shows one strength of the method used here in that one gets the expected range of improved multilinear estimates virtually for free out of the machinery developed.', 'math-0402192-2-22-2': 'We will also discuss why these multilinear estimates are sharp, by testing them on a multilinear analog of the Knapp counterexamples introduced above.', 'math-0402192-2-23-0': 'In an appendix to this paper, we provide a proof of the angularly regular endpoints in the [MATH] regime based on an idea of Igor Rodnianski.', 'math-0402192-2-23-1': 'The proof there is essentially independent of the machinery we develop here (modulo a somewhat similar setup in terms of multilinear estimates) and instead relies on a direct calculation involving the energy-momentum tensor for the wave equation.', 'math-0402192-2-24-0': '# Basic notation', 'math-0402192-2-25-0': 'We list here some of the basic notation used throughout this paper.', 'math-0402192-2-25-1': 'For quantities [MATH] and [MATH], we denote by [MATH] to mean that [MATH] for some large constant [MATH].', 'math-0402192-2-25-2': 'The constant [MATH] may change from line to line, but will always remain fixed for any given instance where this notation appears.', 'math-0402192-2-25-3': 'We will also use the notation [MATH] if there exists a constant [MATH] such that [MATH].', 'math-0402192-2-26-0': 'For a given function of tow variables, say [MATH], we denote the mixed Lebesgue spaces norms [MATH] of [MATH] via the formulas: [EQUATION]', 'math-0402192-2-26-1': 'For a given function of the spatial variable only, we denote its Fourier transform as: [EQUATION]', 'math-0402192-2-26-2': 'With this normalization of the Fourier variables, the Plancherel theorem becomes [MATH], and one has the Fourier inversion formula: [EQUATION]', 'math-0402192-2-26-3': 'Using the Fourier transform, we define the homogeneous Sobolev space of order [MATH] via the identity [MATH].', 'math-0402192-2-27-0': 'For every integer [MATH] we define the spatial Littlewood-Paley cutoff operator by the formula: [EQUATION] where [MATH] and [MATH] is a positive smooth bump function, [MATH] on the interval [MATH] and zero off the interval [MATH].', 'math-0402192-2-27-1': 'With this notation we have the following consequence of the Littlewood-Paley theorem for [MATH] and [MATH]: [EQUATION]', 'math-0402192-2-28-0': 'For a given function of the spatial variable only, we denote its forward and backward wave propagation via the formulas: [EQUATION]', 'math-0402192-2-28-1': 'If [MATH] is an arbitrary solution to the homogeneous wave equation, [MATH], with initial data [MATH] and [MATH], we may decompose it into forward and backward wave propagation in the following way: [EQUATION]', 'math-0402192-2-28-2': 'Notice that in the case where [MATH] is unit frequency, the functions: [EQUATION] have [MATH] norm comparable to the [MATH] norm of [MATH].', 'math-0402192-2-28-3': 'Therefore, in the sequel, we will always assume that our unit frequency waves are of the form [MATH], and we replace: [EQUATION]', 'math-0402192-2-29-0': '# Some results from analysis on the sphere.', 'math-0402192-2-30-0': 'We list here some basic results from Fourier analysis is spherical coordinates which will be used in our proof of Theorem [REF].', 'math-0402192-2-30-1': 'We have already introduced the two basic differential elements of analysis on the sphere, the rotation vector fields [MATH] and the spherical Laplacian [MATH].', 'math-0402192-2-30-2': 'For every integer [MATH], there exists a finite dimensional set of functions [MATH] on the sphere [MATH] with satisfy the equation: [EQUATION]', 'math-0402192-2-30-3': 'These sets of functions exhaust the set of eigenfunctions of [MATH] and can in fact be identified with the homogeneous polynomials [MATH] on [MATH] of degree [MATH] which satisfy: [EQUATION]', 'math-0402192-2-30-4': 'Setting [MATH], and introducing the natural inner product on [MATH]: [EQUATION] we have the following basic properties of the [MATH]:', 'math-0402192-2-31-0': '[Basic properties of spherical harmonics (see e.g. [CITATION]]', 'math-0402192-2-32-0': 'The dimension of the space [MATH] is [MATH].', 'math-0402192-2-32-1': 'The spaces [MATH] are mutually orthogonal.', 'math-0402192-2-32-2': 'That is [MATH] for [MATH].', 'math-0402192-2-32-3': 'Let [MATH] be any orthonormal basis of [MATH], then one has the following identity for all [MATH]: [MATH].', 'math-0402192-2-32-4': 'For each [MATH], we have the identity:', 'math-0402192-2-33-0': '[MATH]', 'math-0402192-2-34-0': '>From now on, we fix an orthonormal basis [MATH] for each [MATH].', 'math-0402192-2-34-1': 'For a given function [MATH], we may expand it in the [MATH] sense along this basis as follows: [EQUATION]', 'math-0402192-2-34-2': 'Using [REF], we can define the action of [MATH] on this [MATH] as follows: [EQUATION]', 'math-0402192-2-34-3': 'Then using item (2) and (4) of Lemma [REF], and the fact that [MATH] is a homogeneous harmonic polynomial of degree [MATH] on [MATH]), we see that we have the identity: [EQUATION]', 'math-0402192-2-34-4': 'Using this, we see that there is, for every test function [MATH] on [MATH] an equivalence of norms: [EQUATION] where [MATH] is the norm from line [REF].', 'math-0402192-2-34-5': 'We also make the definition: [EQUATION]', 'math-0402192-2-34-6': 'Now, using the fact that [MATH] is of the form [MATH], we have the following standard interpolation result (see [CITATION]):', 'math-0402192-2-35-0': 'We will also use here some Littlewood-Paley theory in the angular variable.', 'math-0402192-2-35-1': 'We proceed in analogy with [REF] and let [MATH] be a smooth bump function such that [MATH] on the interval [MATH] and vanishing off the interval [MATH].', 'math-0402192-2-35-2': 'For each [MATH] we denote its dyadic rescaling as [MATH].', 'math-0402192-2-35-3': 'Defining [MATH] and using the decomposition formula [REF], we define the angular frequency dyadic projections of a function [MATH] on the sphere as: [EQUATION]', 'math-0402192-2-35-4': 'We define [MATH] to be the constant [MATH], which is the average of [MATH] over the sphere.', 'math-0402192-2-35-5': 'A formula similar to [REF] can be used to define [MATH] for a function on the whole of [MATH].', 'math-0402192-2-36-0': "For a given function [MATH] on the sphere, we use line (1) from [REF], as well as the fact that [MATH] when [MATH] to prove Bernstein's inequality for the sphere: [EQUATION]", 'math-0402192-2-36-1': 'By rescaling [REF] to spheres of various radii, we have the following result on all of [MATH] (for [MATH], we of course replace the [MATH] on the right hand side by [MATH]): [EQUATION]', 'math-0402192-2-36-2': 'Here [MATH] denotes the sphere of radius [MATH] centered at the origin.', 'math-0402192-2-37-0': 'Next, we record the basic result which allows us to generate certain square function expressions in terms of the [MATH] for [MATH] spaces when [MATH]:', 'math-0402192-2-38-0': 'Using Proposition [REF] along with the decomposition [REF], we have the following estimates for functions on space-time, for [MATH] and [MATH]: [EQUATION]', 'math-0402192-2-39-0': '## Reduction of Theorem [REF] to a frequency localized endpoint', 'math-0402192-2-40-0': 'We now use the setup we have introduced above to reduce the proof of Theorem [REF] to the following frequency localized "endpoint" estimate:', 'math-0402192-2-41-0': 'In the case of [MATH] spatial dimensions, we will also prove the following:', 'math-0402192-2-42-0': 'Assuming now the validity of Proposition [REF], we prove Theorem [REF] as follows: Given exponents [MATH] such that [MATH] and [MATH], we first reduce things to the case where [MATH].', 'math-0402192-2-42-1': 'We define [MATH] and [MATH] via the formulas: [EQUATION]', 'math-0402192-2-42-2': 'Notice that we can find such a [MATH] and [MATH] due to the range of [MATH].', 'math-0402192-2-42-3': 'Therefore, interpolating with the energy estimate (using Proposition [REF]), it suffices to prove [REF] for indices [MATH] with [MATH].', 'math-0402192-2-43-0': 'Next, using the decomposition [REF] an rescaling the spatial frequency of each term in the resulting sum, it suffices to show that: [EQUATION]', 'math-0402192-2-43-1': 'To do this, we choose: [EQUATION] such that there exists an [MATH] with the property that: [EQUATION]', 'math-0402192-2-43-2': 'That such choices are possible follows from our assumptions on the range of [MATH] and the identity: [EQUATION] where we have set [MATH].', 'math-0402192-2-43-3': 'Because of the range [REF], we see that is is also possible to choose an [MATH] with the property that: [EQUATION]', 'math-0402192-2-43-4': 'Furthermore, using Proposition [REF] and possibly a Sobolev embedding, we see that we have the estimate: [EQUATION]', 'math-0402192-2-43-5': 'Interpolating this last line with the [MATH] endpoint of [REF] we have achieved [REF].', 'math-0402192-2-43-6': 'Therefore, in the sequel, we will concentrate on the proof of [REF].', 'math-0402192-2-44-0': '## The Hankel transform', 'math-0402192-2-45-0': 'Finally, to wrap things up for this section, we record here the following formula for the action of the inverse Fourier transform on the decomposition: [EQUATION]', 'math-0402192-2-45-1': 'As is well known, this is given by a series of Hankel transforms.', 'math-0402192-2-45-2': 'The formula is (see [CITATION]): [EQUATION]', 'math-0402192-2-45-3': 'Here [MATH] is the Bessel function of order [MATH].', 'math-0402192-2-45-4': 'For [MATH], this is given by the integral formula: [EQUATION]', 'math-0402192-2-46-0': '# The Hankel-[MATH] transform', 'math-0402192-2-47-0': 'As we see from the formula [REF] of the last subsection, it is possible to expand the expression [MATH] in terms of spherical harmonics as: [EQUATION] where the coefficients [MATH] are given by the Hankel transform formula: [EQUATION]', 'math-0402192-2-47-1': 'Here, as in the previous subsection, the [MATH] are the coefficients in the spherical harmonic expansion of [MATH].', 'math-0402192-2-47-2': 'Also, the coefficients [MATH] should not be confused with the inverse Fourier transform of the [MATH].', 'math-0402192-2-48-0': 'We would now like to be able to localize the expressions [REF] in a manner analogous to the localization of the integral [REF].', 'math-0402192-2-48-1': 'This would be a relatively simple matter, if we could show that the asymptotic [REF] held uniformly in [MATH].', 'math-0402192-2-48-2': 'That is, if there was an asymptotic of the form [REF] for [MATH] which held uniformly as [MATH].', 'math-0402192-2-48-3': 'Unfortunately, it is well known that this is only the case for the region [MATH] (see [CITATION]).', 'math-0402192-2-48-4': 'In the transition regions, that is when [MATH], the asymptotic for [MATH] becomes quite complicated.', 'math-0402192-2-48-5': 'Roughly speaking, it begins to loose oscillations in [MATH] while it gains decay in the parameter [MATH].', 'math-0402192-2-48-6': 'Because of this, it does not seem feasible to try and compute an approximate formula for [MATH] and then substitute it into the integrals [REF].', 'math-0402192-2-48-7': 'Instead we will use a more straight forward approach, by first localizing the [MATH] in frequency as a Fourier series, just as we had done for [MATH] in the integral [REF], and then computing the integral [REF] directly by using appropriate integral representations for the [MATH].', 'math-0402192-2-49-0': 'Since we are assuming that the initial data in Theorem [REF] is unit frequency, we will assume that all of the coefficient functions [MATH] in the integrals [REF] are supported on the interval [MATH].', 'math-0402192-2-49-1': 'We may take their Fourier series developments on the interval [MATH], and we record these as: [EQUATION]', 'math-0402192-2-49-2': 'Expanding the integral [REF] in terms of the above formula, we see that: [EQUATION] where: [EQUATION]', 'math-0402192-2-49-3': 'In the above formula [MATH] is a smooth bump function on the interval [MATH].', 'math-0402192-2-49-4': 'Notice that this is not necessarily equal to [MATH] on any interval because we have absorbed the volume element into our definition of [MATH].', 'math-0402192-2-49-5': 'We call the right hand side of [REF] the Hankel-[MATH] transform of the function [MATH].', 'math-0402192-2-49-6': 'We would now like to be able to give a precise bound on how well localized the functions [MATH] are in physical space for the various values of the half-integer parameter [MATH] and the real variable [MATH].', 'math-0402192-2-49-7': 'This brings us to the main result of this subsection:', 'math-0402192-2-50-0': 'The downside of the above asymptotic is of course the region governed by [REF].', 'math-0402192-2-50-1': 'When [MATH], one can see that the extra factor [REF] will allow this asymptotic to look like [REF].', 'math-0402192-2-50-2': 'Notice that this is consistent with the fact that one has the asymptotic [REF] for [MATH] in this region, and is what is responsible for the good localization [REF] for spherically symmetric waves.', 'math-0402192-2-50-3': 'Unfortunately, it does not seem like one can do much to improve [REF] in the region where [MATH] (except for the extra factor of [MATH]).', 'math-0402192-2-50-4': 'In fact, if one assumes there is an asymptotic for [MATH] in this region which is of the form [REF], and one sets the [MATH] in the sum [REF] for a fixed [MATH], by putting absolute values around the sum [REF] one would get an asymptotic that looks like: [MATH].', 'math-0402192-2-50-5': 'But in this case, [MATH] corresponds to a delta function along the radial variable in Fourier space, say supported at the point [MATH].', 'math-0402192-2-50-6': 'Therefore we would have shown a bound like [MATH] uniform in [MATH].', 'math-0402192-2-50-7': 'This violates the well known asymptotic for Bessel functions: [MATH] (see [CITATION]).', 'math-0402192-2-50-8': 'It would be interesting to know if there is a more coherent decomposition of the Hankel transform that could eliminate this problem.', 'math-0402192-2-51-0': '[proof of Proposition [REF]] The asymptotics [REF]-[REF] follow more or less directly from appropriate integral formulas for the [MATH].', 'math-0402192-2-51-1': 'We will need to split the proof into the two cases: [MATH] or [MATH] and [MATH].', 'math-0402192-2-52-0': 'Case 1: [MATH] or [MATH].', 'math-0402192-2-53-0': 'Here we use a standard integral representation for Bessel functions which differs from [REF].', 'math-0402192-2-53-1': 'For [MATH], one has the following formula: [EQUATION]', 'math-0402192-2-53-2': 'This can be proved by a simple recursive argument (see [CITATION], Chapter 4, Lemma 3.1).', 'math-0402192-2-53-3': 'All one has to is to show that for both the integral formulas [REF] and [REF], the function [MATH] satisfies the recursive relation: [EQUATION] for [MATH].', 'math-0402192-2-53-4': 'In light of [REF], the equality of [REF] and [REF] is reduced to showing that it is true when [MATH].', 'math-0402192-2-53-5': 'This can be achieved directly through a change of variables.', 'math-0402192-2-54-0': 'Now using periodicity, integrating over an adjacent interval of length [MATH], and averaging, we see that for [MATH] the following integral representation also holds: [EQUATION]', 'math-0402192-2-54-1': 'Moreover, by a direct calculation, its not hard to see that the recursive relation [REF] is satisfied by both the integrals [REF] and [REF] whenever [MATH].', 'math-0402192-2-54-2': 'Therefore, throughout the sequel, we may assume that [REF] is our definition of the Bessel function that appears in the integral formula [REF] for the function [MATH] in dimension [MATH].', 'math-0402192-2-54-3': 'Making this substitution yields: [EQUATION]', 'math-0402192-2-55-0': 'We begin by proving the asymptotic [REF].', 'math-0402192-2-55-1': 'In fact, we will prove a bit more.', 'math-0402192-2-55-2': 'We will show that the asymptotic [REF] holds for [MATH].', 'math-0402192-2-55-3': 'Our first step will be to pick up the decay in the [MATH] parameter by integrating by parts the expression [REF] [MATH] times.', 'math-0402192-2-55-4': "The resulting expression looks like: [EQUATION] where the [MATH] in the above formula denote some specific trigonometric polynomials of degree [MATH] who's exact form is not important for our analysis.", 'math-0402192-2-55-5': 'The next step is to gain the decay in [MATH] in conjunction with the damping in terms of inverse powers of [MATH].', 'math-0402192-2-55-6': 'To get this, we Taylor expand each [MATH], where [EQUATION] around the point [MATH].', 'math-0402192-2-55-7': 'Notice that this is consistent with the fact that we are investigating the region where [MATH] is bounded.', 'math-0402192-2-55-8': 'We now define the dimensional constant [MATH], we record this Taylor expansion as: [EQUATION] where [MATH] is some smooth function such that [MATH].', 'math-0402192-2-55-9': 'Substituting the Taylor expansion [REF] into the integral [REF], we see that we may write: [EQUATION] where: [EQUATION] and: [EQUATION]', 'math-0402192-2-55-10': 'We can further estimate the term [MATH] above by using the fact that [MATH] and all of its derivatives have rapid decay away from the origin.', 'math-0402192-2-55-11': 'This follows immediately from the fact that [MATH] is the inverse Fourier transform of a smooth [MATH] bump function.', 'math-0402192-2-55-12': 'We record this observation as: [EQUATION]', 'math-0402192-2-55-13': 'By adding things up and introducing a large enough constant, this allows us to write: [EQUATION]', 'math-0402192-2-55-14': 'Recalling now that we have set [MATH] and that we also have [MATH], we see that in order to achieve the bound [REF], all we need to do is to control the expression for [MATH].', 'math-0402192-2-55-15': 'This is easy to do because a moments inspection shows that in fact one has [MATH].', 'math-0402192-2-55-16': 'This can be readily seen for even dimensions, that is when [MATH] is even, because in this case the trigonometric polynomials under the integral sign in the expression for [MATH] are of degree strictly less than [MATH].', 'math-0402192-2-55-17': 'By orthogonality, the whole expression then integrates to zero.', 'math-0402192-2-55-18': 'In the case of odd dimension, each integral is still zero thanks to the fact that [MATH] is a half integer expression, where as the term: [EQUATION] is a trigonometric polynomial of integer degree.', 'math-0402192-2-55-19': 'Since we are integrating over the double torus, [MATH], a rescaling turns the expression under the integral sign in [MATH] into a product of even degree trigonometric polynomials and odd degree trigonometric polynomials.', 'math-0402192-2-55-20': 'Therefore one has the needed orthogonality.', 'math-0402192-2-55-21': 'This completes the proof of [REF].', 'math-0402192-2-56-0': 'We now turn our attention to proving the asymptotic [REF] for the regime where [MATH].', 'math-0402192-2-56-1': 'For the remainder of this section we will assume that [MATH] is positive, as the other case can be dealt with by a similar argument.', 'math-0402192-2-56-2': 'Furthermore, using the bound that we proved in the previous discussion, we can without loss of generality assume that [MATH].', 'math-0402192-2-57-0': 'Our first step is to split the integral on the right hand side of [REF] smoothly into the regions where [MATH] and otherwise.', 'math-0402192-2-57-1': 'To realize this split, we restrict the integral [REF] to the regions: [EQUATION]', 'math-0402192-2-57-2': 'Notice that by symmetry, we only need to consider the regions [MATH] and [MATH].', 'math-0402192-2-57-3': 'On these regions, a bit of explicit computation using Taylor expansions shows that: [EQUATION] where the function [MATH] in the first line above satisfies the bound: [EQUATION]', 'math-0402192-2-57-4': 'Notice that to get [REF] we have used the condition [MATH].', 'math-0402192-2-57-5': 'Next, using another simple calculation involving the Taylor series of trigonometric functions, as well as the estimates [REF], [REF] and [REF], we see that one may write for [MATH]: [EQUATION]', 'math-0402192-2-57-6': 'We are now in a position to bound the integral [REF] on the region [MATH].', 'math-0402192-2-57-7': 'To do this, we will employ a smooth cutoff function [MATH] such that: [EQUATION]', 'math-0402192-2-57-8': 'We are trying to prove the estimate [REF] for the expression: [EQUATION]', 'math-0402192-2-57-9': 'Integrating by parts [MATH] times in the above integral and using the bound [REF] (along with the fact that we are in dimension [MATH]) we compute: [EQUATION]', 'math-0402192-2-57-10': 'Thus, we have proved [REF] for this portion of things.', 'math-0402192-2-58-0': 'It remains to prove the estimate [REF] for the region [MATH].', 'math-0402192-2-58-1': 'We suggestively (c.f. [REF]) denote the cutoff here by [MATH].', 'math-0402192-2-58-2': 'The calculation is essentially the same as what was done above, except that here we can afford to be more careless about the powers of [MATH] which come up through integration by parts.', 'math-0402192-2-58-3': 'Integrating by parts [MATH] times and using the bound [REF] in conjunction with the estimate [REF], we see that: [EQUATION]', 'math-0402192-2-58-4': 'This completes the proof of [REF] and ends the demonstration of case 1.', 'math-0402192-2-59-0': 'Case 2: [MATH].', 'math-0402192-2-60-0': 'In this section, the main difficulty will be for us to incorporate the remainder term [REF] into our asymptotic.', 'math-0402192-2-60-1': 'In order to motivate the steps we will take here, we argue heuristically as follows: We first define the auxiliary function: [EQUATION]', 'math-0402192-2-60-2': 'It is clear that the main contribution to the integral [REF] integrating around points where the function [MATH] defined above vanishes.', 'math-0402192-2-60-3': 'By symmetry, we need only consider such points which are positive.', 'math-0402192-2-60-4': "We'll call the one closest to zero [MATH].", 'math-0402192-2-60-5': 'Now, expanding [MATH] around this zero gives: [EQUATION] where again [MATH].', 'math-0402192-2-60-6': 'Therefore, it is natural to expect that: [EQUATION] where the [MATH], [MATH], are the (unique) set of derivatives and derivatives of [MATH] which vanish at infinity.', 'math-0402192-2-60-7': 'These satisfy the bounds similar to [REF], even if [MATH] is positive: [EQUATION]', 'math-0402192-2-60-8': 'This, of course, comes from the fact that [MATH] is the inverse Fourier transform of a unit frequency function, and is therefore supported away from the origin in Fourier space.', 'math-0402192-2-60-9': 'Using [REF] we easily get the following bound for arbitrary non-negative [MATH]: [EQUATION]', 'math-0402192-2-61-0': 'In order to make the previous argument rigorous, we need to justify the two approximations used in the lines directly above [REF].', 'math-0402192-2-61-1': 'We will not be able to do this completely, which will be responsible for the extra term in the formula [REF] for [MATH].', 'math-0402192-2-61-2': 'As in the previous two subsections, we argue by isolating the interval of integration.', 'math-0402192-2-61-3': 'By symmetry, and the decay bound [REF], we can without loss of generality assume that we are integrating [REF] over the interval [MATH].', 'math-0402192-2-61-4': 'We will first need to go a little further and chop some more off of the left hand side of this interval.', 'math-0402192-2-61-5': "What we'll do is take [MATH] (we're still keeping [MATH]) to be such that: [EQUATION]", 'math-0402192-2-61-6': 'Taylor expanding [MATH] gives us the bound: [EQUATION]', 'math-0402192-2-61-7': 'Furthermore, we also have the bound: [EQUATION]', 'math-0402192-2-61-8': 'Now, using [REF] and [REF] above, in conjunction with the asymptotic [REF], we immediately see that: [EQUATION]', 'math-0402192-2-61-9': 'This is enough to give [REF] for this portion of things.', 'math-0402192-2-61-10': 'Therefore, it remains to compute the integral: [EQUATION]', 'math-0402192-2-61-11': 'Keeping the Taylor expansion [REF] in mind, we now make the following change of variable for [MATH]: [EQUATION]', 'math-0402192-2-61-12': 'Using this, we can write: [EQUATION]', 'math-0402192-2-61-13': 'Where [MATH] is the interval: [EQUATION]', 'math-0402192-2-61-14': 'The desired result will now follow by integrating by parts as many times as necessary the integral [REF].', 'math-0402192-2-61-15': 'However, some care needs to be taken in order to control terms involving [MATH], which can be as big as [MATH].', 'math-0402192-2-61-16': 'Also, one needs to know that the higher derivatives of [MATH] possess some decay in order to deal with terms where the derivatives fall on [MATH] instead of the exponential factor.', 'math-0402192-2-61-17': 'We address these issues now.', 'math-0402192-2-61-18': 'The first observation we make is that we may integrate the bound: [EQUATION] using the fact that [MATH] to get that: [EQUATION] over the range of integration.', 'math-0402192-2-61-19': 'However, this is not enough to address the issue of a product of the form [MATH].', 'math-0402192-2-61-20': 'To handle this, notice that one has the bound: [EQUATION] as can be seen from the fact that this bound is true for [MATH], and that upon differentiating both sides of this expression one is reduced to showing (the inequality for the derivatives is trivial for [MATH]): [EQUATION]', 'math-0402192-2-61-21': 'This then follows immediately from the increasing nature of [MATH] on this interval and from the identity [MATH].', 'math-0402192-2-61-22': 'Now, combining the bounds [REF], [REF] and [REF], we see that for any positive integers [MATH], one has the estimates: [EQUATION]', 'math-0402192-2-61-23': 'Finally, we record the fact that the higher derivatives of [MATH] satisfy the following simple bounds, which can be verified through direct calculation: [EQUATION]', 'math-0402192-2-62-0': 'We are now ready to bound [REF].', 'math-0402192-2-62-1': 'First, we integrate as many times as necessary, letting the derivatives fall on the term [MATH].', 'math-0402192-2-62-2': 'The resulting expression looks like: [EQUATION]', 'math-0402192-2-62-3': 'Expanding out the derivatives in the first term of the above expression, and using the bounds [REF] and [REF], we see that we can bound: [EQUATION]', 'math-0402192-2-62-4': 'Therefore, the integral in the first term above does not cause us any trouble.', 'math-0402192-2-62-5': 'As far as the boundary values are concerned, we can use the bounds [REF], [REF], [REF], and [REF] to show that: [EQUATION]', 'math-0402192-2-63-0': 'It remains to bound [REF] where we integrate by parts and let the derivatives fall on the exponential factor.', 'math-0402192-2-63-1': 'Doing this [MATH] times yields: [EQUATION]', 'math-0402192-2-63-2': 'To control the first term above, we bound: [EQUATION] where [MATH] is a constant, and the other [MATH] denote a homogeneous expression of weight [MATH] in the variables [MATH], where each [MATH] is given weight [MATH].', 'math-0402192-2-63-3': 'Therefore, using the bound [REF] (which in particular implies that [MATH]) as well as the bounds [REF], we see that we can estimate: [EQUATION]', 'math-0402192-2-63-4': 'Finally, to deal with the boundary terms on the right hand side of [REF] we use the bounds [REF] and [REF] and simply estimate: [EQUATION]', 'math-0402192-2-64-0': 'Combining the above estimates together, we have shown that: [EQUATION]', 'math-0402192-2-64-1': 'This completes the proof of the asymptotic [REF], and the demonstration of Proposition [REF].', 'math-0402192-2-65-0': '# Some [MATH] dispersive estimates for the wave equation; linear and bilinear estimates', 'math-0402192-2-66-0': 'We are now ready to directly proceed with the proof of estimate [REF].', 'math-0402192-2-66-1': 'As we have mentioned previously, we may assume that [MATH] is of the form [MATH], for some unit frequency and dyadic angular frequency function [MATH].', 'math-0402192-2-66-2': 'For this [MATH], we the formulas from lines [REF], [REF], [REF], [REF], and [REF] to expand its wave propagation into harmonics and Hankel-[MATH] transforms as follows: [EQUATION]', 'math-0402192-2-66-3': 'In the above expression, we have absorbed the constants [MATH] into each [MATH].', 'math-0402192-2-66-4': 'By orthogonality of everything in sight, we have that: [EQUATION]', 'math-0402192-2-66-5': 'We now state our main result as follows:', 'math-0402192-2-67-0': 'Before proceeding with the proof, let us first show briefly how the above estimate may be used to show [REF]-[REF].', 'math-0402192-2-67-1': 'We begin with [REF].', 'math-0402192-2-67-2': 'What we need to do is to show that for every [MATH], there exists an [MATH], such that the following estimate holds: [EQUATION] where the implicit constants depend on both [MATH] and [MATH].', 'math-0402192-2-67-3': 'We also need [MATH] to approach [MATH] as [MATH].', 'math-0402192-2-67-4': 'Now, interpolating [REF] with the energy estimate [REF] gives the following result for [MATH]: [EQUATION]', 'math-0402192-2-67-5': 'Choosing [MATH] according to the formula [MATH], we see that we have the following identity holds: [MATH].', 'math-0402192-2-67-6': 'Therefore, for this choice of [MATH], we may square [REF] and integrate directly in time to achieve [REF].', 'math-0402192-2-67-7': 'Also note that [MATH] as [MATH].', 'math-0402192-2-68-0': 'In the case of estimate [REF], that is [MATH] spatial dimensions, we use Holders inequality to see that for every [MATH] the following estimate holds: [EQUATION]', 'math-0402192-2-68-1': 'Therefore, of we choose [MATH] a direct integration and the energy bound [REF] shows that we have: [EQUATION]', 'math-0402192-2-68-2': 'We are now reduced to proving the estimate [REF].', 'math-0402192-2-69-0': '[proof of Proposition [REF]] Writing everything out, and using a Cauchy-Schwartz in the sum over [MATH] for each fixed [MATH] we have that: [EQUATION]', 'math-0402192-2-69-1': 'Now, using the formula in item (3) of Lemma [REF], and the fact that [MATH] for [MATH], we have: [EQUATION] where the last line above follows from a Cauchy-Schwartz and the bound [MATH].', 'math-0402192-2-69-2': 'The proposition will now be shown if we can prove that: [EQUATION]', 'math-0402192-2-69-3': 'We now use the asymptotics of Proposition [REF] and a little computation to split: [EQUATION] where [MATH] is supported where [MATH], and: [EQUATION]', 'math-0402192-2-69-4': 'Here [MATH] is the function from line [REF].', 'math-0402192-2-69-5': 'Substituting [MATH] into the left hand side of the sum [REF], and using a Cauchy-Schwartz immediately yields the desired result for this half of things.', 'math-0402192-2-69-6': 'In order to finish up, then, we only need substitute to [MATH] into the right hand side.', 'math-0402192-2-69-7': 'Doing this and using a Cauchy-Schwartz and re-indexing, we see that it is enough to show: [EQUATION] uniform in [MATH] and [MATH].', 'math-0402192-2-69-8': 'This can be done by comparing things to the appropriate integrals and is left to the reader.', 'math-0402192-2-69-9': 'Notice that the convergence factor [MATH] avoids any logarithmic divergences in this sum.', 'math-0402192-2-69-10': 'This completes the proof of [REF] and therefore the proof of Theorem [REF].', 'math-0402192-2-70-0': '# Bilinear estimates for angularly regular data', 'math-0402192-2-71-0': 'As is well known, for applications in the lower dimensional setting, linear estimates of the form [REF] are not sufficient.', 'math-0402192-2-71-1': 'What is needed are multilinear versions.', 'math-0402192-2-71-2': 'An extremely versatile method for building these type of estimates is based on the "fine and coarse scale" machine of T. Tao (see [CITATION]).', 'math-0402192-2-71-3': 'The basic idea is to fix a scale, say [MATH] for [MATH], and then decompose the domain of spatial variable into cubes with side lengths of this scale.', 'math-0402192-2-71-4': 'Then, one replaces the usual [MATH] norm in the spatial variable with [MATH], where the [MATH] norm is taken on the "fine" scale of each individual cube, while the [MATH] norm represents the "coarse" scale which is summation over all cubes.', 'math-0402192-2-71-5': 'One reason this method is so important, is that it allows one to use the bilinear construction process directly in an iteration procedure where resorting to the canned estimates that this method ultimately provides may be unduly burdensome.', 'math-0402192-2-71-6': 'Also, the way these estimates are constructed will allow us to generate bilinear estimates where only one term in the product contains extra angular regularity.', 'math-0402192-2-71-7': 'These type of estimates are extremely important for applications when one proves inductive estimates via an iteration procedure where it is necessary to consider estimates for products with an angular derivative falling on one or the other term in the product.', 'math-0402192-2-71-8': 'Because of these considerations, we will content ourselves here with the dual scale estimates themselves, and not bother with listing out the various multilinear estimates which follow from them.', 'math-0402192-2-71-9': 'In the [MATH] and higher dimensions, the usual dual scale estimates read:', 'math-0402192-2-72-0': '["Improved" Strichartz estimates] Let [MATH] be the number of spatial dimensions.', 'math-0402192-2-72-1': 'Let [MATH] be given, and let [MATH] be a partition of [MATH] into cubes of side length [MATH].', 'math-0402192-2-72-2': 'Then if [MATH] is a unit frequency solution to the equation [MATH], the following estimates hold: [EQUATION] where [MATH].', 'math-0402192-2-73-0': 'We show here that both the range of [REF] and the power of [MATH] that appearers there can be significantly improved.', 'math-0402192-2-73-1': 'In what follows, we will only bother proving an estimate which is the analog of [REF].', 'math-0402192-2-73-2': 'More general estimates which involve various amounts of angular regularity can then be gained by interpolating this estimate with [REF].', 'math-0402192-2-73-3': 'Also, it is not so easy to work out the Littlewood-Paley theory in the angular variable for localized norms like those that appear on the left hand side of [REF].', 'math-0402192-2-73-4': 'However, since we are already loosing a small amount of angular regularity, one may simply replace the Littlewood-Paley sum [REF] by an [MATH] sum.', 'math-0402192-2-73-5': 'As we have noted before, this has no bearing on applications.', 'math-0402192-2-74-0': '["Improved" frequency localized Strichartz estimates for angularly regular data; endpoint case] Let [MATH] be the number of spatial dimensions, and let [MATH] be a unit frequency and angular frequency localized solution to the homogeneous wave equation [MATH].', 'math-0402192-2-74-1': 'Let [MATH] be given, and let [MATH] be a partition of [MATH] into cubes of side length [MATH].', 'math-0402192-2-74-2': 'Then for every [MATH], there is a [MATH] and [MATH] depending on [MATH], such that [MATH] as [MATH] such that the following estimate holds: [EQUATION]', 'math-0402192-2-75-0': 'Up to the small loss in [MATH] and [MATH], the estimate [REF] is sharp when tested against the bilinear analog of the Knapp counterexamples [REF].', 'math-0402192-2-75-1': 'We construct these briefly as follows.', 'math-0402192-2-75-2': 'We consider (frequency) initial data sets [MATH] which along with being highly localized in the angular variable, are also well localized in the radial variable.', 'math-0402192-2-75-3': 'That is, we are now assuming [MATH] is supported on a small square of dimensions [MATH], lying along the [MATH] axis between [MATH].', 'math-0402192-2-75-4': 'A quick calculation then shows that the integral: [EQUATION] behaves like [MATH] on the space-time region [MATH]: [EQUATION]', 'math-0402192-2-75-5': 'Choosing our cubes [MATH] with side lengths [MATH], we see that for any [MATH] (in fact for any [MATH]) the following is true: [EQUATION]', 'math-0402192-2-75-6': 'Therefore, using [REF], we see that [REF] is indeed sharp for this sequence of initial data.', 'math-0402192-2-75-7': 'Of course the condition [MATH] cannot be improved, even for spherically symmetric initial data.', 'math-0402192-2-75-8': 'In fact, one can also see for these type of waves (say with the asymptotic [REF]), [REF] is again sharp.', 'math-0402192-2-75-9': 'We leave this simple calculation to the interested reader.', 'math-0402192-2-75-10': 'ret', 'math-0402192-2-76-0': '[proof of estimate [REF]] The proof will be similar in spirit to that of [REF].', 'math-0402192-2-76-1': 'However, we will need to use orthogonality in a more fundamental way here.', 'math-0402192-2-76-2': 'This is because we will not be able to rely solely on [MATH] as we did in the proof of [REF].', 'math-0402192-2-76-3': 'Ultimately, this has to do with the fact that the wave packets [MATH] are not well localized in physical space when [MATH] and [MATH] (see the remark after Proposition [REF]).', 'math-0402192-2-77-0': 'In order to prove estimate [REF], we begin by writing the basic energy estimate [REF] in the following way (we are still using the notation from line [REF]): [EQUATION]', 'math-0402192-2-77-1': 'Our next step is to prove the following spatially localized fixed time estimate: [EQUATION]', 'math-0402192-2-78-0': 'As in the previous section, choosing [MATH] by the identity, [MATH], we have that [MATH].', 'math-0402192-2-78-1': 'Therefore, squaring [REF] and integrating directly in time, we will have achieved [REF].', 'math-0402192-2-78-2': 'Therefore, we now concentrate on proving [REF].', 'math-0402192-2-79-0': 'It suffices to show [REF] for a fixed [MATH].', 'math-0402192-2-79-1': 'Therefore, we will now assume that we are on a fixed cube [MATH].', 'math-0402192-2-79-2': 'Our first step is foliate [MATH] with the hypersurfaces [MATH], where [MATH] is the sphere of radius [MATH] centered at the origin.', 'math-0402192-2-79-3': 'Notice that for each [MATH], one has the area estimate [MATH].', 'math-0402192-2-79-4': 'Therefore, using a Cauchy-Schwartz, it suffices to prove the estimate: [EQUATION]', 'math-0402192-2-79-5': 'Next, we chop [MATH] into at most [MATH] dyadic pieces, which are of the form [MATH], where [MATH] is the radial dyadic region of size [MATH].', 'math-0402192-2-79-6': 'We only need to do this for [MATH], i.e we keep the ball of bounded radius (say radius [MATH]) as a single region [MATH].', 'math-0402192-2-79-7': 'Therefore, to show [REF], it suffices to estimate: [EQUATION]', 'math-0402192-2-79-8': 'Now fix [MATH].', 'math-0402192-2-79-9': 'To prove estimate [REF] here, we run the decomposition [REF] and decompose the resulting sum into the sum of pieces: [EQUATION] where: [EQUATION]', 'math-0402192-2-79-10': 'To estimate [REF] on the sum [MATH], we simply keep it as a whole object and use the Bernstein inequality [REF] to estimate: [EQUATION]', 'math-0402192-2-79-11': 'As was to be shown.', 'math-0402192-2-79-12': 'Therefore, we are reduced to bounding [MATH].', 'math-0402192-2-79-13': 'Notice that for each fixed [MATH], and for each term in this sum, we have [MATH].', 'math-0402192-2-79-14': 'Therefore, we can use the well localized asymptotic [REF] and a Cauchy-Schwartz in the [MATH] summation (in a way similar to the computation started on line [REF]) to bound this term in [MATH] by: [EQUATION]', 'math-0402192-2-79-15': 'Squaring this last expression, and integrating each term in the sum with respect to [MATH] (notice there is no extra volume element because we took the sup on [MATH]), we see that we have proved [REF] for this portion of things.', 'math-0402192-2-79-16': 'This then completes the proof of estimate [REF], and therefore the proof of [REF].', 'math-0402192-2-80-0': '# Appendix', 'math-0402192-2-81-0': 'We present here a simplified proof of Theorem [REF] and Theorem [REF].', 'math-0402192-2-81-1': 'First, we note that in order to prove both [REF] [EQUATION] and [REF] [EQUATION] with [MATH] and arbitrary small [MATH], it suffices to prove the estimate [REF].', 'math-0402192-2-81-2': 'Here [MATH] is a unit frequency solution of the wave equation [MATH] of angular frequency [MATH] and the norm [EQUATION] where [MATH] is a partition of [MATH] into cubes [MATH] of side length of [MATH].', 'math-0402192-2-82-0': 'To see this, we choose a partition [MATH] of size [MATH] and compute, using the Sobolev embedding on [MATH], that for any [MATH]: [EQUATION]', 'math-0402192-2-82-1': 'Therefore, we see that it suffices to deal with the estimate [REF].', 'math-0402192-2-82-2': 'First, using the Sobolev inequality [MATH] on the unit sphere [MATH] for angular frequency localized functions, we have the following estimate for any tiling of [MATH] by cubes [MATH] of side length [MATH]: [EQUATION]', 'math-0402192-2-82-3': 'Thus, [EQUATION]', 'math-0402192-2-82-4': 'Interpolating this with the trivial estimate [EQUATION] we arrive at the following estimate which will be our point of departure: [EQUATION] where [MATH].', 'math-0402192-2-82-5': 'Therefore, using [REF] and that [MATH] is another unit frequency solution of the wave equation localized at the angular frequency [MATH], we see that in order to prove [REF], it suffices to prove the following space-time Morawetz type estimate for unit frequency solutions of the homogeneous wave equation when [MATH]: [EQUATION]', 'math-0402192-2-82-6': 'In the above estimate, the implicit constant depends on [MATH] and the dimension, and from now on we will keep this dependence implicit.', 'math-0402192-2-82-7': "Estimate [REF], as well as as its version with [MATH] and a logarithmic loss in time, was proved in the work of Keel-Smith-Sogge [CITATION] in dimension three with the help of the sharp Huygen's principle.", 'math-0402192-2-82-8': 'We will prove [REF] directly using essentially nothing but an integration by part argument.', 'math-0402192-2-82-9': 'Before we continue, let us make one more reduction.', 'math-0402192-2-82-10': 'It turns out that [REF] is more naturally proved for the spatial gradient [MATH].', 'math-0402192-2-82-11': 'Since [MATH] is unit frequency, this reduction does not effect the validity of [REF].', 'math-0402192-2-82-12': 'Indeed, suppose we have the following estimate: [EQUATION] for any unit frequency solution [MATH].', 'math-0402192-2-82-13': 'Let [MATH] be another unit frequency solution with the property that [MATH].', 'math-0402192-2-82-14': 'Then for any [MATH], [EQUATION] and [REF] follows from [REF] by choosing a sufficiently small [MATH].', 'math-0402192-2-82-15': 'Therefore, we may now assume that we are trying to achieve estimate [REF].', 'math-0402192-2-82-16': 'The fact that [REF] contains the gradient [MATH] will allow us to prove it for an arbitrary solution to the homogeneous wave equation.', 'math-0402192-2-82-17': 'This will be done using some more or less standard energy-momentum tensor techniques.', 'math-0402192-2-82-18': 'Let [MATH] be a solution to the homogeneous wave equation: [EQUATION] and let [MATH] be its energy-momentum tensor: [EQUATION]', 'math-0402192-2-82-19': 'Here the Greek indices run on the set [MATH], and [MATH] is the standard Minkowski metric.', 'math-0402192-2-82-20': 'The key feature of [MATH] is that it is space-time divergence free: [EQUATION] where [MATH] denotes the Levi-Civita connection of [MATH].', 'math-0402192-2-82-21': 'We now contract [MATH] a radial vector-field: [EQUATION]', 'math-0402192-2-82-22': 'Define the momentum density: [EQUATION]', 'math-0402192-2-82-23': 'A quick calculation shows that the divergence of [MATH] satisfies the identity: [EQUATION] where: [EQUATION] is the deformation tensor of [MATH].', 'math-0402192-2-82-24': 'Introducing the orthonormal frame [MATH], where the [MATH] form an orthonormal frame on each [MATH] sphere [MATH], one can easily calculate this quantity to be: [EQUATION] where [MATH] denotes the metric on the spheres [MATH].', 'math-0402192-2-82-25': 'In particular, we see that: [EQUATION]', 'math-0402192-2-82-26': 'Next, using [REF] and [REF], a direct computation shows that: [EQUATION]', 'math-0402192-2-82-27': 'In the above formula, [MATH] denotes the angular portion of the spatial gradient [MATH].', 'math-0402192-2-82-28': 'Since, [MATH], if we define the modified momentum density: [EQUATION] we end up with the identity: [EQUATION]', 'math-0402192-2-83-0': 'Integrating [REF] over a time slab, we arrive at the following a-priori estimate for [MATH]: [EQUATION] where: [EQUATION] with an identical expression for the time [MATH] boundary piece on the right hand side of [REF] above.', 'math-0402192-2-83-1': 'In particular, using [REF], one has that if [MATH] then: [EQUATION] where in the last line we used the Hardy inequality.', 'math-0402192-2-83-2': 'Similar estimate holds for the other boundary term [MATH].', 'math-0402192-2-84-0': 'We now use [REF] to derive [REF] by choosing the weight function [MATH].', 'math-0402192-2-84-1': '[EQUATION] for some [MATH].', 'math-0402192-2-84-2': 'A direct calculation shows that, with this choice of [MATH], we have [EQUATION] in dimensions [MATH].', 'math-0402192-2-84-3': 'Therefore this gives the a-priori estimate: [EQUATION] with an implicit constant in [MATH] independent of [MATH].', 'math-0402192-2-84-4': 'In particular, choosing [MATH] with an integer [MATH], we obtain [EQUATION].', 'math-0402192-2-84-5': 'Dividing the above inequality by [MATH] and summing over [MATH] immediately yields the desired estimate [REF].', 'math-0402192-2-84-6': 'ret'}

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/math/0402192

cond-mat-0206183

{'cond-mat-0206183-1-0-0': 'We analyse the behavior of the density of states in a singlet [MATH]-wave superconductor with weak magnetic impurities in the clean limit.', 'cond-mat-0206183-1-0-1': 'By using the method of optimal fluctuation and treating the order parameter self-consistently we show that the density of states is finite everywhere in the superconducting gap, and that it varies as [MATH] near the mean field gap edge [MATH] in a [MATH]-dimensional superconductor.', 'cond-mat-0206183-1-0-2': 'In contrast to most studied cases the optimal fluctuation is strongly anisotropic.', 'cond-mat-0206183-1-1-0': 'One of the most intriguing problems in modern condensed matter physics is the combined effect of correlations and disorder on the ground state and the electronic properties of solids.', 'cond-mat-0206183-1-1-1': "Interplay of superconductivity and impurity scattering is an example of a situation where such an effect is relevant experimentally, and where it has been studied for over forty years, beginning with the seminal papers by Anderson [CITATION] and Abrikosov and Gor'kov (AG) [CITATION].", 'cond-mat-0206183-1-1-2': 'Despite such a long history, however, the spectral properties even in the simplest case of a singlet [MATH]-wave superconductor continue to attract attention.', 'cond-mat-0206183-1-2-0': 'Since nonmagnetic impurities do not break the time-reversal symmetry, the "hard" gap, [MATH], in the single particle excitation spectrum in unaffected by weak potential scattering [CITATION].', 'cond-mat-0206183-1-2-1': 'At the same time scattering by magnetic impurities destroys the phase coherence in the superconducting state, and, consequently, leads to the suppression of the gap, and of the superconducting order parameter.', 'cond-mat-0206183-1-2-2': 'AG analysed magnetic scattering using self-consistent Born approximation, and concluded that a hard gap in the energy spectrum survives up to a critical (average) concentration of weak magnetic impurities; it is followed, upon increasing impurity concentration, by a narrow region of gapless superconductivity, and then by destruction of the superconducting condensate.', 'cond-mat-0206183-1-3-0': 'Resurgence of interest in this problem started with the paper of Balatsky and Trugman [CITATION], who argued that rare regions, where impurity concentration is sufficient to locally destroy superconductivity, yield a finite density of states (DOS) at the Fermi level.', 'cond-mat-0206183-1-3-1': 'Their work was followed by the analyses of several other authors, most notably by Simons and co-workers [CITATION].', 'cond-mat-0206183-1-3-2': 'Our aim here is to elucidate the nature of the subgap states, and to provide a quantitative analysis of the energy profile of the DOS in [MATH]-wave superconductors with weak magnetic impurities.', 'cond-mat-0206183-1-4-0': 'In the AG theory the effect of disorder is controlled by a dimensionless parameter, [MATH], where [MATH] is the scattering time due to magnetic impurities, and [MATH] is the value of the superconducting order parameter.', 'cond-mat-0206183-1-4-1': 'In particular, the single particle spectral gap is [MATH], indicating the onset of the gapless superconductivity at [MATH].', 'cond-mat-0206183-1-4-2': 'In the regime studied here, [MATH], the AG theory predicts a gapped quasiparticle spectrum.', 'cond-mat-0206183-1-5-0': 'These results are obtained by carrying out a standard impurity averaging procedure.', 'cond-mat-0206183-1-5-1': 'It is clear, however, that, among all the realizations of the impurity distribution, there exist regions where the resulting potential generates localized quasiparticle states at an arbitrary energy below the gap edge.', 'cond-mat-0206183-1-5-2': 'Such localized states were extensively studied in doped semiconductors [CITATION].', 'cond-mat-0206183-1-5-3': 'For a particular energy, [MATH], the most probable (albeit still rare) configuration of impurities that creates a state at [MATH], and therefore contributes the most to the DOS, [MATH], is called the optimal fluctuation (OF) [CITATION].', 'cond-mat-0206183-1-5-4': 'Such rare regions provide nonperturbative corrections to the DOS determined in the framework of self-consistent Born approximation.', 'cond-mat-0206183-1-6-0': 'We employ the method of optimal fluctuation in a singlet [MATH]-wave superconductor with magnetic impurities.', 'cond-mat-0206183-1-6-1': 'We include in our analysis the effect of the self-consistent suppression of the superconducting order parameter; this is motivated in large part by the numerical evidence that in [MATH]-wave superconductors such suppression significantly affects the low-energy density of states [CITATION].', 'cond-mat-0206183-1-7-0': 'Our main finding is that there is a nonvanishing density of states everywhere in the superconducting gap.', 'cond-mat-0206183-1-7-1': 'In the most relevant region, just below the AG gap edge, the behavior of the density of states is given by [MATH], where [MATH] is the normal state density of states, and [MATH] is the dimension of the problem.', 'cond-mat-0206183-1-7-2': 'Unlike most of the studied cases, the optimal fluctuation is anisotropic, with its transverse size much smaller than the longitudinal extent.', 'cond-mat-0206183-1-8-0': 'We consider a singlet [MATH]-wave superconductor.', 'cond-mat-0206183-1-8-1': 'In the 4-space of the wave functions [MATH] the mean field hamiltonian is [EQUATION]', 'cond-mat-0206183-1-8-2': 'Here [MATH] is the kinetic energy of a quasiparticle with respect to the Fermi level, [MATH] is the chemical potential, and [MATH] and [MATH] are the Pauli matrices in the particle-hole and the spin space respectively, so that [MATH] should be understood as a 4[MATH]4 direct product.', 'cond-mat-0206183-1-8-3': 'The integration is over the [MATH] spatial dimensions, and [MATH] is the potential due to impurities.', 'cond-mat-0206183-1-9-0': 'In general [MATH] includes both potential and spin-flip scattering processes.', 'cond-mat-0206183-1-9-1': 'In the regime, when the potential scattering is dominant, i.e. when the motion of quasiparticles in the OF is diffusive, the properties of the low-energy states have been explored by Lamacraft and Simons [CITATION].', 'cond-mat-0206183-1-9-2': 'However, there exists experimental evidence that in some situations the magnetic scattering is dominant: upon increasing the concentration of impurities the increase in residual resistivity ratio correlates with the suppression of the superconducting transition temperature [CITATION].', 'cond-mat-0206183-1-9-3': 'Guided by this experimental insight, here we consider only the magnetic scattering, and expect our results to remain valid for as long as it is stronger than, or is of the order of, potential scattering.', 'cond-mat-0206183-1-9-4': 'Hence we write [MATH], where [MATH] is the electron spin operator, [MATH], [MATH] is the exchange constant, and [MATH] is the localized impurity spin at a site [MATH].', 'cond-mat-0206183-1-10-0': 'The main physical difference between our analysis and that of Ref. [CITATION] lies in this choice of the scattering potential.', 'cond-mat-0206183-1-10-1': 'In the regime studied here the mean free path significantly exceeds the coherence length, and hence the motion of the quasiparticles in the optimal fluctuation is ballistic, leading to a substantially different physical picture of the tail states and the optimal fluctuation, and to a different energy dependence of the density of states.', 'cond-mat-0206183-1-11-0': 'The states with energy [MATH] exist in rare regions where the amplitude of the impurity potential differs significantly from its typical value.', 'cond-mat-0206183-1-11-1': 'Therefore in determining [MATH] it is sufficient to consider only such configurations of the impurity potential, for which [MATH] is the lowest quantum mechanical energy level; fluctuations where [MATH] is the energy of a higher bound state are exponentially less probable [CITATION].', 'cond-mat-0206183-1-11-2': 'Also, in essentially all the energy range below the gap the size of the optimal fluctuation is significantly greater than the distance between impurities, so that the exact impurity potential can be replaced by a smooth function, averaged over regions containing many impurities, but smaller than the characteristic size of the wave function in the optimal fluctuation [CITATION].', 'cond-mat-0206183-1-12-0': 'Hence we consider an uncorrelated potential with a gaussian probability density [EQUATION] where [MATH] is related to the scattering time via [MATH], and [MATH] is the average impurity concentration.', 'cond-mat-0206183-1-12-1': 'Such a choice means that we ignore interactions between the magnetic impurities: it was shown in Ref. [CITATION] that the RKKY interaction and glassy behavior of impurity spins modify the AG results very weakly.', 'cond-mat-0206183-1-12-2': 'We also do not include quantum dynamics of the impurity spins, and therefore cannot account for the Kondo effect.', 'cond-mat-0206183-1-12-3': 'This is justified either when the Kondo temperature of individual impurity sites is much smaller than the superconducting transition temperature, [MATH] (in that case the depletion of states at the Fermi level prevents screening of the local moment), or in the opposite limit, [MATH], when the moments are quenched already in the normal state [CITATION].', 'cond-mat-0206183-1-13-0': 'The density of localized states in the fluctuation region of the spectrum is then [CITATION] [EQUATION] where [MATH] is the lowest energy eigenstate in the realization [MATH] of the impurity potential.', 'cond-mat-0206183-1-13-1': 'For rare configurations (exponentially small [MATH]) the integral can be evaluated using saddle point approximation to give [MATH], where the optimal fluctuation is obtained by minimizing the functional [EQUATION] with respect to both the potential [MATH] and the Lagrange multiplier [MATH].', 'cond-mat-0206183-1-13-2': 'The technical difficulty in minimizing the action lies in the nonlinear nature of the equations: optimal potential [MATH] itself depends on the wave function of the particle in this potential, as we will see below.', 'cond-mat-0206183-1-14-0': 'Nonetheless, the method of optimal fluctuation allows for a simple physical analysis.', 'cond-mat-0206183-1-14-1': 'Consider first a semiconductor.', 'cond-mat-0206183-1-14-2': 'In a potential well of depth [MATH] (all energies are measured from the band edge) and size [MATH] the energy of the localized state is of the order of [MATH] (we put [MATH]).', 'cond-mat-0206183-1-14-3': 'In the optimal fluctuation we expect [MATH], so that the action (negative logarithm of probability) for realization of such fluctuation is [MATH], or [MATH].', 'cond-mat-0206183-1-14-4': 'This is exactly the result obtained by Lifshits, and is confirmed by the solution of the nonlinear equations for minimization of action in Eq. ([REF]) [CITATION].', 'cond-mat-0206183-1-15-0': 'The difference between a potential well in a doped semiconductor and in a superconductor is twofold.', 'cond-mat-0206183-1-15-1': 'First, because of particle-hole mixing the hamiltonian Eq. ([REF]) is a matrix in particle-hole and spin space.', 'cond-mat-0206183-1-15-2': 'Second, we are concerned with quasiparticles close to the Fermi energy.', 'cond-mat-0206183-1-16-0': 'We first note that ferromagnetic fluctuation maximizes the effect of the impurity potential.', 'cond-mat-0206183-1-16-1': 'Consequently, we consider such a fluctuation, and choose the direction of the impurity spins along the [MATH]-axis, so that [MATH].', 'cond-mat-0206183-1-16-2': 'Performing a spin rotation, [MATH] in Eq. ([REF]), we obtain a hamiltonian diagonal in the spin space, [EQUATION]', 'cond-mat-0206183-1-16-3': 'It is therefore sufficient to consider only one spin orientation.', 'cond-mat-0206183-1-16-4': 'Let us again consider the problem qualitatively, and concentrate first on the one dimensional case ignoring the suppression of the order parameter.', 'cond-mat-0206183-1-16-5': 'We linearize the kinetic energy near the Fermi surface, [MATH].', 'cond-mat-0206183-1-16-6': 'Then the energy of a quasiparticle in the optimal fluctuation (measured from the Fermi energy) is [MATH].', 'cond-mat-0206183-1-16-7': 'For the energies close to the superconducting gap, [MATH], the optimal fluctuation is large (compared to the coherence length, [MATH]) and shallow ([MATH]), so that [MATH].', 'cond-mat-0206183-1-16-8': 'Introducing the dimensionless energy [MATH], we obtain, in analogy with the arguments above, [MATH].', 'cond-mat-0206183-1-16-9': 'Notice that the size of the fluctuation is indeed [MATH].', 'cond-mat-0206183-1-16-10': 'As a result, we find [MATH].', 'cond-mat-0206183-1-16-11': 'From the definition of [MATH] it follows that [EQUATION]', 'cond-mat-0206183-1-16-12': 'The energy dependence in Eq. ([REF]) is identical to the result of Lifshits in [MATH].', 'cond-mat-0206183-1-16-13': 'This follows from the expansion in [MATH]: even though the bare energy is linear in [MATH], the expansion starts from the quadratic term.', 'cond-mat-0206183-1-17-0': 'We now verify these estimates by a considering the energy of a particle in hamiltonian Eq.([REF]) for spin "up" [EQUATION] where [MATH] is the normalized wave function of the particle.', 'cond-mat-0206183-1-17-1': 'Minimization of Eq.([REF]) with respect to [MATH] gives [EQUATION] while minimization with respect to [MATH] dictates [MATH].', 'cond-mat-0206183-1-17-2': 'We first ignore the self-consistent suppression of the gap, which means [MATH], where [MATH] denotes the vector product in particle-hole space.', 'cond-mat-0206183-1-17-3': 'Then the Schrodinger equation takes the form [EQUATION]', 'cond-mat-0206183-1-17-4': 'This equation is solved by introducing the bilinear forms [MATH], and yields the optimal fluctuation [EQUATION] which corresponds to the value of the action [EQUATION]', 'cond-mat-0206183-1-17-5': 'We immediately notice that for [MATH] the length scale of the optimal fluctuation is [MATH], its depth is [MATH], and the action [MATH], in complete agreement with our estimates above.', 'cond-mat-0206183-1-18-0': 'To demonstrate the validity of our qualitative analysis we need to show that the self-consistent suppression of the order parameter does not appreciably change our result.', 'cond-mat-0206183-1-18-1': 'Such self-consistency is achieved by including the variation of the gap into the variational derivative [MATH].', 'cond-mat-0206183-1-18-2': 'We first notice that, at [MATH], uniform potential [MATH] does not suppress superconductivity.', 'cond-mat-0206183-1-18-3': 'Consequently, [MATH] depends on the gradient of the potential.', 'cond-mat-0206183-1-18-4': 'For [MATH], the potential varies smoothly, so that [MATH] can be accounted for perturbatively.', 'cond-mat-0206183-1-18-5': 'We find the leading local correction to the gap to be [EQUATION] and perturbatively determine the correction to the optimal fluctuation and to the action for [MATH], [EQUATION]', 'cond-mat-0206183-1-18-6': 'At lower energies the self-consistent [MATH] has to be computed numerically.', 'cond-mat-0206183-1-18-7': 'The results are presented in Figs. 1 and 2.', 'cond-mat-0206183-1-18-8': 'It is clear that the suppression of the gap, even for the states with [MATH] is incomplete; since [MATH], the gap remains at a significant fraction of the bulk amplitude throughout the size of the OF.', 'cond-mat-0206183-1-18-9': 'This observation justifies the expansion in the bare energy, [MATH], in our qualitative analysis.', 'cond-mat-0206183-1-18-10': 'Consequently, the optimal action computed self-consistently differs at most by 10% from the action computed assuming a uniform gap, see Fig. 2.', 'cond-mat-0206183-1-19-0': 'Having demonstrated that the qualitative consideration are in excellent agreement with the full solution of the problem in [MATH], we discuss the multidimensional case.', 'cond-mat-0206183-1-19-1': 'In a doped semiconductor the OF in any [MATH] is spherically symmetric [CITATION].', 'cond-mat-0206183-1-19-2': 'This is a consequence of the balance between lowering the energy of a particle in a large and deep fluctuation, and the probability cost of creating a fluctuation with a large volume.', 'cond-mat-0206183-1-20-0': 'Since electrons in a superconductor move with the Fermi velocity, the wave function of the subgap state is concentrated along the quasiclassical trajectory, which is a chord in a potential of an arbitrary shape.', 'cond-mat-0206183-1-20-1': 'Consequently, there is essentially no energy cost in reducing the size of the optimal fluctuation in the "transverse" direction, while the smaller volume makes such fluctuations more probable.', 'cond-mat-0206183-1-20-2': 'As a result, the optimal fluctuation is anisotropic, and strongly elongated in one direction.', 'cond-mat-0206183-1-20-3': 'Choosing this direction as the [MATH]-axis, we can write the wave function of the subgap state as [MATH], where [MATH] denotes the transverse [MATH] coordinates, and [MATH] is a slowly varying function.', 'cond-mat-0206183-1-20-4': 'Therefore the kinetic energy of the quasiparticle is [EQUATION]', 'cond-mat-0206183-1-20-5': 'The transverse size of the fluctuation can therefore be reduced until the second term becomes comparable to the first, i.e. [MATH], where [MATH] is the Fermi wavelength.', 'cond-mat-0206183-1-20-6': 'Consequently, we find still [MATH] and [MATH], and [EQUATION] where [MATH] is the Fermi energy.', 'cond-mat-0206183-1-20-7': 'Eq. ([REF]) is the main result presented here.', 'cond-mat-0206183-1-21-0': 'It is easy to verify that the action for the anisotropic fluctuation is smaller than that for an isotropic OF, by a factor of [MATH], so that the corresponding DOS is exponentially higher.', 'cond-mat-0206183-1-22-0': 'There are two limitations on the validity of the results obtained here.', 'cond-mat-0206183-1-22-1': 'First, since the optimal fluctuation is a result of a saddle point approximation for the functional integral, Eq. ([REF]), it is only valid when [MATH], or [EQUATION]', 'cond-mat-0206183-1-22-2': 'In particular, for [MATH] this condition becomes [MATH], while for [MATH] it does not depend on the gap amplitude, [MATH], where [MATH] is the mean free path.', 'cond-mat-0206183-1-22-3': 'In [MATH] the region of validity is extended by almost an order of magnitude in comparison to this estimate as the action has a large numerical factor [MATH].', 'cond-mat-0206183-1-23-0': 'Second, when the characteristic size of the fluctuation becomes of the order of [MATH], our assumption about the ballistic motion in the fluctuation is invalid, and a crossover to the diffusive regime studied by Lamacraft and Simons [CITATION] occurs.', 'cond-mat-0206183-1-23-1': 'This occurs for [MATH] in any dimension.', 'cond-mat-0206183-1-23-2': 'When [MATH] for [MATH] the saddle point approximation becomes invalid before the diffusive regime is reached.', 'cond-mat-0206183-1-23-3': 'On the other hand, since typically [MATH], in higher dimensions the method of optimal fluctuation works up to the crossover.', 'cond-mat-0206183-1-24-0': 'To summarize, we have analysed the density of subgap states in an s-wave superconductor with weak magnetic impurities using the method of the optimal fluctuation.', 'cond-mat-0206183-1-24-1': 'We concentrated on the clean limit, [MATH], when the motion of particles in the optimal potential is ballistic.', 'cond-mat-0206183-1-24-2': 'We find that the optimal fluctuation in this case is strongly anisotropic, and that the density of states varies as a stretched exponential below the gap edge, with a power that depends on the dimension.', 'cond-mat-0206183-1-24-3': 'The investigation on how the results change for increasing strength of individual impurities is now in progress.', 'cond-mat-0206183-1-25-0': 'This research was supported by the DOE under contract W-7405-ENG-36, by the NSF Grant PHY-94-07194, and by Pappalardo Fellowship.', 'cond-mat-0206183-1-25-1': 'We are grateful to the ITP Santa Barbara, where this work was conceived, for hospitality and support.'}

{'cond-mat-0206183-2-0-0': 'We analyse the behavior of the density of states in a singlet [MATH]-wave superconductor with weak magnetic impurities in the clean limit.', 'cond-mat-0206183-2-0-1': 'By using the method of optimal fluctuation and treating the order parameter self-consistently we show that the density of states is finite everywhere in the superconducting gap, and that it varies as [MATH] near the mean field gap edge [MATH] in a [MATH]-dimensional superconductor.', 'cond-mat-0206183-2-0-2': 'In contrast to most studied cases the optimal fluctuation is strongly anisotropic.', 'cond-mat-0206183-2-1-0': 'One of the most intriguing problems in modern condensed matter physics is the combined effect of correlations and disorder on the ground state and the electronic properties of solids.', 'cond-mat-0206183-2-1-1': "Interplay of superconductivity and impurity scattering is an example of a situation where such an effect is relevant experimentally, and where it has been studied for over forty years, beginning with the seminal papers by Anderson [CITATION] and Abrikosov and Gor'kov (AG) [CITATION].", 'cond-mat-0206183-2-1-2': 'Despite such a long history, however, the spectral properties even in the simplest case of a singlet [MATH]-wave superconductor continue to attract attention.', 'cond-mat-0206183-2-2-0': 'Since nonmagnetic impurities do not break the time-reversal symmetry, the "hard" gap, [MATH], in the single particle excitation spectrum is unaffected by weak potential scattering [CITATION].', 'cond-mat-0206183-2-2-1': 'At the same time scattering by magnetic impurities destroys the phase coherence in the superconducting state, and, consequently, leads to the suppression of the gap, and of the superconducting order parameter.', 'cond-mat-0206183-2-2-2': 'AG analysed magnetic scattering using self-consistent Born approximation, and concluded that a hard gap in the energy spectrum survives up to a critical (average) concentration of weak magnetic impurities; it is followed, upon increasing impurity concentration, by a narrow region of gapless superconductivity, and then by destruction of the superconducting condensate.', 'cond-mat-0206183-2-3-0': 'After some early work [CITATION], resurgence of interest in this problem started with the paper of Balatsky and Trugman [CITATION], who argued that rare regions, where impurity concentration is sufficient to locally destroy superconductivity, yield a finite density of states (DOS) at the Fermi level.', 'cond-mat-0206183-2-3-1': 'Their work was followed by other analyses [CITATION].', 'cond-mat-0206183-2-3-2': 'Here we elucidate the nature of the subgap states, and provide a quantitative analysis of the energy profile of the DOS in [MATH]-wave superconductors with weak magnetic impurities.', 'cond-mat-0206183-2-4-0': 'In the AG theory the effect of disorder is controlled by a dimensionless parameter, [MATH], where [MATH] is the scattering time due to magnetic impurities, and [MATH] is the value of the superconducting order parameter.', 'cond-mat-0206183-2-4-1': 'In particular, the single particle spectral gap is [MATH], indicating the onset of the gapless superconductivity at [MATH].', 'cond-mat-0206183-2-4-2': 'In the regime studied here, [MATH], the AG theory predicts a gapped quasiparticle spectrum.', 'cond-mat-0206183-2-5-0': 'These results are obtained by carrying out a standard impurity averaging procedure.', 'cond-mat-0206183-2-5-1': 'It is clear, however, that, among all the realizations of the impurity distribution, there exist regions where the resulting potential generates localized quasiparticle states at an arbitrary energy below the gap edge.', 'cond-mat-0206183-2-5-2': 'Such localized states were extensively studied in doped semiconductors [CITATION].', 'cond-mat-0206183-2-5-3': 'For a particular energy, [MATH], the most probable (albeit still rare) configuration of impurities that creates a state at [MATH], and therefore contributes the most to the DOS, [MATH], is called the optimal fluctuation (OF) [CITATION].', 'cond-mat-0206183-2-5-4': 'Such rare regions provide nonperturbative corrections to the DOS determined in the framework of self-consistent Born approximation.', 'cond-mat-0206183-2-6-0': 'We employ the OF method in a singlet [MATH]-wave superconductor with magnetic impurities.', 'cond-mat-0206183-2-6-1': 'We also consider the self-consistent suppression of the superconducting order parameter; it is known that in [MATH]-wave superconductors it significantly affects the low-energy DOS [CITATION].', 'cond-mat-0206183-2-6-2': 'We find that the density of states is finite everywhere in the superconducting gap.', 'cond-mat-0206183-2-6-3': 'Just below the AG gap edge, the DOS is [MATH], where [MATH] is the normal state DOS, and [MATH] is the number of spatial dimensions.', 'cond-mat-0206183-2-6-4': 'In contrast to other known cases, the OF is anisotropic, with its transverse size much smaller than the longitudinal extent.', 'cond-mat-0206183-2-7-0': 'We consider a singlet [MATH]-wave superconductor.', 'cond-mat-0206183-2-7-1': 'In the 4-space of the wave functions [MATH] the mean field hamiltonian is [EQUATION]', 'cond-mat-0206183-2-7-2': 'Here [MATH] is the kinetic energy of a quasiparticle with respect to the Fermi level, [MATH], and [MATH] and [MATH] are the Pauli matrices in the particle-hole and the spin space respectively, so that [MATH] is a 4[MATH]4 direct product.', 'cond-mat-0206183-2-8-0': 'Potential due to impurities, [MATH], includes both potential and spin-flip scattering processes.', 'cond-mat-0206183-2-8-1': 'When the potential scattering is dominant (motion of quasiparticles in the OF is diffusive) properties of the low-energy states were explored in Ref. [CITATION].', 'cond-mat-0206183-2-8-2': 'However, there exists experimental evidence that in some situations the magnetic scattering is dominant: upon increasing the impurity concentration the increase in residual resistivity ratio correlates with the suppression of the superconducting transition temperature [CITATION].', 'cond-mat-0206183-2-8-3': 'Guided by this insight, we consider only the magnetic scattering, and expect our results to remain valid for as long as it is stronger than, or is of the order of, potential scattering.', 'cond-mat-0206183-2-8-4': 'Hence we write [MATH], where [MATH] is the electron spin operator, [MATH], [MATH] is the exchange constant, and [MATH] is the localized impurity spin at a site [MATH].', 'cond-mat-0206183-2-9-0': 'The main physical difference between our analysis and that of Ref. [CITATION] lies in this choice of the scattering potential.', 'cond-mat-0206183-2-9-1': 'In the regime studied here the mean free path significantly exceeds the coherence length, and hence the motion of the quasiparticles in the optimal fluctuation is ballistic, leading to a substantially different physical picture of the tail states and the optimal fluctuation, and to a different energy dependence of the density of states.', 'cond-mat-0206183-2-10-0': 'The states with energy [MATH] exist in rare regions where the amplitude of the impurity potential differs significantly from its typical value.', 'cond-mat-0206183-2-10-1': 'Therefore in determining [MATH] it is sufficient to consider only such configurations of the impurity potential, for which [MATH] is the lowest quantum mechanical energy level; fluctuations where [MATH] is the energy of a higher bound state are exponentially less probable [CITATION].', 'cond-mat-0206183-2-10-2': 'Also, in essentially all the energy range below the gap the size of the optimal fluctuation is significantly greater than the distance between impurities, so that the exact impurity potential can be replaced by a smooth function, averaged over regions containing many impurities, but smaller than the characteristic size of the wave function in the optimal fluctuation [CITATION].', 'cond-mat-0206183-2-11-0': 'Hence we consider an uncorrelated potential with a gaussian probability density [EQUATION] where [MATH] is related to the scattering time via [MATH], and [MATH] is the average impurity concentration.', 'cond-mat-0206183-2-11-1': 'We ignore interactions between the magnetic impurities: it was shown in Ref. [CITATION] that the RKKY interaction and glassy behavior of impurity spins modify the AG results very weakly.', 'cond-mat-0206183-2-11-2': 'We also do not include quantum dynamics of the impurity spins, and therefore cannot account for the Kondo effect.', 'cond-mat-0206183-2-11-3': 'This is justified either when the Kondo temperature of individual impurity sites is much smaller than the superconducting transition temperature, [MATH] (and depletion of states at the Fermi level prevents screening of the local moment), or in the opposite limit, [MATH], when the moments are quenched already in the normal state [CITATION].', 'cond-mat-0206183-2-12-0': 'The density of localized states in the fluctuation region of the spectrum is then [CITATION] [EQUATION] where [MATH] is the lowest energy eigenstate in the realization [MATH] of the impurity potential.', 'cond-mat-0206183-2-12-1': 'For rare configurations the integral is evaluated using saddle point approximation to give [MATH], where the optimal fluctuation is obtained by minimizing the functional [EQUATION] with respect to both [MATH] and the Lagrange multiplier [MATH].', 'cond-mat-0206183-2-12-2': 'The difficulty in minimizing the action is in the nonlinear nature of the equations: optimal potential [MATH] depends on the wave function of the particle in this potential.', 'cond-mat-0206183-2-13-0': 'The method of optimal fluctuation allows for a simple physical analysis.', 'cond-mat-0206183-2-13-1': 'Consider first a semiconductor.', 'cond-mat-0206183-2-13-2': 'In a potential well of depth [MATH] (all energies are measured from the band edge) and size [MATH] the energy of the localized state is of the order of [MATH]).', 'cond-mat-0206183-2-13-3': 'In the optimal fluctuation [MATH], so that the action for such fluctuation is [MATH], or [MATH].', 'cond-mat-0206183-2-13-4': 'This is exactly the result obtained by Lifshits, and is confirmed by the solution of the nonlinear equations for minimization of action in Eq. ([REF]) [CITATION].', 'cond-mat-0206183-2-14-0': 'The difference between a potential well in a doped semiconductor and in a superconductor is twofold.', 'cond-mat-0206183-2-14-1': 'First, because of particle-hole mixing the hamiltonian Eq. ([REF]) is a matrix in particle-hole and spin space.', 'cond-mat-0206183-2-14-2': 'Second, we are concerned with quasiparticles close to the Fermi energy.', 'cond-mat-0206183-2-15-0': 'We assume that ferromagnetic fluctuation maximizes the effect of the impurity potential [CITATION].', 'cond-mat-0206183-2-15-1': 'Consequently, we consider such a fluctuation, and choose the direction of the impurity spins along the [MATH]-axis, so that [MATH].', 'cond-mat-0206183-2-15-2': 'Performing a spin rotation, [MATH] in Eq. ([REF]), we obtain a hamiltonian diagonal in the spin space, [EQUATION]', 'cond-mat-0206183-2-15-3': 'It is therefore sufficient to consider only one spin orientation.', 'cond-mat-0206183-2-15-4': 'Let us again consider the problem qualitatively, and concentrate first on the one dimensional case ignoring the suppression of the order parameter.', 'cond-mat-0206183-2-15-5': 'We linearize the kinetic energy near the Fermi surface, [MATH], so that typical kinetic energy in an OF of size [MATH] is [MATH].', 'cond-mat-0206183-2-15-6': 'Then the energy of a quasiparticle in the optimal fluctuation (measured from the Fermi energy) is [MATH].', 'cond-mat-0206183-2-15-7': 'For the energies close to the superconducting gap, [MATH], the OF is large ([MATH]) and shallow ([MATH]), so that [MATH].', 'cond-mat-0206183-2-15-8': 'Introducing the dimensionless energy [MATH], we obtain, in analogy with the arguments above, [MATH].', 'cond-mat-0206183-2-15-9': 'Notice that the size of the fluctuation is indeed [MATH].', 'cond-mat-0206183-2-15-10': 'As a result, we find [MATH].', 'cond-mat-0206183-2-15-11': 'From the definition of [MATH] it follows that [EQUATION]', 'cond-mat-0206183-2-15-12': 'The energy dependence in Eq. ([REF]) is identical to the result of Lifshits in [MATH].', 'cond-mat-0206183-2-15-13': 'This follows from the expansion in [MATH]: even though [MATH], the expansion is in [MATH] .', 'cond-mat-0206183-2-16-0': 'We now verify these estimates by a considering the energy of a particle in hamiltonian Eq.([REF]) for spin "up" [EQUATION] where [MATH] is the normalized wave function of the particle.', 'cond-mat-0206183-2-16-1': 'Minimization of Eq.([REF]) with respect to [MATH] gives [EQUATION] while minimization with respect to [MATH] dictates [MATH].', 'cond-mat-0206183-2-16-2': 'We first ignore the self-consistent suppression of the gap, which means [MATH], where [MATH] denotes the scalar product in particle-hole space.', 'cond-mat-0206183-2-16-3': 'Then the Schrodinger equation takes the form [EQUATION]', 'cond-mat-0206183-2-16-4': 'This equation is solved by introducing the bilinear forms [MATH], and yields the optimal fluctuation [EQUATION] which corresponds to the value of the action [EQUATION]', 'cond-mat-0206183-2-16-5': 'We immediately notice that for [MATH] the length scale of the optimal fluctuation is [MATH], its depth is [MATH], and the action [MATH], in complete agreement with our estimates above.', 'cond-mat-0206183-2-17-0': 'We now show that the self-consistent suppression of the order parameter does not appreciably change our result.', 'cond-mat-0206183-2-17-1': 'Self-consistency is achieved by including the variation of the gap into the variational derivative [MATH].', 'cond-mat-0206183-2-17-2': 'We notice that, at [MATH], uniform [MATH] does not suppress superconductivity.', 'cond-mat-0206183-2-17-3': 'Consequently, [MATH] depends on the gradient of the potential.', 'cond-mat-0206183-2-17-4': 'For [MATH], the potential varies smoothly, so that [MATH] can be accounted for perturbatively.', 'cond-mat-0206183-2-17-5': 'The leading local correction to the gap is [EQUATION] and the correction to the action for [MATH] is [MATH].', 'cond-mat-0206183-2-17-6': 'At lower energies the self-consistent [MATH] has to be computed numerically.', 'cond-mat-0206183-2-17-7': 'The results are presented in Figs. 1 and 2.', 'cond-mat-0206183-2-17-8': 'It is clear that the suppression of the gap, even for the states with [MATH] is incomplete; since [MATH], the gap remains at a significant fraction of [MATH] throughout the OF.', 'cond-mat-0206183-2-17-9': 'This justifies the expansion in the bare energy, [MATH], in our qualitative analysis.', 'cond-mat-0206183-2-17-10': 'Consequently, the optimal action computed self-consistently differs at most by 10% from that computed assuming a uniform gap, see Fig. 2.', 'cond-mat-0206183-2-18-0': 'Having demonstrated that the qualitative consideration are in excellent agreement with the full solution of the problem in [MATH], we discuss the multidimensional case.', 'cond-mat-0206183-2-18-1': 'In a doped semiconductor the OF in any [MATH] is spherically symmetric [CITATION].', 'cond-mat-0206183-2-18-2': 'This results from the balance between lowering the particle energy in a large and deep fluctuation, and the probability cost of such an OF.', 'cond-mat-0206183-2-19-0': 'Since electrons in a superconductor move with the Fermi velocity, the wave function of the subgap state is concentrated along the quasiclassical trajectory, which is a chord in a potential of any shape.', 'cond-mat-0206183-2-19-1': 'Consequently, there is little energy cost in reducing the size of the OF in the "transverse" direction, while the smaller volume makes such fluctuations more probable.', 'cond-mat-0206183-2-19-2': 'As a result, the optimal fluctuation is anisotropic, and strongly elongated in one direction.', 'cond-mat-0206183-2-19-3': 'Choosing this direction as the [MATH]-axis, we can write the wave function of the subgap state as [MATH], where [MATH] denotes the transverse [MATH] coordinates, and [MATH] is a slowly varying function.', 'cond-mat-0206183-2-19-4': 'Therefore the kinetic energy of the quasiparticle is [EQUATION]', 'cond-mat-0206183-2-19-5': 'The transverse size of the fluctuation can therefore be reduced until the second term becomes comparable to the first, i.e. [MATH], where [MATH] is the Fermi wavelength.', 'cond-mat-0206183-2-19-6': 'Consequently, we find [MATH] and [MATH], and [EQUATION] where [MATH] is the Fermi energy.', 'cond-mat-0206183-2-19-7': 'Eq. ([REF]) is the main result presented here.', 'cond-mat-0206183-2-19-8': 'The action for the anisotropic fluctuation is smaller than that for an isotropic OF, by a factor of [MATH], so that the corresponding DOS is exponentially higher.', 'cond-mat-0206183-2-20-0': 'There are two limitations on the validity of the results obtained here.', 'cond-mat-0206183-2-20-1': 'First, since the optimal fluctuation is a result of a saddle point approximation for the functional integral, Eq. ([REF]), it is only valid when [MATH], or [EQUATION]', 'cond-mat-0206183-2-20-2': 'For [MATH] this condition becomes [MATH], while for [MATH] it does not depend on the gap amplitude, [MATH], where [MATH] is the mean free path.', 'cond-mat-0206183-2-20-3': 'In [MATH] the region of validity is extended by almost an order of magnitude in comparison to this estimate as the action has a large numerical factor [MATH].', 'cond-mat-0206183-2-21-0': 'Second, when the characteristic size of the OF [MATH], our assumption about the ballistic motion in the fluctuation is invalid, and a crossover to the diffusive regime studied in Ref. [CITATION] occurs for [MATH] in any dimension.', 'cond-mat-0206183-2-21-1': 'When [MATH] for [MATH] the saddle point approximation becomes invalid before the diffusive regime is reached.', 'cond-mat-0206183-2-21-2': 'For [MATH] and typical [MATH], the OF method works up to the crossover.', 'cond-mat-0206183-2-21-3': 'Taking [MATH], we find that our results hold to within 1% of [MATH], while the expansion in [MATH] is quantitatively valid for [MATH], and qualitatively for [MATH], providing a significant window of applicability for our DOS.', 'cond-mat-0206183-2-22-0': 'Experimental verification of our results, and the underlying physical picture of scattering on randomly distributed impurities, requires averaging the tunneling conductance over regions containing many impurities, and is best done at energies just below [MATH].', 'cond-mat-0206183-2-22-1': 'We suggest averaging the tunneling spectra, obtained from Scanning Tunneling Spectroscopy, over several distinct areas of the sample, each containing a large number of impurities.', 'cond-mat-0206183-2-23-0': 'To summarize, we have analysed the density of subgap states in an s-wave superconductor with weak magnetic impurities using the method of the optimal fluctuation.', 'cond-mat-0206183-2-23-1': 'We concentrated on the clean limit, [MATH], when the motion of particles in the optimal potential is ballistic.', 'cond-mat-0206183-2-23-2': 'We find that the optimal fluctuation in this case is strongly anisotropic, and that the density of states varies as a stretched exponential below the gap edge, with a power that depends on the dimension.', 'cond-mat-0206183-2-24-0': 'This research was supported by the DOE under contract W-7405-ENG-36, by the NSF Grant PHY-94-07194, and by Pappalardo Fellowship.', 'cond-mat-0206183-2-24-1': 'We are grateful to the ITP Santa Barbara for hospitality and support.'}

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[]

[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0206183

1709.08987

{'1709.08987-1-0-0': 'Radiation pressure, electrostriction, and photothermal forces have been investigated to evidence backaction, non-linearities and quantum phenomena in cavity optomechanics.', '1709.08987-1-0-1': 'We show here that optoelectronic forces involving real carrier excitation and deformation potential interaction are the strongest mechanism of light-to-sound transduction when exciting with pulsed lasers close to the GaAs optical gap of semiconductor GaAs/AlAs distributed Bragg reflector optomechanical resonators.', '1709.08987-1-0-2': 'We demonstrate that the ultrafast spatial redistribution of the photoexcited carriers in microcavities with massive GaAs spacers leads to an enhanced coupling to the fundamental 20 GHz vertically polarized mechanical breathing mode.', '1709.08987-1-0-3': 'The carrier diffusion can be limited by embedding GaAs quantum wells in the cavity spacer, a strategy used here to prove and engineer the role of optoelectronic forces in phonon generation with real carriers.', '1709.08987-1-0-4': 'The optical forces associated to the different intervening mechanisms and their relevance for dynamical backaction in optomechanics is evaluated using finite-element methods.', '1709.08987-1-0-5': 'The results presented open the path to the study of hitherto seldom investigated dynamical backaction in optomechanical solid-state resonators in the presence of optoelectronic forces.', '1709.08987-1-1-0': '# Motivation', '1709.08987-1-2-0': 'Backaction in cavity optomechanics has shown to lead to novel physical phenomena including laser cooling, self-oscillation, and non-linear dynamics in systems that go from kilometer size interferometers to single or few trapped ions.', '1709.08987-1-2-1': '[CITATION] Briefly, a resonant photon field in a cavity exerts a force and induces a mechanical motion on the mirrors, which in turn leads to a delayed modification of the resonant condition of the trapped field.', '1709.08987-1-2-2': 'Such coupled dynamics can be exploited for a large variety of applications that span for example from gravitational wave detection [CITATION] to the study of quantum motion states in mesoscopic mechanical systems.', '1709.08987-1-2-3': '[CITATION] How light exerts force on matter is at the center of this investigations.', '1709.08987-1-2-4': 'Photons can apply stress through radiation pressure, transferring impulse when reflected from the mirrors.', '1709.08987-1-2-5': '[CITATION] Related mechanisms also deriving from the same fundamental interaction (Lorentz forces) are gradient forces (also exploited in optical tweezers) [CITATION], and electrostriction.', '1709.08987-1-2-6': 'The latter, linked to the material´s photoelasticity, has been shown to play a role that can be of the same magnitude as radiation pressure, [CITATION] or even larger if optical resonances are exploited in direct bandgap materials as for example GaAs.', '1709.08987-1-2-7': '[CITATION]', '1709.08987-1-3-0': 'In the presence of radiation pressure forces, the energy [MATH] of the photon is shifted by the Doppler effect by an amount of the order [MATH], where [MATH] and [MATH] are the mirror and light velocities, respectively.', '1709.08987-1-3-1': 'The mechanical energy transferred from the photon to the mirror is thus very small.', '1709.08987-1-3-2': 'Electrostriction leads to Raman-like processes, for which the transferred energy [MATH] corresponds to the involved vibrational frequency.', '1709.08987-1-3-3': 'This leads to inelastic scattering side-bands.', '1709.08987-1-3-4': 'Again, tipically [MATH].', '1709.08987-1-3-5': 'Contrastingly, if the photon is absorbed in the process of interaction, all its energy is transferred to the mirror.', '1709.08987-1-3-6': 'This fundamental difference has been used in cavity optomechanics to demonstrate strongly enhanced light-matter interactions based on photothermal forces.', '1709.08987-1-3-7': '[CITATION] In materials displaying optical resonances, the photons can be resonantly absorbed with the consequent transfer of electrons to excited states.', '1709.08987-1-3-8': 'Photoexcited carrier-mediated optomechanical interactions have been reported in semiconductor modulation-doped heterostructure-cantilever cavity-less hybrid systems.', '1709.08987-1-3-9': 'Efficient cavity-less optomechanical transduction involving opto-piezoelectric backaction from the bound photoexcited electron-hole pairs have been demonstrated in these systems, including self-feedback cooling, amplification of the thermomechanical motion, and control of the mechanical quality factor through carrier excitation.', '1709.08987-1-3-10': '[CITATION] The change of electronic landscape produced by photoexcited carriers also induces a stress in the structure through electron-phonon coupling mediated by deformation potential interaction.', '1709.08987-1-3-11': 'This stress can be identified as an optoelectronic force, and should have the same kind of temporal behavior (with different time-scales and details depending on the carrier dynamics) and amplified strength as observed for photothermal forces.', '1709.08987-1-3-12': '[CITATION] Recently optical cooling of mechanical modes of a GaAs nanomembrane forming part of an optical cavity was reported [CITATION] and its relation to optoelectronic stress via the deformation potential was analysed.', '1709.08987-1-3-13': 'Because of the very fast relaxation rate of excited carriers due to surface recombination in such nanometer size structures, it was concluded that thermal (and not optoelectronic) stress was the primary cause of cooling in that case.', '1709.08987-1-3-14': 'We will demonstrate here that the carrier dynamics can be fundamentally modified when the nanometer size GaAs layer is embedded in a monolithic microcavity, making optoelectronic forces the main mechanism of interaction of light with vibrations in such semiconductor devices.', '1709.08987-1-3-15': 'The diffusion of the photoexcited carriers thus assume a central role, a dynamics that can be engineered using embedded quantum wells (QWs).', '1709.08987-1-4-0': 'Because of their optoelectronic properties, semiconductor GaAs/AlAs microcavities are interesting candidates for novel functionalities in cavity optomechanics.', '1709.08987-1-4-1': 'Perfect photon-phonon overlap, and access to electronically resonant coupling in addition to radiation pressure could lead to strong optomechanical interactions of photoelastic origin.', '1709.08987-1-4-2': '[CITATION] The vibrational frequencies in these microresonators are determined by the vertical layering of the device (fabricated with the ultra-high quality of molecular beam epitaxy), and not by the lateral pattering (defined by the more limited performance of microfabrication techniques).', '1709.08987-1-4-3': 'This has allowed access to much higher frequencies for the optomechanical vibrational modes, in the 20-100 GHz range, without significant reduction of the mechanical Q-factors.', '1709.08987-1-4-4': '[CITATION] In addition, these optomechanical resonators allow for the conception of hybrid architectures involving artificial atoms (semiconductor excitons) coupled to the optical cavity mode, and thus combining the physics of cavity optomechanics with cavity quantum electrodynamics.', '1709.08987-1-4-5': '[CITATION] Our motivation here is to search for evidence of optoelectronic forces in these devices, and for that purpose we study the light-sound coupling involving the resonator mechanical modes and with the optical cavity at resonance with the material exciton transition energy.', '1709.08987-1-4-6': 'We demonstrate that the main phonon generation mechanism using pulsed lasers close to resonance in these devices involves indeed the real excitation of carriers and the deformation potential mechanism.', '1709.08987-1-4-7': '[CITATION] We show that in microcavities with "bulk" GaAs spacers (i.e. cavity spacers constituted by a thick [MATH] layer of GaAs) ultrafast carrier redistribution leads to an enhanced coupling to the more uniformly distributed fundamental 20 GHz cavity vibrational mode.', '1709.08987-1-4-8': 'An engineering of the structure taking into account this effect and using embedded quantum wells is used to limit the carrier diffusion allowing for a conceptual design tailored to tune and enhance the optomechanical coupling.', '1709.08987-1-5-0': '# Results', '1709.08987-1-6-0': 'We consider two planar microcavity structures, specifically a "bulk" GaAs and a multiple quantum well (MQW) resonator.', '1709.08987-1-6-1': 'The "bulk" GaAs microcavity is made of a [MATH] GaAs-spacer enclosed by ([MATH] Al[MATH]Ga[MATH]As /AlAs DBRs, 20 pairs of layers on the bottom, 18 on top, grown on a GaAs substrate (a scheme of the structure is presented in Fig. [REF](b)).', '1709.08987-1-6-2': '[CITATION] The DBRs stop band is centered at the design wavelength of [MATH].', '1709.08987-1-6-3': 'As we have demonstrated previously, this structure performs as an optomechanical resonator that simultaneously confines photons and acoustic phonons of the same wavelength.', '1709.08987-1-6-4': '[CITATION] In the MQW microcavity the [MATH] spacer is constituted by six [MATH] GaAs QWs separated by [MATH] AlAs barriers.', '1709.08987-1-6-5': 'To further enhance the light-sound coupling the second and fourth [MATH] DBR alloy layers on each side are also replaced by three GaAs/AlAs QWs.', '1709.08987-1-6-6': 'The reason for this design will become clear below.', '1709.08987-1-6-7': 'The DBRs in this case are ([MATH] Al[MATH]Ga[MATH]As /AlAs multilayers, 27 pairs on the bottom, 23 on top, grown again on a GaAs substrate.', '1709.08987-1-6-8': 'A scheme of this structure is displayed in Fig. [REF](c).', '1709.08987-1-6-9': 'The number of DBR periods in both structures is designed to assure an optical Q-factor [MATH] (cavity photon lifetime [MATH] ps).', '1709.08987-1-7-0': 'Reflection-type degenerate pump-probe experiments were performed with the laser wavelength tuned with the optical cavity mode (see the scheme in Fig. [REF](a)).', '1709.08987-1-7-1': '[CITATION] Picosecond pulses ([MATH] ps) from a mode-locked Ti:Sapphire laser, with repetition rate 80 MHz, were split into cross polarized pump (power 20 mW) and probe (1 mW) pulses.', '1709.08987-1-7-2': 'Both pulses were focused onto superimposed [MATH]m-diameter spots.', '1709.08987-1-7-3': 'To couple the light to the microcavity the probe beam propagates close to the sample normal and is tuned to the high derivative flank of the reflectivity dip, while the pump incidence angle is set for resonant condition precisely at the cavity mode.', '1709.08987-1-7-4': '[CITATION] The laser wavelength was also set so that the phonon generation and detection would be a few meV below resonance with the direct bandgap of the GaAs "bulk" cavity spacer ([MATH] eV) or of the QWs ([MATH] eV).', '1709.08987-1-7-5': 'To accomplish this resonant excitation the temperature was also used as a tuning parameter; the "bulk" cavity experiments were done at room temperature, while the MQW structure was studied at 80 K. Light is thus coupled resonantly with the optical cavity mode and the semiconductor excitonic resonance.', '1709.08987-1-7-6': 'Acoustic phonons confined in the same space as the optical cavity mode are selectively generated within the resonator, and are detected through their modulation of the optical cavity mode frequency.', '1709.08987-1-7-7': '[CITATION]', '1709.08987-1-8-0': 'These pump-probe ultrafast laser experiments are conceptually similar to the ring-down techniques recently exploited in the cavity optomechanics domain, [CITATION] but more appropriate to the study of ultra-high frequency vibrations (GHz-THz range.)', '1709.08987-1-8-1': 'The pulsed laser phonon generation efficiency can be described as: [CITATION] [EQUATION]', '1709.08987-1-8-2': 'Here [MATH] is the phonon frequency, [MATH] describes the elastic strain eigenstates, [MATH] is the spatially dependent pump laser induced perturbation and [MATH] is an effective material-dependent generation parameter that considers different light-matter couplings.', '1709.08987-1-8-3': 'All parameters are implicitly assumed to depend on the laser wavelength.', '1709.08987-1-8-4': 'We will be interested here in the relative intensity of the vibrational modes, not in their absolute values so that the main physical ingredients are expressed in the functional form of Eq. [REF].', '1709.08987-1-8-5': 'Equation [REF] reflects the spatial overlap of the strain eigenstates with the light-induced stress.', '1709.08987-1-8-6': 'The latter can be written (independently of the mechanism involved) as [MATH].', '1709.08987-1-8-7': '[CITATION] Here [MATH] is the function describing the temporal evolution of the light-induced perturbation.', '1709.08987-1-8-8': 'Typically it is a delta-like function for radiation pressure and electrostriction forces, and a step-like function for the photothermal and optoelectronic mechanisms, broadened by the time-delay of the mechanism involved.', '1709.08987-1-8-9': 'The spatial distribution of the optical excitation [MATH] corresponds to the cavity confined electric field [MATH] for radiation pressure and electrostriction forces, but can be different from it for the other two mechanisms depending on the spatial distribution and dynamics of excited charges and laser-induced temperature variations.', '1709.08987-1-8-10': 'As we argue next, this will be a way to identify the main optical force under play in the studied devices.', '1709.08987-1-9-0': 'Figures [REF] a-b present the case of the "bulk" GaAs cavity.', '1709.08987-1-9-1': 'Panel (a) in the figure displays the experimental spectrum, compared with calculations assuming that [MATH] reproduces the instantaneous spatial distribution of the light intensity ([MATH]), or that it corresponds to a flat photoexcited carrier distribution within the GaAs-spacer ("relaxed").', '1709.08987-1-9-2': 'That is, it assumes that within the time in which the pump laser-induced perturbation is effective (of the order of the half period of the vibrational frequencies involved), the photoexcited carrier spatial distribution relaxes extending their presence throughout the full width of the GaAs cavity spacer material.', '1709.08987-1-9-3': 'Three peaks are clearly visible at [MATH], [MATH] and [MATH] GHz, corresponding to the fundamental, third and fifth overtones of the z-polarized cavity confined breathing mechanical modes.', '1709.08987-1-9-4': '[CITATION] The associated spatial distribution of the strain fields [MATH], together with that of the light induced optoelectronic stress, are shown in panel (b) of the figure.', '1709.08987-1-9-5': '[MATH] is assumed to be non-zero only in GaAs.', '1709.08987-1-9-6': 'It reflects the physical ingredient that photoelastic (electrostrictive) coupling is resonantly enhanced in GaAs, and that photons are only absorbed in GaAs, all other materials being fully transparent at the involved wavelengths.', '1709.08987-1-9-7': 'The solid curves in [MATH] thus represent the region where the light-induced stress is finite, either reflecting the excitation field (yellow), or the relaxed situation (red).', '1709.08987-1-9-8': "It is clear in the experiment that the vibrational mode's amplitude decreases systematically with increasing frequency of the mode, something that according to the calculations is only compatible with the carriers having rapidly spread filling the full width of the cavity spacer.", '1709.08987-1-9-9': 'The explanation is straightforward considering the overlap integral given by Eq. [REF], and the involved spatial distributions depicted in panel(b) Fig. [REF].', '1709.08987-1-9-10': 'And it clearly excludes radiation pressure and electrostriction as the possible driving mechanisms.', '1709.08987-1-10-0': 'Based on the above discussion it is also clear from the top curve in Fig. [REF](a) that the relative weight of the higher frequency 60 GHz mode could be enhanced if the spatial distribution of the optical perturbation [MATH] could be forced to map-out the cavity field intensity [MATH].', '1709.08987-1-10-1': 'If, as argued, the main generation mechanism is indeed governed by optoelectronic forces, one could accomplished this task by artificially limiting the vertical diffusion of the photoexcited carriers.', '1709.08987-1-10-2': 'A natural way to do this is through an adequate engineering of the cavity spacer, e.g. by introducing GaAs/AlAs MQWs.', '1709.08987-1-10-3': 'This case is shown in Figs. [REF](c-d).', '1709.08987-1-10-4': 'Again panel (c) shows the experiment and calculation of the coherent phonon spectral intensity.', '1709.08987-1-10-5': 'Panel (d) displays the spatial distribution of the photoexcited stress, and that of the strain related to the involved vibrational modes.', '1709.08987-1-10-6': 'In this case relaxing the carrier distribution to fill the full width of the QWs, or maintaining the exact laser excitation pattern, makes no observable difference in the calculated spectra.', '1709.08987-1-10-7': 'Interestingly, by tailoring the spatial distribution of the photoexcited carriers using quantum wells we are able to confirm the role of optoelectronic stress as the main optomechanical coupling, and furthermore we have pushed the main vibrational frequency of these optomechanical resonators from the fundamental mode at 20 GHz to the third overtone at 60 GHz.', '1709.08987-1-10-8': 'These frequencies are one order of magnitude larger than the record frequencies demonstrated in other cavity optomechanics approaches.', '1709.08987-1-10-9': '[CITATION] From Fig. [REF](d) it also becomes clear why additional QWs were introduced in the design at the second and fourth DBR periods, and not at the first and third: because of a change of sign of the strain fields, the latter contribute to the overlap integral in Eq. [REF] with the sign reversed respect to the cavity spacer.', '1709.08987-1-11-0': 'Having demonstrated the central role of optoelectronic forces in the generation of GHz phonons in semiconductor microcavities upon resonant pulsed excitation, we address next the potentiality of this mechanism for the observation of laser cooling and optically induced self-oscillation.', '1709.08987-1-11-1': 'The fundamental concepts describing backaction dynamics in cavity optomechanics are grasped by the delayed force model which, for the optically modified vibrational damping rate [MATH] gives [CITATION] [EQUATION] with [MATH] and [MATH] the unperturbed mechanical quality factor and damping, respectively.', '1709.08987-1-11-2': '[MATH] and [MATH] are respectively the unperturbed mechanical mode frequency and stiffness, while [MATH] represents the change in the steady-state optical force for a small displacement [MATH] of the mechanical resonator around the equilibrium position.', '1709.08987-1-11-3': 'To lead to backaction optical forces need to respond with a delay to fluctuations that change the frequency of the optical mode.', '1709.08987-1-11-4': 'The simplest delay function to consider is an exponential function [MATH], with [MATH] the corresponding time-delay.', '1709.08987-1-11-5': 'Typically [MATH] would correspond to the cavity photon lifetime [MATH], but in general and particularly for photothermal and optoelectronic forces, it can be significantly longer than [MATH].', '1709.08987-1-11-6': '[MATH] in the presence of dynamical backaction can thus increase if [MATH], leading to laser cooling, or decrease and eventually attain zero (self-oscillation) if [MATH].', '1709.08987-1-11-7': 'The magnitude of this effect is proportional to the gradient of the optical force, which is a function of the deposited optical energy and the involved optomechanical coupling mechanism (that is, how this deposited energy is translated into a mechanical deformation).', '1709.08987-1-11-8': 'It is also proportional to a delay tuning factor [MATH].', '1709.08987-1-11-9': '[MATH] has a maximum for [MATH], which reflects the intuitive fact that the force fluctuations are more effective to induce vibrations if their time-constant is neither to short, nor too long, but tuned to the vibrational frequency so that [MATH].', '1709.08987-1-12-0': 'Table [REF] presents the different factors intervening in Eq. [REF] for the studied DBR microcavities and considered mechanisms, namely, the corresponding magnitude of the involved optical forces and delay tuning factor [MATH].', '1709.08987-1-12-1': 'The optical forces are given per photon (trapped in the cavity or absorbed depending on the mechanism).', '1709.08987-1-12-2': 'They have been evaluated using finite-element methods, [CITATION] and considering a micro-pillar with 20 period DBRs of 2 micrometer diameter.', '1709.08987-1-12-3': 'As argued above, the main difference between photothermal and optoelectronic forces respect to radiation pressure and electrostriction, is the larger amount of deposited energy per trapped cavity photon.', '1709.08987-1-12-4': 'Similarly to what is observed in GaAs microdisks, [CITATION] well below the gap (far from the excitonic resonances) both geometric (radiation-pressure) and photoelastic contributions to the optomechanical coupling factor have similar values.', '1709.08987-1-12-5': 'Ultra-strong resonant enhancement of the photoelastic coupling has been experimentally demonstrated in bare GaAs/AlAs MQWs.', '1709.08987-1-12-6': '[CITATION] In Table [REF] we provide the magnitude of the electrostrictive force per incident photon considering the two situations, far from resonance and at resonance as used in our experiments, assuming that the room temperature values of the photoelastic constant given in Ref. [CITATION] are valid for a similar MQW embedded in a pillar microcavity.', '1709.08987-1-12-7': 'Concerning the photothermal coupling, a quantitative evaluation of its relevance in semiconductor GaAs membranes and microdisks indicates that it can be significant.', '1709.08987-1-12-8': '[CITATION] It is also significant in our microcavities, as evidenced in Table [REF].', '1709.08987-1-12-9': 'What excludes it as a potentially relevant optomechanical force is its slow dynamics, which for the high frequency vibrations considered here leads to delay tuning factors [MATH] even for the fundamental breathing mode at 20 GHz and considering a relatively fast thermal relaxation [MATH] of the order of a [MATH]s.', '1709.08987-1-12-10': 'The delay tuning factor is close to its maximum value 0.5 for the impulsive radiation pressure and electrostrictive mechanisms, considering a cavity photon lifetime of a few picoseconds as in our vertical microcavities.', '1709.08987-1-12-11': 'But is also reasonably well tuned for the optoelectronic forces and 200 ps recombination times as demonstrated for pillars of a few microns lateral size in Ref. [CITATION].', '1709.08987-1-12-12': 'In fact, [MATH] is precisely the same value as attained for optimized cantilliver resonators that have evidenced strong optical cooling and self-oscillation dynamics based on photothermal forces.', '1709.08987-1-12-13': '[CITATION] The factor [MATH] decrease of [MATH] respect to radiation pressure and electrostriction is overly compensated by the much larger efficiency of optoelectronic forces.', '1709.08987-1-12-14': 'Note that the physical mechanism at the base of optoelectronic forces is the same as in photoelastic coupling, namely, deformation potential interaction.', '1709.08987-1-12-15': 'The qualitative difference is that a photon is scattered in the latter, while it is absorbed with the subsequent creation of real electron-hole pairs in the former.', '1709.08987-1-13-0': '# Conclusions and Outlook', '1709.08987-1-14-0': 'In conclusion, we have shown that optoelectronic deformation potential interactions are at the origin of the optical forces acting for excitation close to the semiconductor gaps in GaAs/AlAs microcavities.', '1709.08987-1-14-1': 'The carriers dynamics following photoexcitation is determinant of the emitted coherent acoustic phonon spectrum, and can be tailored using quantum wells to push the vibrational optomechanic frequencies from 20 up to 60 GHz, an order of magnitude larger than the highest standards in cavity optomechanics.', '1709.08987-1-14-2': 'The strong potentiality of optoelectronic forces for the demonstration of backaction dynamical effects in semiconductor microcavities was addressed.', '1709.08987-1-14-3': 'This could open the way to ultra-high frequency cavity optomechanics, and through it to quantum measurements and applications at higher temperatures than currently accessible.', '1709.08987-1-15-0': '# Funding Information', '1709.08987-1-16-0': 'This work was partially supported by the ANPCyT Grants PICT 2012-1661 and 2013-2047, the Labex NanoSaclay, and the international franco-argentinean laboratory LIFAN (CNRS-CONICET).'}

{'1709.08987-2-0-0': 'Radiation pressure, electrostriction, and photothermal forces have been investigated to evidence backaction, non-linearities and quantum phenomena in cavity optomechanics.', '1709.08987-2-0-1': 'We show here that optoelectronic forces involving real carrier excitation and deformation potential interaction are the strongest mechanism of light-to-sound transduction when exciting with pulsed lasers close to the GaAs optical gap of semiconductor GaAs/AlAs distributed Bragg reflector optomechanical resonators.', '1709.08987-2-0-2': 'We demonstrate that the ultrafast spatial redistribution of the photoexcited carriers in microcavities with massive GaAs spacers leads to an enhanced coupling to the fundamental 20 GHz vertically polarized mechanical breathing mode.', '1709.08987-2-0-3': 'The carrier diffusion can be limited by embedding GaAs quantum wells in the cavity spacer, a strategy used here to prove and engineer the role of optoelectronic forces in phonon generation with real carriers.', '1709.08987-2-0-4': 'The optical forces associated to the different intervening mechanisms and their relevance for dynamical backaction in optomechanics is evaluated using finite-element methods.', '1709.08987-2-0-5': 'The results presented open the path to the study of hitherto seldom investigated dynamical backaction in optomechanical solid-state resonators in the presence of optoelectronic forces.', '1709.08987-2-1-0': '# Motivation', '1709.08987-2-2-0': 'Backaction in cavity optomechanics has shown to lead to novel physical phenomena including laser cooling, self-oscillation, and non-linear dynamics in systems that go from kilometer size interferometers to single or few trapped ions.', '1709.08987-2-2-1': '[CITATION] Briefly, a resonant photon field in a cavity exerts a force and induces a mechanical motion on the mirrors, which in turn leads to a delayed modification of the resonant condition of the trapped field.', '1709.08987-2-2-2': 'Such coupled dynamics can be exploited for a large variety of applications that span for example from gravitational wave detection [CITATION] to the study of quantum motion states in mesoscopic mechanical systems.', '1709.08987-2-2-3': '[CITATION] How light exerts force on matter is at the center of this investigations.', '1709.08987-2-2-4': 'Photons can apply stress through radiation pressure, transferring impulse when reflected from the mirrors.', '1709.08987-2-2-5': '[CITATION] Related mechanisms also deriving from the same fundamental interaction (Lorentz forces) are gradient forces (also exploited in optical tweezers) [CITATION], and electrostriction.', '1709.08987-2-2-6': 'The latter, linked to the material´s photoelasticity, has been shown to play a role that can be of the same magnitude as radiation pressure, [CITATION] or even larger if optical resonances are exploited in direct bandgap materials as for example GaAs.', '1709.08987-2-2-7': '[CITATION]', '1709.08987-2-3-0': 'In the presence of radiation pressure forces, the energy [MATH] of the photon is shifted by the Doppler effect by an amount of the order [MATH], where [MATH] and [MATH] are the mirror and light velocities, respectively.', '1709.08987-2-3-1': 'The mechanical energy transferred from the photon to the mirror is thus very small.', '1709.08987-2-3-2': 'Electrostriction leads to Raman-like processes, for which the transferred energy [MATH] corresponds to the involved vibrational frequency.', '1709.08987-2-3-3': 'This leads to inelastic scattering side-bands.', '1709.08987-2-3-4': 'Again, tipically [MATH].', '1709.08987-2-3-5': 'Contrastingly, if the photon is absorbed in the process of interaction, all its energy is transferred to the mirror.', '1709.08987-2-3-6': 'This fundamental difference has been used in cavity optomechanics to demonstrate strongly enhanced light-matter interactions based on photothermal forces.', '1709.08987-2-3-7': '[CITATION] In materials displaying optical resonances, the photons can be resonantly absorbed with the consequent transfer of electrons to excited states.', '1709.08987-2-3-8': 'Photoexcited carrier-mediated optomechanical interactions have been reported in semiconductor modulation-doped heterostructure-cantilever cavity-less hybrid systems.', '1709.08987-2-3-9': 'Efficient cavity-less optomechanical transduction involving opto-piezoelectric backaction from the bound photoexcited electron-hole pairs have been demonstrated in these systems, including self-feedback cooling, amplification of the thermomechanical motion, and control of the mechanical quality factor through carrier excitation.', '1709.08987-2-3-10': '[CITATION] The change of electronic landscape produced by photoexcited carriers also induces a stress in the structure through electron-phonon coupling mediated by deformation potential interaction.', '1709.08987-2-3-11': 'This stress can be identified as an optoelectronic force, and should have the same kind of temporal behavior (with different time-scales and details depending on the carrier dynamics) and amplified strength as observed for photothermal forces.', '1709.08987-2-3-12': '[CITATION] Recently optical cooling of mechanical modes of a GaAs nanomembrane forming part of an optical cavity was reported [CITATION] and its relation to optoelectronic stress via the deformation potential was analysed.', '1709.08987-2-3-13': 'Because of the very fast relaxation rate of excited carriers due to surface recombination in such nanometer size structures, it was concluded that thermal (and not optoelectronic) stress was the primary cause of cooling in that case.', '1709.08987-2-3-14': 'We will demonstrate here that the carrier dynamics can be fundamentally modified when the nanometer size GaAs layer is embedded in a monolithic microcavity, making optoelectronic forces the main mechanism of interaction of light with vibrations in such semiconductor devices.', '1709.08987-2-3-15': 'The diffusion of the photoexcited carriers thus assumes a central role, a dynamics that can be engineered using embedded quantum wells (QWs).', '1709.08987-2-4-0': 'Because of their optoelectronic properties, semiconductor GaAs/AlAs microcavities are interesting candidates for novel functionalities in cavity optomechanics.', '1709.08987-2-4-1': 'Perfect photon-phonon overlap, and access to electronically resonant coupling in addition to radiation pressure could lead to strong optomechanical interactions of photoelastic origin.', '1709.08987-2-4-2': '[CITATION] The vibrational frequencies in these microresonators are determined by the vertical layering of the device (fabricated with the ultra-high quality of molecular beam epitaxy), and not by the lateral pattering (defined by the more limited performance of microfabrication techniques).', '1709.08987-2-4-3': 'This has allowed access to much higher frequencies for the optomechanical vibrational modes, in the 20-100 GHz range, without significant reduction of the mechanical Q-factors.', '1709.08987-2-4-4': '[CITATION] In addition, these optomechanical resonators enable the conception of hybrid architectures involving artificial atoms (semiconductor excitons) coupled to the optical cavity mode, and thus combining the physics of cavity optomechanics with cavity quantum electrodynamics.', '1709.08987-2-4-5': '[CITATION] Our motivation here is to search for evidence of optoelectronic forces in these devices, and for that purpose we study the light-sound coupling involving the resonator mechanical modes and with the optical cavity at resonance with the material exciton transition energy.', '1709.08987-2-4-6': 'We demonstrate that the main phonon generation mechanism using pulsed lasers close to resonance in these devices involves indeed the real excitation of carriers and the deformation potential mechanism.', '1709.08987-2-4-7': '[CITATION] We show that in microcavities with "bulk" GaAs spacers (i.e. cavity spacers constituted by a thick [MATH] layer of GaAs) ultrafast carrier redistribution leads to an enhanced coupling to the more uniformly distributed fundamental 20 GHz cavity vibrational mode.', '1709.08987-2-4-8': 'An engineering of the structure taking into account this effect and using embedded quantum wells is used to limit the carrier diffusion and enhance the optomechanical coupling.', '1709.08987-2-4-9': 'This opens the way to new opportunities to the field of optomechanics.', '1709.08987-2-5-0': '# Results', '1709.08987-2-6-0': 'We consider two planar microcavity structures, specifically a "bulk" GaAs and a multiple quantum well (MQW) resonator.', '1709.08987-2-6-1': 'The "bulk" GaAs microcavity is made of a [MATH] GaAs-spacer enclosed by ([MATH] Al[MATH]Ga[MATH]As /AlAs DBRs, 20 pairs of layers on the bottom, 18 on top, grown on a GaAs substrate (a scheme of the structure is presented in Fig. [REF](b)).', '1709.08987-2-6-2': '[CITATION] The DBRs stop band is centered at the design wavelength of [MATH].', '1709.08987-2-6-3': 'As we have demonstrated previously, this structure performs as an optomechanical resonator that simultaneously confines photons and acoustic phonons of the same wavelength.', '1709.08987-2-6-4': '[CITATION] In the MQW microcavity the [MATH] spacer is constituted by six [MATH] GaAs QWs separated by [MATH] AlAs barriers.', '1709.08987-2-6-5': 'To further enhance the light-sound coupling the second and fourth [MATH] DBR alloy layers on each side are also replaced by three GaAs/AlAs QWs.', '1709.08987-2-6-6': 'The reason for this design will become clear below.', '1709.08987-2-6-7': 'The DBRs in this case are ([MATH] Al[MATH]Ga[MATH]As /AlAs multilayers, 27 pairs on the bottom, 23 on top, grown again on a GaAs substrate.', '1709.08987-2-6-8': 'A scheme of this structure is displayed in Fig. [REF](c).', '1709.08987-2-6-9': 'The number of DBR periods in both structures is designed to assure an optical Q-factor [MATH] (cavity photon lifetime [MATH] ps).', '1709.08987-2-7-0': 'Reflection-type degenerate pump-probe experiments were performed with the laser wavelength tuned with the optical cavity mode (see the scheme in Fig. [REF](a)).', '1709.08987-2-7-1': '[CITATION] Picosecond pulses ([MATH] ps) from a mode-locked Ti:Sapphire laser, with repetition rate 80 MHz, were split into cross polarized pump (power 20 mW) and probe (1 mW) pulses.', '1709.08987-2-7-2': 'Both pulses were focused onto superimposed [MATH]m-diameter spots.', '1709.08987-2-7-3': 'To couple the light to the microcavity the probe beam propagates close to the sample normal and is tuned to the high derivative flank of the reflectivity dip, while the pump incidence angle is set for resonant condition precisely at the cavity mode.', '1709.08987-2-7-4': '[CITATION] The laser wavelength was also set so that the phonon generation and detection would be a few meV below resonance with the direct bandgap of the GaAs "bulk" cavity spacer ([MATH] eV) or of the QWs ([MATH] eV).', '1709.08987-2-7-5': 'To accomplish this resonant excitation the temperature was also used as a tuning parameter; the "bulk" cavity experiments were done at room temperature, while the MQW structure was studied at 80 K. Light is thus coupled resonantly with the optical cavity mode and the semiconductor excitonic resonance.', '1709.08987-2-7-6': 'Acoustic phonons confined in the same space as the optical cavity mode are selectively generated within the resonator, and are detected through their modulation of the optical cavity mode frequency.', '1709.08987-2-7-7': '[CITATION]', '1709.08987-2-8-0': 'These pump-probe ultrafast laser experiments are conceptually similar to the ring-down techniques recently exploited in the cavity optomechanics domain, [CITATION] but more appropriate to the study of ultra-high frequency vibrations (GHz-THz range.)', '1709.08987-2-8-1': 'The pulsed laser phonon generation efficiency can be described as: [CITATION] [EQUATION]', '1709.08987-2-8-2': 'Here [MATH] is the phonon frequency, [MATH] describes the elastic strain eigenstates, [MATH] is the spatially dependent pump laser induced perturbation and [MATH] is an effective material-dependent generation parameter that considers different light-matter couplings.', '1709.08987-2-8-3': 'All parameters are implicitly assumed to depend on the laser wavelength.', '1709.08987-2-8-4': 'We will be interested here in the relative intensity of the vibrational modes, not in their absolute values so that the main physical ingredients are expressed in the functional form of Eq. [REF].', '1709.08987-2-8-5': 'Equation [REF] reflects the spatial overlap of the strain eigenstates with the light-induced stress.', '1709.08987-2-8-6': 'The latter can be written (independently of the mechanism involved) as [MATH].', '1709.08987-2-8-7': '[CITATION] Here [MATH] is the function describing the temporal evolution of the light-induced perturbation.', '1709.08987-2-8-8': 'Typically it is a delta-like function for radiation pressure and electrostriction forces, and a step-like function for the photothermal and optoelectronic mechanisms, broadened by the time-delay of the mechanism involved.', '1709.08987-2-8-9': 'The spatial distribution of the optical excitation [MATH] corresponds to the cavity confined electric field [MATH] for radiation pressure and electrostriction forces, but can be different from it for the other two mechanisms depending on the spatial distribution and dynamics of excited charges and laser-induced temperature variations.', '1709.08987-2-8-10': 'As we argue next, this will be a way to identify the main optical force under play in the studied devices.', '1709.08987-2-9-0': 'Figures [REF] a-b present the case of the "bulk" GaAs cavity.', '1709.08987-2-9-1': 'Panel (a) in the figure displays the experimental spectrum, compared with calculations assuming that [MATH] reproduces the instantaneous spatial distribution of the light intensity ([MATH]), or that it corresponds to a flat photoexcited carrier distribution within the GaAs-spacer ("relaxed").', '1709.08987-2-9-2': 'That is, it assumes that within the time in which the pump laser-induced perturbation is effective (of the order of the half period of the vibrational frequencies involved), the photoexcited carrier spatial distribution relaxes extending their presence throughout the full width of the GaAs cavity spacer material.', '1709.08987-2-9-3': 'Three peaks are clearly visible at [MATH], [MATH] and [MATH] GHz, corresponding to the fundamental, third and fifth overtones of the z-polarized cavity confined breathing mechanical modes.', '1709.08987-2-9-4': '[CITATION] The associated spatial distribution of the strain fields [MATH], together with that of the light induced optoelectronic stress, are shown in panel (b) of the figure.', '1709.08987-2-9-5': '[MATH] is assumed to be non-zero only in GaAs.', '1709.08987-2-9-6': 'It reflects the physical ingredient that photoelastic (electrostrictive) coupling is resonantly enhanced in GaAs, and that photons are only absorbed in GaAs, all other materials being fully transparent at the involved wavelengths.', '1709.08987-2-9-7': 'The solid curves in [MATH] thus represent the region where the light-induced stress is finite, either reflecting the excitation field (yellow), or the relaxed situation (red).', '1709.08987-2-9-8': "It is clear in the experiment that the vibrational mode's amplitude decreases systematically with increasing frequency of the mode, something that according to the calculations is only compatible with the carriers having rapidly spread filling the full width of the cavity spacer.", '1709.08987-2-9-9': 'The explanation is straightforward considering the overlap integral given by Eq. [REF], and the involved spatial distributions depicted in panel(b) Fig. [REF].', '1709.08987-2-9-10': 'And it clearly excludes radiation pressure and electrostriction as the possible driving mechanisms.', '1709.08987-2-10-0': 'Based on the above discussion it is also clear from the top curve in Fig. [REF](a) that the relative weight of the higher frequency 60 GHz mode could be enhanced if the spatial distribution of the optical perturbation [MATH] could be forced to map-out the cavity field intensity [MATH].', '1709.08987-2-10-1': 'If, as argued, the main generation mechanism is indeed governed by optoelectronic forces, one could accomplished this task by artificially limiting the vertical diffusion of the photoexcited carriers.', '1709.08987-2-10-2': 'A natural way to do this is through an adequate engineering of the cavity spacer, e.g. by introducing GaAs/AlAs MQWs.', '1709.08987-2-10-3': 'This case is shown in Figs. [REF](c-d).', '1709.08987-2-10-4': 'Again panel (c) shows the experiment and calculation of the coherent phonon spectral intensity.', '1709.08987-2-10-5': 'Panel (d) displays the spatial distribution of the photoexcited stress, and that of the strain related to the involved vibrational modes.', '1709.08987-2-10-6': 'In this case relaxing the carrier distribution to fill the full width of the QWs, or maintaining the exact laser excitation pattern, makes no observable difference in the calculated spectra.', '1709.08987-2-10-7': 'Interestingly, by tailoring the spatial distribution of the photoexcited carriers using quantum wells we are able to confirm the role of optoelectronic stress as the main optomechanical coupling, and furthermore we have pushed the main vibrational frequency of these optomechanical resonators from the fundamental mode at 20 GHz to the third overtone at 60 GHz.', '1709.08987-2-10-8': 'These frequencies are one order of magnitude larger than the record frequencies demonstrated in other cavity optomechanics approaches.', '1709.08987-2-10-9': '[CITATION] From Fig. [REF](d) it also becomes clear why additional QWs were introduced in the design at the second and fourth DBR periods, and not at the first and third: because of a change of sign of the strain fields, the latter contribute to the overlap integral in Eq. [REF] with the sign reversed respect to the cavity spacer.', '1709.08987-2-11-0': 'Having demonstrated the central role of optoelectronic forces in the generation of GHz phonons in semiconductor microcavities upon resonant pulsed excitation, we address next the potentiality of this mechanism for the observation of laser cooling and optically induced self-oscillation.', '1709.08987-2-11-1': 'The fundamental concepts describing backaction dynamics in cavity optomechanics are grasped by the delayed force model which, for the optically modified vibrational damping rate [MATH] gives [CITATION] [EQUATION] with [MATH] and [MATH] the unperturbed mechanical quality factor and damping, respectively.', '1709.08987-2-11-2': '[MATH] and [MATH] are respectively the unperturbed mechanical mode frequency and stiffness, while [MATH] represents the change in the steady-state optical force for a small displacement [MATH] of the mechanical resonator around the equilibrium position.', '1709.08987-2-11-3': 'To lead to backaction optical forces need to respond with a delay to fluctuations that change the frequency of the optical mode.', '1709.08987-2-11-4': 'The simplest delay function to consider is an exponential function [MATH], with [MATH] the corresponding time-delay.', '1709.08987-2-11-5': 'Typically [MATH] would correspond to the cavity photon lifetime [MATH], but in general and particularly for photothermal and optoelectronic forces, it can be significantly longer than [MATH].', '1709.08987-2-11-6': '[MATH] in the presence of dynamical backaction can thus increase if [MATH], leading to laser cooling, or decrease and eventually attain zero (self-oscillation) if [MATH].', '1709.08987-2-11-7': 'The magnitude of this effect is proportional to the gradient of the optical force, which is a function of the deposited optical energy and the involved optomechanical coupling mechanism (that is, how this deposited energy is translated into a mechanical deformation).', '1709.08987-2-11-8': 'It is also proportional to a delay tuning factor [MATH].', '1709.08987-2-11-9': '[MATH] has a maximum for [MATH], which reflects the intuitive fact that the force fluctuations are more effective to induce vibrations if their time-constant is neither too short, nor too long, but tuned to the vibrational frequency so that [MATH].', '1709.08987-2-12-0': 'Table [REF] presents the different factors intervening in Eq. [REF] for the studied DBR microcavities and considered mechanisms, namely, the corresponding magnitude of the involved optical forces and delay tuning factor [MATH].', '1709.08987-2-12-1': 'The optical forces are given per photon (trapped in the cavity or absorbed depending on the mechanism).', '1709.08987-2-12-2': 'They have been evaluated using finite-element methods, [CITATION] and considering a micro-pillar with 20 period DBRs of 2 micrometer diameter.', '1709.08987-2-12-3': 'As argued above, the main difference between photothermal and optoelectronic forces respect to radiation pressure and electrostriction, is the larger amount of deposited energy per trapped cavity photon.', '1709.08987-2-12-4': 'Similarly to what is observed in GaAs microdisks, [CITATION] well below the gap (far from the excitonic resonances) both geometric (radiation-pressure) and photoelastic contributions to the optomechanical coupling factor have similar values.', '1709.08987-2-12-5': 'Ultra-strong resonant enhancement of the photoelastic coupling has been experimentally demonstrated in bare GaAs/AlAs MQWs.', '1709.08987-2-12-6': '[CITATION] In Table [REF] we provide the magnitude of the electrostrictive force per incident photon considering the two situations, far from resonance and at resonance as used in our experiments, assuming that the room temperature values of the photoelastic constant given in Ref. [CITATION] are valid for a similar MQW embedded in a pillar microcavity.', '1709.08987-2-12-7': 'Concerning the photothermal coupling, a quantitative evaluation of its relevance in semiconductor GaAs membranes and microdisks indicates that it can be significant.', '1709.08987-2-12-8': '[CITATION] It is also significant in our microcavities, as evidenced in Table [REF].', '1709.08987-2-12-9': 'What excludes it as a potentially relevant optomechanical force is its slow dynamics, which for the high frequency vibrations considered here leads to delay tuning factors [MATH] even for the fundamental breathing mode at 20 GHz and considering a relatively fast thermal relaxation [MATH] of the order of a [MATH]s.', '1709.08987-2-12-10': 'The delay tuning factor is close to its maximum value 0.5 for the impulsive radiation pressure and electrostrictive mechanisms, considering a cavity photon lifetime of a few picoseconds as in our vertical microcavities.', '1709.08987-2-12-11': 'It is also reasonably well tuned for the optoelectronic forces and 200 ps recombination times as demonstrated for pillars of a few microns lateral size in Ref. [CITATION].', '1709.08987-2-12-12': 'In fact, [MATH] is precisely the same value as attained for optimized cantilever resonators that have evidenced strong optical cooling and self-oscillation dynamics based on photothermal forces.', '1709.08987-2-12-13': '[CITATION] The factor [MATH] decrease of [MATH] with respect to radiation pressure and electrostriction is overly compensated by the much larger efficiency of optoelectronic forces.', '1709.08987-2-12-14': 'Note that the physical mechanism at the base of optoelectronic forces is the same as in photoelastic coupling, namely, deformation potential interaction.', '1709.08987-2-12-15': 'The qualitative difference is that a photon is scattered in the latter, while it is absorbed with the subsequent creation of real electron-hole pairs in the former.', '1709.08987-2-13-0': '# Conclusions and Outlook', '1709.08987-2-14-0': 'In conclusion, we have shown that optoelectronic deformation potential interactions are at the origin of the optical forces acting for excitation close to the semiconductor gaps in GaAs/AlAs microcavities.', '1709.08987-2-14-1': 'The carriers dynamics following photoexcitation is determinant of the emitted coherent acoustic phonon spectrum, and can be tailored using quantum wells to push the vibrational optomechanic frequencies from 20 up to 60 GHz, an order of magnitude larger than the highest standards in cavity optomechanics.', '1709.08987-2-14-2': 'The strong potentiality of optoelectronic forces for the demonstration of backaction dynamical effects in semiconductor microcavities was addressed.', '1709.08987-2-14-3': 'This could open the way to ultra-high frequency cavity optomechanics, and through it to quantum measurements and applications at higher temperatures than currently accessible.', '1709.08987-2-15-0': '# Funding Information', '1709.08987-2-16-0': 'This work was partially supported by the ANPCyT Grants PICT 2012-1661 and 2013-2047, the Labex NanoSaclay, and the international franco-argentinean laboratory LIFAN (CNRS-CONICET).'}

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[]

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['1709.08987-1-2-7', '1709.08987-1-3-4', '1709.08987-1-7-7', '1709.08987-2-2-7', '1709.08987-2-3-4', '1709.08987-2-7-7', '1709.08987-3-2-7', '1709.08987-3-3-4', '1709.08987-3-7-7', '1709.08987-3-19-2', '1709.08987-3-21-7', '1709.08987-4-2-7', '1709.08987-4-3-4', '1709.08987-4-7-7', '1709.08987-4-21-2', '1709.08987-4-23-7']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1709.08987

{'1709.08987-3-0-0': 'Radiation pressure, electrostriction, and photothermal forces have been investigated to evidence backaction, non-linearities and quantum phenomena in cavity optomechanics.', '1709.08987-3-0-1': 'We show here that optoelectronic forces involving real carrier excitation and deformation potential interaction are the strongest mechanism of light-to-sound transduction when exciting with pulsed lasers close to the GaAs optical gap of semiconductor GaAs/AlAs distributed Bragg reflector optomechanical resonators.', '1709.08987-3-0-2': 'We demonstrate that the ultrafast spatial redistribution of the photoexcited carriers in microcavities with massive GaAs spacers leads to an enhanced coupling to the fundamental 20 GHz vertically polarized mechanical breathing mode.', '1709.08987-3-0-3': 'The carrier diffusion can be limited by embedding GaAs quantum wells in the cavity spacer, a strategy used here to prove and engineer the role of optoelectronic forces in phonon generation with real carriers.', '1709.08987-3-0-4': 'The optical forces associated to the different intervening mechanisms and their relevance for dynamical backaction in optomechanics is evaluated using finite-element methods.', '1709.08987-3-0-5': 'The results presented open the path to the study of hitherto seldom investigated dynamical backaction in optomechanical solid-state resonators in the presence of optoelectronic forces.', '1709.08987-3-1-0': '# Motivation', '1709.08987-3-2-0': 'Backaction in cavity optomechanics has shown to lead to novel physical phenomena including laser cooling, self-oscillation, and non-linear dynamics in systems that go from kilometer size interferometers to single or few trapped ions.', '1709.08987-3-2-1': '[CITATION] Briefly, a resonant photon field in a cavity exerts a force and induces a mechanical motion on the mirrors, which in turn leads to a delayed modification of the resonant condition of the trapped field.', '1709.08987-3-2-2': 'Such coupled dynamics can be exploited for a large variety of applications that span for example from gravitational wave detection [CITATION] to the study of quantum motion states in mesoscopic mechanical systems.', '1709.08987-3-2-3': '[CITATION] How light exerts force on matter is at the center of this investigations.', '1709.08987-3-2-4': 'Photons can apply stress through radiation pressure, transferring impulse when reflected from the mirrors.', '1709.08987-3-2-5': '[CITATION] Related mechanisms also deriving from the same fundamental interaction (Lorentz forces) are gradient forces (also exploited in optical tweezers) [CITATION], and electrostriction.', '1709.08987-3-2-6': 'The latter, linked to the material´s photoelasticity, has been shown to play a role that can be of the same magnitude as radiation pressure, [CITATION] or even larger if optical resonances are exploited in direct bandgap materials as for example GaAs.', '1709.08987-3-2-7': '[CITATION]', '1709.08987-3-3-0': 'In the presence of radiation pressure forces, the energy [MATH] of the photon is shifted by the Doppler effect by an amount of the order [MATH], where [MATH] and [MATH] are the mirror and light velocities, respectively.', '1709.08987-3-3-1': 'The mechanical energy transferred from the photon to the mirror is thus very small.', '1709.08987-3-3-2': 'Electrostriction leads to Raman-like processes, for which the transferred energy [MATH] corresponds to the involved vibrational frequency.', '1709.08987-3-3-3': 'This leads to inelastic scattering side-bands.', '1709.08987-3-3-4': 'Again, tipically [MATH].', '1709.08987-3-3-5': 'Contrastingly, if the photon is absorbed in the process of interaction, all its energy is transferred to the mirror.', '1709.08987-3-3-6': 'This fundamental difference has been used in cavity optomechanics to demonstrate strongly enhanced light-matter interactions based on photothermal forces.', '1709.08987-3-3-7': '[CITATION] In materials displaying optical resonances, the photons can be resonantly absorbed with the consequent transfer of electrons to excited states.', '1709.08987-3-3-8': 'Photoexcited carrier-mediated optomechanical interactions have been reported in semiconductor modulation-doped heterostructure-cantilever cavity-less hybrid systems.', '1709.08987-3-3-9': 'Efficient cavity-less optomechanical transduction involving opto-piezoelectric backaction from the bound photoexcited electron-hole pairs have been demonstrated in these systems, including self-feedback cooling, amplification of the thermomechanical motion, and control of the mechanical quality factor through carrier excitation.', '1709.08987-3-3-10': '[CITATION] The change of electronic landscape produced by photoexcited carriers also induces a stress in the structure through electron-phonon coupling mediated by deformation potential interaction.', '1709.08987-3-3-11': 'This stress can be identified as an optoelectronic force, and should have the same kind of temporal behavior (with different time-scales and details depending on the carrier dynamics) and amplified strength as observed for photothermal forces.', '1709.08987-3-3-12': '[CITATION] Recently optical cooling of mechanical modes of a GaAs nanomembrane forming part of an optical cavity was reported [CITATION] and its relation to optoelectronic stress via the deformation potential was analysed.', '1709.08987-3-3-13': 'Because of the very fast relaxation rate of excited carriers due to surface recombination in such nanometer size structures, it was concluded that thermal (and not optoelectronic) stress was the primary cause of cooling in that case.', '1709.08987-3-3-14': 'We will demonstrate here that the carrier dynamics can be fundamentally modified when the nanometer size GaAs layer is embedded in a monolithic microcavity, making optoelectronic forces the main mechanism of interaction of light with vibrations in such semiconductor devices.', '1709.08987-3-3-15': 'The diffusion of the photoexcited carriers thus assumes a central role, a dynamics that can be engineered using embedded quantum wells (QWs).', '1709.08987-3-4-0': 'Because of their optoelectronic properties, semiconductor GaAs/AlAs microcavities are interesting candidates for novel functionalities in cavity optomechanics.', '1709.08987-3-4-1': 'Perfect photon-phonon overlap, and access to electronically resonant coupling in addition to radiation pressure could lead to strong optomechanical interactions of photoelastic origin.', '1709.08987-3-4-2': '[CITATION] The vibrational frequencies in these microresonators are determined by the vertical layering of the device (fabricated with the ultra-high quality of molecular beam epitaxy), and not by the lateral pattering (defined by the more limited performance of microfabrication techniques).', '1709.08987-3-4-3': 'This has allowed access to much higher frequencies for the optomechanical vibrational modes, in the 20-100 GHz range, without significant reduction of the mechanical Q-factors.', '1709.08987-3-4-4': '[CITATION] In addition, these optomechanical resonators enable the conception of hybrid architectures involving artificial atoms (semiconductor excitons) coupled to the optical cavity mode, and thus combining the physics of cavity optomechanics with cavity quantum electrodynamics.', '1709.08987-3-4-5': '[CITATION] Our motivation here is to search for evidence of optoelectronic forces in these devices, and for that purpose we study the light-sound coupling involving the resonator mechanical modes and with the optical cavity at resonance with the material exciton transition energy.', '1709.08987-3-4-6': 'We demonstrate that the main phonon generation mechanism using pulsed lasers close to resonance in these devices involves indeed the real excitation of carriers and the deformation potential mechanism.', '1709.08987-3-4-7': '[CITATION] We show that in microcavities with "bulk" GaAs spacers (i.e. cavity spacers constituted by a thick [MATH] layer of GaAs) ultrafast carrier redistribution leads to an enhanced coupling to the more uniformly distributed fundamental 20 GHz cavity vibrational mode.', '1709.08987-3-4-8': 'An engineering of the structure taking into account this effect and using embedded quantum wells is used to limit the carrier diffusion and enhance the optomechanical coupling.', '1709.08987-3-4-9': 'This opens the way to new opportunities to the field of optomechanics.', '1709.08987-3-5-0': '# Results', '1709.08987-3-6-0': 'We consider two planar microcavity structures, specifically a "bulk" GaAs and a multiple quantum well (MQW) resonator.', '1709.08987-3-6-1': 'The "bulk" GaAs microcavity is made of a [MATH] GaAs-spacer enclosed by ([MATH] Al[MATH]Ga[MATH]As /AlAs DBRs, 20 pairs of layers on the bottom, 18 on top, grown on a GaAs substrate (a scheme of the structure is presented in Fig. [REF](b)).', '1709.08987-3-6-2': '[CITATION] The DBRs stop band is centered at the design wavelength of [MATH].', '1709.08987-3-6-3': 'As we have demonstrated previously, this structure performs as an optomechanical resonator that simultaneously confines photons and acoustic phonons of the same wavelength.', '1709.08987-3-6-4': '[CITATION] In the MQW microcavity the [MATH] spacer is constituted by six [MATH] GaAs QWs separated by [MATH] AlAs barriers.', '1709.08987-3-6-5': 'To further enhance the light-sound coupling the second and fourth [MATH] DBR alloy layers on each side are also replaced by three GaAs/AlAs QWs.', '1709.08987-3-6-6': 'The reason for this design will become clear below.', '1709.08987-3-6-7': 'The DBRs in this case are ([MATH] Al[MATH]Ga[MATH]As /AlAs multilayers, 27 pairs on the bottom, 23 on top, grown again on a GaAs substrate.', '1709.08987-3-6-8': 'A scheme of this structure is displayed in Fig. [REF](c).', '1709.08987-3-6-9': 'The number of DBR periods in both structures is designed to assure an optical Q-factor [MATH] (cavity photon lifetime [MATH] ps).', '1709.08987-3-7-0': 'Reflection-type degenerate pump-probe experiments were performed with the laser wavelength tuned with the optical cavity mode (see the scheme in Fig. [REF](a)).', '1709.08987-3-7-1': '[CITATION] Picosecond pulses ([MATH] ps) from a mode-locked Ti:Sapphire laser, with repetition rate 80 MHz, were split into cross polarized pump (power 20 mW) and probe (1 mW) pulses.', '1709.08987-3-7-2': 'Both pulses were focused onto superimposed [MATH]m-diameter spots.', '1709.08987-3-7-3': 'To couple the light to the microcavity the probe beam propagates close to the sample normal and is tuned to the high derivative flank of the reflectivity dip, while the pump incidence angle is set for resonant condition precisely at the cavity mode.', '1709.08987-3-7-4': '[CITATION] The laser wavelength was also set so that the phonon generation and detection would be a few meV below resonance with the direct bandgap of the GaAs "bulk" cavity spacer ([MATH] eV) or of the QWs ([MATH] eV).', '1709.08987-3-7-5': 'To accomplish this resonant excitation the temperature was also used as a tuning parameter; the "bulk" cavity experiments were done at room temperature, while the MQW structure was studied at 80 K. Light is thus coupled resonantly with the optical cavity mode and the semiconductor excitonic resonance.', '1709.08987-3-7-6': 'Acoustic phonons confined in the same space as the optical cavity mode are selectively generated within the resonator, and are detected through their modulation of the optical cavity mode frequency.', '1709.08987-3-7-7': '[CITATION]', '1709.08987-3-8-0': 'These pump-probe ultrafast laser experiments are conceptually similar to the ring-down techniques recently exploited in the cavity optomechanics domain, [CITATION] but more appropriate to the study of ultra-high frequency vibrations (GHz-THz range.)', '1709.08987-3-8-1': 'The pulsed laser phonon generation efficiency can be described as: [CITATION] [EQUATION]', '1709.08987-3-8-2': 'Here [MATH] is the phonon frequency, [MATH] describes the elastic strain eigenstates, [MATH] is the spatially dependent pump laser induced perturbation and [MATH] is an effective material-dependent generation parameter that considers different light-matter couplings.', '1709.08987-3-8-3': 'All parameters are implicitly assumed to depend on the laser wavelength.', '1709.08987-3-8-4': 'We will be interested here in the relative intensity of the vibrational modes, not in their absolute values so that the main physical ingredients are expressed in the functional form of Eq. [REF].', '1709.08987-3-8-5': 'Equation [REF] reflects the spatial overlap of the strain eigenstates with the light-induced stress.', '1709.08987-3-8-6': 'The latter can be written (independently of the mechanism involved) as [MATH].', '1709.08987-3-8-7': '[CITATION] Here [MATH] is the function describing the temporal evolution of the light-induced perturbation.', '1709.08987-3-8-8': 'Typically it is a delta-like function for radiation pressure and electrostriction forces, and a step-like function for the photothermal and optoelectronic mechanisms, broadened by the time-delay of the mechanism involved.', '1709.08987-3-8-9': 'The spatial distribution of the optical excitation [MATH] corresponds to the cavity confined electric field [MATH] for radiation pressure and electrostriction forces, but can be different from it for the other two mechanisms depending on the spatial distribution and dynamics of excited charges and laser-induced temperature variations.', '1709.08987-3-8-10': 'As we argue next, this will be a way to identify the main optical force under play in the studied devices.', '1709.08987-3-9-0': 'Figures [REF] a-b present the case of the "bulk" GaAs cavity.', '1709.08987-3-9-1': 'Panel (a) in the figure displays the experimental spectrum, compared with calculations assuming that [MATH] reproduces the instantaneous spatial distribution of the light intensity ([MATH]), or that it corresponds to a flat photoexcited carrier distribution within the GaAs-spacer ("relaxed").', '1709.08987-3-9-2': 'That is, it assumes that within the time in which the pump laser-induced perturbation is effective (of the order of the half period of the vibrational frequencies involved), the photoexcited carrier spatial distribution relaxes extending their presence throughout the full width of the GaAs cavity spacer material.', '1709.08987-3-9-3': 'Three peaks are clearly visible at [MATH], [MATH] and [MATH] GHz, corresponding to the fundamental, third and fifth overtones of the z-polarized cavity confined breathing mechanical modes.', '1709.08987-3-9-4': '[CITATION] The associated spatial distribution of the strain fields [MATH], together with that of the light induced optoelectronic stress, are shown in panel (b) of the figure.', '1709.08987-3-9-5': '[MATH] is assumed to be non-zero only in GaAs.', '1709.08987-3-9-6': 'It reflects the physical ingredient that photoelastic (electrostrictive) coupling is resonantly enhanced in GaAs, and that photons are only absorbed in GaAs, all other materials being fully transparent at the involved wavelengths.', '1709.08987-3-9-7': 'The solid curves in [MATH] thus represent the region where the light-induced stress is finite, either reflecting the excitation field (yellow), or the relaxed situation (red).', '1709.08987-3-9-8': "It is clear in the experiment that the vibrational mode's amplitude decreases systematically with increasing frequency of the mode, something that according to the calculations is only compatible with the carriers having rapidly spread filling the full width of the cavity spacer.", '1709.08987-3-9-9': 'The explanation is straightforward considering the overlap integral given by Eq. [REF], and the involved spatial distributions depicted in panel(b) Fig. [REF].', '1709.08987-3-9-10': 'And it clearly excludes radiation pressure and electrostriction as the possible driving mechanisms.', '1709.08987-3-10-0': 'Based on the above discussion it is also clear from the top curve in Fig. [REF](a) that the relative weight of the higher frequency 60 GHz mode could be enhanced if the spatial distribution of the optical perturbation [MATH] could be forced to map-out the cavity field intensity [MATH].', '1709.08987-3-10-1': 'If, as argued, the main generation mechanism is indeed governed by optoelectronic forces, one could accomplished this task by artificially limiting the vertical diffusion of the photoexcited carriers.', '1709.08987-3-10-2': 'A natural way to do this is through an adequate engineering of the cavity spacer, e.g. by introducing GaAs/AlAs MQWs.', '1709.08987-3-10-3': 'This case is shown in Figs. [REF](c-d).', '1709.08987-3-10-4': 'Again panel (c) shows the experiment and calculation of the coherent phonon spectral intensity.', '1709.08987-3-10-5': 'Panel (d) displays the spatial distribution of the photoexcited stress, and that of the strain related to the involved vibrational modes.', '1709.08987-3-10-6': 'In this case relaxing the carrier distribution to fill the full width of the QWs, or maintaining the exact laser excitation pattern, makes no observable difference in the calculated spectra.', '1709.08987-3-10-7': 'Interestingly, by tailoring the spatial distribution of the photoexcited carriers using quantum wells we are able to confirm the role of optoelectronic stress as the main optomechanical coupling, and furthermore we have pushed the main vibrational frequency of these optomechanical resonators from the fundamental mode at 20 GHz to the third overtone at 60 GHz.', '1709.08987-3-10-8': 'These frequencies are one order of magnitude larger than the record frequencies demonstrated in other cavity optomechanics approaches.', '1709.08987-3-10-9': '[CITATION] From Fig. [REF](d) it also becomes clear why additional QWs were introduced in the design at the second and fourth DBR periods, and not at the first and third: because of a change of sign of the strain fields, the latter contribute to the overlap integral in Eq. [REF] with the sign reversed respect to the cavity spacer.', '1709.08987-3-11-0': 'Having demonstrated the central role of optoelectronic forces in the generation of GHz phonons in semiconductor microcavities upon resonant pulsed excitation, we address next the potentiality of this mechanism for the observation of laser cooling and optically induced self-oscillation.', '1709.08987-3-11-1': 'The fundamental concepts describing backaction dynamics in cavity optomechanics are grasped by the delayed force model which, for the optically modified vibrational damping rate [MATH] gives [CITATION] [EQUATION] with [MATH] and [MATH] the unperturbed mechanical quality factor and damping, respectively.', '1709.08987-3-11-2': '[MATH] and [MATH] are respectively the unperturbed mechanical mode frequency and stiffness, while [MATH] represents the change in the steady-state optical force for a small displacement [MATH] of the mechanical resonator around the equilibrium position.', '1709.08987-3-11-3': 'To lead to backaction optical forces need to respond with a delay to fluctuations that change the frequency of the optical mode.', '1709.08987-3-11-4': 'The simplest delay function to consider is an exponential function [MATH], with [MATH] the corresponding time-delay.', '1709.08987-3-11-5': 'Typically [MATH] would correspond to the cavity photon lifetime [MATH], but in general and particularly for photothermal and optoelectronic forces, it can be significantly longer than [MATH].', '1709.08987-3-11-6': '[MATH] in the presence of dynamical backaction can thus increase if [MATH], leading to laser cooling, or decrease and eventually attain zero (self-oscillation) if [MATH].', '1709.08987-3-11-7': 'The magnitude of this effect is proportional to the gradient of the optical force, which is a function of the deposited optical energy and the involved optomechanical coupling mechanism (that is, how this deposited energy is translated into a mechanical deformation).', '1709.08987-3-11-8': 'It is also proportional to a delay tuning factor [MATH].', '1709.08987-3-11-9': '[MATH] has a maximum for [MATH], which reflects the intuitive fact that the force fluctuations are more effective to induce vibrations if their time-constant is neither too short, nor too long, but tuned to the vibrational frequency so that [MATH].', '1709.08987-3-12-0': 'Table [REF] presents the different factors intervening in Eq. [REF] for the studied DBR microcavities and considered mechanisms, namely, the corresponding magnitude of the involved optical forces and delay tuning factor [MATH].', '1709.08987-3-12-1': 'The optical forces are given per photon (trapped in the cavity or absorbed depending on the mechanism).', '1709.08987-3-12-2': 'They have been evaluated using finite-element methods (see Appendix [REF]), [CITATION] and considering a micro-pillar with 20 period DBRs of 2 micrometer diameter.', '1709.08987-3-12-3': 'As argued above, the main difference between photothermal and optoelectronic forces respect to radiation pressure and electrostriction, is the larger amount of deposited energy per trapped cavity photon.', '1709.08987-3-12-4': 'Similarly to what is observed in GaAs microdisks, [CITATION] well below the gap (far from the excitonic resonances) both geometric (radiation-pressure) and photoelastic contributions to the optomechanical coupling factor have similar values.', '1709.08987-3-12-5': 'Ultra-strong resonant enhancement of the photoelastic coupling has been experimentally demonstrated in bare GaAs/AlAs MQWs.', '1709.08987-3-12-6': '[CITATION] In Table [REF] we provide the magnitude of the electrostrictive force per incident photon considering the two situations, far from resonance and at resonance as used in our experiments, assuming that the room temperature values of the photoelastic constant given in Ref. [CITATION] are valid for a similar MQW embedded in a pillar microcavity.', '1709.08987-3-12-7': 'Concerning the photothermal coupling, a quantitative evaluation of its relevance in semiconductor GaAs membranes and microdisks indicates that it can be significant.', '1709.08987-3-12-8': '[CITATION] It is also significant in our microcavities, as evidenced in Table [REF].', '1709.08987-3-12-9': 'What excludes it as a potentially relevant optomechanical force is its slow dynamics, which for the high frequency vibrations considered here leads to delay tuning factors [MATH] even for the fundamental breathing mode at 20 GHz and considering a relatively fast thermal relaxation [MATH] of the order of a [MATH]s.', '1709.08987-3-12-10': 'The delay tuning factor is close to its maximum value 0.5 for the impulsive radiation pressure and electrostrictive mechanisms, considering a cavity photon lifetime of a few picoseconds as in our vertical microcavities.', '1709.08987-3-12-11': 'It is also reasonably well tuned for the optoelectronic forces and 200 ps recombination times as demonstrated for pillars of a few microns lateral size in Ref. [CITATION].', '1709.08987-3-12-12': 'In fact, [MATH] is precisely the same value as attained for optimized cantilever resonators that have evidenced strong optical cooling and self-oscillation dynamics based on photothermal forces.', '1709.08987-3-12-13': '[CITATION] The factor [MATH] decrease of [MATH] with respect to radiation pressure and electrostriction is overly compensated by the much larger efficiency of optoelectronic forces.', '1709.08987-3-12-14': 'Note that the physical mechanism at the base of optoelectronic forces is the same as in photoelastic coupling, namely, deformation potential interaction.', '1709.08987-3-12-15': 'The qualitative difference is that a photon is scattered in the latter, while it is absorbed with the subsequent creation of real electron-hole pairs in the former.', '1709.08987-3-13-0': '# Conclusions and Outlook', '1709.08987-3-14-0': 'In conclusion, we have shown that optoelectronic deformation potential interactions are at the origin of the optical forces acting for excitation close to the semiconductor gaps in GaAs/AlAs microcavities.', '1709.08987-3-14-1': 'The carriers dynamics following photoexcitation is determinant of the emitted coherent acoustic phonon spectrum, and can be tailored using quantum wells to push the vibrational optomechanic frequencies from 20 up to 60 GHz, an order of magnitude larger than the highest standards in cavity optomechanics.', '1709.08987-3-14-2': 'The strong potentiality of optoelectronic forces for the demonstration of backaction dynamical effects in semiconductor microcavities was addressed.', '1709.08987-3-14-3': 'This could open the way to ultra-high frequency cavity optomechanics, and through it to quantum measurements and applications at higher temperatures than currently accessible.', '1709.08987-3-15-0': '# Funding Information', '1709.08987-3-16-0': 'This work was partially supported by the ANPCyT Grants PICT 2012-1661 and 2013-2047, the Labex NanoSaclay, and the international franco-argentinean laboratory LIFAN (CNRS-CONICET).', '1709.08987-3-17-0': '# Details regarding the calculation of the optical forces.', '1709.08987-3-18-0': 'We describe the optomechanical coupling of a fundamental optical cavity mode with the fundamental acoustic cavity mode.', '1709.08987-3-18-1': '[CITATION] For a normalized displacement mode [MATH], we can parametrize the profile as [MATH].', '1709.08987-3-18-2': 'The effective mass is obtained by the requirement that the potential energy of this parametrized oscillator is equal to the actual potential energy: [EQUATION]', '1709.08987-3-18-3': 'The effective mass [MATH] is, [EQUATION] where [MATH] is the scalar density distribution field for the structure.', '1709.08987-3-18-4': 'As discussed in Ref. [CITATION], we consider the normalization for the mechanical modes such that the position [MATH] (known as the reduction point and chosen so that the displacement is maximum) satisfies [MATH].', '1709.08987-3-18-5': 'In our system, the reduction point lies at the interfaces of the cavity spacer of GaAs.', '1709.08987-3-18-6': 'The equation of motion for the cavity breathing mode is modeled as an oscillator described by the displacement [MATH], with an effective mass m[MATH], mechanical damping [MATH], and stiffness constant [MATH], given by[CITATION] [EQUATION]', '1709.08987-3-18-7': 'The right-hand side corresponds to a sum over all the optical forces that drive the mechanical system: geometrical related to radiation pressure ([MATH]),[CITATION] photoelastic ([MATH]), thermoelastic ([MATH]) and optoelectronic ([MATH]).', '1709.08987-3-19-0': 'We proceed to the evaluation of the forces.', '1709.08987-3-19-1': 'For the electrostrictive and geometrical forces we can obtain the corresponding values computing [MATH], where [MATH] is the geometrical or photoelastic optomechanical coupling constant.', '1709.08987-3-19-2': '[CITATION]', '1709.08987-3-20-0': 'For the calculation of the geometric optomechanical coupling factor [MATH], we follow the analysis proposed by Johnson et al,[CITATION] implementing a finite element-method to obtain the electric and acoustic fields.', '1709.08987-3-20-1': "We generalize the approach presented in Refs. [CITATION] to compute the effects induced by the multiple interfaces at the DBR's boundaries, [EQUATION] where [MATH] is the optical angular frequency at resonance, [MATH] the normalized displacement field, [MATH] the unitary normal-surface vector corresponding to the interface, [MATH] the difference between the dielectric constants of the materials involved, [MATH], [MATH] is the component of the electric-field parallel to the interface surface and [MATH] is the normal component of the displacement field [MATH].", '1709.08987-3-20-2': 'The index [MATH] runs over every distinct interface [MATH].', '1709.08987-3-21-0': 'The photoelastic contribution to the optomechanical coupling occurs due to the strain-field modulation of the dielectric properties, i.e. [MATH], and is given by,[CITATION] [EQUATION] where [MATH] is the dielectric function.', '1709.08987-3-21-1': 'Due to the resonant character[CITATION] we consider [MATH] to be non-vanishing only in the GaAs spacer.', '1709.08987-3-21-2': 'Only three different components for this tensor are non-zero due to the cubic symmetry.', '1709.08987-3-21-3': 'Since the non-diagonal component of the strain [MATH] is non-zero we also take [MATH] into account.[', '1709.08987-3-21-4': '[CITATION] Through Raman experiments, B. Jusserand et al have determined the wavelength and temperature dependence for the GaAs photoelastic constant [MATH].[', '1709.08987-3-21-5': '[CITATION] On the contrary, there is no similar reported information for [MATH].', '1709.08987-3-21-6': 'However, because of the prevalent z-polarized character of the acoustic modes, it turns out that the contribution to [MATH] is dominated by [MATH], which reaches [MATH] for [MATH]eV at room temperature.', '1709.08987-3-21-7': '[CITATION]', '1709.08987-3-22-0': 'For thermoelastic and optoelectronic effects we used the expression [EQUATION] where [MATH] is the strain field tensor related to the mechanical breathing mode [MATH], and [MATH] is the stress field tensor related to the driving mechanism.', '1709.08987-3-22-1': 'The stress tensor for the thermoelasticity can be determined by [CITATION] [EQUATION] where [MATH] is the heat capacity, [MATH] is the Gruneisen coefficient [CITATION] and [MATH] the lattice temperature variation.', '1709.08987-3-22-2': 'Assuming a complete transfer of energy from the electronic to the phononic system, considering intraband and non-radiative interband relaxation processes for excited electron-hole pairs we can determine [MATH][CITATION].', '1709.08987-3-22-3': 'For the intraband decay channel, the temperature variation can be computed as [MATH], where [MATH] is the driving energy and [MATH] represents the photoexcited carriers population density.', '1709.08987-3-22-4': 'The non-radiative interband relaxation processes give a temperature variation that can be accounted as [MATH] when the excitation is resonant with the bandgap energy [MATH].', '1709.08987-3-23-0': 'For the optoelectronic contribution we consider the limiting case in which electron-hole pairs are excited resonantly with the bandgap energy, and the dominant term in the electronic self-energy due to optical excitation corresponds to the variation in [MATH].', '1709.08987-3-23-1': 'This stress can be summarized as follows,[CITATION] [EQUATION] where [MATH] is the deformation potential coefficient ([MATH]eV for GaAs.[', '1709.08987-3-24-0': 'For the geometric and photoelastic effects the forces are given per number of photons.[', '1709.08987-3-24-1': '[CITATION] In order to compare, for the thermoelastic and optoelectronic cases the forces are given per number of excited electron-hole pairs.', '1709.08987-3-24-2': 'For this purpose [MATH] is considered equal to [MATH] (the inverse of the GaAs spacer volume where carriers are optically excited).', '1709.08987-3-25-0': 'In Table [REF] we present relevant optomechanical parameters and the magnitude of the calculated forces involved.', '1709.08987-3-25-1': 'We conclude that the optoelectronic forces have similar magnitude as the thermal forces and are several orders of magnitude greater than the photoelastic and radiation-pressure mechanisms ([MATH]).'}

{'1709.08987-4-0-0': 'Radiation pressure, electrostriction, and photothermal forces have been investigated to evidence backaction, non-linearities and quantum phenomena in cavity optomechanics.', '1709.08987-4-0-1': 'We show here through a detailed study of the relative intensity of the cavity mechanical modes observed when exciting with pulsed lasers close to the GaAs optical gap that optoelectronic forces involving real carrier excitation and deformation potential interaction are the strongest mechanism of light-to-sound transduction in semiconductor GaAs/AlAs distributed Bragg reflector optomechanical resonators.', '1709.08987-4-0-2': 'We demonstrate that the ultrafast spatial redistribution of the photoexcited carriers in microcavities with massive GaAs spacers leads to an enhanced coupling to the fundamental 20 GHz vertically polarized mechanical breathing mode.', '1709.08987-4-0-3': 'The carrier diffusion along the growth axis of the device can be enhanced by increasing the laser power, or limited by embedding GaAs quantum wells in the cavity spacer, a strategy used here to prove and engineer the optoelectronic forces in phonon generation with real carriers.', '1709.08987-4-0-4': 'The wavelength dependence of the observed phenomena provide further proof of the role of optoelectronic forces.', '1709.08987-4-0-5': 'The optical forces associated to the different intervening mechanisms and their relevance for dynamical backaction in optomechanics are evaluated using finite-element methods.', '1709.08987-4-0-6': 'The results presented open the path to the study of hitherto seldom investigated dynamical backaction in optomechanical solid-state resonators in the presence of optoelectronic forces.', '1709.08987-4-1-0': '# Motivation', '1709.08987-4-2-0': 'Backaction in cavity optomechanics has shown to lead to novel physical phenomena including laser cooling, self-oscillation, and non-linear dynamics in systems that go from kilometer size interferometers to single or few trapped ions.', '1709.08987-4-2-1': '[CITATION] Briefly, a resonant photon field in a cavity exerts a force and induces a mechanical motion on the mirrors, which in turn leads to a delayed modification of the resonant condition of the trapped field.', '1709.08987-4-2-2': 'Such coupled dynamics can be exploited for a large variety of applications that span for example from gravitational wave detection [CITATION] to the study of quantum motion states in mesoscopic mechanical systems.', '1709.08987-4-2-3': '[CITATION] How light exerts force on matter is at the center of these investigations.', '1709.08987-4-2-4': 'Photons can apply stress through radiation pressure, transferring impulse when reflected from the mirrors.', '1709.08987-4-2-5': '[CITATION] Related mechanisms also derived from the same fundamental interaction (Lorentz forces) are gradient forces (also exploited in optical tweezers) [CITATION], and electrostriction.', '1709.08987-4-2-6': "The latter, linked to the material's photoelasticity, has been shown to play a role that can be of the same magnitude as radiation pressure, [CITATION] or even larger if optical resonances are exploited in direct bandgap materials as for example GaAs.", '1709.08987-4-2-7': '[CITATION]', '1709.08987-4-3-0': 'In the presence of radiation pressure forces, the energy [MATH] of the photon is shifted by the Doppler effect by an amount of the order [MATH], where [MATH] and [MATH] are the mirror and light velocities, respectively.', '1709.08987-4-3-1': 'The mechanical energy transferred from the photon to the mirror is thus very small.', '1709.08987-4-3-2': 'Electrostriction leads to Raman-like processes, for which the transferred energy [MATH] corresponds to the involved vibrational frequency.', '1709.08987-4-3-3': 'This leads to inelastic scattering sidebands.', '1709.08987-4-3-4': 'Again, typically [MATH].', '1709.08987-4-3-5': 'Contrastingly, if the photon is absorbed in the process of interaction, all its energy is transferred to the mirror.', '1709.08987-4-3-6': 'This fundamental difference has been used in cavity optomechanics to demonstrate strongly enhanced light-matter interactions based on photothermal forces.', '1709.08987-4-3-7': '[CITATION] In materials displaying optical resonances, the photons can be resonantly absorbed with the consequent transfer of electrons to excited states.', '1709.08987-4-3-8': 'Photoexcited carrier-mediated optomechanical interactions have been reported in semiconductor modulation-doped heterostructure-cantilever hybrid systems.', '1709.08987-4-3-9': 'Efficient cavity-less optomechanical transduction involving opto-piezoelectric backaction from the bound photoexcited electron-hole pairs has been demonstrated in these systems, including self-feedback cooling, amplification of the thermomechanical motion, and control of the mechanical quality factor through carrier excitation.', '1709.08987-4-3-10': '[CITATION] The change in the electronic landscape produced by photoexcited carriers also induces a stress in the structure through electron-phonon coupling mediated by deformation potential interaction.', '1709.08987-4-3-11': 'This stress can be identified as an optoelectronic force, and should have the same kind of temporal behavior (with different time-scales and details depending on the carrier dynamics) and amplified strength as observed for photothermal forces.', '1709.08987-4-3-12': '[CITATION] Recently optical cooling of mechanical modes of a GaAs nanomembrane forming part of an optical cavity was reported, [CITATION] and its relation to optoelectronic stress via the deformation potential was analysed.', '1709.08987-4-3-13': 'Because of the very fast relaxation rate of excited carriers due to surface recombination in such nanometer size structures, it was concluded that thermal (and not optoelectronic) stress was the primary cause of cooling in that case.', '1709.08987-4-3-14': 'We will demonstrate here that the carrier dynamics can be fundamentally modified when the nanometer size GaAs layer is embedded in a monolithic microcavity, making optoelectronic forces the main mechanism of interaction of light with vibrations in such semiconductor devices.', '1709.08987-4-3-15': 'The diffusion of the photoexcited carriers thus assumes a central role, a dynamics that can be engineered using embedded quantum wells (QWs).', '1709.08987-4-4-0': 'Because of their optoelectronic properties, semiconductor GaAs/AlAs microcavities are interesting candidates for novel functionalities in cavity optomechanics.', '1709.08987-4-4-1': 'Perfect photon-phonon overlap, and access to electronically resonant coupling in addition to radiation pressure could lead to strong optomechanical interactions of photoelastic origin.', '1709.08987-4-4-2': '[CITATION] The vibrational frequencies in these microresonators are determined by the vertical layering of the device (fabricated with the ultra-high quality of molecular beam epitaxy), and not by the lateral pattering (defined by the more limited performance of microfabrication techniques).', '1709.08987-4-4-3': 'This has allowed access to much higher frequencies for the optomechanical vibrational modes, in the 20-100 GHz range, without significant reduction of the mechanical Q-factors.', '1709.08987-4-4-4': '[CITATION] In addition, these optomechanical resonators enable the conception of hybrid architectures involving artificial atoms (semiconductor excitons) coupled to the optical cavity mode, and thus combining the physics of cavity optomechanics with cavity quantum electrodynamics.', '1709.08987-4-4-5': '[CITATION] Our motivation here is to search for optoelectronic forces in these devices, and for that purpose we study the light-sound coupling involving the resonator mechanical modes and the optical cavity at resonance with the material exciton transition energy.', '1709.08987-4-4-6': 'Clear evidence of the role of optoelectronic forces emerges from new studies based on the spectral dependence and the relative intensity of the observed mechanical cavity modes.', '1709.08987-4-4-7': 'We demonstrate based on these experiments and model calculations that the main phonon generation mechanism using pulsed lasers close to resonance in these devices involves indeed the real excitation of carriers and the deformation potential mechanism.', '1709.08987-4-4-8': '[CITATION] We show that in microcavities with "bulk" GaAs spacers (i.e. cavity spacers constituted by a thick [MATH] layer of GaAs) ultrafast carrier redistribution leads to an enhanced coupling to the more uniformly distributed fundamental 20 GHz cavity vibrational mode.', '1709.08987-4-4-9': 'The relative intensity of the modes in these structures varies with laser power, consistently with a more uniform distribution of carrier being attained at higher concentrations.', '1709.08987-4-4-10': 'An engineering of the structure taking into account this effect and using embedded quantum wells is used to limit the carrier diffusion, leading to mechanical modes with a relative intensity consistent with the spatial distribution of the cavity optical field.', '1709.08987-4-4-11': 'The demonstration of optoelectronic forces and the possibility to tune the coupling to specific vibrations using quantum wells opens the way to new opportunities in the field of optomechanics.', '1709.08987-4-5-0': '# Results', '1709.08987-4-6-0': 'We consider two planar microcavity structures, specifically a "bulk" GaAs and a multiple quantum well (MQW) resonator.', '1709.08987-4-6-1': 'The "bulk" GaAs microcavity is made of a [MATH] GaAs-spacer enclosed by ([MATH] Al[MATH]Ga[MATH]As /AlAs DBRs, 20 pairs of layers on the bottom, 18 on top, grown on a GaAs substrate (a scheme of the structure is presented in Fig. [REF](b)).', '1709.08987-4-6-2': '[CITATION] As we have demonstrated previously, this structure works as an optomechanical resonator that simultaneously confines photons and acoustic phonons of the same wavelength.', '1709.08987-4-6-3': '[CITATION] In the MQW microcavity the [MATH] spacer is constituted by six [MATH] GaAs QWs separated by [MATH] AlAs barriers.', '1709.08987-4-6-4': 'To further enhance the light-sound coupling the second and fourth [MATH] DBR alloy layers on each side are also replaced by three GaAs/AlAs QWs.', '1709.08987-4-6-5': 'The reason for this design will become clear below.', '1709.08987-4-6-6': 'The DBRs in this case are ([MATH] Al[MATH]Ga[MATH]As /AlAs multilayers, 27 pairs on the bottom, 23 on top, grown again on a GaAs substrate.', '1709.08987-4-6-7': 'A scheme of this structure is displayed in Fig. [REF](c).', '1709.08987-4-6-8': 'The number of DBR periods in both structures is designed to assure an optical Q-factor [MATH] (cavity photon lifetime [MATH] ps).', '1709.08987-4-6-9': 'The samples have a thickness gradient so that the energy of the optical cavity mode, and its detuning respect to the bulk GaAs and MQW gaps, can be varied by displacing the laser spot on the surface.', '1709.08987-4-7-0': 'Reflection-type degenerate pump-probe experiments were performed with the laser wavelength tuned with the optical cavity mode (see the scheme in Fig. [REF](a)).', '1709.08987-4-7-1': '[CITATION] Picosecond pulses ([MATH] ps) from a mode-locked Ti:Sapphire laser, with repetition rate 80 MHz, were split into cross polarized pump (power 20 mW) and probe (1 mW) pulses.', '1709.08987-4-7-2': 'Both pulses were focused onto superimposed [MATH]m-diameter spots.', '1709.08987-4-7-3': 'To couple the light to the microcavity the probe beam propagates close to the sample normal and is tuned to the high derivative flank of the cavity mode reflectivity dip, while the pump incidence angle is set for resonant condition precisely at the cavity mode.', '1709.08987-4-7-4': '[CITATION] The laser wavelength was also set so that the phonon generation and detection would be close to resonance with the direct bandgap of the GaAs "bulk" cavity spacer ([MATH] eV) or of the QWs ([MATH] eV).', '1709.08987-4-7-5': 'To accomplish this resonant excitation the temperature was also used as a tuning parameter; the "bulk" cavity experiments were done at room temperature, while the MQW structure was studied at 80 K. Light is thus coupled resonantly with the optical cavity mode and the semiconductor excitonic resonance.', '1709.08987-4-7-6': 'Acoustic phonons confined in the same space as the optical cavity mode are selectively generated within the resonator, and are detected through their modulation of the optical cavity mode frequency.', '1709.08987-4-7-7': '[CITATION]', '1709.08987-4-8-0': 'These pump-probe ultrafast laser experiments are conceptually similar to the ring-down techniques recently exploited in the cavity optomechanics domain, [CITATION] but more appropriate to the study of ultra-high frequency vibrations (GHz-THz range.)', '1709.08987-4-8-1': 'The pulsed laser phonon generation efficiency can be described as: [CITATION] [EQUATION]', '1709.08987-4-8-2': 'Here [MATH] is the phonon frequency, [MATH] describes the elastic strain eigenstates, [MATH] is the spatially dependent perturbation induced by the pump laser, and [MATH] is an effective material-dependent generation parameter that considers different light-matter couplings.', '1709.08987-4-8-3': 'All parameters are implicitly assumed to depend on the laser wavelength.', '1709.08987-4-8-4': 'We will be interested here in the relative intensity of the vibrational modes, not in their absolute values, so the main physical ingredients are expressed in the functional form of Eq. [REF].', '1709.08987-4-8-5': 'This equation reflects the spatial overlap of the strain eigenstates with the light-induced stress.', '1709.08987-4-8-6': 'The latter can be written (independently of the mechanism involved) as [MATH].', '1709.08987-4-8-7': '[CITATION] Here [MATH] is the function describing the temporal evolution of the light-induced perturbation.', '1709.08987-4-8-8': 'Typically it is a delta-like function for radiation pressure and electrostriction forces, and a step-like function for the photothermal and optoelectronic mechanisms, broadened by the time-delay of the mechanism involved.', '1709.08987-4-8-9': 'The spatial distribution of the optical excitation [MATH] along the growth axis ([MATH]) corresponds to the cavity confined electric field [MATH] for radiation pressure and electrostriction forces, but can be different from it for the other two mechanisms depending on the spatial distribution and dynamics of excited charges and laser-induced temperature variations.', '1709.08987-4-8-10': 'As we argue next, this will be a way to identify the main optical force under play in the studied devices.', '1709.08987-4-9-0': 'Figures [REF] a-b present the case of the "bulk" GaAs cavity.', '1709.08987-4-9-1': 'For the experiments reported here the laser was set around 10 meV below the energy of the bulk GaAs gap.', '1709.08987-4-9-2': 'Panel (a) in the figure displays the experimental spectrum, compared with calculations assuming that [MATH] reproduces the instantaneous spatial distribution of the light intensity [MATH] ("unrelaxed"), or that it corresponds to a photoexcited carrier distribution within the GaAs-spacer that is flat along the growth direction ("relaxed").', '1709.08987-4-9-3': 'That is, it assumes that within the time in which the pump laser-induced perturbation is effective (of the order of the half period of the vibrational frequencies involved), the spatial distribution of the photoexcited carriers relaxes extending their presence throughout the full width of the GaAs cavity spacer material.', '1709.08987-4-9-4': 'Three peaks are clearly visible at [MATH], [MATH] and [MATH] GHz, corresponding to the fundamental, second and fourth overtones of the z-polarized cavity confined breathing mechanical modes.', '1709.08987-4-9-5': '[CITATION] The associated spatial distribution of the strain fields [MATH], together with that of the light-induced optoelectronic stress, are shown in panel (b) of the figure.', '1709.08987-4-9-6': 'The solid yellow curves in the top panel corresponds to the unrelaxed spatial pattern of the optical stress.', '1709.08987-4-9-7': 'The red step-like solid line indicates the stress for the relaxed situation.', '1709.08987-4-9-8': 'The grey dashed curve shows how the cavity confined field is distributed, but there is no optical force in the dashed regions because photoelastic (electrostrictive) coupling is resonantly enhanced in GaAs, and photons are only absorbed in GaAs, all other materials being fully transparent at the involved wavelengths.', '1709.08987-4-9-9': 'That is, [MATH] is assumed to be non-zero only in GaAs.', '1709.08987-4-9-10': 'The solid curves in [MATH] thus represent the region where the light-induced stress is finite, either reflecting the excitation field (yellow), or the relaxed situation (red).', '1709.08987-4-9-11': "It is clear in the experiment that the vibrational mode's amplitude decreases systematically with increasing frequency of the mode, something that according to the calculations is only compatible with the carriers having rapidly spread filling the full width of the cavity spacer along the growth direction.", '1709.08987-4-9-12': 'The explanation is straightforward considering the overlap integral given by Eq. [REF], and the involved spatial distributions depicted in panel (b) of Fig. [REF].', '1709.08987-4-9-13': 'It clearly excludes radiation pressure and electrostriction as the possible driving mechanisms.', '1709.08987-4-10-0': 'Based on the above discussion it is also clear from the top curve in Fig. [REF](a) that the relative weight of the higher frequency 60 GHz mode could be enhanced if the spatial distribution of the optical perturbation [MATH] could be forced to map-out the cavity field intensity [MATH].', '1709.08987-4-10-1': 'If, as argued, the main generation mechanism is indeed governed by optoelectronic forces, one could accomplish this task by artificially limiting the diffusion of the photoexcited carriers along the growth axis.', '1709.08987-4-10-2': 'A natural way to do this is through an adequate engineering of the cavity spacer, e.g. by introducing GaAs/AlAs MQWs.', '1709.08987-4-10-3': 'This case is shown in Figs. [REF](c-d).', '1709.08987-4-10-4': 'Again panel (c) shows the experiment and calculation of the coherent phonon spectral intensity, with the laser energy set this time around 10 meV below the MQW exciton transition energy.', '1709.08987-4-10-5': 'Panel (d) displays the spatial distribution of the photoexcited stress, and that of the strain related to the involved vibrational modes.', '1709.08987-4-10-6': 'In this case relaxing the carrier distribution to fill the full width of the QWs (red step-like curves), or maintaining the exact laser excitation pattern (yellow solid curve), makes no observable difference in the calculated spectra.', '1709.08987-4-10-7': 'Interestingly, by tailoring the spatial distribution of the photoexcited carriers using quantum wells we are able to confirm the role of optoelectronic stress as the main optomechanical coupling, and furthermore we have pushed the main vibrational frequency of these optomechanical resonators from the fundamental mode at 20 GHz to the second overtone at 60 GHz.', '1709.08987-4-10-8': 'These frequencies are one order of magnitude larger than the record frequencies demonstrated in other cavity optomechanics approaches.', '1709.08987-4-10-9': '[CITATION] From Fig. [REF](d) it also becomes clear why additional QWs were introduced in the design at the second and fourth DBR periods, and not at the first and third: because of a change of sign of the strain fields, the latter contribute to the overlap integral in Eq. [REF] with the sign reversed respect to the cavity spacer.', '1709.08987-4-11-0': 'The role of photoexcited carriers in the optical forces involved in the described coherent phonon generation with pulsed lasers can be furthermore verified by investigating the laser wavelength dependence of the mechanical mode intensity.', '1709.08987-4-11-1': 'This is shown for the 60 GHz mode of the bulk-GaAs cavity in Fig. [REF].', '1709.08987-4-11-2': 'Note that each point in this figure corresponds to an experiment in which the laser energy is varied and the position of the spot in the tapered structure is accordingly changed so that the tuning with the optical cavity mode remains the same.', '1709.08987-4-11-3': 'The wavelength limits of the experiment are thus defined by the thickness gradient existent in the microcavity structure.', '1709.08987-4-11-4': 'A threshold-like behaviour with spectral intensity tending to zero in the transparency region of the structure, and finite intensity only close and above the gap of GaAs, is observed.', '1709.08987-4-11-5': 'This is indicative of real electron-hole pairs being the mediators of the involved optical force.', '1709.08987-4-12-0': 'Note that although the spectrum calculated for the relaxed situation of the bulk-GaAs microcavity in Fig. [REF]a shows the same tendency as the experimental result, a quantitative difference between experiment and theory still remains.', '1709.08987-4-12-1': 'Namely, the relative intensity of the 60 GHz mode is slightly larger than predicted.', '1709.08987-4-12-2': 'The most natural explanation for this remaining difference between the bulk-GaAs cavity experiments and the relaxed calculation is that relaxation throughout the whole thick GaAs spacer of the cavity is not complete within the relevant times involved in the coherent generation process (times typically of the order of half a period, i.e., around 7 ps for the 60 GHz mode).', '1709.08987-4-12-3': 'Immediately after excitation, the optical force has to reproduce the spatial pattern of the cavity confined optical field (such force leads to larger intensity for the 60 GHz mode).', '1709.08987-4-12-4': 'This pattern rapidly relaxes towards a uniform distribution along the growth axis within the GaAs material (a distribution in the optical force that provides larger intensity for the 20 GHz mode, as shown in Fig. [REF]).', '1709.08987-4-12-5': 'If this relaxation is fast in comparison with the mechanical period, the latter will apply.', '1709.08987-4-12-6': 'If the relaxation is not complete, one can expect a behaviour in between the two limits as observed experimentally.', '1709.08987-4-12-7': 'One way to test this hypothesis is to perform experiments for a varying concentration of photoexcited carriers, as shown in Fig. [REF], where the relative intensity of the 60 and 20 GHz modes as a function of pump laser power is displayed.', '1709.08987-4-12-8': 'One can expect that due to electron-electron Coulomb interactions higher carrier densities tend to accelerate the homogenisation of carriers within their available space.', '1709.08987-4-12-9': 'This is indeed what is evidenced in the experiments, with a progressive trend towards the relaxed situation, i.e. a flat distribution of the optical stress consistent with the 20 GHz mode being stronger than the 60 GHz one, as the pump laser power is increased.', '1709.08987-4-13-0': 'Having demonstrated the central role of optoelectronic forces in the generation of GHz phonons in semiconductor microcavities upon resonant pulsed excitation, we address next the potential of this mechanism for the observation of laser cooling and optically induced self-oscillation.', '1709.08987-4-13-1': 'The fundamental concepts describing backaction dynamics in cavity optomechanics are grasped by the delayed force model which, for the optically modified vibrational damping rate [MATH], gives [CITATION] [EQUATION] with [MATH] and [MATH] the unperturbed mechanical quality factor and damping, respectively.', '1709.08987-4-13-2': '[MATH] and [MATH] are respectively the unperturbed mechanical mode frequency and stiffness, while [MATH] represents the change in the steady-state optical force for a small displacement [MATH] of the mechanical resonator around the equilibrium position.', '1709.08987-4-13-3': 'To lead to backaction optical forces need to respond with a delay to fluctuations that change the frequency of the optical mode.', '1709.08987-4-13-4': 'The simplest delay function to consider is an exponential function [MATH], with [MATH] the corresponding time-delay.', '1709.08987-4-13-5': 'Typically [MATH] would correspond to the cavity photon lifetime [MATH], but in general and particularly for photothermal and optoelectronic forces, it can be significantly longer than [MATH].', '1709.08987-4-13-6': '[MATH] in the presence of dynamical backaction can thus increase if [MATH], leading to laser cooling, or decrease and eventually attain zero (self-oscillation) if [MATH].', '1709.08987-4-13-7': 'The magnitude of this effect is proportional to the gradient of the optical force, which is a function of the deposited optical energy and the involved optomechanical coupling mechanism (that is, how this deposited energy is translated into a mechanical deformation).', '1709.08987-4-13-8': 'It is also proportional to a delay tuning factor [MATH].', '1709.08987-4-13-9': '[MATH] has a maximum for [MATH], which reflects the intuitive fact that the force fluctuations are more effective to induce vibrations if their time-constant is neither too short, nor too long, but tuned to the vibrational frequency so that [MATH].', '1709.08987-4-14-0': 'Table [REF] presents the different factors intervening in Eq. [REF] for the studied DBR microcavities and the considered mechanisms, namely, the corresponding magnitude of the involved optical forces and the delay tuning factor [MATH].', '1709.08987-4-14-1': 'The optical forces are given per photon (trapped in the cavity or absorbed depending on the mechanism).', '1709.08987-4-14-2': 'They have been evaluated using finite-element methods (see Appendix [REF]), [CITATION] and considering a micro-pillar with 20 period DBRs of 2 micrometer diameter.', '1709.08987-4-14-3': 'As argued above, the main difference between photothermal and optoelectronic forces respect to radiation pressure and electrostriction, is the larger amount of deposited energy per trapped cavity photon.', '1709.08987-4-14-4': 'Similarly to what is observed in GaAs microdisks, [CITATION] well below the gap (far from the excitonic resonances) both geometric (radiation-pressure) and photoelastic contributions to the optomechanical coupling factor have similar values.', '1709.08987-4-14-5': 'Ultra-strong resonant enhancement of the photoelastic coupling has been experimentally demonstrated in bare GaAs/AlAs MQWs.', '1709.08987-4-14-6': '[CITATION] In Table [REF] we provide the magnitude of the electrostrictive force per incident photon considering the two situations, far from resonance and at resonance as used in our experiments, assuming that the room temperature values of the photoelastic constant given in Ref. [CITATION] are valid for a similar MQW embedded in a pillar microcavity.', '1709.08987-4-14-7': 'Concerning the photothermal coupling, a quantitative evaluation of its relevance in semiconductor GaAs membranes and microdisks indicates that it can be significant.', '1709.08987-4-14-8': '[CITATION] It is also significant in our microcavities, as evidenced in Table [REF].', '1709.08987-4-14-9': 'What excludes it as a potentially relevant optomechanical force is its slow dynamics, which for the high frequency vibrations considered here leads to delay tuning factors [MATH] even for the fundamental breathing mode at 20 GHz and considering a relatively fast thermal relaxation [MATH] of the order of a [MATH]s. Note that due to the deformation potential interaction in GaAs, optoelectronic forces have the same sign as the thermal stress and contribute to expand the crystal (they have the reverse sign in Si).', '1709.08987-4-14-10': '[CITATION] The delay tuning factor is close to its maximum value 0.5 for the impulsive radiation pressure and electrostrictive mechanisms, considering a cavity photon lifetime of a few picoseconds as in our vertical microcavities.', '1709.08987-4-14-11': 'It is also reasonably well tuned for the optoelectronic forces and 200 ps recombination times as demonstrated for pillars of a few microns lateral size in Ref. [CITATION].', '1709.08987-4-14-12': 'In fact, [MATH] is precisely the same value as attained for optimized cantilever resonators that have evidenced strong optical cooling and self-oscillation dynamics based on photothermal forces.', '1709.08987-4-14-13': '[CITATION] The factor [MATH] decrease of [MATH] with respect to radiation pressure and electrostriction is overly compensated by the larger efficiency of optoelectronic forces.', '1709.08987-4-14-14': 'Note that the physical mechanism at the base of optoelectronic forces is the same as in photoelastic coupling, namely, deformation potential interaction.', '1709.08987-4-14-15': 'The qualitative difference is that a photon is scattered in the latter, while it is absorbed with the subsequent creation of real electron-hole pairs in the former.', '1709.08987-4-15-0': '# Conclusions and Outlook', '1709.08987-4-16-0': 'In conclusion, we have shown that optoelectronic deformation potential interactions are at the origin of the optical forces acting for excitation with pulsed lasers close to the semiconductor gaps in GaAs/AlAs microcavities.', '1709.08987-4-16-1': 'The carrier dynamics following photoexcitation is determinant in the emitted coherent acoustic phonon spectrum, and can be tailored using quantum wells to push the vibrational optomechanic frequencies from 20 up to 60 GHz, an order of magnitude larger than the highest standards in cavity optomechanics.', '1709.08987-4-16-2': 'The strong potential of optoelectronical forces for the demonstration of backaction dynamical effects in semiconductor microcavities was addressed.', '1709.08987-4-16-3': 'This could open the way to ultra-high frequency cavity optomechanics, and through it to quantum measurements and applications at higher temperatures than currently accessible.', '1709.08987-4-17-0': '# Funding Information', '1709.08987-4-18-0': 'This work was partially supported by the ANPCyT Grants PICT 2012-1661 and 2013-2047, the Labex NanoSaclay, and the international franco-argentinean laboratory LIFAN (CNRS-CONICET).', '1709.08987-4-19-0': '# Details regarding the calculation of the optical forces.', '1709.08987-4-20-0': 'We describe the optomechanical coupling of a fundamental optical cavity mode with the fundamental acoustic cavity mode.', '1709.08987-4-20-1': '[CITATION] For a normalized displacement mode [MATH], we can parametrize the profile as [MATH].', '1709.08987-4-20-2': 'The effective mass is obtained by the requirement that the potential energy of this parametrized oscillator is equal to the actual potential energy: [EQUATION]', '1709.08987-4-20-3': 'The effective mass [MATH] is, [EQUATION] where [MATH] is the scalar density distribution field for the structure.', '1709.08987-4-20-4': 'As discussed in Ref. [CITATION], we consider the normalization for the mechanical modes such that the position [MATH] (known as the reduction point and chosen so that the displacement is maximum) satisfies [MATH].', '1709.08987-4-20-5': 'In our system, the reduction point lies at the interfaces of the cavity spacer of GaAs.', '1709.08987-4-20-6': 'The equation of motion for the cavity breathing mode is modeled as an oscillator described by the displacement [MATH], with an effective mass m[MATH], mechanical damping [MATH], and stiffness constant [MATH], given by[CITATION] [EQUATION]', '1709.08987-4-20-7': 'The right-hand side corresponds to a sum over all the optical forces that drive the mechanical system: geometrical related to radiation pressure ([MATH]),[CITATION] photoelastic ([MATH]), thermoelastic ([MATH]) and optoelectronic ([MATH]).', '1709.08987-4-21-0': 'We proceed to the evaluation of the forces.', '1709.08987-4-21-1': 'For the electrostrictive and geometrical forces we can obtain the corresponding values computing [MATH], where [MATH] is the geometrical or photoelastic optomechanical coupling constant.', '1709.08987-4-21-2': '[CITATION]', '1709.08987-4-22-0': 'For the calculation of the geometric optomechanical coupling factor [MATH], we follow the analysis proposed by Johnson et al,[CITATION] implementing a finite element-method to obtain the electric and acoustic fields.', '1709.08987-4-22-1': "We generalize the approach presented in Refs. [CITATION] to compute the effects induced by the multiple interfaces at the DBR's boundaries, [EQUATION] where [MATH] is the optical angular frequency at resonance, [MATH] the normalized displacement field, [MATH] the unitary normal-surface vector corresponding to the interface, [MATH] the difference between the dielectric constants of the materials involved, [MATH], [MATH] is the component of the electric-field parallel to the interface surface and [MATH] is the normal component of the displacement field [MATH].", '1709.08987-4-22-2': 'The index [MATH] runs over every distinct interface [MATH].', '1709.08987-4-23-0': 'The photoelastic contribution to the optomechanical coupling occurs due to the strain-field modulation of the dielectric properties, i.e. [MATH], and is given by,[CITATION] [EQUATION] where [MATH] is the dielectric function.', '1709.08987-4-23-1': 'Due to the resonant character[CITATION] we consider [MATH] to be non-vanishing only in the GaAs spacer.', '1709.08987-4-23-2': 'Only three different components for this tensor are non-zero due to the cubic symmetry.', '1709.08987-4-23-3': 'Since the non-diagonal component of the strain [MATH] is non-zero we also take [MATH] into account.[', '1709.08987-4-23-4': '[CITATION] Through Raman experiments, B. Jusserand et al have determined the wavelength and temperature dependence for the GaAs photoelastic constant [MATH].[', '1709.08987-4-23-5': '[CITATION] On the contrary, there is no similar reported information for [MATH].', '1709.08987-4-23-6': 'However, because of the prevalent z-polarized character of the acoustic modes, it turns out that the contribution to [MATH] is dominated by [MATH], which reaches [MATH] for [MATH]eV at room temperature.', '1709.08987-4-23-7': '[CITATION]', '1709.08987-4-24-0': 'For thermoelastic and optoelectronic effects we used the expression [EQUATION] where [MATH] is the strain field tensor related to the mechanical breathing mode [MATH], and [MATH] is the stress field tensor related to the driving mechanism.', '1709.08987-4-24-1': 'The stress tensor for the thermoelasticity can be determined by [CITATION] [EQUATION] where [MATH] is the heat capacity, [MATH] is the Gruneisen coefficient [CITATION] and [MATH] the lattice temperature variation.', '1709.08987-4-24-2': 'Assuming a complete transfer of energy from the electronic to the phononic system, considering intraband and non-radiative interband relaxation processes for excited electron-hole pairs we can determine [MATH][CITATION].', '1709.08987-4-24-3': 'For the intraband decay channel, the temperature variation can be computed as [MATH], where [MATH] is the driving energy and [MATH] represents the photoexcited carriers population density.', '1709.08987-4-24-4': 'The non-radiative interband relaxation processes give a temperature variation that can be accounted as [MATH] when the excitation is resonant with the bandgap energy [MATH].', '1709.08987-4-25-0': 'For the optoelectronic contribution we consider the limiting case in which electron-hole pairs are excited resonantly with the bandgap energy, and the dominant term in the electronic self-energy due to optical excitation corresponds to the variation in [MATH].', '1709.08987-4-25-1': 'This stress can be summarized as follows,[CITATION] [EQUATION] where [MATH] is the deformation potential coefficient ([MATH]eV for GaAs.[', '1709.08987-4-26-0': 'For the geometric and photoelastic effects the forces are given per number of photons.[', '1709.08987-4-26-1': '[CITATION] In order to compare, for the thermoelastic and optoelectronic cases the forces are given per number of excited electron-hole pairs.', '1709.08987-4-26-2': 'For this purpose [MATH] is considered equal to [MATH] (the inverse of the GaAs spacer volume where carriers are optically excited).', '1709.08987-4-27-0': 'In Table [REF] we present relevant optomechanical parameters and the magnitude of the calculated forces involved.', '1709.08987-4-27-1': 'We conclude that the optoelectronic forces have similar magnitude as the thermal forces and are several orders of magnitude greater than the photoelastic and radiation-pressure mechanisms ([MATH]).'}

1109.6162

{'1109.6162-1-0-0': 'Given a closed subgroup [MATH] and a number [MATH] we can form the homogeneous space [MATH], and it follows from the Stone-Weierstrass theorem that [MATH] is the algebra generated by the last [MATH] rows of coordinates on [MATH].', '1109.6162-1-0-1': "In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case.", '1109.6162-1-0-2': 'We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type [MATH].', '1109.6162-1-1-0': '# Introduction', '1109.6162-1-2-0': 'The notion of "noncommutative space" goes back to an old theorem of Gelfand, stating that any commutative [MATH]-algebra must be of the form [MATH], with [MATH] compact space.', '1109.6162-1-2-1': 'In view of this fundamental result, a "noncommutative compact space" should be just the abstract categorical dual of an arbitrary [MATH]-algebra.', '1109.6162-1-3-0': 'The interesting examples of such spaces abound.', '1109.6162-1-3-1': 'A very general notion of "noncommutative Riemannian manifold" was axiomatized by Connes in [CITATION].', '1109.6162-1-3-2': 'This notion covers a huge number of interesting situations, for instance the Standard Model one.', '1109.6162-1-3-3': 'See [CITATION].', '1109.6162-1-4-0': 'Another class of interesting examples is provided by the compact quantum groups, axiomatized by Woronowicz in [CITATION], [CITATION], [CITATION].', '1109.6162-1-4-1': 'This class covers for instance all duals of discrete groups, as well as the compact forms of the Lie-type algebras of Drinfeld [CITATION] and Jimbo [CITATION], at [MATH].', '1109.6162-1-4-2': 'The Drinfeld-Jimbo quantum groups at [MATH] are not very far either, because they are known to correspond to tensor [MATH]-categories [CITATION].', '1109.6162-1-4-3': 'In general, the compact quantum groups are noncommutative manifolds in the sense of Connes [CITATION].', '1109.6162-1-5-0': 'Yet another class of interesting examples are provided by the noncommutative homogeneous spaces.', '1109.6162-1-5-1': 'For some pioneering axiomatization work and for various results in this direction, see Podles [CITATION] and Boca [CITATION], Tomatsu [CITATION], Vaes [CITATION], Kasprzak [CITATION].', '1109.6162-1-5-2': "A number of key examples, mostly related to the various noncommutative spheres, were systematically investigated in a series of papers by Connes, D'Andrea, Dabrowski, Dubois-Violette, Khalkhali, Landi, van Suijlekom and Wagner [CITATION], [CITATION], [CITATION], [CITATION].", '1109.6162-1-6-0': 'The starting point for the present paper is the following result:', '1109.6162-1-7-0': 'Theorem.', '1109.6162-1-7-1': 'Let [MATH] be a closed subgroup, let [MATH], and set [MATH], where the embedding [MATH] is given by [MATH].', '1109.6162-1-7-2': 'Then [MATH], viewed as subalgebra of [MATH], is generated by the coordinate functions [MATH], with [MATH].', '1109.6162-1-8-0': 'This result follows indeed from the Stone-Weierstrass theorem, because the subalgebra [MATH] generated by [MATH] separates the points of [MATH].', '1109.6162-1-9-0': 'As a basic application, in the case [MATH] and [MATH] the above result tells us, modulo a standard symmetry argument, that each row of coordinates [MATH] on [MATH] can be identified with the standard coordinates [MATH] on the sphere [MATH].', '1109.6162-1-10-0': "In the quantum group case now, an analogue of the above statement can be formulated, in terms of Wang's free unitary quantum group [MATH] [CITATION].", '1109.6162-1-10-1': 'More precisely, if [MATH] is a quantum subgroup and [MATH], we can construct the quantum group [MATH], and then the algebra [MATH].', '1109.6162-1-10-2': 'Also, if we denote by [MATH] the algebra generated by the coordinates [MATH], then we have an inclusion [MATH], and the problem is whether this inclusion is an isomorphism or not.', '1109.6162-1-11-0': 'Our first result is a complete answer to this question in the group dual case:', '1109.6162-1-12-0': 'Theorem A. Assume that [MATH] is a discrete group dual, with [MATH], and with the embedding [MATH] given by [MATH], where [MATH] and [MATH].', '1109.6162-1-12-1': 'Then [MATH] is an isomorphism iff [MATH], where [MATH].', '1109.6162-1-13-0': 'This result is actually a consequence of a slightly more precise statement, saying that the inclusion [MATH] coincides with the inclusion [MATH], where [MATH] is the normal closure of [MATH].', '1109.6162-1-13-1': 'Observe also that in the case of diagonal embeddings ([MATH]) the normality condition in the statement is simply [MATH].', '1109.6162-1-14-0': 'With this theoretical result in hand, we will focus then on the case of "easy quantum groups" [MATH], axiomatized in [CITATION], and studied in [CITATION], [CITATION], [CITATION], [CITATION], [CITATION].', '1109.6162-1-14-1': 'The point is that the "row algebras" of type [MATH] are of particular interest in the easy case:', '1109.6162-1-15-0': 'First, [MATH] is a subgroup of [MATH], and the inclusion [MATH] itself is a quite interesting object: we will prove here, as part of Theorem B below, that [MATH] is equivalent to a certain key combinatorial condition, introduced in [CITATION].', '1109.6162-1-16-0': 'Second, even in classical case, the quotient spaces of type [MATH] are quite subtle.', '1109.6162-1-16-1': 'For instance one strategy in the exact computation of polynomial integrals over [MATH] is that of examining the spaces [MATH], with [MATH] increasing.', '1109.6162-1-16-2': 'See [CITATION].', '1109.6162-1-17-0': 'At the quantum level now, the simplest case is [MATH] and [MATH], corresponding to the "free spheres" studied in [CITATION].', '1109.6162-1-17-1': 'Also, the "free hypergeometric laws" in [CITATION] seem to come from spaces of type [MATH], not axiomatized yet.', '1109.6162-1-18-0': 'Summarizing, the problematics in the easy case, be it classical or free, is related to a number of recent computations and considerations.', '1109.6162-1-18-1': 'Our second result in this paper, dealing with the inclusions [MATH] in the easy case, is as follows:', '1109.6162-1-19-0': 'Theorem B.', '1109.6162-1-19-1': 'For an easy quantum group [MATH] we have [MATH] for any [MATH] if and only if the category of partitions for [MATH] is stable by removing blocks.', '1109.6162-1-19-2': 'If so is the case, and if [MATH] is free (i.e. [MATH]), then [MATH] is in general proper.', '1109.6162-1-20-0': 'We refer to sections 3-4 below for the precise statement here.', '1109.6162-1-20-1': 'The proof of the first assertion is purely combinatorial, using the results in [CITATION], and the proof of the second assertion is inspired by Theorem A, by using a suitable group dual subgroup of [MATH].', '1109.6162-1-21-0': 'The above result raises the question of locating [MATH] inside [MATH].', '1109.6162-1-21-1': 'We do not have results here, but as a further step in investigating [MATH], we will construct and study a certain universal algebra [MATH], having [MATH] as quotient.', '1109.6162-1-22-0': 'Our result here is a bit technical.', '1109.6162-1-22-1': 'Let us recall from [CITATION] that in the free case there are exactly 6 easy quantum groups, namely [MATH], which satisfy the combinatorial condition in Theorem B, plus two more quantum groups [MATH], which do not satisfy it.', '1109.6162-1-22-2': 'So, the 4 quantum groups that we are interested in are the orthogonal, symmetric, hyperoctahedral and bistochastic groups [MATH].', '1109.6162-1-22-3': 'These are defined by the fact that their fundamental corepresentation is orthogonal, magic, cubic and bistochastic, see [CITATION].', '1109.6162-1-22-4': 'Now with these notions in hand, our third result is as follows:', '1109.6162-1-23-0': 'Theorem C.', '1109.6162-1-23-1': 'For [MATH] let [MATH] be the universal algebra generated by the entries of a transposed [MATH] orthogonal, magic, cubic, stochastic isometry.', '1109.6162-1-23-2': 'Then [MATH] has the same abelianization and reduced version as [MATH].', '1109.6162-1-24-0': 'We refer to section 5 below for the precise statement.', '1109.6162-1-24-1': 'The proof uses the integration formula of Collins-Sniady [CITATION], as extended in [CITATION], and a method from [CITATION].', '1109.6162-1-25-0': 'The problem of computing the kernel of [MATH], which would be useful in connection with (1,2,3), seems to be a difficult linear algebra one, somehow in the spirit of those solved in [CITATION].', '1109.6162-1-25-1': 'There might be also some connections with [CITATION], [CITATION].', '1109.6162-1-26-0': 'We also believe that the situation [MATH] can appear in more general contexts, and might help for the general understanding of noncommutative homogeneous spaces.', '1109.6162-1-26-1': 'We have several questions here, formulated in section 6 below.', '1109.6162-1-27-0': 'The paper is organized as follows: 1-2 contain generalities about the spaces of type [MATH], including the group dual case result, and in 3-4 we investigate the "easy" case.', '1109.6162-1-27-1': 'The final sections, 5-6, contain further results, and a few concluding remarks.'}

{'1109.6162-2-0-0': 'Given a quantum subgroup [MATH] and a number [MATH] we can form the homogeneous space [MATH], and it follows from the Stone-Weierstrass theorem that [MATH] is the algebra generated by the last [MATH] rows of coordinates on [MATH].', '1109.6162-2-0-1': "In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case.", '1109.6162-2-0-2': 'We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type [MATH].', '1109.6162-2-1-0': '# Introduction', '1109.6162-2-2-0': 'The notion of "noncommutative space" goes back to an old theorem of Gelfand, stating that any commutative unital [MATH]-algebra must be of the form [MATH], with [MATH] compact Hausdorff space.', '1109.6162-2-2-1': 'In view of this fundamental result, a "noncommutative compact space" should be just the abstract categorical dual of an arbitrary unital [MATH]-algebra.', '1109.6162-2-3-0': 'The interesting examples of such spaces abound.', '1109.6162-2-3-1': 'Connes proposed in [CITATION] an axiomatic framework for the noncommutative (spin) manifolds.', '1109.6162-2-3-2': 'This notion covers a huge number of interesting situations, for instance the Standard Model one.', '1109.6162-2-3-3': 'See [CITATION], [CITATION].', '1109.6162-2-4-0': 'Another class of interesting examples is provided by the compact quantum groups, axiomatized by Woronowicz in [CITATION], [CITATION], [CITATION].', '1109.6162-2-4-1': 'This class covers for instance all the duals of discrete groups, as well as the compact forms of the Lie-type algebras of Drinfeld [CITATION] and Jimbo [CITATION], at [MATH].', '1109.6162-2-4-2': 'Note that by [CITATION] these latter quantum groups have a Dirac operator in the sense of Connes.', '1109.6162-2-4-3': 'The Drinfeld-Jimbo quantum groups at [MATH] are not very far either, because they are known to correspond to tensor [MATH]-categories [CITATION].', '1109.6162-2-5-0': 'Yet another class of interesting examples are provided by the noncommutative homogeneous spaces.', '1109.6162-2-5-1': 'For some pioneering axiomatization work and for various results in this direction, see Podles [CITATION] and Boca [CITATION], Tomatsu [CITATION], Vaes [CITATION], Kasprzak [CITATION].', '1109.6162-2-5-2': "A number of key examples, mostly related to the various noncommutative spheres, were systematically investigated in a series of papers by Connes, D'Andrea, Dabrowski, Dubois-Violette, Khalkhali, Landi, van Suijlekom and Wagner [CITATION], [CITATION], [CITATION], [CITATION].", '1109.6162-2-6-0': 'The starting point for the present paper is the following result:', '1109.6162-2-7-0': 'Theorem.', '1109.6162-2-7-1': 'Let [MATH] be a closed subgroup, let [MATH], and set [MATH], where the embedding [MATH] is given by [MATH].', '1109.6162-2-7-2': 'Then [MATH], viewed as subalgebra of [MATH], is generated by the coordinate functions [MATH], with [MATH].', '1109.6162-2-8-0': 'This result follows indeed from the Stone-Weierstrass theorem, because the subalgebra [MATH] generated by [MATH] separates the points of [MATH].', '1109.6162-2-9-0': 'As a basic application, in the case [MATH] and [MATH] the above result tells us, modulo a standard symmetry argument, that each row of coordinates [MATH] on [MATH] can be identified with the standard coordinates [MATH] on the sphere [MATH].', '1109.6162-2-10-0': "In the quantum group case now, an analogue of the above statement can be formulated, in terms of Wang's quantum group [MATH] [CITATION].", '1109.6162-2-10-1': 'More precisely, if [MATH] is a quantum subgroup and [MATH], we can construct the quantum group [MATH], and then the algebra [MATH].', '1109.6162-2-10-2': 'Also, if we denote by [MATH] the [MATH]-algebra generated by the coordinates [MATH], then we have an inclusion [MATH], and the problem is whether this inclusion is an isomorphism or not.', '1109.6162-2-11-0': 'Our first result is a complete answer to this question in the group dual case:', '1109.6162-2-12-0': 'Theorem A. Assume that [MATH] is a discrete group dual, with [MATH], and with the embedding [MATH] given by [MATH], where [MATH] and [MATH].', '1109.6162-2-12-1': 'Then [MATH] is an isomorphism iff [MATH], where [MATH].', '1109.6162-2-13-0': 'This result is actually a consequence of a slightly more precise statement, saying that the inclusion [MATH] coincides with the inclusion [MATH], where [MATH] is the normal closure of [MATH].', '1109.6162-2-13-1': 'Observe also that in the case of diagonal embeddings ([MATH]) the normality condition in the statement is simply [MATH].', '1109.6162-2-14-0': 'With this theoretical result in hand, we will focus then on the case of "easy quantum groups" [MATH], axiomatized in [CITATION], and studied in [CITATION], [CITATION], [CITATION], [CITATION], [CITATION].', '1109.6162-2-14-1': 'The point is that the "row algebras" of type [MATH] are of particular interest in the easy case:', '1109.6162-2-15-0': 'First, [MATH] is a quantum subgroup of [MATH], and the inclusion [MATH] itself is a quite interesting object: we will prove here, as part of Theorem B below, that [MATH] is equivalent to a certain key combinatorial condition, introduced in [CITATION].', '1109.6162-2-16-0': 'Second, even in classical case, the quotient spaces of type [MATH] are quite subtle.', '1109.6162-2-16-1': 'For instance one strategy in the exact computation of polynomial integrals over [MATH] is that of examining the spaces [MATH], with [MATH] increasing.', '1109.6162-2-16-2': 'See [CITATION].', '1109.6162-2-17-0': 'At the quantum level now, the simplest case is [MATH] and [MATH], corresponding to the "free spheres" studied in [CITATION].', '1109.6162-2-17-1': 'Also, the "free hypergeometric laws" in [CITATION] seem to come from spaces of type [MATH], not axiomatized yet.', '1109.6162-2-18-0': 'Summarizing, the problematics in the easy case, be it classical or free, is related to a number of recent computations and considerations.', '1109.6162-2-18-1': 'Our second result in this paper, dealing with the inclusions [MATH] in the easy case, is as follows:', '1109.6162-2-19-0': 'Theorem B.', '1109.6162-2-19-1': 'For an easy quantum group [MATH] we have [MATH] for any [MATH] if and only if the category of partitions for [MATH] is stable by removing blocks.', '1109.6162-2-19-2': 'If so is the case, and if [MATH] is free (i.e. [MATH]), then [MATH] is in general proper.', '1109.6162-2-20-0': 'We refer to sections 3-4 below for the precise statement here.', '1109.6162-2-20-1': 'The proof of the first assertion is purely combinatorial, using the results in [CITATION], and the proof of the second assertion is inspired by Theorem A, by using a suitable group dual subgroup of [MATH].', '1109.6162-2-21-0': 'The above result raises the question of locating [MATH] inside [MATH].', '1109.6162-2-21-1': 'We do not have results here, but as a further step in investigating [MATH], we will construct and study a certain universal algebra [MATH], having [MATH] as quotient.', '1109.6162-2-22-0': 'Our result here is a bit technical.', '1109.6162-2-22-1': 'Let us recall from [CITATION] that in the free case there are exactly 6 easy quantum groups, namely [MATH], which satisfy the combinatorial condition in Theorem B, plus two more quantum groups [MATH], which do not satisfy it.', '1109.6162-2-22-2': 'So, the 4 quantum groups that we are interested in are the orthogonal, symmetric, hyperoctahedral and bistochastic groups [MATH].', '1109.6162-2-22-3': 'These are defined by the fact that their fundamental corepresentation is orthogonal, magic, cubic and bistochastic, see [CITATION].', '1109.6162-2-22-4': 'Now with these notions in hand, our third result is as follows:', '1109.6162-2-23-0': 'Theorem C.', '1109.6162-2-23-1': 'For [MATH] let [MATH] be the universal [MATH]-algebra generated by the entries of a transposed [MATH] orthogonal, magic, cubic, stochastic isometry.', '1109.6162-2-23-2': 'Then [MATH] has the same abelianization and reduced version as [MATH].', '1109.6162-2-24-0': 'We refer to section 5 below for the precise statement.', '1109.6162-2-24-1': 'The proof uses the integration formula of Collins-Sniady [CITATION], as extended in [CITATION], and a method from [CITATION].', '1109.6162-2-25-0': 'The problem of computing the kernel of [MATH], which would be useful in connection with (1,2,3), seems to be a difficult linear algebra one, somehow in the spirit of those solved in [CITATION].', '1109.6162-2-25-1': 'There might be also some connections with [CITATION], [CITATION].', '1109.6162-2-26-0': 'We also believe that the situation [MATH] can appear in more general contexts, and might help for the general understanding of noncommutative homogeneous spaces.', '1109.6162-2-26-1': 'We have several questions here, formulated in section 6 below.', '1109.6162-2-27-0': 'The paper is organized as follows: 1-2 contain generalities about the spaces of type [MATH], including the group dual case result, and in 3-4 we investigate the "easy" case.', '1109.6162-2-27-1': 'The final sections, 5-6, contain further results, and a few concluding remarks.'}

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[]

[['1109.6162-1-4-3', '1109.6162-2-4-2']]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1109.6162

{'1109.6162-3-0-0': 'Given a quantum subgroup [MATH] and a number [MATH] we can form the homogeneous space [MATH], and it follows from the Stone-Weierstrass theorem that [MATH] is the algebra generated by the last [MATH] rows of coordinates on [MATH].', '1109.6162-3-0-1': "In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case.", '1109.6162-3-0-2': 'We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type [MATH].', '1109.6162-3-1-0': '# Introduction', '1109.6162-3-2-0': 'The notion of "noncommutative space" goes back to an old theorem of Gelfand, stating that any commutative unital [MATH]-algebra must be of the form [MATH], with [MATH] compact Hausdorff space.', '1109.6162-3-2-1': 'In view of this fundamental result, a "noncommutative compact space" should be just the abstract categorical dual of an arbitrary unital [MATH]-algebra.', '1109.6162-3-3-0': 'The interesting examples of such spaces abound.', '1109.6162-3-3-1': 'Connes proposed in [CITATION] an axiomatic framework for the noncommutative (spin) manifolds.', '1109.6162-3-3-2': 'This notion covers a huge number of interesting situations, for instance the Standard Model one.', '1109.6162-3-3-3': 'See [CITATION], [CITATION].', '1109.6162-3-4-0': 'Another class of interesting examples is provided by the compact quantum groups, axiomatized by Woronowicz in [CITATION], [CITATION], [CITATION].', '1109.6162-3-4-1': 'This class covers for instance all the duals of discrete groups, as well as the compact forms of the Lie-type algebras of Drinfeld [CITATION] and Jimbo [CITATION], at [MATH].', '1109.6162-3-4-2': 'Note that by [CITATION] these latter quantum groups have a Dirac operator in the sense of Connes.', '1109.6162-3-4-3': 'The Drinfeld-Jimbo quantum groups at [MATH] are not very far either, because they are known to correspond to tensor [MATH]-categories [CITATION].', '1109.6162-3-5-0': 'Yet another class of interesting examples are provided by the noncommutative homogeneous spaces.', '1109.6162-3-5-1': 'For some pioneering axiomatization work and for various results in this direction, see Podles [CITATION] and Boca [CITATION], Tomatsu [CITATION], Vaes [CITATION], Kasprzak [CITATION].', '1109.6162-3-5-2': "A number of key examples, mostly related to the various noncommutative spheres, were systematically investigated in a series of papers by Connes, D'Andrea, Dabrowski, Dubois-Violette, Khalkhali, Landi, van Suijlekom and Wagner [CITATION], [CITATION], [CITATION], [CITATION].", '1109.6162-3-6-0': 'The starting point for the present paper is the following result:', '1109.6162-3-7-0': 'Theorem.', '1109.6162-3-7-1': 'Let [MATH] be a closed subgroup, let [MATH], and set [MATH], where the embedding [MATH] is given by [MATH].', '1109.6162-3-7-2': 'Then [MATH], viewed as subalgebra of [MATH], is generated by the coordinate functions [MATH], with [MATH].', '1109.6162-3-8-0': 'This result follows indeed from the Stone-Weierstrass theorem, because the subalgebra [MATH] generated by [MATH] separates the points of [MATH].', '1109.6162-3-9-0': 'As a basic application, in the case [MATH] and [MATH] the above result tells us, modulo a standard symmetry argument, that each row of coordinates [MATH] on [MATH] can be identified with the standard coordinates [MATH] on the sphere [MATH].', '1109.6162-3-10-0': "In the quantum group case now, an analogue of the above statement can be formulated, in terms of Wang's quantum group [MATH] [CITATION].", '1109.6162-3-10-1': 'More precisely, if [MATH] is a quantum subgroup and [MATH], we can construct the quantum group [MATH], and then the algebra [MATH].', '1109.6162-3-10-2': 'Also, if we denote by [MATH] the [MATH]-algebra generated by the coordinates [MATH], then we have an inclusion [MATH], and the problem is whether this inclusion is an isomorphism or not.', '1109.6162-3-11-0': 'Our first result is a complete answer to this question in the group dual case:', '1109.6162-3-12-0': 'Theorem A. Assume that [MATH] is a discrete group dual, with [MATH], and with the embedding [MATH] given by [MATH], where [MATH] and [MATH].', '1109.6162-3-12-1': 'Then [MATH] is an isomorphism iff [MATH], where [MATH].', '1109.6162-3-13-0': 'This result is actually a consequence of a slightly more precise statement, saying that the inclusion [MATH] coincides with the inclusion [MATH], where [MATH] is the normal closure of [MATH].', '1109.6162-3-13-1': 'Observe also that in the case of diagonal embeddings ([MATH]) the normality condition in the statement is simply [MATH].', '1109.6162-3-14-0': 'With this theoretical result in hand, we will focus then on the case of "easy quantum groups" [MATH], axiomatized in [CITATION], and studied in [CITATION], [CITATION], [CITATION], [CITATION], [CITATION].', '1109.6162-3-14-1': 'The point is that the "row algebras" of type [MATH] are of particular interest in the easy case:', '1109.6162-3-15-0': 'First, [MATH] is a quantum subgroup of [MATH], and the inclusion [MATH] itself is a quite interesting object: we will prove here, as part of Theorem B below, that [MATH] is equivalent to a certain key combinatorial condition, introduced in [CITATION].', '1109.6162-3-16-0': 'Second, even in classical case, the quotient spaces of type [MATH] are quite subtle.', '1109.6162-3-16-1': 'For instance one strategy in the exact computation of polynomial integrals over [MATH] is that of examining the spaces [MATH], with [MATH] increasing.', '1109.6162-3-16-2': 'See [CITATION].', '1109.6162-3-17-0': 'At the quantum level now, the simplest case is [MATH] and [MATH], corresponding to the "free spheres" studied in [CITATION].', '1109.6162-3-17-1': 'Also, the "free hypergeometric laws" in [CITATION] seem to come from spaces of type [MATH], not axiomatized yet.', '1109.6162-3-18-0': 'Summarizing, the problematics in the easy case, be it classical or free, is related to a number of recent computations and considerations.', '1109.6162-3-18-1': 'Our second result in this paper, dealing with the inclusions [MATH] in the easy case, is as follows:', '1109.6162-3-19-0': 'Theorem B.', '1109.6162-3-19-1': 'For an easy quantum group [MATH] we have [MATH] for any [MATH] if and only if the category of partitions for [MATH] is stable by removing blocks.', '1109.6162-3-19-2': 'If so is the case, and if [MATH] is free (i.e. [MATH]), then [MATH] is in general proper.', '1109.6162-3-20-0': 'We refer to sections 3-4 below for the precise statement here.', '1109.6162-3-20-1': 'The proof of the first assertion is purely combinatorial, using the results in [CITATION], and the proof of the second assertion is inspired by Theorem A, by using a suitable group dual subgroup of [MATH].', '1109.6162-3-21-0': 'The above result raises the question of locating [MATH] inside [MATH].', '1109.6162-3-21-1': 'We do not have results here, but as a further step in investigating [MATH], we will construct and study a certain universal algebra [MATH], having [MATH] as quotient.', '1109.6162-3-22-0': 'Our result here is a bit technical.', '1109.6162-3-22-1': 'Let us recall from [CITATION] that in the free case there are exactly 6 easy quantum groups, namely [MATH], which satisfy the combinatorial condition in Theorem B, plus two more quantum groups [MATH], which do not satisfy it.', '1109.6162-3-22-2': 'So, the 4 quantum groups that we are interested in are the orthogonal, symmetric, hyperoctahedral and bistochastic groups [MATH].', '1109.6162-3-22-3': 'These are defined by the fact that their fundamental corepresentation is orthogonal, magic, cubic and bistochastic, see [CITATION].', '1109.6162-3-22-4': 'Now with these notions in hand, our third result is as follows:', '1109.6162-3-23-0': 'Theorem C.', '1109.6162-3-23-1': 'For [MATH] let [MATH] be the universal [MATH]-algebra generated by the entries of a transposed [MATH] orthogonal, magic, cubic, stochastic isometry.', '1109.6162-3-23-2': 'Then [MATH] has the same abelianization and reduced version as [MATH].', '1109.6162-3-24-0': 'We refer to section 5 below for the precise statement.', '1109.6162-3-24-1': 'The proof uses the integration formula of Collins-Sniady [CITATION], as extended in [CITATION], and a method from [CITATION].', '1109.6162-3-25-0': 'The problem of computing the kernel of [MATH], which would be useful in connection with (1,2,3), seems to be a difficult linear algebra one, somehow in the spirit of those solved in [CITATION].', '1109.6162-3-25-1': 'There might be also some connections with [CITATION], [CITATION].', '1109.6162-3-26-0': 'We also believe that the situation [MATH] can appear in more general contexts, and might help for the general understanding of noncommutative homogeneous spaces.', '1109.6162-3-26-1': 'We have several questions here, formulated in section 6 below.', '1109.6162-3-27-0': 'The paper is organized as follows: 1-2 contain generalities about the spaces of type [MATH], including the group dual case result, and in 3-4 we investigate the "easy" case.', '1109.6162-3-27-1': 'The final sections, 5-6, contain further results, and a few concluding remarks.'}

1205.2316

{'1205.2316-1-0-0': 'A fraction of the extragalactic near-infrared (near-IR) background light involves redshifted photons from the ultraviolet (UV) emission from galaxies present during reionization at redshifts above 6.', '1205.2316-1-0-1': 'The absolute intensity and the anisotropies of the near-IR background provide an observational probe of the first-light galaxies and their spatial distribution.', '1205.2316-1-0-2': 'We estimate the extragalactic background light intensity during reionization by accounting for the stellar and nebular emission from first-light galaxies.', '1205.2316-1-0-3': 'We require the UV photon density from these galaxies to generate a reionization history that is consistent with the optical depth to electron scattering from cosmic microwave background measurements.', '1205.2316-1-0-4': 'We also require the bright-end luminosity function of galaxies in our models to reproduce the measured Lyman drop-out luminosity functions at redshifts of 6 to 8.', '1205.2316-1-0-5': 'The absolute intensity is about 0.1 to 0.3 nW m[MATH] sr[MATH] at the peak of its spectrum at [MATH] 1.1 [MATH]m.', '1205.2316-1-0-6': 'We also discuss the anisotropy power spectrum of the near-IR background using a halo model to describe the galaxy distribution.', '1205.2316-1-0-7': 'We compare our predictions for the anisotropy power spectrum to existing measurements from deep near-IR imaging data from Spitzer/IRAC, Hubble/NICMOS, and AKARI.', '1205.2316-1-0-8': 'The predicted rms fluctuations at tens of arcminute angular scales are roughly an order of magnitude smaller than the existing measurements.', '1205.2316-1-0-9': 'While strong arguments have been made that the measured fluctuations do not have an origin involving faint low-redshift galaxies, we find that measurements in the literature are also incompatible with galaxies present during the era of reionization.', '1205.2316-1-0-10': 'The measured near-IR background anisotropies remain unexplained with an unknown origin.', '1205.2316-1-1-0': '# Introduction', '1205.2316-1-2-0': 'The optical and UV radiation from sources present during reionization is expected to leave a signature in the extragalactic background light (EBL) at near-IR wavelengths (e.g., Santos et al. 2002; Salvaterra Ferrara 2003; Cooray Yoshida 2004; Fernandez Komatsu 2006; Raue 2009).', '1205.2316-1-2-1': 'Such radiation is not expected to be present in the background light at UV and optical wavelengths due to the redshifted Lyman limit.', '1205.2316-1-2-2': 'The exact intensity from first-light galaxies present during reionization is currently unknown.', '1205.2316-1-2-3': 'The first predictions suggested an intensity as high as 10 to 30 nW m[MATH] sr[MATH] (Santos et al. 2002; Salvaterra Ferrara 2003).', '1205.2316-1-2-4': 'These estimates were partly motivated by the need to explain the difference between DIRBE EBL measurements (e.g., Cambresy et al. 2001) and the integrated galaxy light (IGL) from deep galaxy counts (Madau Pozzetti 2000; Totani et al. 2001).', '1205.2316-1-3-0': 'These predictions with high backgrounds were questioned by Madau Silk (2005) based on existing limits related to metal content at high redshifts and the X-ray background produced by stellar end-products such as black holes.', '1205.2316-1-3-1': 'They suggest an intensity less than about 2.5 nW m[MATH] sr[MATH] in the J-band from a galaxy population made up of population III stars during reionization (Madau Silk 2005).', '1205.2316-1-3-2': 'With a combination of Population II stars and changes to the lifetime of stars, Fernandez Komatsu (2006) argued that the background could be as high as 4 to 8 nW m[MATH] sr[MATH].', '1205.2316-1-3-3': 'Even in such a scenario a simple estimate of the UV photon density at [MATH] shows that there are roughly an order of magnitude higher number of H-ionizing photons per baryon during reionization than necessary to explain the reionization history.', '1205.2316-1-3-4': 'Since one does not expect more than a few H-ionizing photons per baryon during reionization, a first-order estimate suggests that the background intensity cannot be larger than a few tenths nW m[MATH] sr[MATH] between 1 and 2 [MATH]m.', '1205.2316-1-4-0': 'Unfortunately a direct search for the integrated intensity of galaxies present during reionization based on absolute background measurements has been problematic due to the confusion with the Zodiacal foreground.', '1205.2316-1-4-1': 'At 1 AU Zodiacal light is two to three orders of magnitude brighter than the [MATH] 10 nW m[MATH] sr[MATH] intensity produced by extragalactic sources.', '1205.2316-1-4-2': 'While challenging, techniques have been devised to estimate the Zodiacal dust column density based on the line strengths of Fraunhofer lines seen in the dust-scattered Solar spectrum (e.g., Bernstein Dyson 2003).', '1205.2316-1-4-3': 'Instead of the absolute background, in Cooray et al. (2004; also Kashlinsky et al. 2004), it was proposed that the galaxies present during reionization can be studied with anisotropies of the near-IR background.', '1205.2316-1-4-4': 'The anisotropy studies have the potential to probe deeper than the absolute experiments and could study a galaxy population present during reionization that leads to an intensity well below 0.1 nW m[MATH] sr[MATH] (Cooray et al. 2004).', '1205.2316-1-5-0': 'This suggestion has motivated experimental measurements on the near-IR anisotropy power spectrum with data from Spitzer/IRAC, HST/NICMOS, and AKARI.', '1205.2316-1-5-1': 'After a deep removal of point sources, Kashlinsky et al. (2005, 2007, 2012) claimed a detection of first-light galaxy fluctuations at [MATH].', '1205.2316-1-5-2': 'The detected signal is an excess of clustering power above shot-noise on the largest angular scales.', '1205.2316-1-5-3': 'A similar suggestion was also made by an AKARI group (Matsumoto et al. 2011), but an analysis of the HST/NICMOS Ultra Deep Field led to an opposite conclusion that the sources contributing to the near-IR excess fluctuations are at [MATH] (Thompson et al. 2007).', '1205.2316-1-5-4': 'Due to the limited areas of existing deep surveys near-IR background anisotropy measurements are limited to angular scales less than about one degree.', '1205.2316-1-5-5': 'The limited field of view is especially a problem for existing NICMOS UDF measurements (Thompson et al. 2007), where the fluctuations are limited to angular scales less than 5[MATH].', '1205.2316-1-5-6': 'Separately, a joint analysis of IRAC and HST/ACS data in the same GOODS fields as studied by Kashlinsky et al. (2007) led to the suggestion that up to 50 of the excess fluctuations at 3.6 [MATH]m could come from faint dwarf galaxies at [MATH] (Cooray et al. 2007; Chary et al. 2008).', '1205.2316-1-5-7': 'Through detailed models combined with more recent measurements of faint galaxy clustering, Helgason et al. (2012) has lowered this low-redshift contribution to 3.6 [MATH]m intensity fluctuations to be at most 20.', '1205.2316-1-5-8': 'The rest of the anisotropies continue to be interpreted as originating from first-light galaxies during reionization (Kashlinsky et al. 2012).', '1205.2316-1-6-0': 'While there are still uncertainties on the exact intensity and the amplitude of intensity fluctuations in experimental measurements, the situation is no different on the theory side.', '1205.2316-1-6-1': 'The first estimates on the anisotropy power spectrum made use of linear theory clustering (Cooray et al. 2004).', '1205.2316-1-6-2': 'Fernandez et al. (2010) used numerical simulations of reionization to predict the expected power spectrum during reionization.', '1205.2316-1-6-3': 'Their power spectrum has a shape in the form of a power-law with [MATH] between ten arcminute to arcsecond angular scales.', '1205.2316-1-6-4': 'Fernandez et al. (2010) suggested that the power-law behavior arises from significant non-linear biasing of dark matter halos at high redshifts.', '1205.2316-1-6-5': 'Due to the limited box sizes of existing reionization simulations of the order 100 to 140 Mpc on the side, numerical studies are limited to angular scales of 30 arcminutes and below at [MATH].', '1205.2316-1-7-0': 'With the availability of WFC3 on the Hubble Space Telescope, dedicated IR background experiments (e.g., CIBER sounding-rocket experiment; Bock et al. 2006), and plans for a future space-based absolute intensity measurement (ZEBRA; Cooray et al. 2009) there is now a clear need to revisit theoretical predictions on both the absolute intensity and the anisotropy power spectrum from galaxies present during reionization.', '1205.2316-1-7-1': 'While current multi-band Spitzer and AKARI measurements do not overlap in the same fields, the combination of IRAC and WFC3 data on some of the same well-studied fields on the sky (e.g., fields covered by the CANDELS survey; Grogin et al. 2011; Koekemoer et al. 2011) will soon allow the spectral energy distribution (SED) of intensity fluctuations be studied uniformly.', '1205.2316-1-7-2': 'Separately CIBER is conducting spectral imaging absolute measurements between 0.8 and 1.6 [MATH]m in wide 4 deg.', '1205.2316-1-7-3': '[MATH] fields instantaneously using multiple sounding rocket flights (Zemcov et al. 2012).', '1205.2316-1-7-4': 'The combination of IRAC and CIBER is capable of extending anisotropy measurements out to angular scales of more than a degree from optical to 4.5 [MATH]m.', '1205.2316-1-8-0': 'In this work we establish both the mean intensity and the anisotropy power spectrum of galaxies present during reionization.', '1205.2316-1-8-1': 'We update Cooray et al. (2004) by taking into account recent developments in the study of reionization, and by introducing a halo model to calculate the non-linear clustering of the IR background intensity.', '1205.2316-1-8-2': 'The stellar and nebular emission from first light galaxies follow the calculations presented in Fernandez Komatsu (2006), but we specifically require that the UV photon background produced by the galaxy population present from [MATH] to 30 is consistent with the optical depth to electron scattering as measured by the WMAP polarization data with a value of 0.088 [MATH] (Komatsu et al. 2011).', '1205.2316-1-8-3': 'We account for the current uncertainty in the optical depth by introducing variations to the fiducial model so that the optical depth to electron scattering varies between 0.07 and 0.1.', '1205.2316-1-8-4': 'Our models are also designed to reproduce the bright-end galaxy luminosity functions (LFs) in deep HST/WFC3 surveys at [MATH] involving the Lyman-dropout galaxy samples.', '1205.2316-1-8-5': 'This normalization at the bright-end of galaxy luminosities puts strong constraints on the intensity.', '1205.2316-1-9-0': 'This paper is organized as follows.', '1205.2316-1-9-1': 'In Section 2 we outline our model for the reionization galaxies including stellar nebulae and the IGM emission.', '1205.2316-1-9-2': 'Section 3 presents the calculation related to luminosity mass density of these galaxies.', '1205.2316-1-9-3': 'In Section 4, 5 and 6 we outline the background intensity and spatial anisotropy power spectrum calculations, respectively.', '1205.2316-1-9-4': 'In Section 7 we discuss our results related to the intensity and angular power spectrum and present a comparison to existing measurements.', '1205.2316-1-9-5': 'We conclude with a summary in Section 8.', '1205.2316-1-9-6': 'We assume the flat [MATH]CDM model with [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] throughout the paper.', '1205.2316-1-10-0': '# Emission from stars and the intergalactic medium', '1205.2316-1-11-0': 'We first describe the emission from stars in first-light galaxies present during reionization.', '1205.2316-1-11-1': 'Following Fernandez Komatsu (2006), we consider two stellar populations in this calculation.', '1205.2316-1-11-2': 'The first, referred to as Pop II stars, are metal-poor stars with metallicity [MATH], and the second, Pop III stars, are metal-free stars with [MATH].', '1205.2316-1-12-0': 'To describe the stellar initial mass function (IMF) we make use of two descriptions.', '1205.2316-1-12-1': 'For Pop II stars, we adopt the IMF given by Salpeter (1995) [EQUATION] with mass range from 3 to 150 [MATH].', '1205.2316-1-12-2': 'For Pop III stars we use the IMF obtained by Larson (1999), which takes the form as [EQUATION] where [MATH], and the mass range is from 3 to 500 [MATH].', '1205.2316-1-13-0': 'we utilize the fitting results from [CITATION] and [CITATION] to calculate other stellar parameters, such as the intrinsic bolometric luminosity [MATH], the effective temperature [MATH], the main-sequence lifetime [MATH], and the time-averaged hydrogen photoionization rate [MATH].', '1205.2316-1-13-1': 'The fitting forms of these parameters are different for Pop II and Pop III stars.', '1205.2316-1-13-2': 'For Pop II stars, they are given as [EQUATION] where [MATH] and for Pop III stars, they are [EQUATION]', '1205.2316-1-13-3': 'From these expressions the stellar radius [MATH] is [EQUATION] where [MATH] is the Stefan-Boltzmann constant.', '1205.2316-1-13-4': 'The stellar radius is useful for the calculation related to the stellar emission spectrum (see Section 3.1).', '1205.2316-1-14-0': 'The ionization volume in the nebulae surrounding the stars (Str[MATH]mgren sphere) can be derived if assuming ionization equilibrium where the ionization rate equals recombination rate [EQUATION] where [MATH] is the hydrogen case B recombination coefficient which depends on the gas temperature [MATH] (assumed to be [MATH] K), and we will discuss it in detail in the next Section.', '1205.2316-1-14-1': 'Here, [MATH] and [MATH] are the local number density of electron and HII in the stellar nebulae where we assume [MATH].', '1205.2316-1-15-0': 'For the IGM, the hydrogen density is lower than that of the stellar nebulae, so we no longer assume the ionization equilibrium.', '1205.2316-1-15-1': 'We estimate the ionization volume by a redshift-dependent form as [EQUATION] where [MATH] is the mean hydrogen number density for [MATH] assumed in this work (Shull et al. 2011).', '1205.2316-1-16-0': 'These quantities discussed here would now be used to estimate the luminosity mass density, near-IR background intensity SED, and the anisotropy power spectrum.', '1205.2316-1-16-1': 'We make use of emission from the stellar nebulae and the IGM for both Pop II and Pop III stars in galaxies present during reionization.', '1205.2316-1-17-0': '# Luminosity mass density of the sources', '1205.2316-1-18-0': 'In this Section, we calculate the luminosity per stellar mass at frequency [MATH], i.e. luminosity mass density, for several sources that contribute to the infrared background, such as the direct emission from the stars, Lyman-[MATH] line, and free-free, free-bound and two photon processes.', '1205.2316-1-18-1': 'The luminosity mass density takes the central role in our estimation of the near-IR background intensity spectrum.', '1205.2316-1-19-0': '## Stellar spectrum', '1205.2316-1-20-0': 'For simplicity, we assume the stellar spectrum is a Planckian truncated at [MATH] eV.', '1205.2316-1-20-1': 'Thus, the stellar luminosity at frequency [MATH] can be expressed as [EQUATION] where [MATH] is the stellar radius, [MATH] is the stellar mass and [MATH] is the Planck spectrum [EQUATION]', '1205.2316-1-20-2': 'Note that we have ignored the absorption lines of the Lyman-alpha (Ly[MATH]) series here.', '1205.2316-1-20-3': 'The emission with [MATH] eV cannot be approximated by a black-body spectrum, thus we use the fitting formulae for time-averaged photoionization rate, [MATH], for Pop II and Pop III stars to calculate the emission at higher energies.', '1205.2316-1-21-0': '## Lyman alpha emission', '1205.2316-1-22-0': 'The luminosity of Ly[MATH] emission at a frequency [MATH] is [EQUATION] where [MATH] is the frequency of Ly[MATH] photons, and [MATH] is the emission volume that can be estimated by Eq. ([REF]) and Eq. ([REF]) for the stellar nebulae and the IGM respectively.', '1205.2316-1-22-1': 'Here [MATH] is the Ly[MATH] recombination emission rate per [MATH], which is given by [EQUATION] where [MATH] is the electron number density, [MATH] is the HII number density, [MATH] is the hydrogen case B recombination coefficient, and [MATH] is the fraction of the Ly[MATH] photons produced in the case B recombination.', '1205.2316-1-22-2': 'This fraction can be estimated through the fitting formula [EQUATION] where [MATH]K, which is accurate to 0.1% for [MATH].', '1205.2316-1-22-3': 'Note that this fraction is actually not sensitive to the temperature, so it can be treated as a constant [MATH] in the most cases.', '1205.2316-1-22-4': 'The hydrogen case B recombination coefficient [MATH] we use here is from [CITATION], which is fitted by [CITATION] as [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1205.2316-1-22-5': 'We assume a gas temperature of [MATH] K in our calculation to obtain [MATH].', '1205.2316-1-23-0': 'The [MATH] is the collisional emission rate per [MATH] given by [EQUATION] where [MATH] is the neutral hydrogen number density and [MATH] is the effective collisional excitation coefficient.', '1205.2316-1-23-1': 'It has the form [EQUATION]', '1205.2316-1-23-2': 'Here we take into account the excitation up to energy level [MATH] to produce Ly-[MATH] photons.', '1205.2316-1-23-3': 'The higher level emission can be neglected given the high temperature we consider for this calculation.', '1205.2316-1-23-4': 'The excitation collisional rate [MATH], in cm[MATH] per second, can be written as [EQUATION] where [MATH] is the energy difference between lower level [MATH] and higher level [MATH], [MATH] is the statistic weight for level [MATH], and [MATH] is the effective collision strength calculated using the fitting formulae from [CITATION].', '1205.2316-1-24-0': 'The [MATH] is the Ly[MATH] line profile, and we use the result from [CITATION] where they fitted the simulated Ly[MATH] line profile of [CITATION] for a homogeneous and expanding IGM: [EQUATION] where [MATH], and [EQUATION].', '1205.2316-1-24-1': 'Here [MATH], and we assume the flat [MATH]CDM model and take [MATH].', '1205.2316-1-25-0': '## Free-free and free-bound emission', '1205.2316-1-26-0': 'For free-free and free-bound emission we again follow the same approach as Fernandez Komatsu (2006).', '1205.2316-1-26-1': 'Following their derivation, the continuum luminosity of these two processes at frequency [MATH] has the same form with [EQUATION] where [MATH] is the specific emission coefficient for free-free and free-bound emission [EQUATION]', '1205.2316-1-26-2': 'Here [MATH] is the proton number density, [MATH] is the gas temperature, and [MATH] is effective Gaunt factor for free-free and free-bound emission, which takes the form as [EQUATION] where [MATH] is the thermal averaged Gaunt factor of free-free emission and [MATH] is the free-bound emission Gaunt factor for a different energy level [MATH].', '1205.2316-1-26-3': 'These values have an accuracy of 10% .', '1205.2316-1-26-4': 'In above [MATH], where [MATH] is around 10 for the parameter space we are interested in .', '1205.2316-1-26-5': 'The energy level [MATH] is determined by the emission photon frequency [MATH].', '1205.2316-1-26-6': 'If [MATH], and then [MATH] where [MATH] is the Rydberg constant.', '1205.2316-1-26-7': 'Note that the [MATH] here starts at [MATH], since the photons from [MATH] can be easily absorbed by other neutral hydrogen atoms and be ionized instantly.', '1205.2316-1-27-0': '## Two-photon emission', '1205.2316-1-28-0': 'For the two-photon process we also follow the approach of Fernandez Komatsu (2006) and write the luminosity as [EQUATION] where [MATH] is the two-photon emission rate per [MATH], and [MATH].', '1205.2316-1-28-1': 'The [MATH] is the normalized probability of generating one photon in the range [MATH] from per two-photon decay.', '1205.2316-1-28-2': 'We use the fitting formula derived in [CITATION] [EQUATION] where [MATH], which is a good fit to the data given in [CITATION].', '1205.2316-1-29-0': '## Luminosity mass density and total emission', '1205.2316-1-30-0': 'Then following Fernandez Komatsu (2006) and Fernandez et al. (2010), we can derive the mean luminosity mass density for each emission source by integrating over the IMF for the Pop II or Pop III stars [EQUATION] where the ranges of the integral are from 3 to 150 [MATH] with the IMF given by Salpeter (1955) for Pop II stars, and from 3 to 500 [MATH] with the IMF in Larson (1998) for Pop III stars.', '1205.2316-1-30-1': 'Here, the average mass [MATH] is given as [EQUATION] where [MATH] is the normalized IMF with [EQUATION].', '1205.2316-1-31-0': 'Note that this expression is only valid when the main sequence lifetime is larger than the star formation time scale.', '1205.2316-1-31-1': 'Otherwise, it should be evaluated by [EQUATION]', '1205.2316-1-31-2': 'Here [MATH] is the star formation time scale which is given by [EQUATION] where [MATH] is the stellar mass density at [MATH], which is related with the comoving star formation rate density (SFRD) as [MATH].', '1205.2316-1-31-3': 'We use the halo mass function to calculate [MATH] and [MATH] and the details are described in the next Section.', '1205.2316-1-31-4': 'We note that the [MATH] is important for the estimation of the [MATH] as discussed in [CITATION].', '1205.2316-1-32-0': 'Finally we obtain the total luminosity mass density from the stellar nebulae [EQUATION] and the same from the IGM [EQUATION] where [MATH] is the escape fraction of the ionization photons that propagate into the IGM from the nebulae surrounding the stars.', '1205.2316-1-33-0': 'In Fig. [REF], we show the total luminosity mass density as a function of the rest-frame wavelength [MATH] from the stellar nebula and the IGM for Pop II and Pop III stars respectively.', '1205.2316-1-33-1': 'The contributions from the different sources we consider are also shown for the stellar nebula case.', '1205.2316-1-33-2': 'Here we set [MATH] and [MATH] for both Pop II and Pop III cases.', '1205.2316-1-34-0': 'With these parameters we find that the stellar spectrum is dominant for Pop II stars while the "background" spectrum, such as Ly-[MATH] and free-bound, are comparable with or even larger than the stellar spectrum for Pop III stars.', '1205.2316-1-34-1': 'Also, as can be seen, the [MATH] from the IGM is much lower than that from the stellar nebula for both Pop II and Pop III cases, and the total [MATH] from the Pop III stars is similar to that from the Pop II stars.', '1205.2316-1-34-2': 'These results are already discussed in Fernandez Komatsu (2006).', '1205.2316-1-35-0': '# Reionization History and UV Luminosity Density', '1205.2316-1-36-0': 'To test if the reionization history associated with our stellar model is consistent with that of the current observations, such as WMAP 7-year results , we need to calculate the hydrogen reionization fraction [MATH] as a function of redshift.', '1205.2316-1-36-1': 'Following Madau et al. (1998), [MATH] can be estimated as [EQUATION] where [MATH] is the redshift of the beginning of the reionization epoch (we take [MATH]), [MATH] is the comoving star formation rate density (SFRD), and the function [MATH] is defined as [MATH].', '1205.2316-1-36-2': 'Here [MATH] is the average stellar lifetime which is given by [MATH], and [MATH] is the volume averaged recombination time, which can be written as [EQUATION] where [MATH] is the clumping factor of ionized hydrogen.', '1205.2316-1-36-3': 'Here we adopt the simulation result from Trac Cen (2007).', '1205.2316-1-36-4': '[MATH] and [MATH] are the mass fractions of hydrogen and helium, respectively.', '1205.2316-1-36-5': 'Note that we have already considered the escape fraction [MATH], so the [MATH] here is the clumping factor excluding the halos with star formation.', '1205.2316-1-37-0': 'For the comoving SFRD we consider the ongoing star formation model : [EQUATION] where [MATH] is the star formation efficiency which denotes the fraction of baryons converted to stars, [MATH] is the halo mass function , and [MATH] is the threshold mass for a dark matter halo to form a galaxy during reionization.', '1205.2316-1-37-1': 'This minimum mass is taken to be a free parameter and is varied to fit a combination of the WMAP 7-year optical depth and the galaxy LF, as we discuss in Section 6.', '1205.2316-1-38-0': 'We also need a stellar population evolution model to describe [MATH] with the relative fraction of the Pop II and Pop III stars at different redshifts.', '1205.2316-1-38-1': 'In principle there should be a cutoff at some redshift for Pop III stars as they are not expected to form at low redshifts once the gas is polluted by metals.', '1205.2316-1-38-2': 'We assume this cutoff is not lower than [MATH] when the universe is fully ionized.', '1205.2316-1-38-3': 'We assume that the Pop III stars are mainly dominant for [MATH] while Pop II stars are for [MATH].', '1205.2316-1-38-4': 'Then we use the error function to denote the population fraction as [EQUATION] where [MATH] is the population transition width.', '1205.2316-1-38-5': 'Then the term [MATH] in Eq. ([REF]) can be expressed by [EQUATION]', '1205.2316-1-38-6': 'We estimate the optical depth to electron scattering with the reionization fraction [MATH] as [EQUATION] where [MATH] is the Thompson scattering cross-section, and [MATH] is the electron number density of the Universe at redshift [MATH], and we assume the helium is singly ionized for [MATH]) and doubly ionized for [MATH]) .', '1205.2316-1-39-0': 'In Fig. [REF], we plot the hydrogen reionization fraction [MATH] vs. [MATH] for three optical depth [MATH] with three stellar models.', '1205.2316-1-39-1': 'The blue solid line denotes the [MATH] with [MATH] which is close to the result of WMAP 7-year data with [MATH] .', '1205.2316-1-39-2': 'In this case reionization ends around [MATH] which is consistent with the current studies, and we find Pop III stars can ionize the Universe to [MATH] by [MATH] and Pop II stars is responsible for the rest of the reionization over an interval [MATH].', '1205.2316-1-40-0': 'For [MATH] and 0.09, we find [MATH], shown with a dashed line and [MATH], shown with a dotted line, respectively.', '1205.2316-1-40-1': 'These two reionization histories are such that [MATH] and 9, respectively.', '1205.2316-1-40-2': 'For case with [MATH], the Pop III stars ionize [MATH] of the Universe and the Pop II stars are needed over the interval [MATH] to complete reionization.', '1205.2316-1-40-3': 'For the third case with [MATH], Pop III stars ionize [MATH] of the Universe at [MATH] with Pop II stars completing the rest [MATH] in an interval of [MATH]', '1205.2316-1-41-0': 'In Fig. [REF], we also show the dependence of the optical depth on the minimum redshift [MATH] in Eq. ([REF]).', '1205.2316-1-41-1': 'The blue solid, dashed and dotted lines are for the three stellar models with [MATH], 0.074 and 0.093 for [MATH].', '1205.2316-1-41-2': 'The WMAP 7-year result with [MATH] is also shown with a red solid line and a yellow [MATH] error region.', '1205.2316-1-41-3': 'We find the slope of the curves is steeper for [MATH] and flatter for [MATH] which is caused by the Pop III to Pop II transition around [MATH] in our model.', '1205.2316-1-41-4': 'Note that this transition is arbitrarily chosen.', '1205.2316-1-41-5': 'We varied the transition redshift and also cases where PopII and PopIII stars are mixed in with different fractions at different redshifts.', '1205.2316-1-41-6': 'In all these cases we found results that are generally consistent with each other.', '1205.2316-1-41-7': 'Thus, the three choices related to the reionization history that we show here to keep this presentation simple are not biased with respect to the final result related to the IR background intensity that we are trying to estimate in this paper.', '1205.2316-1-42-0': 'We also estimate the total number of ionizing photons per baryon required to maintain the ionized IGM between [MATH] and z, [EQUATION] where [MATH] is the redshift of the end of the reionization, and if we assume the helium is also totally singly ionized at [MATH] we can get [MATH].', '1205.2316-1-42-1': 'In fig. [REF], we show the [MATH] at different [MATH] for three [MATH] cases.', '1205.2316-1-42-2': 'We find the number keeps going up until around [MATH] and becomes constant [MATH] at higher redshift for three cases of the reionization histories.', '1205.2316-1-43-0': '# The near-IR EBL Intensity from Reionization', '1205.2316-1-44-0': 'The mean cosmic infrared background can be estimated by [EQUATION] where [MATH], and we take [MATH] and [MATH] in the calculation.', '1205.2316-1-44-1': 'This redshift range can fully take account of the emission from the Pop III and early Pop II stars, which is redshifted into the near-IR band.', '1205.2316-1-44-2': 'We take this to be the form of [EQUATION] where [MATH] and [MATH] are the comoving specific emission coefficients [EQUATION] where [MATH] is the luminosity mass density at [MATH], [MATH] is the mean stellar lifetime of each of the stellar type and [MATH] is the comoving SFRD given by Eq. ([REF]).', '1205.2316-1-45-0': 'In Fig. [REF], we show the spectrum [MATH] of the near-IR background light intensity from both Pop II and Pop III stars.', '1205.2316-1-45-1': 'We assume that the reionization is ending around [MATH] and integrate up to [MATH] to determine [MATH].', '1205.2316-1-45-2': 'Similar to Fig. [REF], we plot the total spectrum from the stellar nebula and the IGM for both Pop II and Pop III stars.', '1205.2316-1-45-3': 'The contributions from different sources we consider are also shown in colored thin lines.', '1205.2316-1-46-0': 'Here we still take the same value for [MATH] and [MATH] as in Fig. 1 related to the [MATH] calculation.', '1205.2316-1-46-1': 'Similar to the luminosity mass density, the spectrum from stellar nebula is much larger than that from the IGM, and the stellar spectrum is higher for Pop II stars while the "background" spectrum is higher for Pop III stars.', '1205.2316-1-46-2': 'However, different from the luminosity mass density, we now find the spectrum from Pop II stars is larger than that from Pop III stars.', '1205.2316-1-46-3': 'This is basically because the typical lifetime of Pop II stars is longer than that of the Pop III stars.', '1205.2316-1-47-0': '# Angular power spectrum', '1205.2316-1-48-0': 'The angular cross power spectrum of the infrared emission at observed frequencies [MATH] and [MATH] for a multipole [MATH] is [EQUATION] where [MATH] is the comoving angular diameter distance, [MATH] is the scale factor, and [MATH] is the mean emission per comoving volume at frequency [MATH] and redshift [MATH].', '1205.2316-1-48-1': 'If we just take account of the flux lower than a upper cut [MATH], and then [MATH] can be written as [EQUATION]', '1205.2316-1-48-2': 'Here [MATH] is the source flux and [MATH] is the number of sources.', '1205.2316-1-48-3': 'This quantity is just the comoving specific emission coefficient we derive in the last section.', '1205.2316-1-48-4': 'The [MATH] is the galaxy power spectrum at wavenumber [MATH] and redshift [MATH], and we will make use of the model of halo occupation distribution to calculate the [MATH].', '1205.2316-1-49-0': '## First Galaxy Clustering', '1205.2316-1-50-0': 'To calculate the [MATH] we extend the linear theory model of Cooray et al. (2004) and make use of the halo occupation distribution (HOD) for first-light galaxies during reionization.', '1205.2316-1-50-1': 'The galaxy power spectrum can be written as [EQUATION]', '1205.2316-1-50-2': 'Where [MATH] and [MATH] denote the power spectrum contributed by galaxies in a single dark matter halo and galaxies in two different dark matter halos respectively.', '1205.2316-1-50-3': 'Then we can write [EQUATION]', '1205.2316-1-50-4': 'Here [MATH] is the halo mass, [MATH] is the halo mass function, [MATH] is the Fourier transform of the NFW halo density profile , [MATH] when [MATH] and [MATH] otherwise , [MATH] is the halo bias , and [MATH] is the linear matter power spectrum .', '1205.2316-1-50-5': 'The [MATH] is the mean number density of galaxies, which is given by [EQUATION]', '1205.2316-1-50-6': 'The [MATH] is the mean number of galaxies in a halo with mass [MATH], which is the sum of number of central galaxies and satellite galaxies [EQUATION] where we define [EQUATION] and [EQUATION]', '1205.2316-1-50-7': 'In this definition, the [MATH] denotes the mass of a halo that has 50% probability to host a central galaxy, and [MATH] is the transition width.', '1205.2316-1-50-8': 'For the satellite galaxies, [MATH] is the truncation mass for satellites, [MATH] is the normalization mass and [MATH] denotes the slope of the power-law relation about the halo mass [MATH].', '1205.2316-1-50-9': 'We assume [MATH] is always larger than [MATH] since there should not be satellites without central galaxy, and assume [MATH].', '1205.2316-1-50-10': 'We take [MATH], [MATH], and [MATH] in this paper.', '1205.2316-1-50-11': 'If assuming a Poisson distribution for satellite galaxies, we can get [EQUATION]', '1205.2316-1-50-12': 'This expression could take account of the case [MATH] and is consistent with our definitions for the [MATH] and [MATH].', '1205.2316-1-51-0': '## Poisson Fluctuations', '1205.2316-1-52-0': 'The clustering measurements are affected by the Poisson fluctuations associated with the shot-noise caused by the discrete and finite number of galaxies from which clustering is measured.', '1205.2316-1-52-1': 'Assuming a Poisson distribution the [MATH]-independent shot-noise power spectrum is [EQUATION]', '1205.2316-1-52-2': 'To estimate [MATH], we first define the luminosity mass density for the mass of the dark matter halos at frequency [MATH] [EQUATION] where [MATH] is the luminosity mass density for the stellar mass discussed in Section 3.', '1205.2316-1-52-3': 'We derive the 3-D shot-noise power spectrum by assuming [MATH] is proportional to the halo mass [MATH], i.e. [MATH] is independent on [MATH] [EQUATION]', '1205.2316-1-52-4': 'Then the 2-D shot-noise power spectrum can be written as [EQUATION]', '1205.2316-1-53-0': '## Band-Averaged Intensity Power Spectrum', '1205.2316-1-54-0': 'For a specific near-IR observation with a band frequency from [MATH] to [MATH], we can define a band-averaged luminosity mass density as [EQUATION]', '1205.2316-1-54-1': 'Then we can derive the band-averaged comoving specific emission coefficient [MATH] using Eq. ([REF]) and the 3-D shot-noise power spectrum [MATH] using Eq. ([REF]) respectively.', '1205.2316-1-55-0': 'Finally we find the band-averaged angular cross power spectrum and shot-noise power spectrum are [EQUATION] and [EQUATION] respectively.', '1205.2316-1-55-1': 'Note that here we have a factor [MATH] instead of [MATH] in [MATH] and [MATH] and this dependence has been explained in the Appendix of [CITATION].', '1205.2316-1-56-0': '# Results and Discussion', '1205.2316-1-57-0': 'In this Section, we first estimate the infrared background intensity and then discuss the angular power spectrum as derived previously.', '1205.2316-1-57-1': 'We also compare our estimation with the observational data and discuss the dependence of the result on the parameters in the model.', '1205.2316-1-58-0': '## Near-IR EBL from Reionization', '1205.2316-1-59-0': 'In Fig. [REF], we show the spectrum of the near-IR intensity vs. the observed frequency for three cases of the reionization history with [MATH], 0.087 and 0.093 corresponding to [MATH], 0.07 and 0.09.', '1205.2316-1-59-1': 'Here the spectrum is the total spectrum of the sum of that from both stellar nebula and the IGM, which are calculated by putting Eq. ([REF]) into Eq. ([REF]).', '1205.2316-1-59-2': 'Also, we set [MATH] for both Pop II and Pop III stars.', '1205.2316-1-59-3': 'The right panel of Fig. [REF] shows three cases with [MATH] varied at fixed [MATH].', '1205.2316-1-59-4': 'In both panels, for comparison, we plot the observational data in terms of the excess of EBL relative to the integrated galaxy light (IGL) of known galaxy populations at low redshifts (Madau Pozzetti 2000).', '1205.2316-1-59-5': 'The excess EBL data plotted here are the same as those shown in Santos et al. (2002) and involves measurements mainly from DIRBE with various models for zodiacal light and Galactic stellar contribution.', '1205.2316-1-60-0': 'We find the emission from the Pop II stars dominates the spectrum for all of the three cases.', '1205.2316-1-60-1': 'As we have just discussed in the last Section, this is because the lifetime of the Pop II is longer than the Pop III stars.', '1205.2316-1-60-2': 'In the spectrum, the shorter and longer wavelength parts are mainly contributed by the "background" spectrum from Pop III stars while the medium part by Pop II stellar spectrum, so if just the Pop III stars get longer lifetime, only the "background" spectrum can be effectively reinforced in the total spectrum.', '1205.2316-1-61-0': 'In any case, regardless of assumptions related to the stellar type, we find that the EBL intensity from reionization is no more than 0.3 nW m[MATH] sr[MATH].', '1205.2316-1-61-1': 'Such an intensity is significantly smaller than the previous predictions that attempted to explain almost all or a significant fraction of the excess EBL seen in DIRBE data relative to IGL estimates.', '1205.2316-1-61-2': 'An intensity larger than about 2.5 nW m[MATH] sr[MATH] in the J-band could be in conflict with metal production considerations and the X-ray background (Madau Silk 2005), though they do not necessarily require high efficiency factors to generate the required star formation (Fernandez Komatsu 2006).', '1205.2316-1-61-3': 'The difference between our calculation and the previous ones is that we primarily require the reionization model to generate a reionization history consistent with the WMAP optical depth.', '1205.2316-1-61-4': 'This limits the number of H-ionizing photons per baryon during reionization to be less than 3.', '1205.2316-1-61-5': 'Previous estimates ignored such a constraint and either focused on explaining all of the DIRBE excess (Santos et al. 2002; Salvaterra Ferrara 2003) or amplitude of the measured near-IR background anisotropies (e.g., Fernandez et al. 2010).', '1205.2316-1-62-0': '## Bright-end Galaxy Luminosity Functions', '1205.2316-1-63-0': 'In order to relate our galaxy population responsible for reionization to the observations, we also compare our model to observations of [MATH] galaxies, focusing on the UV LFs.', '1205.2316-1-63-1': 'To obtain the UV LF, instead of the occupation number which is luminosity independent, we make use of the conditional luminosity function (CLF) approach .', '1205.2316-1-63-2': 'To compute the CLF we map galaxy rest-frame UV luminosity to halo mass with some scatter added similar to the case of low-redshift galaxy populations (Cooray 2005).', '1205.2316-1-63-3': "The total luminosity of a halo is taken to be [EQUATION] and we assume that this total luminosity can be ascribed to the central galaxy when [MATH], following the' earlier description related to the occupation number.", '1205.2316-1-63-4': 'When [MATH] we introduce satellites with central galaxy luminosity kept fixed at [MATH].', '1205.2316-1-63-5': 'However, when comparing to the existing measurements we found that all of the rest-UV LF measurements are in the range where central galaxies dominate the LF and thus our comparison to the measured LFs is independent of assumptions related to the exact form of the satellite occupation number or conditional luminosity function.', '1205.2316-1-64-0': 'To compare with existing rest-UV LF measurements, we convert the luminosity of each galaxy to the AB absolute magnitude via the relation [MATH] .', '1205.2316-1-64-1': 'In the left panel of Fig. [REF], we show the SFRD as a function of redshift derived from Eq. ([REF]) for three reionization histories with [MATH], 0.087 and 0.093 which are obtained by setting [MATH], 0.07 and 0.09, respectively.', '1205.2316-1-64-2': 'The red points are the data from HUDF09+ERS+CANDELS observations .', '1205.2316-1-64-3': 'We find the SFRD of the three cases are higher than the existing measurements, especially at the high redshifts.', '1205.2316-1-64-4': 'This difference is mainly due to the fact that existing SFRD estimates are limited to galaxies with [MATH], while the bulk of the reionization UV density budget is contained in the galaxies at the faint-end of the LF.', '1205.2316-1-64-5': 'This is especially the case at high redshift since the faint-end slope of the LF is steep with values reaching close to -2 already.', '1205.2316-1-65-0': 'In the right panel of Fig. [REF], we show our rest-UV LF (corresponding to [MATH]) at [MATH]6, 7 and 8.', '1205.2316-1-65-1': 'The central thick solid curves are the luminosity function derived from our default model with [MATH] and [MATH], and the thin dotted lines are obtained by [MATH] (lower) and 0.09 (upper) which could match the [MATH] errors of the data given in .', '1205.2316-1-65-2': 'The three values of the faint-end slope [MATH], -2.0 and -1.5 are shown, which indicates the slope of our model is between -2.5 and -2.0.', '1205.2316-1-65-3': 'Also, we find the star formation time scale [MATH] is around [MATH] yrs at [MATH], 7 and 8 when we calculate the luminosity mass function here.', '1205.2316-1-66-0': 'We also explore the dependence of the galaxy bias on the [MATH] in Fig. [REF].', '1205.2316-1-66-1': 'We can define a effective galaxy bias here from the HOD model as [EQUATION]', '1205.2316-1-66-2': 'In the plot we show the [MATH] as a function of [MATH] at [MATH]6, 7 and 8 for the [MATH].', '1205.2316-1-66-3': 'We find that the galaxy bias increases as the redshift increases and decreases as the [MATH] increases.', '1205.2316-1-66-4': 'The reason is obvious that the galaxy number density [MATH] defined by Eq. ([REF]) becomes smaller at higher redshift and bigger at larger [MATH] (the larger [MATH] means smaller halo mass [MATH]).', '1205.2316-1-67-0': '## Anisotropy Power Spectrum', '1205.2316-1-68-0': 'In Fig. [REF] we show the near-IR background anisotropy angular power spectrum at [MATH].', '1205.2316-1-68-1': 'The clustering power spectra with non-linear power spectrum from the HOD model are in solid lines.', '1205.2316-1-68-2': 'The left panel shows the case with [MATH], the minimum mass to host a galaxy, is changed from [MATH] to [MATH] M[MATH].', '1205.2316-1-68-3': 'For comparison, we also plot the linear power spectrum which has a turnover around [MATH].', '1205.2316-1-68-4': 'When calculating the clustering power spectrum, we set the parameters of the HOD with [MATH] values as listed in the figure with [MATH].', '1205.2316-1-68-5': 'The corresponding values on the optical depth to electron scattering are also listed in the figure.', '1205.2316-1-68-6': 'We note that the shot-noise amplitude is larger for the case with [MATH] M[MATH] in comparison to the case with, say, [MATH] M[MATH], though in those two cases the clustering amplitude is higher with [MATH] M[MATH].', '1205.2316-1-68-7': 'This is because the shot-noise amplitude is sensitive to the second flux-moment of the number counts.', '1205.2316-1-68-8': 'By keeping the minimum mass higher we force the overall counts to be restricted to brighter sources than the case with a lower minimum halo mass.', '1205.2316-1-68-9': 'On the other hand the clustering power spectrum reflects the total background intensity.', '1205.2316-1-68-10': 'With the minimum mass lowered, both the overall number density of galaxies and the background intensity are increased.', '1205.2316-1-69-0': 'The right panel of Fig. [REF] shows the case where we vary [MATH] to highlight the fact [MATH] amplitude is inversely proportional to [MATH].', '1205.2316-1-69-1': 'However, one cannot arbitrarily reduce [MATH] to a small value since this results in a low optical depth to electron scattering.', '1205.2316-1-69-2': 'If [MATH] to 10[MATH], such that one does not need to be concerned of effects due to feedback negatively impacting the formation of galaxies in lower mass halos, then we find that [MATH] must be at the level of 0.3 or more, under the assumptions related to the IMFs for PopII and PopIII stars we use in this paper.', '1205.2316-1-70-0': 'In Fig. [REF], we plot the [MATH] at different redshift bins.', '1205.2316-1-70-1': 'As can be seen, the main contribution of the [MATH] comes from the lowest redshift range of 6 to 10.', '1205.2316-1-70-2': 'For all practical purposes one can assume that the near-IR background is probing the end of reionization not the first objects to form in the universe at the beginning of reionization.', '1205.2316-1-71-0': 'To compare with the total near-IR intensity spectrum, we plot the ratio of the square root of [MATH] to the [MATH] in Fig. [REF].', '1205.2316-1-71-1': 'We integrate from [MATH] to 30 with the same HOD model described above.', '1205.2316-1-71-2': 'For easy comparison, we restrict the predictions to be with [MATH] M[MATH] such that [MATH] to remain consistent with WMAP 7-year result.', '1205.2316-1-71-3': 'We find the ratio of [MATH] to [MATH] is about [MATH] for [MATH] and the clustering fluctuations amplitude is below 100% of the intensity at all angular scales.', '1205.2316-1-71-4': 'This shows that large fluctuations of the background intensity is not expected and the background behaves in a manner that is smooth and not clumpy as in the case if spatial variations are dominated by rare, bright galaxies.', '1205.2316-1-72-0': 'In Fig. [REF], we show the near-IR background intensity power spectrum (solid curves), shot-noise power spectrum (dotted curves) with a comparison to existing observational measurements at several wavelengths: [MATH], 2.4, 3.6, and 4.5 [MATH] .', '1205.2316-1-72-1': 'Note that for the data from Kashlinsky et al. (2012) at 3.6 and 4.5 [MATH], we removed the IRAC beam transfer function [MATH] so the comparison to theoretical predictions is easier.', '1205.2316-1-72-2': 'An accounting of the beam is essential since realistic instruments do not have perfect resolution causing a loss of power on the smallest scales.', '1205.2316-1-72-3': 'This effect can be explicitly measured from the point spread function.', '1205.2316-1-72-4': 'We calculate the beam transfer function [MATH] by taking the power spectrum of the measured point spread function for the Spitzer/IRAC instrument, which is publicly available and compute [MATH] (for more information see Smidt et al. 2012).', '1205.2316-1-73-0': 'We also show the results from the simulation in [CITATION] for comparison.', '1205.2316-1-73-1': 'The red solid and magenta dashed lines are for the simulation with mass resolution [MATH] and [MATH], respectively.', '1205.2316-1-73-2': 'These two cases are what is described in their work as not having a suppression of the small mass halos (ie star-formation present in halos with mass between 10[MATH] and 10[MATH] M[MATH], where one expects photo-ionization heating to suppress star-formation).', '1205.2316-1-73-3': 'The reionization histories of these two cases involve PopII stars with [MATH].', '1205.2316-1-74-0': 'For more clarity, we show the square root of the [MATH] at [MATH] and [MATH] together with the data as a function of the wavelength in Fig. [REF].', '1205.2316-1-74-1': 'The data points are from the data sets shown in Fig. [REF] which are around [MATH] or [MATH].', '1205.2316-1-74-2': 'We find the curve of the square root of the [MATH] has similar shape as [MATH], but the amplitude of rms fluctuations in our models is lower than the existing measurements.', '1205.2316-1-74-3': 'We capture the uncertainties related to [MATH], [MATH], and [MATH] by considering a low and high range for our prediction related to [MATH] with [MATH] falling within the 1 [MATH] uncertainty range of the WMAP 7-year result when these parameters are varied.', '1205.2316-1-74-4': 'Even with parameter uncertainties accounted for, we find that the existing measurements are at least an order of magnitude larger than our model predictions.', '1205.2316-1-75-0': 'We attempted additional model variations but failed to find a scenario where [MATH] is consistent with WMAP 7-year result and the existing LFs leading to a model consistent with existing near-IR background.', '1205.2316-1-75-1': 'One can, in principle, model fit the near-IR fluctuation power spectra by increasing the photon output of first galaxies.', '1205.2316-1-75-2': 'This results in an optical depth that is higher than the WMAP value and a LF that has brighter galaxies than observed in existing deep HST/WFC3 images.', '1205.2316-1-75-3': 'The existing measurements require a background intensity that is at least 3 nW m[MATH] sr[MATH] at 1.6 [MATH]m so that the model predictions become consistent with measurements within the 1[MATH] uncertainties of the measurements.', '1205.2316-1-75-4': 'Since 0.3 nW m[MATH] sr[MATH] at 1.6 [MATH]m is associated with 2.5 H-ionizing photons per baryon during reionization, if reionization is to explain the background fluctuations then we are dealing with a situation where [MATH] 25 H-ionizing photons per baryon are present.', '1205.2316-1-75-5': 'A possibility is to introduce a spectrum for the emission that has a rest-frame cut-off in UV at wavelengths shortward of the Lyman-limit that is not associated with reionization but by the emission mechanism itself.', '1205.2316-1-75-6': 'Unfortunately we have not been able to come up with such an emission spectrum.', '1205.2316-1-76-0': 'As discussed in Helgason et al. (2012) faint, low-redshift galaxies do not have a clustering shape consistent with the anisotropy power spectrum measurements.', '1205.2316-1-76-1': 'The existing measurements show a clear excess in clustering at 30 arcsec to few tens arcminute angular scales that differ from faint galaxy clustering power spectrum.', '1205.2316-1-76-2': 'This difference is statistically significant and one can only explain at most 20% of the near-IR fluctuations to be associated with faint galaxies, reducing the previous estimate in Cooray et al. (2007; Chary et al. 2008) that suggested a contribution at the 50% level.', '1205.2316-1-76-3': 'We are now left with an unexplained set of measurements since neither the low redshift faint galaxies nor the high redshift reionization galaxies can explain them.', '1205.2316-1-76-4': 'One possibility is that the existing fluctuation excess in the data has a non-astrophysical origin, perhaps either involving systematics in the data or fluctuations in the Zodiacal light.', '1205.2316-1-76-5': 'While arguments have been made that these effects are insignificant with a set of well-coordinated multi-wavelength measurements we plan to further address the origin of near-IR fluctuations in our upcoming papers.', '1205.2316-1-77-0': '# Summary', '1205.2316-1-78-0': 'The UV emission from stars which are formed in the early Universe from [MATH] to 30 can contribute to the near-IR background light.', '1205.2316-1-78-1': 'By measuring the intensity and anisotropies of the near-IR background, we can investigate the properties of these early stars and the epoch of reionization.', '1205.2316-1-78-2': 'In this work we discuss several sources that contribute to the near-IR background intensity, including the emission of stars, Ly[MATH], free-free, free-bound and two-photon.', '1205.2316-1-78-3': 'We first estimate the frequency spectrum separately for Pop II and Pop III star with the redshift range [MATH].', '1205.2316-1-78-4': 'By using the initial stellar mass spectrum and the fitting model of time-averaged hydrogen photoionization rate we calculate the luminosity mass density.', '1205.2316-1-78-5': 'We find that although the luminosity mass density of the Pop III stars is a bit larger than that of Pop II stars, the near-IR intensity spectrum from the Pop II stars is stronger than that from Pop III stars, which is caused by the longer lifetime of Pop II stars.', '1205.2316-1-79-0': 'In order to check the consistence of our stellar model and the reionization history, we derive the hydrogen reionization fraction at different redshifts and calculate the optical depth by assuming the Pop II stars are dominant for [MATH] while Pop III stars for [MATH].', '1205.2316-1-79-1': 'We find that if we set the star formation efficiency [MATH], the Universe would be totally reionized around [MATH] with the optical depth [MATH], which is well consistent with the result from the WMAP 7-year data.', '1205.2316-1-79-2': 'Also, we explore the other possible models with [MATH] and [MATH] and get [MATH] and 0.093, respectively.', '1205.2316-1-79-3': 'The total number of the ionizing photon per baryon required to maintain the ionized IGM, [MATH], is also evaluated for the three cases, and we find the [MATH] becomes a constant with a value around 2.5 after [MATH].', '1205.2316-1-80-0': 'To compare with existing bright-end luminosity function measurements, we evaluate the UV luminosity function from our model at [MATH] for [MATH]6, 7 and 8 from [MATH] to [MATH].', '1205.2316-1-80-1': 'We compare to the measurements of Bouwens et al. (2012).', '1205.2316-1-80-2': 'We find our derived luminosity function is consistent with the data with the slope of the faint-end [MATH].', '1205.2316-1-80-3': 'This is a steep slope and existing measurements do suggest that the slope is steeper than for LFs at low redshifts.', '1205.2316-1-80-4': 'We then define an effective galaxy bias [MATH] from the HOD model, and find the [MATH] becomes bigger at high redshift and bright luminosities because of the lower number density of the galaxies.', '1205.2316-1-81-0': 'Finally, we calculate the angular power spectrum of the near-IR background by making use of a halo model.', '1205.2316-1-81-1': 'The non-linearities provided by the 1-halo term increase the clustering strength at multipoles greater than about [MATH].', '1205.2316-1-81-2': 'The suggest turn-over at this scale with the linear power spectrum alone in Cooray et al. (2004) is no longer present.', '1205.2316-1-81-3': 'We find the shot-noise power spectrum of the Pop III stars is greater than that of Pop II stars because of the larger luminosity mass density for Pop III stars.', '1205.2316-1-81-4': 'By making use of the stellar population evolution model, we calculate the near-IR anisotropy power spectra [MATH] at different wavelengths and compare to the observational data.', '1205.2316-1-81-5': 'We find our results are lower than the observational data by at least an order of magnitude.', '1205.2316-1-81-6': 'There are now strong arguments that the near-IR anisotropies cannot originate from low redshift faint galaxies (Helgason et al. 2012).', '1205.2316-1-81-7': 'We have failed to explain the alternative origin of near-IR background anisotropies involving galaxies present during reionization, contrary to suggestions in the literature (Kashlinsky et al. 2012).', '1205.2316-1-81-8': 'In future works, using additional measurements with Spitzer and Hubble/WFC3, we plan to further discuss the near-IR fluctuations and to explain if the origin is astrophysical or whether it is associated with yet-unknown systematic effect in the data.', '1205.2316-1-82-0': 'This work was supported by NSF CAREER AST-0645427 and NASA NNX10AD42G at UCI.', '1205.2316-1-82-1': 'We thank CIBER, SDWFS, and CANDELS teams for helpful discussions and questions that motivated this paper.'}

{'1205.2316-2-0-0': 'A fraction of the extragalactic near-infrared (near-IR) background light involves redshifted photons from the ultraviolet (UV) emission from galaxies present during reionization at redshifts above 6.', '1205.2316-2-0-1': 'The absolute intensity and the anisotropies of the near-IR background provide an observational probe of the first-light galaxies and their spatial distribution.', '1205.2316-2-0-2': 'We estimate the extragalactic background light intensity during reionization by accounting for the stellar and nebular emission from first-light galaxies.', '1205.2316-2-0-3': 'We require the UV photon density from these galaxies to generate a reionization history that is consistent with the optical depth to electron scattering from cosmic microwave background measurements.', '1205.2316-2-0-4': 'We also require the bright-end luminosity function of galaxies in our models to reproduce the measured Lyman drop-out luminosity functions at redshifts of 6 to 8.', '1205.2316-2-0-5': 'The absolute intensity is about 0.1 to 0.3 nW m[MATH] sr[MATH] at the peak of its spectrum at [MATH] 1.1 [MATH]m.', '1205.2316-2-0-6': 'We also discuss the anisotropy power spectrum of the near-IR background using a halo model to describe the galaxy distribution.', '1205.2316-2-0-7': 'We compare our predictions for the anisotropy power spectrum to existing measurements from deep near-IR imaging data from Spitzer/IRAC, Hubble/NICMOS, and AKARI.', '1205.2316-2-0-8': 'The predicted rms fluctuations at tens of arcminute angular scales are roughly an order of magnitude smaller than the existing measurements.', '1205.2316-2-0-9': 'While strong arguments have been made that the measured fluctuations do not have an origin involving faint low-redshift galaxies, we find that measurements in the literature are also incompatible with galaxies present during the era of reionization.', '1205.2316-2-0-10': 'The measured near-IR background anisotropies remain unexplained with an unknown origin.', '1205.2316-2-1-0': '# Introduction', '1205.2316-2-2-0': 'The optical and UV radiation from sources present during reionization is expected to leave a signature in the extragalactic background light (EBL) at near-IR wavelengths (e.g., Santos et al. 2002; Salvaterra Ferrara 2003; Cooray Yoshida 2004; Fernandez Komatsu 2006; Raue 2009).', '1205.2316-2-2-1': 'Such radiation is not expected to be present in the background light at UV and optical wavelengths due to the redshifted Lyman limit.', '1205.2316-2-2-2': 'The exact intensity from first-light galaxies present during reionization is currently unknown.', '1205.2316-2-2-3': 'The first predictions suggested an intensity as high as 10 to 30 nW m[MATH] sr[MATH] (Santos et al. 2002; Salvaterra Ferrara 2003).', '1205.2316-2-2-4': 'These estimates were partly motivated by the need to explain the difference between DIRBE EBL measurements (e.g., Cambresy et al. 2001) and the integrated galaxy light (IGL) from deep galaxy counts (Madau Pozzetti 2000; Totani et al. 2001).', '1205.2316-2-3-0': 'These predictions with high backgrounds were questioned by Madau Silk (2005) based on existing limits related to metal content at high redshifts and the X-ray background produced by stellar end-products such as black holes.', '1205.2316-2-3-1': 'They suggest an intensity less than about 2.5 nW m[MATH] sr[MATH] in the J-band from a galaxy population made up of population III stars during reionization (Madau Silk 2005).', '1205.2316-2-3-2': 'With a combination of Population II stars and changes to the lifetime of stars, Fernandez Komatsu (2006) argued that the background could be as high as 4 to 8 nW m[MATH] sr[MATH].', '1205.2316-2-3-3': 'Even in such a scenario a simple estimate of the UV photon density at [MATH] shows that there are roughly an order of magnitude higher number of H-ionizing photons per baryon during reionization than necessary to explain the reionization history.', '1205.2316-2-3-4': 'Since one does not expect more than a few H-ionizing photons per baryon during reionization, a first-order estimate suggests that the background intensity cannot be larger than a few tenths nW m[MATH] sr[MATH] between 1 and 2 [MATH]m.', '1205.2316-2-4-0': 'Unfortunately a direct search for the integrated intensity of galaxies present during reionization based on absolute background measurements has been problematic due to the confusion with the Zodiacal foreground.', '1205.2316-2-4-1': 'At 1 AU Zodiacal light is two to three orders of magnitude brighter than the [MATH] 10 nW m[MATH] sr[MATH] intensity produced by extragalactic sources.', '1205.2316-2-4-2': 'While challenging, techniques have been devised to estimate the Zodiacal dust column density based on the line strengths of Fraunhofer lines seen in the dust-scattered Solar spectrum (e.g., Bernstein Dyson 2003).', '1205.2316-2-4-3': 'Instead of the absolute background, in Cooray et al. (2004; also Kashlinsky et al. 2004), it was proposed that the galaxies present during reionization can be studied with anisotropies of the near-IR background.', '1205.2316-2-4-4': 'The anisotropy studies have the potential to probe deeper than the absolute experiments and could study a galaxy population present during reionization that leads to an intensity well below 0.1 nW m[MATH] sr[MATH] (Cooray et al. 2004).', '1205.2316-2-5-0': 'This suggestion has motivated experimental measurements on the near-IR anisotropy power spectrum with data from Spitzer/IRAC, HST/NICMOS, and AKARI.', '1205.2316-2-5-1': 'After a deep removal of point sources, Kashlinsky et al. (2005, 2007, 2012) claimed a detection of first-light galaxy fluctuations at [MATH].', '1205.2316-2-5-2': 'The detected signal is an excess of clustering power above shot-noise on the largest angular scales.', '1205.2316-2-5-3': 'A similar suggestion was also made by an AKARI group (Matsumoto et al. 2011), but an analysis of the HST/NICMOS Ultra Deep Field led to an opposite conclusion that the sources contributing to the near-IR excess fluctuations are at [MATH] (Thompson et al. 2007).', '1205.2316-2-5-4': 'Due to the limited areas of existing deep surveys near-IR background anisotropy measurements are limited to angular scales less than about one degree.', '1205.2316-2-5-5': 'The limited field of view is especially a problem for existing NICMOS UDF measurements (Thompson et al. 2007), where the fluctuations are limited to angular scales less than 5[MATH].', '1205.2316-2-5-6': 'Separately, a joint analysis of IRAC and HST/ACS data in the same GOODS fields as studied by Kashlinsky et al. (2007) led to the suggestion that up to 50 of the excess fluctuations at 3.6 [MATH]m could come from faint dwarf galaxies at [MATH] (Cooray et al. 2007; Chary et al. 2008).', '1205.2316-2-5-7': 'Through detailed models combined with more recent measurements of faint galaxy clustering, Helgason et al. (2012) has lowered this low-redshift contribution to 3.6 [MATH]m intensity fluctuations to be at most 20.', '1205.2316-2-5-8': 'The rest of the anisotropies continue to be interpreted as originating from first-light galaxies during reionization (Kashlinsky et al. 2012).', '1205.2316-2-6-0': 'While there are still uncertainties on the exact intensity and the amplitude of intensity fluctuations in experimental measurements, the situation is no different on the theory side.', '1205.2316-2-6-1': 'The first estimates on the anisotropy power spectrum made use of linear theory clustering (Cooray et al. 2004).', '1205.2316-2-6-2': 'Fernandez et al. (2010) used numerical simulations of reionization to predict the expected power spectrum during reionization.', '1205.2316-2-6-3': 'Their power spectrum has a shape in the form of a power-law with [MATH] between ten arcminute to arcsecond angular scales.', '1205.2316-2-6-4': 'Fernandez et al. (2010) suggested that the power-law behavior arises from significant non-linear biasing of dark matter halos at high redshifts.', '1205.2316-2-6-5': 'Due to the limited box sizes of existing reionization simulations of the order 100 to 140 Mpc on the side, numerical studies are limited to angular scales of 30 arcminutes and below at [MATH].', '1205.2316-2-7-0': 'With the availability of WFC3 on the Hubble Space Telescope, dedicated IR background experiments (e.g., CIBER sounding-rocket experiment; Bock et al. 2006), and plans for a future space-based absolute intensity measurement (ZEBRA; Cooray et al. 2009) there is now a clear need to revisit theoretical predictions on both the absolute intensity and the anisotropy power spectrum from galaxies present during reionization.', '1205.2316-2-7-1': 'While current multi-band Spitzer and AKARI measurements do not overlap in the same fields, the combination of IRAC and WFC3 data on some of the same well-studied fields on the sky (e.g., fields covered by the CANDELS survey; Grogin et al. 2011; Koekemoer et al. 2011) will soon allow the spectral energy distribution (SED) of intensity fluctuations be studied uniformly.', '1205.2316-2-7-2': 'Separately CIBER is conducting spectral imaging absolute measurements between 0.8 and 1.6 [MATH]m in wide 4 deg.', '1205.2316-2-7-3': '[MATH] fields instantaneously using multiple sounding rocket flights (Zemcov et al. 2012).', '1205.2316-2-7-4': 'The combination of IRAC and CIBER is capable of extending anisotropy measurements out to angular scales of more than a degree from optical to 4.5 [MATH]m.', '1205.2316-2-8-0': 'In this work we establish both the mean intensity and the anisotropy power spectrum of galaxies present during reionization.', '1205.2316-2-8-1': 'We update Cooray et al. (2004) by taking into account recent developments in the study of reionization, and by introducing a halo model to calculate the non-linear clustering of the IR background intensity.', '1205.2316-2-8-2': 'The stellar and nebular emission from first light galaxies follow the calculations presented in Fernandez Komatsu (2006), but we specifically require that the UV photon background produced by the galaxy population present from [MATH] to 30 is consistent with the optical depth to electron scattering as measured by the WMAP polarization data with a value of 0.088 [MATH] (Komatsu et al. 2011).', '1205.2316-2-8-3': 'We account for the current uncertainty in the optical depth by introducing variations to the fiducial model so that the optical depth to electron scattering varies between 0.07 and 0.1.', '1205.2316-2-8-4': 'Our models are also designed to reproduce the bright-end galaxy luminosity functions (LFs) in deep HST/WFC3 surveys at [MATH] involving the Lyman-dropout galaxy samples.', '1205.2316-2-8-5': 'This normalization at the bright-end of galaxy luminosities puts strong constraints on the intensity.', '1205.2316-2-9-0': 'This paper is organized as follows.', '1205.2316-2-9-1': 'In Section 2 we outline our model for the reionization galaxies including stellar nebulae and the IGM emission.', '1205.2316-2-9-2': 'Section 3 presents the calculation related to luminosity mass density of these galaxies.', '1205.2316-2-9-3': 'In Section 4, 5 and 6 we outline the background intensity and spatial anisotropy power spectrum calculations, respectively.', '1205.2316-2-9-4': 'In Section 7 we discuss our results related to the intensity and angular power spectrum and present a comparison to existing measurements.', '1205.2316-2-9-5': 'We conclude with a summary in Section 8.', '1205.2316-2-9-6': 'We assume the flat [MATH]CDM model with [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] throughout the paper.', '1205.2316-2-10-0': '# Emission from stars and the intergalactic medium', '1205.2316-2-11-0': 'We first describe the emission from stars in first-light galaxies present during reionization.', '1205.2316-2-11-1': 'Following Fernandez Komatsu (2006), we consider two stellar populations in this calculation.', '1205.2316-2-11-2': 'The first, referred to as Pop II stars, are metal-poor stars with metallicity [MATH], and the second, Pop III stars, are metal-free stars with [MATH].', '1205.2316-2-12-0': 'To describe the stellar initial mass function (IMF) we make use of two descriptions.', '1205.2316-2-12-1': 'For Pop II stars, we adopt the IMF given by Salpeter (1995) [EQUATION] with mass range from 3 to 150 [MATH].', '1205.2316-2-12-2': 'For Pop III stars we use the IMF obtained by Larson (1999), which takes the form as [EQUATION] where [MATH], and the mass range is from 3 to 500 [MATH].', '1205.2316-2-13-0': 'we utilize the fitting results from [CITATION] and [CITATION] to calculate other stellar parameters, such as the intrinsic bolometric luminosity [MATH], the effective temperature [MATH], the main-sequence lifetime [MATH], and the time-averaged hydrogen photoionization rate [MATH].', '1205.2316-2-13-1': 'The fitting forms of these parameters are different for Pop II and Pop III stars.', '1205.2316-2-13-2': 'For Pop II stars, they are given as [EQUATION] where [MATH] and for Pop III stars, they are [EQUATION]', '1205.2316-2-13-3': 'From these expressions the stellar radius [MATH] is [EQUATION] where [MATH] is the Stefan-Boltzmann constant.', '1205.2316-2-13-4': 'The stellar radius is useful for the calculation related to the stellar emission spectrum (see Section 3.1).', '1205.2316-2-14-0': 'The ionization volume in the nebulae surrounding the stars (Str[MATH]mgren sphere) can be derived if assuming ionization equilibrium where the ionization rate equals recombination rate [EQUATION] where [MATH] is the hydrogen case B recombination coefficient which depends on the gas temperature [MATH] (assumed to be [MATH] K), and we will discuss it in detail in the next Section.', '1205.2316-2-14-1': 'Here, [MATH] and [MATH] are the local number density of electron and HII in the stellar nebulae where we assume [MATH].', '1205.2316-2-15-0': 'For the IGM, the hydrogen density is lower than that of the stellar nebulae, so we no longer assume the ionization equilibrium.', '1205.2316-2-15-1': 'We estimate the ionization volume by a redshift-dependent form as [EQUATION] where [MATH] is the mean hydrogen number density for [MATH] assumed in this work (Shull et al. 2011).', '1205.2316-2-16-0': 'These quantities discussed here would now be used to estimate the luminosity mass density, near-IR background intensity SED, and the anisotropy power spectrum.', '1205.2316-2-16-1': 'We make use of emission from the stellar nebulae and the IGM for both Pop II and Pop III stars in galaxies present during reionization.', '1205.2316-2-17-0': '# Luminosity mass density of the sources', '1205.2316-2-18-0': 'In this Section, we calculate the luminosity per stellar mass at frequency [MATH], i.e. luminosity mass density, for several sources that contribute to the infrared background, such as the direct emission from the stars, Lyman-[MATH] line, and free-free, free-bound and two photon processes.', '1205.2316-2-18-1': 'The luminosity mass density takes the central role in our estimation of the near-IR background intensity spectrum.', '1205.2316-2-19-0': '## Stellar spectrum', '1205.2316-2-20-0': 'For simplicity, we assume the stellar spectrum is a Planckian truncated at [MATH] eV.', '1205.2316-2-20-1': 'Thus, the stellar luminosity at frequency [MATH] can be expressed as [EQUATION] where [MATH] is the surface area of the star, [MATH] is the stellar radius, [MATH] is the stellar mass and [MATH] is the Planck spectrum [EQUATION]', '1205.2316-2-20-2': 'Note that we have ignored the absorption lines of the Lyman-alpha (Ly[MATH]) series here.', '1205.2316-2-20-3': 'The emission with [MATH] eV cannot be approximated by a black-body spectrum, thus we use the fitting formulae for time-averaged photoionization rate, [MATH], for Pop II and Pop III stars to calculate the emission at higher energies.', '1205.2316-2-21-0': '## Lyman alpha emission', '1205.2316-2-22-0': 'The luminosity of Ly[MATH] emission at a frequency [MATH] is [EQUATION] where [MATH] is the frequency of Ly[MATH] photons, and [MATH] is the emission volume that can be estimated by Eq. ([REF]) and Eq. ([REF]) for the stellar nebulae and the IGM respectively.', '1205.2316-2-22-1': 'Here [MATH] is the Ly[MATH] recombination emission rate per [MATH], which is given by [EQUATION] where [MATH] is the electron number density, [MATH] is the HII number density, [MATH] is the hydrogen case B recombination coefficient, and [MATH] is the fraction of the Ly[MATH] photons produced in the case B recombination.', '1205.2316-2-22-2': 'This fraction can be estimated through the fitting formula [EQUATION] where [MATH]K, which is accurate to 0.1% for [MATH].', '1205.2316-2-22-3': 'Note that this fraction is actually not sensitive to the temperature, so it can be treated as a constant [MATH] in the most cases.', '1205.2316-2-22-4': 'The hydrogen case B recombination coefficient [MATH] we use here is from [CITATION], which is fitted by [CITATION] as [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1205.2316-2-22-5': 'We assume a gas temperature of [MATH] K in our calculation to obtain [MATH].', '1205.2316-2-23-0': 'The [MATH] is the collisional emission rate per [MATH] given by [EQUATION] where [MATH] is the neutral hydrogen number density and [MATH] is the effective collisional excitation coefficient.', '1205.2316-2-23-1': 'It has the form [EQUATION]', '1205.2316-2-23-2': 'Here we take into account the excitation up to energy level [MATH] to produce Ly-[MATH] photons.', '1205.2316-2-23-3': 'The higher level emission can be neglected given the high temperature we consider for this calculation.', '1205.2316-2-23-4': 'The excitation collisional rate [MATH], in cm[MATH] per second, can be written as [EQUATION] where [MATH] is the energy difference between lower level [MATH] and higher level [MATH], [MATH] is the statistic weight for level [MATH], and [MATH] is the effective collision strength calculated using the fitting formulae from [CITATION].', '1205.2316-2-24-0': 'The [MATH] is the Ly[MATH] line profile, and we use the result from [CITATION] where they fitted the simulated Ly[MATH] line profile of [CITATION] for a homogeneous and expanding IGM: [EQUATION] where [MATH], and [EQUATION].', '1205.2316-2-24-1': 'Here [MATH], and we assume the flat [MATH]CDM model and take [MATH].', '1205.2316-2-25-0': '## Free-free and free-bound emission', '1205.2316-2-26-0': 'For free-free and free-bound emission we again follow the same approach as Fernandez Komatsu (2006).', '1205.2316-2-26-1': 'Following their derivation, the continuum luminosity of these two processes at frequency [MATH] has the same form with [EQUATION] where [MATH] is the specific emission coefficient for free-free and free-bound emission [EQUATION]', '1205.2316-2-26-2': 'Here [MATH] is the proton number density, [MATH] is the gas temperature, and [MATH] is effective Gaunt factor for free-free and free-bound emission, which takes the form as [EQUATION] where [MATH] is the thermal averaged Gaunt factor of free-free emission and [MATH] is the free-bound emission Gaunt factor for a different energy level [MATH].', '1205.2316-2-26-3': 'These values have an accuracy of 10% .', '1205.2316-2-26-4': 'In above [MATH], where [MATH] is around 10 for the parameter space we are interested in .', '1205.2316-2-26-5': 'The energy level [MATH] is determined by the emission photon frequency [MATH].', '1205.2316-2-26-6': 'If [MATH], and then [MATH] where [MATH] is the Rydberg constant.', '1205.2316-2-26-7': 'Note that the [MATH] here starts at [MATH], since the photons from [MATH] can be easily absorbed by other neutral hydrogen atoms and be ionized instantly.', '1205.2316-2-27-0': '## Two-photon emission', '1205.2316-2-28-0': 'For the two-photon process we also follow the approach of Fernandez Komatsu (2006) and write the luminosity as [EQUATION] where [MATH] is the two-photon emission rate per [MATH], and [MATH].', '1205.2316-2-28-1': 'The [MATH] is the normalized probability of generating one photon in the range [MATH] from per two-photon decay.', '1205.2316-2-28-2': 'We use the fitting formula derived in [CITATION] [EQUATION] where [MATH], which is a good fit to the data given in [CITATION].', '1205.2316-2-29-0': '## Luminosity mass density and total emission', '1205.2316-2-30-0': 'Then following Fernandez Komatsu (2006) and Fernandez et al. (2010), we can derive the mean luminosity mass density for each emission source by integrating over the IMF for the Pop II or Pop III stars [EQUATION] where the ranges of the integral are from 3 to 150 [MATH] with the IMF given by Salpeter (1955) for Pop II stars, and from 3 to 500 [MATH] with the IMF in Larson (1998) for Pop III stars.', '1205.2316-2-30-1': 'Here, the average mass [MATH] is given as [EQUATION] where [MATH] is the normalized IMF with [EQUATION].', '1205.2316-2-31-0': 'Note that this expression is only valid when the main sequence lifetime is larger than the star formation time scale.', '1205.2316-2-31-1': 'Otherwise, it should be evaluated by [EQUATION]', '1205.2316-2-31-2': 'Here [MATH] is the star formation time scale which is given by [EQUATION] where [MATH] is the stellar mass density at [MATH], which is related with the comoving star formation rate density (SFRD) as [MATH].', '1205.2316-2-31-3': 'We use the halo mass function to calculate [MATH] and [MATH] and the details are described in the next Section.', '1205.2316-2-31-4': 'We note that the [MATH] is important for the estimation of the [MATH] as discussed in [CITATION].', '1205.2316-2-32-0': 'Finally we obtain the total luminosity mass density from the stellar nebulae [EQUATION] and the same from the IGM [EQUATION] where [MATH] is the escape fraction of the ionization photons that propagate into the IGM from the nebulae surrounding the stars.', '1205.2316-2-33-0': 'In Fig. [REF], we show the total luminosity mass density as a function of the rest-frame wavelength [MATH] from the stellar nebula and the IGM for Pop II and Pop III stars respectively.', '1205.2316-2-33-1': 'The contributions from the different sources we consider are also shown for the stellar nebula case.', '1205.2316-2-33-2': 'Here we set [MATH] and [MATH] for both Pop II and Pop III cases.', '1205.2316-2-34-0': 'With these parameters we find that the stellar spectrum is dominant for Pop II stars while the "background" spectrum, such as Ly-[MATH] and free-bound, are comparable with or even larger than the stellar spectrum for Pop III stars.', '1205.2316-2-34-1': 'Also, as can be seen, the [MATH] from the IGM is much lower than that from the stellar nebula for both Pop II and Pop III cases, and the total [MATH] from the Pop III stars is similar to that from the Pop II stars.', '1205.2316-2-34-2': 'These results are already discussed in Fernandez Komatsu (2006).', '1205.2316-2-35-0': '# Reionization History and UV Luminosity Density', '1205.2316-2-36-0': 'To test if the reionization history associated with our stellar model is consistent with that of the current observations, such as WMAP 7-year results , we need to calculate the hydrogen reionization fraction [MATH] as a function of redshift.', '1205.2316-2-36-1': 'Following Madau et al. (1998), [MATH] can be estimated as [EQUATION] where [MATH] is the redshift of the beginning of the reionization epoch (we take [MATH]), [MATH] is the comoving star formation rate density (SFRD), and the function [MATH] is defined as [MATH].', '1205.2316-2-36-2': 'Here [MATH] is the average stellar lifetime which is given by [MATH], and [MATH] is the volume averaged recombination time, which can be written as [EQUATION] where [MATH] is the clumping factor of ionized hydrogen.', '1205.2316-2-36-3': 'Here we adopt the simulation result from Trac Cen (2007).', '1205.2316-2-36-4': '[MATH] and [MATH] are the mass fractions of hydrogen and helium, respectively.', '1205.2316-2-36-5': 'Note that we have already considered the escape fraction [MATH], so the [MATH] here is the clumping factor excluding the halos with star formation.', '1205.2316-2-37-0': 'For the comoving SFRD we consider the ongoing star formation model : [EQUATION] where [MATH] is the star formation efficiency which denotes the fraction of baryons converted to stars, [MATH] is the halo mass function , and [MATH] is the threshold mass for a dark matter halo to form a galaxy during reionization.', '1205.2316-2-37-1': 'This minimum mass is taken to be a free parameter and is varied to fit a combination of the WMAP 7-year optical depth and the galaxy LF, as we discuss in Section 6.', '1205.2316-2-38-0': 'We also need a stellar population evolution model to describe [MATH] with the relative fraction of the Pop II and Pop III stars at different redshifts.', '1205.2316-2-38-1': 'In principle there should be a cutoff at some redshift for Pop III stars as they are not expected to form at low redshifts once the gas is polluted by metals.', '1205.2316-2-38-2': 'We assume this cutoff is not lower than [MATH] when the universe is fully ionized.', '1205.2316-2-38-3': 'We assume that the Pop III stars are mainly dominant for [MATH] while Pop II stars are for [MATH].', '1205.2316-2-38-4': 'Then we use the error function to denote the population fraction as [EQUATION] where [MATH] is the population transition width.', '1205.2316-2-38-5': 'Then the term [MATH] in Eq. ([REF]) can be expressed by [EQUATION]', '1205.2316-2-38-6': 'We estimate the optical depth to electron scattering with the reionization fraction [MATH] as [EQUATION] where [MATH] is the Thompson scattering cross-section, and [MATH] is the electron number density of the Universe at redshift [MATH], and we assume the helium is singly ionized for [MATH]) and doubly ionized for [MATH]) .', '1205.2316-2-39-0': 'In Fig. [REF], we plot the hydrogen reionization fraction [MATH] vs. [MATH] for three optical depth [MATH] with three stellar models.', '1205.2316-2-39-1': 'The blue solid line denotes the [MATH] with [MATH] which is close to the result of WMAP 7-year data with [MATH] .', '1205.2316-2-39-2': 'In this case reionization ends around [MATH] which is consistent with the current studies, and we find Pop III stars can ionize the Universe to [MATH] by [MATH] and Pop II stars is responsible for the rest of the reionization over an interval [MATH].', '1205.2316-2-40-0': 'For [MATH] and 0.04, we find [MATH], shown with a dashed line and [MATH], shown with a dotted line, respectively.', '1205.2316-2-40-1': 'These two reionization histories are such that [MATH] and 9, respectively.', '1205.2316-2-40-2': 'For case with [MATH], the Pop III stars ionize [MATH] of the Universe and the Pop II stars are needed over the interval [MATH] to complete reionization.', '1205.2316-2-40-3': 'For the third case with [MATH], Pop III stars ionize [MATH] of the Universe at [MATH] with Pop II stars completing the rest [MATH] in an interval of [MATH]', '1205.2316-2-41-0': 'In Fig. [REF], we also show the dependence of the optical depth on the minimum redshift [MATH] in Eq. ([REF]).', '1205.2316-2-41-1': 'The blue solid, dashed and dotted lines are for the three stellar models with [MATH], 0.077 and 0.099 for [MATH].', '1205.2316-2-41-2': 'The WMAP 7-year result with [MATH] is also shown with a red solid line and a yellow [MATH] error region.', '1205.2316-2-41-3': 'We find the slope of the curves is steeper for [MATH] and flatter for [MATH] which is caused by the Pop III to Pop II transition around [MATH] in our model.', '1205.2316-2-41-4': 'Note that this transition is arbitrarily chosen.', '1205.2316-2-41-5': 'We varied the transition redshift and also cases where PopII and PopIII stars are mixed in with different fractions at different redshifts.', '1205.2316-2-41-6': 'In all these cases we found results that are generally consistent with each other.', '1205.2316-2-41-7': 'Thus, the three choices related to the reionization history that we show here to keep this presentation simple are not biased with respect to the final result related to the IR background intensity that we are trying to estimate in this paper.', '1205.2316-2-42-0': 'We also estimate the total number of ionizing photons per baryon required to maintain the ionized IGM between [MATH] and z, [EQUATION] where [MATH] is the redshift of the end of the reionization, and if we assume the helium is also totally singly ionized at [MATH] we can get [MATH].', '1205.2316-2-42-1': 'In fig. [REF], we show the [MATH] at different [MATH] for three [MATH] cases.', '1205.2316-2-42-2': 'We find the number keeps going up until around [MATH] and becomes constant [MATH] at higher redshift for three cases of the reionization histories.', '1205.2316-2-43-0': '# The near-IR EBL Intensity from Reionization', '1205.2316-2-44-0': 'The mean cosmic infrared background can be estimated by [EQUATION] where [MATH], and we take [MATH] and [MATH] in the calculation.', '1205.2316-2-44-1': 'This redshift range can fully take account of the emission from the Pop III and early Pop II stars, which is redshifted into the near-IR band.', '1205.2316-2-44-2': 'We take this to be the form of [EQUATION] where [MATH] and [MATH] are the comoving specific emission coefficients [EQUATION] where [MATH] is the luminosity mass density at [MATH], [MATH] is the mean stellar lifetime of each of the stellar type and [MATH] is the comoving SFRD given by Eq. ([REF]).', '1205.2316-2-45-0': 'In Fig. [REF], we show the spectrum [MATH] of the near-IR background light intensity from both Pop II and Pop III stars.', '1205.2316-2-45-1': 'We assume that the reionization is ending around [MATH] and integrate up to [MATH] to determine [MATH].', '1205.2316-2-45-2': 'Similar to Fig. [REF], we plot the total spectrum from the stellar nebula and the IGM for both Pop II and Pop III stars.', '1205.2316-2-45-3': 'The contributions from different sources we consider are also shown in colored thin lines.', '1205.2316-2-46-0': 'Here we still take the same value for [MATH] and [MATH] as in Fig. 1 related to the [MATH] calculation.', '1205.2316-2-46-1': 'Similar to the luminosity mass density, the spectrum from stellar nebula is much larger than that from the IGM, and the stellar spectrum is higher for Pop II stars while the "background" spectrum is higher for Pop III stars.', '1205.2316-2-46-2': 'However, different from the luminosity mass density, we now find the spectrum from Pop II stars is larger than that from Pop III stars.', '1205.2316-2-46-3': 'This is basically because the typical lifetime of Pop II stars is longer than that of the Pop III stars.', '1205.2316-2-47-0': '# Angular power spectrum', '1205.2316-2-48-0': 'The angular cross power spectrum of the infrared emission at observed frequencies [MATH] and [MATH] for a multipole [MATH] is [EQUATION] where [MATH] is the comoving angular diameter distance, [MATH] is the scale factor, and [MATH] is the mean emission per comoving volume at frequency [MATH] and redshift [MATH].', '1205.2316-2-48-1': 'If we just take account of the flux lower than a upper cut [MATH], and then [MATH] can be written as [EQUATION]', '1205.2316-2-48-2': 'Here [MATH] is the source flux and [MATH] is the number of sources.', '1205.2316-2-48-3': 'This quantity is just the comoving specific emission coefficient we derive in the last section.', '1205.2316-2-48-4': 'The [MATH] is the galaxy power spectrum at wavenumber [MATH] and redshift [MATH], and we will make use of the model of halo occupation distribution to calculate the [MATH].', '1205.2316-2-49-0': '## First Galaxy Clustering', '1205.2316-2-50-0': 'To calculate the [MATH] we extend the linear theory model of Cooray et al. (2004) and make use of the halo occupation distribution (HOD) for first-light galaxies during reionization.', '1205.2316-2-50-1': 'The galaxy power spectrum can be written as [EQUATION]', '1205.2316-2-50-2': 'Where [MATH] and [MATH] denote the power spectrum contributed by galaxies in a single dark matter halo and galaxies in two different dark matter halos respectively.', '1205.2316-2-50-3': 'Then we can write [EQUATION]', '1205.2316-2-50-4': 'Here [MATH] is the halo mass, [MATH] is the halo mass function, [MATH] is the Fourier transform of the NFW halo density profile , [MATH] when [MATH] and [MATH] otherwise , [MATH] is the halo bias , and [MATH] is the linear matter power spectrum .', '1205.2316-2-50-5': 'The [MATH] is the mean number density of galaxies, which is given by [EQUATION]', '1205.2316-2-50-6': 'The [MATH] is the mean number of galaxies in a halo with mass [MATH], which is the sum of number of central galaxies and satellite galaxies [EQUATION] where we define [EQUATION] and [EQUATION]', '1205.2316-2-50-7': 'In this definition, the [MATH] denotes the mass of a halo that has 50% probability to host a central galaxy, and [MATH] is the transition width.', '1205.2316-2-50-8': 'For the satellite galaxies, [MATH] is the truncation mass for satellites, [MATH] is the normalization mass and [MATH] denotes the slope of the power-law relation about the halo mass [MATH].', '1205.2316-2-50-9': 'We assume [MATH] is always larger than [MATH] since there should not be satellites without central galaxy, and assume [MATH].', '1205.2316-2-50-10': 'We take [MATH], [MATH], and [MATH] in this paper.', '1205.2316-2-50-11': 'If assuming a Poisson distribution for satellite galaxies, we can get [EQUATION]', '1205.2316-2-50-12': 'This expression could take account of the case [MATH] and is consistent with our definitions for the [MATH] and [MATH].', '1205.2316-2-51-0': '## Poisson Fluctuations', '1205.2316-2-52-0': 'The clustering measurements are affected by the Poisson fluctuations associated with the shot-noise caused by the discrete and finite number of galaxies from which clustering is measured.', '1205.2316-2-52-1': 'Assuming a Poisson distribution the [MATH]-independent shot-noise power spectrum is [EQUATION]', '1205.2316-2-52-2': 'To estimate [MATH], we first define the luminosity mass density for the mass of the dark matter halos at frequency [MATH] [EQUATION] where [MATH] is the luminosity mass density for the stellar mass discussed in Section 3.', '1205.2316-2-52-3': 'We derive the 3-D shot-noise power spectrum by assuming [MATH] is proportional to the halo mass [MATH], i.e. [MATH] is independent on [MATH] [EQUATION]', '1205.2316-2-52-4': 'Then the 2-D shot-noise power spectrum can be written as [EQUATION]', '1205.2316-2-53-0': '## Band-Averaged Intensity Power Spectrum', '1205.2316-2-54-0': 'For a specific near-IR observation with a band frequency from [MATH] to [MATH], we can define a band-averaged luminosity mass density as [EQUATION]', '1205.2316-2-54-1': 'Then we can derive the band-averaged comoving specific emission coefficient [MATH] using Eq. ([REF]) and the 3-D shot-noise power spectrum [MATH] using Eq. ([REF]) respectively.', '1205.2316-2-55-0': 'Finally we find the band-averaged angular cross power spectrum and shot-noise power spectrum are [EQUATION] and [EQUATION] respectively.', '1205.2316-2-55-1': 'Note that here we have a factor [MATH] instead of [MATH] in [MATH] and [MATH] and this dependence has been explained in the Appendix of [CITATION].', '1205.2316-2-56-0': '# Results and Discussion', '1205.2316-2-57-0': 'In this Section, we first estimate the infrared background intensity and then discuss the angular power spectrum as derived previously.', '1205.2316-2-57-1': 'We also compare our estimation with the observational data and discuss the dependence of the result on the parameters in the model.', '1205.2316-2-58-0': '## Near-IR EBL from Reionization', '1205.2316-2-59-0': 'In Fig. [REF], we show the spectrum of the near-IR intensity vs. the observed frequency for three cases of the reionization history with [MATH], 0.090 and 0.099 corresponding to [MATH], 0.03 and 0.04.', '1205.2316-2-59-1': 'Here the spectrum is the total spectrum of the sum of that from both stellar nebula and the IGM, which are calculated by putting Eq. ([REF]) into Eq. ([REF]).', '1205.2316-2-59-2': 'Also, we set [MATH] for both Pop II and Pop III stars.', '1205.2316-2-59-3': 'The right panel of Fig. [REF] shows three cases with [MATH] varied at fixed [MATH].', '1205.2316-2-59-4': 'In both panels, for comparison, we plot the observational data in terms of the excess of EBL relative to the integrated galaxy light (IGL) of known galaxy populations at low redshifts (Madau Pozzetti 2000).', '1205.2316-2-59-5': 'The excess EBL data plotted here are the same as those shown in Santos et al. (2002) and involves measurements mainly from DIRBE with various models for zodiacal light and Galactic stellar contribution.', '1205.2316-2-60-0': 'We find the emission from the Pop II stars dominates the spectrum for all of the three cases.', '1205.2316-2-60-1': 'As we have just discussed in the last Section, this is because the lifetime of the Pop II is longer than the Pop III stars.', '1205.2316-2-60-2': 'In the spectrum, the shorter and longer wavelength parts are mainly contributed by the "background" spectrum from Pop III stars while the medium part by Pop II stellar spectrum, so if just the Pop III stars get longer lifetime, only the "background" spectrum can be effectively reinforced in the total spectrum.', '1205.2316-2-61-0': 'In any case, regardless of assumptions related to the stellar type, we find that the EBL intensity from reionization is no more than 0.3 nW m[MATH] sr[MATH].', '1205.2316-2-61-1': 'Such an intensity is significantly smaller than the previous predictions that attempted to explain almost all or a significant fraction of the excess EBL seen in DIRBE data relative to IGL estimates.', '1205.2316-2-61-2': 'An intensity larger than about 2.5 nW m[MATH] sr[MATH] in the J-band could be in conflict with metal production considerations and the X-ray background (Madau Silk 2005), though they do not necessarily require high efficiency factors to generate the required star formation (Fernandez Komatsu 2006).', '1205.2316-2-61-3': 'The difference between our calculation and the previous ones is that we primarily require the reionization model to generate a reionization history consistent with the WMAP optical depth.', '1205.2316-2-61-4': 'This limits the number of H-ionizing photons per baryon during reionization to be less than 3.', '1205.2316-2-61-5': 'Previous estimates ignored such a constraint and either focused on explaining all of the DIRBE excess (Santos et al. 2002; Salvaterra Ferrara 2003) or amplitude of the measured near-IR background anisotropies (e.g., Fernandez et al. 2010).', '1205.2316-2-62-0': '## Bright-end Galaxy Luminosity Functions', '1205.2316-2-63-0': 'In order to relate our galaxy population responsible for reionization to the observations, we also compare our model to observations of [MATH] galaxies, focusing on the UV LFs.', '1205.2316-2-63-1': 'To obtain the UV LF, instead of the occupation number which is luminosity independent, we make use of the conditional luminosity function (CLF) approach .', '1205.2316-2-63-2': 'To compute the CLF we map galaxy rest-frame UV luminosity to halo mass with some scatter added similar to the case of low-redshift galaxy populations (Cooray 2005).', '1205.2316-2-63-3': "The total luminosity of a halo is taken to be [EQUATION] and we assume that this total luminosity can be ascribed to the central galaxy when [MATH], following the' earlier description related to the occupation number.", '1205.2316-2-63-4': 'When [MATH] we introduce satellites with central galaxy luminosity kept fixed at [MATH].', '1205.2316-2-63-5': 'However, when comparing to the existing measurements we found that all of the rest-UV LF measurements are in the range where central galaxies dominate the LF and thus our comparison to the measured LFs is independent of assumptions related to the exact form of the satellite occupation number or conditional luminosity function.', '1205.2316-2-64-0': 'To compare with existing rest-UV LF measurements, we convert the luminosity of each galaxy to the AB absolute magnitude via the relation [MATH] .', '1205.2316-2-64-1': 'In the left panel of Fig. [REF], we show the SFRD as a function of redshift derived from Eq. ([REF]) for three reionization histories with [MATH], 0.090 and 0.099 which are obtained by setting [MATH], 0.03 and 0.04, respectively.', '1205.2316-2-64-2': 'The red points are the data from HUDF09+ERS+CANDELS observations .', '1205.2316-2-64-3': 'We find the SFRD of the three cases are higher than the existing measurements, especially at the high redshifts.', '1205.2316-2-64-4': 'This difference is mainly due to the fact that existing SFRD estimates are limited to galaxies with [MATH], while the bulk of the reionization UV density budget is contained in the galaxies at the faint-end of the LF.', '1205.2316-2-64-5': 'This is especially the case at high redshift since the faint-end slope of the LF is steep with values reaching close to -2 already.', '1205.2316-2-65-0': 'In the right panel of Fig. [REF], we show our rest-UV LF (corresponding to [MATH]) at [MATH]6, 7 and 8.', '1205.2316-2-65-1': 'The central thick solid curves are the luminosity function derived from our default model with [MATH] and [MATH], and the thin dotted lines are obtained by [MATH] (lower) and 0.04 (upper) which could match the [MATH] errors of the data given in .', '1205.2316-2-65-2': 'The three values of the faint-end slope [MATH], -2.0 and -1.5 are shown, which indicates the slope of our model is between -2.5 and -2.0.', '1205.2316-2-65-3': 'Also, we find the star formation time scale [MATH] is around [MATH] yrs at [MATH], 7 and 8 when we calculate the luminosity mass function here.', '1205.2316-2-66-0': 'We also explore the dependence of the galaxy bias on the [MATH] in Fig. [REF].', '1205.2316-2-66-1': 'We can define a effective galaxy bias here from the HOD model as [EQUATION]', '1205.2316-2-66-2': 'In the plot we show the [MATH] as a function of [MATH] at [MATH]6, 7 and 8 for the [MATH].', '1205.2316-2-66-3': 'We find that the galaxy bias increases as the redshift increases and decreases as the [MATH] increases.', '1205.2316-2-66-4': 'The reason is obvious that the galaxy number density [MATH] defined by Eq. ([REF]) becomes smaller at higher redshift and bigger at larger [MATH] (the larger [MATH] means smaller halo mass [MATH]).', '1205.2316-2-67-0': '## Anisotropy Power Spectrum', '1205.2316-2-68-0': 'In Fig. [REF] we show the near-IR background anisotropy angular power spectrum at [MATH].', '1205.2316-2-68-1': 'The clustering power spectra with non-linear power spectrum from the HOD model are in solid lines.', '1205.2316-2-68-2': 'The left panel shows the case with [MATH], the minimum mass to host a galaxy, is changed from [MATH] to [MATH] M[MATH].', '1205.2316-2-68-3': 'For comparison, we also plot the linear power spectrum which has a turnover around [MATH].', '1205.2316-2-68-4': 'When calculating the clustering power spectrum, we set the parameters of the HOD with [MATH] values as listed in the figure with [MATH].', '1205.2316-2-68-5': 'The corresponding values on the optical depth to electron scattering are also listed in the figure.', '1205.2316-2-68-6': 'We note that the shot-noise amplitude is larger for the case with [MATH] M[MATH] in comparison to the case with, say, [MATH] M[MATH], though in those two cases the clustering amplitude is higher with [MATH] M[MATH].', '1205.2316-2-68-7': 'This is because the shot-noise amplitude is sensitive to the second flux-moment of the number counts.', '1205.2316-2-68-8': 'By keeping the minimum mass higher we force the overall counts to be restricted to brighter sources than the case with a lower minimum halo mass.', '1205.2316-2-68-9': 'On the other hand the clustering power spectrum reflects the total background intensity.', '1205.2316-2-68-10': 'With the minimum mass lowered, both the overall number density of galaxies and the background intensity are increased.', '1205.2316-2-69-0': 'The right panel of Fig. [REF] shows the case where we vary [MATH] to highlight the fact [MATH] amplitude is inversely proportional to [MATH].', '1205.2316-2-69-1': 'However, one cannot arbitrarily reduce [MATH] to a small value since this results in a low optical depth to electron scattering.', '1205.2316-2-69-2': 'If [MATH] to 10[MATH], such that one does not need to be concerned of effects due to feedback negatively impacting the formation of galaxies in lower mass halos, then we find that [MATH] must be at the level of 0.3 or more, under the assumptions related to the IMFs for PopII and PopIII stars we use in this paper.', '1205.2316-2-70-0': 'In Fig. [REF], we plot the [MATH] at different redshift bins.', '1205.2316-2-70-1': 'As can be seen, the main contribution of the [MATH] comes from the lowest redshift range of 6 to 10.', '1205.2316-2-70-2': 'For all practical purposes one can assume that the near-IR background is probing the end of reionization not the first objects to form in the universe at the beginning of reionization.', '1205.2316-2-71-0': 'To compare with the total near-IR intensity spectrum, we plot the ratio of the square root of [MATH] to the [MATH] in Fig. [REF].', '1205.2316-2-71-1': 'We integrate from [MATH] to 30 with the same HOD model described above.', '1205.2316-2-71-2': 'For easy comparison, we restrict the predictions to be with [MATH] M[MATH] such that [MATH] to remain consistent with WMAP 7-year result.', '1205.2316-2-71-3': 'We find the ratio of [MATH] to [MATH] is about [MATH] for [MATH] and the clustering fluctuations amplitude is below 100% of the intensity at all angular scales.', '1205.2316-2-71-4': 'This shows that large fluctuations of the background intensity is not expected and the background behaves in a manner that is smooth and not clumpy as in the case if spatial variations are dominated by rare, bright galaxies.', '1205.2316-2-72-0': 'In Fig. [REF], we show the near-IR background intensity power spectrum (solid curves), shot-noise power spectrum (dotted curves) with a comparison to existing observational measurements at several wavelengths: [MATH], 2.4, 3.6, and 4.5 [MATH] .', '1205.2316-2-72-1': 'Note that for the data from Kashlinsky et al. (2012) at 3.6 and 4.5 [MATH], we removed the IRAC beam transfer function [MATH] so the comparison to theoretical predictions is easier.', '1205.2316-2-72-2': 'An accounting of the beam is essential since realistic instruments do not have perfect resolution causing a loss of power on the smallest scales.', '1205.2316-2-72-3': 'This effect can be explicitly measured from the point spread function.', '1205.2316-2-72-4': 'We calculate the beam transfer function [MATH] by taking the power spectrum of the measured point spread function for the Spitzer/IRAC instrument, which is publicly available and compute [MATH] (for more information see Smidt et al. 2012).', '1205.2316-2-73-0': 'We also show the results from the simulation in [CITATION] for comparison.', '1205.2316-2-73-1': 'The red solid and magenta dashed lines are for the simulation with mass resolution [MATH] and [MATH], respectively.', '1205.2316-2-73-2': 'These two cases are what is described in their work as not having a suppression of the small mass halos (ie star-formation present in halos with mass between 10[MATH] and 10[MATH] M[MATH], where one expects photo-ionization heating to suppress star-formation).', '1205.2316-2-73-3': 'The reionization histories of these two cases involve PopII stars with [MATH].', '1205.2316-2-74-0': 'For more clarity, we show the square root of the [MATH] at [MATH] and [MATH] together with the data as a function of the wavelength in Fig. [REF].', '1205.2316-2-74-1': 'The data points are from the data sets shown in Fig. [REF] which are around [MATH] or [MATH].', '1205.2316-2-74-2': 'We find the curve of the square root of the [MATH] has similar shape as [MATH], but the amplitude of rms fluctuations in our models is lower than the existing measurements.', '1205.2316-2-74-3': 'We capture the uncertainties related to [MATH], [MATH], and [MATH] by considering a low and high range for our prediction related to [MATH] with [MATH] falling within the 1 [MATH] uncertainty range of the WMAP 7-year result when these parameters are varied.', '1205.2316-2-74-4': 'Even with parameter uncertainties accounted for, we find that the existing measurements are at least an order of magnitude larger than our model predictions.', '1205.2316-2-75-0': 'We attempted additional model variations but failed to find a scenario where [MATH] is consistent with WMAP 7-year result and the existing LFs leading to a model consistent with existing near-IR background.', '1205.2316-2-75-1': 'One can, in principle, model fit the near-IR fluctuation power spectra by increasing the photon output of first galaxies.', '1205.2316-2-75-2': 'This results in an optical depth that is higher than the WMAP value and a LF that has brighter galaxies than observed in existing deep HST/WFC3 images.', '1205.2316-2-75-3': 'The existing measurements require a background intensity that is at least 3 nW m[MATH] sr[MATH] at 1.6 [MATH]m so that the model predictions become consistent with measurements within the 1[MATH] uncertainties of the measurements.', '1205.2316-2-75-4': 'Since 0.3 nW m[MATH] sr[MATH] at 1.6 [MATH]m is associated with 2.5 H-ionizing photons per baryon during reionization, if reionization is to explain the background fluctuations then we are dealing with a situation where [MATH] 25 H-ionizing photons per baryon are present.', '1205.2316-2-75-5': 'A possibility is to introduce a spectrum for the emission that has a rest-frame cut-off in UV at wavelengths shortward of the Lyman-limit that is not associated with reionization but by the emission mechanism itself.', '1205.2316-2-75-6': 'Unfortunately we have not been able to come up with such an emission spectrum.', '1205.2316-2-76-0': 'As discussed in Helgason et al. (2012) faint, low-redshift galaxies do not have a clustering shape consistent with the anisotropy power spectrum measurements.', '1205.2316-2-76-1': 'The existing measurements show a clear excess in clustering at 30 arcsec to few tens arcminute angular scales that differ from faint galaxy clustering power spectrum.', '1205.2316-2-76-2': 'This difference is statistically significant and one can only explain at most 20% of the near-IR fluctuations to be associated with faint galaxies, reducing the previous estimate in Cooray et al. (2007; Chary et al. 2008) that suggested a contribution at the 50% level.', '1205.2316-2-76-3': 'We are now left with an unexplained set of measurements since neither the low redshift faint galaxies nor the high redshift reionization galaxies can explain them.', '1205.2316-2-76-4': 'One possibility is that the existing fluctuation excess in the data has a non-astrophysical origin, perhaps either involving systematics in the data or fluctuations in the Zodiacal light.', '1205.2316-2-76-5': 'While arguments have been made that these effects are insignificant with a set of well-coordinated multi-wavelength measurements we plan to further address the origin of near-IR fluctuations in our upcoming papers.', '1205.2316-2-77-0': '# Summary', '1205.2316-2-78-0': 'The UV emission from stars which are formed in the early Universe from [MATH] to 30 can contribute to the near-IR background light.', '1205.2316-2-78-1': 'By measuring the intensity and anisotropies of the near-IR background, we can investigate the properties of these early stars and the epoch of reionization.', '1205.2316-2-78-2': 'In this work we discuss several sources that contribute to the near-IR background intensity, including the emission of stars, Ly[MATH], free-free, free-bound and two-photon.', '1205.2316-2-78-3': 'We first estimate the frequency spectrum separately for Pop II and Pop III star with the redshift range [MATH].', '1205.2316-2-78-4': 'By using the initial stellar mass spectrum and the fitting model of time-averaged hydrogen photoionization rate we calculate the luminosity mass density.', '1205.2316-2-78-5': 'We find that although the luminosity mass density of the Pop III stars is a bit larger than that of Pop II stars, the near-IR intensity spectrum from the Pop II stars is stronger than that from Pop III stars, which is caused by the longer lifetime of Pop II stars.', '1205.2316-2-79-0': 'In order to check the consistence of our stellar model and the reionization history, we derive the hydrogen reionization fraction at different redshifts and calculate the optical depth by assuming the Pop II stars are dominant for [MATH] while Pop III stars for [MATH].', '1205.2316-2-79-1': 'We find that if we set the star formation efficiency [MATH], the Universe would be totally reionized around [MATH] with the optical depth [MATH], which is well consistent with the result from the WMAP 7-year data.', '1205.2316-2-79-2': 'Also, we explore the other possible models with [MATH] and [MATH] and get [MATH] and 0.099, respectively.', '1205.2316-2-79-3': 'The total number of the ionizing photon per baryon required to maintain the ionized IGM, [MATH], is also evaluated for the three cases, and we find the [MATH] becomes a constant with a value around 2.5 after [MATH].', '1205.2316-2-80-0': 'To compare with existing bright-end luminosity function measurements, we evaluate the UV luminosity function from our model at [MATH] for [MATH]6, 7 and 8 from [MATH] to [MATH].', '1205.2316-2-80-1': 'We compare to the measurements of Bouwens et al. (2012).', '1205.2316-2-80-2': 'We find our derived luminosity function is consistent with the data with the slope of the faint-end [MATH].', '1205.2316-2-80-3': 'This is a steep slope and existing measurements do suggest that the slope is steeper than for LFs at low redshifts.', '1205.2316-2-80-4': 'We then define an effective galaxy bias [MATH] from the HOD model, and find the [MATH] becomes bigger at high redshift and bright luminosities because of the lower number density of the galaxies.', '1205.2316-2-81-0': 'Finally, we calculate the angular power spectrum of the near-IR background by making use of a halo model.', '1205.2316-2-81-1': 'The non-linearities provided by the 1-halo term increase the clustering strength at multipoles greater than about [MATH].', '1205.2316-2-81-2': 'The suggest turn-over at this scale with the linear power spectrum alone in Cooray et al. (2004) is no longer present.', '1205.2316-2-81-3': 'We find the shot-noise power spectrum of the Pop III stars is greater than that of Pop II stars because of the larger luminosity mass density for Pop III stars.', '1205.2316-2-81-4': 'By making use of the stellar population evolution model, we calculate the near-IR anisotropy power spectra [MATH] at different wavelengths and compare to the observational data.', '1205.2316-2-81-5': 'We find our results are lower than the observational data by at least an order of magnitude.', '1205.2316-2-81-6': 'There are now strong arguments that the near-IR anisotropies cannot originate from low redshift faint galaxies (Helgason et al. 2012).', '1205.2316-2-81-7': 'We have failed to explain the alternative origin of near-IR background anisotropies involving galaxies present during reionization, contrary to suggestions in the literature (Kashlinsky et al. 2012).', '1205.2316-2-81-8': 'In future works, using additional measurements with Spitzer and Hubble/WFC3, we plan to further discuss the near-IR fluctuations and to explain if the origin is astrophysical or whether it is associated with yet-unknown systematic effect in the data.', '1205.2316-2-82-0': 'This work was supported by NSF CAREER AST-0645427 and NASA NNX10AD42G at UCI.', '1205.2316-2-82-1': 'We thank CIBER, SDWFS, and CANDELS teams for helpful discussions and questions that motivated this paper.'}

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'1205.2316-3-7-1'], ['1205.2316-2-7-2', '1205.2316-3-7-2'], ['1205.2316-2-7-3', '1205.2316-3-7-3'], ['1205.2316-2-7-4', '1205.2316-3-7-4'], ['1205.2316-2-4-0', '1205.2316-3-4-0'], ['1205.2316-2-4-1', '1205.2316-3-4-1'], ['1205.2316-2-4-2', '1205.2316-3-4-2'], ['1205.2316-2-4-3', '1205.2316-3-4-3'], ['1205.2316-2-4-4', '1205.2316-3-4-4'], ['1205.2316-2-34-0', '1205.2316-3-34-0'], ['1205.2316-2-34-1', '1205.2316-3-34-1'], ['1205.2316-2-34-2', '1205.2316-3-34-2'], ['1205.2316-2-13-0', '1205.2316-3-13-0'], ['1205.2316-2-13-1', '1205.2316-3-13-1'], ['1205.2316-2-13-2', '1205.2316-3-13-2'], ['1205.2316-2-13-3', '1205.2316-3-13-3'], ['1205.2316-2-13-4', '1205.2316-3-13-4'], ['1205.2316-2-14-0', '1205.2316-3-14-0'], ['1205.2316-2-14-1', '1205.2316-3-14-1'], ['1205.2316-2-41-0', '1205.2316-3-42-0'], ['1205.2316-2-41-1', '1205.2316-3-42-1'], ['1205.2316-2-41-2', '1205.2316-3-42-2'], ['1205.2316-2-41-3', '1205.2316-3-42-3'], ['1205.2316-2-41-4', '1205.2316-3-42-4'], ['1205.2316-2-41-5', 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['1205.2316-2-80-4', '1205.2316-3-82-4'], ['1205.2316-2-30-0', '1205.2316-3-30-0'], ['1205.2316-2-30-1', '1205.2316-3-30-1'], ['1205.2316-2-82-0', '1205.2316-3-84-0'], ['1205.2316-2-82-1', '1205.2316-3-84-1'], ['1205.2316-2-64-0', '1205.2316-3-66-0'], ['1205.2316-2-64-1', '1205.2316-3-66-1'], ['1205.2316-2-64-2', '1205.2316-3-66-2'], ['1205.2316-2-64-3', '1205.2316-3-66-3'], ['1205.2316-2-64-4', '1205.2316-3-66-4'], ['1205.2316-2-64-5', '1205.2316-3-66-5'], ['1205.2316-2-48-0', '1205.2316-3-49-0'], ['1205.2316-2-48-1', '1205.2316-3-49-1'], ['1205.2316-2-48-2', '1205.2316-3-49-2'], ['1205.2316-2-48-3', '1205.2316-3-49-3'], ['1205.2316-2-48-4', '1205.2316-3-49-4'], ['1205.2316-2-23-0', '1205.2316-3-23-0'], ['1205.2316-2-23-1', '1205.2316-3-23-1'], ['1205.2316-2-23-2', '1205.2316-3-23-2'], ['1205.2316-2-23-3', '1205.2316-3-23-3'], ['1205.2316-2-23-4', '1205.2316-3-23-4'], ['1205.2316-2-78-0', '1205.2316-3-80-0'], ['1205.2316-2-78-1', '1205.2316-3-80-1'], ['1205.2316-2-78-2', '1205.2316-3-80-2'], ['1205.2316-2-78-3', '1205.2316-3-80-3'], ['1205.2316-2-78-4', '1205.2316-3-80-4'], ['1205.2316-2-78-5', '1205.2316-3-80-5'], ['1205.2316-2-20-0', '1205.2316-3-20-0'], ['1205.2316-2-20-1', '1205.2316-3-20-1'], ['1205.2316-2-75-0', '1205.2316-3-77-0'], ['1205.2316-2-75-1', '1205.2316-3-77-1'], ['1205.2316-2-75-2', '1205.2316-3-77-2'], ['1205.2316-2-75-3', '1205.2316-3-77-3'], ['1205.2316-2-75-4', '1205.2316-3-77-4'], ['1205.2316-2-75-5', '1205.2316-3-77-5'], ['1205.2316-2-75-6', '1205.2316-3-77-6'], ['1205.2316-2-59-0', '1205.2316-3-60-0'], ['1205.2316-2-59-1', '1205.2316-3-60-1'], ['1205.2316-2-59-2', '1205.2316-3-60-2'], ['1205.2316-2-59-3', '1205.2316-3-60-3'], ['1205.2316-2-59-4', '1205.2316-3-60-4'], ['1205.2316-2-59-5', '1205.2316-3-60-5'], ['1205.2316-2-38-0', '1205.2316-3-38-0'], ['1205.2316-2-38-1', '1205.2316-3-38-1'], ['1205.2316-2-38-2', '1205.2316-3-38-2'], ['1205.2316-2-38-5', '1205.2316-3-38-4'], ['1205.2316-2-38-6', '1205.2316-3-39-0']]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1205.2316

{'1205.2316-3-0-0': 'A fraction of the extragalactic near-infrared (near-IR) background light involves redshifted photons from the ultraviolet (UV) emission from galaxies present during reionization at redshifts above 6.', '1205.2316-3-0-1': 'The absolute intensity and the anisotropies of the near-IR background provide an observational probe of the first-light galaxies and their spatial distribution.', '1205.2316-3-0-2': 'We estimate the extragalactic background light intensity during reionization by accounting for the stellar and nebular emission from first-light galaxies.', '1205.2316-3-0-3': 'We require the UV photon density from these galaxies to generate a reionization history that is consistent with the optical depth to electron scattering from cosmic microwave background measurements.', '1205.2316-3-0-4': 'We also require the bright-end luminosity function of galaxies in our models to reproduce the measured Lyman drop-out luminosity functions at redshifts of 6 to 8.', '1205.2316-3-0-5': 'The absolute intensity is about 0.1 to 0.4 nW m[MATH] sr[MATH] at the peak of its spectrum at [MATH] 1.1 [MATH]m.', '1205.2316-3-0-6': 'We also discuss the anisotropy power spectrum of the near-IR background using a halo model to describe the galaxy distribution.', '1205.2316-3-0-7': 'We compare our predictions for the anisotropy power spectrum to existing measurements from deep near-IR imaging data from Spitzer/IRAC, Hubble/NICMOS, and AKARI.', '1205.2316-3-0-8': 'The predicted rms fluctuations at tens of arcminute angular scales are roughly an order of magnitude smaller than the existing measurements.', '1205.2316-3-0-9': 'While strong arguments have been made that the measured fluctuations do not have an origin involving faint low-redshift galaxies, we find that measurements in the literature are also incompatible with galaxies present during the era of reionization.', '1205.2316-3-0-10': 'The measured near-IR background anisotropies remain unexplained with an unknown origin.', '1205.2316-3-1-0': '# Introduction', '1205.2316-3-2-0': 'The optical and UV radiation from sources present during reionization is expected to leave a signature in the extragalactic background light (EBL) at near-IR wavelengths (e.g., Santos et al. 2002; Salvaterra Ferrara 2003; Cooray Yoshida 2004; Fernandez Komatsu 2006; Raue 2009).', '1205.2316-3-2-1': 'Such radiation is not expected to be present in the background light at UV and optical wavelengths due to the redshifted Lyman limit.', '1205.2316-3-2-2': 'The exact intensity from first-light galaxies present during reionization is currently unknown.', '1205.2316-3-2-3': 'The first predictions suggested an intensity as high as 10 to 30 nW m[MATH] sr[MATH] (Santos et al. 2002; Salvaterra Ferrara 2003).', '1205.2316-3-2-4': 'These estimates were partly motivated by the need to explain the difference between DIRBE EBL measurements (e.g., Cambresy et al. 2001) and the integrated galaxy light (IGL) from deep galaxy counts (Madau Pozzetti 2000; Totani et al. 2001).', '1205.2316-3-3-0': 'These predictions with high backgrounds were questioned by Madau Silk (2005) based on existing limits related to metal content at high redshifts and the X-ray background produced by stellar end-products such as black holes.', '1205.2316-3-3-1': 'They suggest an intensity less than about 2.5 nW m[MATH] sr[MATH] in the J-band from a galaxy population made up of population III stars during reionization (Madau Silk 2005).', '1205.2316-3-3-2': 'With a combination of Population II stars and changes to the lifetime of stars, Fernandez Komatsu (2006) argued that the background could be as high as 4 to 8 nW m[MATH] sr[MATH].', '1205.2316-3-3-3': 'Even in such a scenario a simple estimate of the UV photon density at [MATH] shows that there are roughly an order of magnitude higher number of H-ionizing photons per baryon during reionization than necessary to explain the reionization history.', '1205.2316-3-3-4': 'Since one does not expect more than a few H-ionizing photons per baryon during reionization, a first-order estimate suggests that the background intensity cannot be larger than a few tenths nW m[MATH] sr[MATH] between 1 and 2 [MATH]m.', '1205.2316-3-4-0': 'Unfortunately a direct search for the integrated intensity of galaxies present during reionization based on absolute background measurements has been problematic due to the confusion with the Zodiacal foreground.', '1205.2316-3-4-1': 'At 1 AU Zodiacal light is two to three orders of magnitude brighter than the [MATH] 10 nW m[MATH] sr[MATH] intensity produced by extragalactic sources.', '1205.2316-3-4-2': 'While challenging, techniques have been devised to estimate the Zodiacal dust column density based on the line strengths of Fraunhofer lines seen in the dust-scattered Solar spectrum (e.g., Bernstein Dyson 2003).', '1205.2316-3-4-3': 'Instead of the absolute background, in Cooray et al. (2004; also Kashlinsky et al. 2004), it was proposed that the galaxies present during reionization can be studied with anisotropies of the near-IR background.', '1205.2316-3-4-4': 'The anisotropy studies have the potential to probe deeper than the absolute experiments and could study a galaxy population present during reionization that leads to an intensity well below 0.1 nW m[MATH] sr[MATH] (Cooray et al. 2004).', '1205.2316-3-5-0': 'This suggestion has motivated experimental measurements on the near-IR anisotropy power spectrum with data from Spitzer/IRAC, HST/NICMOS, and AKARI.', '1205.2316-3-5-1': 'After a deep removal of point sources, Kashlinsky et al. (2005, 2007, 2012) claimed a detection of first-light galaxy fluctuations at [MATH].', '1205.2316-3-5-2': 'The detected signal is an excess of clustering power above shot-noise on the largest angular scales.', '1205.2316-3-5-3': 'A similar suggestion was also made by an AKARI group (Matsumoto et al. 2011), but an analysis of the HST/NICMOS Ultra Deep Field led to an opposite conclusion that the sources contributing to the near-IR excess fluctuations are at [MATH] (Thompson et al. 2007).', '1205.2316-3-5-4': 'Due to the limited areas of existing deep surveys near-IR background anisotropy measurements are limited to angular scales less than about one degree.', '1205.2316-3-5-5': 'The limited field of view is especially a problem for existing NICMOS UDF measurements (Thompson et al. 2007), where the fluctuations are limited to angular scales less than 5[MATH].', '1205.2316-3-5-6': 'Separately, a joint analysis of IRAC and HST/ACS data in the same GOODS fields as studied by Kashlinsky et al. (2007) led to the suggestion that up to 50 of the excess fluctuations at 3.6 [MATH]m could come from faint dwarf galaxies at [MATH] (Cooray et al. 2007; Chary et al. 2008).', '1205.2316-3-5-7': 'Through detailed models combined with more recent measurements of faint galaxy clustering, Helgason et al. (2012) has lowered this low-redshift contribution to 3.6 [MATH]m intensity fluctuations to be at most 20.', '1205.2316-3-5-8': 'The rest of the anisotropies continue to be interpreted as originating from first-light galaxies during reionization (Kashlinsky et al. 2012).', '1205.2316-3-6-0': 'While there are still uncertainties on the exact intensity and the amplitude of intensity fluctuations in experimental measurements, the situation is no different on the theory side.', '1205.2316-3-6-1': 'The first estimates on the anisotropy power spectrum made use of linear theory clustering (Cooray et al. 2004).', '1205.2316-3-6-2': 'Fernandez et al. (2010) used numerical simulations of reionization to predict the expected power spectrum during reionization.', '1205.2316-3-6-3': 'Their power spectrum has a shape in the form of a power-law with [MATH] between ten arcminute to arcsecond angular scales.', '1205.2316-3-6-4': 'Fernandez et al. (2010) suggested that the power-law behavior arises from significant non-linear biasing of dark matter halos at high redshifts.', '1205.2316-3-6-5': 'Due to the limited box sizes of existing reionization simulations of the order 100 to 140 Mpc on the side, numerical studies are limited to angular scales of 30 arcminutes and below at [MATH].', '1205.2316-3-7-0': 'With the availability of WFC3 on the Hubble Space Telescope, dedicated IR background experiments (e.g., CIBER sounding-rocket experiment; Bock et al. 2006), and plans for a future space-based absolute intensity measurement (ZEBRA; Cooray et al. 2009) there is now a clear need to revisit theoretical predictions on both the absolute intensity and the anisotropy power spectrum from galaxies present during reionization.', '1205.2316-3-7-1': 'While current multi-band Spitzer and AKARI measurements do not overlap in the same fields, the combination of IRAC and WFC3 data on some of the same well-studied fields on the sky (e.g., fields covered by the CANDELS survey; Grogin et al. 2011; Koekemoer et al. 2011) will soon allow the spectral energy distribution (SED) of intensity fluctuations be studied uniformly.', '1205.2316-3-7-2': 'Separately CIBER is conducting spectral imaging absolute measurements between 0.8 and 1.6 [MATH]m in wide 4 deg.', '1205.2316-3-7-3': '[MATH] fields instantaneously using multiple sounding rocket flights (Zemcov et al. 2012).', '1205.2316-3-7-4': 'The combination of IRAC and CIBER is capable of extending anisotropy measurements out to angular scales of more than a degree from optical to 4.5 [MATH]m.', '1205.2316-3-8-0': 'In this work we establish both the mean intensity and the anisotropy power spectrum of galaxies present during reionization.', '1205.2316-3-8-1': 'We update Cooray et al. (2004) by taking into account recent developments in the study of reionization, and by introducing a halo model to calculate the non-linear clustering of the IR background intensity.', '1205.2316-3-8-2': 'The stellar and nebular emission from first light galaxies follow the calculations presented in Fernandez Komatsu (2006), but we specifically require that the UV photon background produced by the galaxy population present from [MATH] to 30 is consistent with the optical depth to electron scattering as measured by the WMAP polarization data with a value of 0.088 [MATH] (Komatsu et al. 2011).', '1205.2316-3-8-3': 'We account for the current uncertainty in the optical depth by introducing variations to the fiducial model so that the optical depth to electron scattering varies between 0.07 and 0.1.', '1205.2316-3-8-4': 'Our models are also designed to reproduce the bright-end galaxy luminosity functions (LFs) in deep HST/WFC3 surveys at [MATH] involving the Lyman-dropout galaxy samples.', '1205.2316-3-8-5': 'This normalization at the bright-end of galaxy luminosities puts strong constraints on the intensity.', '1205.2316-3-9-0': 'This paper is organized as follows.', '1205.2316-3-9-1': 'In Section 2 we outline our model for the reionization galaxies including stellar nebulae and the IGM emission.', '1205.2316-3-9-2': 'Section 3 presents the calculation related to luminosity mass density of these galaxies.', '1205.2316-3-9-3': 'In Section 4, 5 and 6 we outline the background intensity and spatial anisotropy power spectrum calculations, respectively.', '1205.2316-3-9-4': 'In Section 7 we discuss our results related to the intensity and angular power spectrum and present a comparison to existing measurements.', '1205.2316-3-9-5': 'We conclude with a summary in Section 8.', '1205.2316-3-9-6': 'We assume the flat [MATH]CDM model with [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] throughout the paper.', '1205.2316-3-10-0': '# Emission from stars and the intergalactic medium', '1205.2316-3-11-0': 'We first describe the emission from stars in first-light galaxies present during reionization.', '1205.2316-3-11-1': 'Following Fernandez Komatsu (2006), we consider two stellar populations in this calculation.', '1205.2316-3-11-2': 'The first, referred to as Pop II stars, are metal-poor stars with metallicity [MATH], and the second, Pop III stars, are metal-free stars with [MATH].', '1205.2316-3-12-0': 'To describe the stellar initial mass function (IMF) we make use of two descriptions.', '1205.2316-3-12-1': 'For Pop II stars, we adopt the IMF given by Salpeter (1995) [EQUATION] with mass range from 3 to 150 [MATH].', '1205.2316-3-12-2': 'For Pop III stars we use the IMF obtained by Larson (1999), which takes the form as [EQUATION] where [MATH], and the mass range is from 3 to 500 [MATH].', '1205.2316-3-13-0': 'we utilize the fitting results from [CITATION] and [CITATION] to calculate other stellar parameters, such as the intrinsic bolometric luminosity [MATH], the effective temperature [MATH], the main-sequence lifetime [MATH], and the time-averaged hydrogen photoionization rate [MATH].', '1205.2316-3-13-1': 'The fitting forms of these parameters are different for Pop II and Pop III stars.', '1205.2316-3-13-2': 'For Pop II stars, they are given as [EQUATION] where [MATH] and for Pop III stars, they are [EQUATION]', '1205.2316-3-13-3': 'From these expressions the stellar radius [MATH] is [EQUATION] where [MATH] is the Stefan-Boltzmann constant.', '1205.2316-3-13-4': 'The stellar radius is useful for the calculation related to the stellar emission spectrum (see Section 3.1).', '1205.2316-3-14-0': 'The ionization volume in the nebulae surrounding the stars (Str[MATH]mgren sphere) can be derived if assuming ionization equilibrium where the ionization rate equals recombination rate [EQUATION] where [MATH] is the hydrogen case B recombination coefficient which depends on the gas temperature [MATH] (assumed to be [MATH] K), and we will discuss it in detail in the next Section.', '1205.2316-3-14-1': 'Here, [MATH] and [MATH] are the local number density of electron and HII in the stellar nebulae where we assume [MATH].', '1205.2316-3-15-0': 'For the IGM, the hydrogen density is lower than that of the stellar nebulae, so we no longer assume the ionization equilibrium.', '1205.2316-3-15-1': 'We estimate the ionization volume by a redshift-dependent form as [EQUATION] where [MATH] is the mean hydrogen number density for [MATH] assumed in this work (Shull et al. 2011).', '1205.2316-3-16-0': 'These quantities discussed here would now be used to estimate the luminosity mass density, near-IR background intensity SED, and the anisotropy power spectrum.', '1205.2316-3-16-1': 'We make use of emission from the stellar nebulae and the IGM for both Pop II and Pop III stars in galaxies present during reionization.', '1205.2316-3-17-0': '# Luminosity mass density of the sources', '1205.2316-3-18-0': 'In this Section, we calculate the luminosity per stellar mass at frequency [MATH], i.e. luminosity mass density, for several sources that contribute to the infrared background, such as the direct emission from the stars, Lyman-[MATH] line, and free-free, free-bound and two photon processes.', '1205.2316-3-18-1': 'The luminosity mass density takes the central role in our estimation of the near-IR background intensity spectrum.', '1205.2316-3-19-0': '## Stellar spectrum', '1205.2316-3-20-0': 'For simplicity, we assume the stellar spectrum is a Planckian truncated at [MATH] eV.', '1205.2316-3-20-1': 'Thus, the stellar luminosity at frequency [MATH] can be expressed as [EQUATION] where [MATH] is the surface area of the star, [MATH] is the stellar radius, [MATH] is the stellar mass and [MATH] is the Planck spectrum [EQUATION]', '1205.2316-3-20-2': 'Note that for simplicity we have ignored the absorption lines of the Lyman-alpha (Ly[MATH]) series here.', '1205.2316-3-20-3': 'The absorption is not strong enough to affect the shape of the spectrum and hence it is not expected to affect our results especially in the infrared wavelengths .', '1205.2316-3-20-4': 'Our predictions related to the optical background fluctuations may be somewhat overestimated.', '1205.2316-3-20-5': 'Also, the emission with [MATH] eV cannot be approximated by a black-body spectrum, thus we use the fitting formulae for time-averaged photoionization rate, [MATH], for Pop II and Pop III stars to calculate the emission at higher energies.', '1205.2316-3-21-0': '## Lyman alpha emission', '1205.2316-3-22-0': 'The luminosity of Ly[MATH] emission at a frequency [MATH] is [EQUATION] where [MATH] is the frequency of Ly[MATH] photons, and [MATH] is the emission volume that can be estimated by Eq. ([REF]) and Eq. ([REF]) for the stellar nebulae and the IGM respectively.', '1205.2316-3-22-1': 'Here [MATH] is the Ly[MATH] recombination emission rate per [MATH], which is given by [EQUATION] where [MATH] is the electron number density, [MATH] is the HII number density, [MATH] is the hydrogen case B recombination coefficient, and [MATH] is the fraction of the Ly[MATH] photons produced in the case B recombination.', '1205.2316-3-22-2': 'This fraction can be estimated through the fitting formula [EQUATION] where [MATH]K, which is accurate to 0.1% for [MATH].', '1205.2316-3-22-3': 'Note that this fraction is actually not sensitive to the temperature, so it can be treated as a constant [MATH] in the most cases.', '1205.2316-3-22-4': 'The hydrogen case B recombination coefficient [MATH] we use here is from [CITATION], which is fitted by [CITATION] as [EQUATION] where [MATH], [MATH], [MATH] and [MATH].', '1205.2316-3-22-5': 'We assume a gas temperature of [MATH] K in our calculation to obtain [MATH].', '1205.2316-3-23-0': 'The [MATH] is the collisional emission rate per [MATH] given by [EQUATION] where [MATH] is the neutral hydrogen number density and [MATH] is the effective collisional excitation coefficient.', '1205.2316-3-23-1': 'It has the form [EQUATION]', '1205.2316-3-23-2': 'Here we take into account the excitation up to energy level [MATH] to produce Ly-[MATH] photons.', '1205.2316-3-23-3': 'The higher level emission can be neglected given the high temperature we consider for this calculation.', '1205.2316-3-23-4': 'The excitation collisional rate [MATH], in cm[MATH] per second, can be written as [EQUATION] where [MATH] is the energy difference between lower level [MATH] and higher level [MATH], [MATH] is the statistic weight for level [MATH], and [MATH] is the effective collision strength calculated using the fitting formulae from [CITATION].', '1205.2316-3-24-0': 'The [MATH] is the Ly[MATH] line profile, and we use the result from [CITATION] where they fitted the simulated Ly[MATH] line profile of [CITATION] for a homogeneous and expanding IGM: [EQUATION] where [MATH], and [EQUATION].', '1205.2316-3-24-1': 'Here [MATH], and we assume the flat [MATH]CDM model and take [MATH].', '1205.2316-3-25-0': '## Free-free and free-bound emission', '1205.2316-3-26-0': 'For free-free and free-bound emission we again follow the same approach as Fernandez Komatsu (2006).', '1205.2316-3-26-1': 'Following their derivation, the continuum luminosity of these two processes at frequency [MATH] has the same form with [EQUATION] where [MATH] is the specific emission coefficient for free-free and free-bound emission [EQUATION]', '1205.2316-3-26-2': 'Here [MATH] is the proton number density, [MATH] is the gas temperature, and [MATH] is effective Gaunt factor for free-free and free-bound emission, which takes the form as [EQUATION] where [MATH] is the thermal averaged Gaunt factor of free-free emission and [MATH] is the free-bound emission Gaunt factor for a different energy level [MATH].', '1205.2316-3-26-3': 'These values have an accuracy of 10% .', '1205.2316-3-26-4': 'In above [MATH], where [MATH] is around 10 for the parameter space we are interested in .', '1205.2316-3-26-5': 'The energy level [MATH] is determined by the emission photon frequency [MATH].', '1205.2316-3-26-6': 'If [MATH], and then [MATH] where [MATH] is the Rydberg constant.', '1205.2316-3-26-7': 'Note that the [MATH] here starts at [MATH], since the photons from [MATH] can be easily absorbed by other neutral hydrogen atoms and be ionized instantly.', '1205.2316-3-27-0': '## Two-photon emission', '1205.2316-3-28-0': 'For the two-photon process we also follow the approach of Fernandez Komatsu (2006) and write the luminosity as [EQUATION] where [MATH] is the two-photon emission rate per [MATH], and [MATH].', '1205.2316-3-28-1': 'The [MATH] is the normalized probability of generating one photon in the range [MATH] from per two-photon decay.', '1205.2316-3-28-2': 'We use the fitting formula derived in [CITATION] [EQUATION] where [MATH], which is a good fit to the data given in [CITATION].', '1205.2316-3-29-0': '## Luminosity mass density and total emission', '1205.2316-3-30-0': 'Then following Fernandez Komatsu (2006) and Fernandez et al. (2010), we can derive the mean luminosity mass density for each emission source by integrating over the IMF for the Pop II or Pop III stars [EQUATION] where the ranges of the integral are from 3 to 150 [MATH] with the IMF given by Salpeter (1955) for Pop II stars, and from 3 to 500 [MATH] with the IMF in Larson (1998) for Pop III stars.', '1205.2316-3-30-1': 'Here, the average mass [MATH] is given as [EQUATION] where [MATH] is the normalized IMF with [EQUATION].', '1205.2316-3-31-0': 'Note that this expression is only valid when the main sequence lifetime is larger than the star formation time scale.', '1205.2316-3-31-1': 'Otherwise, it should be evaluated by [EQUATION]', '1205.2316-3-31-2': 'Here [MATH] is the star formation time scale which is given by [EQUATION] where [MATH] is the stellar mass density at [MATH], which is related with the comoving star formation rate density (SFRD) as [MATH].', '1205.2316-3-31-3': 'We use the halo mass function to calculate [MATH] and [MATH] and the details are described in the next Section.', '1205.2316-3-31-4': 'We note that the [MATH] is important for the estimation of the [MATH] as discussed in [CITATION].', '1205.2316-3-32-0': 'Finally we obtain the total luminosity mass density from the stellar nebulae [EQUATION] and the same from the IGM [EQUATION] where [MATH] is the escape fraction of the ionization photons that propagate into the IGM from the nebulae surrounding the stars.', '1205.2316-3-33-0': 'In Fig. [REF], we show the total luminosity mass density as a function of the rest-frame wavelength [MATH] from the stellar nebula and the IGM for Pop II and Pop III stars respectively.', '1205.2316-3-33-1': 'The contributions from the different sources we consider are also shown for the stellar nebula case.', '1205.2316-3-33-2': 'Here we set [MATH] and [MATH] for both Pop II and Pop III cases.', '1205.2316-3-34-0': 'With these parameters we find that the stellar spectrum is dominant for Pop II stars while the "background" spectrum, such as Ly-[MATH] and free-bound, are comparable with or even larger than the stellar spectrum for Pop III stars.', '1205.2316-3-34-1': 'Also, as can be seen, the [MATH] from the IGM is much lower than that from the stellar nebula for both Pop II and Pop III cases, and the total [MATH] from the Pop III stars is similar to that from the Pop II stars.', '1205.2316-3-34-2': 'These results are already discussed in Fernandez Komatsu (2006).', '1205.2316-3-35-0': '# Reionization History and UV Luminosity Density', '1205.2316-3-36-0': 'To test if the reionization history associated with our stellar model is consistent with that of the current observations, such as WMAP 7-year results , we need to calculate the hydrogen reionization fraction [MATH] as a function of redshift.', '1205.2316-3-36-1': 'Following Madau et al. (1998), [MATH] can be estimated as [EQUATION] where [MATH] is the redshift of the beginning of the reionization epoch (we take [MATH]), [MATH] is the comoving star formation rate density (SFRD), and the function [MATH] is defined as [MATH].', '1205.2316-3-36-2': 'Here [MATH] is the average stellar lifetime which is given by [MATH], and [MATH] is the volume averaged recombination time, which can be written as [EQUATION] where [MATH] is the clumping factor of ionized hydrogen.', '1205.2316-3-36-3': 'Here we adopt the simulation result from Trac Cen (2007).', '1205.2316-3-36-4': '[MATH] and [MATH] are the mass fractions of hydrogen and helium, respectively.', '1205.2316-3-36-5': 'Note that we have already considered the escape fraction [MATH], so the [MATH] here is the clumping factor excluding the halos with star formation.', '1205.2316-3-37-0': 'For the comoving SFRD we consider the ongoing star formation model : [EQUATION] where [MATH] is the star formation efficiency which denotes the fraction of baryons converted to stars, [MATH] is the halo mass function , and [MATH] is the threshold mass for a dark matter halo to form a galaxy during reionization.', '1205.2316-3-37-1': 'This minimum mass is taken to be a free parameter and is varied to fit a combination of the WMAP 7-year optical depth and the galaxy LF, as we discuss in Section 6.', '1205.2316-3-38-0': 'We also need a stellar population evolution model to describe [MATH] with the relative fraction of the Pop II and Pop III stars at different redshifts.', '1205.2316-3-38-1': 'In principle there should be a cutoff at some redshift for Pop III stars as they are not expected to form at low redshifts once the gas is polluted by metals.', '1205.2316-3-38-2': 'We assume this cutoff is not lower than [MATH] when the universe is fully ionized.', '1205.2316-3-38-3': 'We use the error function to denote the population fraction as [EQUATION] where [MATH] is the population transition width.', '1205.2316-3-38-4': 'Then the term [MATH] in Eq. ([REF]) can be expressed by [EQUATION]', '1205.2316-3-38-5': 'As our default model we assume that the Pop III stars are mainly dominant for [MATH] while Pop II stars are for [MATH] with [MATH].', '1205.2316-3-38-6': 'When we present our results we also show results for three additional values of the transition from [MATH] to 30.', '1205.2316-3-39-0': 'We estimate the optical depth to electron scattering with the reionization fraction [MATH] as [EQUATION] where [MATH] is the Thompson scattering cross-section, and [MATH] is the electron number density of the Universe at redshift [MATH], and we assume the helium is singly ionized for [MATH]) and doubly ionized for [MATH]) .', '1205.2316-3-40-0': 'In Fig. [REF], we plot the hydrogen reionization fraction [MATH] vs. [MATH] for three optical depth [MATH] with three stellar models.', '1205.2316-3-40-1': 'The blue solid line denotes the [MATH] with [MATH] which is close to the result of WMAP 7-year data with [MATH] .', '1205.2316-3-40-2': 'In this case reionization ends around [MATH] which is consistent with the current studies, and we find Pop III stars can ionize the Universe to [MATH] by [MATH] and Pop II stars is responsible for the rest of the reionization over an interval [MATH].', '1205.2316-3-41-0': 'For [MATH] and 0.04, we find [MATH], shown with a dashed line and [MATH], shown with a dotted line, respectively.', '1205.2316-3-41-1': 'These two reionization histories are such that [MATH] and 9, respectively.', '1205.2316-3-41-2': 'For case with [MATH], the Pop III stars ionize [MATH] of the Universe and the Pop II stars are needed over the interval [MATH] to complete reionization.', '1205.2316-3-41-3': 'For the third case with [MATH], Pop III stars ionize [MATH] of the Universe at [MATH] with Pop II stars completing the rest [MATH] in an interval of [MATH]', '1205.2316-3-42-0': 'In Fig. [REF], we also show the dependence of the optical depth on the minimum redshift [MATH] in Eq. ([REF]).', '1205.2316-3-42-1': 'The blue solid, dashed and dotted lines are for the three stellar models with [MATH], 0.077 and 0.099 for [MATH].', '1205.2316-3-42-2': 'The WMAP 7-year result with [MATH] is also shown with a red solid line and a yellow [MATH] error region.', '1205.2316-3-42-3': 'We find the slope of the curves is steeper for [MATH] and flatter for [MATH] which is caused by the Pop III to Pop II transition around [MATH] in our model.', '1205.2316-3-42-4': 'Note that this transition is arbitrarily chosen.', '1205.2316-3-42-5': 'We varied the transition redshift and also cases where PopII and PopIII stars are mixed in with different fractions at different redshifts.', '1205.2316-3-42-6': 'In all these cases we found results that are generally consistent with each other.', '1205.2316-3-42-7': 'Thus, the three choices related to the reionization history that we show here to keep this presentation simple are not biased with respect to the final result related to the IR background intensity that we are trying to estimate in this paper.', '1205.2316-3-43-0': 'We also estimate the total number of ionizing photons per baryon required to maintain the ionized IGM between [MATH] and z, [EQUATION] where [MATH] is the redshift of the end of the reionization, and if we assume the helium is also totally singly ionized at [MATH] we can get [MATH].', '1205.2316-3-43-1': 'In fig. [REF], we show the [MATH] at different [MATH] for three [MATH] cases.', '1205.2316-3-43-2': 'We find the number keeps going up until around [MATH] and becomes constant [MATH] at higher redshift for three cases of the reionization histories.', '1205.2316-3-44-0': '# The near-IR EBL Intensity from Reionization', '1205.2316-3-45-0': 'The mean cosmic infrared background can be estimated by [EQUATION] where [MATH], and we take [MATH] and [MATH] in the calculation.', '1205.2316-3-45-1': 'This redshift range can fully take account of the emission from the Pop III and early Pop II stars, which is redshifted into the near-IR band.', '1205.2316-3-45-2': 'We take this to be the form of [EQUATION] where [MATH] and [MATH] are the comoving specific emission coefficients [EQUATION] where [MATH] is the luminosity mass density at [MATH], [MATH] is the mean stellar lifetime of each of the stellar type and [MATH] is the comoving SFRD given by Eq. ([REF]).', '1205.2316-3-46-0': 'In Fig. [REF], we show the spectrum [MATH] of the near-IR background light intensity from both Pop II and Pop III stars.', '1205.2316-3-46-1': 'We assume that the reionization is ending around [MATH] and integrate up to [MATH] to determine [MATH].', '1205.2316-3-46-2': 'Similar to Fig. [REF], we plot the total spectrum from the stellar nebula and the IGM for both Pop II and Pop III stars.', '1205.2316-3-46-3': 'The contributions from different sources we consider are also shown in colored thin lines.', '1205.2316-3-47-0': 'Here we still take the same value for [MATH] and [MATH] as in Fig. 1 related to the [MATH] calculation.', '1205.2316-3-47-1': 'Similar to the luminosity mass density, the spectrum from stellar nebula is much larger than that from the IGM, and the stellar spectrum is higher for Pop II stars while the "background" spectrum is higher for Pop III stars.', '1205.2316-3-47-2': 'However, different from the luminosity mass density, we now find the spectrum from Pop II stars is larger than that from Pop III stars.', '1205.2316-3-47-3': 'This is basically because the typical lifetime of Pop II stars is longer than that of the Pop III stars.', '1205.2316-3-48-0': '# Angular power spectrum', '1205.2316-3-49-0': 'The angular cross power spectrum of the infrared emission at observed frequencies [MATH] and [MATH] for a multipole [MATH] is [EQUATION] where [MATH] is the comoving angular diameter distance, [MATH] is the scale factor, and [MATH] is the mean emission per comoving volume at frequency [MATH] and redshift [MATH].', '1205.2316-3-49-1': 'If we just take account of the flux lower than a upper cut [MATH], and then [MATH] can be written as [EQUATION]', '1205.2316-3-49-2': 'Here [MATH] is the source flux and [MATH] is the number of sources.', '1205.2316-3-49-3': 'This quantity is just the comoving specific emission coefficient we derive in the last section.', '1205.2316-3-49-4': 'The [MATH] is the galaxy power spectrum at wavenumber [MATH] and redshift [MATH], and we will make use of the model of halo occupation distribution to calculate the [MATH].', '1205.2316-3-50-0': '## First Galaxy Clustering', '1205.2316-3-51-0': 'To calculate the [MATH] we extend the linear theory model of Cooray et al. (2004) and make use of the halo occupation distribution (HOD) for first-light galaxies during reionization.', '1205.2316-3-51-1': 'The galaxy power spectrum can be written as [EQUATION]', '1205.2316-3-51-2': 'Where [MATH] and [MATH] denote the power spectrum contributed by galaxies in a single dark matter halo and galaxies in two different dark matter halos respectively.', '1205.2316-3-51-3': 'Then we can write [EQUATION]', '1205.2316-3-51-4': 'Here [MATH] is the halo mass, [MATH] is the halo mass function, [MATH] is the Fourier transform of the NFW halo density profile , [MATH] when [MATH] and [MATH] otherwise , [MATH] is the halo bias , and [MATH] is the linear matter power spectrum .', '1205.2316-3-51-5': 'The [MATH] is the mean number density of galaxies, which is given by [EQUATION]', '1205.2316-3-51-6': 'The [MATH] is the mean number of galaxies in a halo with mass [MATH], which is the sum of number of central galaxies and satellite galaxies [EQUATION] where we define [EQUATION] and [EQUATION]', '1205.2316-3-51-7': 'In this definition, the [MATH] denotes the mass of a halo that has 50% probability to host a central galaxy, and [MATH] is the transition width.', '1205.2316-3-51-8': 'For the satellite galaxies, [MATH] is the truncation mass for satellites, [MATH] is the normalization mass and [MATH] denotes the slope of the power-law relation about the halo mass [MATH].', '1205.2316-3-51-9': 'We assume [MATH] is always larger than [MATH] since there should not be satellites without central galaxy, and assume [MATH].', '1205.2316-3-51-10': 'We take [MATH], [MATH], and [MATH] in this paper.', '1205.2316-3-51-11': 'If assuming a Poisson distribution for satellite galaxies, we can get [EQUATION]', '1205.2316-3-51-12': 'This expression could take account of the case [MATH] and is consistent with our definitions for the [MATH] and [MATH].', '1205.2316-3-52-0': '## Poisson Fluctuations', '1205.2316-3-53-0': 'The clustering measurements are affected by the Poisson fluctuations associated with the shot-noise caused by the discrete and finite number of galaxies from which clustering is measured.', '1205.2316-3-53-1': 'Assuming a Poisson distribution the [MATH]-independent shot-noise power spectrum is [EQUATION]', '1205.2316-3-53-2': 'To estimate [MATH], we first define the luminosity mass density for the mass of the dark matter halos at frequency [MATH] [EQUATION] where [MATH] is the luminosity mass density for the stellar mass discussed in Section 3.', '1205.2316-3-53-3': 'We derive the 3-D shot-noise power spectrum by assuming [MATH] is proportional to the halo mass [MATH], i.e. [MATH] is independent on [MATH] [EQUATION]', '1205.2316-3-53-4': 'Then the 2-D shot-noise power spectrum can be written as [EQUATION]', '1205.2316-3-54-0': '## Band-Averaged Intensity Power Spectrum', '1205.2316-3-55-0': 'For a specific near-IR observation with a band frequency from [MATH] to [MATH], we can define a band-averaged luminosity mass density as [EQUATION] where [MATH] is the bandwidth.', '1205.2316-3-55-1': 'Then we can derive the band-averaged comoving specific emission coefficient [MATH] using Eq. ([REF]) and the 3-D shot-noise power spectrum [MATH] using Eq. ([REF]) respectively.', '1205.2316-3-56-0': 'Finally we find the band-averaged angular cross power spectrum and shot-noise power spectrum are [EQUATION] and [EQUATION] respectively.', '1205.2316-3-56-1': 'Note that here we have a factor [MATH] instead of [MATH] in [MATH] and [MATH] and this dependence has been explained in the Appendix of [CITATION].', '1205.2316-3-57-0': '# Results and Discussion', '1205.2316-3-58-0': 'In this Section, we first estimate the infrared background intensity and then discuss the angular power spectrum as derived previously.', '1205.2316-3-58-1': 'We also compare our estimation with the observational data and discuss the dependence of the result on the parameters in the model.', '1205.2316-3-59-0': '## Near-IR EBL from Reionization', '1205.2316-3-60-0': 'In Fig. [REF], we show the spectrum of the near-IR intensity vs. the observed frequency for three cases of the reionization history with [MATH], 0.090 and 0.099 corresponding to [MATH], 0.03 and 0.04.', '1205.2316-3-60-1': 'Here the spectrum is the total spectrum of the sum of that from both stellar nebula and the IGM, which are calculated by putting Eq. ([REF]) into Eq. ([REF]).', '1205.2316-3-60-2': 'Also, we set [MATH] for both Pop II and Pop III stars.', '1205.2316-3-60-3': 'The right panel of Fig. [REF] shows three cases with [MATH] varied at fixed [MATH].', '1205.2316-3-60-4': 'In both panels, for comparison, we plot the observational data in terms of the excess of EBL relative to the integrated galaxy light (IGL) of known galaxy populations at low redshifts (Madau Pozzetti 2000).', '1205.2316-3-60-5': 'The excess EBL data plotted here are the same as those shown in Santos et al. (2002) and involves measurements mainly from DIRBE with various models for zodiacal light and Galactic stellar contribution.', '1205.2316-3-61-0': 'We find the emission from the Pop II stars dominates the spectrum for all of the three cases.', '1205.2316-3-61-1': 'As we have just discussed in the last Section, this is because the lifetime of the Pop II is longer than the Pop III stars.', '1205.2316-3-61-2': 'In the spectrum, the shorter and longer wavelength parts are mainly contributed by the "background" spectrum from Pop III stars while the medium part by Pop II stellar spectrum, so if just the Pop III stars get longer lifetime, only the "background" spectrum can be effectively reinforced in the total spectrum.', '1205.2316-3-62-0': 'In any case, regardless of assumptions related to the stellar type, we find that the EBL intensity from reionization is no more than 0.4 nW m[MATH] sr[MATH].', '1205.2316-3-62-1': 'Such an intensity is significantly smaller than the previous predictions that attempted to explain almost all or a significant fraction of the excess EBL seen in DIRBE data relative to IGL estimates.', '1205.2316-3-62-2': 'An intensity larger than about 2.5 nW m[MATH] sr[MATH] in the J-band could be in conflict with metal production considerations and the X-ray background (Madau Silk 2005), though they do not necessarily require high efficiency factors to generate the required star formation (Fernandez Komatsu 2006).', '1205.2316-3-62-3': 'The difference between our calculation and the previous ones is that we primarily require the reionization model to generate a reionization history consistent with the WMAP optical depth.', '1205.2316-3-62-4': 'This limits the number of H-ionizing photons per baryon during reionization to be less than 3.', '1205.2316-3-62-5': 'Previous estimates ignored such a constraint and either focused on explaining all of the DIRBE excess (Santos et al. 2002; Salvaterra Ferrara 2003) or amplitude of the measured near-IR background anisotropies (e.g., Fernandez et al. 2010).', '1205.2316-3-63-0': 'In Fig. [REF], we show the total near-IR background intensity spectrum for different transition redshifts [MATH] from Pop II to Pop III stars with increasing redshift.', '1205.2316-3-63-1': 'Here we take four transition redshifts [MATH], 15, 20 and 30 using Eq. ([REF]).', '1205.2316-3-63-2': 'To maintain the same optical depth to electron scattering [MATH], we find a higher intensity is predicted for higher [MATH].', '1205.2316-3-63-3': 'Correspondingly [MATH] should be increased to values of 0.040, 0.043 and 0.044 for [MATH], 20 and 30, compared to [MATH] for [MATH].', '1205.2316-3-63-4': 'If [MATH] (red dotted line) the integrated intensity is about [MATH] 3 to 4 times greater than the case for [MATH] (blue solid line) when [MATH].', '1205.2316-3-63-5': 'Also, the peak of the integrated intensity spectrum moves to longer wavelengths as [MATH] is increased.', '1205.2316-3-63-6': 'This is because the Pop III stars are generally hotter than the Pop II stars, which can produce more ionizing photons.', '1205.2316-3-63-7': 'When the transition redshift is higher, the longer the Pop II stars dominate the Universe, and less ionizing photons are produced when compared to the case for a low [MATH].', '1205.2316-3-63-8': 'To keep the reionization history unchanged, we need more Pop II stars to generate enough ionizing photons.', '1205.2316-3-63-9': 'This results in a near-IR background intensity that is higher.', '1205.2316-3-64-0': '## Bright-end Galaxy Luminosity Functions', '1205.2316-3-65-0': 'In order to relate our galaxy population responsible for reionization to the observations, we also compare our model to observations of [MATH] galaxies, focusing on the UV LFs.', '1205.2316-3-65-1': 'To obtain the UV LF, instead of the occupation number which is luminosity independent, we make use of the conditional luminosity function (CLF) approach .', '1205.2316-3-65-2': 'To compute the CLF we map galaxy rest-frame UV luminosity to halo mass with some scatter added similar to the case of low-redshift galaxy populations (Cooray 2005).', '1205.2316-3-65-3': "The total luminosity of a halo is taken to be [EQUATION] and we assume that this total luminosity can be ascribed to the central galaxy when [MATH], following the' earlier description related to the occupation number.", '1205.2316-3-65-4': 'When [MATH] we introduce satellites with central galaxy luminosity kept fixed at [MATH].', '1205.2316-3-65-5': 'However, when comparing to the existing measurements we found that all of the rest-UV LF measurements are in the range where central galaxies dominate the LF and thus our comparison to the measured LFs is independent of assumptions related to the exact form of the satellite occupation number or conditional luminosity function.', '1205.2316-3-66-0': 'To compare with existing rest-UV LF measurements, we convert the luminosity of each galaxy to the AB absolute magnitude via the relation [MATH] .', '1205.2316-3-66-1': 'In the left panel of Fig. [REF], we show the SFRD as a function of redshift derived from Eq. ([REF]) for three reionization histories with [MATH], 0.090 and 0.099 which are obtained by setting [MATH], 0.03 and 0.04, respectively.', '1205.2316-3-66-2': 'The red points are the data from HUDF09+ERS+CANDELS observations .', '1205.2316-3-66-3': 'We find the SFRD of the three cases are higher than the existing measurements, especially at the high redshifts.', '1205.2316-3-66-4': 'This difference is mainly due to the fact that existing SFRD estimates are limited to galaxies with [MATH], while the bulk of the reionization UV density budget is contained in the galaxies at the faint-end of the LF.', '1205.2316-3-66-5': 'This is especially the case at high redshift since the faint-end slope of the LF is steep with values reaching close to -2 already.', '1205.2316-3-67-0': 'In the right panel of Fig. [REF], we show our rest-UV LF (corresponding to [MATH]) at [MATH]6, 7 and 8.', '1205.2316-3-67-1': 'The central thick solid curves are the luminosity function derived from our default model with [MATH] and [MATH], and the thin dotted lines are obtained by [MATH] (lower) and 0.04 (upper) which could match the [MATH] errors of the data given in .', '1205.2316-3-67-2': 'The three values of the faint-end slope [MATH], -2.0 and -1.5 are shown, which indicates the slope of our model is between -2.5 and -2.0.', '1205.2316-3-67-3': 'Also, we find the star formation time scale [MATH] is around [MATH] yrs at [MATH], 7 and 8 when we calculate the luminosity mass function here.', '1205.2316-3-68-0': 'We also explore the dependence of the galaxy bias on the [MATH] in Fig. [REF].', '1205.2316-3-68-1': 'We can define a effective galaxy bias here from the HOD model as [EQUATION]', '1205.2316-3-68-2': 'In the plot we show the [MATH] as a function of [MATH] at [MATH]6, 7 and 8 for the [MATH].', '1205.2316-3-68-3': 'We find that the galaxy bias increases as the redshift increases and decreases as the [MATH] increases.', '1205.2316-3-68-4': 'The reason is obvious that the galaxy number density [MATH] defined by Eq. ([REF]) becomes smaller at higher redshift and bigger at larger [MATH] (the larger [MATH] means smaller halo mass [MATH]).', '1205.2316-3-69-0': '## Anisotropy Power Spectrum', '1205.2316-3-70-0': 'In Fig. [REF] we show the near-IR background anisotropy angular power spectrum at [MATH].', '1205.2316-3-70-1': 'The clustering power spectra with non-linear power spectrum from the HOD model are in solid lines.', '1205.2316-3-70-2': 'The left panel shows the case with [MATH], the minimum mass to host a galaxy, is changed from [MATH] to [MATH] M[MATH].', '1205.2316-3-70-3': 'For comparison, we also plot the linear power spectrum which has a turnover around [MATH].', '1205.2316-3-70-4': 'When calculating the clustering power spectrum, we set the parameters of the HOD with [MATH] values as listed in the figure with [MATH].', '1205.2316-3-70-5': 'The corresponding values on the optical depth to electron scattering are also listed in the figure.', '1205.2316-3-70-6': 'We note that the shot-noise amplitude is larger for the case with [MATH] M[MATH] in comparison to the case with, say, [MATH] M[MATH], though in those two cases the clustering amplitude is higher with [MATH] M[MATH].', '1205.2316-3-70-7': 'This is because the shot-noise amplitude is sensitive to the second flux-moment of the number counts.', '1205.2316-3-70-8': 'By keeping the minimum mass higher we force the overall counts to be restricted to brighter sources than the case with a lower minimum halo mass.', '1205.2316-3-70-9': 'On the other hand the clustering power spectrum reflects the total background intensity.', '1205.2316-3-70-10': 'With the minimum mass lowered, both the overall number density of galaxies and the background intensity are increased.', '1205.2316-3-71-0': 'The right panel of Fig. [REF] shows the case where we vary [MATH] to highlight the fact [MATH] amplitude is inversely proportional to [MATH].', '1205.2316-3-71-1': 'However, one cannot arbitrarily reduce [MATH] to a small value since this results in a low optical depth to electron scattering.', '1205.2316-3-71-2': 'If [MATH] to 10[MATH], such that one does not need to be concerned of effects due to feedback negatively impacting the formation of galaxies in lower mass halos, then we find that [MATH] must be at the level of 0.3 or more, under the assumptions related to the IMFs for PopII and PopIII stars we use in this paper.', '1205.2316-3-72-0': 'In Fig. [REF], we plot the [MATH] at different redshift bins.', '1205.2316-3-72-1': 'As can be seen, the main contribution of the [MATH] comes from the lowest redshift range of 6 to 10.', '1205.2316-3-72-2': 'For all practical purposes one can assume that the near-IR background is probing the end of reionization not the first objects to form in the universe at the beginning of reionization.', '1205.2316-3-73-0': 'To compare with the total near-IR intensity spectrum, we plot the ratio of the square root of [MATH] to the [MATH] in Fig. [REF].', '1205.2316-3-73-1': 'We integrate from [MATH] to 30 with the same HOD model described above.', '1205.2316-3-73-2': 'For easy comparison, we restrict the predictions to be with [MATH] M[MATH] such that [MATH] to remain consistent with WMAP 7-year result.', '1205.2316-3-73-3': 'We find the ratio of [MATH] to [MATH] is about [MATH] for [MATH] and the clustering fluctuations amplitude is below 100% of the intensity at all angular scales.', '1205.2316-3-73-4': 'This shows that large fluctuations of the background intensity is not expected and the background behaves in a manner that is smooth and not clumpy as in the case if spatial variations are dominated by rare, bright galaxies.', '1205.2316-3-74-0': 'In Fig. [REF], we show the near-IR background intensity power spectrum (solid curves), shot-noise power spectrum (dotted curves) with a comparison to existing observational measurements at several wavelengths: [MATH], 2.4, 3.6, and 4.5 [MATH] .', '1205.2316-3-74-1': 'Note that for the data from Kashlinsky et al. (2012) at 3.6 and 4.5 [MATH], we removed the IRAC beam transfer function [MATH] so the comparison to theoretical predictions is easier.', '1205.2316-3-74-2': 'An accounting of the beam is essential since realistic instruments do not have perfect resolution causing a loss of power on the smallest scales.', '1205.2316-3-74-3': 'This effect can be explicitly measured from the point spread function.', '1205.2316-3-74-4': 'We calculate the beam transfer function [MATH] by taking the power spectrum of the measured point spread function for the Spitzer/IRAC instrument, which is publicly available and compute [MATH] (for more information see Smidt et al. 2012).', '1205.2316-3-75-0': 'We also show the results from the simulation in [CITATION] for comparison.', '1205.2316-3-75-1': 'The red solid and magenta dashed lines are for the simulation with mass resolution [MATH] and [MATH], respectively.', '1205.2316-3-75-2': 'These two cases are what is described in their work as not having a suppression of the small mass halos (ie star-formation present in halos with mass between 10[MATH] and 10[MATH] M[MATH], where one expects photo-ionization heating to suppress star-formation).', '1205.2316-3-75-3': 'The reionization histories of these two cases involve PopII stars with [MATH].', '1205.2316-3-76-0': 'For more clarity, we show the square root of the [MATH] at [MATH] and [MATH] together with the data as a function of the wavelength in Fig. [REF].', '1205.2316-3-76-1': 'The data points are from the data sets shown in Fig. [REF] which are around [MATH] or [MATH].', '1205.2316-3-76-2': 'We find the curve of the square root of the [MATH] has similar shape as [MATH], but the amplitude of rms fluctuations in our models is lower than the existing measurements.', '1205.2316-3-76-3': 'We capture the uncertainties related to [MATH], [MATH], and [MATH] by considering a low and high range for our prediction related to [MATH] with [MATH] falling within the 1 [MATH] uncertainty range of the WMAP 7-year result when these parameters are varied.', '1205.2316-3-76-4': 'Even with parameter uncertainties accounted for, we find that the existing measurements are at least an order of magnitude larger than our model predictions.', '1205.2316-3-77-0': 'We attempted additional model variations but failed to find a scenario where [MATH] is consistent with WMAP 7-year result and the existing LFs leading to a model consistent with existing near-IR background.', '1205.2316-3-77-1': 'One can, in principle, model fit the near-IR fluctuation power spectra by increasing the photon output of first galaxies.', '1205.2316-3-77-2': 'This results in an optical depth that is higher than the WMAP value and a LF that has brighter galaxies than observed in existing deep HST/WFC3 images.', '1205.2316-3-77-3': 'The existing measurements require a background intensity that is at least 3 nW m[MATH] sr[MATH] at 1.6 [MATH]m so that the model predictions become consistent with measurements within the 1[MATH] uncertainties of the measurements.', '1205.2316-3-77-4': 'Since 0.3 nW m[MATH] sr[MATH] at 1.6 [MATH]m is associated with 2.5 H-ionizing photons per baryon during reionization, if reionization is to explain the background fluctuations then we are dealing with a situation where [MATH] 25 H-ionizing photons per baryon are present.', '1205.2316-3-77-5': 'A possibility is to introduce a spectrum for the emission that has a rest-frame cut-off in UV at wavelengths shortward of the Lyman-limit that is not associated with reionization but by the emission mechanism itself.', '1205.2316-3-77-6': 'Unfortunately we have not been able to come up with such an emission spectrum.', '1205.2316-3-78-0': 'As discussed in Helgason et al. (2012) faint, low-redshift galaxies do not have a clustering shape consistent with the anisotropy power spectrum measurements.', '1205.2316-3-78-1': 'The existing measurements show a clear excess in clustering at 30 arcsec to few tens arcminute angular scales that differ from faint galaxy clustering power spectrum.', '1205.2316-3-78-2': 'This difference is statistically significant and one can only explain at most 20% of the near-IR fluctuations to be associated with faint galaxies, reducing the previous estimate in Cooray et al. (2007; Chary et al. 2008) that suggested a contribution at the 50% level.', '1205.2316-3-78-3': 'We are now left with an unexplained set of measurements since neither the low redshift faint galaxies nor the high redshift reionization galaxies can explain them.', '1205.2316-3-78-4': 'One possibility is that the existing fluctuation excess in the data has a non-astrophysical origin, perhaps either involving systematics in the data or fluctuations in the Zodiacal light.', '1205.2316-3-78-5': 'While arguments have been made that these effects are insignificant with a set of well-coordinated multi-wavelength measurements we plan to further address the origin of near-IR fluctuations in our upcoming papers.', '1205.2316-3-79-0': '# Summary', '1205.2316-3-80-0': 'The UV emission from stars which are formed in the early Universe from [MATH] to 30 can contribute to the near-IR background light.', '1205.2316-3-80-1': 'By measuring the intensity and anisotropies of the near-IR background, we can investigate the properties of these early stars and the epoch of reionization.', '1205.2316-3-80-2': 'In this work we discuss several sources that contribute to the near-IR background intensity, including the emission of stars, Ly[MATH], free-free, free-bound and two-photon.', '1205.2316-3-80-3': 'We first estimate the frequency spectrum separately for Pop II and Pop III star with the redshift range [MATH].', '1205.2316-3-80-4': 'By using the initial stellar mass spectrum and the fitting model of time-averaged hydrogen photoionization rate we calculate the luminosity mass density.', '1205.2316-3-80-5': 'We find that although the luminosity mass density of the Pop III stars is a bit larger than that of Pop II stars, the near-IR intensity spectrum from the Pop II stars is stronger than that from Pop III stars, which is caused by the longer lifetime of Pop II stars.', '1205.2316-3-81-0': 'In order to check the consistence of our stellar model and the reionization history, we derive the hydrogen reionization fraction at different redshifts and calculate the optical depth by assuming the Pop II stars are dominant for [MATH] while Pop III stars for [MATH].', '1205.2316-3-81-1': 'We find that if we set the star formation efficiency [MATH], the Universe would be totally reionized around [MATH] with the optical depth [MATH], which is well consistent with the result from the WMAP 7-year data.', '1205.2316-3-81-2': 'Also, we explore the other possible models with [MATH] and [MATH] and get [MATH] and 0.099, respectively.', '1205.2316-3-81-3': 'The total number of the ionizing photon per baryon required to maintain the ionized IGM, [MATH], is also evaluated for the three cases, and we find the [MATH] becomes a constant with a value around 2.5 after [MATH].', '1205.2316-3-82-0': 'To compare with existing bright-end luminosity function measurements, we evaluate the UV luminosity function from our model at [MATH] for [MATH]6, 7 and 8 from [MATH] to [MATH].', '1205.2316-3-82-1': 'We compare to the measurements of Bouwens et al. (2012).', '1205.2316-3-82-2': 'We find our derived luminosity function is consistent with the data with the slope of the faint-end [MATH].', '1205.2316-3-82-3': 'This is a steep slope and existing measurements do suggest that the slope is steeper than for LFs at low redshifts.', '1205.2316-3-82-4': 'We then define an effective galaxy bias [MATH] from the HOD model, and find the [MATH] becomes bigger at high redshift and bright luminosities because of the lower number density of the galaxies.', '1205.2316-3-83-0': 'Finally, we calculate the angular power spectrum of the near-IR background by making use of a halo model.', '1205.2316-3-83-1': 'The non-linearities provided by the 1-halo term increase the clustering strength at multipoles greater than about [MATH].', '1205.2316-3-83-2': 'The suggest turn-over at this scale with the linear power spectrum alone in Cooray et al. (2004) is no longer present.', '1205.2316-3-83-3': 'We find the shot-noise power spectrum of the Pop III stars is greater than that of Pop II stars because of the larger luminosity mass density for Pop III stars.', '1205.2316-3-83-4': 'By making use of the stellar population evolution model, we calculate the near-IR anisotropy power spectra [MATH] at different wavelengths and compare to the observational data.', '1205.2316-3-83-5': 'We find our results are lower than the observational data by at least an order of magnitude.', '1205.2316-3-83-6': 'There are now strong arguments that the near-IR anisotropies cannot originate from low redshift faint galaxies (Helgason et al. 2012).', '1205.2316-3-83-7': 'We have failed to explain the alternative origin of near-IR background anisotropies involving galaxies present during reionization, contrary to suggestions in the literature (Kashlinsky et al. 2012).', '1205.2316-3-83-8': 'In future works, using additional measurements with Spitzer and Hubble/WFC3, we plan to further discuss the near-IR fluctuations and to explain if the origin is astrophysical or whether it is associated with yet-unknown systematic effect in the data.', '1205.2316-3-84-0': 'This work was supported by NSF CAREER AST-0645427 and NASA NNX10AD42G at UCI.', '1205.2316-3-84-1': 'We thank CIBER, SDWFS, and CANDELS teams for helpful discussions and questions that motivated this paper.'}

1203.4126

{'1203.4126-1-0-0': 'A seminal theorem due to Weyl [CITATION] states that if [MATH] is any sequence of distinct integers, then, for almost every [MATH], the sequence [MATH] is uniformly distributed modulo one.', '1203.4126-1-0-1': 'In particular, for almost every [MATH] in the unit interval, the sequence [MATH] is uniformly distributed modulo one for every computable sequence [MATH] of distinct integers.', '1203.4126-1-0-2': 'Call such an [MATH] UD random.', '1203.4126-1-0-3': 'Here it is shown that every Schnorr-random real is UD random, but there are Kurtz-random reals that are not UD random.', '1203.4126-1-0-4': "On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.", '1203.4126-1-1-0': '# Introduction', '1203.4126-1-2-0': 'Let [MATH] denote the fractional part of a real number [MATH], and let [MATH] denote Lebesgue measure.', '1203.4126-1-2-1': 'A sequence [MATH] of real numbers is said to be uniformly distributed modulo one if for every interval [MATH], [EQUATION].', '1203.4126-1-2-2': 'In words, a sequence is uniformly distributed modulo one if the limiting frequency with which it visits any given interval is what one would expect if the elements of the sequence were chosen at random.', '1203.4126-1-2-3': 'A remarkable theorem by Hermann Weyl [CITATION] states the following:', '1203.4126-1-3-0': 'If [MATH] is any sequence of distinct integers, then for almost every [MATH], [MATH] is uniformly distributed modulo one.', '1203.4126-1-4-0': 'This seminal result lies at the intersection of harmonic analysis, number theory, ergodic theory, and computer science; see, for example, [CITATION].', '1203.4126-1-5-0': 'Now let us restrict attention to computable sequences [MATH] of distinct integers.', '1203.4126-1-5-1': 'Since there are only countably many of these, it follows that almost every [MATH] has the property that [MATH] is uniformly distributed modulo one for each such sequence.', '1203.4126-1-5-2': 'Let us say that such an [MATH] is UD random.', '1203.4126-1-5-3': 'Our goal here is to explore the relationship of UD randomness to other senses in which a real number in the unit interval can be said to be "random."', '1203.4126-1-6-0': 'Section [REF] considers sufficient conditions for UD randomness.', '1203.4126-1-6-1': 'In particular, in the terminology of algorithmic randomness, every Schnorr-random element of [MATH] is UD random, but there are Kurtz-random elements of [MATH] that are not.', '1203.4126-1-7-0': 'Regarding consequences of UD randomness, recall that a real number [MATH] is normal if, when it expressed in any base [MATH], each sequence of [MATH] digits occurs with limiting frequency [MATH].', '1203.4126-1-7-1': 'It is not hard to see that a real number [MATH] is normal with respect to base [MATH] if and only if the sequence [MATH] is uniformly distributed modulo one, so every UD-random real is normal to every base.', '1203.4126-1-7-2': 'In fact, more is true: if [MATH] is UD random and [MATH] is any computable sequence of distinct integers, then the limiting frequency of occurrences of any [MATH]-digit block at positions [MATH] of [MATH] is again [MATH].', '1203.4126-1-7-3': 'Thus a UD-random number looks random, at least in some ways.', '1203.4126-1-8-0': 'In Section [REF], however, I show that there are straightforward ways that a UD-random number can look very nonrandom.', '1203.4126-1-8-1': 'Specifically, consider the set [MATH] of elements [MATH] in [MATH] such that for every [MATH], the [MATH]-th digit in the binary expansion of [MATH] is equal to the [MATH]-st digit.', '1203.4126-1-8-2': "Then Weyl's theorem holds with respect to the natural measure on [MATH], showing that, in particular, almost every element of [MATH] is UD random.", '1203.4126-1-9-0': '# A sufficient condition for UD randomness', '1203.4126-1-10-0': 'The subject of algorithmic randomness [CITATION] aims to characterize different senses in which a real number (or, say, an infinite binary sequence) can be said to be "random."', '1203.4126-1-10-1': 'A subset [MATH] of [MATH] is said to be effectively open if there are computable sequences [MATH] and [MATH] of rational numbers such that [MATH].', '1203.4126-1-10-2': 'An effectively open subset of the unit interval is obtained by restricting [MATH] to [MATH].', '1203.4126-1-10-3': 'A sequence [MATH] of effectively open sets is uniformly effective if the representing sequences [MATH] and [MATH] can be computed by a single algorithm with [MATH] as a parameter.', '1203.4126-1-10-4': 'A Martin-Lof test is a uniformly effective sequence of open sets [MATH] such that for each [MATH], [MATH].', '1203.4126-1-10-5': 'An element [MATH] of [MATH] fails the Martin-Lof test [MATH] if it is an element of [MATH], and passes the test otherwise.', '1203.4126-1-10-6': 'An element [MATH] of [MATH] is Martin-Lof random if it passes every Martin-Lof test.', '1203.4126-1-10-7': 'In other words, [MATH] is Martin-Lof nonrandom if it can be covered, effectively, by arbitrarily small open sets, and is thus contained in an effectively presented null [MATH] set.', '1203.4126-1-11-0': 'One can weaken or strengthen this notion of randomness by restricting or enlarging the class of tests.', '1203.4126-1-11-1': 'For example, a Schnorr test is a Martin-Lof test with the additional property that for each [MATH], the measure of [MATH] is computable.', '1203.4126-1-11-2': 'This has the effect that when enumerating the intervals that cover the nonrandoms with a set of size at most [MATH], one knows how much of the set is yet to come.', '1203.4126-1-11-3': 'An element [MATH] of [MATH] is Schnorr random if it passes every Schnorr test.', '1203.4126-1-12-0': 'An element [MATH] of [MATH] is Kurtz random if it is contained in an effectively closed set of measure [MATH], that is, the complement of an effectively open set of measure [MATH].', '1203.4126-1-12-1': 'Clearly every Martin-Lof-random element of [MATH] is Schnorr random, and every Schnorr-random element is Kurtz random.', '1203.4126-1-12-2': 'The notion of Kurtz randomness is fairly weak.', '1203.4126-1-12-3': 'For example, it is not hard to show that if a real number [MATH] is weakly 1-generic then it is Kurtz random but its binary digits fail to satisfy the law of large of numbers.', '1203.4126-1-12-4': '(See [CITATION] for the definition of weak 1-genericity, and Section 8.11 for the facts just mentioned.)', '1203.4126-1-12-5': 'In particular, this shows that there are Kurtz-random reals that are not UD random.', '1203.4126-1-13-0': 'The main result of this section is this:', '1203.4126-1-14-0': 'Every Schnorr-random element of [MATH] is UD random.', '1203.4126-1-15-0': "The proof of this lemma follows conventional proofs of Weyl's theorem, for example, as presented in [CITATION]; we need only make certain constructive and quantitative aspects of the argument explicit.", '1203.4126-1-15-1': 'To start with, we need an equivalent formulation of uniform distribution modulo one.', '1203.4126-1-16-0': 'For any sequence [MATH] of real numbers, the following are equivalent:', '1203.4126-1-17-0': '[MATH] is uniformly distributed modulo one.', '1203.4126-1-17-1': 'For any continuous function [MATH], [EQUATION]', '1203.4126-1-17-2': 'For any [MATH], [EQUATION].', '1203.4126-1-18-0': 'The third property is known as "the Weyl criterion."', '1203.4126-1-18-1': 'Roughly, the second follows from the first using Riemann sums to approximate the integral; the third follows immediately from the second; and the first follows from the third, using trigonometric sums to approximate the characteristic function of an interval.', '1203.4126-1-18-2': 'For details, see any of the references mentioned above.', '1203.4126-1-19-0': 'It will be convenient to adopt the Vinogradov convention of writing [MATH] for [MATH], thus freeing up the symbol [MATH] to be used as an index.', '1203.4126-1-19-1': 'Note that [MATH] and [MATH] for every [MATH] and [MATH].', '1203.4126-1-19-2': 'Theorem [REF] is an immediate consequence of the next lemma, since if [MATH] is a Schnorr-random element of [MATH] and [MATH] is any computable sequence of distinct integers, the lemma implies that [MATH] is uniformly distributed modulo one.', '1203.4126-1-20-0': 'For each [MATH], let [MATH] be a sequence of distinct integers, such that [MATH] is uniformly computable from [MATH] and [MATH].', '1203.4126-1-20-1': 'Then there is a Schnorr test [MATH] such that for every [MATH] not in [MATH] and every integer [MATH], [MATH] is uniformly distributed modulo one.', '1203.4126-1-21-0': 'For the moment, fix a sequence [MATH] of distinct integers, and for any [MATH] in [MATH] write [EQUATION].', '1203.4126-1-21-1': 'Then we have [EQUATION] and hence [EQUATION] since each term in the sum has integral [MATH] except when [MATH].', '1203.4126-1-21-2': '(A more perspicuous way of carrying out this calculation is to notice that the left hand side is the square [MATH] of the norm of [MATH] in the Hilbert space [MATH], and the functions [MATH] form an orthornormal set.)', '1203.4126-1-21-3': "Markov's inequality implies that for every [MATH] we have [EQUATION].", '1203.4126-1-21-4': 'In other words, for [MATH] large, [MATH] is small for most [MATH].', '1203.4126-1-22-0': 'Now fix a doubly-indexed sequence [MATH] as in the statement of the lemma, and for each [MATH] write [EQUATION].', '1203.4126-1-22-1': 'Without loss of generality we can assume that for every [MATH] and [MATH] the sequence [MATH] already appears as one of the sequences [MATH] for some [MATH], since we can replace the original sequence of sequences with a sequence that includes all such multiples.', '1203.4126-1-22-2': 'By the Weyl criterion, then, it suffices to find a Schnorr test [MATH] such that for each [MATH] not in [MATH] we have [MATH].', '1203.4126-1-23-0': 'A simple calculation shows that for any [MATH] such that [MATH] we have [MATH]; in other words, between [MATH] and [MATH], the averages do not change that much.', '1203.4126-1-23-1': 'This reduces our task to finding a Schnorr test [MATH] such that for each [MATH] not in [MATH] we have [MATH].', '1203.4126-1-24-0': 'For each [MATH], rational [MATH], and [MATH] define [EQUATION].', '1203.4126-1-24-1': 'By the calculation above, we have [EQUATION] which decreases to [MATH] as [MATH] approaches infinity.', '1203.4126-1-24-2': 'Choose an enumeration [MATH] of all pairs [MATH], and for each [MATH] and [MATH] choose [MATH] large enough so that [MATH].', '1203.4126-1-24-3': 'For each [MATH], let [EQUATION].', '1203.4126-1-24-4': 'Then for each [MATH], [MATH].', '1203.4126-1-24-5': 'Moreover, the measure of [MATH] is uniformly computable in [MATH], since for each [MATH] the measure of [MATH] is computable and the measure of [MATH] is at most [MATH].', '1203.4126-1-24-6': 'Thus the sequence [MATH] is a Schnorr test, and need only confirm that it meets the specification above.', '1203.4126-1-25-0': 'So suppose [MATH] passes this test, that is, for some [MATH], [MATH] is not in [MATH].', '1203.4126-1-25-1': 'Given any [MATH] and [MATH], choose [MATH] such that [MATH].', '1203.4126-1-25-2': 'Since [MATH] is not in [MATH], it is not in [MATH].', '1203.4126-1-25-3': 'This means that for every [MATH], we have [MATH].', '1203.4126-1-25-4': 'Since [MATH] and [MATH] were arbitrary, we have that [MATH] approaches [MATH] for every [MATH], as required.', '1203.4126-1-26-0': 'The lemma also has the following nice consequence, to the effect that we can actually compute an element [MATH] that is UD random with respect to a computable list of computable sequences.', '1203.4126-1-27-0': 'For every uniformly computable sequence of sequences [MATH] of distinct integers there is a computable element [MATH] of [MATH] that is UD random for these sequences; that is, such that [MATH] is uniformly distributed modulo one for each [MATH].', '1203.4126-1-28-0': 'The lemma implies that there is a Schnorr test [MATH] such that every [MATH] outside of [MATH], [MATH] has the desired property.', '1203.4126-1-28-1': 'In particular, there is an effectively open set [MATH] with computable measure less than [MATH], such that any [MATH] outside of [MATH] has the desired property.', '1203.4126-1-28-2': 'Then it is not hard to compute an element [MATH] that does not land in [MATH], as follows.', '1203.4126-1-28-3': 'First, divide [MATH] into the two intervals [MATH] and [MATH], and compute enough of [MATH] to be able to select one of the intervals on which the measure of [MATH] is at most [MATH].', '1203.4126-1-28-4': 'Then divide that interval in two, and compute enough of [MATH] to be able to select one on which the measure of [MATH] is at most [MATH].', '1203.4126-1-28-5': 'Continue so that at the [MATH]th state, one has a closed interval of size at most [MATH] on which the measure of [MATH] is at most [MATH].', '1203.4126-1-28-6': 'Let [MATH] be the limit of these nested intervals.', '1203.4126-1-29-0': '# UD random reals with nonrandom properties', '1203.4126-1-30-0': 'The main result of this section is that there are UD random reals that are not Kurtz random.', '1203.4126-1-30-1': 'In fact, I prove something stronger.', '1203.4126-1-30-2': 'Ignoring the countable set of dyadic rationals, every element [MATH] of [MATH] can be uniquely identified with its binary representation, or, equivalently, the element of Cantor space, [MATH], corresponding to the sequence of bits after the (binary) decimal point.', '1203.4126-1-30-3': 'Let [MATH] denote the [MATH]th binary digit of [MATH], and let [EQUATION] that is, the set of binary sequences such that for every [MATH], the [MATH]-th digit is the same as the [MATH]-st digit.', '1203.4126-1-30-4': '(Here and below it is convenient to start the indexing with [MATH], so that the first bit of [MATH] is [MATH] rather than [MATH].)', '1203.4126-1-30-5': 'Clearly [MATH] is an effectively closed null set.', '1203.4126-1-30-6': 'There is a natural bijection [MATH] from [MATH] to [MATH] which takes any infinite binary sequence and inserts the appropriate digits at positions [MATH] for each [MATH].', '1203.4126-1-30-7': 'Let [MATH] be the "uniform" probability measure on [MATH] defined by setting [MATH], where [MATH] is any Borel subset of [MATH] and [MATH] is the uniform measure on [MATH].', '1203.4126-1-30-8': "The following asserts that the analog of Weyl's theorem holds for this measure on [MATH]:", '1203.4126-1-31-0': 'Let [MATH] be any sequence of distinct integers.', '1203.4126-1-31-1': 'Then for every almost every element [MATH] with respect to [MATH], [MATH] is UD-random.', '1203.4126-1-32-0': 'This implies that given any countable collection of sequences of distinct integers, in particular all the computable ones, almost every element [MATH] of [MATH] is UD-random with respect to this collection.', '1203.4126-1-32-1': 'I am grateful to Terence Tao for suggesting this approach.', '1203.4126-1-33-0': 'The values [MATH] are known as the Fourier-Stieltjes coefficients of the measure [MATH].', '1203.4126-1-33-1': "The proof of Weyl's theorem presented in Section [REF] relied on the fact that the Fourier-Stieltjes coefficients [MATH] of Lebesgue measure are [MATH] when [MATH].", '1203.4126-1-33-2': "The next two lemmas show that for the conclusion of Weyl's theorem to hold with respect to a measure [MATH] it suffices to show that the Fourier-Stieltjes coefficients of [MATH] approach [MATH] sufficiently quickly.", '1203.4126-1-33-3': "First, the proof of Weyl's theorem can be modified to obtain following result due to Davenport, Erdos, and LeVeque [CITATION]:", '1203.4126-1-34-0': 'Fix a finite measure [MATH] on [MATH], and let [MATH] be a sequence of bounded, [MATH]-integrable functions.', '1203.4126-1-34-1': 'Let [MATH].', '1203.4126-1-34-2': 'If the series [EQUATION] converges for every [MATH], then the sequence [MATH] is uniformly distributed modulo one for [MATH]-almost every [MATH].', '1203.4126-1-35-0': 'The proof can also be found in Kuipers and Niederreiter [CITATION].', '1203.4126-1-35-1': '(In both places, the result is stated for Lebesgue measure, but the proof establishes the more general result.', '1203.4126-1-35-2': 'See also Loeve [CITATION] for a similar argument in a slightly different context.)', '1203.4126-1-35-3': "Lyons [CITATION] uses this fact to establish the following strengthening of Weyl's theorem:", '1203.4126-1-36-0': 'If the Fourier-Stieltjes coefficients of [MATH] have the property that [EQUATION] converges, then for any sequence [MATH] of distinct integers and [MATH]-almost every [MATH], the sequence [MATH] is uniformly distributed modulo one.', '1203.4126-1-37-0': '(Lyons only states the theorem for strictly increasing sequences [MATH], and shows, more generally, that in the hypothesis one can replace the sequence [MATH] by its nonincreasing rearrangement.', '1203.4126-1-37-1': 'It is easy to check, however, that the proof only requires that the values of the sequence [MATH] be distinct.', '1203.4126-1-37-2': 'See also [CITATION].)', '1203.4126-1-37-3': 'Notice that a mild rate of convergence of [MATH] to [MATH], such as [MATH] for any [MATH], is sufficient to meet the criteria of Lemma [REF].', '1203.4126-1-38-0': 'I will show that with the choice of [MATH] and [MATH] above, we have [MATH], which suffices to prove Theorem [REF].', '1203.4126-1-38-1': 'The intuition behind the theorem is that multiplying [MATH] by some number and testing for membership in an interval only enables one to probe "local" properties of binary expansion of [MATH], and so a sequence [MATH] cannot exploit correlations between bits that are far apart.', '1203.4126-1-38-2': 'This intuition is reflected in the fact that the mass of [MATH] is distributed sufficiently uniformly throughout the interval [MATH] that periodic functions like [MATH] do not detect significant irregularities.', '1203.4126-1-38-3': 'The rest of this section is devoted to carrying out the calculations that back up these intuitions.', '1203.4126-1-39-0': 'Since we are only estimating [MATH] up to a multiplicative constant, we can assume that [MATH].', '1203.4126-1-39-1': 'Find [MATH] such that [MATH].', '1203.4126-1-39-2': 'Viewing [MATH] as a subset of [MATH], write [EQUATION] where:', '1203.4126-1-40-0': "Let [MATH] denote a string of [MATH]'s of length [MATH], let [MATH], and for any [MATH], let [MATH] denote the translate of [MATH] (now viewed as a subset of [MATH]) by [MATH].", '1203.4126-1-40-1': 'For any binary sequence [MATH], let [MATH] denote the binary number denoted by [MATH].', '1203.4126-1-40-2': 'Notice that as [MATH] ranges over [MATH], [MATH] ranges from [MATH] to [MATH], and as [MATH] ranges over [MATH], [MATH] ranges from [MATH] to [MATH].', '1203.4126-1-40-3': 'Thus we can describe [MATH] as [EQUATION].', '1203.4126-1-40-4': 'Using the identity [MATH], the integral we need to evaluate, [MATH], is therefore equal to', '1203.4126-1-41-0': 'Since [MATH] for every [MATH], by the triangle inequality the absolute value of the first term is at most the cardinality of [MATH], which is [MATH].', '1203.4126-1-41-1': 'The second term is a geometric progression, equal to [EQUATION]', '1203.4126-1-41-2': 'Similarly, the third term is equal to [EQUATION]', '1203.4126-1-41-3': 'There is nothing more to say about the fourth term, for the moment.', '1203.4126-1-41-4': 'The absolute value of the last term, the integral, is at most [EQUATION] though we will have to evaluate this term more precisely later on.', '1203.4126-1-42-0': 'We now consider cases, keeping in mind that [MATH] if [MATH] is an integer, and [MATH] if [MATH] is equal to half an odd integer.', '1203.4126-1-42-1': 'If [MATH] is divisible by [MATH] but not [MATH], the second term, and hence the product, is [MATH].', '1203.4126-1-43-0': 'If [MATH] is divisible by [MATH] but not [MATH], the second term is equal to [MATH], the third term is equal to [EQUATION] and the fourth term is equal to [EQUATION] since [MATH] is equal to half an odd integer.', '1203.4126-1-43-1': 'But the product of the third and fourth terms is [EQUATION] since the denominator of the third term is of the form [MATH], where the fourth term is [MATH].', '1203.4126-1-43-2': 'In absolute value, then, the product of these terms is at most [MATH].', '1203.4126-1-43-3': 'Thus the absolute value of [MATH] is at most [EQUATION] since [MATH].', '1203.4126-1-44-0': 'If [MATH] is divisible by [MATH] but not [MATH], then the third term, and hence the product, is equal to [MATH].', '1203.4126-1-44-1': 'If [MATH] is divisible by [MATH] but not [MATH], then [MATH] is an integer and [MATH] is a half integer, in which case the fourth term, and hence the product, is equal to [MATH].', '1203.4126-1-45-0': 'Hence we are left with two cases: first, where [MATH] is not divisible by [MATH], and, second, where [MATH] is divisible by [MATH].', '1203.4126-1-45-1': 'In both cases, we need to further expand the value of [MATH].', '1203.4126-1-45-2': 'Write [EQUATION] where [MATH], [MATH], [MATH] are obtained by replacing [MATH] by [MATH] in the definitions of [MATH], [MATH], and [MATH], and define [MATH] analogously.', '1203.4126-1-45-3': 'As above, [MATH] evaluates to [EQUATION]', '1203.4126-1-46-0': 'In the case where [MATH] is divisible by [MATH], the argument proceeds as above.', '1203.4126-1-46-1': 'Since, by assumption, [MATH], we know that [MATH] is not divisible by [MATH], and the argument terminates after the second new case.', '1203.4126-1-47-0': 'Thus we are left with the situation where [MATH] is not divisible by [MATH], in which case [MATH] evaluates to the following: [EQUATION]', '1203.4126-1-47-1': 'Notice that the denominator of the second fraction is equal to the numerator of the third.', '1203.4126-1-47-2': 'Also, the denominator of the first fraction is of the form [MATH], where the numerator of the second fraction is [MATH].', '1203.4126-1-47-3': 'Simplifying the quotient then leaves a multiplicand with absolute value at most 2; and similarly for the denominator of the third fraction and the numerator of the fourth.', '1203.4126-1-47-4': 'The numerator of the first fraction has absolute value at most 2, as do the fourth and seventh terms in the product.', '1203.4126-1-47-5': 'Finally, the absolute value of the last term is at most [EQUATION].', '1203.4126-1-47-6': 'Thus the absolute value of [MATH] is at most [EQUATION].', '1203.4126-1-47-7': 'All we need to do now is use the upper bound [MATH] on [MATH] to get a lower bound on the denominator of the last fraction, and then use the lower bound [MATH] on [MATH] to show that the resulting expression is comparable to [MATH].', '1203.4126-1-47-8': 'When [MATH] is any real value satisfying [MATH], a Taylor-series approximation shows [EQUATION].', '1203.4126-1-47-9': 'Thus when [MATH], [MATH].', '1203.4126-1-47-10': 'As a result, the absolute value of [MATH] is at most [EQUATION] since we have [MATH].', '1203.4126-1-48-0': '# Final comments', '1203.4126-1-49-0': 'The goal here has been to explore the extent to which real numbers satisfying the conclusion of Weyl\'s theorem for computable sequences look "random."', '1203.4126-1-49-1': 'Generally speaking, the failure of Kurtz randomness is striking, since it can be expressed in terms of observable properties.', '1203.4126-1-49-2': 'That is, the statement that a real number [MATH] is in an effectively closed set [MATH] is a universal property, which means that if [MATH] is not in [MATH], then one can verify this fact by carrying out a finite computation on finitely many bits of [MATH].', '1203.4126-1-49-3': 'Thus, Section [REF] shows that there are real numbers [MATH] that are UD random, but, at any finite level of accuracy, can be seen to satisfy a distinctly nonrandom property.', '1203.4126-1-50-0': 'UD random reals may fail to be random in other striking ways.', '1203.4126-1-50-1': 'For example, one can ask:', '1203.4126-1-51-0': "Is there a UD-random [MATH] such that every initial segment of the binary representation of [MATH] has at least as many 1's as 0's?", '1203.4126-1-51-1': "More generally, does Weyl's theorem hold relative to a suitable measure on this set?", '1203.4126-1-52-0': 'I suspect that the answer to these questions is "yes."', '1203.4126-1-53-0': 'It is interesting to compare the notion of UD randomness to the notion of Church stochasticity.', '1203.4126-1-53-1': 'Roughly speaking, a real number [MATH] is Church stochastic if the zeros in any subsequence of digits of the binary representation of [MATH] obtained by a computable "selection procedure" has a limiting density of one half.', '1203.4126-1-53-2': 'This notion is, in a sense, orthogonal to UD randomness: by a suitable choice of the sequence [MATH], a test for UD randomness can sample bits in any order; but, to compensate, a computable selection procedure is allowed to see the previous bits of [MATH] before deciding whether or not to select the next bit.', '1203.4126-1-53-3': "Ville's theorem [CITATION] (see also [CITATION]) shows that there are real numbers [MATH] that are Church stochastic (in fact, with respect to any countable collection of selection procedures) but have the property that every initial segment of the binary representation of [MATH] has at least as many [MATH]'s as [MATH]'s.", '1203.4126-1-53-4': 'Wang [CITATION] has shown that there are numbers [MATH] that are Schnorr random but not Church stochastic (see also [CITATION]).', '1203.4126-1-53-5': 'By Theorem [REF], this implies that UD randomness does not imply Church stochasticity.', '1203.4126-1-53-6': 'One can ask about the converse direction:', '1203.4126-1-54-0': 'Is there a real number [MATH] that is Church stochastic but not UD random?', '1203.4126-1-55-0': 'Ultimately, I hope the results here suggest that it is interesting and worthwhile to study the relationships between the various notions of randomness that are implicit in ordinary mathematical theorems.'}

{'1203.4126-2-0-0': 'A seminal theorem due to Weyl [CITATION] states that if [MATH] is any sequence of distinct integers, then, for almost every [MATH], the sequence [MATH] is uniformly distributed modulo one.', '1203.4126-2-0-1': 'In particular, for almost every [MATH] in the unit interval, the sequence [MATH] is uniformly distributed modulo one for every computable sequence [MATH] of distinct integers.', '1203.4126-2-0-2': 'Call such an [MATH] UD random.', '1203.4126-2-0-3': 'Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random.', '1203.4126-2-0-4': "On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.", '1203.4126-2-1-0': '# Introduction', '1203.4126-2-2-0': 'Let [MATH] denote the fractional part of a real number [MATH], and let [MATH] denote Lebesgue measure.', '1203.4126-2-2-1': 'A sequence [MATH] of real numbers is said to be uniformly distributed modulo one if for every interval [MATH], [EQUATION].', '1203.4126-2-2-2': 'In words, a sequence is uniformly distributed modulo one if the limiting frequency with which it visits any given interval is what one would expect if the elements of the sequence were chosen at random.', '1203.4126-2-2-3': 'A remarkable theorem by Hermann Weyl [CITATION] states the following:', '1203.4126-2-3-0': 'If [MATH] is any sequence of distinct integers, then for almost every [MATH], [MATH] is uniformly distributed modulo one.', '1203.4126-2-4-0': 'This seminal result lies at the intersection of harmonic analysis, number theory, ergodic theory, and computer science; see, for example, [CITATION].', '1203.4126-2-5-0': 'Now let us restrict attention to computable sequences [MATH] of distinct integers.', '1203.4126-2-5-1': 'Since there are only countably many of these, it follows that almost every [MATH] has the property that [MATH] is uniformly distributed modulo one for each such sequence.', '1203.4126-2-5-2': 'Let us say that such an [MATH] is UD random.', '1203.4126-2-5-3': 'Our goal here is to explore the relationship of UD randomness to other senses in which a real number in the unit interval can be said to be "random."', '1203.4126-2-6-0': 'Section [REF] considers sufficient conditions for UD randomness.', '1203.4126-2-6-1': 'In particular, in the terminology of algorithmic randomness, every Schnorr random element of [MATH] is UD random, but there are Kurtz random elements of [MATH] that are not.', '1203.4126-2-7-0': 'Regarding consequences of UD randomness, recall that a real number [MATH] is normal to a base [MATH] if, when it is expressed in that base, each sequence of [MATH] digits occurs with limiting frequency [MATH].', '1203.4126-2-7-1': 'It is not hard to see that a real number [MATH] is normal with respect to base [MATH] if and only if the sequence [MATH] is uniformly distributed modulo one, so every UD random real is normal to every base.', '1203.4126-2-7-2': 'In fact, more is true: if [MATH] is UD random and [MATH] is any computable sequence of distinct integers, then the limiting frequency of occurrences of any [MATH]-digit block at positions [MATH] of [MATH] is again [MATH].', '1203.4126-2-7-3': 'Thus a UD random number looks random, at least in some ways.', '1203.4126-2-8-0': 'In Section [REF], however, I show that there are straightforward ways that a UD random number can look very nonrandom.', '1203.4126-2-8-1': 'Specifically, consider the set [MATH] of elements [MATH] in [MATH] such that for every [MATH], the [MATH]-th digit in the binary expansion of [MATH] is equal to the [MATH]-st digit.', '1203.4126-2-8-2': "Then Weyl's theorem holds with respect to the natural measure on [MATH], showing that, in particular, almost every element of [MATH] is UD random.", '1203.4126-2-9-0': '# A sufficient condition for UD randomness', '1203.4126-2-10-0': 'The subject of algorithmic randomness [CITATION] aims to characterize different senses in which a real number (or, say, an infinite binary sequence) can be said to be "random."', '1203.4126-2-10-1': 'A subset [MATH] of [MATH] is said to be effectively open if there are computable sequences [MATH] and [MATH] of rational numbers such that [MATH].', '1203.4126-2-10-2': 'An effectively open subset of the unit interval is obtained by restricting [MATH] to [MATH].', '1203.4126-2-10-3': 'A sequence [MATH] of effectively open sets is uniformly effective if the representing sequences [MATH] and [MATH] can be computed by a single algorithm with [MATH] as a parameter.', '1203.4126-2-10-4': 'A Martin-Lof test is a uniformly effective sequence of open sets [MATH] such that for each [MATH], [MATH].', '1203.4126-2-10-5': 'An element [MATH] of [MATH] fails the Martin-Lof test [MATH] if it is an element of [MATH], and passes the test otherwise.', '1203.4126-2-10-6': 'An element [MATH] of [MATH] is Martin-Lof random if it passes every Martin-Lof test.', '1203.4126-2-10-7': 'In other words, [MATH] is Martin-Lof nonrandom if it can be covered, effectively, by arbitrarily small open sets, and is thus contained in an effectively presented null [MATH] set.', '1203.4126-2-11-0': 'One can weaken or strengthen this notion of randomness by restricting or enlarging the class of tests.', '1203.4126-2-11-1': 'For example, a Schnorr test is a Martin-Lof test with the additional property that for each [MATH], the measure of [MATH] is computable.', '1203.4126-2-11-2': 'This has the effect that when enumerating the intervals that cover the nonrandoms with a set of size at most [MATH], one knows how much of the set is yet to come.', '1203.4126-2-11-3': 'An element [MATH] of [MATH] is Schnorr random if it passes every Schnorr test.', '1203.4126-2-12-0': 'An element [MATH] of [MATH] is Kurtz random if it is contained in an effectively closed set of measure [MATH], that is, the complement of an effectively open set of measure [MATH].', '1203.4126-2-12-1': 'Clearly every Martin Lof random element of [MATH] is Schnorr random, and every Schnorr random element is Kurtz random.', '1203.4126-2-12-2': 'The notion of Kurtz randomness is fairly weak.', '1203.4126-2-12-3': 'For example, it is not hard to show that if a real number [MATH] is weakly 1-generic then it is Kurtz random but its binary digits fail to satisfy the law of large of numbers.', '1203.4126-2-12-4': '(See [CITATION] for the definition of weak 1-genericity, and [CITATION] for the facts just mentioned.)', '1203.4126-2-12-5': 'In particular, this shows that there are Kurtz random reals that are not UD random.', '1203.4126-2-13-0': 'The main result of this section is this:', '1203.4126-2-14-0': 'Every Schnorr random element of [MATH] is UD random.', '1203.4126-2-15-0': "The proof of this lemma follows conventional proofs of Weyl's theorem, for example, as presented in [CITATION]; we need only make certain constructive and quantitative aspects of the argument explicit.", '1203.4126-2-15-1': 'To start with, we need an equivalent formulation of uniform distribution modulo one.', '1203.4126-2-16-0': 'For any sequence [MATH] of real numbers, the following are equivalent:', '1203.4126-2-17-0': '[MATH] is uniformly distributed modulo one.', '1203.4126-2-17-1': 'For any continuous function [MATH], [EQUATION]', '1203.4126-2-17-2': 'For any integer [MATH], [EQUATION].', '1203.4126-2-18-0': 'The third property is known as "the Weyl criterion."', '1203.4126-2-18-1': 'Roughly, the second follows from the first using Riemann sums to approximate the integral; the third follows immediately from the second; and the first follows from the third, using trigonometric sums to approximate the characteristic function of an interval.', '1203.4126-2-18-2': 'For details, see any of the references mentioned above.', '1203.4126-2-19-0': 'It will be convenient to adopt the Vinogradov convention of writing [MATH] for [MATH], thus freeing up the symbol [MATH] to be used as an index.', '1203.4126-2-19-1': 'Note that [MATH] and [MATH] for every [MATH] and [MATH].', '1203.4126-2-19-2': 'Theorem [REF] is an immediate consequence of the next lemma, since if [MATH] is a Schnorr random element of [MATH] and [MATH] is any computable sequence of distinct integers, the lemma implies that [MATH] is uniformly distributed modulo one.', '1203.4126-2-20-0': 'For each [MATH], let [MATH] be a sequence of distinct integers, such that [MATH] is computable from [MATH] and [MATH].', '1203.4126-2-20-1': 'Then there is a Schnorr test [MATH] such that for every [MATH] not in [MATH] and every integer [MATH], [MATH] is uniformly distributed modulo one.', '1203.4126-2-21-0': 'For the moment, fix a sequence [MATH] of distinct integers, and for any [MATH] in [MATH] write [EQUATION].', '1203.4126-2-21-1': 'Then we have [EQUATION] and hence [EQUATION] since each term in the sum is [MATH] except when [MATH].', '1203.4126-2-21-2': '(A more perspicuous way of carrying out this calculation is to notice that the left hand side is the square [MATH] of the norm of [MATH] in the Hilbert space [MATH], and the functions [MATH] form an orthornormal set.)', '1203.4126-2-21-3': "Markov's inequality implies that for every [MATH] we have [EQUATION].", '1203.4126-2-21-4': 'In other words, for [MATH] large, [MATH] is small for most [MATH].', '1203.4126-2-22-0': 'Now fix a doubly-indexed sequence [MATH] as in the statement of the lemma, and for each [MATH] write [EQUATION].', '1203.4126-2-22-1': 'Without loss of generality we can assume that for every integer [MATH] and [MATH] the sequence [MATH] already appears as one of the sequences [MATH] for some [MATH], since we can replace the original sequence of sequences with a sequence that includes all such multiples.', '1203.4126-2-22-2': 'By the Weyl criterion, then, it suffices to find a Schnorr test [MATH] such that for each [MATH] not in [MATH] we have [MATH].', '1203.4126-2-23-0': 'A simple calculation shows that for any [MATH] such that [MATH] we have [MATH]; in other words, between [MATH] and [MATH], the averages do not change that much.', '1203.4126-2-23-1': 'This reduces our task to finding a Schnorr test [MATH] such that for each [MATH] not in [MATH] we have [MATH].', '1203.4126-2-24-0': 'For each [MATH], rational [MATH], and [MATH] define [EQUATION].', '1203.4126-2-24-1': 'By the calculation above, we have [EQUATION] which decreases to [MATH] as [MATH] approaches infinity.', '1203.4126-2-24-2': 'Choose an enumeration [MATH] of all pairs [MATH], and for each [MATH] and [MATH] choose [MATH] large enough so that [MATH].', '1203.4126-2-24-3': 'For each [MATH], let [EQUATION].', '1203.4126-2-24-4': 'Then for each [MATH], [MATH].', '1203.4126-2-24-5': 'Moreover, the measure of [MATH] is uniformly computable in [MATH], since for each [MATH] the measure of [MATH] is computable and the measure of [MATH] is at most [MATH].', '1203.4126-2-24-6': 'Thus the sequence [MATH] is a Schnorr test, and we need only confirm that it meets the specification above.', '1203.4126-2-25-0': 'So suppose [MATH] passes this test, that is, for some [MATH], [MATH] is not in [MATH].', '1203.4126-2-25-1': 'Given any [MATH] and [MATH], choose [MATH] such that [MATH].', '1203.4126-2-25-2': 'Since [MATH] is not in [MATH], it is not in [MATH].', '1203.4126-2-25-3': 'This means that for every [MATH], we have [MATH].', '1203.4126-2-25-4': 'Since [MATH] and [MATH] were arbitrary, we have that [MATH] approaches [MATH] for every [MATH], as required.', '1203.4126-2-26-0': 'It is well known that given any Schnorr test, one can find a computable real that passes that test.', '1203.4126-2-26-1': '(See the proof of Proposition 7.1.11 in [CITATION] or the discussion after Definition 3.5.8 in [CITATION].)', '1203.4126-2-26-2': 'As result, the preceding lemma also has the following nice consequence, to the effect that we can actually compute a real number that is UD random with respect to a computable list of computable sequences.', '1203.4126-2-27-0': 'For every uniformly computable sequence of sequences [MATH] of distinct integers there is a computable element [MATH] of [MATH] that is UD random for these sequences; that is, such that [MATH] is uniformly distributed modulo one for each [MATH].', '1203.4126-2-28-0': '# UD random reals with nonrandom properties', '1203.4126-2-29-0': 'The main result of this section is that there are UD random reals that are not Kurtz random.', '1203.4126-2-29-1': 'In fact, I prove something stronger.', '1203.4126-2-29-2': 'Ignoring the countable set of dyadic rationals, every element [MATH] of [MATH] can be uniquely identified with its binary representation, or, equivalently, the element of Cantor space, [MATH], corresponding to the sequence of bits after the (binary) decimal point.', '1203.4126-2-29-3': 'Let [MATH] denote the [MATH]th binary digit of [MATH], and let [EQUATION] that is, the set of binary sequences such that for every [MATH], the [MATH]-th digit is the same as the [MATH]-st digit.', '1203.4126-2-29-4': '(Here and below it is convenient to start the indexing with [MATH], so that the first bit of [MATH] is [MATH] rather than [MATH].)', '1203.4126-2-29-5': 'Clearly [MATH] is an effectively closed null set.', '1203.4126-2-29-6': 'There is a natural bijection [MATH] from [MATH] to [MATH] which takes any infinite binary sequence and inserts the appropriate digits at positions [MATH] for each [MATH].', '1203.4126-2-29-7': 'Let [MATH] be the "uniform" probability measure on [MATH] defined by setting [MATH], where [MATH] is any Borel subset of [MATH] and [MATH] is the uniform measure on [MATH].', '1203.4126-2-29-8': "The following asserts that the analogue of Weyl's theorem holds for this measure on [MATH]:", '1203.4126-2-30-0': 'Let [MATH] be any sequence of distinct integers.', '1203.4126-2-30-1': 'Then for every almost every element [MATH] with respect to [MATH], [MATH] is uniformly distributed modulo one.', '1203.4126-2-31-0': 'This implies that given any countable collection of sequences of distinct integers, in particular all the computable ones, almost every element [MATH] of [MATH] is UD random with respect to this collection.', '1203.4126-2-31-1': 'I am grateful to Terence Tao for suggesting this approach.', '1203.4126-2-32-0': 'We will prove Theorem [REF] as follows.', '1203.4126-2-32-1': "The values [MATH] are known as the Fourier-Stieltjes coefficients of the measure [MATH], and the proof of Weyl's theorem presented in Section [REF] relied on the fact that the Fourier-Stieltjes coefficients [MATH] of Lebesgue measure are [MATH] when [MATH].", '1203.4126-2-32-2': "In fact, the next two lemmas show that for the conclusion of Weyl's theorem to hold with respect to a measure [MATH] it suffices to show that the Fourier-Stieltjes coefficients of [MATH] approach [MATH] sufficiently quickly.", '1203.4126-2-32-3': 'After stating this criterion precisely, we will then show that it is satisfied by the measure [MATH] at hand.', '1203.4126-2-33-0': "First, the proof of Weyl's theorem can be modified to obtain the following result due to Davenport, Erdos, and LeVeque [CITATION]:", '1203.4126-2-34-0': 'Fix a finite measure [MATH] on [MATH], and let [MATH] be a sequence of bounded, [MATH]-integrable functions.', '1203.4126-2-34-1': 'Let [MATH].', '1203.4126-2-34-2': 'If the series [EQUATION] converges for every [MATH], then the sequence [MATH] is uniformly distributed modulo one for [MATH]-almost every [MATH].', '1203.4126-2-35-0': 'The proof can also be found in Kuipers and Niederreiter [CITATION].', '1203.4126-2-35-1': '(In both places, the result is stated for Lebesgue measure, but the proof establishes the more general result.', '1203.4126-2-35-2': 'See also Loeve [CITATION] for a similar argument in a slightly different context.)', '1203.4126-2-35-3': "Lyons [CITATION] uses this fact to establish the following strengthening of Weyl's theorem:", '1203.4126-2-36-0': 'If the Fourier-Stieltjes coefficients of [MATH] have the property that [EQUATION] converges, then for any sequence [MATH] of distinct integers and [MATH]-almost every [MATH], the sequence [MATH] is uniformly distributed modulo one.', '1203.4126-2-37-0': '(Lyons only states the theorem for strictly increasing sequences [MATH], and shows, more generally, that in the hypothesis one can replace the sequence [MATH] by its nonincreasing rearrangement.', '1203.4126-2-37-1': 'It is easy to check, however, that the proof only requires that the values of the sequence [MATH] be distinct.', '1203.4126-2-37-2': 'See also [CITATION].)', '1203.4126-2-37-3': 'Notice that a mild rate of convergence of [MATH] to [MATH], such as [MATH] for any [MATH], is sufficient to meet the criteria of Lemma [REF].', '1203.4126-2-38-0': 'We will show that with the choice of [MATH] and [MATH] above, we have [MATH], which suffices to prove Theorem [REF].', '1203.4126-2-38-1': 'The intuition behind the theorem is that multiplying [MATH] by some number and testing for membership in an interval only enables one to probe "local" properties of binary expansion of [MATH], and so a sequence [MATH] cannot exploit correlations between bits that are far apart.', '1203.4126-2-38-2': 'This intuition is reflected in the fact that the mass of [MATH] is distributed sufficiently uniformly throughout the interval [MATH] that periodic functions like [MATH] do not detect significant irregularities.', '1203.4126-2-38-3': 'The rest of this section is devoted to carrying out the calculations that back up these intuitions.', '1203.4126-2-39-0': 'Since we are only estimating [MATH] up to a multiplicative constant, we can assume that [MATH].', '1203.4126-2-39-1': 'Find [MATH] such that [MATH].', '1203.4126-2-39-2': 'Viewing [MATH] as a subset of [MATH], write [EQUATION] where:', '1203.4126-2-40-0': "Let [MATH] denote a string of [MATH]'s of length [MATH], let [MATH], and for any [MATH], let [MATH] denote the translate of [MATH] (now viewed as a subset of [MATH]) by [MATH].", '1203.4126-2-40-1': 'For any binary sequence [MATH], let [MATH] denote the binary number denoted by [MATH].', '1203.4126-2-40-2': 'Notice that as [MATH] ranges over [MATH], [MATH] ranges from [MATH] to [MATH], and as [MATH] ranges over [MATH], [MATH] ranges from [MATH] to [MATH].', '1203.4126-2-40-3': 'Thus we can describe [MATH] as [EQUATION].', '1203.4126-2-40-4': 'Using the identity [MATH], the integral we need to evaluate, [MATH], is therefore equal to', '1203.4126-2-41-0': 'Since [MATH] for every [MATH], by the triangle inequality the absolute value of the first term is at most the cardinality of [MATH], which is [MATH].', '1203.4126-2-41-1': 'Since [MATH] for every [MATH] and [MATH], the second term is a geometric progression.', '1203.4126-2-41-2': 'If [MATH] divides [MATH], each term in the sum is equal to [MATH].', '1203.4126-2-41-3': 'Otherwise, using the identity [MATH] with [MATH] and [MATH], and computing [EQUATION] we have that the second term is equal to [EQUATION]', '1203.4126-2-41-4': 'A similar calculation shows that the third term is equal to [EQUATION]', '1203.4126-2-41-5': 'There is nothing more to say about the fourth term, for the moment.', '1203.4126-2-41-6': 'The absolute value of the last term, the integral, is at most [EQUATION] though we will have to evaluate this term more precisely later on.', '1203.4126-2-42-0': 'We now consider cases, keeping in mind that [MATH] if [MATH] is an integer, and [MATH] if [MATH] is equal to half an odd integer.', '1203.4126-2-42-1': 'If [MATH] is divisible by [MATH] but not [MATH], the second term, and hence the product, is [MATH].', '1203.4126-2-43-0': 'If [MATH] is divisible by [MATH] but not [MATH], the second term is equal to [MATH], the third term is equal to [EQUATION] and the fourth term is equal to [EQUATION] since [MATH] is equal to half an odd integer.', '1203.4126-2-43-1': 'But the product of the third and fourth terms is [EQUATION] since the denominator of the third term is of the form [MATH], where the fourth term is [MATH].', '1203.4126-2-43-2': 'In absolute value, then, the product of these terms is at most [MATH].', '1203.4126-2-43-3': 'Thus the absolute value of [MATH] is at most [EQUATION] since [MATH].', '1203.4126-2-44-0': 'If [MATH] is divisible by [MATH] but not [MATH], then the third term, and hence the product, is equal to [MATH].', '1203.4126-2-44-1': 'If [MATH] is divisible by [MATH] but not [MATH], then [MATH] is an integer and [MATH] is a half integer, in which case the fourth term, and hence the product, is equal to [MATH].', '1203.4126-2-45-0': 'Hence we are left with two cases: first, where [MATH] is not divisible by [MATH], and, second, where [MATH] is divisible by [MATH].', '1203.4126-2-45-1': 'In both cases, we need to further expand the value of [MATH].', '1203.4126-2-45-2': "Write [EQUATION] where, as before, [MATH] is a string of [MATH]'s of length [MATH], and [MATH], [MATH], [MATH] are obtained by replacing [MATH] by [MATH] in the definitions of [MATH], [MATH], and [MATH], respectively.", '1203.4126-2-45-3': 'Define [MATH] analogously.', '1203.4126-2-45-4': 'As above, [MATH] evaluates to [EQUATION]', '1203.4126-2-46-0': 'In the case where [MATH] is divisible by [MATH], the argument proceeds as before.', '1203.4126-2-46-1': 'Since, by assumption, [MATH], we know that [MATH] is not divisible by [MATH], and the argument terminates after the second new case.', '1203.4126-2-47-0': 'Thus we are left with the situation where [MATH] is not divisible by [MATH], in which case [MATH] evaluates to the following: [EQUATION]', '1203.4126-2-47-1': 'Notice that the denominator of the second fraction is equal to the numerator of the third.', '1203.4126-2-47-2': 'Also, the denominator of the first fraction is of the form [MATH], where the numerator of the second fraction is [MATH].', '1203.4126-2-47-3': 'Simplifying the quotient then leaves a multiplicand with absolute value at most 2; and similarly for the denominator of the third fraction and the numerator of the fourth.', '1203.4126-2-47-4': 'The numerator of the first fraction has absolute value at most 2, as do the fourth and seventh terms in the product.', '1203.4126-2-47-5': 'Finally, the absolute value of the last term is at most [EQUATION].', '1203.4126-2-47-6': 'Thus the absolute value of [MATH] is at most [EQUATION].', '1203.4126-2-47-7': 'All we need to do now is to get a lower bound on the denominator of the last fraction, and then use the lower bound [MATH] on [MATH] to show that the resulting expression is [MATH].', '1203.4126-2-47-8': 'When [MATH] is any real value satisfying [MATH], a Taylor-series approximation shows [EQUATION].', '1203.4126-2-47-9': 'Thus when [MATH], we have [MATH].', '1203.4126-2-47-10': 'As a result, the absolute value of [MATH] is at most [EQUATION] since we have [MATH].', '1203.4126-2-48-0': '# Final comments', '1203.4126-2-49-0': 'The goal here has been to explore the extent to which real numbers satisfying the conclusion of Weyl\'s theorem for computable sequences look "random."', '1203.4126-2-49-1': 'Generally speaking, the failure of Kurtz randomness is striking, since it can be expressed in terms of observable properties.', '1203.4126-2-49-2': 'That is, the statement that a real number [MATH] is in an effectively closed set [MATH] is a universal property, which means that if [MATH] is not in [MATH], then one can verify this fact by carrying out a finite computation on finitely many bits of [MATH].', '1203.4126-2-49-3': 'Thus, Section [REF] shows that there are real numbers [MATH] that are UD random, but, at any finite level of accuracy, can be seen to satisfy a distinctly nonrandom property.', '1203.4126-2-50-0': 'UD random reals may fail to be random in other striking ways.', '1203.4126-2-50-1': 'For example, one can ask:', '1203.4126-2-51-0': "Is there a UD random [MATH] such that every initial segment of the binary representation of [MATH] has at least as many 1's as 0's?", '1203.4126-2-51-1': "More generally, does Weyl's theorem hold relative to a suitable measure on this set?", '1203.4126-2-52-0': 'I suspect that the answer to these questions is "yes."', '1203.4126-2-53-0': 'It is interesting to compare the notion of UD randomness to the notion of Church stochasticity.', '1203.4126-2-53-1': 'Roughly speaking, a real number [MATH] is Church stochastic if the zeros in any subsequence of digits of the binary representation of [MATH] obtained by a computable "selection procedure" has a limiting density of one half.', '1203.4126-2-53-2': 'This notion is, in a sense, orthogonal to UD randomness: by a suitable choice of the sequence [MATH], a test for UD randomness can sample bits in any order; but, to compensate, a computable selection procedure is allowed to see the previous bits of [MATH] before deciding whether or not to select the next bit.', '1203.4126-2-53-3': "Ville's theorem [CITATION] (see also [CITATION]) shows that there are real numbers [MATH] that are Church stochastic (in fact, with respect to any countable collection of selection procedures) but have the property that every initial segment of the binary representation of [MATH] has at least as many [MATH]'s as [MATH]'s.", '1203.4126-2-53-4': 'Wang [CITATION] has shown that there are numbers [MATH] that are Schnorr random but not Church stochastic (see also [CITATION]).', '1203.4126-2-53-5': 'By Theorem [REF], this implies that UD randomness does not imply Church stochasticity.', '1203.4126-2-53-6': 'One can ask about the converse direction:', '1203.4126-2-54-0': 'Is there a real number [MATH] that is Church stochastic but not UD random?', '1203.4126-2-55-0': 'See [CITATION] for various notions of stochasticity and some of their properties.', '1203.4126-2-56-0': 'Ultimately, I hope the results here suggest that it is interesting and worthwhile to study the relationships between notions of randomness that are implicit in ordinary mathematical theorems.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1203.4126

1802.01500

{'1802.01500-1-0-0': 'Deep learning approaches have made tremendous progress in the field of semantic segmentation over the past few years.', '1802.01500-1-0-1': 'However, most current approaches operate in the 2D image space.', '1802.01500-1-0-2': 'Direct semantic segmentation of unstructured 3D point clouds is still an open research problem.', '1802.01500-1-0-3': 'The recently proposed PointNet architecture presents an interesting step ahead in that it can operate on unstructured point clouds, achieving encouraging segmentation results.', '1802.01500-1-0-4': 'However, it subdivides the input points into a grid of blocks and processes each such block individually.', '1802.01500-1-0-5': 'In this paper, we investigate the question how such an architecture can be extended to incorporate larger-scale spatial context.', '1802.01500-1-0-6': 'We build upon PointNet and propose two extensions that enlarge the receptive field over the 3D scene.', '1802.01500-1-0-7': 'We evaluate the proposed strategies on challenging indoor and outdoor datasets and show improved results in both scenarios.', '1802.01500-1-1-0': '# Introduction', '1802.01500-1-2-0': 'Semantic segmentation is an important capability for intelligent vehicles, such as autonomous cars or mobile robots.', '1802.01500-1-2-1': 'Identifying the semantic meaning of the observed 3D structure around the vehicle is a prerequisite for solving subsequent tasks such as navigation or reconstruction [CITATION].', '1802.01500-1-2-2': 'Consequently, the problem has attracted a lot of attention, and notable successes have been achieved with the help of deep learning techniques.', '1802.01500-1-2-3': 'However, most state-of-the-art semantic segmentation approaches operate on 2D images, which naturally lend themselves to processing with Convolutional Neural Networks (CNNs) [CITATION].', '1802.01500-1-3-0': 'Processing unstructured 3D point clouds, such as those obtained from LiDAR or stereo sensors, is a much harder problem, and it is only recently that first successful deep learning approaches have been proposed for this task [CITATION].', '1802.01500-1-3-1': "Such point clouds can be obtained from LiDAR sensors mounted on top of a recording vehicle or they can be obtained from visual SLAM approaches operating on the vehicle's cameras [CITATION].", '1802.01500-1-3-2': 'Finding approaches that can directly operate on point cloud data is highly desirable, since it avoids costly preprocessing and format conversion steps.', '1802.01500-1-3-3': 'However, the question what is the best network architecture to process unstructured 3D point clouds is still largely open.', '1802.01500-1-4-0': 'In this paper, we take inspiration from the recent PointNet work by Qi [CITATION], which currently defines the state of the art in 3D semantic segmentation.', '1802.01500-1-4-1': 'PointNet learns a higher dimensional spatial feature representation for each 3D point and then aggregates all the points within a small 3D volume (typically an occupancy grid cell) in order to bring in some form of 3D neighborhood context.', '1802.01500-1-4-2': 'However, this neighborhood context is very restricted, as each grid cell is processed independently.', '1802.01500-1-5-0': 'In this paper, we investigate possible mechanisms to incorporate context into a point cloud processing architecture.', '1802.01500-1-5-1': 'We focus on spatial context, which has been identified as being very important for semantic segmentation [CITATION].', '1802.01500-1-5-2': 'We introduce two mechanisms to add spatial context to an existing PointNet.', '1802.01500-1-5-3': 'The first mechanism incorporates neighborhood information by processing input data from multiple scales or multiple adjacent regions together (input-level context).', '1802.01500-1-5-4': 'The second mechanism operates on the estimated point descriptors and aims at consolidating them by exchanging information over a larger spatial neighborhood (output-level context).', '1802.01500-1-5-5': 'For both mechanisms, we explore several possible realizations and compare them experimentally.', '1802.01500-1-5-6': 'As our results show, both mechanisms improve semantic segmentation quality.', '1802.01500-1-6-0': 'Contributions.', '1802.01500-1-6-1': 'The key contributions of our work can be summarized as follows: (1) We present two mechanisms that can be used to incorporate spatial context into semantic 3D point cloud segmentation.', '1802.01500-1-6-2': '(2) We show how these mechanisms can be incorporated into the PointNet pipeline.', '1802.01500-1-6-3': '(3) We verify experimentally that our proposed extensions achieve improved results on challenging indoor and outdoor datasets.', '1802.01500-1-7-0': '# Related Work', '1802.01500-1-8-0': 'Unstructured Point Clouds.', '1802.01500-1-8-1': 'A varied number of sensors and setups exist which help to obtain unstructured point clouds: areal data from airborne laser scanners, laser scanners mounted on dynamic setups in a push-broom configuration [CITATION], rotating lasers e.g. Velodyne [CITATION], or static lasers [CITATION].', '1802.01500-1-8-2': 'Additionally, indoor spaces can be scanned using devices such as the Microsoft Kinect [CITATION] or Matterport cameras [CITATION].', '1802.01500-1-8-3': 'All these devices produce point clouds of different quality and density.', '1802.01500-1-8-4': 'We apply our method to indoor data from [CITATION] and to synthetic urban outdoor data from [CITATION].', '1802.01500-1-9-0': 'Traditional Methods.', '1802.01500-1-9-1': 'Hackel et al. [CITATION] use traditional random forest classifiers with 3D features (without color).', '1802.01500-1-9-2': 'Their method is based on eigenvalues and eigenvectors of covariance tensors created by the nearest neighbors of the points.', '1802.01500-1-9-3': 'Their main contribution is an efficient approximate nearest neighbors computation at different scales.', '1802.01500-1-9-4': 'Munoz et al. [CITATION] follow a similar approach but replace the random forest classifier with an associative Markov network.', '1802.01500-1-9-5': 'Random forest classifiers are also used in [CITATION] to classify data from 2D images and 3D point clouds, which they later fuse.', '1802.01500-1-9-6': 'Similarly, Xu et al. [CITATION] fuse camera and LiDAR sensor data.', '1802.01500-1-9-7': 'Xiong et al. [CITATION] propose a sequential parsing procedure that learns the spatial relationships of objects.', '1802.01500-1-9-8': 'Lai et al. [CITATION] introduce a hierarchical sparse coding technique for learning features from synthetic data.', '1802.01500-1-9-9': 'Vosselman et al. [CITATION] combine multiple segmentation and post-processing methods to achieve useful point cloud segmentations.', '1802.01500-1-10-0': 'Deep-learning Methods.', '1802.01500-1-10-1': 'In a deep learning context, point clouds can be represented in a regular volumetric grid in order to apply 3D convolutions [CITATION].', '1802.01500-1-10-2': 'Alternatively, 3D points can be mapped to a 2D representation followed by 2D convolutions [CITATION].', '1802.01500-1-10-3': 'In [CITATION], the authors are performing 2D convolutions in 2D snapshots of a 3D point cloud and then project the labels back to 3D space.', '1802.01500-1-10-4': 'In [CITATION] a deep learning framework learns semantic segmentation by tracking point clouds.', '1802.01500-1-10-5': 'Yi et al. [CITATION] use spectral CNNs on 3D models represented as shape graphs for shape part segmentation.', '1802.01500-1-10-6': 'Recent methods operate directly on raw point clouds with KD-trees [CITATION] or fully convolutional layers [CITATION].', '1802.01500-1-11-0': '# Method', '1802.01500-1-12-0': 'In this section we start by reviewing the PointNet model, then we introduce our mechanisms of extending context and finish by describing our two exemplary architectures.', '1802.01500-1-13-0': '## PointNet', '1802.01500-1-14-0': 'PointNet [CITATION] is a deep neural network that, when used for semantic segmentation, takes as input a point cloud and outputs the per point semantic class labels.', '1802.01500-1-14-1': 'First, it splits a point cloud into 3D blocks, then it takes [MATH] points inside a block and after a series of Multi-Layer-Perceptrons (MLP) per point, the points are mapped into a higher dimensional space [MATH], these are called local point-features.', '1802.01500-1-14-2': 'Max-pooling is applied to aggregate information from all the points resulting in a common global-feature invariant to input permutations.', '1802.01500-1-14-3': 'The global-feature is then concatenated with all the point-features.', '1802.01500-1-14-4': 'After another series of MLPs these combined features are used to predict the [MATH] output class scores.', '1802.01500-1-14-5': 'Figure [REF] shows a simplified model.', '1802.01500-1-15-0': 'Caveats.', '1802.01500-1-15-1': 'The global-features in PointNet summarize the context of a single block (block-feature), as a result the aggregated information is passed only among points inside the same block.', '1802.01500-1-16-0': 'Context outside a block is equally important and could help make more informed class label predictions.', '1802.01500-1-16-1': 'Therefore we introduce two mechanisms to add context: input-level context - which operates directly on the input point clouds - and output-level context - which consolidates the output from the input-level context.', '1802.01500-1-17-0': '## Input-Level Context', '1802.01500-1-18-0': 'In this straightforward addition, we increase the context of the network by considering a group of blocks simultaneously instead of one individual block at a time as done in PointNet.', '1802.01500-1-18-1': 'Context is shared among all blocks in a group.', '1802.01500-1-18-2': 'These groups of blocks are selected either from the same position but at multiple different scales (Multi-Scale Blocks, see Figure [REF], left) or from neighboring cells in a regular grid (Grid Blocks, see Figure [REF], left).', '1802.01500-1-18-3': 'For each input block, we compute a block-feature using the mechanism from PointNet.', '1802.01500-1-18-4': 'For the multi-scale version, we train a block-descriptor for each scale individually to obtain scale-dependent block-features.', '1802.01500-1-18-5': 'In the case of grid blocks, all block features are computed by a shared single-scale block-descriptor.', '1802.01500-1-18-6': 'In the end, both approaches output a set of block-features corresponding to the input blocks.', '1802.01500-1-19-0': '## Output-Level Context', '1802.01500-1-20-0': 'At this stage, we further consolidate the block-features obtained from the previous stage.', '1802.01500-1-20-1': 'Here, we differ between two consolidation approaches:', '1802.01500-1-21-0': 'Consolidation Units (CU) consume a set of point features, transform them into a higher dimensional space using MLPs and apply max-pooling to generate a common block-feature which is again concatenated with each of the high dimensional input features (see Figure [REF], blue box).', '1802.01500-1-21-1': 'This procedure is similar to the block-feature mechanism of PointNet.', '1802.01500-1-21-2': 'The key point is that CUs can be chained together into a sequence CUs forming a deeper network.', '1802.01500-1-21-3': 'The intuition behind this setup is as follows: In the beginning each point sees only its own features.', '1802.01500-1-21-4': 'After appending the block-features, each point is additionally informed about the features of its neighboring points.', '1802.01500-1-21-5': 'By applying CUs multiple times, this shared knowledge is reinforced.', '1802.01500-1-22-0': 'Recurrent Consolidation Units (RCU) are the second type of context consolidation we employ.', '1802.01500-1-22-1': 'RCUs take as input a sequence of block-features originating from spatially nearby blocks and return a sequence of corresponding updated block-features.', '1802.01500-1-22-2': 'The core idea is to create block-features that take into consideration neighboring blocks as well.', '1802.01500-1-22-3': 'In more detail, RCUs are implemented as RNNs, specifically GRUs [CITATION], which are a simpler variation of standard LSTMs [CITATION].', '1802.01500-1-22-4': 'GRUs have the capability to learn long range dependencies.', '1802.01500-1-22-5': 'That range can either be over time (as in speech recognition) or over space as in our case.', '1802.01500-1-22-6': 'The cells of the unrolled GRU are connected in an unsynchronized many-to-many fashion (see Figure [REF], blue box).', '1802.01500-1-22-7': 'This means that the updated block-features are returned only after the GRU has seen the whole input sequence of block-features.', '1802.01500-1-22-8': 'Intuitively, GRU retain relevant information about the scene in their internal memory and update it according to new observations.', '1802.01500-1-22-9': 'We use this memory mechanism to consolidate and share the information across all input blocks.', '1802.01500-1-22-10': 'For example, the decision about whether a point belongs to a chair is changed if the network remembers that it has seen a table further down in the room.', '1802.01500-1-23-0': 'In the following, we describe two exemplary architectures which combine the previously introduced components.', '1802.01500-1-23-1': 'For those, we provide a detailed evaluation and report improved results in Section [REF].', '1802.01500-1-24-0': '## Multi-Scale (MS) Architecture', '1802.01500-1-25-0': 'The full MS architecture is displayed in Figure [REF].', '1802.01500-1-25-1': 'The learned block-features from the multi-scale blocks, (see Section [REF]) are concatenated into one multi-scale block-feature.', '1802.01500-1-25-2': 'This multi-scale block-feature is further concatenated with the transformed input point-features and passed through a series of CUs (see Section [REF]).', '1802.01500-1-25-3': 'Applying a final MLP results in output scores for each input point.', '1802.01500-1-26-0': 'Specific for this architecture is the sampling procedure to select the positions of the multi-scale blocks: We randomly pick a [MATH]-dimensional point from the input point cloud as the center of the blocks and we group together [MATH] randomly selected points that fall within a specified radius.', '1802.01500-1-26-1': 'This procedure is repeated at the same point for multiple radii.', '1802.01500-1-27-0': '## Grid (G) Architecture', '1802.01500-1-28-0': 'Figure [REF] shows the pipeline of the architecture with grid input blocks.', '1802.01500-1-28-1': 'It consists of the following components: The input level context is a group of four blocks from a 2x2 grid-neighborhood (see Section [REF]) is fed into a series of MLPs that transform the point features, with weights shared among all blocks.', '1802.01500-1-28-2': 'These block-features are passed to an RCU that updates the individual block-features with common context from all neighboring blocks.', '1802.01500-1-28-3': 'The updated block-features are then concatenated with the original block-features.', '1802.01500-1-28-4': 'They are then used, along with the local features, for class predictions.', '1802.01500-1-28-5': 'After a series of fully connected layers the output of class scores is computed for each point.', '1802.01500-1-29-0': '# Experiments', '1802.01500-1-30-0': 'For experimental evaluation, we compare our two architectures with PointNet [CITATION], the current state-of-the-art semantic segmentation method directly operating on point clouds.', '1802.01500-1-30-1': 'We produce quantitative results for our models and the baseline on two challenging datasets: Stanford Large-Scale 3D Indoor Spaces (S3DIS) [CITATION] and on Virtual KITTI (vKITTI) [CITATION].', '1802.01500-1-30-2': 'Additionally, we provide qualitative results on point clouds obtained from a Velodyne HDL-64E LiDAR scanner from the KITTI dataset [CITATION].', '1802.01500-1-30-3': 'We will now describe these datasets in more detail.', '1802.01500-1-31-0': 'Stanford Large Scale 3D Indoor Scenes.', '1802.01500-1-31-1': 'This dataset is composed of 6 different large scale indoor areas, mainly conference rooms, personal offices and open spaces.', '1802.01500-1-31-2': 'It contains dense 3D point clouds scanned using a Matterport camera.', '1802.01500-1-31-3': 'Each point is labeled with one of the 13 semantic classes listed in Table [REF].', '1802.01500-1-31-4': 'Using the reference implementation of PointNet, we were able to reproduce the results reported by Qi et al. [CITATION], see Table [REF].', '1802.01500-1-31-5': 'Throughout the paper, we follow the same evaluation protocol used in [CITATION], which is a 6-fold cross validation over all the areas.', '1802.01500-1-32-0': 'Virtual KITTI.', '1802.01500-1-32-1': 'Due to the lack of semantically annotated large-scale outdoor datasets, we rely on the photo-realistic synthetic vKITTI dataset which closely mimics the real-world KITTI dataset.', '1802.01500-1-32-2': 'It consists of 5 different monocular video sequences in urban settings, fully annotated with depth and pixel-level semantic labels.', '1802.01500-1-32-3': 'In total there are 13 semantic class, listed in Table [REF].', '1802.01500-1-32-4': 'For our purposes, we project the given 2D depth into 3D space to obtain semantically annotated 3D point clouds.', '1802.01500-1-32-5': 'Conveniently, this procedure results in point clouds that resemble the varying density of real world point clouds obtained by Velodyne LiDAR scanners (see Figure [REF]).', '1802.01500-1-32-6': 'For test and training, we split the original sequences into 6 non-overlapping subsequences.', '1802.01500-1-32-7': 'The final train-test sets are created by choosing point clouds from each subsequence at regular time-intervals.', '1802.01500-1-32-8': 'For evaluation, we also follow the 6-fold cross validation protocol.', '1802.01500-1-33-0': '## Evaluation Measures', '1802.01500-1-34-0': 'As in [CITATION], we evaluate on the intersection over union (IoU), the average per class accuracy and overall accuracy.', '1802.01500-1-34-1': 'Intersection over union is computed as: [EQUATION] where TP is the number of true positives, FP the number of false positives and FN the number of false negatives.', '1802.01500-1-35-0': '## Quantitative Results', '1802.01500-1-36-0': 'In this section, we analyze the effectiveness of the input-block schemes and the consolidation units exemplary on the two previously introduced models.', '1802.01500-1-36-1': 'As input features, we differentiate between geometry (XYZ) and geometry with color (XYZ+RGB).', '1802.01500-1-37-0': 'Geometry with Color.', '1802.01500-1-37-1': 'First, we compare the grid-blocks in combination with a recurrent consolidation block (G+RCU) to the original PointNet.', '1802.01500-1-37-2': 'Using the same evaluation setup as described in [CITATION] we are able to show improved results over PointNet, see Table [REF] and Table [REF].', '1802.01500-1-37-3': 'This proves our hypothesis that RCU are able to convey context among blocks and thus improving results.', '1802.01500-1-37-4': 'During training, each room is split into blocks of 1x1 m on the ground plane.', '1802.01500-1-37-5': 'Each block extends over the whole room height.', '1802.01500-1-37-6': 'Neighboring blocks are overlapping by 0.5 meters in both directions.', '1802.01500-1-37-7': 'We select four blocks simultaneously from a 2x2 grid-neighborhood (see Figure [REF], left).', '1802.01500-1-37-8': 'Each block contains 4096 points.', '1802.01500-1-37-9': 'The unrolled GRU is 8 cells long (4 input, 4 output).', '1802.01500-1-37-10': "It's memory size is 64.", '1802.01500-1-37-11': 'During testing, the room is split into non-overlapping blocks and evaluated on all 2x2 groups of blocks.', '1802.01500-1-37-12': 'Each block is evaluated only once.', '1802.01500-1-38-0': 'Next, we take a look at the multi-scale input block with consolidation units (MS-CU) model.', '1802.01500-1-38-1': 'To build the multi-scale blocks, we follow the process described in Section [REF].', '1802.01500-1-38-2': 'As radii, we choose [0.25, 0.5, 1.0] m.', '1802.01500-1-38-3': 'As distance metric we choose the Chebyshev-distance which generates axis-aligned rectangular blocks.', '1802.01500-1-38-4': 'The middle scale block is equal to the PointNet block regarding shape and size.', '1802.01500-1-39-0': 'By using sampling (necessary for the multi-scale block construction), we diverge from the previous training procedure so we re-run all experiments under these new conditions.', '1802.01500-1-40-0': "We validate the influence of each of the architecture's components by adding them one-by-one to our pipeline and evaluating after each step, see Table [REF] and Table [REF].", '1802.01500-1-40-1': 'First, we only consider the input-level context i.e the multi-scale block feature (MS) as input to our pipeline while skipping the consolidation units.', '1802.01500-1-40-2': 'This shows some performance benefits over PointNet but not as much as one would expect considering the enlarged input context.', '1802.01500-1-40-3': 'Next, we take only single-scale input blocks and add one consolidation unit (SS+CU(1)).', '1802.01500-1-40-4': 'The results show that the CU outperforms the MS input blocks.', '1802.01500-1-40-5': "It also shows that CUs provide a simple technique to boost the network's performance.", '1802.01500-1-40-6': 'Finally, we combine both the MS blocks and the CU while appending another CU to the network (MS+CC(2)).', '1802.01500-1-40-7': 'This full model is depicted in Figure [REF].', '1802.01500-1-41-0': 'Geometry only.', '1802.01500-1-41-1': 'Until now, each input point was described by a 9-dimensional feature vector [MATH] where [MATH] are the spatial coordinates of a point, [MATH] its color and [MATH] the normalized coordinated based on the size of the environment, see [CITATION] for further details.', '1802.01500-1-41-2': 'Without doubt, color is a very strong input feature in the context of semantic segmentation.', '1802.01500-1-41-3': 'In this section, we pose the question what will happen if no color information is available like it is the case with point clouds obtained from laser scanners.', '1802.01500-1-41-4': 'To simulate the missing colors, we simply discard the color information from the input feature and re-run the experiments.', '1802.01500-1-41-5': 'Table [REF] and [REF] show the obtained results.', '1802.01500-1-41-6': 'See caption for discussion of the results.', '1802.01500-1-42-0': '## Qualitative Results', '1802.01500-1-43-0': 'We present qualitative results of our models applied to indoor scenarios in Figure [REF] and outdoor results in Figure [REF] along with a short discussion.', '1802.01500-1-43-1': 'Additionally, we applied our pre-trained geometry-only model (vKITTI) to real-world laser data.', '1802.01500-1-43-2': 'The results are shown in Figure [REF] and Figure [REF].', '1802.01500-1-44-0': '# Conclusion', '1802.01500-1-45-0': 'In this work, we investigated the question how to incorporate spatial context into a neural network architecture for 3D semantic segmentation.', '1802.01500-1-45-1': 'Building upon PointNet, we proposed two extension (Input-level context and Output-level context) which we successfully applied onto indoor and outdoor datasets.', '1802.01500-1-45-2': 'Still, numerous other combinations remain possible.', '1802.01500-1-45-3': 'The full exploration of the design space is left for future work.', '1802.01500-1-46-0': '-13px [t]1 [t]0.99', '1802.01500-1-47-0': '0.33 PointNet [CITATION]', '1802.01500-1-48-0': '0.33 Ours, MS-CU(2)', '1802.01500-1-49-0': '0.33 Ground Truth', '1802.01500-1-50-0': 'Figure8: Outdoor qualitative results.', '1802.01500-1-50-1': 'Dataset: Virtual KITTI [CITATION].', '1802.01500-1-50-2': 'Results were obtained using only XYZ coordinates as input, no color information was used.', '1802.01500-1-50-3': 'Left: baseline method PointNet.', '1802.01500-1-50-4': 'Center: our results using the MS-CU model as illustrated in Figure [REF].', '1802.01500-1-50-5': 'Right: ground truth semantic labels.', '1802.01500-1-50-6': 'The outputs of our method are less fragmented (cars, houses) and finer structures like street lights and poles are recognized better.', '1802.01500-1-51-0': '[t]0.475', '1802.01500-1-52-0': '3px', '1802.01500-1-53-0': '0.45 -5px Ground truth Labels: top', '1802.01500-1-54-0': '0.45 -5px Our prediction Labels: below', '1802.01500-1-55-0': '1px', '1802.01500-1-56-0': '1px 1px 1px 1px 1px', '1802.01500-1-57-0': "Figure9:Qualitative results on 3DRMS'17 Challenge.", '1802.01500-1-57-1': 'We trained our model on vKITTI point clouds without color and applied it to the 3DRMS laser data.', '1802.01500-1-57-2': 'Training and test datasets do not have the same semantic labels.', '1802.01500-1-57-3': 'Despite that, common classes like trees are successfully segmented and plausible ones are given otherwise (e.g. terrain instead of grass, guardrail instead of obstacle).', '1802.01500-1-58-0': '[t]0.475'}

{'1802.01500-2-0-0': 'Deep learning approaches have made tremendous progress in the field of semantic segmentation over the past few years.', '1802.01500-2-0-1': 'However, most current approaches operate in the 2D image space.', '1802.01500-2-0-2': 'Direct semantic segmentation of unstructured 3D point clouds is still an open research problem.', '1802.01500-2-0-3': 'The recently proposed PointNet architecture presents an interesting step ahead in that it can operate on unstructured point clouds, achieving encouraging segmentation results.', '1802.01500-2-0-4': 'However, it subdivides the input points into a grid of blocks and processes each such block individually.', '1802.01500-2-0-5': 'In this paper, we investigate the question how such an architecture can be extended to incorporate larger-scale spatial context.', '1802.01500-2-0-6': 'We build upon PointNet and propose two extensions that enlarge the receptive field over the 3D scene.', '1802.01500-2-0-7': 'We evaluate the proposed strategies on challenging indoor and outdoor datasets and show improved results in both scenarios.', '1802.01500-2-1-0': '# Introduction', '1802.01500-2-2-0': 'Semantic segmentation is an important capability for intelligent vehicles, such as autonomous cars or mobile robots.', '1802.01500-2-2-1': 'Identifying the semantic meaning of the observed 3D structure around the vehicle is a prerequisite for solving subsequent tasks such as navigation or reconstruction [CITATION].', '1802.01500-2-2-2': 'Consequently, the problem has attracted a lot of attention, and notable successes have been achieved with the help of deep learning techniques.', '1802.01500-2-2-3': 'However, most state-of-the-art semantic segmentation approaches operate on 2D images, which naturally lend themselves to processing with Convolutional Neural Networks (CNNs) [CITATION].', '1802.01500-2-3-0': 'Processing unstructured 3D point clouds, such as those obtained from LiDAR or stereo sensors, is a much harder problem, and it is only recently that first successful deep learning approaches have been proposed for this task [CITATION].', '1802.01500-2-3-1': "Such point clouds can be obtained from LiDAR sensors mounted on top of a recording vehicle or they can be obtained from visual SLAM approaches operating on the vehicle's cameras [CITATION].", '1802.01500-2-3-2': 'Finding approaches that can directly operate on point cloud data is highly desirable, since it avoids costly preprocessing and format conversion steps.', '1802.01500-2-3-3': 'However, the question what is the best network architecture to process unstructured 3D point clouds is still largely open.', '1802.01500-2-4-0': 'In this paper, we take inspiration from the recent PointNet work by Qi [CITATION], which currently defines the state of the art in 3D semantic segmentation.', '1802.01500-2-4-1': 'PointNet learns a higher dimensional spatial feature representation for each 3D point and then aggregates all the points within a small 3D volume (typically an occupancy grid cell) in order to bring in some form of 3D neighborhood context.', '1802.01500-2-4-2': 'However, this neighborhood context is very restricted, as each grid cell is processed independently.', '1802.01500-2-5-0': 'In this paper, we investigate possible mechanisms to incorporate context into a point cloud processing architecture.', '1802.01500-2-5-1': 'We focus on spatial context, which has been identified as being very important for semantic segmentation [CITATION].', '1802.01500-2-5-2': 'We introduce two mechanisms to add spatial context to an existing PointNet.', '1802.01500-2-5-3': 'The first mechanism incorporates neighborhood information by processing input data from multiple scales or multiple adjacent regions together (input-level context).', '1802.01500-2-5-4': 'The second mechanism operates on the estimated point descriptors and aims at consolidating them by exchanging information over a larger spatial neighborhood (output-level context).', '1802.01500-2-5-5': 'For both mechanisms, we explore several possible realizations and compare them experimentally.', '1802.01500-2-5-6': 'As our results show, both mechanisms improve semantic segmentation quality.', '1802.01500-2-6-0': 'Contributions.', '1802.01500-2-6-1': 'The key contributions of our work can be summarized as follows: (1) We present two mechanisms that can be used to incorporate spatial context into semantic 3D point cloud segmentation.', '1802.01500-2-6-2': '(2) We show how these mechanisms can be incorporated into the PointNet pipeline.', '1802.01500-2-6-3': '(3) We verify experimentally that our proposed extensions achieve improved results on challenging indoor and outdoor datasets.', '1802.01500-2-7-0': '# Related Work', '1802.01500-2-8-0': 'Unstructured Point Clouds.', '1802.01500-2-8-1': 'A varied number of sensors and setups exist which help to obtain unstructured point clouds: areal data from airborne laser scanners, laser scanners mounted on dynamic setups in a push-broom configuration [CITATION], rotating lasers e.g. Velodyne [CITATION], or static lasers [CITATION].', '1802.01500-2-8-2': 'Additionally, indoor spaces can be scanned using devices such as the Microsoft Kinect [CITATION] or Matterport cameras [CITATION].', '1802.01500-2-8-3': 'All these devices produce point clouds of different quality and density.', '1802.01500-2-8-4': 'We apply our method to indoor data from [CITATION] and to synthetic urban outdoor data from [CITATION].', '1802.01500-2-9-0': 'Traditional Methods.', '1802.01500-2-9-1': 'Hackel et al. [CITATION] use traditional random forest classifiers with 3D features (without color).', '1802.01500-2-9-2': 'Their method is based on eigenvalues and eigenvectors of covariance tensors created by the nearest neighbors of the points.', '1802.01500-2-9-3': 'Their main contribution is an efficient approximate nearest neighbors computation at different scales.', '1802.01500-2-9-4': 'Munoz et al. [CITATION] follow a similar approach but replace the random forest classifier with an associative Markov network.', '1802.01500-2-9-5': 'Random forest classifiers are also used in [CITATION] to classify data from 2D images and 3D point clouds, which they later fuse.', '1802.01500-2-9-6': 'Similarly, Xu et al. [CITATION] fuse camera and LiDAR sensor data.', '1802.01500-2-9-7': 'Xiong et al. [CITATION] propose a sequential parsing procedure that learns the spatial relationships of objects.', '1802.01500-2-9-8': 'Lai et al. [CITATION] introduce a hierarchical sparse coding technique for learning features from synthetic data.', '1802.01500-2-9-9': 'Vosselman et al. [CITATION] combine multiple segmentation and post-processing methods to achieve useful point cloud segmentations.', '1802.01500-2-10-0': 'Deep-learning Methods.', '1802.01500-2-10-1': 'In a deep learning context, point clouds can be represented in a regular volumetric grid in order to apply 3D convolutions [CITATION].', '1802.01500-2-10-2': 'Alternatively, 3D points can be mapped to a 2D representation followed by 2D convolutions [CITATION].', '1802.01500-2-10-3': 'In [CITATION], the authors are performing 2D convolutions in 2D snapshots of a 3D point cloud and then project the labels back to 3D space.', '1802.01500-2-10-4': 'In [CITATION] a deep learning framework learns semantic segmentation by tracking point clouds.', '1802.01500-2-10-5': 'Yi et al. [CITATION] use spectral CNNs on 3D models represented as shape graphs for shape part segmentation.', '1802.01500-2-10-6': 'Recent methods operate directly on raw point clouds with KD-trees [CITATION] or fully convolutional layers [CITATION].', '1802.01500-2-11-0': '# Method', '1802.01500-2-12-0': 'In this section we start by reviewing the PointNet model, then we introduce our mechanisms of extending context and finish by describing our two exemplary architectures.', '1802.01500-2-13-0': '## PointNet', '1802.01500-2-14-0': 'PointNet [CITATION] is a deep neural network that, when used for semantic segmentation, takes as input a point cloud and outputs the per point semantic class labels.', '1802.01500-2-14-1': 'First, it splits a point cloud into 3D blocks, then it takes [MATH] points inside a block and after a series of Multi-Layer-Perceptrons (MLP) per point, the points are mapped into a higher dimensional space [MATH], these are called local point-features.', '1802.01500-2-14-2': 'Max-pooling is applied to aggregate information from all the points resulting in a common global-feature invariant to input permutations.', '1802.01500-2-14-3': 'The global-feature is then concatenated with all the point-features.', '1802.01500-2-14-4': 'After another series of MLPs these combined features are used to predict the [MATH] output class scores.', '1802.01500-2-14-5': 'Figure [REF] shows a simplified model.', '1802.01500-2-15-0': 'Caveats.', '1802.01500-2-15-1': 'The global-features in PointNet summarize the context of a single block (block-feature), as a result the aggregated information is passed only among points inside the same block.', '1802.01500-2-16-0': 'Context outside a block is equally important and could help make more informed class label predictions.', '1802.01500-2-16-1': 'Therefore we introduce two mechanisms to add context: input-level context - which operates directly on the input point clouds - and output-level context - which consolidates the output from the input-level context.', '1802.01500-2-17-0': '## Input-Level Context', '1802.01500-2-18-0': 'In this straightforward addition, we increase the context of the network by considering a group of blocks simultaneously instead of one individual block at a time as done in PointNet.', '1802.01500-2-18-1': 'Context is shared among all blocks in a group.', '1802.01500-2-18-2': 'These groups of blocks are selected either from the same position but at multiple different scales (Multi-Scale Blocks, see Figure [REF], left) or from neighboring cells in a regular grid (Grid Blocks, see Figure [REF], left).', '1802.01500-2-18-3': 'For each input block, we compute a block-feature using the mechanism from PointNet.', '1802.01500-2-18-4': 'For the multi-scale version, we train a block-descriptor for each scale individually to obtain scale-dependent block-features.', '1802.01500-2-18-5': 'In the case of grid blocks, all block features are computed by a shared single-scale block-descriptor.', '1802.01500-2-18-6': 'In the end, both approaches output a set of block-features corresponding to the input blocks.', '1802.01500-2-19-0': '## Output-Level Context', '1802.01500-2-20-0': 'At this stage, we further consolidate the block-features obtained from the previous stage.', '1802.01500-2-20-1': 'Here, we differ between two consolidation approaches:', '1802.01500-2-21-0': 'Consolidation Units (CU) consume a set of point features, transform them into a higher dimensional space using MLPs and apply max-pooling to generate a common block-feature which is again concatenated with each of the high dimensional input features (see Figure [REF], blue box).', '1802.01500-2-21-1': 'This procedure is similar to the block-feature mechanism of PointNet.', '1802.01500-2-21-2': 'The key point is that CUs can be chained together into a sequence CUs forming a deeper network.', '1802.01500-2-21-3': 'The intuition behind this setup is as follows: In the beginning each point sees only its own features.', '1802.01500-2-21-4': 'After appending the block-features, each point is additionally informed about the features of its neighboring points.', '1802.01500-2-21-5': 'By applying CUs multiple times, this shared knowledge is reinforced.', '1802.01500-2-22-0': 'Recurrent Consolidation Units (RCU) are the second type of context consolidation we employ.', '1802.01500-2-22-1': 'RCUs take as input a sequence of block-features originating from spatially nearby blocks and return a sequence of corresponding updated block-features.', '1802.01500-2-22-2': 'The core idea is to create block-features that take into consideration neighboring blocks as well.', '1802.01500-2-22-3': 'In more detail, RCUs are implemented as RNNs, specifically GRUs [CITATION], which are a simpler variation of standard LSTMs [CITATION].', '1802.01500-2-22-4': 'GRUs have the capability to learn long range dependencies.', '1802.01500-2-22-5': 'That range can either be over time (as in speech recognition) or over space as in our case.', '1802.01500-2-22-6': 'The cells of the unrolled GRU are connected in an unsynchronized many-to-many fashion (see Figure [REF], blue box).', '1802.01500-2-22-7': 'This means that the updated block-features are returned only after the GRU has seen the whole input sequence of block-features.', '1802.01500-2-22-8': 'Intuitively, GRU retain relevant information about the scene in their internal memory and update it according to new observations.', '1802.01500-2-22-9': 'We use this memory mechanism to consolidate and share the information across all input blocks.', '1802.01500-2-22-10': 'For example, the decision about whether a point belongs to a chair is changed if the network remembers that it has seen a table further down in the room.', '1802.01500-2-23-0': 'In the following, we describe two exemplary architectures which combine the previously introduced components.', '1802.01500-2-23-1': 'For those, we provide a detailed evaluation and report improved results in Section [REF].', '1802.01500-2-24-0': '## Multi-Scale (MS) Architecture', '1802.01500-2-25-0': 'The full MS architecture is displayed in Figure [REF].', '1802.01500-2-25-1': 'The learned block-features from the multi-scale blocks, (see Section [REF]) are concatenated into one multi-scale block-feature.', '1802.01500-2-25-2': 'This multi-scale block-feature is further concatenated with the transformed input point-features and passed through a series of CUs (see Section [REF]).', '1802.01500-2-25-3': 'Applying a final MLP results in output scores for each input point.', '1802.01500-2-26-0': 'Specific for this architecture is the sampling procedure to select the positions of the multi-scale blocks: We randomly pick a [MATH]-dimensional point from the input point cloud as the center of the blocks and we group together [MATH] randomly selected points that fall within a specified radius.', '1802.01500-2-26-1': 'This procedure is repeated at the same point for multiple radii.', '1802.01500-2-27-0': '## Grid (G) Architecture', '1802.01500-2-28-0': 'Figure [REF] shows the pipeline of the architecture with grid input blocks.', '1802.01500-2-28-1': 'It consists of the following components: The input level context is a group of four blocks from a 2x2 grid-neighborhood (see Section [REF]) is fed into a series of MLPs that transform the point features, with weights shared among all blocks.', '1802.01500-2-28-2': 'These block-features are passed to an RCU that updates the individual block-features with common context from all neighboring blocks.', '1802.01500-2-28-3': 'The updated block-features are then concatenated with the original block-features.', '1802.01500-2-28-4': 'They are then used, along with the local features, for class predictions.', '1802.01500-2-28-5': 'After a series of fully connected layers the output of class scores is computed for each point.', '1802.01500-2-29-0': '# Experiments', '1802.01500-2-30-0': 'For experimental evaluation, we compare our two architectures with PointNet [CITATION], the current state-of-the-art semantic segmentation method directly operating on point clouds.', '1802.01500-2-30-1': 'We produce quantitative results for our models and the baseline on two challenging datasets: Stanford Large-Scale 3D Indoor Spaces (S3DIS) [CITATION] and on Virtual KITTI (vKITTI) [CITATION].', '1802.01500-2-30-2': 'Additionally, we provide qualitative results on point clouds obtained from a Velodyne HDL-64E LiDAR scanner from the KITTI dataset [CITATION].', '1802.01500-2-30-3': 'We will now describe these datasets in more detail.', '1802.01500-2-31-0': 'Stanford Large Scale 3D Indoor Scenes.', '1802.01500-2-31-1': 'This dataset is composed of 6 different large scale indoor areas, mainly conference rooms, personal offices and open spaces.', '1802.01500-2-31-2': 'It contains dense 3D point clouds scanned using a Matterport camera.', '1802.01500-2-31-3': 'Each point is labeled with one of the 13 semantic classes listed in Table [REF].', '1802.01500-2-31-4': 'Using the reference implementation of PointNet, we were able to reproduce the results reported by Qi et al. [CITATION], see Table [REF].', '1802.01500-2-31-5': 'Throughout the paper, we follow the same evaluation protocol used in [CITATION], which is a 6-fold cross validation over all the areas.', '1802.01500-2-32-0': 'Virtual KITTI.', '1802.01500-2-32-1': 'Due to the lack of semantically annotated large-scale outdoor datasets, we rely on the photo-realistic synthetic vKITTI dataset which closely mimics the real-world KITTI dataset.', '1802.01500-2-32-2': 'It consists of 5 different monocular video sequences in urban settings, fully annotated with depth and pixel-level semantic labels.', '1802.01500-2-32-3': 'In total there are 13 semantic class, listed in Table [REF].', '1802.01500-2-32-4': 'For our purposes, we project the given 2D depth into 3D space to obtain semantically annotated 3D point clouds.', '1802.01500-2-32-5': 'Conveniently, this procedure results in point clouds that resemble the varying density of real world point clouds obtained by Velodyne LiDAR scanners (see Figure [REF]).', '1802.01500-2-32-6': 'For test and training, we split the original sequences into 6 non-overlapping subsequences.', '1802.01500-2-32-7': 'The final train-test sets are created by choosing point clouds from each subsequence at regular time-intervals.', '1802.01500-2-32-8': 'For evaluation, we also follow the 6-fold cross validation protocol.', '1802.01500-2-33-0': '## Evaluation Measures', '1802.01500-2-34-0': 'As in [CITATION], we evaluate on the intersection over union (IoU), the average per class accuracy and overall accuracy.', '1802.01500-2-34-1': 'Intersection over union is computed as: [EQUATION] where TP is the number of true positives, FP the number of false positives and FN the number of false negatives.', '1802.01500-2-35-0': '## Quantitative Results', '1802.01500-2-36-0': 'In this section, we analyze the effectiveness of the input-block schemes and the consolidation units exemplary on the two previously introduced models.', '1802.01500-2-36-1': 'As input features, we differentiate between geometry (XYZ) and geometry with color (XYZ+RGB).', '1802.01500-2-37-0': 'Geometry with Color.', '1802.01500-2-37-1': 'First, we compare the grid-blocks in combination with a recurrent consolidation block (G+RCU) to the original PointNet.', '1802.01500-2-37-2': 'Using the same evaluation setup as described in [CITATION] we are able to show improved results over PointNet, see Table [REF] and Table [REF].', '1802.01500-2-37-3': 'This proves our hypothesis that RCU are able to convey context among blocks and thus improving results.', '1802.01500-2-37-4': 'During training, each room is split into blocks of 1x1 m on the ground plane.', '1802.01500-2-37-5': 'Each block extends over the whole room height.', '1802.01500-2-37-6': 'Neighboring blocks are overlapping by 0.5 meters in both directions.', '1802.01500-2-37-7': 'We select four blocks simultaneously from a 2x2 grid-neighborhood (see Figure [REF], left).', '1802.01500-2-37-8': 'Each block contains 4096 points.', '1802.01500-2-37-9': 'The unrolled GRU is 8 cells long (4 input, 4 output).', '1802.01500-2-37-10': "It's memory size is 64.", '1802.01500-2-37-11': 'During testing, the room is split into non-overlapping blocks and evaluated on all 2x2 groups of blocks.', '1802.01500-2-37-12': 'Each block is evaluated only once.', '1802.01500-2-38-0': 'Next, we take a look at the multi-scale input block with consolidation units (MS-CU) model.', '1802.01500-2-38-1': 'To build the multi-scale blocks, we follow the process described in Section [REF].', '1802.01500-2-38-2': 'As radii, we choose [0.25, 0.5, 1.0] m.', '1802.01500-2-38-3': 'As distance metric we choose the Chebyshev-distance which generates axis-aligned rectangular blocks.', '1802.01500-2-38-4': 'The middle scale block is equal to the PointNet block regarding shape and size.', '1802.01500-2-39-0': 'By using sampling (necessary for the multi-scale block construction), we diverge from the previous training procedure so we re-run all experiments under these new conditions.', '1802.01500-2-40-0': "We validate the influence of each of the architecture's components by adding them one-by-one to our pipeline and evaluating after each step, see Table [REF] and Table [REF].", '1802.01500-2-40-1': 'First, we only consider the input-level context i.e the multi-scale block feature (MS) as input to our pipeline while skipping the consolidation units.', '1802.01500-2-40-2': 'This shows some performance benefits over PointNet but not as much as one would expect considering the enlarged input context.', '1802.01500-2-40-3': 'Next, we take only single-scale input blocks and add one consolidation unit (SS+CU(1)).', '1802.01500-2-40-4': 'The results show that the CU outperforms the MS input blocks.', '1802.01500-2-40-5': "It also shows that CUs provide a simple technique to boost the network's performance.", '1802.01500-2-40-6': 'Finally, we combine both the MS blocks and the CU while appending another CU to the network (MS+CC(2)).', '1802.01500-2-40-7': 'This full model is depicted in Figure [REF].', '1802.01500-2-41-0': 'Geometry only.', '1802.01500-2-41-1': 'Until now, each input point was described by a 9-dimensional feature vector [MATH] where [MATH] are the spatial coordinates of a point, [MATH] its color and [MATH] the normalized coordinated based on the size of the environment, see [CITATION] for further details.', '1802.01500-2-41-2': 'Without doubt, color is a very strong input feature in the context of semantic segmentation.', '1802.01500-2-41-3': 'In this section, we pose the question what will happen if no color information is available like it is the case with point clouds obtained from laser scanners.', '1802.01500-2-41-4': 'To simulate the missing colors, we simply discard the color information from the input feature and re-run the experiments.', '1802.01500-2-41-5': 'Table [REF] and [REF] show the obtained results.', '1802.01500-2-41-6': 'See caption for discussion of the results.', '1802.01500-2-42-0': '## Qualitative Results', '1802.01500-2-43-0': 'We present qualitative results of our models applied to indoor scenarios in Figure [REF] and outdoor results in Figure [REF] along with a short discussion.', '1802.01500-2-43-1': 'Additionally, we applied our pre-trained geometry-only model (vKITTI) to real-world laser data.', '1802.01500-2-43-2': 'The results are shown in Figure [REF] and Figure [REF].', '1802.01500-2-44-0': '# Conclusion', '1802.01500-2-45-0': 'In this work, we investigated the question how to incorporate spatial context into a neural network architecture for 3D semantic segmentation.', '1802.01500-2-45-1': 'Building upon PointNet, we proposed two extension (Input-level context and Output-level context) which we successfully applied onto indoor and outdoor datasets.', '1802.01500-2-45-2': 'Still, numerous other combinations remain possible.', '1802.01500-2-45-3': 'The full exploration of the design space is left for future work.', '1802.01500-2-46-0': '-13px [t]1 [t]0.99', '1802.01500-2-47-0': '0.33 PointNet [CITATION]', '1802.01500-2-48-0': '0.33 Ours, MS-CU(2)', '1802.01500-2-49-0': '0.33 Ground Truth', '1802.01500-2-50-0': 'Figure8: Outdoor qualitative results.', '1802.01500-2-50-1': 'Dataset: Virtual KITTI [CITATION].', '1802.01500-2-50-2': 'Results were obtained using only XYZ coordinates as input, no color information was used.', '1802.01500-2-50-3': 'Left: baseline method PointNet.', '1802.01500-2-50-4': 'Center: our results using the MS-CU model as illustrated in Figure [REF].', '1802.01500-2-50-5': 'Right: ground truth semantic labels.', '1802.01500-2-50-6': 'The outputs of our method are less fragmented (cars, houses) and finer structures like street lights and poles are recognized better.', '1802.01500-2-51-0': '[t]0.475', '1802.01500-2-52-0': '3px', '1802.01500-2-53-0': '0.45 -5px Ground truth Labels: top', '1802.01500-2-54-0': '0.45 -5px Our prediction Labels: below', '1802.01500-2-55-0': '1px', '1802.01500-2-56-0': '1px 1px 1px 1px 1px', '1802.01500-2-57-0': "Figure9:Qualitative results on 3DRMS'17 Challenge.", '1802.01500-2-57-1': 'We trained our model on vKITTI point clouds without color and applied it to the 3DRMS laser data.', '1802.01500-2-57-2': 'Training and test datasets do not have the same semantic labels.', '1802.01500-2-57-3': 'Despite that, common classes like trees are successfully segmented and plausible ones are given otherwise (e.g. terrain instead of grass, guardrail instead of obstacle).', '1802.01500-2-58-0': '[t]0.475'}

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'1802.01500-2-39-0'], ['1802.01500-1-38-0', '1802.01500-2-38-0'], ['1802.01500-1-38-1', '1802.01500-2-38-1'], ['1802.01500-1-38-3', '1802.01500-2-38-3'], ['1802.01500-1-38-4', '1802.01500-2-38-4'], ['1802.01500-1-36-0', '1802.01500-2-36-0'], ['1802.01500-1-36-1', '1802.01500-2-36-1'], ['1802.01500-1-3-0', '1802.01500-2-3-0'], ['1802.01500-1-3-1', '1802.01500-2-3-1'], ['1802.01500-1-3-2', '1802.01500-2-3-2'], ['1802.01500-1-3-3', '1802.01500-2-3-3'], ['1802.01500-1-12-0', '1802.01500-2-12-0'], ['1802.01500-1-30-0', '1802.01500-2-30-0'], ['1802.01500-1-30-1', '1802.01500-2-30-1'], ['1802.01500-1-30-2', '1802.01500-2-30-2'], ['1802.01500-1-30-3', '1802.01500-2-30-3'], ['1802.01500-1-28-0', '1802.01500-2-28-0'], ['1802.01500-1-28-1', '1802.01500-2-28-1'], ['1802.01500-1-28-2', '1802.01500-2-28-2'], ['1802.01500-1-28-3', '1802.01500-2-28-3'], ['1802.01500-1-28-4', '1802.01500-2-28-4'], ['1802.01500-1-28-5', '1802.01500-2-28-5'], ['1802.01500-1-41-1', '1802.01500-2-41-1'], ['1802.01500-1-41-2', '1802.01500-2-41-2'], ['1802.01500-1-41-3', '1802.01500-2-41-3'], ['1802.01500-1-41-4', '1802.01500-2-41-4'], ['1802.01500-1-41-5', '1802.01500-2-41-5'], ['1802.01500-1-41-6', '1802.01500-2-41-6'], ['1802.01500-1-45-0', '1802.01500-2-45-0'], ['1802.01500-1-45-1', '1802.01500-2-45-1'], ['1802.01500-1-45-2', '1802.01500-2-45-2'], ['1802.01500-1-45-3', '1802.01500-2-45-3'], ['1802.01500-1-26-0', '1802.01500-2-26-0'], ['1802.01500-1-26-1', '1802.01500-2-26-1'], ['1802.01500-1-2-0', '1802.01500-2-2-0'], ['1802.01500-1-2-1', '1802.01500-2-2-1'], ['1802.01500-1-2-2', '1802.01500-2-2-2'], ['1802.01500-1-2-3', '1802.01500-2-2-3'], ['1802.01500-1-20-0', '1802.01500-2-20-0'], ['1802.01500-1-0-0', '1802.01500-2-0-0'], ['1802.01500-1-0-1', '1802.01500-2-0-1'], ['1802.01500-1-0-2', '1802.01500-2-0-2'], ['1802.01500-1-0-3', '1802.01500-2-0-3'], ['1802.01500-1-0-4', '1802.01500-2-0-4'], ['1802.01500-1-0-5', '1802.01500-2-0-5'], ['1802.01500-1-0-6', '1802.01500-2-0-6'], ['1802.01500-1-0-7', '1802.01500-2-0-7'], ['1802.01500-1-9-1', '1802.01500-2-9-1'], ['1802.01500-1-9-2', '1802.01500-2-9-2'], ['1802.01500-1-9-3', '1802.01500-2-9-3'], ['1802.01500-1-9-4', '1802.01500-2-9-4'], ['1802.01500-1-9-5', '1802.01500-2-9-5'], ['1802.01500-1-9-6', '1802.01500-2-9-6'], ['1802.01500-1-9-7', '1802.01500-2-9-7'], ['1802.01500-1-9-8', '1802.01500-2-9-8'], ['1802.01500-1-9-9', '1802.01500-2-9-9'], ['1802.01500-1-31-0', '1802.01500-2-31-0'], ['1802.01500-1-31-1', '1802.01500-2-31-1'], ['1802.01500-1-31-2', '1802.01500-2-31-2'], ['1802.01500-1-31-3', '1802.01500-2-31-3'], ['1802.01500-1-31-4', '1802.01500-2-31-4'], ['1802.01500-1-31-5', '1802.01500-2-31-5'], ['1802.01500-1-25-0', '1802.01500-2-25-0'], ['1802.01500-1-25-1', '1802.01500-2-25-1'], ['1802.01500-1-25-2', '1802.01500-2-25-2'], ['1802.01500-1-25-3', '1802.01500-2-25-3'], ['1802.01500-1-4-0', '1802.01500-2-4-0'], ['1802.01500-1-4-1', '1802.01500-2-4-1'], ['1802.01500-1-4-2', '1802.01500-2-4-2'], ['1802.01500-1-32-1', '1802.01500-2-32-1'], ['1802.01500-1-32-2', '1802.01500-2-32-2'], ['1802.01500-1-32-3', '1802.01500-2-32-3'], ['1802.01500-1-32-4', '1802.01500-2-32-4'], ['1802.01500-1-32-5', '1802.01500-2-32-5'], ['1802.01500-1-32-6', '1802.01500-2-32-6'], ['1802.01500-1-32-7', '1802.01500-2-32-7'], ['1802.01500-1-32-8', '1802.01500-2-32-8'], ['1802.01500-1-5-0', '1802.01500-2-5-0'], ['1802.01500-1-5-1', '1802.01500-2-5-1'], ['1802.01500-1-5-2', '1802.01500-2-5-2'], ['1802.01500-1-5-3', '1802.01500-2-5-3'], ['1802.01500-1-5-4', '1802.01500-2-5-4'], ['1802.01500-1-5-5', '1802.01500-2-5-5'], ['1802.01500-1-5-6', '1802.01500-2-5-6'], ['1802.01500-1-43-0', '1802.01500-2-43-0'], ['1802.01500-1-43-1', '1802.01500-2-43-1'], ['1802.01500-1-43-2', '1802.01500-2-43-2'], ['1802.01500-1-21-0', '1802.01500-2-21-0'], ['1802.01500-1-21-1', '1802.01500-2-21-1'], ['1802.01500-1-21-2', '1802.01500-2-21-2'], ['1802.01500-1-21-3', '1802.01500-2-21-3'], ['1802.01500-1-21-4', '1802.01500-2-21-4'], ['1802.01500-1-21-5', '1802.01500-2-21-5'], ['1802.01500-1-16-0', '1802.01500-2-16-0'], ['1802.01500-1-16-1', '1802.01500-2-16-1'], ['1802.01500-1-50-0', '1802.01500-2-50-0'], ['1802.01500-1-50-1', '1802.01500-2-50-1'], ['1802.01500-1-50-2', '1802.01500-2-50-2'], ['1802.01500-1-50-3', '1802.01500-2-50-3'], ['1802.01500-1-50-4', '1802.01500-2-50-4'], ['1802.01500-1-50-5', '1802.01500-2-50-5'], ['1802.01500-1-50-6', '1802.01500-2-50-6'], ['1802.01500-1-23-0', '1802.01500-2-23-0'], ['1802.01500-1-23-1', '1802.01500-2-23-1'], ['1802.01500-1-22-0', '1802.01500-2-22-0'], ['1802.01500-1-22-1', '1802.01500-2-22-1'], ['1802.01500-1-22-2', '1802.01500-2-22-2'], ['1802.01500-1-22-3', '1802.01500-2-22-3'], ['1802.01500-1-22-4', '1802.01500-2-22-4'], ['1802.01500-1-22-5', '1802.01500-2-22-5'], ['1802.01500-1-22-6', '1802.01500-2-22-6'], ['1802.01500-1-22-7', '1802.01500-2-22-7'], ['1802.01500-1-22-8', '1802.01500-2-22-8'], ['1802.01500-1-22-9', '1802.01500-2-22-9'], ['1802.01500-1-22-10', '1802.01500-2-22-10'], ['1802.01500-1-10-1', '1802.01500-2-10-1'], ['1802.01500-1-10-2', '1802.01500-2-10-2'], ['1802.01500-1-10-3', '1802.01500-2-10-3'], ['1802.01500-1-10-4', '1802.01500-2-10-4'], ['1802.01500-1-10-5', '1802.01500-2-10-5'], ['1802.01500-1-10-6', '1802.01500-2-10-6'], ['1802.01500-1-34-0', '1802.01500-2-34-0'], ['1802.01500-1-34-1', '1802.01500-2-34-1']]

[]

[]

[]

[]

['1802.01500-1-6-0', '1802.01500-1-8-0', '1802.01500-1-9-0', '1802.01500-1-10-0', '1802.01500-1-15-0', '1802.01500-1-20-1', '1802.01500-1-32-0', '1802.01500-1-37-0', '1802.01500-1-38-2', '1802.01500-1-41-0', '1802.01500-1-46-0', '1802.01500-1-47-0', '1802.01500-1-48-0', '1802.01500-1-49-0', '1802.01500-1-51-0', '1802.01500-1-52-0', '1802.01500-1-53-0', '1802.01500-1-54-0', '1802.01500-1-55-0', '1802.01500-1-56-0', '1802.01500-1-58-0', '1802.01500-2-6-0', '1802.01500-2-8-0', '1802.01500-2-9-0', '1802.01500-2-10-0', '1802.01500-2-15-0', '1802.01500-2-20-1', '1802.01500-2-32-0', '1802.01500-2-37-0', '1802.01500-2-38-2', '1802.01500-2-41-0', '1802.01500-2-46-0', '1802.01500-2-47-0', '1802.01500-2-48-0', '1802.01500-2-49-0', '1802.01500-2-51-0', '1802.01500-2-52-0', '1802.01500-2-53-0', '1802.01500-2-54-0', '1802.01500-2-55-0', '1802.01500-2-56-0', '1802.01500-2-58-0']

{'1': 'http://creativecommons.org/licenses/by-nc-sa/4.0/', '2': 'http://creativecommons.org/licenses/by-nc-sa/4.0/'}

https://arxiv.org/abs/1802.01500

1505.04499

{'1505.04499-1-0-0': 'The Magellanic clouds are uniquely placed to study the stellar contribution to dust emission.', '1505.04499-1-0-1': 'Individual stars can be resolved in these systems even in the mid-infrared, and they are close enough to allow detection of infrared excess caused by dust.', '1505.04499-1-0-2': 'We have searched the Space Telescope data archive for all Infrared Spectrograph (IRS) staring-mode observations of the Small Magellanic Cloud (SMC) and found that 209 Infrared Array Camera (IRAC) point sources within the footprint of the Surveying the Agents of Galaxy Evolution in the Small Magellanic Cloud (SAGE-SMC) Legacy programme were targeted, within a total of 311 staring mode observations.', '1505.04499-1-0-3': 'We classify these point sources using a decision tree method of object classification, based on infrared spectral features, continuum and spectral energy distribution shape, bolometric luminosity, cluster membership and variability information.', '1505.04499-1-0-4': 'We find 58 asymptotic giant branch (AGB) stars, 51 young stellar objects (YSOs), 4 post-AGB objects, 22 Red Supergiants (RSGs), 27 stars (of which 23 are dusty OB stars), 24 planetary nebulae (PNe), 10 Wolf-Rayet (WR) stars, 3 regions, 3 R Coronae Borealis (R CrB) stars, 1 Blue Supergiant and 6 other objects, including 2 foreground AGB stars.', '1505.04499-1-0-5': 'We use these classifications to evaluate the success of photometric classification methods reported in the literature.', '1505.04499-1-1-0': '# Introduction', '1505.04499-1-2-0': 'The Mega-Surveying the Agents of Galaxy Evolution (Mega-SAGE) project has obtained infrared photometric and spectroscopic inventories of the Magellanic Clouds with the Space Telescope (hereafter ), using and Legacy Programmes.', '1505.04499-1-2-1': 'The initial SAGE survey detected and catalogued [MATH]6.9 million point sources in the Large Magellanic Cloud (LMC), while the SAGE-SMC survey detected and catalogued [MATH]2.2 million point sources in the Small Magellanic Cloud (SMC).', '1505.04499-1-2-2': 'Both surveys used all bands of the Infrared Array Camera and the Multi-Band Imaging Photometer for instruments on board .', '1505.04499-1-2-3': 'The resolution of the IRAC observations is [MATH]2, while for the MIPS bands, three different resolutions apply: 6, 18, and 40 for the 24, 70 and 160 bands respectively .', '1505.04499-1-2-4': "To follow up on these programmes, the SAGE-Spec project obtained 196 staring-mode pointings using 's Infrared Spectrograph of positions selected from the SAGE catalogue.", '1505.04499-1-2-5': 'SAGE-Spec will relate SAGE photometry to the spectral characteristics of different types of objects in both Magellanic Clouds, and ultimately, allow us to classify photometric point sources in both the LMC and SMC.', '1505.04499-1-2-6': 'This characterisation of the point sources observed in the SAGE-Spec survey, and the IRS data archive, builds an inventory of dusty sources and their interrelation in each of the Magellanic Clouds.', '1505.04499-1-2-7': 'In a first step towards this goal, [CITATION] classified the initial 196 LMC point sources, using a decision tree method of object classification, based on infrared (IR) spectral features, continuum and spectral energy distribution (SED) shape, bolometric luminosity, cluster membership and variability information.', '1505.04499-1-2-8': 'The initial classification of LMC objects is being extended to [MATH]1,000 point sources, covering all archival IRS observations within the SAGE footprint (Woods et al. in prep.)', '1505.04499-1-3-0': 'To extend the LMC classifications to the SMC, we have searched the data archive for IRS staring-mode observations and found 311 spectra, yielding 209 unique and genuine point sources with IRS data within the footprint of the SAGE-SMC Legacy programme.', '1505.04499-1-3-1': 'The data used in the classification process are described in Section [REF].', '1505.04499-1-3-2': 'In Section [REF] we discuss the classification method, and in Section [REF] we classify the 209 SMC point sources using the decision tree method.', '1505.04499-1-3-3': 'Finally, in Section [REF] we compare spectral versus colour classifications by means of colour-magnitude diagrams (CMDs).', '1505.04499-1-3-4': 'We use our spectroscopic classifications to test photometric classification methods, e.g. those by [CITATION]; [CITATION] and [CITATION].', '1505.04499-1-3-5': 'The classification of each of these 209 sources is part of the data delivery of the SAGE-Spec Legacy project to the Science Center and the community.', '1505.04499-1-3-6': 'These classifications will also be used to benchmark a colour-classification scheme that will be applied to all point sources in the SAGE and SAGE-SMC surveys (Marengo et al. in prep).', '1505.04499-1-4-0': '# Data preparation', '1505.04499-1-5-0': '## IRS staring mode observations', '1505.04499-1-6-0': 'The IRS on board covers the wavelength range 5-38.', '1505.04499-1-6-1': 'For the low-resolution mode, the spectrum splits in two bands, short-low (SL: 5.2-14.5) and long-low (LL: 14.0-38.0), with almost perpendicular slits.', '1505.04499-1-6-2': 'Each segment splits into a range covered at second order (SL2, LL2), and one at first order (SL1, LL1).', '1505.04499-1-6-3': 'The resolution varies between 60 and 130.', '1505.04499-1-6-4': 'The high resolution mode covers the wavelength range from 10-19.6 (short-high; SH) and from 18.7-37.2 (long-high; LH) with a spectral resolution of [MATH].', '1505.04499-1-7-0': 'We identified 311 IRS low- and high-resolution staring mode observations within the footprint of the SAGE-SMC survey , not necessarily associated with a point source.', '1505.04499-1-7-1': 'We numbered these SMC IRS 1-311, by ordering them by observing program (Project ID; PID) first, and then the Astronomical Observation Request (AOR) number (Table [REF]).', '1505.04499-1-7-2': 'Where available (for SL and LL observations only), the reduced spectra were downloaded from the Cornell Atlas of IRS Sources , in a full resolution grid, using the optimal extraction method.', '1505.04499-1-7-3': 'At the moment, the CASSIS database only contains SL and LL data, but it turns out that there are no point sources in the SMC targeted with only SH and LH, so we will use the SL and LL data only.', '1505.04499-1-7-4': 'The data were downloaded in the Infrared Processing and Analysis Center (IPAC) Table format; the file names are also given in Table [REF].', '1505.04499-1-7-5': 'As an example, the upper left panel of Fig. [REF] shows the spectrum for point source SMC IRS 110, with the SL2 and LL2 data from CASSIS in red and the SL1 and LL1 data in blue.', '1505.04499-1-8-0': "The CASSIS-reduced IRS data header provides the original PI's requested (REQ) position and the field of view (FOV) position, i.e. where the telescope actually pointed (usually, but not always coincident within 1 of the REQ position).", '1505.04499-1-8-1': 'In many cases, however, neither of these two positions is the same as the position at which the spectrum is actually extracted from the slit (EXT), using the optimal extraction method.', '1505.04499-1-8-2': "In order to check the spectrum position for each source, slit images were generated by over-plotting the IRS SL and LL slit positions (recorded in each AOR's BCD FITS header) on a 360 360 image extracted from the SAGE-SMC 8 image.", '1505.04499-1-8-3': 'The REQ and EXT positions were also over-plotted on the image.', '1505.04499-1-8-4': 'In cases where the spectrum was extracted at the FOV position (and thus an EXT position is lacking), the FOV position was overplotted instead (see upper right panel of Fig. [REF] for an example for SMC IRS 110).', '1505.04499-1-8-5': 'The slit images are useful in determining the origin of the emission seen in the IRS spectra.', '1505.04499-1-8-6': 'The coordinates in Table [REF] represent, if available, the EXT position.', '1505.04499-1-8-7': 'The next preference is the FOV position, and if neither of these is available, the REQ position is given.', '1505.04499-1-9-0': '## Photometric matching', '1505.04499-1-10-0': 'In order to find matching photometry for the IRS spectra, we searched the SAGE-SMC Single Frame + Mosaic Photometry (SMP) Archive v1.5 available on Gator, using, in order of preference, the EXT, FOV or REQ spectrum positions.', '1505.04499-1-10-1': 'We searched for IRAC point source matches within 3 of the spectrum positions, which corresponds with the pointing accuracy of the IRS mode on .', '1505.04499-1-10-2': 'In cases where the SAGE-SMC point source catalogue did not provide a match, we also searched the Survey of the Small Magellanic Cloud () catalogue for IRAC matches within 3.', '1505.04499-1-10-3': 'We found three sources in without a SAGE-SMC catalogue counterpart.', '1505.04499-1-10-4': 'Although the data are included in the SAGE-SMC result, both teams used different point source extraction pipelines and the source catalogues therefore do not provide a one-to-one match.', '1505.04499-1-11-0': 'Of the original list of 311 IRS staring mode observations within the SAGE-SMC footprint, we discarded all 44 spectra for which we could not identify an IRAC point source within 3 in either the SAGE-SMC or the surveys.', '1505.04499-1-11-1': 'In cases where multiple matches were present within 3, we manually compared the magnitudes with the flux levels of the spectra, and used the slit images to establish which source was responsible for the spectrum.', '1505.04499-1-11-2': 'We also consolidated duplicate measurements of the spectrum of a given source, as evidenced by their SAGE-SMC or identification, into a single entry in our analysis; this is sufficient for spectral identification purposes.', '1505.04499-1-11-3': 'This further reduced the number by 58 to a list of 209 unique -IRAC point sources, with either SAGE-SMC or identifications, for which IRS staring mode observations are available.', '1505.04499-1-11-4': 'We compiled all relevant information in a table available online.', '1505.04499-1-11-5': 'Table [REF] describes the columns of the online table.', '1505.04499-1-11-6': 'Fig. [REF] shows the distribution of the 209 sources over the SMC.', '1505.04499-1-12-0': 'Preferring the IRAC coordinates over the spectrum coordinates, we then matched the IRAC point sources to a number of other infrared and optical photometric surveys.', '1505.04499-1-12-1': 'We obtained MIPS-[24], [70] and [160] matches, within a search radius of 3, 9, and 20, respectively, from the SAGE-SMC survey , corresponding to half a resolution element in these bands.', '1505.04499-1-12-2': 'We also searched the Wide-Field Infrared Survey Explorer () All-Sky Source Catalog for matches within 3.', '1505.04499-1-12-3': 'We also searched for matches in the N3, N4, S7, S11, L15, and S22 bands within 3 using the catalogue provided by [CITATION].', '1505.04499-1-12-4': 'In the near-infrared, the Two Micron All Sky Survey (2MASS) Long Exposure (6X) survey was searched for matches within 2 of the IRAC positions (SAGE-SMC matches with 2MASS), and we also used this search radius with the InfraRed Survey Facility (IRSF) catalogue .', '1505.04499-1-12-5': 'The Deep Near Infrared Survey (DENIS) of the Southern Sky catalogue was also searched with a 2 search radius.', '1505.04499-1-12-6': 'In the optical, many of our sources have matches in the Magellanic Clouds Photometric Survey and the catalogue published by [CITATION].', '1505.04499-1-12-7': 'In both catalogues we looked for matches within 1.5 of the IRAC position.', '1505.04499-1-12-8': 'Some of the objects in our sample are actually too bright for those two optical surveys, and a search of the TYCHO catalogue with a radius of 3 filled in some of these gaps.', '1505.04499-1-12-9': 'All tabulated photometry is available from the online database (see Table [REF]).', '1505.04499-1-12-10': 'We only provide the magnitudes for the purpose of evaluating the shape of the SED.', '1505.04499-1-12-11': 'Further information, including the photometric uncertainties, can be found in the respective source tables using the designations provided, as well as Appendix [REF].', '1505.04499-1-13-0': '## Bolometric magnitudes, variability and colour classifications', '1505.04499-1-14-0': 'For each source bolometric magnitudes were calculated via a simple trapezoidal integration of the SED, to which a Wien tail was fitted to the short-wavelength data, and a Rayleigh-Jeans tail was fitted to the long-wavelength data.', '1505.04499-1-14-1': 'The following SED combinations were calculated:', '1505.04499-1-15-0': 'MCPS or [CITATION] optical photometry; photometry; and IRAC and MIPS-[24] photometry, all as available (mbolphot in Table [REF]); like (i) but combined with the photometry (mbolphwi); like (i) but combined with the IRS spectrum (mbolphsp); like (i) but combined with both the photometry and the IRS spectrum (mbolpwsp);', '1505.04499-1-16-0': 'For sources where there is little reprocessing of the optical emission, i.e. little infrared excess, bolometric magnitudes were calculated using an SED fitting code .', '1505.04499-1-16-1': 'This code performs a [MATH]-minimisation between the observed SED (corrected for interstellar reddening) and a grid of bt-settl stellar atmosphere models , which are scaled in flux to derive a bolometric luminosity.', '1505.04499-1-16-2': 'This SED fitter only works effectively where a Rayleigh-Jeans tail is a good description of the 3 to 8 region, and provides a better fit to the optical and near-IR photometry than a Planck function.', '1505.04499-1-16-3': "For the most enshrouded stars, fitting the SED with 'naked' stellar photosphere models leads to an underestimation of the temperature and luminosity, due to circumstellar reddening, and hence the integration method (above) for calculating is preferred for very dusty sources.", '1505.04499-1-16-4': 'Experience shows that good fits can be separated from bad fits in the NIR: if the model and observations differ at [MATH], [MATH], [MATH] or by more than a magnitude in any band, the fit is considered bad.', '1505.04499-1-16-5': 'This retains the cases where the difference between model and observations in the MIR or FIR is large, but often in these cases the excess emission is unrelated to the point sources.', '1505.04499-1-16-6': 'In cases where it is related to the point source, making the source very red, the values calculated by trapezoidal integration provide a better estimate of .', '1505.04499-1-16-7': ', and [MATH] for the good fits are included in the online table as teffmcd, mbolmcd and lummcd, respectively (see Table [REF]).', '1505.04499-1-17-0': 'The sample was then matched to the Optical Gravitational Lensing Experiment (OGLE-III) catalogue of long-period variables in the SMC and [CITATION] to obtain variability periods, and the variability information is included in the on-line table (see Table [REF]).', '1505.04499-1-18-0': 'Finally, we included a number of colour classification schemes for comparison.', '1505.04499-1-18-1': 'First, [CITATION] have extended the classification scheme developed by [CITATION] to classify dusty mass-losing evolved stars into subcategories, using IRAC, MIPS and NIR colours.', '1505.04499-1-18-2': 'We checked our source list against their catalogue for matches.', '1505.04499-1-18-3': 'Their classifications (O-AGB, C-AGB, x-AGB, aO-AGB, RSG, RGB and FIR) are included in the on-line table as boyerclass (see Table [REF]).', '1505.04499-1-18-4': 'Definitions of these classes can be found in [CITATION].', '1505.04499-1-18-5': 'Furthermore, we also applied the colour classification scheme proposed by [CITATION] to the sources in our list.', '1505.04499-1-18-6': 'This classification scheme is also designed to distinguish between various kind of very red objects, to estimate the dust production rate.', '1505.04499-1-18-7': 'We applied the cuts described by [CITATION] on our sample and list the classifications that follow from these cuts (O-AGB, C-AGB, RSG) in our online table, as matsuuraclass (see Table [REF]).', '1505.04499-1-18-8': 'The last colour classification scheme we apply is the one proposed by [CITATION] for YSOs, who applied classification cuts in the five different infrared CMDs, followed by visual inspection of images and SED fitting to select YSO candidates from the SAGE-SMC survey.', '1505.04499-1-18-9': "We checked our source list against their catalogue and identified 'high-reliability' and 'probable' YSO candidates accordingly (sewiloclass; Table [REF]).", '1505.04499-1-19-0': '# The Classification Method', '1505.04499-1-20-0': 'To classify our sample of 209 SMC point sources for which IRS staring mode data exist, we follow the method described by [CITATION].', '1505.04499-1-20-1': 'Fig. [REF] shows a restyled version of the classification decision tree.', '1505.04499-1-20-2': 'We made enhancements to the tree, which will be discussed in this section.', '1505.04499-1-21-0': 'A literature search was performed for each object to retrieve other information useful in the process of classification, including (but not limited to) determination of stellar type, luminosity, age of nascent cluster of stars (if the object was found to be a member of a cluster), detections, etc.', '1505.04499-1-21-1': 'This information was used in addition to the spectroscopic data, the photometric matches and derived bolometric luminosity, and the variability data, described in Sec. [REF], to classify the sources.', '1505.04499-1-21-2': 'Any existing classification from the literature was used as a starting point before our spectral classification.', '1505.04499-1-21-3': 'Appendix [REF] provides a brief summary of the literature survey for each object.', '1505.04499-1-22-0': 'As in [CITATION], we adopt the following categories for our point source classification.', '1505.04499-1-22-1': 'Low- and intermediate-mass ([MATH] ) post-main-sequence stars are classified by chemistry (O- or C-rich) and by evolutionary stage (asymptotic giant branch, post-asymptotic giant branch and planetary nebula), hence our groupings O-AGB, O-PAGB, OPN, C-AGB, C-PAGB, C-PN.', '1505.04499-1-22-2': 'We propose an enhancement of the classification tree by [CITATION] to include early-type O-rich AGB stars, namely O-EAGB.', '1505.04499-1-22-3': 'These stars do not show any evidence for dust features in their infrared spectra, but they do show long period variability in OGLE and MACHO and some evidence for continuum infrared excess.', '1505.04499-1-22-4': 'Although these stars are in the early stages of AGB evolution, they are most likely more evolved than genuine early-AGB (E-AGB) stars, which have not yet started helium shell burning.', '1505.04499-1-22-5': 'E-AGB stars do not normally show long period variability.', '1505.04499-1-22-6': 'We assume that O-EAGB stars are thermally-pulsing AGB-type objects prior to the onset of, or just beginning, significant dust formation.', '1505.04499-1-22-7': 'More massive and luminous red supergiants have a class of their own, RSG.', '1505.04499-1-22-8': 'Young stellar objects can be classified phenomenologically into four groups, YSO-1, YSO-2, YSO-3, YSO-4 (see for the definition of these classes).', '1505.04499-1-22-9': 'Stars showing a stellar photosphere, but no additional dust or gas features, or long-period variability, are classified as STAR.', '1505.04499-1-22-10': 'One observational program focussed on stars showing a far-infrared excess in their MIPS photometry , due to the illumination and heating of interstellar dust (the Pleiades effect); these objects are classified as a subcategory of STAR: dusty OB stars.', '1505.04499-1-22-11': 'The tree also distinguishes galaxies (GAL; even though none are actually found in the present work) and regions (HII), and we furthermore have a class for R CrB stars (labeled RCRB in the classification table).', '1505.04499-1-22-12': 'Finally the classification OTHER exists for objects of known type which do not belong in the categories above and do not follow from the classification tree (e.g. B[e] stars).', '1505.04499-1-22-13': 'The nature of these objects are usually identified by other means, and as such reported in the astronomical literature.', '1505.04499-1-23-0': '## Classification Process', '1505.04499-1-24-0': 'Each source was classified independently by at least three of the co-authors.', '1505.04499-1-24-1': 'In cases where the classification was unanimous, it was simply adopted as the final classification, whereas in those cases where some discrepancies occurred, we asked additional co-authors to classify the sources, to settle the issue.', '1505.04499-1-24-2': 'In some cases, a discussion on the nature of the source ensued.', '1505.04499-1-24-3': 'We aimed to reach consensus among the co-authors on the nature of a source in cases where differences in classification occurred.', '1505.04499-1-25-0': 'The lead author of this work (Paul Ruffle) developed an internal web browser-based classification tool, to facilitate the classification process.', '1505.04499-1-25-1': 'The decision tree was built into the tool as a series of questions, and the collected data were available in tabulated form.', '1505.04499-1-25-2': 'For each source the classification tool provided a slit image and plots of its spectrum, SED, log SED and bolometric magnitudes (see Fig. [REF] for examples).', '1505.04499-1-25-3': 'Each author was free to use either the decision tree logic or their own method of classification.', '1505.04499-1-25-4': 'There was also room for the co-authors to add additional comments to the table.', '1505.04499-1-26-0': '# Sage-Spec Point Source Classification', '1505.04499-1-27-0': 'Table [REF] lists the classification types, used in the decision tree shown in Fig. [REF], and counts for a total of 209 SMC point sources, for which -IRS staring mode observations are available.', '1505.04499-1-27-1': 'The classifications are also shown overplotted on the map of the SMC (Fig. [REF]).', '1505.04499-1-28-0': 'Fig. [REF] shows typical spectra of objects classified as one of the four YSO type objects, as well as a typical IRS spectrum of an region.', '1505.04499-1-28-1': 'The numbering 1-4 for the YSO classes represent an evolutionary sequence, with YSO-1 being the most embedded and YSO-4 the most evolved type of YSO, namely Herbig Ae Be (HAeBe) stars.', '1505.04499-1-28-2': 'This is evident from the spectral appearance of the silicate feature, which appears in absorption towards YSO-1 objects, then gradually in self-absorption (YSO-2), until it finally appears in emission (YSO-3, YSO-4), for less embedded objects.', '1505.04499-1-28-3': 'The spectra of the YSO-1 objects also show ice absorption features, for instance the CO[MATH] ice feature at 15.2 , further evidence of their early evolutionary phase.', '1505.04499-1-28-4': 'Polycyclic aromatic hydrocarbons (PAHs) are seen in the spectra of the YSO-3, YSO-4 and classes, indicative of a UV radiation field.', '1505.04499-1-28-5': 'Atomic lines are also seen, particularly in YSO-3 and objects, and the latter category shows a rising continuum indicative of cold dust in the vicinity of the ionizing star.', '1505.04499-1-29-0': 'Fig. [REF] shows typical spectra in the group of oxygen-rich evolved stars.', '1505.04499-1-29-1': 'The earliest type of O-rich AGB stars are shown at the bottom of the plot (O-EAGB), and the spectra shown here do not show any dust features, while the signature of oxygen-rich molecular species may be present in the spectra.', '1505.04499-1-29-2': 'A slight change of slope due to a small infrared excess caused by thermal dust emission, may be visible in the SED, however.', '1505.04499-1-29-3': 'Later type O-AGB stars, red supergiants (RSG) and oxygen-rich post-AGB stars (O-PAGB), share spectroscopic characteristics, such as the presence of silicate emission features, although the detailed shape can be different.', '1505.04499-1-29-4': 'To distinguish between the RSG and O-AGB category, a bolometric luminosity cut of [MATH] is used, while the distinction between O-AGB and O-PAGB is based on the presence of a detached shell where a double-peaked SED is used as a criterion for the latter category.', '1505.04499-1-29-5': 'This is demonstrated in the lower right panel of Fig. [REF] showing the SED of the sole O-PAGB object in the sample, LHA 115-S 38 (SMC IRS 257).', '1505.04499-1-29-6': 'O-rich PNe (O-PN) may still show a distinguishable oxygen-rich chemistry in their dust mineralogy (although not in the case shown), but they are discriminated from C-PN using the presence of PAH lines in the spectra of the carbon-rich objects.', '1505.04499-1-30-0': 'Fig. [REF] gives an overview of the 5-38 spectral appearance of carbon-rich evolved stars.', '1505.04499-1-30-1': 'Tracers of the carbon-rich chemistry are the C[MATH]H[MATH] molecular absorption bands at 5.0, 7.5 and 13.7, the SiC dust feature at 11.3, the 21- feature (which remains unidentified and really peaks at 20.1), and the so-called 30- feature, which has recently been suggested to be due to the same carbonaceous compound that carries the continuum , while the MgS identification is under debate .', '1505.04499-1-30-2': 'Again, the distinction between the C-PAGB and C-AGB categories is based on whether the SED is double-peaked, which is evidence for a detached shell, indicating mass loss has stopped.', '1505.04499-1-30-3': 'Fig. [REF] shows two clear examples of this, namely SMC IRS 243 and SMC IRS 268.', '1505.04499-1-30-4': 'SMC IRS 95 is confirmed to be a C-PAGB star , despite its absence of a double-peaked SED.', '1505.04499-1-30-5': 'C-PAGB and C-PN also show the UV-excited PAH features, and in case of the C-PN, the presence of atomic emission lines.', '1505.04499-1-31-0': 'Fig. [REF] shows typical spectra of a number of star-like categories, namely (from top to bottom) stellar photospheres, with no discernible dust features in the spectrum; R CrB stars, which only appear to show a dust continuum with no spectral substructure; a Blue Supergiant in our sample, which appears to have a strong far-infrared excess on top of a stellar photosphere, and finally Wolf-Rayet stars, which may form dust and show the corresponding infrared excess in the spectrum.', '1505.04499-1-31-1': 'The Blue Supergiant and the Wolf-Rayet star classifications are taken from the literature, and are not based on the infrared spectroscopy.', '1505.04499-1-31-2': 'Similarly, Fig. [REF] shows the spectra of the object types group under OTHER, of which the classifications are taken from the literature (see Appendix [REF]).', '1505.04499-1-31-3': 'The examples shown in Fig. [REF] represent from top to bottom a B[e] star, a foreground AGB star, an S star and a symbiotic star.', '1505.04499-1-31-4': 'Their infrared spectra are not used to achieve this classification, and the IRS data are plotted only for illustration.', '1505.04499-1-32-0': 'Figs. [REF]-[REF] show the 209 classified point sources on the [8.0] versus [MATH] CMD and two different colour-colour diagrams (CCDs), overlayed on the SAGE-SMC point source catalogue in gray scale.', '1505.04499-1-33-0': 'The [8.0] versus [MATH] CMD (Fig. [REF]) shows a large spread for the population of 209 objects.', '1505.04499-1-33-1': 'The stellar atmospheres (STAR) and Wolf-Rayet stars have colours more or less indistinguishable from the bulk of the SAGE-SMC catalogue (with [MATH] 0-1mag), and modest brightness at 8.0, even though these stars are amongst the brightest stars in the optical in the SMC.', '1505.04499-1-33-2': 'All other categories displayed are bright in the IRAC [8.0] band, and often show considerable redness in their [MATH] colour.', '1505.04499-1-33-3': 'In this diagram RSG, O-AGB/O-EAGB, C-AGB and YSOs (all classes combined), are reasonably well separated from each other, although there are some interlopers.', '1505.04499-1-33-4': 'It appears to be difficult to separate PNe and YSOs on the one hand, and the most extreme C-AGB stars and YSOs on the other hand.', '1505.04499-1-33-5': 'Distinguishing between the four YSO classes is also not possible in this diagram.', '1505.04499-1-34-0': 'Fig. [REF] is a CCD composed of the four IRAC bands, namely the [3.6][MATH][4.5] versus [5.8][MATH][8.0] colours.', '1505.04499-1-34-1': 'The advantage of using a CCD is that it is distance independent.', '1505.04499-1-34-2': 'The coverage of the sample of 209 objects fans out nicely over colour-colour space.', '1505.04499-1-34-3': 'In this diagram, the stars and WR stars no longer separate well from the rest of the sample, however it now appears easier to separate YSOs from C-AGB stars on the one hand and PNe on the other hand.', '1505.04499-1-34-4': 'However, C-PAGB and O-PAGB stars probably overlap with the colour-colour space taken up by YSOs, as they transition from the AGB region to the PN region in the diagram.', '1505.04499-1-34-5': 'But as this phase is short-lived, pollution of the colour-selected YSO sample with post-AGB stars is limited.', '1505.04499-1-34-6': 'Subdivision within the YSO category is still not possible, and also the O-(E)AGB objects do not seem to occupy a unique part of colour-colour space.', '1505.04499-1-35-0': 'Finally, Fig. [REF] shows the versus [8.0][MATH][24] CCD.', '1505.04499-1-35-1': 'In this CCD, the C-AGB objects are easily separated from all other types of objects, as their colour quickly increases, with increasing [8.0][MATH][24].', '1505.04499-1-35-2': 'For O-(E)AGB stars and RSGs this increase is less steep, forming a separate branch in the middle of the plot.', '1505.04499-1-35-3': 'O-PN and C-PN group together with YSOs towards the top of the diagram, showing the highest values of [8.0][MATH][24], against modest reddening.', '1505.04499-1-36-0': '# Comparison with existing colour classifications', '1505.04499-1-37-0': 'These 209 spectral classifications will allow us to verify existing infrared photometric classification schemes that have come out of recent studies of the Magellanic Clouds.', '1505.04499-1-37-1': 'We compare our results with three distinct colour classification schemes: a) the colour classification scheme for evolved stars by [CITATION], expanded by [CITATION] to include mid-IR wavelengths; b) the IRAC classification scheme for AGB stars and RSGs by [CITATION], which is based on the previous work by [CITATION] on the LMC; and c) the classification scheme to select YSO candidates by [CITATION], based on earlier work by [CITATION].', '1505.04499-1-38-0': '## [CITATION]', '1505.04499-1-39-0': 'In order to classify all evolved stars in the SAGE-SMC data, [CITATION] devised a classification scheme, based on the 2MASS classification scheme presented by [CITATION].', '1505.04499-1-39-1': 'The basis for this classification scheme is the [MATH] versus CMD (Fig. [REF]), showing the cuts for the RSG, O-AGB and C-AGB object classes, superposed on the SAGE-SMC point sources (grey pixels).', '1505.04499-1-39-2': 'The dotted line refers to the tip of the Red Giant Branch (RGB), and [CITATION] use this line to separate RGB stars from the AGB and RSG categories.', '1505.04499-1-39-3': 'Photometrically classified objects from [CITATION] are shown as coloured pixels.', '1505.04499-1-39-4': 'At the red end of the diagram, some photometrically classified C-AGB stars appear below the diagonal cut, or even below the dotted line corresponding to the tip of the RGB.', '1505.04499-1-39-5': 'These objects are called extreme AGB stars, predominantly carbon-rich, which are defined as [MATH] mag .', '1505.04499-1-39-6': 'Often these objects are so red, that they are not detected in 2MASS , and alternative selection criteria in the mid-infrared are required.', '1505.04499-1-39-7': 'The reader is referred to [CITATION] for a detailed description.', '1505.04499-1-39-8': 'The larger symbols in Fig. [REF] represent the sample of -IRS spectroscopically classified objects described in this work.', '1505.04499-1-40-0': '### C-AGB stars', '1505.04499-1-41-0': 'Objects showing the C-AGB spectral signature are all classified as either C-AGB or x-AGB (most of which are expected to be C-rich), according to the [CITATION] classification, although three of these objects are not included in the catalogue published by [CITATION].', '1505.04499-1-41-1': 'These objects (OGLE SMC-LPV-7488 (SMC IRS 44; SSTISAGEMA J004903.78[MATH]730520.1); 2MASS J01060330[MATH]7222322 (SMC IRS 109; SSTISAGEMA J010603.27[MATH]722232.1); and 2MASS J00561639[MATH]7216413 (SMC IRS 129; SSTISAGEMA J005616.36[MATH]721641.3)) are among a larger set of objects that had not been properly matched between the IRAC Epochs in the mosaicked catalogue (Srinivasan et al. in prep.)', '1505.04499-1-41-2': 'These objects were thus missing photometric measurements in the point source catalogue, in some bands, and were therefore not classified (correctly) by [CITATION].', '1505.04499-1-41-3': 'The on-line table described by Table [REF] shows the correct photometry for these three targets.', '1505.04499-1-42-0': '### Red Supergiants', '1505.04499-1-43-0': 'Most of the 21 spectroscopically classified RSGs indeed fall within the RSG strip defined by [CITATION].', '1505.04499-1-43-1': 'Only three of the objects classified as an RSG by us were classified differently by [CITATION] using only the photometry: HV 11417 (SMC IRS 115) is designated as a FIR object by [CITATION]; Massey SMC 55188 (SMC IRS 232) as an O-AGB star; and IRAS F00483[MATH]7347 (SMC IRS 98) as an x-AGB star.', '1505.04499-1-43-2': 'HV 11417 is a variable star with a period of 1092 days, and an amplitude in the I band of 1.9 mag .', '1505.04499-1-43-3': 'Due to this large amplitude, the timing of the observations affects the IR colours of the object significantly, which is why the photometry measurements show a small positive slope between 8 and 24 ([MATH]mag), causing it to be classified as a FIR object by [CITATION], while the slope of the IRS spectrum is distinctly negative.', '1505.04499-1-43-4': 'If we disregard the FIR colour cut, which is meant to separate AGB stars from objects such as YSOs, PNe and background galaxies, this object would have been classified as an O-AGB star, which deviates from our determination of RSG, and from past classifications .', '1505.04499-1-43-5': 'The bolometric luminosity of this object is close to the RSG/O-AGB boundary, and could be affected by the variability of the object too.', '1505.04499-1-43-6': 'The large amplitude favours a classification as luminous AGB star over a RSG.', '1505.04499-1-43-7': 'RSGs were separated from O-AGB stars using the classical AGB bolometric magnitude limit (Fig. [REF]), but this boundary is not absolute.', '1505.04499-1-43-8': 'AGB stars undergoing hot bottom burning can be brighter than this limit, while less-evolved RSGs can be fainter.', '1505.04499-1-43-9': 'This causes some disagreement in classifications of stars near the boundary.', '1505.04499-1-44-0': 'Alternatively, the luminosity boundary between RSG and O-AGB may not be properly represented by the cuts in the vs diagram from [CITATION].', '1505.04499-1-44-1': 'Similary, the misclassification of Massey SMC 55188 also suggests that the bolometric cut we applied to distinguish between O-AGB stars and RSGs does not correspond to the boundaries between these two categories in the vs CMD.', '1505.04499-1-44-2': 'IRAS F00483[MATH]7347 clearly shows an oxygen-rich chemistry in its spectrum, with the presence of the amorphous silicate bands at 9.7 and 18.', '1505.04499-1-44-3': 'The object is heavily embedded, with a very red SED, and the 9.7- feature starts to show signs of self-absorption.', '1505.04499-1-44-4': 'IRAS F00483[MATH]7347 demonstrates that not all x-AGB objects are in fact carbon-rich AGB stars.', '1505.04499-1-45-0': '### O-AGB stars', '1505.04499-1-46-0': 'All objects classified as O-EAGB in this work are similarly classified as O-AGB by [CITATION].', '1505.04499-1-46-1': 'The spectrally-confirmed O-AGB objects, however, show a much wider spread in colour-magnitude space than the defined strip, and they encroach on the photometrically-defined RSG, C-AGB and FIR categories.', '1505.04499-1-46-2': 'In particular, only two spectroscopically classified O-AGB stars, HV 12149 (SMC IRS 45) and 2MASS J00444463[MATH]7314076 (SMC IRS 259) are classified as O-AGB based on photometry.', '1505.04499-1-46-3': 'One object (HV 1375; SMC IRS 110) is classified as a C-AGB based on its photometric colours, and a further six objects (RAW 631, 2MASS J00463159[MATH]7328464, 2MASS J00445256[MATH]7318258, BMB-B 75, IRAS F01066[MATH]7332 and HV 12956) are apparently sufficiently red ([MATH] mag) to be classified as FIR, even though they are not background galaxies, YSOs or PNe.', '1505.04499-1-46-4': 'These six objects are heavily embedded, and show the effect of a high optical depth in the relative strength of the 18 silicate band with respect to the 9.7- band.', '1505.04499-1-46-5': 'They also show evidence for the presence of crystalline silicates in their spectrum, another sign of high optical depth .', '1505.04499-1-46-6': 'Finally, HV 2232 (SMC IRS 230) and HV 11464 (SMC IRS 309) are photometrically classified as Red Supergiants (see the table described by Table [REF]).', '1505.04499-1-47-0': '### Additional sources', '1505.04499-1-48-0': 'The further nine interlopers in the C-AGB/x-AGB section of the CMD defined by [CITATION] are a mixed bag of objects, all but one falling in the x-AGB category.', '1505.04499-1-48-1': 'These objects reflect selection biases and include rare categories of previously known types such as R CrB stars (2MASS J00461632[MATH]7411135, 2MASS J00571814[MATH]7242352 and OGLE SMC-SC10 107856), an S star (BFM 1) and a B[e] star (Lin 250).', '1505.04499-1-48-2': 'These object types are not included in the [CITATION] classification scheme, and could therefore never have agreed with the spectral classification.', '1505.04499-1-48-3': 'True misclassifications are the objects thought to be C-rich AGB stars, only to be revealed to be something else based on their IRS spectroscopy.', '1505.04499-1-48-4': 'These include O-PAGB star LHA 115-S 38, YSO-2 object 2MASS J01050732[MATH]7159427, and RSG IRAS F00483[MATH]7347.', '1505.04499-1-48-5': 'NGC 346:KWBBe 200 is a special case, as it was thought to be a B[e] supergiant , but its classification has recently been revised to a YSO , in line with our classification of YSO-3 (See Appendix [REF]).', '1505.04499-1-49-0': 'The FIR category defined by [CITATION] was introduced to exclude YSOs, compact regions, PNe and background galaxies from the AGB/RSG sample, by applying the [MATH] mag cut, corresponding to a rising continuum.', '1505.04499-1-49-1': 'Indeed, 14 out of 23 FIR objects are either C-PN (three objects) or YSOs (11 objects), according to their IRS spectra.', '1505.04499-1-49-2': 'The remaining nine FIR objects include the six O-AGB stars discussed in Sec. [REF], but also a symbiotic star (SSTISAGEMA J005419.21[MATH]722909.7; SMC IRS 260) and a C-PAGB (2MASS J00364631[MATH]7331351; SMC IRS 95), both of which are classified based on their IRS spectra.', '1505.04499-1-50-0': '## [CITATION]', '1505.04499-1-51-0': 'The second classification scheme was proposed by [CITATION] to identify mass-losing O-rich and C-rich AGB stars, as well as Red Supergiants, in order to estimate the dust production budget in the LMC, separated out by C-rich and O-rich chemistries.', '1505.04499-1-51-1': 'The scheme is based on the IRAC bands, using the [8.0] vs [3.6][MATH][8.0] CMD, and separates out the O-rich AGB stars and RSGs on the one hand, and C-rich AGB stars on the other hand (Fig. [REF]).', '1505.04499-1-51-2': '[CITATION] used only this diagram to classify the IRAC point sources in the LMC, but for the SMC, [CITATION] added an extra step to overcome pollution between the categories for redder [3.6][MATH][8.0] sources.', '1505.04499-1-51-3': 'Indeed, from Fig. [REF] it is clear that the red part of the C-AGB section is dominated by sources identified as YSOs (tan diamonds), while a significant fraction of the O-AGB objects (blue diamonds) also fall within the C-AGB section.', '1505.04499-1-51-4': 'In the second step, MIPS and NIR data are included, and a complex series of cuts is made in the vs CCD .', '1505.04499-1-51-5': 'In cases where the classification using the vs diagram (Fig. [REF]) deviates from the [8.0] vs [3.6][MATH][8.0] classification, the vs classification takes preference.', '1505.04499-1-51-6': 'Most spectral classifications agree with the photometric classification in the second step (see Fig. [REF]).', '1505.04499-1-51-7': 'In case where sources were only classified in one of the two steps, we used that classification.', '1505.04499-1-51-8': 'We have executed this classification method for our 209 sources, and included the results in the table described by Table [REF].', '1505.04499-1-52-0': '### Carbon-rich AGB stars', '1505.04499-1-53-0': 'The two-step classification described by [CITATION] correctly identifies most of the spectroscopically classified C-AGB stars in our sample as such.', '1505.04499-1-53-1': 'In only three cases was an object photometrically classified as being oxygen-rich (RSG/O-AGB), while the spectroscopy shows the carbon-rich nature of the source.', '1505.04499-1-53-2': 'These sources are 2MASS J00515018[MATH]7250496 (SMC IRS 103); 2MASS J00524017[MATH]7247276 (SMC IRS 127) and IRAS 00350[MATH]7436 (SMC IRS 238).', '1505.04499-1-53-3': 'IRAS 00350[MATH]7436 is among the brightest mid-infrared objects in the SMC, and falls above the C-AGB cut in Fig. 4 of [CITATION], due to its exceptional brightness.', '1505.04499-1-54-0': 'Furthermore, the classification of several C-AGB stars by [CITATION] disagrees with the spectral classification.', '1505.04499-1-54-1': 'These include: a C-rich post-AGB star (2MASS J00364631[MATH]7331351; SMC IRS 95), four O-rich early-type AGB stars (HV 1366, HV 11303, HV 838 and HV 12122; SMC IRS 36, 38, 39 and 116, respectively), a WR star (HD 5980; SMC IRS 281), a RSG (HV 11262; SMC IRS 111), a YSO-3 object (NGC 346:KWBBe 200; SMC IRS 19), the three R CrB stars (2MASS J00461632[MATH]7411135, 2MASS J00571814[MATH]7242352 and OGLE SMC-SC10 107856; SMC IRS 94, 114 and 245, respectively), the two foreground O-rich AGB stars NGC 362 SAW V16 and HV 206, the S star BFM 1 and the symbiotic star SSTISAGEMA J005419.21[MATH]722909.7.', '1505.04499-1-54-2': 'The last four objects are all in the OTHER category, which contains subclasses the work by [CITATION] does not seek to classify.', '1505.04499-1-54-3': 'The four O-rich early-type objects have actually long been recognized as such, in some cases their nature was already known in the 1980s, and they fall only slightly outside the boundaries in Fig. 5 of [CITATION].', '1505.04499-1-54-4': 'The RSG HV 11262 has very similar and colours to the four O-EAGB objects.', '1505.04499-1-54-5': '2MASS J00364631[MATH]7331351 is not properly filtered out by Fig. 5, as it falls just below the = 8 mag cutoff that is meant to exclude YSO, C-PN, and C-PAGB objects from the C-AGB category.', '1505.04499-1-54-6': 'The R CrB stars represent a rare class which, not surprisingly, overlaps in infrared colours with the C-AGB stars, as it is believed that the dust in these stars is carbonaceous .', '1505.04499-1-54-7': 'HD 5980 is the only WR star that is classified by [CITATION].', '1505.04499-1-54-8': 'Although other WR stars do appear in Fig. [REF], they do not receive a classification as they are too faint in the [8.0] band.', '1505.04499-1-54-9': 'HD 5980 is considerably brighter in the [8.0], and is the only WR star with a MIPS-[24] detection in the SMC .', '1505.04499-1-54-10': 'Closer inspection of the IRS spectrum seems to suggest that a chance superposition with a compact region gives rise to this 24 detection, and is actually not a detection of the WR star itself.', '1505.04499-1-54-11': 'Thus, the position of HD 5980 in Fig. [REF] and the classification following from it, should be disregarded.', '1505.04499-1-54-12': 'Finally, NGC 346:KWBBe 200 is a curious object, which, from its IRS spectrum appears to be a YSO, but has characteristics in common with B[e] stars (See Appendix [REF]).', '1505.04499-1-54-13': 'It falls well outside the box defined for YSO, C-PN and C-PAGB, perhaps due to its unusual nature.', '1505.04499-1-55-0': '### Oxygen-rich AGB stars and RSGs', '1505.04499-1-56-0': 'Since the main purpose of the classification scheme by [CITATION] is to determine the dust production by evolved stars distinguished by carbon-rich and oxygen-rich chemistry, the subdivision in types of evolved stars within the oxygen-rich class is less important.', '1505.04499-1-56-1': 'In fact, in the first step, all types of oxygen-rich evolved stars (AGB stars and RSGs) are lumped together as RSG/O-AGB (Fig. [REF]), while in the second step a subdivision is made between O-AGB (which also includes RSG) and O-AGB/O-PAGB (Fig. [REF]).', '1505.04499-1-56-2': 'The latter category contains the more evolved O-AGB stars.', '1505.04499-1-56-3': 'Thus, a total of three partially overlapping categories exist.', '1505.04499-1-56-4': 'The RSG/O-AGB category contains objects that are classified as such in the first step, but remained unclassified in the second step.', '1505.04499-1-56-5': 'This class contains only three objects (2MASS J00515018[MATH]7250496, 2MASS J00524017[MATH]7247276 and IRAS 00350[MATH]7436), which are all C-AGB according to their IRS spectroscopy, and are already discussed in Sec. [REF].', '1505.04499-1-57-0': 'The 24 objects classified as O-AGB in the second step are indeed all O-AGB or RSG objects according to their IRS spectroscopy.', '1505.04499-1-57-1': 'However, the category O-PAGB/O-AGB contains a more diverse range of objects.', '1505.04499-1-57-2': 'Of the eight SMC IRS objects classified by the method of [CITATION] to be in this category only four are genuine O-rich evolved stars: IRAS F00483[MATH]7347 (SMC IRS 98) is a RSG according to our classification, LHA 115-S 38 (SMC IRS 257) is found to be a O-PAGB object, and 2MASS J00463159[MATH]7328464 (SMC IRS 121) and HV 12956 (SMC IRS 277) are O-AGB stars.', '1505.04499-1-57-3': 'The other objects in this category include two carbon-rich evolved stars, 2MASS J00444111[MATH]7321361 (SMC IRS 268), a C-PAGB object, and Lin 343 (SMC IRS 161), a C-PN.', '1505.04499-1-57-4': 'Both of these objects should have fallen in the YSO/C-PN/C-PAGB, but with just below 8mag they both just miss the cutoff.', '1505.04499-1-57-5': 'Since Fig. [REF] does not show any O-rich objects in the vicinity of the K-[24] = 8mag cutoff between C-rich and O-rich post-AGB objects, a case can be made to lower the cutoff slightly.', '1505.04499-1-57-6': 'Finally, the remaining two misclassified objects in the O-PAGB/O-AGB category by [CITATION] are spectrally classified as B[e] stars in the OTHER category.', '1505.04499-1-57-7': 'They are RMC 50 (SMC IRS 193) and Lin 250 (SMC IRS 262).', '1505.04499-1-58-0': 'When checking the reverse direction, we find that all objects classified by us as O-AGB are indeed identified as either O-AGB or as O-PAGB/O-AGB according to the classification scheme by [CITATION].', '1505.04499-1-58-1': 'However, this classification scheme misclassifies all objects identified as O-EAGB by us.', '1505.04499-1-58-2': 'According to the scheme by [CITATION], they are either C-AGB (See Sect. [REF]; four out of eight objects), or remain unclassified, as they have [MATH]mag, rendering them unclassifiable in Fig. [REF], and have [MATH]mag and [MATH]mag, which causes them to fall outside all boundaries in Fig. [REF].', '1505.04499-1-58-3': 'This also happens to six of the 22 objects classified by us to be RSGs; one is misclassified as a C-AGB (See Sect. [REF]), and the other five do not receive a classification because their colours are too blue (i.e. their mass-loss rates are too low).', '1505.04499-1-59-0': 'Our sample of 209 targets with IRS spectra contains only one oxygen-rich post-AGB star, which agrees with the O-PAGB/O-AGB classification from [CITATION].', '1505.04499-1-60-0': '## [CITATION]', '1505.04499-1-61-0': 'The third classification scheme that we compare with is the method developed by [CITATION] and refined by [CITATION] to select YSO candidates.', '1505.04499-1-61-1': 'From the set of CMDs used by [CITATION], [CITATION] selected a combination of five different CMDs to select YSO candidates in the SMC.', '1505.04499-1-61-2': 'Two of these diagrams are reproduced in this work, with the 209 IRS point sources overplotted (Figs. [REF] and [REF]).', '1505.04499-1-61-3': 'After the initial colour selection of YSO candidates, [CITATION] performed additional tests, including a visual inspection of the imaging, a check against the SIMBAD and other catalogues for known non-YSO sources, and fitting against the YSO SED grid calculated by [CITATION].', '1505.04499-1-61-4': "[CITATION] arrive at a list of approximately 1000 'high-reliability' and 'probable' YSOs in the SMC.", '1505.04499-1-62-0': 'Although Figs. [REF] and [REF] appear to show considerable amount of pollution from non-YSO IRS staring mode targets, as well as many confirmed YSOs based on the IRS data straying out of the defined boxes, it is the combination of the five CMDs, along with the additional non-colour based checks that yields a highly reliable YSO candidate list.', '1505.04499-1-62-1': 'Indeed, the [CITATION] classification favours reliability over completeness, and CMD areas with significant pollution due to other sources (background galaxies, PNe) have been excluded.', '1505.04499-1-62-2': 'The list of 209 IRS staring mode point sources contains nine objects classified as probable YSOs by [CITATION], and 45 high-reliability YSO candidates.', '1505.04499-1-62-3': 'Of the high-reliability YSOs for which IRS observations are available, we find that the vast majority are indeed YSOs (covering all four classes).', '1505.04499-1-62-4': 'Only two objects turn out to be something different: region IRAS 00436[MATH]7321 (SMC IRS 310) and C-PAGB object 2MASS J01054645[MATH]7147053 (SMC IRS 243).', '1505.04499-1-62-5': 'Among the probable YSO candidates, the success rate is lower: four out of nine are not YSOs, upon inspection of their IRS spectra.', '1505.04499-1-62-6': 'RMC 50 (SMC IRS 193) is a B[e] star, while SMP SMC 11 (SMC IRS 32), LHA 115-N 43 (SMC IRS 155) and Lin 49 (SMC IRS 292) are actually C-PN objects.', '1505.04499-1-63-0': 'Furthermore, there are three YSO-3 type objects, as spectrally classified, that are not identified as YSO candidates by [CITATION].', '1505.04499-1-63-1': 'These are NGC 346:KWBBe 200 (SMC IRS 19), 2MASS J00465185[MATH]7315248 (SMC IRS 269) and LHA 115-N 8 (SMC IRS 274).', '1505.04499-1-64-0': '# Summary', '1505.04499-1-65-0': 'We have analysed all 311 -IRS staring mode observations within the SAGE-SMC IRAC and MIPS coverage of the SMC .', '1505.04499-1-65-1': 'After removing IRS observations of extended emission, blank sky and duplicate observations, we find that 209 unique IRAC point sources were targeted.', '1505.04499-1-65-2': 'We applied the infrared spectroscopic classification method devised by [CITATION], with the addition of one more category, namely O-EAGB stars (early-type oxygen-rich AGB stars).', '1505.04499-1-65-3': 'We find that the -IRS staring mode sample of point sources in the SMC contains 51 YSOs, subdivided in 14 embedded YSOs (YSO-1), 5 less-embedded YSOs (YSO-2), 22 evolved YSOs (YSO-3) and 10 HAeBe type objects (YSO-4).', '1505.04499-1-65-4': 'Furthermore, we find the sample contains 46 oxygen-rich evolved stars: 8 O-EAGB stars, 11 O-AGB stars, 22 RSGs, 1 O-PAGB object and 4 O-PN objects.', '1505.04499-1-65-5': '62 objects turn out to be carbon-rich evolved stars, namely 39 C-AGB stars, 3 C-PAGB objects and 20 C-PN objects.', '1505.04499-1-65-6': 'The sample also includes 3 R CrB stars, 1 BSG and 10 WR stars.', '1505.04499-1-65-7': '27 objects show stellar photospheres, 23 of which were selected based on their MIPS-[24] excess, and labelled by us as dusty OB stars.', '1505.04499-1-65-8': 'It turns out that the 24- emission is in general not related to the host OB star .', '1505.04499-1-65-9': 'Finally, the sample includes a small number of other objects (OTHER), which do not follow from the classification method.', '1505.04499-1-65-10': 'These include 2 B[e] stars, 2 foreground oxygen-rich AGB stars, an S star, and a symbiotic star.', '1505.04499-1-66-0': 'In this work, we have compared the resulting spectral classifications with the outcome of photometric classification schemes.', '1505.04499-1-66-1': 'It should be noted that the spectroscopic observations are obtained in 14 different observing programs, with a diverse range of science goals.', '1505.04499-1-66-2': 'Thus, there is no homogeneous coverage of colour-magnitude space, and it will be impossible to quantify the goodness of any given photometric classification method.', '1505.04499-1-66-3': 'Furthermore, the observing programs tend to target rare types of sources in a disproportionate amount, and some of these rare type of sources (R CrB stars, WR stars, B[e] stars, etc.) are not included in photometric classification schemes, precisely because they are rare.', '1505.04499-1-66-4': 'Thus, these objects tend to pollute the classification schemes discussed here, but statistically they are rather insignificant.', '1505.04499-1-67-0': 'We reviewed three different photometric classification schemes for infrared sources in the SMC: the schemes by [CITATION] and [CITATION] for evolved stars, and the classification scheme to select candidate YSOs by [CITATION].', '1505.04499-1-67-1': 'The latter scheme is not a pure photometric classification, as it includes additional steps, such as visual inspection of the direct environment of the point source in imaging, checks against existing catalogues and fitting against the grid of YSO SEDs calculated by [CITATION].', '1505.04499-1-67-2': 'However, as discussed in Sec. [REF], the 54 overlapping sources from this work with the resulting YSO candidate list are mostly correctly classified.', '1505.04499-1-67-3': 'Only a few sources are misclassified in either direction, i.e. three spectroscopically confirmed YSOs were not on the candidate list by [CITATION], and six sources on the candidate list were found to be something else upon inspection of their IRS spectroscopy.', '1505.04499-1-67-4': 'All-in-all we conclude that the YSO candidate list produced by [CITATION] is reliable, with 48/54 sources indeed being YSOs and the high-reliability sources doing better than the probable sources.', '1505.04499-1-67-5': 'Only three spectroscopically confirmed YSOs were missed due to unusual infrared colors.', '1505.04499-1-68-0': 'The two photometric classification methods for evolved stars can be directly compared to each other.', '1505.04499-1-68-1': 'The method by [CITATION] has its focus on identifying the entire dusty evolved star population, while the [CITATION] method is mainly driven by the motivation to determine the carbon-rich and oxygen-rich dust production rates.', '1505.04499-1-68-2': 'Thus, in the later method, correct identification of lower mass-loss rate stars is not so important.', '1505.04499-1-68-3': 'Both methods can be used to estimate the integrated dust production rate.', '1505.04499-1-68-4': 'Due to the low metallicity of the SMC, carbon-rich evolved stars are more numerous , and the carbon stars have, on average, the highest mass-loss rates.', '1505.04499-1-68-5': 'Thus, identifying carbon stars correctly is important as they dominate the dust budget .', '1505.04499-1-68-6': 'Apart from rare object classes, [CITATION] very efficiently separate C-AGB stars from the other classes, only classifying one non-C-AGB star as C-AGB star, while classifying all genuine C-AGB objects as C-AGB.', '1505.04499-1-68-7': '[CITATION] do slightly worse, with three genuine C-AGB objects being classified as something else, and a number of objects, including some rare types, incorrectly classified as C-AGB star.', '1505.04499-1-69-0': 'On the oxygen-rich side, we find that [CITATION] perform better for the high-mass loss rate objects (O-AGB stars), but that they perform rather poorly on the low-mass rate objects (O-EAGB stars), while the performance of the [CITATION] classification method is reversed because of overlap between high-mass AGB stars and RSGs near the classical AGB limit.', '1505.04499-1-70-0': 'Finally, we note that of the two steps involved in the classification method by [CITATION], the second step (Fig. [REF]) matches very well with the actual spectroscopic classification.', '1505.04499-1-70-1': 'It has been introduced by [CITATION] as a correction on the first step from [CITATION], but almost all photometric classifications currently included in the online table described by Table [REF] correspond to the second step, and most of the time match the spectroscopic classification, rendering the first step practically unnecessary.', '1505.04499-1-70-2': 'Thus, in case of the [CITATION] classification scheme, applying only the second step, as demonstrated in Fig. [REF], would suffice for dusty sources.'}

{'1505.04499-2-0-0': 'The Magellanic clouds are uniquely placed to study the stellar contribution to dust emission.', '1505.04499-2-0-1': 'Individual stars can be resolved in these systems even in the mid-infrared, and they are close enough to allow detection of infrared excess caused by dust.', '1505.04499-2-0-2': 'We have searched the Space Telescope data archive for all Infrared Spectrograph (IRS) staring-mode observations of the Small Magellanic Cloud (SMC) and found that 209 Infrared Array Camera (IRAC) point sources within the footprint of the Surveying the Agents of Galaxy Evolution in the Small Magellanic Cloud (SAGE-SMC) Legacy programme were targeted, within a total of 311 staring mode observations.', '1505.04499-2-0-3': 'We classify these point sources using a decision tree method of object classification, based on infrared spectral features, continuum and spectral energy distribution shape, bolometric luminosity, cluster membership and variability information.', '1505.04499-2-0-4': 'We find 58 asymptotic giant branch (AGB) stars, 51 young stellar objects (YSOs), 4 post-AGB objects, 22 Red Supergiants (RSGs), 27 stars (of which 23 are dusty OB stars), 24 planetary nebulae (PNe), 10 Wolf-Rayet (WR) stars, 3 regions, 3 R Coronae Borealis (R CrB) stars, 1 Blue Supergiant and 6 other objects, including 2 foreground AGB stars.', '1505.04499-2-0-5': 'We use these classifications to evaluate the success of photometric classification methods reported in the literature.', '1505.04499-2-1-0': '# Introduction', '1505.04499-2-2-0': 'The Mega-Surveying the Agents of Galaxy Evolution (Mega-SAGE) project has obtained infrared photometric and spectroscopic inventories of the Magellanic Clouds with the Space Telescope (hereafter ), using and Legacy Programmes.', '1505.04499-2-2-1': 'The initial SAGE survey detected and catalogued [MATH]6.9 million point sources in the Large Magellanic Cloud (LMC), while the SAGE-SMC survey detected and catalogued [MATH]2.2 million point sources in the Small Magellanic Cloud (SMC).', '1505.04499-2-2-2': 'Both surveys used all bands of the Infrared Array Camera and the Multi-Band Imaging Photometer for instruments on board .', '1505.04499-2-2-3': 'The resolution of the IRAC observations is [MATH]2, while for the MIPS bands, three different resolutions apply: 6, 18, and 40 for the 24, 70 and 160 bands respectively .', '1505.04499-2-2-4': "To follow up on these programmes, the SAGE-Spec project obtained 196 staring-mode pointings using 's Infrared Spectrograph of positions selected from the SAGE catalogue.", '1505.04499-2-2-5': 'SAGE-Spec will relate SAGE photometry to the spectral characteristics of different types of objects in both Magellanic Clouds, and ultimately, allow us to classify photometric point sources in both the LMC and SMC.', '1505.04499-2-2-6': 'This characterisation of the point sources observed in the SAGE-Spec survey, and the IRS data archive, builds an inventory of dusty sources and their interrelation in each of the Magellanic Clouds.', '1505.04499-2-2-7': 'In a first step towards this goal, [CITATION] classified the initial 196 LMC point sources, using a decision tree method of object classification, based on infrared (IR) spectral features, continuum and spectral energy distribution (SED) shape, bolometric luminosity, cluster membership and variability information.', '1505.04499-2-2-8': 'The initial classification of LMC objects is being extended to [MATH]1,000 point sources, covering all archival IRS observations within the SAGE footprint (Woods et al. in prep.)', '1505.04499-2-3-0': 'To extend the LMC classifications to the SMC, we have searched the data archive for IRS staring-mode observations and found 311 spectra, yielding 209 unique and genuine point sources with IRS data within the footprint of the SAGE-SMC Legacy programme.', '1505.04499-2-3-1': 'The data used in the classification process are described in Section [REF].', '1505.04499-2-3-2': 'In Section [REF] we discuss the classification method, and in Section [REF] we classify the 209 SMC point sources using the decision tree method.', '1505.04499-2-3-3': 'Finally, in Section [REF] we compare spectral versus colour classifications by means of colour-magnitude diagrams (CMDs).', '1505.04499-2-3-4': 'We use our spectroscopic classifications to test photometric classification methods, e.g. those by [CITATION]; [CITATION] and [CITATION].', '1505.04499-2-3-5': 'The classification of each of these 209 sources is part of the data delivery of the SAGE-Spec Legacy project to the Science Center and the community.', '1505.04499-2-3-6': 'These classifications will also be used to benchmark a colour-classification scheme that will be applied to all point sources in the SAGE and SAGE-SMC surveys (Marengo et al. in prep).', '1505.04499-2-4-0': '# Data preparation', '1505.04499-2-5-0': '## IRS staring mode observations', '1505.04499-2-6-0': 'The IRS on board covers the wavelength range 5-38.', '1505.04499-2-6-1': 'For the low-resolution mode, the spectrum splits in two bands, short-low (SL: 5.2-14.5) and long-low (LL: 14.0-38.0), with almost perpendicular slits.', '1505.04499-2-6-2': 'Each segment splits into a range covered at second order (SL2, LL2), and one at first order (SL1, LL1).', '1505.04499-2-6-3': 'The resolution varies between 60 and 130.', '1505.04499-2-6-4': 'The high resolution mode covers the wavelength range from 10-19.6 (short-high; SH) and from 18.7-37.2 (long-high; LH) with a spectral resolution of [MATH].', '1505.04499-2-7-0': 'We identified 311 IRS low- and high-resolution staring mode observations within the footprint of the SAGE-SMC survey , not necessarily associated with a point source.', '1505.04499-2-7-1': 'We numbered these SMC IRS 1-311, by ordering them by observing program (Project ID; PID) first, and then the Astronomical Observation Request (AOR) number (Table [REF]).', '1505.04499-2-7-2': 'Where available (for SL and LL observations only), the reduced spectra were downloaded from the Cornell Atlas of IRS Sources , in a full resolution grid, using the optimal extraction method.', '1505.04499-2-7-3': 'At the moment, the CASSIS database only contains SL and LL data, but it turns out that there are no point sources in the SMC targeted with only SH and LH, so we will use the SL and LL data only.', '1505.04499-2-7-4': 'The data were downloaded in the Infrared Processing and Analysis Center (IPAC) Table format; the file names are also given in Table [REF].', '1505.04499-2-7-5': 'As an example, the upper left panel of Fig. [REF] shows the spectrum for point source SMC IRS 110, with the SL2 and LL2 data from CASSIS in red and the SL1 and LL1 data in blue.', '1505.04499-2-8-0': "The CASSIS-reduced IRS data header provides the original PI's requested (REQ) position and the field of view (FOV) position, i.e. where the telescope actually pointed (usually, but not always coincident within 1 of the REQ position).", '1505.04499-2-8-1': 'In many cases, however, neither of these two positions is the same as the position at which the spectrum is actually extracted from the slit (EXT), using the optimal extraction method.', '1505.04499-2-8-2': "In order to check the spectrum position for each source, slit images were generated by over-plotting the IRS SL and LL slit positions (recorded in each AOR's BCD FITS header) on a 360 360 image extracted from the SAGE-SMC 8 image.", '1505.04499-2-8-3': 'The REQ and EXT positions were also over-plotted on the image.', '1505.04499-2-8-4': 'In cases where the spectrum was extracted at the FOV position (and thus an EXT position is lacking), the FOV position was overplotted instead (see upper right panel of Fig. [REF] for an example for SMC IRS 110).', '1505.04499-2-8-5': 'The slit images are useful in determining the origin of the emission seen in the IRS spectra.', '1505.04499-2-8-6': 'The coordinates in Table [REF] represent, if available, the EXT position.', '1505.04499-2-8-7': 'The next preference is the FOV position, and if neither of these is available, the REQ position is given.', '1505.04499-2-9-0': '## Photometric matching', '1505.04499-2-10-0': 'In order to find matching photometry for the IRS spectra, we searched the SAGE-SMC Single Frame + Mosaic Photometry (SMP) Archive v1.5 available on Gator, using, in order of preference, the EXT, FOV or REQ spectrum positions.', '1505.04499-2-10-1': 'We searched for IRAC point source matches within 3 of the spectrum positions, which corresponds with the pointing accuracy of the IRS mode on .', '1505.04499-2-10-2': 'In cases where the SAGE-SMC point source catalogue did not provide a match, we also searched the Survey of the Small Magellanic Cloud () catalogue for IRAC matches within 3.', '1505.04499-2-10-3': 'We found three sources in without a SAGE-SMC catalogue counterpart.', '1505.04499-2-10-4': 'Although the data are included in the SAGE-SMC result, both teams used different point source extraction pipelines and the source catalogues therefore do not provide a one-to-one match.', '1505.04499-2-11-0': 'Of the original list of 311 IRS staring mode observations within the SAGE-SMC footprint, we discarded all 44 spectra for which we could not identify an IRAC point source within 3 in either the SAGE-SMC or the surveys.', '1505.04499-2-11-1': 'In cases where multiple matches were present within 3, we manually compared the magnitudes with the flux levels of the spectra, and used the slit images to establish which source was responsible for the spectrum.', '1505.04499-2-11-2': 'We also consolidated duplicate measurements of the spectrum of a given source, as evidenced by their SAGE-SMC or identification, into a single entry in our analysis; this is sufficient for spectral identification purposes.', '1505.04499-2-11-3': 'This further reduced the number by 58 to a list of 209 unique -IRAC point sources, with either SAGE-SMC or identifications, for which IRS staring mode observations are available.', '1505.04499-2-11-4': 'We compiled all relevant information in a table available online.', '1505.04499-2-11-5': 'Table [REF] describes the columns of the online table.', '1505.04499-2-11-6': 'Fig. [REF] shows the distribution of the 209 sources over the SMC.', '1505.04499-2-12-0': 'Preferring the IRAC coordinates over the spectrum coordinates, we then matched the IRAC point sources to a number of other infrared and optical photometric surveys.', '1505.04499-2-12-1': 'We obtained MIPS-[24], [70] and [160] matches, within a search radius of 3, 9, and 20, respectively, from the SAGE-SMC survey , corresponding to half a resolution element in these bands.', '1505.04499-2-12-2': 'We also searched the Wide-Field Infrared Survey Explorer () All-Sky Source Catalog for matches within 3.', '1505.04499-2-12-3': 'We also searched for matches in the N3, N4, S7, S11, L15, and S22 bands within 3 using the catalogue provided by [CITATION].', '1505.04499-2-12-4': 'In the near-infrared, the Two Micron All Sky Survey (2MASS) Long Exposure (6X) survey was searched for matches within 2 of the IRAC positions (SAGE-SMC matches with 2MASS), and we also used this search radius with the InfraRed Survey Facility (IRSF) catalogue .', '1505.04499-2-12-5': 'The Deep Near Infrared Survey (DENIS) of the Southern Sky catalogue was also searched with a 2 search radius.', '1505.04499-2-12-6': 'In the optical, many of our sources have matches in the Magellanic Clouds Photometric Survey and the catalogue published by [CITATION].', '1505.04499-2-12-7': 'In both catalogues we looked for matches within 1.5 of the IRAC position.', '1505.04499-2-12-8': 'Some of the objects in our sample are actually too bright for those two optical surveys, and a search of the TYCHO catalogue with a radius of 3 filled in some of these gaps.', '1505.04499-2-12-9': 'All tabulated photometry is available from the online database (see Table [REF]).', '1505.04499-2-12-10': 'We only provide the magnitudes for the purpose of evaluating the shape of the SED.', '1505.04499-2-12-11': 'Further information, including the photometric uncertainties, can be found in the respective source tables using the designations provided, as well as Appendix [REF].', '1505.04499-2-13-0': '## Bolometric magnitudes, variability and colour classifications', '1505.04499-2-14-0': 'For each source bolometric magnitudes were calculated via a simple trapezoidal integration of the SED, to which a Wien tail was fitted to the short-wavelength data, and a Rayleigh-Jeans tail was fitted to the long-wavelength data.', '1505.04499-2-14-1': 'The following SED combinations were calculated:', '1505.04499-2-15-0': 'MCPS or [CITATION] optical photometry; photometry; and IRAC and MIPS-[24] photometry, all as available (mbolphot in Table [REF]); like (i) but combined with the photometry (mbolphwi); like (i) but combined with the IRS spectrum (mbolphsp); like (i) but combined with both the photometry and the IRS spectrum (mbolpwsp);', '1505.04499-2-16-0': 'For sources where there is little reprocessing of the optical emission, i.e. little infrared excess, bolometric magnitudes were calculated using an SED fitting code .', '1505.04499-2-16-1': 'This code performs a [MATH]-minimisation between the observed SED (corrected for interstellar reddening) and a grid of bt-settl stellar atmosphere models , which are scaled in flux to derive a bolometric luminosity.', '1505.04499-2-16-2': 'This SED fitter only works effectively where a Rayleigh-Jeans tail is a good description of the 3 to 8 region, and provides a better fit to the optical and near-IR photometry than a Planck function.', '1505.04499-2-16-3': "For the most enshrouded stars, fitting the SED with 'naked' stellar photosphere models leads to an underestimation of the temperature and luminosity, due to circumstellar reddening, and hence the integration method (above) for calculating is preferred for very dusty sources.", '1505.04499-2-16-4': 'Experience shows that good fits can be separated from bad fits in the NIR: if the model and observations differ at [MATH], [MATH], [MATH] or by more than a magnitude in any band, the fit is considered bad.', '1505.04499-2-16-5': 'This retains the cases where the difference between model and observations in the MIR or FIR is large, but often in these cases the excess emission is unrelated to the point sources.', '1505.04499-2-16-6': 'In cases where it is related to the point source, making the source very red, the values calculated by trapezoidal integration provide a better estimate of .', '1505.04499-2-16-7': ', and [MATH] for the good fits are included in the online table as teffmcd, mbolmcd and lummcd, respectively (see Table [REF]).', '1505.04499-2-17-0': 'The sample was then matched to the Optical Gravitational Lensing Experiment (OGLE-III) catalogue of long-period variables in the SMC and [CITATION] to obtain variability periods, and the variability information is included in the on-line table (see Table [REF]).', '1505.04499-2-18-0': 'Finally, we included a number of colour classification schemes for comparison.', '1505.04499-2-18-1': 'First, [CITATION] have extended the classification scheme developed by [CITATION] to classify dusty mass-losing evolved stars into subcategories, using IRAC, MIPS and NIR colours.', '1505.04499-2-18-2': 'We checked our source list against their catalogue for matches.', '1505.04499-2-18-3': 'Their classifications (O-AGB, C-AGB, x-AGB, aO-AGB, RSG, RGB and FIR) are included in the on-line table as boyerclass (see Table [REF]).', '1505.04499-2-18-4': 'Definitions of these classes can be found in [CITATION].', '1505.04499-2-18-5': 'Furthermore, we also applied the colour classification scheme proposed by [CITATION] to the sources in our list.', '1505.04499-2-18-6': 'This classification scheme is also designed to distinguish between various kind of very red objects, to estimate the dust production rate.', '1505.04499-2-18-7': 'We applied the cuts described by [CITATION] on our sample and list the classifications that follow from these cuts (O-AGB, C-AGB, RSG) in our online table, as matsuuraclass (see Table [REF]).', '1505.04499-2-18-8': 'The last colour classification scheme we apply is the one proposed by [CITATION] for YSOs, who applied classification cuts in the five different infrared CMDs, followed by visual inspection of images and SED fitting to select YSO candidates from the SAGE-SMC survey.', '1505.04499-2-18-9': "We checked our source list against their catalogue and identified 'high-reliability' and 'probable' YSO candidates accordingly (sewiloclass; Table [REF]).", '1505.04499-2-19-0': '# The Classification Method', '1505.04499-2-20-0': 'To classify our sample of 209 SMC point sources for which IRS staring mode data exist, we follow the method described by [CITATION].', '1505.04499-2-20-1': 'Fig. [REF] shows a restyled version of the classification decision tree.', '1505.04499-2-20-2': 'We made enhancements to the tree, which will be discussed in this section.', '1505.04499-2-21-0': 'A literature search was performed for each object to retrieve other information useful in the process of classification, including (but not limited to) determination of stellar type, luminosity, age of nascent cluster of stars (if the object was found to be a member of a cluster), detections, etc.', '1505.04499-2-21-1': 'This information was used in addition to the spectroscopic data, the photometric matches and derived bolometric luminosity, and the variability data, described in Sec. [REF], to classify the sources.', '1505.04499-2-21-2': 'Any existing classification from the literature was used as a starting point before our spectral classification.', '1505.04499-2-21-3': 'Appendix [REF] provides a brief summary of the literature survey for each object.', '1505.04499-2-22-0': 'As in [CITATION], we adopt the following categories for our point source classification.', '1505.04499-2-22-1': 'Low- and intermediate-mass ([MATH] ) post-main-sequence stars are classified by chemistry (O- or C-rich) and by evolutionary stage (asymptotic giant branch, post-asymptotic giant branch and planetary nebula), hence our groupings O-AGB, O-PAGB, OPN, C-AGB, C-PAGB, C-PN.', '1505.04499-2-22-2': 'We propose an enhancement of the classification tree by [CITATION] to include early-type O-rich AGB stars, namely O-EAGB.', '1505.04499-2-22-3': 'These stars do not show any evidence for dust features in their infrared spectra, but they do show long period variability in OGLE and MACHO and some evidence for continuum infrared excess.', '1505.04499-2-22-4': 'Although these stars are in the early stages of AGB evolution, they are most likely more evolved than genuine early-AGB (E-AGB) stars, which have not yet started helium shell burning.', '1505.04499-2-22-5': 'E-AGB stars do not normally show long period variability.', '1505.04499-2-22-6': 'We assume that O-EAGB stars are thermally-pulsing AGB-type objects prior to the onset of, or just beginning, significant dust formation.', '1505.04499-2-22-7': 'More massive and luminous red supergiants have a class of their own, RSG.', '1505.04499-2-22-8': 'Young stellar objects can be classified phenomenologically into four groups, YSO-1, YSO-2, YSO-3, YSO-4 (see for the definition of these classes).', '1505.04499-2-22-9': 'Stars showing a stellar photosphere, but no additional dust or gas features, or long-period variability, are classified as STAR.', '1505.04499-2-22-10': 'One observational program focussed on stars showing a far-infrared excess in their MIPS photometry , due to the illumination and heating of interstellar dust (the Pleiades effect); these objects are classified as a subcategory of STAR: dusty OB stars.', '1505.04499-2-22-11': 'The tree also distinguishes galaxies (GAL; even though none are actually found in the present work) and regions (HII), and we furthermore have a class for R CrB stars (labeled RCRB in the classification table).', '1505.04499-2-22-12': 'Finally the classification OTHER exists for objects of known type which do not belong in the categories above and do not follow from the classification tree (e.g. B[e] stars).', '1505.04499-2-22-13': 'The nature of these objects are usually identified by other means, and as such reported in the astronomical literature.', '1505.04499-2-23-0': '## Classification Process', '1505.04499-2-24-0': 'Each source was classified independently by at least three of the co-authors.', '1505.04499-2-24-1': 'In cases where the classification was unanimous, it was simply adopted as the final classification, whereas in those cases where some discrepancies occurred, we asked additional co-authors to classify the sources, to settle the issue.', '1505.04499-2-24-2': 'In some cases, a discussion on the nature of the source ensued.', '1505.04499-2-24-3': 'We aimed to reach consensus among the co-authors on the nature of a source in cases where differences in classification occurred.', '1505.04499-2-25-0': 'The lead author of this work (Paul Ruffle) developed an internal web browser-based classification tool, to facilitate the classification process.', '1505.04499-2-25-1': 'The decision tree was built into the tool as a series of questions, and the collected data were available in tabulated form.', '1505.04499-2-25-2': 'For each source the classification tool provided a slit image and plots of its spectrum, SED, log SED and bolometric magnitudes (see Fig. [REF] for examples).', '1505.04499-2-25-3': 'Each author was free to use either the decision tree logic or their own method of classification.', '1505.04499-2-25-4': 'There was also room for the co-authors to add additional comments to the table.', '1505.04499-2-26-0': '# Sage-Spec Point Source Classification', '1505.04499-2-27-0': 'Table [REF] lists the classification types, used in the decision tree shown in Fig. [REF], and counts for a total of 209 SMC point sources, for which -IRS staring mode observations are available.', '1505.04499-2-27-1': 'The classifications are also shown overplotted on the map of the SMC (Fig. [REF]).', '1505.04499-2-28-0': 'Fig. [REF] shows typical spectra of objects classified as one of the four YSO type objects, as well as a typical IRS spectrum of an region.', '1505.04499-2-28-1': 'The numbering 1-4 for the YSO classes represent an evolutionary sequence, with YSO-1 being the most embedded and YSO-4 the most evolved type of YSO, namely Herbig Ae Be (HAeBe) stars.', '1505.04499-2-28-2': 'This is evident from the spectral appearance of the silicate feature, which appears in absorption towards YSO-1 objects, then gradually in self-absorption (YSO-2), until it finally appears in emission (YSO-3, YSO-4), for less embedded objects.', '1505.04499-2-28-3': 'The spectra of the YSO-1 objects also show ice absorption features, for instance the CO[MATH] ice feature at 15.2 , further evidence of their early evolutionary phase.', '1505.04499-2-28-4': 'Polycyclic aromatic hydrocarbons (PAHs) are seen in the spectra of the YSO-3, YSO-4 and classes, indicative of a UV radiation field.', '1505.04499-2-28-5': 'Atomic lines are also seen, particularly in YSO-3 and objects, and the latter category shows a rising continuum indicative of cold dust in the vicinity of the ionizing star.', '1505.04499-2-29-0': 'Fig. [REF] shows typical spectra in the group of oxygen-rich evolved stars.', '1505.04499-2-29-1': 'The earliest type of O-rich AGB stars are shown at the bottom of the plot (O-EAGB), and the spectra shown here do not show any dust features, while the signature of oxygen-rich molecular species may be present in the spectra.', '1505.04499-2-29-2': 'A slight change of slope due to a small infrared excess caused by thermal dust emission, may be visible in the SED, however.', '1505.04499-2-29-3': 'Later type O-AGB stars, red supergiants (RSG) and oxygen-rich post-AGB stars (O-PAGB), share spectroscopic characteristics, such as the presence of silicate emission features, although the detailed shape can be different.', '1505.04499-2-29-4': 'To distinguish between the RSG and O-AGB category, a bolometric luminosity cut of [MATH] is used, while the distinction between O-AGB and O-PAGB is based on the presence of a detached shell where a double-peaked SED is used as a criterion for the latter category.', '1505.04499-2-29-5': 'This is demonstrated in the lower right panel of Fig. [REF] showing the SED of the sole O-PAGB object in the sample, LHA 115-S 38 (SMC IRS 257).', '1505.04499-2-29-6': 'O-rich PNe (O-PN) may still show a distinguishable oxygen-rich chemistry in their dust mineralogy (although not in the case shown), but they are discriminated from C-PN using the presence of PAH lines in the spectra of the carbon-rich objects.', '1505.04499-2-30-0': 'Fig. [REF] gives an overview of the 5-38 spectral appearance of carbon-rich evolved stars.', '1505.04499-2-30-1': 'Tracers of the carbon-rich chemistry are the C[MATH]H[MATH] molecular absorption bands at 5.0, 7.5 and 13.7, the SiC dust feature at 11.3, the 21- feature (which remains unidentified and really peaks at 20.1), and the so-called 30- feature, which has recently been suggested to be due to the same carbonaceous compound that carries the continuum , while the MgS identification is under debate .', '1505.04499-2-30-2': 'Again, the distinction between the C-PAGB and C-AGB categories is based on whether the SED is double-peaked, which is evidence for a detached shell, indicating mass loss has stopped.', '1505.04499-2-30-3': 'Fig. [REF] shows two clear examples of this, namely SMC IRS 243 and SMC IRS 268.', '1505.04499-2-30-4': 'SMC IRS 95 is confirmed to be a C-PAGB star , despite its absence of a double-peaked SED.', '1505.04499-2-30-5': 'C-PAGB and C-PN also show the UV-excited PAH features, and in case of the C-PN, the presence of atomic emission lines.', '1505.04499-2-31-0': 'Fig. [REF] shows typical spectra of a number of star-like categories, namely (from top to bottom) stellar photospheres, with no discernible dust features in the spectrum; R CrB stars, which only appear to show a dust continuum with no spectral substructure; a Blue Supergiant in our sample, which appears to have a strong far-infrared excess on top of a stellar photosphere, and finally Wolf-Rayet stars, which may form dust and show the corresponding infrared excess in the spectrum.', '1505.04499-2-31-1': 'The Blue Supergiant and the Wolf-Rayet star classifications are taken from the literature, and are not based on the infrared spectroscopy.', '1505.04499-2-31-2': 'Similarly, Fig. [REF] shows the spectra of the object types group under OTHER, of which the classifications are taken from the literature (see Appendix [REF]).', '1505.04499-2-31-3': 'The examples shown in Fig. [REF] represent from top to bottom a B[e] star, a foreground AGB star, an S star and a symbiotic star.', '1505.04499-2-31-4': 'Their infrared spectra are not used to achieve this classification, and the IRS data are plotted only for illustration.', '1505.04499-2-32-0': 'Figs. [REF]-[REF] show the 209 classified point sources on the [8.0] versus [MATH] CMD and two different colour-colour diagrams (CCDs), overlayed on the SAGE-SMC point source catalogue in gray scale.', '1505.04499-2-33-0': 'The [8.0] versus [MATH] CMD (Fig. [REF]) shows a large spread for the population of 209 objects.', '1505.04499-2-33-1': 'The stellar atmospheres (STAR) and Wolf-Rayet stars have colours more or less indistinguishable from the bulk of the SAGE-SMC catalogue (with [MATH] 0-1mag), and modest brightness at 8.0, even though these stars are amongst the brightest stars in the optical in the SMC.', '1505.04499-2-33-2': 'All other categories displayed are bright in the IRAC [8.0] band, and often show considerable redness in their [MATH] colour.', '1505.04499-2-33-3': 'In this diagram RSG, O-AGB/O-EAGB, C-AGB and YSOs (all classes combined), are reasonably well separated from each other, although there are some interlopers.', '1505.04499-2-33-4': 'It appears to be difficult to separate PNe and YSOs on the one hand, and the most extreme C-AGB stars and YSOs on the other hand.', '1505.04499-2-33-5': 'Distinguishing between the four YSO classes is also not possible in this diagram.', '1505.04499-2-34-0': 'Fig. [REF] is a CCD composed of the four IRAC bands, namely the [3.6][MATH][4.5] versus [5.8][MATH][8.0] colours.', '1505.04499-2-34-1': 'The advantage of using a CCD is that it is distance independent.', '1505.04499-2-34-2': 'The coverage of the sample of 209 objects fans out nicely over colour-colour space.', '1505.04499-2-34-3': 'In this diagram, the stars and WR stars no longer separate well from the rest of the sample, however it now appears easier to separate YSOs from C-AGB stars on the one hand and PNe on the other hand.', '1505.04499-2-34-4': 'However, C-PAGB and O-PAGB stars probably overlap with the colour-colour space taken up by YSOs, as they transition from the AGB region to the PN region in the diagram.', '1505.04499-2-34-5': 'But as this phase is short-lived, pollution of the colour-selected YSO sample with post-AGB stars is limited.', '1505.04499-2-34-6': 'Subdivision within the YSO category is still not possible, and also the O-(E)AGB objects do not seem to occupy a unique part of colour-colour space.', '1505.04499-2-35-0': 'Finally, Fig. [REF] shows the versus [8.0][MATH][24] CCD.', '1505.04499-2-35-1': 'In this CCD, the C-AGB objects are easily separated from all other types of objects, as their colour quickly increases, with increasing [8.0][MATH][24].', '1505.04499-2-35-2': 'For O-(E)AGB stars and RSGs this increase is less steep, forming a separate branch in the middle of the plot.', '1505.04499-2-35-3': 'O-PN and C-PN group together with YSOs towards the top of the diagram, showing the highest values of [8.0][MATH][24], against modest reddening.', '1505.04499-2-36-0': '# Comparison with existing colour classifications', '1505.04499-2-37-0': 'These 209 spectral classifications will allow us to verify existing infrared photometric classification schemes that have come out of recent studies of the Magellanic Clouds.', '1505.04499-2-37-1': 'We compare our results with three distinct colour classification schemes: a) the colour classification scheme for evolved stars by [CITATION], expanded by [CITATION] to include mid-IR wavelengths; b) the IRAC classification scheme for AGB stars and RSGs by [CITATION], which is based on the previous work by [CITATION] on the LMC; and c) the classification scheme to select YSO candidates by [CITATION], based on earlier work by [CITATION].', '1505.04499-2-38-0': '## [CITATION]', '1505.04499-2-39-0': 'In order to classify all evolved stars in the SAGE-SMC data, [CITATION] devised a classification scheme, based on the 2MASS classification scheme presented by [CITATION].', '1505.04499-2-39-1': 'The basis for this classification scheme is the [MATH] versus CMD (Fig. [REF]), showing the cuts for the RSG, O-AGB and C-AGB object classes, superposed on the SAGE-SMC point sources (grey pixels).', '1505.04499-2-39-2': 'The dotted line refers to the tip of the Red Giant Branch (RGB), and [CITATION] use this line to separate RGB stars from the AGB and RSG categories.', '1505.04499-2-39-3': 'Photometrically classified objects from [CITATION] are shown as coloured pixels.', '1505.04499-2-39-4': 'At the red end of the diagram, some photometrically classified C-AGB stars appear below the diagonal cut, or even below the dotted line corresponding to the tip of the RGB.', '1505.04499-2-39-5': 'These objects are called extreme AGB stars, predominantly carbon-rich, which are defined as [MATH] mag .', '1505.04499-2-39-6': 'Often these objects are so red, that they are not detected in 2MASS , and alternative selection criteria in the mid-infrared are required.', '1505.04499-2-39-7': 'The reader is referred to [CITATION] for a detailed description.', '1505.04499-2-39-8': 'The larger symbols in Fig. [REF] represent the sample of -IRS spectroscopically classified objects described in this work.', '1505.04499-2-40-0': '### C-AGB stars', '1505.04499-2-41-0': 'Objects showing the C-AGB spectral signature are all classified as either C-AGB or x-AGB (most of which are expected to be C-rich), according to the [CITATION] classification, although three of these objects are not included in the catalogue published by [CITATION].', '1505.04499-2-41-1': 'These objects (OGLE SMC-LPV-7488 (SMC IRS 44; SSTISAGEMA J004903.78[MATH]730520.1); 2MASS J01060330[MATH]7222322 (SMC IRS 109; SSTISAGEMA J010603.27[MATH]722232.1); and 2MASS J00561639[MATH]7216413 (SMC IRS 129; SSTISAGEMA J005616.36[MATH]721641.3)) are among a larger set of objects that had not been properly matched between the IRAC Epochs in the mosaicked catalogue (Srinivasan et al. in prep.)', '1505.04499-2-41-2': 'These objects were thus missing photometric measurements in the point source catalogue, in some bands, and were therefore not classified (correctly) by [CITATION].', '1505.04499-2-41-3': 'The on-line table described by Table [REF] shows the correct photometry for these three targets.', '1505.04499-2-42-0': '### Red Supergiants', '1505.04499-2-43-0': 'Most of the 21 spectroscopically classified RSGs indeed fall within the RSG strip defined by [CITATION].', '1505.04499-2-43-1': 'Only three of the objects classified as an RSG by us were classified differently by [CITATION] using only the photometry: HV 11417 (SMC IRS 115) is designated as a FIR object by [CITATION]; Massey SMC 55188 (SMC IRS 232) as an O-AGB star; and IRAS F00483[MATH]7347 (SMC IRS 98) as an x-AGB star.', '1505.04499-2-43-2': 'HV 11417 is a variable star with a period of 1092 days, and an amplitude in the I band of 1.9 mag .', '1505.04499-2-43-3': 'Due to this large amplitude, the timing of the observations affects the IR colours of the object significantly, which is why the photometry measurements show a small positive slope between 8 and 24 ([MATH]mag), causing it to be classified as a FIR object by [CITATION], while the slope of the IRS spectrum is distinctly negative.', '1505.04499-2-43-4': 'If we disregard the FIR colour cut, which is meant to separate AGB stars from objects such as YSOs, PNe and background galaxies, this object would have been classified as an O-AGB star, which deviates from our determination of RSG, and from past classifications .', '1505.04499-2-43-5': 'The bolometric luminosity of this object is close to the RSG/O-AGB boundary, and could be affected by the variability of the object too.', '1505.04499-2-43-6': 'The large amplitude favours a classification as luminous AGB star over a RSG.', '1505.04499-2-43-7': 'RSGs were separated from O-AGB stars using the classical AGB bolometric magnitude limit (Fig. [REF]), but this boundary is not absolute.', '1505.04499-2-43-8': 'AGB stars undergoing hot bottom burning can be brighter than this limit, while less-evolved RSGs can be fainter.', '1505.04499-2-43-9': 'This causes some disagreement in classifications of stars near the boundary.', '1505.04499-2-44-0': 'Alternatively, the luminosity boundary between RSG and O-AGB may not be properly represented by the cuts in the vs diagram from [CITATION].', '1505.04499-2-44-1': 'Similary, the misclassification of Massey SMC 55188 also suggests that the bolometric cut we applied to distinguish between O-AGB stars and RSGs does not correspond to the boundaries between these two categories in the vs CMD.', '1505.04499-2-44-2': 'IRAS F00483[MATH]7347 clearly shows an oxygen-rich chemistry in its spectrum, with the presence of the amorphous silicate bands at 9.7 and 18.', '1505.04499-2-44-3': 'The object is heavily embedded, with a very red SED, and the 9.7- feature starts to show signs of self-absorption.', '1505.04499-2-44-4': 'IRAS F00483[MATH]7347 demonstrates that not all x-AGB objects are in fact carbon-rich AGB stars.', '1505.04499-2-45-0': '### O-AGB stars', '1505.04499-2-46-0': 'All objects classified as O-EAGB in this work are similarly classified as O-AGB by [CITATION].', '1505.04499-2-46-1': 'The spectrally-confirmed O-AGB objects, however, show a much wider spread in colour-magnitude space than the defined strip, and they encroach on the photometrically-defined RSG, C-AGB and FIR categories.', '1505.04499-2-46-2': 'In particular, only two spectroscopically classified O-AGB stars, HV 12149 (SMC IRS 45) and 2MASS J00444463[MATH]7314076 (SMC IRS 259) are classified as O-AGB based on photometry.', '1505.04499-2-46-3': 'One object (HV 1375; SMC IRS 110) is classified as a C-AGB based on its photometric colours, and a further six objects (RAW 631, 2MASS J00463159[MATH]7328464, 2MASS J00445256[MATH]7318258, BMB-B 75, IRAS F01066[MATH]7332 and HV 12956) are apparently sufficiently red ([MATH] mag) to be classified as FIR, even though they are not background galaxies, YSOs or PNe.', '1505.04499-2-46-4': 'These six objects are heavily embedded, and show the effect of a high optical depth in the relative strength of the 18 silicate band with respect to the 9.7- band.', '1505.04499-2-46-5': 'They also show evidence for the presence of crystalline silicates in their spectrum, another sign of high optical depth .', '1505.04499-2-46-6': 'Finally, HV 2232 (SMC IRS 230) and HV 11464 (SMC IRS 309) are photometrically classified as Red Supergiants (see the table described by Table [REF]).', '1505.04499-2-47-0': '### Additional sources', '1505.04499-2-48-0': 'The further nine interlopers in the C-AGB/x-AGB section of the CMD defined by [CITATION] are a mixed bag of objects, all but one falling in the x-AGB category.', '1505.04499-2-48-1': 'These objects reflect selection biases and include rare categories of previously known types such as R CrB stars (2MASS J00461632[MATH]7411135, 2MASS J00571814[MATH]7242352 and OGLE SMC-SC10 107856), an S star (BFM 1) and a B[e] star (Lin 250).', '1505.04499-2-48-2': 'These object types are not included in the [CITATION] classification scheme, and could therefore never have agreed with the spectral classification.', '1505.04499-2-48-3': 'True misclassifications are the objects thought to be C-rich AGB stars, only to be revealed to be something else based on their IRS spectroscopy.', '1505.04499-2-48-4': 'These include O-PAGB star LHA 115-S 38, YSO-2 object 2MASS J01050732[MATH]7159427, and RSG IRAS F00483[MATH]7347.', '1505.04499-2-48-5': 'NGC 346:KWBBe 200 is a special case, as it was thought to be a B[e] supergiant , but its classification has recently been revised to a YSO , in line with our classification of YSO-3 (See Appendix [REF]).', '1505.04499-2-49-0': 'The FIR category defined by [CITATION] was introduced to exclude YSOs, compact regions, PNe and background galaxies from the AGB/RSG sample, by applying the [MATH] mag cut, corresponding to a rising continuum.', '1505.04499-2-49-1': 'Indeed, 14 out of 23 FIR objects are either C-PN (three objects) or YSOs (11 objects), according to their IRS spectra.', '1505.04499-2-49-2': 'The remaining nine FIR objects include the six O-AGB stars discussed in Sec. [REF], but also a symbiotic star (SSTISAGEMA J005419.21[MATH]722909.7; SMC IRS 260) and a C-PAGB (2MASS J00364631[MATH]7331351; SMC IRS 95), both of which are classified based on their IRS spectra.', '1505.04499-2-50-0': '## [CITATION]', '1505.04499-2-51-0': 'The second classification scheme was proposed by [CITATION] to identify mass-losing O-rich and C-rich AGB stars, as well as Red Supergiants, in order to estimate the dust production budget in the LMC, separated out by C-rich and O-rich chemistries.', '1505.04499-2-51-1': 'The scheme is based on the IRAC bands, using the [8.0] vs [3.6][MATH][8.0] CMD, and separates out the O-rich AGB stars and RSGs on the one hand, and C-rich AGB stars on the other hand (Fig. [REF]).', '1505.04499-2-51-2': '[CITATION] used only this diagram to classify the IRAC point sources in the LMC, but for the SMC, [CITATION] added an extra step to overcome pollution between the categories for redder [3.6][MATH][8.0] sources.', '1505.04499-2-51-3': 'Indeed, from Fig. [REF] it is clear that the red part of the C-AGB section is dominated by sources identified as YSOs (tan diamonds), while a significant fraction of the O-AGB objects (blue diamonds) also fall within the C-AGB section.', '1505.04499-2-51-4': 'In the second step, MIPS and NIR data are included, and a complex series of cuts is made in the vs CCD .', '1505.04499-2-51-5': 'In cases where the classification using the vs diagram (Fig. [REF]) deviates from the [8.0] vs [3.6][MATH][8.0] classification, the vs classification takes preference.', '1505.04499-2-51-6': 'Most spectral classifications agree with the photometric classification in the second step (see Fig. [REF]).', '1505.04499-2-51-7': 'In case where sources were only classified in one of the two steps, we used that classification.', '1505.04499-2-51-8': 'We have executed this classification method for our 209 sources, and included the results in the table described by Table [REF].', '1505.04499-2-52-0': '### Carbon-rich AGB stars', '1505.04499-2-53-0': 'The two-step classification described by [CITATION] correctly identifies most of the spectroscopically classified C-AGB stars in our sample as such.', '1505.04499-2-53-1': 'In only three cases was an object photometrically classified as being oxygen-rich (RSG/O-AGB), while the spectroscopy shows the carbon-rich nature of the source.', '1505.04499-2-53-2': 'These sources are 2MASS J00515018[MATH]7250496 (SMC IRS 103); 2MASS J00524017[MATH]7247276 (SMC IRS 127) and IRAS 00350[MATH]7436 (SMC IRS 238).', '1505.04499-2-53-3': 'IRAS 00350[MATH]7436 is among the brightest mid-infrared objects in the SMC, and falls above the C-AGB cut in Fig. 4 of [CITATION], due to its exceptional brightness.', '1505.04499-2-54-0': 'Furthermore, the classification of several C-AGB stars by [CITATION] disagrees with the spectral classification.', '1505.04499-2-54-1': 'These include: a C-rich post-AGB star (2MASS J00364631[MATH]7331351; SMC IRS 95), four O-rich early-type AGB stars (HV 1366, HV 11303, HV 838 and HV 12122; SMC IRS 36, 38, 39 and 116, respectively), a WR star (HD 5980; SMC IRS 281), a RSG (HV 11262; SMC IRS 111), a YSO-3 object (NGC 346:KWBBe 200; SMC IRS 19), the three R CrB stars (2MASS J00461632[MATH]7411135, 2MASS J00571814[MATH]7242352 and OGLE SMC-SC10 107856; SMC IRS 94, 114 and 245, respectively), the two foreground O-rich AGB stars NGC 362 SAW V16 and HV 206, the S star BFM 1 and the symbiotic star SSTISAGEMA J005419.21[MATH]722909.7.', '1505.04499-2-54-2': 'The last four objects are all in the OTHER category, which contains subclasses the work by [CITATION] does not seek to classify.', '1505.04499-2-54-3': 'The four O-rich early-type objects have actually long been recognized as such, in some cases their nature was already known in the 1980s, and they fall only slightly outside the boundaries in Fig. 5 of [CITATION].', '1505.04499-2-54-4': 'The RSG HV 11262 has very similar and colours to the four O-EAGB objects.', '1505.04499-2-54-5': '2MASS J00364631[MATH]7331351 is not properly filtered out by Fig. 5, as it falls just below the = 8 mag cutoff that is meant to exclude YSO, C-PN, and C-PAGB objects from the C-AGB category.', '1505.04499-2-54-6': 'The R CrB stars represent a rare class which, not surprisingly, overlaps in infrared colours with the C-AGB stars, as it is believed that the dust in these stars is carbonaceous .', '1505.04499-2-54-7': 'HD 5980 is the only WR star that is classified by [CITATION].', '1505.04499-2-54-8': 'Although other WR stars do appear in Fig. [REF], they do not receive a classification as they are too faint in the [8.0] band.', '1505.04499-2-54-9': 'HD 5980 is considerably brighter in the [8.0], and is the only WR star with a MIPS-[24] detection in the SMC .', '1505.04499-2-54-10': 'Closer inspection of the IRS spectrum seems to suggest that a chance superposition with a compact region gives rise to this 24 detection, and is actually not a detection of the WR star itself.', '1505.04499-2-54-11': 'Thus, the position of HD 5980 in Fig. [REF] and the classification following from it, should be disregarded.', '1505.04499-2-54-12': 'Finally, NGC 346:KWBBe 200 is a curious object, which, from its IRS spectrum appears to be a YSO, but has characteristics in common with B[e] stars (See Appendix [REF]).', '1505.04499-2-54-13': 'It falls well outside the box defined for YSO, C-PN and C-PAGB, perhaps due to its unusual nature.', '1505.04499-2-55-0': '### Oxygen-rich AGB stars and RSGs', '1505.04499-2-56-0': 'Since the main purpose of the classification scheme by [CITATION] is to determine the dust production by evolved stars distinguished by carbon-rich and oxygen-rich chemistry, the subdivision in types of evolved stars within the oxygen-rich class is less important.', '1505.04499-2-56-1': 'In fact, in the first step, all types of oxygen-rich evolved stars (AGB stars and RSGs) are lumped together as RSG/O-AGB (Fig. [REF]), while in the second step a subdivision is made between O-AGB (which also includes RSG) and O-AGB/O-PAGB (Fig. [REF]).', '1505.04499-2-56-2': 'The latter category contains the more evolved O-AGB stars.', '1505.04499-2-56-3': 'Thus, a total of three partially overlapping categories exist.', '1505.04499-2-56-4': 'The RSG/O-AGB category contains objects that are classified as such in the first step, but remained unclassified in the second step.', '1505.04499-2-56-5': 'This class contains only three objects (2MASS J00515018[MATH]7250496, 2MASS J00524017[MATH]7247276 and IRAS 00350[MATH]7436), which are all C-AGB according to their IRS spectroscopy, and are already discussed in Sec. [REF].', '1505.04499-2-57-0': 'The 24 objects classified as O-AGB in the second step are indeed all O-AGB or RSG objects according to their IRS spectroscopy.', '1505.04499-2-57-1': 'However, the category O-PAGB/O-AGB contains a more diverse range of objects.', '1505.04499-2-57-2': 'Of the eight SMC IRS objects classified by the method of [CITATION] to be in this category only four are genuine O-rich evolved stars: IRAS F00483[MATH]7347 (SMC IRS 98) is a RSG according to our classification, LHA 115-S 38 (SMC IRS 257) is found to be a O-PAGB object, and 2MASS J00463159[MATH]7328464 (SMC IRS 121) and HV 12956 (SMC IRS 277) are O-AGB stars.', '1505.04499-2-57-3': 'The other objects in this category include two carbon-rich evolved stars, 2MASS J00444111[MATH]7321361 (SMC IRS 268), a C-PAGB object, and Lin 343 (SMC IRS 161), a C-PN.', '1505.04499-2-57-4': 'Both of these objects should have fallen in the YSO/C-PN/C-PAGB, but with just below 8mag they both just miss the cutoff.', '1505.04499-2-57-5': 'Since Fig. [REF] does not show any O-rich objects in the vicinity of the K-[24] = 8mag cutoff between C-rich and O-rich post-AGB objects, a case can be made to lower the cutoff slightly.', '1505.04499-2-57-6': 'Finally, the remaining two misclassified objects in the O-PAGB/O-AGB category by [CITATION] are spectrally classified as B[e] stars in the OTHER category.', '1505.04499-2-57-7': 'They are RMC 50 (SMC IRS 193) and Lin 250 (SMC IRS 262).', '1505.04499-2-58-0': 'When checking the reverse direction, we find that all objects classified by us as O-AGB are indeed identified as either O-AGB or as O-PAGB/O-AGB according to the classification scheme by [CITATION].', '1505.04499-2-58-1': 'However, this classification scheme misclassifies all objects identified as O-EAGB by us.', '1505.04499-2-58-2': 'According to the scheme by [CITATION], they are either C-AGB (See Sect. [REF]; four out of eight objects), or remain unclassified, as they have [MATH]mag, rendering them unclassifiable in Fig. [REF], and have [MATH]mag and [MATH]mag, which causes them to fall outside all boundaries in Fig. [REF].', '1505.04499-2-58-3': 'This also happens to six of the 22 objects classified by us to be RSGs; one is misclassified as a C-AGB (See Sect. [REF]), and the other five do not receive a classification because their colours are too blue (i.e. their mass-loss rates are too low).', '1505.04499-2-59-0': 'Our sample of 209 targets with IRS spectra contains only one oxygen-rich post-AGB star, which agrees with the O-PAGB/O-AGB classification from [CITATION].', '1505.04499-2-60-0': '## [CITATION]', '1505.04499-2-61-0': 'The third classification scheme that we compare with is the method developed by [CITATION] and refined by [CITATION] to select YSO candidates.', '1505.04499-2-61-1': 'From the set of CMDs used by [CITATION], [CITATION] selected a combination of five different CMDs to select YSO candidates in the SMC.', '1505.04499-2-61-2': 'Two of these diagrams are reproduced in this work, with the 209 IRS point sources overplotted (Figs. [REF] and [REF]).', '1505.04499-2-61-3': 'After the initial colour selection of YSO candidates, [CITATION] performed additional tests, including a visual inspection of the imaging, a check against the SIMBAD and other catalogues for known non-YSO sources, and fitting against the YSO SED grid calculated by [CITATION].', '1505.04499-2-61-4': "[CITATION] arrive at a list of approximately 1000 'high-reliability' and 'probable' YSOs in the SMC.", '1505.04499-2-62-0': 'Although Figs. [REF] and [REF] appear to show considerable amount of pollution from non-YSO IRS staring mode targets, as well as many confirmed YSOs based on the IRS data straying out of the defined boxes, it is the combination of the five CMDs, along with the additional non-colour based checks that yields a highly reliable YSO candidate list.', '1505.04499-2-62-1': 'Indeed, the [CITATION] classification favours reliability over completeness, and CMD areas with significant pollution due to other sources (background galaxies, PNe) have been excluded.', '1505.04499-2-62-2': 'The list of 209 IRS staring mode point sources contains nine objects classified as probable YSOs by [CITATION], and 45 high-reliability YSO candidates.', '1505.04499-2-62-3': 'Of the high-reliability YSOs for which IRS observations are available, we find that the vast majority are indeed YSOs (covering all four classes).', '1505.04499-2-62-4': 'Only two objects turn out to be something different: region IRAS 00436[MATH]7321 (SMC IRS 310) and C-PAGB object 2MASS J01054645[MATH]7147053 (SMC IRS 243).', '1505.04499-2-62-5': 'Among the probable YSO candidates, the success rate is lower: four out of nine are not YSOs, upon inspection of their IRS spectra.', '1505.04499-2-62-6': 'RMC 50 (SMC IRS 193) is a B[e] star, while SMP SMC 11 (SMC IRS 32), LHA 115-N 43 (SMC IRS 155) and Lin 49 (SMC IRS 292) are actually C-PN objects.', '1505.04499-2-63-0': 'Furthermore, there are three YSO-3 type objects, as spectrally classified, that are not identified as YSO candidates by [CITATION].', '1505.04499-2-63-1': 'These are NGC 346:KWBBe 200 (SMC IRS 19), 2MASS J00465185[MATH]7315248 (SMC IRS 269) and LHA 115-N 8 (SMC IRS 274).', '1505.04499-2-64-0': '# Summary', '1505.04499-2-65-0': 'We have analysed all 311 -IRS staring mode observations within the SAGE-SMC IRAC and MIPS coverage of the SMC .', '1505.04499-2-65-1': 'After removing IRS observations of extended emission, blank sky and duplicate observations, we find that 209 unique IRAC point sources were targeted.', '1505.04499-2-65-2': 'We applied the infrared spectroscopic classification method devised by [CITATION], with the addition of one more category, namely O-EAGB stars (early-type oxygen-rich AGB stars).', '1505.04499-2-65-3': 'We find that the -IRS staring mode sample of point sources in the SMC contains 51 YSOs, subdivided in 14 embedded YSOs (YSO-1), 5 less-embedded YSOs (YSO-2), 22 evolved YSOs (YSO-3) and 10 HAeBe type objects (YSO-4).', '1505.04499-2-65-4': 'Furthermore, we find the sample contains 46 oxygen-rich evolved stars: 8 O-EAGB stars, 11 O-AGB stars, 22 RSGs, 1 O-PAGB object and 4 O-PN objects.', '1505.04499-2-65-5': '62 objects turn out to be carbon-rich evolved stars, namely 39 C-AGB stars, 3 C-PAGB objects and 20 C-PN objects.', '1505.04499-2-65-6': 'The sample also includes 3 R CrB stars, 1 BSG and 10 WR stars.', '1505.04499-2-65-7': '27 objects show stellar photospheres, 23 of which were selected based on their MIPS-[24] excess, and labelled by us as dusty OB stars.', '1505.04499-2-65-8': 'It turns out that the 24- emission is in general not related to the host OB star .', '1505.04499-2-65-9': 'Finally, the sample includes a small number of other objects (OTHER), which do not follow from the classification method.', '1505.04499-2-65-10': 'These include 2 B[e] stars, 2 foreground oxygen-rich AGB stars, an S star, and a symbiotic star.', '1505.04499-2-66-0': 'In this work, we have compared the resulting spectral classifications with the outcome of photometric classification schemes.', '1505.04499-2-66-1': 'It should be noted that the spectroscopic observations are obtained in 14 different observing programs, with a diverse range of science goals.', '1505.04499-2-66-2': 'Thus, there is no homogeneous coverage of colour-magnitude space, and it will be impossible to quantify the goodness of any given photometric classification method.', '1505.04499-2-66-3': 'Furthermore, the observing programs tend to target rare types of sources in a disproportionate amount, and some of these rare type of sources (R CrB stars, WR stars, B[e] stars, etc.) are not included in photometric classification schemes, precisely because they are rare.', '1505.04499-2-66-4': 'Thus, these objects tend to pollute the classification schemes discussed here, but statistically they are rather insignificant.', '1505.04499-2-67-0': 'We reviewed three different photometric classification schemes for infrared sources in the SMC: the schemes by [CITATION] and [CITATION] for evolved stars, and the classification scheme to select candidate YSOs by [CITATION].', '1505.04499-2-67-1': 'The latter scheme is not a pure photometric classification, as it includes additional steps, such as visual inspection of the direct environment of the point source in imaging, checks against existing catalogues and fitting against the grid of YSO SEDs calculated by [CITATION].', '1505.04499-2-67-2': 'However, as discussed in Sec. [REF], the 54 overlapping sources from this work with the resulting YSO candidate list are mostly correctly classified.', '1505.04499-2-67-3': 'Only a few sources are misclassified in either direction, i.e. three spectroscopically confirmed YSOs were not on the candidate list by [CITATION], and six sources on the candidate list were found to be something else upon inspection of their IRS spectroscopy.', '1505.04499-2-67-4': 'All-in-all we conclude that the YSO candidate list produced by [CITATION] is reliable, with 48/54 sources indeed being YSOs and the high-reliability sources doing better than the probable sources.', '1505.04499-2-67-5': 'Only three spectroscopically confirmed YSOs were missed due to unusual infrared colors.', '1505.04499-2-68-0': 'The two photometric classification methods for evolved stars can be directly compared to each other.', '1505.04499-2-68-1': 'The method by [CITATION] has its focus on identifying the entire dusty evolved star population, while the [CITATION] method is mainly driven by the motivation to determine the carbon-rich and oxygen-rich dust production rates.', '1505.04499-2-68-2': 'Thus, in the later method, correct identification of lower mass-loss rate stars is not so important.', '1505.04499-2-68-3': 'Both methods can be used to estimate the integrated dust production rate.', '1505.04499-2-68-4': 'Due to the low metallicity of the SMC, carbon-rich evolved stars are more numerous , and the carbon stars have, on average, the highest mass-loss rates.', '1505.04499-2-68-5': 'Thus, identifying carbon stars correctly is important as they dominate the dust budget .', '1505.04499-2-68-6': 'Apart from rare object classes, [CITATION] very efficiently separate C-AGB stars from the other classes, only classifying one non-C-AGB star as C-AGB star, while classifying all genuine C-AGB objects as C-AGB.', '1505.04499-2-68-7': '[CITATION] do slightly worse, with three genuine C-AGB objects being classified as something else, and a number of objects, including some rare types, incorrectly classified as C-AGB star.', '1505.04499-2-69-0': 'On the oxygen-rich side, we find that [CITATION] perform better for the high-mass loss rate objects (O-AGB stars), but that they perform rather poorly on the low-mass rate objects (O-EAGB stars), while the performance of the [CITATION] classification method is reversed because of overlap between high-mass AGB stars and RSGs near the classical AGB limit.', '1505.04499-2-70-0': 'Finally, we note that of the two steps involved in the classification method by [CITATION], the second step (Fig. [REF]) matches very well with the actual spectroscopic classification.', '1505.04499-2-70-1': 'It has been introduced by [CITATION] as a correction on the first step from [CITATION], but almost all photometric classifications currently included in the online table described by Table [REF] correspond to the second step, and most of the time match the spectroscopic classification, rendering the first step practically unnecessary.', '1505.04499-2-70-2': 'Thus, in case of the [CITATION] classification scheme, applying only the second step, as demonstrated in Fig. [REF], would suffice for dusty sources.'}

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[]

[]

[]

[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1505.04499

1110.1226

{'1110.1226-1-0-0': 'If quark stars exist, they may be enveloped in thin electron layers (electron seas), which uniformly surround the entire star.', '1110.1226-1-0-1': 'These layers will be affected by the magnetic fields of quark stars in such a way that the electron seas would transmit hydromagnetic cyclotron waves, as studied in this paper.', '1110.1226-1-0-2': 'Particular attention is devoted to vortex hydrodynamical oscillations of the electron sea.', '1110.1226-1-0-3': 'The frequency spectrum of these oscillations is derived in analytic form.', '1110.1226-1-0-4': 'If the thermal X-ray spectra of quark stars are modulated by vortex hydrodynamical vibrations, the thermal spectra of compact stars, foremost cental compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally.', '1110.1226-1-0-5': 'The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model.', '1110.1226-1-1-0': '# Introduction', '1110.1226-1-2-0': 'The spectral features of thermal X-ray emission are essential for us to understand the real nature of pulsar-like compact stars.', '1110.1226-1-3-0': 'Calculations show that atomic spectral lines form in the atmospheres of neutron stars.', '1110.1226-1-3-1': 'From the detection and identification of atomic lines in thermal X-ray spectra one can infer neutron star masses ([MATH]) and radii ([MATH]), since the redshift and broadening of the spectral lines depend on [MATH] and [MATH], respectively.', '1110.1226-1-4-0': 'Atomic features are expected to be detectable with the spectrographs on board of Chandra and XMM-Newton.', '1110.1226-1-4-1': 'No atomic features have yet been discovered with certainty, however.', '1110.1226-1-4-2': 'This may have it origin in the very strong magnetic fields carried by neutron stars.', '1110.1226-1-4-3': 'An alternative explanation could be that the underlying compact star is not a neutron star but a bare strange (quark matter) star [CITATION].', '1110.1226-1-4-4': 'The surface of such an object does not consist of atomic nuclei/ions, as it is the case for a neutron star, but of a sea of electrons which envelopes the quark matter.', '1110.1226-1-5-0': 'Strange stars are quark stars made of absolutely stable strange quark matter [CITATION].', '1110.1226-1-5-1': 'They consist of essentially equal numbers of up, down and strange quarks as well as of electrons [CITATION].', '1110.1226-1-5-2': 'The latter are needed to neutralize the electric charges of the quarks, rendering the interior of strange stars electrically neutral.', '1110.1226-1-6-0': 'Quark matter is bound by the strong interaction, while electrons are bound to quark matter by the electromagnetic interaction.', '1110.1226-1-6-1': 'Since the latter is long-range, some of the electrons in the surface region of a quark star reside outside of the quark matter boundary, leading to a quark matter core which is surrounded by a fairly thin (thousands of femtometer thick) sea of electrons [CITATION].', '1110.1226-1-7-0': 'Due to the enormous advances in X-ray astronomy, more and more so-called dead pulsars are discovered, whose thermal radiation dominates over a very weak or negligibly small magnetospheric activity.', '1110.1226-1-8-0': 'The best absorption features (at [MATH] keV and [MATH] keV) were detected for the central compact object (CCO) 1E 1207.4-5209 in the center of supernova remnant PKS 1209-51/52 (see Table [REF]).', '1110.1226-1-8-1': 'Initially, these', '1110.1226-1-9-0': 'features were thought to be associated with the atomic transitions of ionized helium in a stellar atmosphere where a strong magnetic field is present [CITATION].', '1110.1226-1-9-1': 'Soon thereafter, however, it was noted that these lines are of electron-cyclotron origin [CITATION].', '1110.1226-1-10-0': 'The spectrum of 1E 1207.4-5209 shows two more features that may be caused by resonant cyclotron absorption, one at [MATH] keV and another, but of lower significance, at [MATH] keV [CITATION].', '1110.1226-1-10-1': "These features vary in phase with the star's rotation.", '1110.1226-1-11-0': 'Although the detailed mechanism which causes the absorption features is still a matter of debate, timing observations predict a rather weak magnetic field for this CCO, in agreement to what is obtained under the assumption that the lowest-energy line at 0.7 keV is the electron-cyclotron fundamental, favoring the electron-cyclotron interpretation [CITATION].', '1110.1226-1-12-0': 'Besides 1E 1207.4-5209, broad absorption lines have also been discovered in other dead pulsars (listed in Table [REF]), especially in so-called X-ray dim isolated neutron stars (XDINSs), between about 0.3 and 0.7 keV [CITATION].', '1110.1226-1-13-0': 'In this paper, we re-investigate the physics of these absorption features.', '1110.1226-1-13-1': 'The key assumption that we make here is that these features originate from the electron seas on quarks stars rather than from neutron stars, whose surface properties are radically different from those of strange stars [CITATION].', '1110.1226-1-13-2': 'Of key importance is the magnetic field carried by a quark star, which critically affects the global properties (hydrodynamic surface fluctuations) of the electron sea at the surface of the star.', '1110.1226-1-14-0': 'We study this problem in the framework of classical electrodynamics in terms of cyclotron resonances of electrons in weak magnetic fields, since the magnetic fields of dead pulsars are much lower than the critical field, [MATH] G, at which the quantization of the cyclotron orbits of electrons into Landau levels occurs.', '1110.1226-1-15-0': '# Hydro-cyclotron waves', '1110.1226-1-16-0': '## Governing equations', '1110.1226-1-17-0': 'For what follows we restrict ourselves to a discussion of the large-scale oscillations of the electron sea.', '1110.1226-1-17-1': 'We will be applying the semi-classical approach of classical electron theory of metals and making use of standard equations of fluid-mechanics.', '1110.1226-1-17-2': 'The equations of motions of a viscous electron fluid are given by [CITATION] [EQUATION] where [MATH] and [MATH] are the change and number density of electrons, respectively.', '1110.1226-1-18-0': "We emphasize that [MATH] stands for the convective current density and not for Ampere's [MATH], as it is the case for magneto-hydrodynamics.", '1110.1226-1-18-1': 'This means that the hydrodynamic oscillations in question are of non-Alfven type.', '1110.1226-1-18-2': 'In Eq. ([REF]), [MATH] denotes the effective viscosity of the electron fluid, which originates from collisions of electrons with the magnetic field lines at the stellar surface.', '1110.1226-1-19-0': 'The governing equation, Eq. ([REF]), can be represented as', '1110.1226-1-20-0': '[EQUATION] where [MATH] is the cyclotron frequency and [MATH] stands for the effective viscosity of the electron fluid, originating from collisions between electrons.', '1110.1226-1-20-1': 'In the Appendix, we show that the electron sea can transmit macroscopic perturbations in the form of rotational hydro-cyclotron waves which are characterized by the following dispersion relation,', '1110.1226-1-21-0': '[EQUATION] where [MATH] and [MATH] denote the frequency and wave vector of the perturbations, respectively.', '1110.1226-1-21-1': 'In the collision-free regime, [MATH], the hydro-cyclotron electron wave is described as a transverse, circularly polarized wave whose dispersion relation and propagation speed are given by', '1110.1226-1-22-0': '[EQUATION] respectively.', '1110.1226-1-22-1': 'Here, [MATH] is the angle between the magnetic field [MATH] and the wave vector [MATH].', '1110.1226-1-22-2': 'If [MATH] one has [MATH].', '1110.1226-1-22-3': 'In metals these kind of oscillations are observed as electron-cyclotron resonances.', '1110.1226-1-22-4': 'There are two possible resonance states, one for [MATH] and the other for [MATH].', '1110.1226-1-22-5': 'These resonances correspond to the two opposite orientations of circularly polarized electron cyclotron waves.', '1110.1226-1-23-0': '## Hydro-cyclotron oscillations of electrons on bare strange quark stars', '1110.1226-1-24-0': 'We restrict our analysis to the collision-free regime of vortex hydro-cyclotron oscillations.', '1110.1226-1-24-1': 'Using spherical coordinates, equation ([REF]) then takes the form [EQUATION]', '1110.1226-1-24-2': 'Taking the curl of both sides of Eq. ([REF]), we obtain', '1110.1226-1-25-0': '[EQUATION]', '1110.1226-1-25-1': 'Let the magnetic field [MATH] be directed along the [MATH]-axis, so that in Cartesian coordinates [MATH].', '1110.1226-1-25-2': 'We then have [EQUATION]', '1110.1226-1-25-3': "From a mathematical point of view, the problem can be considerably simplified if one expresses the velocity [MATH], which obeys the condition [MATH], in terms of Stokes' stream function, [MATH].", '1110.1226-1-25-4': 'This leads to [EQUATION]', '1110.1226-1-25-5': 'The depth of the electron layer near the star is much smaller than the stellar radius so that [MATH] to a very good approximation.', '1110.1226-1-25-6': 'Equation ([REF]) then simplifies to [EQUATION] with the radial component of the vortex given in terms of [MATH], [EQUATION]', '1110.1226-1-25-7': 'Substituting Eq. ([REF]) and Eq. ([REF]) into Eq. ([REF]) leads to', '1110.1226-1-26-0': '[EQUATION]', '1110.1226-1-26-1': 'The fact that free electrons undergo cyclotron oscillations in the planes perpendicular to the magnetic field suggests that the stream function [MATH] can be written in the following separable form,', '1110.1226-1-27-0': '[EQUATION]', '1110.1226-1-27-1': 'The "[MATH]" sign allows for cyclotron oscillations which are induced by the clockwise polarized wave, and the "[MATH]" sign allows for cyclotron oscillations induced by the count-clockwise polarized wave.', '1110.1226-1-27-2': 'Substituting ([REF]) into ([REF]) leads to', '1110.1226-1-28-0': '[EQUATION]', '1110.1226-1-28-1': 'In the reference frame where the polar axis is fixed, Eq. ([REF]) is identical to the Legendre equation for the surface spherical function, [EQUATION] where [MATH] denotes the Legendre polynomial of degree [MATH].', '1110.1226-1-28-2': 'Hence, setting [MATH] we obtain [EQUATION]', '1110.1226-1-28-3': 'From this relation we can read off the frequency of a surface hydro-cyclotron oscillation of a given order [MATH].', '1110.1226-1-29-0': '## Characteristic features of cyclotron frequencies', '1110.1226-1-30-0': 'Let us consider the spectrum of the positive branch [MATH] of Eq. ([REF]),', '1110.1226-1-31-0': '[EQUATION]', '1110.1226-1-31-1': 'From [EQUATION] it follows that this ratio becomes a constant for [MATH].', '1110.1226-1-32-0': 'Such a spectral feature is notably different from the one of electron-cyclotron resonances of transitions between different Landau levels.', '1110.1226-1-33-0': 'From the energy eigenvalues, [MATH], of an electron in a strong magnetic field, which are found by solving the Dirac equation (see Ref. [CITATION]), one may approximate the value of [MATH] for a relatively weak magnetic field, [MATH], by [EQUATION]', '1110.1226-1-33-1': 'Therefore, in the framework of a single-particle approximation, the emission/absorption frequencies, which are given by [MATH], should occur at [EQUATION]', '1110.1226-1-33-2': 'Table [REF] compares the results of Eq. ([REF]) with the results of Eq. ([REF]) obtained for the hydro-cyclotron wave model.', '1110.1226-1-34-0': 'Most notably, it follows that for the hydro-cyclotron wave model one obtains [MATH], in contrast to the cyclotron resonance model for single electrons which predicts this ratio for [MATH].', '1110.1226-1-35-0': '# 1E 1207.4-5209 and other compact objects', '1110.1226-1-36-0': 'As already mentioned in the Introduction, 1E 1207.4-5209 (or J1210-5226) in PKS 1209-51/52 is one of the central compact objects in supernova remnants [CITATION], where broad absorption lines, near (0.7, 1.4) keV [CITATION], and possibly near (2.1, 2.8) keV [CITATION] were detected for the first time.', '1110.1226-1-37-0': 'The interpretation of the absorption feature at [MATH] keV is currently a matter of debate, in contrast to the feature at [MATH] keV which is essentially unexplained.', '1110.1226-1-37-1': 'Intriguingly, an absorption feature with the same energy, 2.1 keV, has also been detected in the accretion-driven X-ray pulsar 4U 1538-52 [CITATION].', '1110.1226-1-38-0': 'For what follows, we assume that 1E 1207.4-5209 is a strange quark star and that (some of) these absorption features are produced by the hydro-cyclotron oscillations of the electron sea at the surface of such an object.', '1110.1226-1-38-1': 'Assuming a magnetic surface field of [MATH] G and thus [MATH] keV, we obtain the oscillation frequencies shown in Table [REF].', '1110.1226-1-39-0': 'A magnetic field of [MATH] G is compatible with the magnetic fields inferred for 1E 1207.4-5209 from timing solutions [CITATION] ([MATH] G or [MATH] G), since 1E 1207.4-5209 shows no magnetospheric activity and the [MATH]-value would be overestimated if one applies the spin-down power of magnetic-dipole radiation [CITATION].', '1110.1226-1-39-1': 'We note that the absorption feature at [MATH] keV shown in Table [REF] may not be detectable since the stellar temperature is only [MATH] keV (see Table [REF]), which will suppress any thermal feature in that energy range.', '1110.1226-1-40-0': 'Aside from 1E 1207.4-5209, one may ask what would be the magnetic fields of other dead pulsars (e.g., radio-quiet compact objects) if their spectral absorption features would also be of hydro-cyclotron origin?', '1110.1226-1-40-1': 'Intriguingly, the hydro-cyclotron wave model predicts magnetic fields that are twice as large as those derived from the electron cyclotron model if the absorption feature is at [MATH]; these fields could be [MATH] times greater (see Table [REF]) if the absorption feature is at [MATH] or [MATH].', '1110.1226-1-41-0': 'The absorption lines at [MATH] keV may indicate that the fields of XDINSs are on the order of [MATH] to [MATH] G, if oscillation modes with [MATH] are not significant.', '1110.1226-1-42-0': 'As noted in [CITATION], unique absorption features on compact stars are only detectable with Chandra and XMM-Newton if the stellar magnetic fields are relatively weak ([MATH] G to [MATH] G), since the stellar temperatures are only a few 0.1 keV.', '1110.1226-1-43-0': 'The fields of many pulsar-like objects are generally greater than this value, with the exception of old millisecond pulsars whose fields are on the order of [MATH] G. Central compact objects, on the other hand, seem to have sufficiently weak magnetic fields (see Table [REF]) so that absorption features originating from their surfaces should be detectable by Chandra and XMM-Newton.', '1110.1226-1-43-1': 'Arguments favoring the interpretation of compact central objects as strange quark stars have been put forward in [CITATION], where it was shown that the magnetic field observed for some CCOs could be generated by small amounts of differential rotation between the quark matter core and the electron sea.', '1110.1226-1-44-0': '# Summary', '1110.1226-1-45-0': 'In this paper, we study the global motion of the electron seas on the surfaces of hypothetical strange quark stars.', '1110.1226-1-45-1': 'It is found that such electron seas may undergo hydro-cyclotron oscillations whose frequencies are given by [MATH], where [MATH] and [MATH] the cyclotron frequency.', '1110.1226-1-46-0': 'We propose that some of the absorption features detected in the thermal X-ray spectra of dead (e.g., radio silent) compact objects may have their origin in excitations of these hydro-cyclotron oscillations of the electron sea, provided these stellar objects are interpreted as strange quark stars.', '1110.1226-1-47-0': 'The central compact object 1E 1207.4-5209 appears particularly interesting.', '1110.1226-1-47-1': 'It shows an absorption feature at 0.7 keV which is not much stronger than the another absorption feature observed at 1.4 keV.', '1110.1226-1-47-2': 'This can be readily explained in the framework of the hydro-cyclotron oscillation model, since two lines with [MATH] and [MATH] could essentially have the same intensity.', '1110.1226-1-47-3': 'This is very different for the electron-cyclotron model, for which the oscillator strength of the first harmonic is much smaller than the oscillator strength of the fundamental.', '1110.1226-1-48-0': '# Appendix', '1110.1226-1-49-0': 'Here we derive the dispersion relation characterizing the propagation of hydro-cyclotron electron wave in the slab-geometry approximation.', '1110.1226-1-50-0': 'The governing equation of viscous electron fluid under the action of Lorentz force is given by [EQUATION] which can be written as [EQUATION] where [EQUATION] [MATH] is the cyclotron frequency, and [MATH] stands for the effective viscosity of an electron fluid originating from collisions between electrons.', '1110.1226-1-50-1': 'To make the problem analytically tractable, we treat the electron sea as an incompressible fluid and assume a uniform magnetic field.', '1110.1226-1-50-2': 'Equation ([REF]) can then be written as [EQUATION]', '1110.1226-1-50-3': 'Upon applying to Eq. ([REF]) the operator [MATH], we arrive at [EQUATION] where [MATH], and [EQUATION] where [MATH][MATH].', '1110.1226-1-51-0': 'Considering a perturbation in the form of [MATH], we have [EQUATION] where [MATH].', '1110.1226-1-51-1': 'It is convenient to rewrite the last equation as [EQUATION]', '1110.1226-1-51-2': 'Multiplication of both sides of Eq. ([REF]) with [MATH] leads to [EQUATION]', '1110.1226-1-51-3': 'Inserting the left-hand-side of Eq. ([REF]) into the right-hand-side of Eq. ([REF]) gives [EQUATION] or [EQUATION] which is Eq. ([REF]).'}

{'1110.1226-2-0-0': 'If quark stars exist, they may be enveloped in thin electron layers (electron seas), which uniformly surround the entire star.', '1110.1226-2-0-1': 'These layers will be affected by the magnetic fields of quark stars in such a way that the electron seas would transmit hydromagnetic cyclotron waves, as studied in this paper.', '1110.1226-2-0-2': 'Particular attention is devoted to vortex hydrodynamical oscillations of the electron sea.', '1110.1226-2-0-3': 'The frequency spectrum of these oscillations is derived in analytic form.', '1110.1226-2-0-4': 'If the thermal X-ray spectra of quark stars are modulated by vortex hydrodynamical vibrations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally.', '1110.1226-2-0-5': 'The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model.', '1110.1226-2-1-0': '# Introduction', '1110.1226-2-2-0': 'The spectral features of thermal X-ray emission are essential for us to understand the real nature of pulsar-like compact stars.', '1110.1226-2-3-0': 'Calculations show that atomic spectral lines form in the atmospheres of neutron stars.', '1110.1226-2-3-1': 'From the detection and identification of atomic lines in thermal X-ray spectra one can infer neutron star masses ([MATH]) and radii ([MATH]), since the redshift and broadening of the spectral lines depend on [MATH] and [MATH], respectively.', '1110.1226-2-4-0': 'Atomic features are expected to be detectable with the spectrographs on board of Chandra and XMM-Newton.', '1110.1226-2-4-1': 'No atomic features have yet been discovered with certainty, however.', '1110.1226-2-4-2': 'This may have it origin in the very strong magnetic fields carried by neutron stars.', '1110.1226-2-4-3': 'An alternative explanation could be that the underlying compact star is not a neutron star but a bare strange (quark matter) star [CITATION].', '1110.1226-2-4-4': 'The surface of such an object does not consist of atomic nuclei/ions, as it is the case for a neutron star, but of a sea of electrons which envelopes the quark matter.', '1110.1226-2-5-0': 'Strange stars are quark stars made of absolutely stable strange quark matter [CITATION].', '1110.1226-2-5-1': 'They consist of essentially equal numbers of up, down and strange quarks as well as of electrons [CITATION].', '1110.1226-2-5-2': 'The latter are needed to neutralize the electric charges of the quarks, rendering the interior of strange stars electrically neutral.', '1110.1226-2-6-0': 'Quark matter is bound by the strong interaction, while electrons are bound to quark matter by the electromagnetic interaction.', '1110.1226-2-6-1': 'Since the latter is long-range, some of the electrons in the surface region of a quark star reside outside of the quark matter boundary, leading to a quark matter core which is surrounded by a fairly thin (thousands of femtometer thick) sea of electrons [CITATION].', '1110.1226-2-7-0': 'Due to the enormous advances in X-ray astronomy, more and more so-called dead pulsars are discovered, whose thermal radiation dominates over a very weak or negligibly small magnetospheric activity.', '1110.1226-2-8-0': 'The best absorption features (at [MATH] keV and [MATH] keV) were detected for the central compact object (CCO) 1E 1207.4-5209 in the center of supernova remnant PKS 1209-51/52 (see Table [REF]).', '1110.1226-2-8-1': 'Initially, these', '1110.1226-2-9-0': 'features were thought to be associated with the atomic transitions of ionized helium in a stellar atmosphere where a strong magnetic field is present [CITATION].', '1110.1226-2-9-1': 'Soon thereafter, however, it was noted that these lines are of electron-cyclotron origin [CITATION].', '1110.1226-2-10-0': 'The spectrum of 1E 1207.4-5209 shows two more features that may be caused by resonant cyclotron absorption, one at [MATH] keV and another, but of lower significance, at [MATH] keV [CITATION].', '1110.1226-2-10-1': "These features vary in phase with the star's rotation.", '1110.1226-2-11-0': 'Although the detailed mechanism which causes the absorption features is still a matter of debate, timing observations predict a rather weak magnetic field for this CCO, in agreement to what is obtained under the assumption that the lowest-energy line at 0.7 keV is the electron-cyclotron fundamental, favoring the electron-cyclotron interpretation [CITATION].', '1110.1226-2-12-0': 'Besides 1E 1207.4-5209, broad absorption lines have also been discovered in other dead pulsars (listed in Table [REF]), especially in so-called X-ray dim isolated neutron stars (XDINSs), between about 0.3 and 0.7 keV [CITATION].', '1110.1226-2-13-0': 'In this paper, we re-investigate the physics of these absorption features.', '1110.1226-2-13-1': 'The key assumption that we make here is that these features originate from the electron seas on quarks stars rather than from neutron stars, whose surface properties are radically different from those of strange stars [CITATION].', '1110.1226-2-13-2': 'Of key importance is the magnetic field carried by a quark star, which critically affects the global properties (hydrodynamic surface fluctuations) of the electron sea at the surface of the star.', '1110.1226-2-14-0': 'We study this problem in the framework of classical electrodynamics in terms of cyclotron resonances of electrons in weak magnetic fields, since the magnetic fields of dead pulsars are much lower than the critical field, [MATH] G, at which the quantization of the cyclotron orbits of electrons into Landau levels occurs.', '1110.1226-2-15-0': '# Hydro-cyclotron waves', '1110.1226-2-16-0': '## Governing equations', '1110.1226-2-17-0': 'For what follows we restrict ourselves to a discussion of the large-scale oscillations of an electron sea subjected to a stellar magnetic field.', '1110.1226-2-17-1': 'We will be applying the semi-classical approach of classical electron theory of metals and making use of standard equations of fluid-mechanics.', '1110.1226-2-18-0': 'The electrons are viewed as a viscous fluid of uniform density [MATH] (where [MATH] is the electron number density) whose oscillations are given in terms of the mean electron flow velocity [MATH].', '1110.1226-2-18-1': 'This implies that the fluctuation current-carrying flow is described by the density of the convective current [MATH], where [MATH] is the electron charge density.', '1110.1226-2-19-0': 'The equations of motions of a viscous electron fluid are then given by [CITATION] [EQUATION] where [MATH] and [MATH] are the change and number densities of electrons, respectively.', '1110.1226-2-20-0': "We emphasize that [MATH] stands for the convective current density and not for Ampere's [MATH], as it is the case for magneto-hydrodynamics.", '1110.1226-2-20-1': 'This means that the hydrodynamic oscillations in question are of non-Alfven type.', '1110.1226-2-20-2': 'In Eq. ([REF]), [MATH] denotes the effective viscosity of the electron fluid, which originates from collisions of electrons with the magnetic field lines at the stellar surface.', '1110.1226-2-21-0': "It is worth noting that the cyclotron waves can be regarded as an analogue of the inertial waves in a rotating incompressible fluid, as presented in Eq. (III.56) of Chandrasekhar's book [CITATION].", '1110.1226-2-22-0': 'The governing equation, Eq. ([REF]), can be represented as', '1110.1226-2-23-0': '[EQUATION] where [MATH] is the cyclotron frequency.', '1110.1226-2-23-1': 'In the Appendix, we show that the electron sea can transmit macroscopic perturbations in the form of rotational hydro-cyclotron waves which are characterized by the following dispersion relation,', '1110.1226-2-24-0': '[EQUATION] where [MATH] and [MATH] denote the frequency and wave vector of the perturbations, respectively.', '1110.1226-2-25-0': 'The Larmor radius of an electron in a strong magnetic field, [MATH], is very small for pulsar-like compact stars, and we neglect the viscosity term in the following analysis of the motion of collective electrons.', '1110.1226-2-26-0': 'In the collision-free regime, [MATH], the hydro-cyclotron electron wave is described as a transverse, circularly polarized wave whose dispersion relation and propagation speed are given by', '1110.1226-2-27-0': '[EQUATION] respectively.', '1110.1226-2-27-1': 'Here, [MATH] is the angle between the magnetic field [MATH] and the wave vector [MATH].', '1110.1226-2-27-2': 'If [MATH] one has [MATH].', '1110.1226-2-27-3': 'In metals these kind of oscillations are observed as electron-cyclotron resonances.', '1110.1226-2-27-4': 'There are two possible resonance states, one for [MATH] and the other for [MATH].', '1110.1226-2-27-5': 'These resonances correspond to the two opposite orientations of circularly polarized electron cyclotron waves.', '1110.1226-2-28-0': '## Hydro-cyclotron oscillations of electrons on bare strange quark stars', '1110.1226-2-29-0': 'We restrict our analysis to the collision-free regime of vortex hydro-cyclotron oscillations.', '1110.1226-2-29-1': 'Using spherical coordinates, equation ([REF]) then takes the form [EQUATION]', '1110.1226-2-29-2': 'Taking the curl of both sides of Eq. ([REF]), we obtain', '1110.1226-2-30-0': '[EQUATION]', '1110.1226-2-30-1': 'Let the magnetic field [MATH] be directed along the [MATH]-axis, so that in Cartesian coordinates [MATH].', '1110.1226-2-30-2': 'We then have [EQUATION]', '1110.1226-2-30-3': "From a mathematical point of view, the problem can be considerably simplified if one expresses the velocity [MATH], which obeys the condition [MATH], in terms of Stokes' stream function, [MATH].", '1110.1226-2-30-4': 'This leads to [EQUATION]', '1110.1226-2-30-5': 'The depth of the electron layer near the star is much smaller than the stellar radius so that [MATH] to a very good approximation.', '1110.1226-2-30-6': 'Equation ([REF]) then simplifies to [EQUATION] with the radial component of the vortex given in terms of [MATH], [EQUATION]', '1110.1226-2-30-7': 'Substituting Eq. ([REF]) and Eq. ([REF]) into Eq. ([REF]) leads to', '1110.1226-2-31-0': '[EQUATION]', '1110.1226-2-31-1': 'The fact that free electrons undergo cyclotron oscillations in the planes perpendicular to the magnetic field suggests that the stream function [MATH] can be written in the following separable form,', '1110.1226-2-32-0': '[EQUATION]', '1110.1226-2-32-1': 'The "[MATH]" sign allows for cyclotron oscillations which are induced by the clockwise polarized wave, and the "[MATH]" sign allows for cyclotron oscillations induced by the count-clockwise polarized wave.', '1110.1226-2-32-2': 'Substituting ([REF]) into ([REF]) leads to', '1110.1226-2-33-0': '[EQUATION]', '1110.1226-2-33-1': 'In the reference frame where the polar axis is fixed, Eq. ([REF]) is identical to the Legendre equation for the surface spherical function, [EQUATION] where [MATH] denotes the Legendre polynomial of degree [MATH].', '1110.1226-2-33-2': 'Hence, setting [MATH] we obtain [EQUATION]', '1110.1226-2-33-3': 'From this relation we can read off the frequency of a surface hydro-cyclotron oscillation of a given order [MATH].', '1110.1226-2-34-0': '## Characteristic features of cyclotron frequencies', '1110.1226-2-35-0': 'Let us consider the spectrum of the positive branch [MATH] of Eq. ([REF]),', '1110.1226-2-36-0': '[EQUATION]', '1110.1226-2-36-1': 'From [EQUATION] it follows that this ratio becomes a constant for [MATH].', '1110.1226-2-37-0': 'Such a spectral feature is notably different from the one of electron-cyclotron resonances of transitions between different Landau levels.', '1110.1226-2-38-0': 'From the energy eigenvalues, [MATH], of an electron in a strong magnetic field, which are found by solving the Dirac equation (see Ref. [CITATION]), one may approximate the value of [MATH] for a relatively weak magnetic field, [MATH], by [EQUATION]', '1110.1226-2-38-1': 'Therefore, in the framework of a single-particle approximation, the emission/absorption frequencies, which are given by [MATH], should occur at [EQUATION]', '1110.1226-2-38-2': 'Table [REF] compares the results of Eq. ([REF]) with the results of Eq. ([REF]) obtained for the hydro-cyclotron wave model.', '1110.1226-2-39-0': 'Most notably, it follows that for the hydro-cyclotron wave model one obtains [MATH], in contrast to the cyclotron resonance model for single electrons which predicts this ratio for [MATH].', '1110.1226-2-40-0': '# 1E 1207.4-5209 and other compact objects', '1110.1226-2-41-0': 'As already mentioned in the Introduction, 1E 1207.4-5209 (or J1210-5226) in PKS 1209-51/52 is one of the central compact objects in supernova remnants [CITATION], where broad absorption lines, near (0.7, 1.4) keV [CITATION], and possibly near (2.1, 2.8) keV [CITATION] were detected for the first time.', '1110.1226-2-42-0': 'The interpretation of the absorption feature at [MATH] keV is currently a matter of debate, in contrast to the feature at [MATH] keV which is essentially unexplained.', '1110.1226-2-42-1': 'Intriguingly, an absorption feature with the same energy, 2.1 keV, has also been detected in the accretion-driven X-ray pulsar 4U 1538-52 [CITATION].', '1110.1226-2-43-0': 'For what follows, we assume that 1E 1207.4-5209 is a strange quark star and that (some of) these absorption features are produced by the hydro-cyclotron oscillations of the electron sea at the surface of such an object.', '1110.1226-2-43-1': 'Assuming a magnetic surface field of [MATH] G and thus [MATH] keV, we obtain the oscillation frequencies shown in Table [REF].', '1110.1226-2-44-0': 'A magnetic field of [MATH] G is compatible with the magnetic fields inferred for 1E 1207.4-5209 from timing solutions [CITATION] ([MATH] G or [MATH] G), since 1E 1207.4-5209 shows no magnetospheric activity and the [MATH]-value would be overestimated if one applies the spin-down power of magnetic-dipole radiation [CITATION].', '1110.1226-2-44-1': 'We note that the absorption feature at [MATH] keV shown in Table [REF] may not be detectable since the stellar temperature is only [MATH] keV (see Table [REF]), which will suppress any thermal feature in that energy range.', '1110.1226-2-45-0': 'Aside from 1E 1207.4-5209, one may ask what would be the magnetic fields of other dead pulsars (e.g., radio-quiet compact objects) if their spectral absorption features would also be of hydro-cyclotron origin?', '1110.1226-2-45-1': 'Intriguingly, the hydro-cyclotron wave model predicts magnetic fields that are twice as large as those derived from the electron cyclotron model if the absorption feature is at [MATH]; these fields could be [MATH] times greater (see Table [REF]) if the absorption feature is at [MATH] or [MATH].', '1110.1226-2-46-0': 'The absorption lines at [MATH] keV may indicate that the fields of XDINSs are on the order of [MATH] to [MATH] G, if oscillation modes with [MATH] are not significant.', '1110.1226-2-47-0': 'As noted in [CITATION], unique absorption features on compact stars are only detectable with Chandra and XMM-Newton if the stellar magnetic fields are relatively weak ([MATH] G to [MATH] G), since the stellar temperatures are only a few 0.1 keV.', '1110.1226-2-48-0': 'The fields of many pulsar-like objects are generally greater than this value, with the exception of old millisecond pulsars whose fields are on the order of [MATH] G. Central compact objects, on the other hand, seem to have sufficiently weak magnetic fields (see Table [REF]) so that absorption features originating from their surfaces should be detectable by Chandra and XMM-Newton.', '1110.1226-2-48-1': 'Arguments favoring the interpretation of compact central objects as strange quark stars have been put forward in [CITATION], where it was shown that the magnetic field observed for some CCOs could be generated by small amounts of differential rotation between the quark matter core and the electron sea.', '1110.1226-2-49-0': 'Besides dead pulsars, anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are enigmatic objects which have become hot topics of modern astrophysics.', '1110.1226-2-50-0': 'Whether they are magnetars/quark stars is an open question [CITATION].', '1110.1226-2-51-0': 'In case that AXPs/SGRSs should be bare strange stars, the absorption lines detected from SGR 1806-20 could be understood in the framework of the hydro-cyclotron oscillation model.', '1110.1226-2-52-0': 'Assuming a normal magnetic field of [MATH] G so that [MATH] keV, one sees that the oscillation model predicts hydro-cyclotron frequencies which coincide with the observed listed in Table [REF].', '1110.1226-2-53-0': '# Summary', '1110.1226-2-54-0': 'In this paper, we study the global motion of the electron seas on the surfaces of hypothetical strange quark stars.', '1110.1226-2-54-1': 'It is found that such electron seas may undergo hydro-cyclotron oscillations whose frequencies are given by [MATH], where [MATH] and [MATH] the cyclotron frequency.', '1110.1226-2-55-0': 'We propose that some of the absorption features detected in the thermal X-ray spectra of dead (e.g., radio silent) compact objects may have their origin in excitations of these hydro-cyclotron oscillations of the electron sea, provided these stellar objects are interpreted as strange quark stars.', '1110.1226-2-56-0': 'The central compact object 1E 1207.4-5209 appears particularly interesting.', '1110.1226-2-56-1': 'It shows an absorption feature at 0.7 keV which is not much stronger than the another absorption feature observed at 1.4 keV.', '1110.1226-2-56-2': 'This can be readily explained in the framework of the hydro-cyclotron oscillation model, since two lines with [MATH] and [MATH] could essentially have the same intensity.', '1110.1226-2-56-3': 'This is very different for the electron-cyclotron model, for which the oscillator strength of the first harmonic is much smaller than the oscillator strength of the fundamental.', '1110.1226-2-57-0': '# Appendix', '1110.1226-2-58-0': 'Here we derive the dispersion relation characterizing the propagation of hydro-cyclotron electron wave in the slab-geometry approximation.', '1110.1226-2-59-0': 'The governing equation of viscous electron fluid under the action of Lorentz force is given by [EQUATION] which can be written as [EQUATION] where [EQUATION] [MATH] is the cyclotron frequency, and [MATH] stands for the effective viscosity of an electron fluid originating from collisions between electrons.', '1110.1226-2-59-1': 'To make the problem analytically tractable, we treat the electron sea as an incompressible fluid and assume a uniform magnetic field.', '1110.1226-2-59-2': 'Equation ([REF]) can then be written as [EQUATION]', '1110.1226-2-59-3': 'Upon applying to Eq. ([REF]) the operator [MATH], we arrive at [EQUATION] where [MATH], and [EQUATION] where [MATH][MATH].', '1110.1226-2-60-0': 'Considering a perturbation in the form of [MATH], we have [EQUATION] where [MATH].', '1110.1226-2-60-1': 'It is convenient to rewrite the last equation as [EQUATION]', '1110.1226-2-60-2': 'Multiplication of both sides of Eq. ([REF]) with [MATH] leads to [EQUATION]', '1110.1226-2-60-3': 'Inserting the left-hand-side of Eq. ([REF]) into the right-hand-side of Eq. ([REF]) gives [EQUATION] or [EQUATION] which is Eq. ([REF]).'}

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[]

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[]

['1110.1226-1-8-1', '1110.1226-1-19-0', '1110.1226-1-20-1', '1110.1226-1-22-0', '1110.1226-1-25-0', '1110.1226-1-26-0', '1110.1226-1-26-1', '1110.1226-1-27-0', '1110.1226-1-28-0', '1110.1226-1-30-0', '1110.1226-1-31-0', '1110.1226-2-8-1', '1110.1226-2-22-0', '1110.1226-2-23-0', '1110.1226-2-23-1', '1110.1226-2-27-0', '1110.1226-2-30-0', '1110.1226-2-31-0', '1110.1226-2-31-1', '1110.1226-2-32-0', '1110.1226-2-33-0', '1110.1226-2-35-0', '1110.1226-2-36-0', '1110.1226-2-50-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1110.1226

hep-lat-0004010

{'hep-lat-0004010-1-0-0': 'We present results for light quark masses from a systematic calculation in lattice QCD with two degenerate flavors of dynamical quarks.', 'hep-lat-0004010-1-0-1': 'Simulations are made with a renormalization-group improved gauge action and a mean-field improved clover quark action for sea quark masses corresponding to [MATH]-0.6 and the lattice spacing [MATH]-0.11 fm.', 'hep-lat-0004010-1-0-2': 'In the continuum limit we find [MATH] MeV using the [MATH] and [MATH] meson masses as physical input, and [MATH] MeV or [MATH] MeV with the [MATH] or [MATH] meson mass as additional input.', 'hep-lat-0004010-1-0-3': 'Compared to quenched results, two flavors of dynamical quarks reduce the quark masses by about 25%.', 'hep-lat-0004010-1-1-0': ']', 'hep-lat-0004010-1-2-0': 'Masses of light quarks belong to the most fundamental parameters of the Standard Model[CITATION], and yet their precise values have been difficult to determine due to quark confinement.', 'hep-lat-0004010-1-2-1': 'Lattice QCD provides a fundamental approach to overcome this problem [CITATION] since it enables first principle calculations of hadron masses as functions of quark masses.', 'hep-lat-0004010-1-2-2': 'This approach has progressed considerably recently [CITATION] through high statistics calculations that have allowed the continuum limit to be taken for quark masses, and the development of non-perturbative renormalization techniques for a reliable conversion of lattice bare quark masses to those in the continuum.', 'hep-lat-0004010-1-3-0': 'These studies, however, have been carried out within the quenched approximation which ignores effects of sea quarks.', 'hep-lat-0004010-1-3-1': 'A limitation, shown in Ref. [CITATION], is that the strange quark mass cannot be consistently determined, with the values differing by 20% depending on the choice of meson mass taken for input.', 'hep-lat-0004010-1-3-2': 'It has been suspected[CITATION], furthermore, that dynamical sea quark effects sizably reduce the values of light quark masses.', 'hep-lat-0004010-1-4-0': 'Clearly, systematic full QCD studies incorporating dynamical sea quark effects are needed for progress in the determination of quark masses.', 'hep-lat-0004010-1-4-1': 'A recent attempt has been reported in [CITATION].', 'hep-lat-0004010-1-4-2': 'In this Letter we present results of our investigation in which the [MATH] and [MATH] quarks, assumed degenerate, are simulated dynamically while the [MATH] quark is treated in the quenched approximation[CITATION].', 'hep-lat-0004010-1-5-0': 'Full QCD simulations are computationally much more demanding than quenched simulations.', 'hep-lat-0004010-1-5-1': 'We can deal with this problem with the use of improved actions.', 'hep-lat-0004010-1-5-2': 'Because of reduced cutoff errors, they should allow continuum extrapolations from coarser lattices, and hence require smaller lattice sizes, and smaller computational costs, for making simulations with the same physical lattice size.', 'hep-lat-0004010-1-6-0': 'We employ improved actions both for the gluon part and the quark part.', 'hep-lat-0004010-1-6-1': 'The gluon action consists of [MATH] and [MATH] Wilson loops whose coefficients are determined by an approximate renormalization-group analysis[CITATION].', 'hep-lat-0004010-1-6-2': 'For the quark part we choose the "clover" improvement of the Wilson action[CITATION], adopting, for the clover coefficient, a mean-field value [MATH] with the one-loop result [MATH][CITATION] substituted for the plaquette [MATH].', 'hep-lat-0004010-1-7-0': 'The improved action described here was tested in our preparatory full QCD study[CITATION].', 'hep-lat-0004010-1-7-1': 'We found that hadron masses exhibit a good scaling behavior from a coarse lattice spacing of [MATH] GeV, which is much improved over [MATH]>[MATH][MATH] GeV needed for the standard plaquette and Wilson quark actions.', 'hep-lat-0004010-1-7-2': 'We therefore aim at a continuum extrapolation from simulations made at [MATH]-2 GeV.', 'hep-lat-0004010-1-8-0': 'We make runs at three values of the coupling [MATH] to cover this range, employing lattices of a similar physical spatial size [MATH] fm, as listed in Table [REF].', 'hep-lat-0004010-1-8-1': 'For each [MATH], gauge configurations are generated by the hybrid Monte Carlo algorithm at four values of the sea quark hopping parameter [MATH] corresponding to the pseudoscalar (PS) to vector (V) meson mass ratio of [MATH], 0.75, 0.7 and 0.6.', 'hep-lat-0004010-1-8-2': 'For each sea quark mass, we calculate hadron masses at five values of the valence quark hopping parameter [MATH] corresponding to [MATH], 0.75, 0.7, 0.6 and 0.5, taking unequal as well as equal quark mass cases.', 'hep-lat-0004010-1-8-3': 'Masses are extracted from hadron propagators with the standard correlated [MATH] fit.', 'hep-lat-0004010-1-8-4': 'Errors are estimated with the jackknife procedure with a bin size of 50 trajectories, derived from an autocorrelation study.', 'hep-lat-0004010-1-9-0': 'For the Wilson-type quark action including the clover action, the definition of quark mass is not unique at finite lattice spacings due to explicit breaking of chiral symmetry.', 'hep-lat-0004010-1-9-1': 'We employ three definitions in the present work, checking consistency among them for reliability of results: (i) using the Ward identity for axial vector currents [MATH], one may define, [EQUATION] with [MATH] the pseudoscalar density; (ii) another possibility, suggested by the Ward identity for vector currents and naturally appearing in perturbative analyses, reads [MATH] where [MATH] represents the critical hopping parameter at which the PS meson mass vanishes; (iii) replacing [MATH] in (ii) by the "partially quenched" critical value [MATH] where the PS meson mass vanishes as a function of the valence hopping parameter [MATH] when [MATH] for sea quark is fixed to the physical point of [MATH] and [MATH] quark[CITATION].', 'hep-lat-0004010-1-9-2': 'We denote this quark mass as [MATH].', 'hep-lat-0004010-1-10-0': 'We express the PS meson mass [MATH] in terms of quark masses by a general quadratic ansatz of the form, [EQUATION]', 'hep-lat-0004010-1-10-1': 'Here [MATH] with [MATH] the bare mass of the valence quark and antiquark of the PS meson, and [MATH] the mass of sea quark.', 'hep-lat-0004010-1-10-2': 'We mention that details vary depending on the definitions, e.g., terms depending only on [MATH] are absent for the case of AWI mass as follows from ([REF]), and a cross term [MATH] is found to be necessary in the case of VWI mass.', 'hep-lat-0004010-1-11-0': 'The vector meson mass [MATH] is written in a similar manner, adopting, however, PS meson masses as independent variables.', 'hep-lat-0004010-1-11-1': 'The formula we use, being universally applicable for various definitions of quark mass, reads, [EQUATION]', 'hep-lat-0004010-1-11-2': 'Here [MATH] represents the average of PS meson mass squared [MATH] made of a degenerate quark-antiquark pair of the valence hopping parameter [MATH], and [MATH].', 'hep-lat-0004010-1-12-0': 'The bare lattice value of the average [MATH] and [MATH] quark mass [MATH] is determined by fixing the valence and sea quark masses to be degenerate in ([REF]) and ([REF]), and requiring the experimental value for the ratio [MATH].', 'hep-lat-0004010-1-12-1': 'For the [MATH] quark mass [MATH] we use either [MATH] or [MATH] while keeping the sea quark mass [MATH] at the value [MATH] determined above for [MATH] and [MATH] quarks.', 'hep-lat-0004010-1-12-2': 'The lattice scale [MATH] is set using [MATH] GeV as input.', 'hep-lat-0004010-1-13-0': 'We convert bare quark masses calculated above to renormalized quark masses in the [MATH] scheme at [MATH] through [MATH] and [MATH].', 'hep-lat-0004010-1-13-1': 'While not written explicitly, the [MATH] improvement terms with the coefficients [MATH], [MATH] and [MATH] are also included in these relations.', 'hep-lat-0004010-1-13-2': 'For renormalization factors one-loop perturbative values for massless quark[CITATION] are used.', 'hep-lat-0004010-1-13-3': 'For the coupling constant we adopt a mean-field improved value in the [MATH] scheme appropriate for the RG-improved gluon action: [MATH] where measured values extrapolated to zero sea quark mass are substituted for the Wilson loops.', 'hep-lat-0004010-1-13-4': 'The results for quark masses are finally run from [MATH] to [MATH] GeV using the 3-loop beta function[CITATION].', 'hep-lat-0004010-1-13-5': 'Numerical values of quark masses at each [MATH] are given in Table [REF].', 'hep-lat-0004010-1-14-0': 'In Fig. [REF] we show our two-flavor full QCD results for [MATH] with filled symbols.', 'hep-lat-0004010-1-14-1': 'The values for the three definitions, while differing sizably at finite lattice spacings[CITATION], tend to converge toward the continuum limit.', 'hep-lat-0004010-1-14-2': 'A similar trend has been seen in the quenched data for the standard action[CITATION], reproduced in Fig. [REF] with thin open symbols.', 'hep-lat-0004010-1-14-3': 'We take this as a basis to apply a combined continuum extrapolation of the results for the three definitions, requiring a unique value in the continuum limit.', 'hep-lat-0004010-1-15-0': 'The choice of the fitting function requires some consideration.', 'hep-lat-0004010-1-15-1': 'The leading scaling violation with our improved action is [MATH].', 'hep-lat-0004010-1-15-2': 'The coefficient is expected to be small since the clover coefficient [MATH] calculated to one-loop order[CITATION] differs from our mean-field improved choice only by [MATH].', 'hep-lat-0004010-1-15-3': 'We suspect that such a small term is masked by higher order corrections in [MATH] and [MATH] in the range of our coarse lattice spacing.', 'hep-lat-0004010-1-15-4': 'Indeed our results in Fig. [REF], particularly those from the VWI definitions, exhibit noticeable scaling violation of up to about 30 %.', 'hep-lat-0004010-1-15-5': 'Since fits including higher order corrections in addition to [MATH] are quite unstable, and a fit linear in [MATH] is not entirely justifiable in our range of lattice spacings, we make an effective continuum extrapolation linear in [MATH].', 'hep-lat-0004010-1-16-0': 'We then calculate the central value in the continuum limit with a combined linear fit.', 'hep-lat-0004010-1-16-1': 'A systematic error of this fit is estimated by examining the spread of values obtained by separate linear fits of results for the three definitions (see Table [REF] for numerical results of these fits).', 'hep-lat-0004010-1-16-2': 'With this procedure, we obtain [EQUATION]', 'hep-lat-0004010-1-16-3': 'We remark that a fit linear in [MATH] yields a result in agreement with this value within the quoted error.', 'hep-lat-0004010-1-17-0': 'Another source of systematic error is the use of one-loop perturbative values for the renormalization factors.', 'hep-lat-0004010-1-17-1': 'We estimate this error by shifting the matching scale [MATH] from [MATH] to [MATH].', 'hep-lat-0004010-1-17-2': 'After the continuum extrapolation we find the change in the central value to be +6%, which is within the error quoted in ([REF]).', 'hep-lat-0004010-1-18-0': 'Our full QCD result for the average [MATH]-[MATH] quark mass is considerably lower than our previous quenched result for the standard action[CITATION] given by [EQUATION]', 'hep-lat-0004010-1-18-1': 'In order to confirm that the decrease is a dynamical sea quark effect, we carry out a quenched simulation using the same improved gluon and quark actions as for the full QCD runs.', 'hep-lat-0004010-1-18-2': 'This simulation is made at 10 values of [MATH] chosen so that the string tension matches that of two-flavor full QCD for each simulated value of sea quark mass and for the chiral limit at [MATH] and 2.1.', 'hep-lat-0004010-1-19-0': 'Analyses leading from hadron masses to quark masses parallel those for full QCD.', 'hep-lat-0004010-1-19-1': 'In particular we employ polynomial chiral expansions of the form ([REF])-([REF]), except that terms referring to sea quark masses are dropped.', 'hep-lat-0004010-1-19-2': 'As a cross-check we also make an analysis parallel to the one in [CITATION], employing quenched chiral perturbation theory formulae, and obtain consistent results.', 'hep-lat-0004010-1-20-0': 'We plot results of this quenched analysis with thick open symbols in Fig. [REF].', 'hep-lat-0004010-1-20-1': 'Good consistency is observed between the continuum values for the standard and improved actions.', 'hep-lat-0004010-1-20-2': 'We also note that scaling violations are visibly reduced for the latter.', 'hep-lat-0004010-1-20-3': 'Making a combined linear extrapolation, we obtain in the continuum limit, [EQUATION] where the error includes estimates of systematic errors due to the continuum extrapolation.', 'hep-lat-0004010-1-20-4': 'From this analysis we conclude that two dynamical quarks influence [MATH] to decrease by about 25%.', 'hep-lat-0004010-1-21-0': 'In Fig. [REF] we show results for the strange quark mass [MATH] determined from the [MATH] meson mass.', 'hep-lat-0004010-1-21-1': 'A parallel figure obtained with the [MATH] meson mass is given in Fig. [REF].', 'hep-lat-0004010-1-21-2': 'The strange quark is heavy enough so that the difference between [MATH] and [MATH] has only small effects on the VWI masses in full QCD.', 'hep-lat-0004010-1-21-3': 'Employing a combined linear continuum extrapolation in [MATH], and estimating the systematic error in the same way as for [MATH] we obtain [EQUATION]', 'hep-lat-0004010-1-21-4': 'With ([REF]) this gives [MATH] input) and [MATH] input), to be compared with 24.4(1.5)[CITATION] computed from chiral perturbation theory to one loop.', 'hep-lat-0004010-1-22-0': 'Similar analyses for quenched QCD lead to the results [EQUATION] where the values for the standard action[CITATION] are also quoted for comparison.', 'hep-lat-0004010-1-23-0': 'The quenched values for the standard and improved actions are mutually consistent for each choice of the input.', 'hep-lat-0004010-1-23-1': 'This confirms the existence of a systematic uncertainty of 20-30 in the value of [MATH] in quenched QCD[CITATION].', 'hep-lat-0004010-1-24-0': 'One of our important results for the strange quark mass is that the magnitude of this uncertainty is reduced to a 10 level by the inclusion of two flavors of sea quarks.', 'hep-lat-0004010-1-24-1': 'The better consistency reflects the closer agreement of the [MATH] and [MATH] mass splittings with experiment in our two-flavor QCD results than in the quenched case[CITATION].', 'hep-lat-0004010-1-25-0': 'Another important result is that dynamical quark effects reduce the value of [MATH] significantly, from the range 110-140 MeV in quenched QCD to 90-100 MeV for two-flavor full QCD.', 'hep-lat-0004010-1-26-0': 'It will be interesting to see whether the latter range of values converge to a single point once one includes dynamical effects of the strange quark itself.', 'hep-lat-0004010-1-26-1': 'Presumably the corresponding value for [MATH], and also the value for [MATH], would be smaller than the one for the case of two dynamical flavors.', 'hep-lat-0004010-1-26-2': 'Elucidating the magnitude of decrease is an important issue to settle, as our two-flavor results are already close to the lower bounds estimated from the positivity of spectral functions[CITATION].', 'hep-lat-0004010-1-26-3': 'A potential implication concerns the interpretation of the recent experimental results on direct CP violation[CITATION] which favors a small value for the strange quark mass.', 'hep-lat-0004010-1-27-0': 'Clearly, establishing the values of light quark masses will be one of the main tasks of future full QCD simulations incorporating three flavors of dynamical quarks.', 'hep-lat-0004010-1-28-0': 'This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 09304029, 10640246, 10640248, 10740107, 11640250, 11640294, 11740162).', 'hep-lat-0004010-1-28-1': 'AAK and TM are supported by the JSPS Research for the Future Program (No. JSPS-RFTF 97P01102).', 'hep-lat-0004010-1-28-2': 'GB, SE, KN and HPS are JSPS Research Fellows.', 'hep-lat-0004010-1-29-0': 'reviews Early studies were reviewed in, A. Ukawa, Nucl.', 'hep-lat-0004010-1-29-1': 'For a recent review, see R. Gupta and T. Bhattacharya, Nucl.'}

{'hep-lat-0004010-2-0-0': 'We present results for light quark masses from a systematic lattice QCD study with two degenerate flavors of dynamical quarks.', 'hep-lat-0004010-2-0-1': 'Simulations are made with a renormalization-group improved gauge action and a mean-field improved clover quark action for sea quark masses corresponding to [MATH]-0.6 and the lattice spacing [MATH]-0.11 fm.', 'hep-lat-0004010-2-0-2': 'In the continuum limit we find [MATH] MeV using the [MATH] and [MATH] meson masses as physical input, and [MATH] MeV or [MATH] MeV with the [MATH] or [MATH] meson mass as additional input.', 'hep-lat-0004010-2-0-3': 'The quoted errors represent statistical and systematic combined, the latter including those from continuum and chiral extrapolations, and from renormalization factors.', 'hep-lat-0004010-2-0-4': 'Compared to quenched results, two flavors of dynamical quarks reduce quark masses by about 25%.', 'hep-lat-0004010-2-1-0': ']', 'hep-lat-0004010-2-2-0': 'Masses of light quarks belong to the most fundamental parameters of the Standard Model[CITATION], and yet their precise values have been difficult to determine due to quark confinement.', 'hep-lat-0004010-2-2-1': 'Lattice QCD provides a fundamental approach to overcome this problem[CITATION] since it enables first principle calculations of hadron masses as functions of quark masses.', 'hep-lat-0004010-2-2-2': 'This approach has progressed considerably recently[CITATION] through high statistics calculations that have allowed the continuum limit to be taken for quark masses, and the development of non-perturbative renormalization techniques for a reliable conversion of lattice bare quark masses to those in the continuum.', 'hep-lat-0004010-2-3-0': 'These studies, however, have been carried out within the quenched approximation which ignores effects of sea quarks.', 'hep-lat-0004010-2-3-1': 'A limitation, shown in Ref. [CITATION], is that the strange quark mass cannot be consistently determined, with the values differing by 20% depending on the choice of meson mass taken for input.', 'hep-lat-0004010-2-3-2': 'It has been suspected[CITATION], furthermore, that dynamical sea quark effects sizably reduce the values of light quark masses.', 'hep-lat-0004010-2-4-0': 'Clearly, systematic full QCD studies incorporating dynamical sea quark effects are needed for progress in the determination of quark masses.', 'hep-lat-0004010-2-4-1': 'A recent attempt has been reported in[CITATION].', 'hep-lat-0004010-2-4-2': 'In this Letter we present results of our investigation in which the [MATH] and [MATH] quarks, assumed degenerate, are simulated dynamically while the [MATH] quark is treated in the quenched approximation[CITATION].', 'hep-lat-0004010-2-5-0': 'Full QCD simulations are computationally much more demanding than quenched simulations.', 'hep-lat-0004010-2-5-1': 'This problem can be significantly eased by the use of improved actions.', 'hep-lat-0004010-2-5-2': 'Because of reduced cutoff errors, they should allow continuum extrapolations from coarser lattices, and hence require smaller lattice sizes, and smaller computational costs, for simulations with the same physical lattice size.', 'hep-lat-0004010-2-6-0': 'We employ improved actions both for the gluon part and the quark part.', 'hep-lat-0004010-2-6-1': 'The gluon action consists of [MATH] and [MATH] Wilson loops whose coefficients are determined by an approximate renormalization-group analysis[CITATION].', 'hep-lat-0004010-2-6-2': 'For the quark part we choose the "clover" improvement of the Wilson action[CITATION], adopting, for the clover coefficient, a mean-field value [MATH].', 'hep-lat-0004010-2-6-3': 'We substitute the one-loop result [MATH][CITATION] for the plaquette [MATH], which agrees within 8% with the values measured in our runs.', 'hep-lat-0004010-2-6-4': 'The one-loop result of [MATH][CITATION] is found to be close to our choice.', 'hep-lat-0004010-2-7-0': 'The improved action described here was tested in our preparatory full QCD study[CITATION].', 'hep-lat-0004010-2-7-1': 'We found that scaling violation in hadron masses is small with this action already at [MATH] GeV, as compared to [MATH]>[MATH][MATH] GeV needed for the standard plaquette and Wilson quark actions.', 'hep-lat-0004010-2-7-2': 'We therefore aim at a continuum extrapolation from simulations made at [MATH]-2 GeV.', 'hep-lat-0004010-2-8-0': 'We make runs at three values of the coupling [MATH] to cover this range, employing lattices of a similar physical spatial size [MATH] fm, as listed in Table [REF].', 'hep-lat-0004010-2-8-1': 'For each [MATH], gauge configurations are generated by the hybrid Monte Carlo algorithm at four values of the sea quark hopping parameter [MATH] corresponding to the pseudoscalar (PS) to vector (V) meson mass ratio of [MATH], 0.75, 0.7 and 0.6.', 'hep-lat-0004010-2-8-2': 'For each sea quark mass, we calculate hadron masses at five values of the valence quark hopping parameter [MATH] corresponding to [MATH], 0.75, 0.7, 0.6 and 0.5, taking unequal as well as equal quark mass cases.', 'hep-lat-0004010-2-8-3': 'Masses are extracted from hadron propagators with the standard correlated [MATH] fit.', 'hep-lat-0004010-2-8-4': 'Errors are estimated with the jackknife procedure with a bin size of 50 trajectories, derived from an autocorrelation study.', 'hep-lat-0004010-2-9-0': 'For the Wilson-type quark action including the clover action, different definitions of quark masses lead to results that differ at finite lattice spacing due to explicit breaking of chiral symmetry.', 'hep-lat-0004010-2-9-1': 'We employ three definitions in the present work, checking consistency among them for reliability of results: (i) using the axial vector Ward identity, we define, [EQUATION] with [MATH] the pseudoscalar density and [MATH] the axial vector current improved to [MATH]; (ii) another possibility, suggested by the vector Ward identity and naturally appearing in perturbative analyses, reads [MATH] where [MATH] represents the critical hopping parameter at which the PS meson mass vanishes [MATH]; (iii) a third possibility, suggested in[CITATION] and denoted by [MATH], replaces [MATH] in (ii) by the "partially quenched" critical value [MATH] where the PS meson mass vanishes as a function of the valence hopping parameter [MATH] when [MATH] for sea quark is fixed to the physical point of [MATH] and [MATH] quark, [MATH].', 'hep-lat-0004010-2-10-0': 'We express the PS meson mass [MATH] in terms of quark masses by a general quadratic ansatz of the form, [EQUATION]', 'hep-lat-0004010-2-10-1': 'Here [MATH] with [MATH] the bare mass of the valence quark and antiquark of the PS meson, and [MATH] the mass of sea quark.', 'hep-lat-0004010-2-10-2': 'We mention that details vary depending on the definitions, e.g., terms depending only on [MATH] are absent for the case of AWI mass since with ([REF]) [MATH] is zero for vanishing [MATH], and a cross term [MATH] is found to be necessary in the case of VWI mass.', 'hep-lat-0004010-2-11-0': 'The vector meson mass [MATH] is written in a similar manner, adopting, however, PS meson masses as independent variables.', 'hep-lat-0004010-2-11-1': 'We fit data with the formula [EQUATION]', 'hep-lat-0004010-2-11-2': 'Here [MATH] represents the average of PS meson mass squared [MATH] made of a degenerate quark-antiquark pair of the valence hopping parameter [MATH], and [MATH].', 'hep-lat-0004010-2-12-0': 'Fitting our hadron mass data with ([REF]) and ([REF]) we find reasonable results with [MATH] in the range 0.6-2.3 (except for ([REF]) for VWI quark mass at [MATH] for which [MATH]).', 'hep-lat-0004010-2-12-1': 'We then determine the bare lattice value of the average [MATH] and [MATH] quark mass [MATH] by fixing the valence and sea quark masses to be degenerate in ([REF]) and ([REF]), and requiring the experimental value for the ratio [MATH].', 'hep-lat-0004010-2-12-2': 'For the [MATH] quark mass [MATH] we use either [MATH] or [MATH] while keeping the sea quark mass [MATH] at the value [MATH] determined above.', 'hep-lat-0004010-2-12-3': 'The lattice scale [MATH] is set using [MATH] GeV as input.', 'hep-lat-0004010-2-13-0': 'We convert bare quark masses calculated above to renormalized quark masses in the [MATH] scheme at [MATH] through [MATH] and [MATH].', 'hep-lat-0004010-2-13-1': 'In these relations, the [MATH] improvement terms with the coefficients [MATH], [MATH] and [MATH] are also included.', 'hep-lat-0004010-2-13-2': 'For renormalization factors and improvement coefficients, including that for [MATH], one-loop perturbative values for massless quark[CITATION] are used.', 'hep-lat-0004010-2-13-3': 'For the coupling constant we adopt a mean-field improved value in the [MATH] scheme appropriate for the RG-improved gluon action (see Table [REF] for numerical values): [MATH] where measured values extrapolated to zero sea quark mass are substituted for the Wilson loops.', 'hep-lat-0004010-2-13-4': 'The results for quark masses are run from [MATH] to [MATH] GeV using the 3-loop beta function for [MATH][CITATION].', 'hep-lat-0004010-2-13-5': 'Numerical values of quark masses at each [MATH] are given in Table [REF].', 'hep-lat-0004010-2-14-0': 'In Fig. [REF] we show our two-flavor full QCD results for [MATH] with filled symbols.', 'hep-lat-0004010-2-14-1': 'The values for the three definitions, while differing sizably at finite lattice spacings[CITATION], tend to converge toward the continuum limit.', 'hep-lat-0004010-2-14-2': 'A similar trend has been seen in the quenched data for the standard action[CITATION], reproduced in Fig. [REF] with thin open symbols.', 'hep-lat-0004010-2-15-0': 'For our choice of the improved action, the scaling violation starts at [MATH] for the quark masses at the scale [MATH] GeV.', 'hep-lat-0004010-2-15-1': 'We therefore make a continuum extrapolation linear in [MATH].', 'hep-lat-0004010-2-15-2': 'The results are given in Table [REF].', 'hep-lat-0004010-2-15-3': 'The masses for the three definitions are consistent with each other at two-sigma level of statistics.', 'hep-lat-0004010-2-15-4': 'Hence we carry out a combined linear fit, as shown in Fig. [REF] by dashed lines, obtaining [MATH] with [MATH] where the error is only statistical.', 'hep-lat-0004010-2-15-5': 'The systematic error of the continuum extrapolation is estimated from the spread of values obtained by separate fits of data for the three definitions.', 'hep-lat-0004010-2-15-6': 'The fractional error thus calculated is given in Table [REF].', 'hep-lat-0004010-2-16-0': 'This table lists our estimate for two more systematic errors that we need to incorporate.', 'hep-lat-0004010-2-16-1': 'One is an uncertainty due to chiral extrapolations.', 'hep-lat-0004010-2-16-2': 'We estimate this error from the change of the combined linear fit in the continuum limit when the quadratic term [MATH] in the vector mass formula ([REF]) is replaced by [MATH] or cubic terms [MATH] are included in the PS mass formula ([REF]).', 'hep-lat-0004010-2-16-3': 'Another is the error due to the use of one-loop perturbative values for the renormalization factors.', 'hep-lat-0004010-2-16-4': 'As non-perturbative values are not yet available, we estimate the effect of higher order contributions by recalculating masses while either shifting the matching scale from [MATH] to [MATH] or using an alternative definition of coupling given by [MATH] using only the plaquette.', 'hep-lat-0004010-2-17-0': 'Combining the statistical error and the systematic errors listed in Table [REF] by quadrature to obtain the total error, we find for our final value, [EQUATION]', 'hep-lat-0004010-2-17-1': 'Our full QCD result for the average [MATH]-[MATH] quark mass is considerably lower than our previous quenched result for the standard action given in[CITATION] as [EQUATION] where the error is only statistical.', 'hep-lat-0004010-2-17-2': 'In order to confirm that the decrease is a dynamical sea quark effect, we carry out a quenched simulation using the same improved gluon and quark actions as for the full QCD runs.', 'hep-lat-0004010-2-17-3': 'This simulation is made at 10 values of [MATH] chosen so that the string tension matches that of two-flavor full QCD for each simulated value of sea quark mass and for the chiral limit at [MATH] and 2.1.', 'hep-lat-0004010-2-18-0': 'Analyses leading from hadron masses to quark masses parallel those for full QCD.', 'hep-lat-0004010-2-18-1': 'In particular we employ polynomial chiral expansions of the form ([REF])-([REF]), except that terms referring to sea quark masses are dropped.', 'hep-lat-0004010-2-18-2': 'As a cross-check we also make an analysis parallel to the one in[CITATION], employing quenched chiral perturbation theory formulae, and obtain consistent results.', 'hep-lat-0004010-2-19-0': 'We plot results of this quenched analysis with thick open symbols in Fig. [REF].', 'hep-lat-0004010-2-19-1': 'Good consistency is observed between the continuum values for the standard and improved actions.', 'hep-lat-0004010-2-19-2': 'We also note that scaling violations are visibly reduced for the latter.', 'hep-lat-0004010-2-19-3': 'Making a combined linear extrapolation we obtain in the continuum limit, [EQUATION] where the error is estimated in a similar way as for full QCD.', 'hep-lat-0004010-2-19-4': 'From this analysis we conclude that the effect of two dynamical quarks is to decrease [MATH] by about 25%.', 'hep-lat-0004010-2-20-0': 'In Fig. [REF] we show results for the strange quark mass [MATH] determined from the [MATH] meson mass.', 'hep-lat-0004010-2-20-1': 'A parallel figure obtained with the [MATH] meson mass is given in Fig. [REF].', 'hep-lat-0004010-2-20-2': 'Using [MATH] instead of [MATH] gives the same results within 1%.', 'hep-lat-0004010-2-20-3': 'The strange quark is heavy enough so that the difference between [MATH] and [MATH] has only small effects on the VWI masses in full QCD.', 'hep-lat-0004010-2-20-4': 'Employing combined linear continuum extrapolations in [MATH] (with [MATH] and 3.0, respectively), and estimating the error in the same way as for [MATH] (see Table [REF] for details), we obtain [EQUATION]', 'hep-lat-0004010-2-20-5': 'With ([REF]) this gives [MATH] to be compared with 24.4(1.5)[CITATION] computed from chiral perturbation theory to one loop.', 'hep-lat-0004010-2-21-0': 'Similar analyses for quenched QCD lead to the results [EQUATION] where the values for the standard action[CITATION] are also quoted for comparison.', 'hep-lat-0004010-2-22-0': 'The quenched values for the standard and improved actions are mutually consistent for each choice of the input.', 'hep-lat-0004010-2-22-1': 'This confirms the existence of a systematic uncertainty of 20-30 in the value of [MATH] in quenched QCD[CITATION].', 'hep-lat-0004010-2-23-0': 'One of our important results for the strange quark mass is that this uncertainty disappears within an error of 10 by the inclusion of two flavors of sea quarks.', 'hep-lat-0004010-2-23-1': 'The consistency reflects a closer agreement of the [MATH] and [MATH] mass splittings with experiment in our two-flavor QCD results compared to the quenched case[CITATION].', 'hep-lat-0004010-2-24-0': 'Another important result is that dynamical quark effects reduce the value of [MATH] significantly, from the range 110-140 MeV in quenched QCD to 90 MeV for two-flavor full QCD.', 'hep-lat-0004010-2-24-1': 'It will be interesting to see whether the inclusion of dynamical effects of the strange quark itself would decrease the value of [MATH] even further.', 'hep-lat-0004010-2-24-2': 'This is an important issue to settle as our two-flavor results are already close to the lower bounds estimated from the positivity of spectral functions[CITATION].', 'hep-lat-0004010-2-25-0': 'Clearly, establishing the values of light quark masses incorporating three flavors of dynamical quarks will be one of the main tasks of future lattice QCD calculations.', 'hep-lat-0004010-2-26-0': 'This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 09304029, 10640246, 10640248, 10740107, 11640250, 11640294, 11740162).', 'hep-lat-0004010-2-26-1': 'AAK and TM are supported by the JSPS Research for the Future Program (No. JSPS-RFTF 97P01102).', 'hep-lat-0004010-2-26-2': 'GB, SE, TK, KN and HPS are JSPS Research Fellows.', 'hep-lat-0004010-2-27-0': 'reviews Early studies were reviewed in, A. Ukawa, Nucl.', 'hep-lat-0004010-2-27-1': 'For a recent review, see R. Gupta and T. Bhattacharya, Nucl.'}

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[]

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[['hep-lat-0004010-1-16-1', 'hep-lat-0004010-2-15-5']]

['hep-lat-0004010-1-1-0', 'hep-lat-0004010-1-28-0', 'hep-lat-0004010-1-28-1', 'hep-lat-0004010-2-1-0', 'hep-lat-0004010-2-26-0', 'hep-lat-0004010-2-26-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-lat/0004010

1901.06015

{'1901.06015-1-0-0': '# Introduction', '1901.06015-1-1-0': 'The BLAS [CITATION] defines the general matrix-matrix multiplication (gemm) operation to support any of the following computations: [EQUATION] where [MATH] is [MATH], the left-hand matrix product operand ([MATH], [MATH], or [MATH]) is [MATH], the right-hand matrix product operand ([MATH], [MATH], or [MATH]) is [MATH], and [MATH] and [MATH] are scalars.', '1901.06015-1-2-0': 'This matrix multiplication functionality is made available to software developers via the following application programming interfaces, or APIs:', '1901.06015-1-3-0': 'sgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-1-4-0': 'dgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-1-5-0': 'cgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-1-6-0': 'zgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-1-7-0': 'The first letter of the routine name uniquely encodes the datatype-that is, the domain and precision-of the matrix and scalar operands as well as the computation: single-precision real (s); double-precision real (d); single-precision complex (c); and double-precision complex (z).', '1901.06015-1-7-1': 'The parameters transa and transb indicate if [MATH] and/or [MATH], respectively, should be computed upon as if they were transposed or conjugate-transposed.', '1901.06015-1-7-2': 'The interfaces implicitly require that matrices be stored in column-major order.', '1901.06015-1-7-3': 'Accordingly, the parameters ldA, ldB, and ldC convey the so-called "leading dimensions" of the arrays A, B, and C, respectively-that is, the number of elements that separate matrix element [MATH] from element [MATH] in memory.', '1901.06015-1-8-0': 'While this interface has served the HPC community well, it has also become constraining.', '1901.06015-1-8-1': 'For example, when computing tensor contractions (which often resemble matrix multiplications), one may need to refer to a sub-tensor that cannot be represented with column-major storage without making a temporary copy.', '1901.06015-1-8-2': 'Similarly, some situations may call for conjugating (but not transposing) a matrix operand.', '1901.06015-1-8-3': 'Indeed, such functionality is already supported by the BLIS framework, which exports BLAS-like operations and APIs [CITATION].', '1901.06015-1-8-4': 'However, even BLIS only supports computation on operands with identical datatypes.', '1901.06015-1-8-5': 'Consider the following:', '1901.06015-1-9-0': 'Thus, there is likely a fair amount of pent-up demand for high-performance implementations to datatype-flexible BLAS-like APIs.', '1901.06015-1-10-0': 'As alluded to above, the naive approach to supporting mixed-datatype functionality within the gemm operation comes with obvious memory, performance, and productivity drawbacks: typecasting matrix operands to a common domain and/or precision outside of the original implementation requires considerable workspace; the memory access patterns engendered by monolithic casting almost surely acts as a drag on performance; and programming an ad-hoc solution in terms of the BLAS gemm interfaces sometimes requires non-trivial skills.', '1901.06015-1-10-1': 'Indeed, some would find providing the full combinatoric space of functionality daunting, and in response might attempt to survey the community and then only implement those cases for which interest was expressed.', '1901.06015-1-10-2': 'Instead, our goal from the outset is to implement support for all cases, and to do so in a manner that delivers high or near-high performance.', '1901.06015-1-11-0': 'Our approach, which builds on the BLIS framework, yields a comprehensive reference implementation with which consumers of this functionality can explore the benefits of mixed-domain and mixed-precision gemm computation without being constrained by limitations in the interface, incomplete coverage within the implementation, or unnecessarily inefficient performance.', '1901.06015-1-12-0': '## Notation', '1901.06015-1-13-0': 'Our notation should be mostly self-evident to most readers of high-performance dense linear algebra literature.', '1901.06015-1-13-1': 'We use uppercase Roman letters, such as [MATH], [MATH], and [MATH] to denote matrices, and lowercase Greek letters [MATH] and [MATH] to represent scalars.', '1901.06015-1-14-0': 'Real and complex domains are indicated by [MATH] and [MATH], respectively.', '1901.06015-1-14-1': 'Occasionally, we refer to the real part of a matrix or matrix expression [MATH] with [MATH] and to the imaginary part with [MATH].', '1901.06015-1-14-2': 'In other places, such as where this notation would be too cumbersome, we use superscripts, such as [MATH] and [MATH] for the real and imaginary components, respectively, of a scalar [MATH].', '1901.06015-1-15-0': 'When representing elements within a matrix, we use a subscript to encode the row and column indices.', '1901.06015-1-15-1': 'For example, a scalar [MATH] would reference the element located in the 2nd row and 4th column of a matrix [MATH].', '1901.06015-1-16-0': '# Background', '1901.06015-1-17-0': 'In this section, we review the approach to matrix multiplication taken within the BLIS framework as well as some related implementation details that will provide important context to discussions later in this article.', '1901.06015-1-18-0': '## Matrix Multiplication in BLIS', '1901.06015-1-19-0': 'The GotoBLAS algorithm [CITATION] for performing matrix multiplication underlies the highest-performing BLAS libraries for current general-purpose microprocessors.', '1901.06015-1-19-1': 'The BLAS-like Library Instantiation Software (BLIS) framework, which implements gemm and other matrix-matrix operations, refactors the GotoBLAS algorithm as pictured in Figure [REF].', '1901.06015-1-19-2': 'BLIS isolates the code that needs to be optimized (in assembly code or with vector intrinsics) for different architectures in a microkernel that updates a very small submatrix, or microtile, of [MATH] with a sequence of rank-1 updates that are accumulated in registers [CITATION].', '1901.06015-1-19-3': 'All other loops and supporting kernels are implemented portably in C99.', '1901.06015-1-19-4': "By contrast, Goto's implementation-also adopted by the OpenBLAS fork [CITATION] of the GotoBLAS library-casts the computation into a larger assembly-coded kernel.", '1901.06015-1-19-5': 'This larger unit of code, which corresponds to what BLIS refers to as the macrokernel, consists of the microkernel plus the logic that falls within the first two loops around the microkernel.', '1901.06015-1-20-0': 'A key element of the GotoBLAS algorithm is that high-performance implementation of gemm incorporates the packing of submatrices of [MATH] (into buffer [MATH]) and of [MATH] (into buffer [MATH]) to improve data locality during the execution of the microkernel.', '1901.06015-1-21-0': 'This feature has been used in the past to consolidate other functionality into the same framework: implementation of other matrix-matrix operations (level-3 BLAS) [CITATION], fusing sequences of matrix operations of importance to Machine Learning [CITATION], and implementation of practical Fast Matrix Multiplication (Strassen-like) algorithms [CITATION].', '1901.06015-1-21-1': 'A final insight comes from Van Zee [CITATION], in which it is shown how complex matrix multiplication can be cast in terms of only microkernels designed for real domain gemm without a significant performance penalty.', '1901.06015-1-22-0': 'Given a target architecture, instantiating the traditional functionality of gemm with BLIS requires only two microkernels, one each for single- and double-precision real domain computations, with the insight from [CITATION] inducing the functionality typically provided by complex domain microkernels.', '1901.06015-1-22-1': 'It also requires packing functions, which by default take the form of architecture-agnostic (C99) implementations provided by the framework, as well as architecture-specific cache and register blocking parameters.', '1901.06015-1-22-2': 'BLIS exposes several other configure-time options, though they all default to values that typically need no further tweaking.', '1901.06015-1-23-0': '## Managing complexity in BLIS', '1901.06015-1-24-0': 'The combinatoric complexity of the gemm operation in BLIS is mitigated in several ways.', '1901.06015-1-25-0': '### Storage', '1901.06015-1-26-0': 'BLIS tracks separate row and column strides for each matrix object.', '1901.06015-1-27-0': 'Using these two stride parameters, BLIS supports three matrix storage formats in its end-user APIs: row-major storage (where the column stride is unit); column-major storage (where the row stride is unit); and general storage (where neither stride is unit).', '1901.06015-1-28-0': 'Each gemm operand in BLIS may individually be stored in any of the three aforementioned storage formats.', '1901.06015-1-28-1': 'If we consider all possible variations of general storage as a single format, this results in a total of [MATH] different storage combinations.', '1901.06015-1-28-2': 'The gemm operation supports these [MATH] storage combinations as follows: the packing function is written generically to allow it to read from any of the three storage formats when reading matrices [MATH] and [MATH] (during the packing of [MATH] and [MATH]), and the microkernel is required to handle input/output of [MATH] in any of the three supported formats.', '1901.06015-1-29-0': '### Transposition', '1901.06015-1-30-0': 'BLIS easily accommodates transposition of matrices [MATH] or [MATH] by swapping the row and column strides (and their corresponding dimensions) just prior to packing.', '1901.06015-1-30-1': 'This technique merely affects how the matrices are traversed rather than how they are stored; thus, we call this an "induced" (or logical) transposition, as no matrix elements are actually copied or moved.', '1901.06015-1-31-0': '### Conjugation', '1901.06015-1-32-0': 'In the case of complex matrices, optional conjugation', '1901.06015-1-33-0': 'of [MATH] and [MATH] is handled during the packing into [MATH] and [MATH].', '1901.06015-1-34-0': '### Multithreaded parallelization', '1901.06015-1-35-0': 'The authors of [CITATION] discuss how BLIS exposes many loops, each of which can be parallelized.', '1901.06015-1-35-1': 'Crucially, the complexity over matrix storage datatypes (domain and precision), transposition/conjugation parameters (transA and transb), and matrix storage formats (row, column, generalized) are orthogonal to the issues pertaining to extracting multithreaded parallelism from gemm.', '1901.06015-1-35-2': 'Therefore, the insights of [CITATION] carry over to the parallelization when mixing domains and/or precisions.', '1901.06015-1-36-0': '### Optimizing input/output on [MATH]', '1901.06015-1-37-0': 'High-performance microkernels accumulate their intermediate results using vector registers.', '1901.06015-1-37-1': 'Thus, the microkernel author must decide whether to semantically assign the vector registers to contain contiguous rows or contiguous columns of the microtile submatrix.', '1901.06015-1-37-2': "We refer to this as the microkernel's register orientation.", '1901.06015-1-37-3': 'Interestingly, the register orientation necessarily biases the microkernel towards loading and storing elements of [MATH] as either rows or columns, since performing IO on elements in the opposite orientation would require a sequence of costly permutation instructions.', '1901.06015-1-37-4': "BLIS tracks this intrinsic property, or IO preference, of the microkernel so that the framework can transform the matrix problem to the microkernel's liking.", '1901.06015-1-37-5': 'For example, if our microkernel is row-preferential and the gemm implementation is executed on a column-stored matrix [MATH], BLIS will employ a high-level transposition of the entire operation (to [MATH]) so that, from the perspective of the microkernel, [MATH] appears to be stored in its preferred format-that is, a manner consistent with its vector register orientation.', '1901.06015-1-38-0': 'Thus, regardless of whether the microkernel is row- or column-preferential, the gemm implementation will, generally-speaking, yield similar performance on row- and column-stored matrices [MATH].', '1901.06015-1-39-0': '## For the busy reader', '1901.06015-1-40-0': 'We acknowledge that some readers may wish for a very short synopsis of the insights presented later in this article.', '1901.06015-1-40-1': 'We sum up the techniques that allow implementing all cases of mixed-domain and mixed-precision gemm within the BLIS framework as follows: The mixing of domains can be handled by enumerating the eight cases (six of them new), which largely reduce to either manipulating matrix metadata, and/or exposing the real and imaginary elements in a complex matrix in such a way that a real matrix multiplication may be performed to induce the desired result, in part motivated by insights from the 1m method [CITATION].', '1901.06015-1-40-2': 'The mixing of precisions can be handled by typecasting between precisions, as needed, during the packing [MATH] and [MATH] (into [MATH] and [MATH]), while the typecasting during the accumulation of the intermediate product [MATH] may occur in special code wrappers that update [MATH] appropriately.', '1901.06015-1-41-0': '# API Considerations', '1901.06015-1-42-0': 'In the spirit of the BLAS API, a more complete interface that supports this richer, mixed-datatype environment could take the form of', '1901.06015-1-43-0': 'where transX indicates whether [MATH] should be computed upon as if it were (optionally) transposed, conjugate-transposed, or conjugated without transposition; X is the address where matrix [MATH] is stored; domainX indicates whether [MATH] is stored as a matrix of real or complex elements; precX indicates the precision in which elements of [MATH] are stored (e.g., half-, single-, double-, or quad-precision); rstrideX and cstrideX indicate the row and column strides, respectively, for storing [MATH]; and precAB indicates the precision in which the matrix multiplication takes place (possibly implying promotion or demotion from the storage precision of either [MATH] or [MATH]).', '1901.06015-1-43-1': 'Note that column-major order and row-major order are the special cases where rstrideX is unit and cstrideX is unit, respectively.', '1901.06015-1-44-0': 'The hypothetical API shown in Figure [REF] plainly exposes all of the dimensions of functionality along which the library developer must provide implementations.', '1901.06015-1-44-1': 'However, we feel that such an interface is not very useful towards inspiring those implementations, as it subtly nudges the developer towards extending the use of separate interfaces for each parameter combination down the function stack, leading to solutions that are vertically siloed from each other, even if various subsets share many similarities.', '1901.06015-1-45-0': 'BLIS preempted this problem by starting with an object-based foundation for encoding and expressing matrix (and vector) operands.', '1901.06015-1-45-1': 'Each linear algebra entity (such as a matrix or vector) is encapsulated within a data structure, or more specifically, a custom struct type.', '1901.06015-1-45-2': 'For example, BLIS currently exports the following object-based function prototype for invoking the gemm operation:', '1901.06015-1-46-0': 'The function bligemm() takes five arguments of type objt*, each of which corresponds to the address of a struct representing the floating-point operands traditionally passed into the gemm operation.', '1901.06015-1-46-1': 'The function exposes no other arguments, because all of the conventional parameters (such as transA and transB) may be interpreted as properties of one of the floating-point operands.', '1901.06015-1-47-0': 'A simplified version of the objt type definition may be given as:', '1901.06015-1-48-0': 'Here, dimt, inct, dofft, sizt, and objbitst represent various integer types defined within BLIS for representing dimensions, strides and increments, diagonal offsets, byte sizes, and object property bitfields, respectively.', '1901.06015-1-48-1': 'The idea behind the struct example in Figure [REF] is that matrices may be represented by a collection of properties, or metadata, and that these properties may be set-for example, when the object is initialized and its underlying data buffer is allocated-and then subsequently queried or modified using a collection of object-based accessor functions.', '1901.06015-1-48-2': 'Encapsulating matrix properties within objects helps hide details that need not be exposed at certain levels of the implementation.', '1901.06015-1-49-0': 'The key observation to make now is that the domainX and precX arguments shown in Figure [REF] can be completely hidden within the object API of BLIS.', '1901.06015-1-49-1': 'Indeed, the current definition of objt within the framework already includes domain and precision bits within the info bitfield.', '1901.06015-1-49-2': 'We only need to add an additional parameter, or designate additional bits within the info property, to support the computation precision (labeled in Figure [REF] as precAB).', '1901.06015-1-49-3': 'Thus, it is possible to add mixed-datatype support to gemm without any modification to the function interface to bligemm().', '1901.06015-1-50-0': "A more thorough walkthrough of BLIS's object API is well beyond the scope of this article.", '1901.06015-1-51-0': 'The main takeaway from this discussion is that the original author of BLIS designed the framework around an object-based core with the keen understanding that additional APIs of arbitrary format, including (but not limited to) those in the style of the BLAS, could always be layered on top of this more general abstraction.', '1901.06015-1-51-1': 'Consequently, any such APIs built above and beyond the underlying object layer are only incidental to the framework; they merely constitute syntactic re-expressions of some subset of the functionality made possible by the object foundation.', '1901.06015-1-52-0': '# Supporting Mixed Domain Computation', '1901.06015-1-53-0': 'We consider the storage domain (real or complex) of the matrix to be orthogonal to the storage precision (single, double, etc).', '1901.06015-1-53-1': 'In this section, we consider how to handle mixing matrix operands of different domains.', '1901.06015-1-53-2': 'For now, the reader should assume that the storage precision is held constant across all matrix operands, and therefore can be ignored.', '1901.06015-1-53-3': 'We also ignore scalars [MATH] and [MATH] for the time being, which simplifies the general matrix multiplication operation to [MATH].', '1901.06015-1-54-0': '## The 1m method', '1901.06015-1-55-0': 'The author of [CITATION] recently presented a novel method of computing complex matrix multiplication without relying upon kernels that explicitly perform complex arithmetic at the scalar level, as is typically the case in high-performance BLAS libraries.', '1901.06015-1-55-1': 'Instead, the so-called 1m method relies only upon matrix primitives (kernels) that compute real matrix multiplication.', '1901.06015-1-55-2': 'And unlike the older and more easily understood 4m method [CITATION], 1m replaces each complex matrix multiplication with only a single real matrix multiplication.', '1901.06015-1-56-0': 'The key to 1m is a pair of special packing formats, which Van Zee denotes 1e and 1r.', '1901.06015-1-56-1': 'The author illustrates the role of these two packing formats using the following example of complex matrix multiplication [MATH] where [MATH], [MATH], and [MATH].', '1901.06015-1-56-2': '[EQUATION]', '1901.06015-1-56-3': 'In this example, it is assumed that matrix [MATH] is column-stored, which prescribes that the 1e format be applied to [MATH] and the 1r format be applied to [MATH].', '1901.06015-1-56-4': 'This can be confirmed by inspection: applying a real matrix multiplication to the left- and right-hand matrix product operands would not correctly compute the complex matrix multiplication if [MATH] were row-stored because the intermediate elements of [MATH] would update the wrong elements of [MATH].', '1901.06015-1-56-5': 'However, 1m provides a cure to this situation.', '1901.06015-1-56-6': 'Namely, symmetry in the method allows for a row-oriented variant where [MATH] is row-stored.', '1901.06015-1-56-7': '[EQUATION]', '1901.06015-1-56-8': 'In this case, the application of the packing formats is reversed, such that 1e is applied to [MATH] and 1r is applied to [MATH].', '1901.06015-1-56-9': 'Van Zee later points out that it is not actually the storage of [MATH] that determines whether Eq. [REF] or [REF] is employed, but rather the SIMD register orientation of the underlying real domain microkernel-which determines the input/output preference of the microtile and thus the natural method of performing SIMD reads and write instructions on [MATH].', '1901.06015-1-57-0': 'While the 1m method was originally articulated as a way to implement complex domain matrix multiplication on operands of identical datatypes, we will soon show that not only can it be extended to mixed-precision complex domain computation, but it also indirectly supports a particular combination of mixed-domain operands.', '1901.06015-1-58-0': '## Enumerating the cases', '1901.06015-1-59-0': 'Consider for simplicity [MATH], ignoring the scaling factors [MATH] and [MATH].', '1901.06015-1-59-1': 'Each of [MATH] can be stored in either the real or complex domains, leading to [MATH] different combinations.', '1901.06015-1-59-2': 'Figure [REF] enumerates and names each possible case.', '1901.06015-1-59-3': 'We will now discuss each case, how it is interpreted, and how it is implemented within the BLIS framework.', '1901.06015-1-60-0': '### Cases 0 , 3', '1901.06015-1-61-0': 'The trivial case where all matrices are stored in the real domain, which we refer to as Case 0, is already supported by the framework via algorithms based on real domain microkernels.', '1901.06015-1-61-1': 'Similarly, Case 3, which applies when all matrices are stored in the complex domain, is also already supported.', '1901.06015-1-61-2': 'Support for Case 3 is provided in BLIS via conventional algorithms based on complex domain microkernels as well as via the 1m method, which is particularly useful when complex microkernels are not available.', '1901.06015-1-61-3': 'Cases 0 and 3 incur [MATH] and [MATH] flops, respectively.', '1901.06015-1-62-0': '### Cases 1a , 1b', '1901.06015-1-63-0': 'Case 1a captures situations where [MATH] and [MATH] are real while [MATH] is complex.', '1901.06015-1-63-1': 'We interpret such an operation as [MATH].', '1901.06015-1-63-2': 'Implementing this case in BLIS is rather straightforward: we ignore the imaginary part of [MATH] and compute as if all matrices were real.', '1901.06015-1-63-3': 'Because BLIS tracks both row and column strides for each matrix operand, ignoring the imaginary elements amounts to a temporary change to the dimension and stride metadata contained within the object representing [MATH].', '1901.06015-1-63-4': 'Case 1b involves a complex matrix [MATH] and real matrices [MATH] and [MATH], but is otherwise handled similarly.', '1901.06015-1-63-5': 'Since these case are ultimately implemented in terms of Case 0, they both performs [MATH] flops.', '1901.06015-1-64-0': '### Case 2ab', '1901.06015-1-65-0': 'Case 2ab is applicable when [MATH] and [MATH] are complex while the matrix to which they accumulate, [MATH], is real.', '1901.06015-1-65-1': 'We interpret this somewhat curious scenario as a matrix product that takes place in the complex domain, but one for which the imaginary result is discarded: [MATH].', '1901.06015-1-65-2': 'Since [MATH] is not needed, only [MATH] flops need to be performed.', '1901.06015-1-65-3': 'Thus, this case provides an opportunity for computational savings when properly implemented.', '1901.06015-1-65-4': 'BLIS implements 2ab by borrowing the 1r packing format used by the 1m method.', '1901.06015-1-66-0': 'Specifically, BLIS packs both matrices [MATH] and [MATH] using the 1r format while simultaneously conjugating [MATH].', '1901.06015-1-66-1': 'This has the effect of allowing a subsequent real matrix multiplication over the packed matrices to correctly compute only the real half of the complex matrix multiplication update.', '1901.06015-1-67-0': '### Case 1c', '1901.06015-1-68-0': 'The opposite of 2ab-Case 1c-refers to settings in which matrix [MATH] is complex while [MATH] and [MATH] are real.', '1901.06015-1-68-1': 'Since the matrix product takes place entirely in the real domain, the natural interpretation is that [MATH] updates only [MATH], and [MATH] is left untouched.', '1901.06015-1-68-2': 'Generally speaking, BLIS implements 1c using a strategy similar to the one used with 1a and 1b.', '1901.06015-1-68-3': 'That is, a temporary change to the object metadata describing matrix [MATH] allows us to isolate [MATH], which once again reduces the problem to Case 0.', '1901.06015-1-68-4': 'Accordingly, this case requires only [MATH] flops.', '1901.06015-1-69-0': '### Cases 2ac , 2bc', '1901.06015-1-70-0': 'Consider Case 2ac, in which matrices [MATH] and [MATH] are complex and matrix [MATH] is real.', '1901.06015-1-70-1': 'We interpret this situation as performing a complex matrix product [MATH] to update both real and imaginary parts of [MATH].', '1901.06015-1-70-2': 'However, all computation involving the imaginary part of [MATH], which is implicitly zero, may be ignored.', '1901.06015-1-70-3': 'This means that the computation requires only [MATH] flops.', '1901.06015-1-70-4': 'Now, if [MATH] and [MATH] were guaranteed to be column-stored, BLIS could handle this case with a simple change of metadata that recasts those complex matrices as real, with the real and imaginary elements treated equally and indistinguishably.', '1901.06015-1-70-5': 'However, BLIS also allows row storage (and general storage) for all matrices, and thus the solution is not quite so simple.', '1901.06015-1-70-6': 'Instead, BLIS handles 2ac as follows.', '1901.06015-1-71-0': 'If the original problem fits into Case 2ac and the microkernel is column-preferential, [MATH] is packed as a real matrix (with imaginary elements stored traditionally, in element-wise interleaved fashion) and [MATH] is packed normally.', '1901.06015-1-71-1': 'A real domain macrokernel (Case 0) is then executed, which will properly update [MATH].', '1901.06015-1-71-2': 'However, if the microkernel is row-preferential, the operation is logically transposed and Case 2bc is executed instead (whereby matrices [MATH] and [MATH] are complex and [MATH] is real).', '1901.06015-1-71-3': 'Thus, the effective case employed, 2ac or 2bc, depends on the register orientation of the microkernel, not the storage of [MATH], and does so for the same reason that the 1m method, discussed previously in Section [REF], depends on the same property.', '1901.06015-1-72-0': 'After the aforementioned logical transposition is applied (or not), the microkernel input/output preference may differ from the storage of matrix [MATH].', '1901.06015-1-72-1': 'If this is the case, then BLIS calls a virtual microkernel , instead of calling the microkernel directly, allowing for logic that will use a very small amount of temporary storage-equivalent to one microtile-in order to facilitate the proper use of the microkernel (whether it be row- or column-preferential) and then copy and/or accumulate the temporary microtile result back to the appropriate location within the output matrix [MATH].', '1901.06015-1-73-0': '## Computation domain', '1901.06015-1-74-0': 'Unlike the computation precision, which is discussed in the next section, the computation domain is implied according to the case-specific interpretations covered in Section [REF] and summarized in Figure [REF].', '1901.06015-1-74-1': 'Alternate interpretations exist, however.', '1901.06015-1-74-2': 'For example, the computation of Case 2ab could be interpreted as taking place in the real domain, which would result in [MATH] and [MATH] being ignored.', '1901.06015-1-74-3': 'Similar interpretations-all of which would change the update to [MATH]-could be applied to Cases 2ac, 2bc, or even 3.', '1901.06015-1-74-4': 'However, we do not immediately see significant utility in exposing these cases.', '1901.06015-1-75-0': 'Thus, our implementation in BLIS does not presently allow the caller to explicitly specify the computation domain.', '1901.06015-1-76-0': '# Supporting Mixed Precision Computation', '1901.06015-1-77-0': '# Optimizations', '1901.06015-1-78-0': '# Handling scalars', '1901.06015-1-79-0': 'Before concluding our discussion of how to implement and support mixed-datatype gemm, we turn our attention to scalars [MATH] and [MATH], which have been omitted from our discussion thus far.', '1901.06015-1-80-0': '## Mixed precision', '1901.06015-1-81-0': 'If the precision of [MATH] differs from the computation precision, a decision must be made as to how to proceed.', '1901.06015-1-81-1': 'Numerous possible policies exist for handling such situations.', '1901.06015-1-81-2': 'Three examples follow:', '1901.06015-1-82-0': 'Typecast [MATH] to match the computation precision.', '1901.06015-1-82-1': 'Typecast the computation precision to match that of [MATH].', '1901.06015-1-82-2': 'Unconditionally promote the lower precision value to the precision of the other (higher precision) value.', '1901.06015-1-83-0': 'Our mixed-datatype extension to BLIS currently opts for option (1).', '1901.06015-1-84-0': 'A similar decision must be made for handling the precision of [MATH].', '1901.06015-1-84-1': 'The choices here are even more numerous, because while we consider the precision of [MATH] and the storage precision of [MATH] to be the main inputs to the runtime logic, one could also argue for considering the disagreement with the computation precision that governs the computation of [MATH].', '1901.06015-1-84-2': 'A few possible policies are:', '1901.06015-1-85-0': 'Typecast [MATH] to match the storage precision of [MATH].', '1901.06015-1-85-1': 'Perform the scaling [MATH] in the higher precision of the two values, typecasting back to the storage precision of [MATH], as necessary.', '1901.06015-1-85-2': 'Typecast both [MATH] and [MATH] to the computation precision so that all suboperations within [MATH] can occur in the same precision before being typecast back to the storage precision of [MATH].', '1901.06015-1-86-0': 'Once again, our solution in BLIS opts for the relatively simple solution alluded to in (1), with [MATH] being typecast to the storage precision on accumulation to [MATH].', '1901.06015-1-87-0': '## Mixed domain', '1901.06015-1-88-0': 'Real values of [MATH] are easily handled for all eight cases enumerated in Section [REF].', '1901.06015-1-89-0': 'For Cases 0, 1a, 1b, and 1c, complex [MATH] may be projected into the real domain (which discards [MATH] entirely) since, with [MATH], [MATH] cannot change the final result.', '1901.06015-1-89-1': 'Similarly, complex values of [MATH] are handled as expected in the four cases where [MATH].', '1901.06015-1-89-2': 'Specifically, Case 3 already handles complex [MATH] while Cases 2ab, 2ac, and 2bc support [MATH] via extra logic in the virtual microkernel.', '1901.06015-1-90-0': 'Real and complex values of [MATH] are already handled in Cases 0 and 3, respectively.', '1901.06015-1-90-1': 'Real [MATH] are also already handled by Case 3 since [MATH].', '1901.06015-1-90-2': 'In Case 0, a complex [MATH] can be projected to the real domain since [MATH] would not change the computation, even if we had storage in which to save the final result.', '1901.06015-1-91-0': "For Cases 1a, 1b, 1c, 2ab, 2ac, and 2bc, a non-zero imaginary component in [MATH] could presumably change the final computation under a literal interpretation of the computation-that is, one in which all five operands' domains are taken at face value.", '1901.06015-1-91-1': 'However, implementing this logic is non-trival.', '1901.06015-1-91-2': 'For example, consider our approach to handling Case 1a, as discussed in Section [REF].', '1901.06015-1-91-3': 'In this case, we perform the computation according to Case 0, as if the imaginary components of [MATH] were zero.', '1901.06015-1-91-4': "That case's handling is completely consistent with its mathematics, since in that scenario [MATH] is the only operand with non-zero imaginary values, and thus they would have no impact on the final result.", '1901.06015-1-91-5': 'However, if both [MATH] and [MATH] are complex, then the imaginary components could combine to change the real component of the scalar-matrix product [MATH].', '1901.06015-1-91-6': 'Adjusting for this new possible use case would require a different approach in the implementation, perhaps using temporary workspace to store a copy of [MATH] while it is scaled by [MATH], after which the imaginary components may be ignored (assuming they were even computed to begin with).', '1901.06015-1-92-0': "However, while it is clear that going through such motions would maintain deeper fidelity to the literal mathematics expressed in the mixed-domain scenario, it's not clear to us that this additional functionality would be vital for most applications.", '1901.06015-1-92-1': 'As we continue to solicit feedback from the community, we will pay close attention to whether users expect or request support for non-zero imaginary values of [MATH] in Cases 1a, 1b, 1c, 2ab, 2ac, and 2bc.', '1901.06015-1-92-2': 'For now, our mixed-datatype solution supports only real values of [MATH] for those six cases, and prints an error message if the scalar is given with a non-zero imaginary component.', '1901.06015-1-93-0': '# Performance', '1901.06015-1-94-0': 'In this section, we discuss performance results for our mixed-datatype implementations on two servers with modern hardware architectures.', '1901.06015-1-95-0': '## Platform and implementation details', '1901.06015-1-96-0': '### Marvell ThunderX2', '1901.06015-1-97-0': 'The first system upon which we measured performance is a single compute node consisting of two 28-core Marvell ThunderX2 CN9975 processors.', '1901.06015-1-98-0': 'Each core, running at a clock rate of 2.2 GHz, provides a single-core peak performance of 17.6 gigaflops (GFLOPS) in double precision and 35.2 GFLOPS in single precision.', '1901.06015-1-98-1': 'Each of the two sockets has a 32MB L3 cache that is shared among its local cores, and each core has a private 256KB L2 cache and 32KB L1 (data) cache.', '1901.06015-1-98-2': 'The installed operating system was Ubuntu 16.04 running the Linux 4.15.0 kernel.', '1901.06015-1-98-3': 'Source code was compiled by the GNU C compiler (gcc) version 7.3.0.', '1901.06015-1-99-0': '### Intel Xeon Platinum', '1901.06015-1-100-0': 'The second system is a single node consisting of two 26-core Intel Xeon Platinum 8167M processors.', '1901.06015-1-101-0': 'Each core ran at a clock rate of 2.0 GHz, providing single-core peak performance of 64 gigaflops (GFLOPS) in double precision and 128 GFLOPS in single precision.', '1901.06015-1-101-1': 'Each of the two sockets has a 35.75MB L3 cache that is shared among its local cores, and each core has a private 1MB L2 cache and 32KB L1 (data) cache.', '1901.06015-1-101-2': 'The installed operating system was Ubuntu 18.04.1 running the Linux 4.15.0 kernel.', '1901.06015-1-101-3': 'Source code was compiled by the GNU C compiler (gcc) version 7.3.0.', '1901.06015-1-102-0': '### Implementations', '1901.06015-1-103-0': 'On both the ThunderX2 and the Xeon Platinum, the version of BLIS used was based on an inter-version release that preceded 0.5.1.', '1901.06015-1-103-1': 'In both cases, BLIS was configured with OpenMP-based multithreading.', '1901.06015-1-103-2': 'Architecture-specific configuration, which determines settings such as kernel sets and cache blocksizes, was performed automatically via the auto target to the configure script.', '1901.06015-1-104-0': 'We showcase two (or, in some cases, three) implementations, with a third (or fourth) provided for reference:', '1901.06015-1-105-0': '## Results', '1901.06015-1-106-0': '### Scope and Conventions', '1901.06015-1-107-0': 'Performance data for sequential, multithreaded within one socket (28 threads), and multithreaded across two sockets (56 threads) was gathered on the ThunderX2.', '1901.06015-1-107-1': 'Similarly, sequential, single-socket (26 threads), and dual-socket (52 threads) data was gathered on the Xeon Platinum.', '1901.06015-1-107-2': 'This produced a rather large set of data, which we formatted into 768 individual graphs.', '1901.06015-1-107-3': 'As a practical matter, we have relegated this complete set of performance results to Online Appendix [REF].', '1901.06015-1-107-4': 'Here, in the main body of the article, we limit our presentation to a slice of data that we feel is broadly representative of the full set.', '1901.06015-1-108-0': 'For any given graph, the [MATH]-axis denotes the problem size (where [MATH]), the [MATH]-axis shows observed floating-point performance in units of GFLOPS per core, and the theoretical peak performance of the hardware coincides with the top of the graph.', '1901.06015-1-108-1': 'Problem sizes for sequential instances of gemm were run from 40 to 2000 in increments of 40 while multithreaded executions were run from 120 to 6000 in increments of 120.', '1901.06015-1-109-0': 'The data points in all performance graphs report the best of three trials.', '1901.06015-1-110-0': 'Individual graphs are labeled according to the mixed-datatype case of its Internal and Ad-hoc implementations.', '1901.06015-1-110-1': 'The datatypes are encoded as [MATH], where the characters [MATH], [MATH], and [MATH] encode the storage datatypes of [MATH], [MATH], and [MATH], respectively, while [MATH] encodes the computation precision.', '1901.06015-1-110-2': 'For example, a case labeled "zcsd" would refer to mixed-domain Case 2ac, where matrices [MATH] and [MATH] are stored in single-precision (complex and real, respectively), matrix [MATH] is double-precision complex, and the computation occurs in double-precision arithmetic.', '1901.06015-1-111-0': 'All experiments reflect the use of randomized, column-stored matrices with gemm scalars [MATH] and [MATH].', '1901.06015-1-112-0': '### Exposition', '1901.06015-1-113-0': 'Figure [REF] (top) reports sequential performance on the Marvell ThunderX2.', '1901.06015-1-113-1': 'The six graphs on the left half of Figure [REF] (top) report performance for six select mixed-datatype cases that we felt are interesting: sdds, ddds, dsss, ccss, cscs, and csss.', '1901.06015-1-113-2': 'All of these mixed-datatype cases perform their computation in single-precision.', '1901.06015-1-113-3': 'On the right half, we display graphs that correspond to the precision-toggled analogues of the graphs on the left-dssd, sssd, sddd, zzdd, zdzd, and zddd-all of which perform their computation in double-precision.', '1901.06015-1-113-4': 'Both Internal implementations (with and without extra memory) are shown in four of the six graphs in each group, with the remaining two graphs displaying performance only for the implementation where extra memory is disabled, since the optimization is not applicable for those cases.', '1901.06015-1-114-0': 'The legends are shown once for each group of six graphs.', '1901.06015-1-114-1': 'Within the legend, the curves labeled "Intern (+xm)" and "Intern (-xm)" refer to the Internal implementations with and without extra memory, respectively.', '1901.06015-1-114-2': 'Additionally, the label for the Reference implementation is augmented, in parentheses, with the conventional gemm routine that serves as the reference curve within that group of six graphs.', '1901.06015-1-115-0': 'Organized identically as those in the top, the graphs in Figure [REF] (bottom) report multithreaded performance on one socket (28 threads) of the ThunderX2.', '1901.06015-1-116-0': 'Finally, Figure [REF] (top) and (bottom) report sequential and single-socket (26 threads) performance, respectively, on the Intel Xeon Platinum using the same mixed-datatype cases and organization as shown in Figure [REF].', '1901.06015-1-117-0': '### Observations and Analysis', '1901.06015-1-118-0': 'Let us turn first to the sequential performance results from the ThunderX2.', '1901.06015-1-119-0': 'The first thing we notice is that, in five of the six mixed-datatype cases, the Ad-hoc approach is capable of performing quite well relative to the Internal implementations.', '1901.06015-1-119-1': 'The sixth case, which falls within mixed-domain Case 2bc, suffers because, unlike with 2ac, we are unable to cast the problem in terms of sgemm() or dgemm() by manipulating matrix metadata, and therefore must resort to using cgemm() and zgemm().', '1901.06015-1-119-2': 'And because [MATH], this approach necessarily wastes half of the floating-point computations.', '1901.06015-1-120-0': 'Next, we notice that employing [MATH] often affords a modest but noticeable increase in performance relative to forgoing extra memory.', '1901.06015-1-120-1': 'This is expected for all of the reasons described in Sections [REF]-[REF].', '1901.06015-1-121-0': 'Turning to the multithreaded performance on ThunderX2, we notice that the effect of using extra memory in the Internal implementation is not only reversed, but also magnified.', '1901.06015-1-121-1': 'Here, the additional costs incurred within the virtual microkernel are overwhelmed by the cost of the accumulation of [MATH] back to [MATH] that must be performed after the gemm operation, which is not parallelized.', '1901.06015-1-122-0': 'The performance of the Ad-hoc implementation also suffers, for similar reasons to that of the Internal implementation using [MATH].', '1901.06015-1-122-1': 'The effect is even worse for Ad-hoc, however, because that implementation must make whole copies of matrices up-front, and does so sequentially, before executing the underlying gemm operation.', '1901.06015-1-122-2': 'By contrast, the Internal implementations benefit from typecasting [MATH] and [MATH] during packing, which is already parallelized.', '1901.06015-1-123-0': 'Overall, the extra-memory-avoiding Internal implementation performs quite well relative to its sgemm() and dgemm() benchmarks.', '1901.06015-1-124-0': 'Turning to the Intel Xeon Platinum results in Figure [REF], we find the data largely tells the same story.', '1901.06015-1-124-1': 'Here, the multithreaded performance degradation caused by employing extra memory is even more severe, and the Ad-hoc performance is similarly attenuated.', '1901.06015-1-124-2': 'Once again, in both sequential and multithreaded cases, one of the Internal implementations matches or exceeds (sometimes by a large margin) that of the Ad-hoc approach.', '1901.06015-1-124-3': 'However, for some datatype cases, even the memory-avoiding Internal implementation lags noticeably behind its sgemm() or dgemm() benchmark.', '1901.06015-1-124-4': 'The cause of this is not immediately clear, but may be related to memory bandwidth becoming strained in cases where the the virtual microkernel is relied upon to update the output matrix in two steps, using a temporary microtile as intermediate storage.', '1901.06015-1-125-0': 'These results strongly suggest that, in general, BLIS should employ the use of Internal implementation with [MATH] for sequential invocations of gemm, but avoid [MATH] in the case of many-threaded execution.', '1901.06015-1-125-1': 'As a function of the number of threads, the crossover point between the two Internal implementations will likely depend on the amount of parallelism that can be extracted within the accumulation of [MATH] back to [MATH] before memory bandwidth is saturated.', '1901.06015-1-125-2': 'We leave this topic for future exploration.', '1901.06015-1-126-0': '# Measuring the impact on code size', '1901.06015-1-127-0': 'Prior to reading the article, a casual reader might have been skeptical of the practicality of our solution.', '1901.06015-1-127-1': 'However, Sections [REF] and [REF] decompose the problem into mostly orthogonal use cases, giving hope that the ultimate impact on library code size is much more manageable.', '1901.06015-1-128-0': 'Indeed, by multiple measures, the BLIS library grew only modestly after introducing mixed-datatype support.', '1901.06015-1-129-0': 'The second column in Figure [REF] shows the total number of lines', '1901.06015-1-130-0': 'of code present in the BLIS framework proper-which excludes other components such as the build system, kernels, and the testsuite-before and after mixed-datatype support was added to the gemm operation.', '1901.06015-1-131-0': 'The fourth column shows the total size in kilobytes of the source code.', '1901.06015-1-131-1': 'The change between the "before" and "after" values for total lines and total size are shown in the third and fifth columns, respectively.', '1901.06015-1-131-2': 'Mixed-datatype support for the gemm operation adds approximately 4% each to the total number of lines and total bytes of source code.', '1901.06015-1-132-0': "Figure [REF] lists similar metrics for the BLIS testsuite, which is capable of testing the vast majority of BLIS's computational operations.", '1901.06015-1-132-1': 'Here, the support for testing all combinations of mixed-datatype execution, with any combination of matrix storage storage or transposition and/or combinations, increases the source code footprint by approximately 6% in both lines and total size.', '1901.06015-1-133-0': 'Finally, Figure [REF] shows the object (binary) code size for three build products: BLIS built as a static library; BLIS built as a shared library; and the BLIS testsuite linked against the static library.', '1901.06015-1-134-0': 'This figure shows three rows of values: before mixed-datatype support for gemm was added (667d3929); after mixed-datatype support was added (5fec95b9) where the feature was disabled at configure-time; and after mixed-datatype support was added (5fec95b9) where the feature was enabled at configure-time.', '1901.06015-1-134-1': '(In the the latter two cases, the "Change" columns represent the change from the "before" state.)', '1901.06015-1-134-2': 'With mixed-datatype support present and enabled, the size of the static library increases by only 255KB, or 8 of the original library size.', '1901.06015-1-134-3': 'In the case of the shared library, the increase is just under 9.', '1901.06015-1-134-4': 'And a statically-linked instance of the BLIS testsuite increases by about 11.', '1901.06015-1-135-0': 'Thus, no matter the metric, the increase in code footprint is quite modest relative to the scope of functionality added.', '1901.06015-1-136-0': '# Final Thoughts', '1901.06015-1-137-0': '# Complete performance results', '1901.06015-1-138-0': 'This section contains complete performance results using the same hardware, implementations, and test parameters discussed in Section [REF].', '1901.06015-1-138-1': 'We report 128 performance graphs for each combination of sequential, single-socket, and dual-socket execution on both the Marvell ThunderX2 and Intel Xeon Platinum, resulting in a total of [MATH] graphs.', '1901.06015-1-139-0': 'Performance graphs are organized into one set for each mixed-domain case, with each set containing the 16 possible precision combinations within that case.', '1901.06015-1-139-1': 'The mixed-domain sets of graphs appear in pairs (top and bottom) across Figures [REF] to [REF].', '1901.06015-1-139-2': 'Within a figure, graphs in the left and center-left columns report performance using a computation precision of single precision while those in the right and center-right columns show performance using a computation precision of double precision.', '1901.06015-1-139-3': 'Furthermore, the graphs are organized such that, for any given graph in the single-precision computation subgroup, the graph located two spots to its right corresponds to the experiments where all precisions are toggled (from single to double or vice versa).', '1901.06015-1-140-0': 'Within the graphs for any given mixed-domain set, the legends are omitted from all except the top-right graph within each computation precision subgroup-that is, the top graph in the center-left and right columns.', '1901.06015-1-140-1': 'As with the graphs presented in Section [REF], the "Reference" curves are listed as "Ref (?', '1901.06015-1-140-2': 'gemm)", where the ?', '1901.06015-1-140-3': 'indicates one of s,d,c,z.', '1901.06015-1-140-4': 'This added detail serves to remind the reader which datatype-specific variant of BLIS\'s conventional (datatype-homogeneous) gemm is the most appropriate curve against which to judge the "Internal" and "Ad-hoc" mixed-datatype implementations.', '1901.06015-1-140-5': 'In general, the graphs in the left and right halves of Figures [REF]-[REF] (top and bottom) use sgemm() and dgemm() as reference curves, respectively, except for the mixed-domain Case 3 results in the bottom halves of Figures [REF], [REF], [REF], [REF], [REF], and [REF], which compare against cgemm() and zgemm() in the left and right halves, respectively.', '1901.06015-1-141-0': 'We provide the following table to aid the reader in finding the performance graphs associated with any given mixed-domain case, for each hardware and threading configuration.'}

{'1901.06015-2-0-0': '# Introduction', '1901.06015-2-1-0': 'The BLAS [CITATION] defines the general matrix-matrix multiplication (gemm) operation to support any of the following computations: [EQUATION] where [MATH] is [MATH], the left-hand matrix product operand ([MATH], [MATH], or [MATH]) is [MATH], the right-hand matrix product operand ([MATH], [MATH], or [MATH]) is [MATH], and [MATH] and [MATH] are scalars.', '1901.06015-2-2-0': 'This matrix multiplication functionality is made available to software developers via the following application programming interfaces, or APIs:', '1901.06015-2-3-0': 'sgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-2-4-0': 'dgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-2-5-0': 'cgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-2-6-0': 'zgemm( transa, transb, m, n, k, alpha, A, ldA, B, ldB, beta, C, ldC )', '1901.06015-2-7-0': 'The first letter of the routine name uniquely encodes the datatype-that is, the domain and precision-of the matrix and scalar operands as well as the computation: single-precision real (s); double-precision real (d); single-precision complex (c); and double-precision complex (z).', '1901.06015-2-7-1': 'The parameters transa and transb indicate if [MATH] and/or [MATH], respectively, should be computed upon as if they were transposed or conjugate-transposed.', '1901.06015-2-7-2': 'The interfaces implicitly require that matrices be stored in column-major order.', '1901.06015-2-7-3': 'Accordingly, the parameters ldA, ldB, and ldC convey the so-called "leading dimensions" of the arrays A, B, and C, respectively-that is, the number of elements that separate matrix element [MATH] from element [MATH] in memory.', '1901.06015-2-8-0': 'While this interface has served the HPC community well, it has also become constraining.', '1901.06015-2-8-1': 'For example, when computing tensor contractions (which often resemble matrix multiplications), one may need to refer to a sub-tensor that cannot be represented with column-major storage without making a temporary copy.', '1901.06015-2-8-2': 'Similarly, some situations may call for conjugating (but not transposing) a matrix operand.', '1901.06015-2-8-3': 'Indeed, such functionality is already supported by the BLIS framework, which exports BLAS-like operations and APIs [CITATION].', '1901.06015-2-8-4': 'However, even BLIS only supports computation on operands with identical datatypes.', '1901.06015-2-8-5': 'Consider the following:', '1901.06015-2-9-0': 'Thus, there is likely a fair amount of pent-up demand for high-performance implementations to datatype-flexible BLAS-like APIs.', '1901.06015-2-10-0': 'As alluded to above, the naive approach to supporting mixed-datatype functionality within the gemm operation comes with obvious memory, performance, and productivity drawbacks: typecasting matrix operands to a common domain and/or precision outside of the original implementation requires considerable workspace; the memory access patterns engendered by monolithic casting almost surely acts as a drag on performance; and programming an ad-hoc solution in terms of the BLAS gemm interfaces sometimes requires non-trivial skills.', '1901.06015-2-10-1': 'Indeed, some would find providing the full combinatoric space of functionality daunting, and in response might attempt to survey the community and then only implement those cases for which interest was expressed.', '1901.06015-2-10-2': 'Instead, our goal from the outset is to implement support for all cases, and to do so in a manner that delivers high or near-high performance.', '1901.06015-2-11-0': 'Our approach, which builds on the BLIS framework, yields a comprehensive reference implementation with which consumers of this functionality can explore the benefits of mixed-domain and mixed-precision gemm computation without being constrained by limitations in the interface, incomplete coverage within the implementation, or unnecessarily inefficient performance.', '1901.06015-2-12-0': '## Notation', '1901.06015-2-13-0': 'Our notation should be mostly self-evident to most readers of high-performance dense linear algebra literature.', '1901.06015-2-13-1': 'We use uppercase Roman letters, such as [MATH], [MATH], and [MATH] to denote matrices, and lowercase Greek letters [MATH] and [MATH] to represent scalars.', '1901.06015-2-14-0': 'Real and complex domains are indicated by [MATH] and [MATH], respectively.', '1901.06015-2-14-1': 'Occasionally, we refer to the real part of a matrix or matrix expression [MATH] with [MATH] and to the imaginary part with [MATH].', '1901.06015-2-14-2': 'In other places, such as where this notation would be too cumbersome, we use superscripts, such as [MATH] and [MATH] for the real and imaginary components, respectively, of a scalar [MATH].', '1901.06015-2-15-0': 'When representing elements within a matrix, we use a subscript to encode the row and column indices.', '1901.06015-2-15-1': 'For example, a scalar [MATH] would reference the element located in the 2nd row and 4th column of a matrix [MATH].', '1901.06015-2-16-0': '# Background', '1901.06015-2-17-0': 'In this section, we review the approach to matrix multiplication taken within the BLIS framework as well as some related implementation details that will provide important context to discussions later in this article.', '1901.06015-2-18-0': '## Matrix Multiplication in BLIS', '1901.06015-2-19-0': 'The GotoBLAS algorithm [CITATION] for performing matrix multiplication underlies the highest-performing BLAS libraries for current general-purpose microprocessors.', '1901.06015-2-19-1': 'The BLAS-like Library Instantiation Software (BLIS) framework, which implements gemm and other matrix-matrix operations, refactors the GotoBLAS algorithm as pictured in Figure [REF].', '1901.06015-2-19-2': 'BLIS isolates the code that needs to be optimized (in assembly code or with vector intrinsics) for different architectures in a microkernel that updates a very small submatrix, or microtile, of [MATH] with a sequence of rank-1 updates that are accumulated in registers [CITATION].', '1901.06015-2-19-3': 'All other loops and supporting kernels are implemented portably in C99.', '1901.06015-2-19-4': "By contrast, Goto's implementation-also adopted by the OpenBLAS fork [CITATION] of the GotoBLAS library-casts the computation into a larger assembly-coded kernel.", '1901.06015-2-19-5': 'This larger unit of code, which corresponds to what BLIS refers to as the macrokernel, consists of the microkernel plus the logic that falls within the first two loops around the microkernel.', '1901.06015-2-20-0': 'A key element of the GotoBLAS algorithm is that high-performance implementation of gemm incorporates the packing of submatrices of [MATH] (into buffer [MATH]) and of [MATH] (into buffer [MATH]) to improve data locality during the execution of the microkernel.', '1901.06015-2-21-0': 'This feature has been used in the past to consolidate other functionality into the same framework: implementation of other matrix-matrix operations (level-3 BLAS) [CITATION], fusing sequences of matrix operations of importance to Machine Learning [CITATION], and implementation of practical Fast Matrix Multiplication (Strassen-like) algorithms [CITATION].', '1901.06015-2-21-1': 'A final insight comes from Van Zee [CITATION], in which it is shown how complex matrix multiplication can be cast in terms of only microkernels designed for real domain gemm without a significant performance penalty.', '1901.06015-2-22-0': 'Given a target architecture, instantiating the traditional functionality of gemm with BLIS requires only two microkernels, one each for single- and double-precision real domain computations, with the insight from [CITATION] inducing the functionality typically provided by complex domain microkernels.', '1901.06015-2-22-1': 'It also requires packing functions, which by default take the form of architecture-agnostic (C99) implementations provided by the framework, as well as architecture-specific cache and register blocking parameters.', '1901.06015-2-22-2': 'BLIS exposes several other configure-time options, though they all default to values that typically need no further tweaking.', '1901.06015-2-23-0': '## Managing complexity in BLIS', '1901.06015-2-24-0': 'The combinatoric complexity of the gemm operation in BLIS is mitigated in several ways.', '1901.06015-2-25-0': '### Storage', '1901.06015-2-26-0': 'BLIS tracks separate row and column strides for each matrix object.', '1901.06015-2-27-0': 'Using these two stride parameters, BLIS supports three matrix storage formats in its end-user APIs: row-major storage (where the column stride is unit); column-major storage (where the row stride is unit); and general storage (where neither stride is unit).', '1901.06015-2-28-0': 'Each gemm operand in BLIS may individually be stored in any of the three aforementioned storage formats.', '1901.06015-2-28-1': 'If we consider all possible variations of general storage as a single format, this results in a total of [MATH] different storage combinations.', '1901.06015-2-28-2': 'The gemm operation supports these [MATH] storage combinations as follows: the packing function is written generically to allow it to read from any of the three storage formats when reading matrices [MATH] and [MATH] (during the packing of [MATH] and [MATH]), and the microkernel is required to handle input/output of [MATH] in any of the three supported formats.', '1901.06015-2-29-0': '### Transposition', '1901.06015-2-30-0': 'BLIS easily accommodates transposition of matrices [MATH] or [MATH] by swapping the row and column strides (and their corresponding dimensions) just prior to packing.', '1901.06015-2-30-1': 'This technique merely affects how the matrices are traversed rather than how they are stored; thus, we call this an "induced" (or logical) transposition, as no matrix elements are actually copied or moved.', '1901.06015-2-31-0': '### Conjugation', '1901.06015-2-32-0': 'In the case of complex matrices, optional conjugation', '1901.06015-2-33-0': 'of [MATH] and [MATH] is handled during the packing into [MATH] and [MATH].', '1901.06015-2-34-0': '### Multithreaded parallelization', '1901.06015-2-35-0': 'The authors of [CITATION] discuss how BLIS exposes many loops, each of which can be parallelized.', '1901.06015-2-35-1': 'Crucially, the complexity over matrix storage datatypes (domain and precision), transposition/conjugation parameters (transA and transb), and matrix storage formats (row, column, generalized) are orthogonal to the issues pertaining to extracting multithreaded parallelism from gemm.', '1901.06015-2-35-2': 'Therefore, the insights of [CITATION] carry over to the parallelization when mixing domains and/or precisions.', '1901.06015-2-36-0': '### Optimizing input/output on [MATH]', '1901.06015-2-37-0': 'High-performance microkernels accumulate their intermediate results using vector registers.', '1901.06015-2-37-1': 'Thus, the microkernel author must decide whether to semantically assign the vector registers to contain contiguous rows or contiguous columns of the microtile submatrix.', '1901.06015-2-37-2': "We refer to this as the microkernel's register orientation.", '1901.06015-2-37-3': 'Interestingly, the register orientation necessarily biases the microkernel towards loading and storing elements of [MATH] as either rows or columns, since performing IO on elements in the opposite orientation would require a sequence of costly permutation instructions.', '1901.06015-2-37-4': "BLIS tracks this intrinsic property, or IO preference, of the microkernel so that the framework can transform the matrix problem to the microkernel's liking.", '1901.06015-2-37-5': 'For example, if our microkernel is row-preferential and the gemm implementation is executed on a column-stored matrix [MATH], BLIS will employ a high-level transposition of the entire operation (to [MATH]) so that, from the perspective of the microkernel, [MATH] appears to be stored in its preferred format-that is, a manner consistent with its vector register orientation.', '1901.06015-2-38-0': 'Thus, regardless of whether the microkernel is row- or column-preferential, the gemm implementation will, generally-speaking, yield similar performance on row- and column-stored matrices [MATH].', '1901.06015-2-39-0': '## For the busy reader', '1901.06015-2-40-0': 'We acknowledge that some readers may wish for a very short synopsis of the insights presented later in this article.', '1901.06015-2-40-1': 'We sum up the techniques that allow implementing all cases of mixed-domain and mixed-precision gemm within the BLIS framework as follows: The mixing of domains can be handled by enumerating the eight cases (six of them new), which largely reduce to either manipulating matrix metadata, and/or exposing the real and imaginary elements in a complex matrix in such a way that a real matrix multiplication may be performed to induce the desired result, in part motivated by insights from the 1m method [CITATION].', '1901.06015-2-40-2': 'The mixing of precisions can be handled by typecasting between precisions, as needed, during the packing [MATH] and [MATH] (into [MATH] and [MATH]), while the typecasting during the accumulation of the intermediate product [MATH] may occur in special code wrappers that update [MATH] appropriately.', '1901.06015-2-41-0': '# API Considerations', '1901.06015-2-42-0': 'In the spirit of the BLAS API, a more complete interface that supports this richer, mixed-datatype environment could take the form of', '1901.06015-2-43-0': 'where transX indicates whether [MATH] should be computed upon as if it were (optionally) transposed, conjugate-transposed, or conjugated without transposition; X is the address where matrix [MATH] is stored; domainX indicates whether [MATH] is stored as a matrix of real or complex elements; precX indicates the precision in which elements of [MATH] are stored (e.g., half-, single-, double-, or quad-precision); rstrideX and cstrideX indicate the row and column strides, respectively, for storing [MATH]; and precAB indicates the precision in which the matrix multiplication takes place (possibly implying promotion or demotion from the storage precision of either [MATH] or [MATH]).', '1901.06015-2-43-1': 'Note that column-major order and row-major order are the special cases where rstrideX is unit and cstrideX is unit, respectively.', '1901.06015-2-44-0': 'The hypothetical API shown in Figure [REF] plainly exposes all of the dimensions of functionality along which the library developer must provide implementations.', '1901.06015-2-44-1': 'However, we feel that such an interface is not very useful towards inspiring those implementations, as it subtly nudges the developer towards extending the use of separate interfaces for each parameter combination down the function stack, leading to solutions that are vertically siloed from each other, even if various subsets share many similarities.', '1901.06015-2-45-0': 'BLIS preempted this problem by starting with an object-based foundation for encoding and expressing matrix (and vector) operands.', '1901.06015-2-45-1': 'Each linear algebra entity (such as a matrix or vector) is encapsulated within a data structure, or more specifically, a custom struct type.', '1901.06015-2-45-2': 'For example, BLIS currently exports the following object-based function prototype for invoking the gemm operation:', '1901.06015-2-46-0': 'The function bligemm() takes five arguments of type objt*, each of which corresponds to the address of a struct representing the floating-point operands traditionally passed into the gemm operation.', '1901.06015-2-46-1': 'The function exposes no other arguments, because all of the conventional parameters (such as transA and transB) may be interpreted as properties of one of the floating-point operands.', '1901.06015-2-47-0': 'A simplified version of the objt type definition may be given as:', '1901.06015-2-48-0': 'Here, dimt, inct, dofft, sizt, and objbitst represent various integer types defined within BLIS for representing dimensions, strides and increments, diagonal offsets, byte sizes, and object property bitfields, respectively.', '1901.06015-2-48-1': 'The idea behind the struct example in Figure [REF] is that matrices may be represented by a collection of properties, or metadata, and that these properties may be set-for example, when the object is initialized and its underlying data buffer is allocated-and then subsequently queried or modified using a collection of object-based accessor functions.', '1901.06015-2-48-2': 'Encapsulating matrix properties within objects helps hide details that need not be exposed at certain levels of the implementation.', '1901.06015-2-49-0': 'The key observation to make now is that the domainX and precX arguments shown in Figure [REF] can be completely hidden within the object API of BLIS.', '1901.06015-2-49-1': 'Indeed, the current definition of objt within the framework already includes domain and precision bits within the info bitfield.', '1901.06015-2-49-2': 'We only need to add an additional parameter, or designate additional bits within the info property, to support the computation precision (labeled in Figure [REF] as precAB).', '1901.06015-2-49-3': 'Thus, it is possible to add mixed-datatype support to gemm without any modification to the function interface to bligemm().', '1901.06015-2-50-0': "A more thorough walkthrough of BLIS's object API is well beyond the scope of this article.", '1901.06015-2-51-0': 'The main takeaway from this discussion is that the original author of BLIS designed the framework around an object-based core with the keen understanding that additional APIs of arbitrary format, including (but not limited to) those in the style of the BLAS, could always be layered on top of this more general abstraction.', '1901.06015-2-51-1': 'Consequently, any such APIs built above and beyond the underlying object layer are only incidental to the framework; they merely constitute syntactic re-expressions of some subset of the functionality made possible by the object foundation.', '1901.06015-2-52-0': '# Supporting Mixed Domain Computation', '1901.06015-2-53-0': 'We consider the storage domain (real or complex) of the matrix to be orthogonal to the storage precision (single, double, etc).', '1901.06015-2-53-1': 'In this section, we consider how to handle mixing matrix operands of different domains.', '1901.06015-2-53-2': 'For now, the reader should assume that the storage precision is held constant across all matrix operands, and therefore can be ignored.', '1901.06015-2-53-3': 'We also ignore scalars [MATH] and [MATH] for the time being, which simplifies the general matrix multiplication operation to [MATH].', '1901.06015-2-54-0': '## The 1m method', '1901.06015-2-55-0': 'The author of [CITATION] recently presented a novel method of computing complex matrix multiplication without relying upon kernels that explicitly perform complex arithmetic at the scalar level, as is typically the case in high-performance BLAS libraries.', '1901.06015-2-55-1': 'Instead, the so-called 1m method relies only upon matrix primitives (kernels) that compute real matrix multiplication.', '1901.06015-2-55-2': 'And unlike the older and more easily understood 4m method [CITATION], 1m replaces each complex matrix multiplication with only a single real matrix multiplication.', '1901.06015-2-56-0': 'The key to 1m is a pair of special packing formats, which Van Zee denotes 1e and 1r.', '1901.06015-2-56-1': 'The author illustrates the role of these two packing formats using the following example of complex matrix multiplication [MATH] where [MATH], [MATH], and [MATH].', '1901.06015-2-56-2': '[EQUATION]', '1901.06015-2-56-3': 'In this example, it is assumed that matrix [MATH] is column-stored, which prescribes that the 1e format be applied to [MATH] and the 1r format be applied to [MATH].', '1901.06015-2-56-4': 'This can be confirmed by inspection: applying a real matrix multiplication to the left- and right-hand matrix product operands would not correctly compute the complex matrix multiplication if [MATH] were row-stored because the intermediate elements of [MATH] would update the wrong elements of [MATH].', '1901.06015-2-56-5': 'However, 1m provides a cure to this situation.', '1901.06015-2-56-6': 'Namely, symmetry in the method allows for a row-oriented variant where [MATH] is row-stored.', '1901.06015-2-56-7': '[EQUATION]', '1901.06015-2-56-8': 'In this case, the application of the packing formats is reversed, such that 1e is applied to [MATH] and 1r is applied to [MATH].', '1901.06015-2-56-9': 'Van Zee later points out that it is not actually the storage of [MATH] that determines whether Eq. [REF] or [REF] is employed, but rather the SIMD register orientation of the underlying real domain microkernel-which determines the input/output preference of the microtile and thus the natural method of performing SIMD reads and write instructions on [MATH].', '1901.06015-2-57-0': 'While the 1m method was originally articulated as a way to implement complex domain matrix multiplication on operands of identical datatypes, we will soon show that not only can it be extended to mixed-precision complex domain computation, but it also indirectly supports a particular combination of mixed-domain operands.', '1901.06015-2-58-0': '## Enumerating the cases', '1901.06015-2-59-0': 'Consider for simplicity [MATH], ignoring the scaling factors [MATH] and [MATH].', '1901.06015-2-59-1': 'Each of [MATH] can be stored in either the real or complex domains, leading to [MATH] different combinations.', '1901.06015-2-59-2': 'Figure [REF] enumerates and names each possible case.', '1901.06015-2-59-3': 'We will now discuss each case, how it is interpreted, and how it is implemented within the BLIS framework.', '1901.06015-2-60-0': '### Cases 0 , 3', '1901.06015-2-61-0': 'The trivial case where all matrices are stored in the real domain, which we refer to as Case 0, is already supported by the framework via algorithms based on real domain microkernels.', '1901.06015-2-61-1': 'Similarly, Case 3, which applies when all matrices are stored in the complex domain, is also already supported.', '1901.06015-2-61-2': 'Support for Case 3 is provided in BLIS via conventional algorithms based on complex domain microkernels as well as via the 1m method, which is particularly useful when complex microkernels are not available.', '1901.06015-2-61-3': 'Cases 0 and 3 incur [MATH] and [MATH] flops, respectively.', '1901.06015-2-62-0': '### Cases 1a , 1b', '1901.06015-2-63-0': 'Case 1a captures situations where [MATH] and [MATH] are real while [MATH] is complex.', '1901.06015-2-63-1': 'We interpret such an operation as [MATH].', '1901.06015-2-63-2': 'Implementing this case in BLIS is rather straightforward: we ignore the imaginary part of [MATH] and compute as if all matrices were real.', '1901.06015-2-63-3': 'Because BLIS tracks both row and column strides for each matrix operand, ignoring the imaginary elements amounts to a temporary change to the dimension and stride metadata contained within the object representing [MATH].', '1901.06015-2-63-4': 'Case 1b involves a complex matrix [MATH] and real matrices [MATH] and [MATH], but is otherwise handled similarly.', '1901.06015-2-63-5': 'Since these case are ultimately implemented in terms of Case 0, they both performs [MATH] flops.', '1901.06015-2-64-0': '### Case 2ab', '1901.06015-2-65-0': 'Case 2ab is applicable when [MATH] and [MATH] are complex while the matrix to which they accumulate, [MATH], is real.', '1901.06015-2-65-1': 'We interpret this somewhat curious scenario as a matrix product that takes place in the complex domain, but one for which the imaginary result is discarded: [MATH].', '1901.06015-2-65-2': 'Since [MATH] is not needed, only [MATH] flops need to be performed.', '1901.06015-2-65-3': 'Thus, this case provides an opportunity for computational savings when properly implemented.', '1901.06015-2-65-4': 'BLIS implements 2ab by borrowing the 1r packing format used by the 1m method.', '1901.06015-2-66-0': 'Specifically, BLIS packs both matrices [MATH] and [MATH] using the 1r format while simultaneously conjugating [MATH].', '1901.06015-2-66-1': 'This has the effect of allowing a subsequent real matrix multiplication over the packed matrices to correctly compute only the real half of the complex matrix multiplication update.', '1901.06015-2-67-0': '### Case 1c', '1901.06015-2-68-0': 'The opposite of 2ab-Case 1c-refers to settings in which matrix [MATH] is complex while [MATH] and [MATH] are real.', '1901.06015-2-68-1': 'Since the matrix product takes place entirely in the real domain, the natural interpretation is that [MATH] updates only [MATH], and [MATH] is left untouched.', '1901.06015-2-68-2': 'Generally speaking, BLIS implements 1c using a strategy similar to the one used with 1a and 1b.', '1901.06015-2-68-3': 'That is, a temporary change to the object metadata describing matrix [MATH] allows us to isolate [MATH], which once again reduces the problem to Case 0.', '1901.06015-2-68-4': 'Accordingly, this case requires only [MATH] flops.', '1901.06015-2-69-0': '### Cases 2ac , 2bc', '1901.06015-2-70-0': 'Consider Case 2ac, in which matrices [MATH] and [MATH] are complex and matrix [MATH] is real.', '1901.06015-2-70-1': 'We interpret this situation as performing a complex matrix product [MATH] to update both real and imaginary parts of [MATH].', '1901.06015-2-70-2': 'However, all computation involving the imaginary part of [MATH], which is implicitly zero, may be ignored.', '1901.06015-2-70-3': 'This means that the computation requires only [MATH] flops.', '1901.06015-2-70-4': 'Now, if [MATH] and [MATH] were guaranteed to be column-stored, BLIS could handle this case with a simple change of metadata that recasts those complex matrices as real, with the real and imaginary elements treated equally and indistinguishably.', '1901.06015-2-70-5': 'However, BLIS also allows row storage (and general storage) for all matrices, and thus the solution is not quite so simple.', '1901.06015-2-70-6': 'Instead, BLIS handles 2ac as follows.', '1901.06015-2-71-0': 'If the original problem fits into Case 2ac and the microkernel is column-preferential, [MATH] is packed as a real matrix (with imaginary elements stored traditionally, in element-wise interleaved fashion) and [MATH] is packed normally.', '1901.06015-2-71-1': 'A real domain macrokernel (Case 0) is then executed, which will properly update [MATH].', '1901.06015-2-71-2': 'However, if the microkernel is row-preferential, the operation is logically transposed and Case 2bc is executed instead (whereby matrices [MATH] and [MATH] are complex and [MATH] is real).', '1901.06015-2-71-3': 'Thus, the effective case employed, 2ac or 2bc, depends on the register orientation of the microkernel, not the storage of [MATH], and does so for the same reason that the 1m method, discussed previously in Section [REF], depends on the same property.', '1901.06015-2-72-0': 'After the aforementioned logical transposition is applied (or not), the microkernel input/output preference may differ from the storage of matrix [MATH].', '1901.06015-2-72-1': 'If this is the case, then BLIS calls a virtual microkernel , instead of calling the microkernel directly, allowing for logic that will use a very small amount of temporary storage-equivalent to one microtile-in order to facilitate the proper use of the microkernel (whether it be row- or column-preferential) and then copy and/or accumulate the temporary microtile result back to the appropriate location within the output matrix [MATH].', '1901.06015-2-73-0': '## Computation domain', '1901.06015-2-74-0': 'Unlike the computation precision, which is discussed in the next section, the computation domain is implied according to the case-specific interpretations covered in Section [REF] and summarized in Figure [REF].', '1901.06015-2-74-1': 'Alternate interpretations exist, however.', '1901.06015-2-74-2': 'For example, the computation of Case 2ab could be interpreted as taking place in the real domain, which would result in [MATH] and [MATH] being ignored.', '1901.06015-2-74-3': 'Similar interpretations-all of which would change the update to [MATH]-could be applied to Cases 2ac, 2bc, or even 3.', '1901.06015-2-74-4': 'However, we do not immediately see significant utility in exposing these cases.', '1901.06015-2-75-0': 'Thus, our implementation in BLIS does not presently allow the caller to explicitly specify the computation domain.', '1901.06015-2-76-0': '# Supporting Mixed Precision Computation', '1901.06015-2-77-0': '# Optimizations', '1901.06015-2-78-0': '# Handling scalars', '1901.06015-2-79-0': 'Before concluding our discussion of how to implement and support mixed-datatype gemm, we turn our attention to scalars [MATH] and [MATH], which have been omitted from our discussion thus far.', '1901.06015-2-80-0': '## Mixed precision', '1901.06015-2-81-0': 'If the precision of [MATH] differs from the computation precision, a decision must be made as to how to proceed.', '1901.06015-2-81-1': 'Numerous possible policies exist for handling such situations.', '1901.06015-2-81-2': 'Three examples follow:', '1901.06015-2-82-0': 'Typecast [MATH] to match the computation precision.', '1901.06015-2-82-1': 'Typecast the computation precision to match that of [MATH].', '1901.06015-2-82-2': 'Unconditionally promote the lower precision value to the precision of the other (higher precision) value.', '1901.06015-2-83-0': 'Our mixed-datatype extension to BLIS currently opts for option (1).', '1901.06015-2-84-0': 'A similar decision must be made for handling the precision of [MATH].', '1901.06015-2-84-1': 'The choices here are even more numerous, because while we consider the precision of [MATH] and the storage precision of [MATH] to be the main inputs to the runtime logic, one could also argue for considering the disagreement with the computation precision that governs the computation of [MATH].', '1901.06015-2-84-2': 'A few possible policies are:', '1901.06015-2-85-0': 'Typecast [MATH] to match the storage precision of [MATH].', '1901.06015-2-85-1': 'Perform the scaling [MATH] in the higher precision of the two values, typecasting back to the storage precision of [MATH], as necessary.', '1901.06015-2-85-2': 'Typecast both [MATH] and [MATH] to the computation precision so that all suboperations within [MATH] can occur in the same precision before being typecast back to the storage precision of [MATH].', '1901.06015-2-86-0': 'Once again, our solution in BLIS opts for the relatively simple solution alluded to in (1), with [MATH] being typecast to the storage precision on accumulation to [MATH].', '1901.06015-2-87-0': '## Mixed domain', '1901.06015-2-88-0': 'Real values of [MATH] are easily handled for all eight cases enumerated in Section [REF].', '1901.06015-2-89-0': 'For Cases 0, 1a, 1b, and 1c, complex [MATH] may be projected into the real domain (which discards [MATH] entirely) since, with [MATH], [MATH] cannot change the final result.', '1901.06015-2-89-1': 'Similarly, complex values of [MATH] are handled as expected in the four cases where [MATH].', '1901.06015-2-89-2': 'Specifically, Case 3 already handles complex [MATH] while Cases 2ab, 2ac, and 2bc support [MATH] via extra logic in the virtual microkernel.', '1901.06015-2-90-0': 'Real and complex values of [MATH] are already handled in Cases 0 and 3, respectively.', '1901.06015-2-90-1': 'Real [MATH] are also already handled by Case 3 since [MATH].', '1901.06015-2-90-2': 'In Case 0, a complex [MATH] can be projected to the real domain since [MATH] would not change the computation, even if we had storage in which to save the final result.', '1901.06015-2-91-0': "For Cases 1a, 1b, 1c, 2ab, 2ac, and 2bc, a non-zero imaginary component in [MATH] could presumably change the final computation under a literal interpretation of the computation-that is, one in which all five operands' domains are taken at face value.", '1901.06015-2-91-1': 'However, implementing this logic is non-trival.', '1901.06015-2-91-2': 'For example, consider our approach to handling Case 1a, as discussed in Section [REF].', '1901.06015-2-91-3': 'In this case, we perform the computation according to Case 0, as if the imaginary components of [MATH] were zero.', '1901.06015-2-91-4': "That case's handling is completely consistent with its mathematics, since in that scenario [MATH] is the only operand with non-zero imaginary values, and thus they would have no impact on the final result.", '1901.06015-2-91-5': 'However, if both [MATH] and [MATH] are complex, then the imaginary components could combine to change the real component of the scalar-matrix product [MATH].', '1901.06015-2-91-6': 'Adjusting for this new possible use case would require a different approach in the implementation, perhaps using temporary workspace to store a copy of [MATH] while it is scaled by [MATH], after which the imaginary components may be ignored (assuming they were even computed to begin with).', '1901.06015-2-92-0': "However, while it is clear that going through such motions would maintain deeper fidelity to the literal mathematics expressed in the mixed-domain scenario, it's not clear to us that this additional functionality would be vital for most applications.", '1901.06015-2-92-1': 'As we continue to solicit feedback from the community, we will pay close attention to whether users expect or request support for non-zero imaginary values of [MATH] in Cases 1a, 1b, 1c, 2ab, 2ac, and 2bc.', '1901.06015-2-92-2': 'For now, our mixed-datatype solution supports only real values of [MATH] for those six cases, and prints an error message if the scalar is given with a non-zero imaginary component.', '1901.06015-2-93-0': '# Performance', '1901.06015-2-94-0': 'In this section, we discuss performance results for our mixed-datatype implementations on two servers with modern hardware architectures.', '1901.06015-2-95-0': '## Platform and implementation details', '1901.06015-2-96-0': '### Marvell ThunderX2', '1901.06015-2-97-0': 'The first system upon which we measured performance is a single compute node consisting of two 28-core Marvell ThunderX2 CN9975 processors.', '1901.06015-2-98-0': 'Each core, running at a clock rate of 2.2 GHz, provides a single-core peak performance of 17.6 gigaflops (GFLOPS) in double precision and 35.2 GFLOPS in single precision.', '1901.06015-2-98-1': 'Each of the two sockets has a 32MB L3 cache that is shared among its local cores, and each core has a private 256KB L2 cache and 32KB L1 (data) cache.', '1901.06015-2-98-2': 'The installed operating system was Ubuntu 16.04 running the Linux 4.15.0 kernel.', '1901.06015-2-98-3': 'Source code was compiled by the GNU C compiler (gcc) version 7.3.0.', '1901.06015-2-99-0': '### Intel Xeon Platinum', '1901.06015-2-100-0': 'The second system is a single node consisting of two 26-core Intel Xeon Platinum 8167M processors.', '1901.06015-2-101-0': 'Each core ran at a clock rate of 2.0 GHz, providing single-core peak performance of 64 gigaflops (GFLOPS) in double precision and 128 GFLOPS in single precision.', '1901.06015-2-101-1': 'Each of the two sockets has a 35.75MB L3 cache that is shared among its local cores, and each core has a private 1MB L2 cache and 32KB L1 (data) cache.', '1901.06015-2-101-2': 'The installed operating system was Ubuntu 18.04.1 running the Linux 4.15.0 kernel.', '1901.06015-2-101-3': 'Source code was compiled by the GNU C compiler (gcc) version 7.3.0.', '1901.06015-2-102-0': '### Implementations', '1901.06015-2-103-0': 'On both the ThunderX2 and the Xeon Platinum, the version of BLIS used was based on an inter-version release that preceded 0.5.1.', '1901.06015-2-103-1': 'In both cases, BLIS was configured with OpenMP-based multithreading.', '1901.06015-2-103-2': 'Architecture-specific configuration, which determines settings such as kernel sets and cache blocksizes, was performed automatically via the auto target to the configure script.', '1901.06015-2-104-0': 'We showcase two (or, in some cases, three) implementations, with a third (or fourth) provided for reference:', '1901.06015-2-105-0': '## Results', '1901.06015-2-106-0': '### Scope and Conventions', '1901.06015-2-107-0': 'Performance data for sequential, multithreaded within one socket (28 threads), and multithreaded across two sockets (56 threads) was gathered on the ThunderX2.', '1901.06015-2-107-1': 'Similarly, sequential, single-socket (26 threads), and dual-socket (52 threads) data was gathered on the Xeon Platinum.', '1901.06015-2-107-2': 'This produced a rather large set of data, which we formatted into 768 individual graphs.', '1901.06015-2-107-3': 'As a practical matter, we have relegated this complete set of performance results to Online Appendix [REF].', '1901.06015-2-107-4': 'Here, in the main body of the article, we limit our presentation to a slice of data that we feel is broadly representative of the full set.', '1901.06015-2-108-0': 'For any given graph, the [MATH]-axis denotes the problem size (where [MATH]), the [MATH]-axis shows observed floating-point performance in units of GFLOPS per core, and the theoretical peak performance of the hardware coincides with the top of the graph.', '1901.06015-2-108-1': 'Problem sizes for sequential instances of gemm were run from 40 to 2000 in increments of 40 while multithreaded executions were run from 120 to 6000 in increments of 120.', '1901.06015-2-109-0': 'The data points in all performance graphs report the best of three trials.', '1901.06015-2-110-0': 'Individual graphs are labeled according to the mixed-datatype case of its Internal and Ad-hoc implementations.', '1901.06015-2-110-1': 'The datatypes are encoded as [MATH], where the characters [MATH], [MATH], and [MATH] encode the storage datatypes of [MATH], [MATH], and [MATH], respectively, while [MATH] encodes the computation precision.', '1901.06015-2-110-2': 'For example, a case labeled "zcsd" would refer to mixed-domain Case 2ac, where matrices [MATH] and [MATH] are stored in single-precision (complex and real, respectively), matrix [MATH] is double-precision complex, and the computation occurs in double-precision arithmetic.', '1901.06015-2-111-0': 'All experiments reflect the use of randomized, column-stored matrices with gemm scalars [MATH] and [MATH].', '1901.06015-2-112-0': '### Exposition', '1901.06015-2-113-0': 'Figure [REF] (top) reports sequential performance on the Marvell ThunderX2.', '1901.06015-2-113-1': 'The six graphs on the left half of Figure [REF] (top) report performance for six select mixed-datatype cases that we felt are interesting: sdds, ddds, dsss, ccss, cscs, and csss.', '1901.06015-2-113-2': 'All of these mixed-datatype cases perform their computation in single-precision.', '1901.06015-2-113-3': 'On the right half, we display graphs that correspond to the precision-toggled analogues of the graphs on the left-dssd, sssd, sddd, zzdd, zdzd, and zddd-all of which perform their computation in double-precision.', '1901.06015-2-113-4': 'Both Internal implementations (with and without extra memory) are shown in four of the six graphs in each group, with the remaining two graphs displaying performance only for the implementation where extra memory is disabled, since the optimization is not applicable for those cases.', '1901.06015-2-114-0': 'The legends are shown once for each group of six graphs.', '1901.06015-2-114-1': 'Within the legend, the curves labeled "Intern (+xm)" and "Intern (-xm)" refer to the Internal implementations with and without extra memory, respectively.', '1901.06015-2-114-2': 'Additionally, the label for the Reference implementation is augmented, in parentheses, with the conventional gemm routine that serves as the reference curve within that group of six graphs.', '1901.06015-2-115-0': 'Organized identically as those in the top, the graphs in Figure [REF] (bottom) report multithreaded performance on one socket (28 threads) of the ThunderX2.', '1901.06015-2-116-0': 'Finally, Figure [REF] (top) and (bottom) report sequential and single-socket (26 threads) performance, respectively, on the Intel Xeon Platinum using the same mixed-datatype cases and organization as shown in Figure [REF].', '1901.06015-2-117-0': '### Observations and Analysis', '1901.06015-2-118-0': 'Let us turn first to the sequential performance results from the ThunderX2.', '1901.06015-2-119-0': 'The first thing we notice is that, in five of the six mixed-datatype cases, the Ad-hoc approach is capable of performing quite well relative to the Internal implementations.', '1901.06015-2-119-1': 'The sixth case, which falls within mixed-domain Case 2bc, suffers because, unlike with 2ac, we are unable to cast the problem in terms of sgemm() or dgemm() by manipulating matrix metadata, and therefore must resort to using cgemm() and zgemm().', '1901.06015-2-119-2': 'And because [MATH], this approach necessarily wastes half of the floating-point computations.', '1901.06015-2-120-0': 'Next, we notice that employing [MATH] often affords a modest but noticeable increase in performance relative to forgoing extra memory.', '1901.06015-2-120-1': 'This is expected for all of the reasons described in Sections [REF]-[REF].', '1901.06015-2-121-0': 'Turning to the multithreaded performance on ThunderX2, we notice that the effect of using extra memory in the Internal implementation is not only reversed, but also magnified.', '1901.06015-2-121-1': 'Here, the additional costs incurred within the virtual microkernel are overwhelmed by the cost of the accumulation of [MATH] back to [MATH] that must be performed after the gemm operation, which is not parallelized.', '1901.06015-2-122-0': 'The performance of the Ad-hoc implementation also suffers, for similar reasons to that of the Internal implementation using [MATH].', '1901.06015-2-122-1': 'The effect is even worse for Ad-hoc, however, because that implementation must make whole copies of matrices up-front, and does so sequentially, before executing the underlying gemm operation.', '1901.06015-2-122-2': 'By contrast, the Internal implementations benefit from typecasting [MATH] and [MATH] during packing, which is already parallelized.', '1901.06015-2-123-0': 'Overall, the extra-memory-avoiding Internal implementation performs quite well relative to its sgemm() and dgemm() benchmarks.', '1901.06015-2-124-0': 'Turning to the Intel Xeon Platinum results in Figure [REF], we find the data largely tells the same story.', '1901.06015-2-124-1': 'Here, the multithreaded performance degradation caused by employing extra memory is even more severe, and the Ad-hoc performance is similarly attenuated.', '1901.06015-2-124-2': 'Once again, in both sequential and multithreaded cases, one of the Internal implementations matches or exceeds (sometimes by a large margin) that of the Ad-hoc approach.', '1901.06015-2-124-3': 'However, for some datatype cases, even the memory-avoiding Internal implementation lags noticeably behind its sgemm() or dgemm() benchmark.', '1901.06015-2-124-4': 'The cause of this is not immediately clear, but may be related to memory bandwidth becoming strained in cases where the the virtual microkernel is relied upon to update the output matrix in two steps, using a temporary microtile as intermediate storage.', '1901.06015-2-125-0': 'These results strongly suggest that, in general, BLIS should employ the use of Internal implementation with [MATH] for sequential invocations of gemm, but avoid [MATH] in the case of many-threaded execution.', '1901.06015-2-125-1': 'As a function of the number of threads, the crossover point between the two Internal implementations will likely depend on the amount of parallelism that can be extracted within the accumulation of [MATH] back to [MATH] before memory bandwidth is saturated.', '1901.06015-2-125-2': 'We leave this topic for future exploration.', '1901.06015-2-126-0': '# Measuring the impact on code size', '1901.06015-2-127-0': 'Prior to reading the article, a casual reader might have been skeptical of the practicality of our solution.', '1901.06015-2-127-1': 'However, Sections [REF] and [REF] decompose the problem into mostly orthogonal use cases, giving hope that the ultimate impact on library code size is much more manageable.', '1901.06015-2-128-0': 'Indeed, by multiple measures, the BLIS library grew only modestly after introducing mixed-datatype support.', '1901.06015-2-129-0': 'The second column in Figure [REF] shows the total number of lines', '1901.06015-2-130-0': 'of code present in the BLIS framework proper-which excludes other components such as the build system, kernels, and the testsuite-before and after mixed-datatype support was added to the gemm operation.', '1901.06015-2-131-0': 'The fourth column shows the total size in kilobytes of the source code.', '1901.06015-2-131-1': 'The change between the "before" and "after" values for total lines and total size are shown in the third and fifth columns, respectively.', '1901.06015-2-131-2': 'Mixed-datatype support for the gemm operation adds approximately 4% each to the total number of lines and total bytes of source code.', '1901.06015-2-132-0': "Figure [REF] lists similar metrics for the BLIS testsuite, which is capable of testing the vast majority of BLIS's computational operations.", '1901.06015-2-132-1': 'Here, the support for testing all combinations of mixed-datatype execution, with any combination of matrix storage storage or transposition and/or combinations, increases the source code footprint by approximately 6% in both lines and total size.', '1901.06015-2-133-0': 'Finally, Figure [REF] shows the object (binary) code size for three build products: BLIS built as a static library; BLIS built as a shared library; and the BLIS testsuite linked against the static library.', '1901.06015-2-134-0': 'This figure shows three rows of values: before mixed-datatype support for gemm was added (667d3929); after mixed-datatype support was added (5fec95b9) where the feature was disabled at configure-time; and after mixed-datatype support was added (5fec95b9) where the feature was enabled at configure-time.', '1901.06015-2-134-1': '(In the the latter two cases, the "Change" columns represent the change from the "before" state.)', '1901.06015-2-134-2': 'With mixed-datatype support present and enabled, the size of the static library increases by only 255KB, or 8 of the original library size.', '1901.06015-2-134-3': 'In the case of the shared library, the increase is just under 9.', '1901.06015-2-134-4': 'And a statically-linked instance of the BLIS testsuite increases by about 11.', '1901.06015-2-135-0': 'Thus, no matter the metric, the increase in code footprint is quite modest relative to the scope of functionality added.', '1901.06015-2-136-0': '# Final Thoughts'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1901.06015

astro-ph-0006226

{'astro-ph-0006226-1-0-0': 'We propose an approximate description of basic parameters (radius, mass and oblateness) of general relativistic compact rotating objects in terms of the parameters of the static configuration and of the angular velocity only.', 'astro-ph-0006226-1-0-1': 'The representation in terms of static properties is derived using the condition of stationary equilibrium together with some phenomenological assumptions.', 'astro-ph-0006226-1-0-2': 'The predicted radius and mass of rotating neutron star (described by some realistic equations of state) and strange star (described by the bag model equation of state) are compared with data obtained by numerical integration of gravitational field equation.The obtained formulae also allow a simple derivation of the "empirical" equation relating the maximum rotation frequency [MATH] of uniformly rotating star models to the mass and radius of the maximum allowable mass configuration of the non-rotating model.', 'astro-ph-0006226-1-1-0': 'Pacs Numbers:', 'astro-ph-0006226-1-2-0': '# INTRODUCTION', 'astro-ph-0006226-1-3-0': 'Rotation is a basic physical property of astrophysical objects.', 'astro-ph-0006226-1-3-1': 'Study of rotational properties of stars can lead to some restrictions imposed on the nuclear equation of state of dense matter at densities larger than nuclear density.Oscillations of rapidly rotating stars can become unstable hence producing detectable gravitational wave emissions.', 'astro-ph-0006226-1-4-0': 'In the past decades there were numerous attempts to construct analytic models for rotating perfect fluid bodies in general relativity.', 'astro-ph-0006226-1-4-1': 'But obtaining an exact interior solution for a rotating body proved to be a formidable task.The first step in this direction was the fundamental result of Kerr [1] who obtained the solution for the vacuum domain, outside the rotating star.', 'astro-ph-0006226-1-4-2': 'It took no less than three decades of investigations to obtain the first, highly idealized model of a general relativistic, thin rotating disk of dust, by Neugebauer and Meinel [2].', 'astro-ph-0006226-1-4-3': 'Various schemes have been developed for obtaining stationary and axisymmetric perfect fluid solutions of the gravitational equations like Petrov type D, local rotational symmetry, fluid kinematics, non-trivial Killing tensor, vanishing Simon tensor, electric magnetic Weyl curvature, lagrangian or static- stationary symmetry, geodesic eigenrays etc. (for a recent review of rotating perfect fluid models in general relativity see [3]).', 'astro-ph-0006226-1-5-0': 'If in the case of rapidly rotating stellar configurations there are still many unsolved problems, a remarkable progress has been made in the study of slowly and rigidly rotating perfect fluid configurations.', 'astro-ph-0006226-1-5-1': 'By casting the metric in the form [EQUATION]', 'astro-ph-0006226-1-5-2': 'Hartle [4] obtained a formalism that proved to be very useful in many investigations of the rotational properties of stars.', 'astro-ph-0006226-1-5-3': 'However, this model, which considers only first order corrections in [MATH], can not be used to compute models of rapidly rotating relativistic stars with sufficient accuracy.', 'astro-ph-0006226-1-6-0': 'On the other hand there has been recently a considerable advance in the numerical understanding of rotating stars.', 'astro-ph-0006226-1-6-1': 'Several high precision numerical codes are now avalaible and it has been shown that they agree with each other to remarkable accuracy (see [5] for a review of recent developments in numerical study of rotatation, nonaxisymmetric oscillations and instabilities of general relativistic stars).', 'astro-ph-0006226-1-7-0': 'The non-sphericity of rapidly rotating stationary stellar configurations and the complicated character of the interplay of the effects of rotation and of those of general relativity seem not to permit a simple universal description of rotating compact objects.', 'astro-ph-0006226-1-7-1': 'However, Haensel and Zdunik [6] and Friedman, Ipser and Parker [7] have found a simple relation connecting the maximum rotation frequency [MATH] with the maximum mass [MATH] and radius [MATH] of the static configuration: [EQUATION] with [MATH] a constant which does not depend on the equation of state of the dense matter.', 'astro-ph-0006226-1-7-2': 'The value of the constant [MATH] has been obtained by fitting equation ([REF]) with data obtained by numerically integrating the gravitational field equations.', 'astro-ph-0006226-1-7-3': 'It is given by [MATH] [7], [MATH] [8] or by [MATH] [9].', 'astro-ph-0006226-1-7-4': 'The empirical relation ([REF]) has been checked for many realistic equations of state for neutron stars [8], [10]-[12] and for the case of strange stars where the empirical formula also holds with a very good precision, the relative deviations do not exceeding 2% [9].', 'astro-ph-0006226-1-7-5': 'Equation ([REF]), obtained on the basis of analyzing numerical solutions of the gravitational field equations, provides an enormous simplification of the problem of dynamical effects of rotation because the solutions of the complicate general relativistic equations for a rotating star can be replaced with the solutions of the much simpler TOV equation.', 'astro-ph-0006226-1-7-6': 'Some attempts to explain this empirical relation were not concluded with satisfactory result.', 'astro-ph-0006226-1-7-7': 'Weber and Glendenning [13], [14] used numerical models of slowly rotating relativistic stars to show that the formula still hold, but with [MATH].', 'astro-ph-0006226-1-7-8': 'Also in the slow rotation limit Glendenning and Weber [15] derived a formula relating [MATH] to [MATH], in terms of the mass, equatorial radius, moment of inertia, angular momentum and quadrupole moment of the maximally rotating configuration only.', 'astro-ph-0006226-1-7-9': 'But it is not clear how the formula ([REF]) follows from their results.Up to now a clear physical understanding of this relation is still missing.', 'astro-ph-0006226-1-8-0': 'On the other hand a universal relation between the maximum mass and radius of non-rotating neutron star configuration and the mass [MATH] and radius [MATH] of the configuration rotating with [MATH] of the form [EQUATION] has also been found [12].', 'astro-ph-0006226-1-8-1': 'In ([REF]) [MATH] and [MATH] are specific equation of state dependent constants, whose values have been calculated, for a broad set of realistic EOS, in [12].', 'astro-ph-0006226-1-8-2': 'The empirical constant [MATH] can be obtained, within an approximation better than [MATH] , from the formula [12] [EQUATION]', 'astro-ph-0006226-1-8-3': 'The general validity of ([REF]) and ([REF]) suggests the possibility that all basic physical parameters of general relativistic rotating stellar objects (like mass and radius) can be somehow related to the similar parameters of the static configuration.', 'astro-ph-0006226-1-8-4': 'It is the purpose of the present paper to propose a general description of basic physical parameters (mass, radius and oblateness) of compact general relativistic objects in terms of the physical properties of the static configuration and of the angular velocity only.', 'astro-ph-0006226-1-8-5': 'To obtain the representation in terms of static physical parameters we use only the general relativistic conditions of equilibrium for static and rotating stars and some phenomenological assumptions.The resulted approximate mass and radius formulae are compared with data obtained from numerical integration of gravitational field equations in case of neutron stars described by realistic equations of state and strange stars described by the bag model equation of state.', 'astro-ph-0006226-1-9-0': 'The present paper is organized as follows.The general formalism allowing to obtain mass and radius formulae for rotating general relativistic stars is presented in Section 2.', 'astro-ph-0006226-1-9-1': 'In Section 3 we apply our results to the case of neutron stars described by realistic equations of state.', 'astro-ph-0006226-1-9-2': 'In Section 4 we consider the case of strange stars.', 'astro-ph-0006226-1-9-3': 'We discuss and conclude our results in Section 5.', 'astro-ph-0006226-1-10-0': '# THE GENERAL FORMALISM', 'astro-ph-0006226-1-11-0': 'For a static equilibrium stellar type configuration, with interior described by the metric [EQUATION] the condition of the hydrostatic equilibrium, which follows from the Bianchi identities, can be written as [EQUATION] with [MATH] and [MATH] the energy density and the pressure of the matter respectively ( in the present paper we shall use units so that [MATH]).', 'astro-ph-0006226-1-12-0': 'We shall also make the assumption that the matter EOS is a one parameter one, [EQUATION] with [MATH] the proper baryon density.', 'astro-ph-0006226-1-12-1': 'For such an equation of state the heat function defined by [EQUATION] is a regular function of [MATH].', 'astro-ph-0006226-1-12-2': 'It can always be written in the form [16] [EQUATION] where [EQUATION] is the enthalpy per baryon.', 'astro-ph-0006226-1-13-0': 'As applied at the center and at the surface of the star respectively, the hydrostatic equilibrium condition yields [EQUATION]', 'astro-ph-0006226-1-13-1': 'At the vacuum boundary of the static star the Schwarzschild exterior solution gives the metric and we have [17] [EQUATION] with [MATH] and [MATH] the mass and radius of the static stellar configuration.', 'astro-ph-0006226-1-13-2': 'We denote by [MATH] and [MATH] the baryon density at the center of the star and at the surface, respectively.', 'astro-ph-0006226-1-13-3': 'Therefore from ([REF]) and ([REF]) we obtain the following general and exact expression for the mass-radius ratio of the static star: [EQUATION] where we have denoted [EQUATION] and [EQUATION]', 'astro-ph-0006226-1-13-4': 'For a given equation of state [MATH] is a function of the central density only.', 'astro-ph-0006226-1-13-5': 'From a physical point of view [MATH] can be related via the relation [MATH] to the redshift [MATH] of a photon emitted from the center of the star.', 'astro-ph-0006226-1-14-0': 'For the mass of the star we can obtain another general representation by assuming [EQUATION] where [MATH] is a specific density that can be arbitrarily chosen (for example it is the central density of the minimum mass configuration) and [MATH] is a function describing the effects of the variation of the central density on the basic parameters of the stellar configuration.', 'astro-ph-0006226-1-15-0': 'The two unknown functions [MATH] and [MATH] can be determined by fitting equations ([REF]) and ([REF]) with the exact values of the mass and radius obtained by numerically integrating the TOV equation for a given equation of state.', 'astro-ph-0006226-1-15-1': 'Their knowledge allows us to construct exact mass and radius formulae for sequences of static general relativistic stars having different central densities.', 'astro-ph-0006226-1-15-2': 'From equations ([REF]) and ([REF]) we obtain the radius and the mass of the star in the form [EQUATION]', 'astro-ph-0006226-1-15-3': 'Solving the equations [MATH] and [MATH] for the value of the central density [MATH] will give, with the use of ([REF]) and ([REF]) the maximum values of the radius and mass of the static star, respectively.', 'astro-ph-0006226-1-16-0': 'To describe the interior of the rotating general relativistic star we shall adopt the formalism presented in [16] and [9].', 'astro-ph-0006226-1-16-1': 'Under the hypothesis of stationarity, axial symmetry and purely azimuthal motion a coordinate system [MATH] can be chosen so that inside the star the line element takes the form [EQUATION] where [MATH],[MATH],[MATH] and [MATH] are functions of [MATH] and [MATH] only.', 'astro-ph-0006226-1-16-2': 'As measured by the locally non-rotating observer the fluid 3-velocity is given by [MATH] , where [MATH] is the angular velocity of a fluid element moving in the [MATH]-direction (physically it is the angular velocity as measured by an observer at spatial infinity) [9],[16].', 'astro-ph-0006226-1-17-0': 'From the point of view of our phenomenological approach the most important result is the equation of the stationary motion, which results from the Bianchi identities and which for a rotating perfect fluid reduces to [16] [EQUATION] where [MATH] , [MATH] and [MATH].', 'astro-ph-0006226-1-17-1': 'If [MATH] (case called uniform or rigid rotation) equation ([REF]) can be integrated to give the following fundamental result describing the stationary equilibrium of a rotating general relativistic star: [EQUATION]', 'astro-ph-0006226-1-17-2': 'Equation ([REF]) is just the generalization to the case of rotation of the well-known static Bianchi identity [MATH] [17] we have already used to describe static stellar configurations.', 'astro-ph-0006226-1-18-0': 'We assume that the vacuum boundary of the rotating star is described by the Kerr metric [1] in the Boyer-Lindquist coordinates, which has the form [18]: [EQUATION]', 'astro-ph-0006226-1-18-1': 'In this form the Kerr metric is manifestly axially symmetric and closely resembles the Schwarzschild solution in its standard form.', 'astro-ph-0006226-1-18-2': '[MATH] is the mass of the source and the parameter [MATH] is the ratio between the angular momentum [MATH] and the mass[MATH] of the rotating star.', 'astro-ph-0006226-1-19-0': "Let's apply equation ([REF]) at two points: at the center of the dense core and at the pole of the rotating star.", 'astro-ph-0006226-1-19-1': 'We denote by [MATH] the value of the metric tensor component [MATH] at the center of the star.', 'astro-ph-0006226-1-19-2': 'At the polar point [MATH] and [MATH], where [MATH] is the polar radius of the star.', 'astro-ph-0006226-1-19-3': 'We also have [MATH] .', 'astro-ph-0006226-1-19-4': 'Therefore at this point the line element is given by [EQUATION]', 'astro-ph-0006226-1-19-5': 'Consequently from equations ([REF]) and ([REF]) we obtain the following exact mass-polar radius relation for the rotating general relativistic configuration: [EQUATION]', 'astro-ph-0006226-1-19-6': 'The function [MATH] is identical to that of the static case.', 'astro-ph-0006226-1-19-7': "Let's apply now equation ([REF]) for two points situated in the equatorial plan of the rotating star: at the center of the star and at the equator respectively.", 'astro-ph-0006226-1-19-8': 'At the equator [MATH] and [MATH] is the equatorial radius of the star).', 'astro-ph-0006226-1-19-9': 'For a uniform rotation the rotation angle of the source/observer at the equator is [MATH] .', 'astro-ph-0006226-1-19-10': 'Taking into account these results we obtain the Kerr metric at the equator of the rotating star in the form: [EQUATION]', 'astro-ph-0006226-1-19-11': 'With the use of ([REF]) and ([REF]) we obtain the following mass-equatorial radius relation: [EQUATION] where a new function [EQUATION] has also been defined.', 'astro-ph-0006226-1-20-0': 'From Eqs. ([REF]) and ([REF]) we obtain the ratio [MATH] of the polar and equatorial radius (the oblateness) of the star in the form: [EQUATION]', 'astro-ph-0006226-1-20-1': 'In the case of the static star we have also proposed the alternative Eq. ([REF]) for providing another mass-radius relation.', 'astro-ph-0006226-1-20-2': 'We shall generalize this equation to the rotating case by assuming that the following formula relates the mass of the rotating star to its equatorial radius: [EQUATION] with [MATH] a function describing the combined general relativistic effects of rotation and central density on the mass of the star and with [MATH] the same specific density as used in the static case.', 'astro-ph-0006226-1-20-3': 'For [MATH], [MATH] and Eq. ([REF]) must reduce to the static case equation ([REF]).', 'astro-ph-0006226-1-21-0': 'Equations ([REF]),([REF]) and ([REF])give a complete and exact description of the mass and radius of the rotating general relativistic star.', 'astro-ph-0006226-1-21-1': 'But unfortunately the present approach, which is basically thermodynamic in its essence, can not predict the exact form and the values of the three unknown functions entering in the formalism.', 'astro-ph-0006226-1-21-2': 'The only thing we can do is to assume, also based on the static case, some empirical forms for the functions [MATH] and to check whether the resulting formulae can give a satisfactory description of rotating star configurations.', 'astro-ph-0006226-1-21-3': 'Therefore in the following we shall use the following five approximations:', 'astro-ph-0006226-1-22-0': 'i) As a first approximation we shall define the moment of inertia [MATH] of the rotating compact relativistic object via the Newtonian expression [EQUATION]', 'astro-ph-0006226-1-22-1': 'In fact over a wide range of [MATH] and [MATH] the corrections added by general relativistic effects to the moment of inertia can be approximated by [MATH] [15], but we shall not use this representation.', 'astro-ph-0006226-1-22-2': 'By adopting the Newtonian formula we obtain [EQUATION] ii) We suppose that the mass of the star can be expressed as a function of the equatorial radius by means of equation ([REF]) with [MATH]: [EQUATION] [MATH] is the function corresponding to the static case.', 'astro-ph-0006226-1-23-0': 'iii) We assume that the function [MATH], describing the metric tensor component [MATH] at the center of the rotating star is given by [EQUATION] with [MATH] an EOS dependent function given by [EQUATION] and [MATH] a non-negative constant.', 'astro-ph-0006226-1-23-1': '[MATH] is also the function corresponding to the static case.', 'astro-ph-0006226-1-24-0': 'iv) We assume that inside the rotating star [MATH] can be represented in the following phenomenological form: [EQUATION] v) We assume that: [EQUATION]', 'astro-ph-0006226-1-24-1': 'Equivalently,this condition can be written as [EQUATION]', 'astro-ph-0006226-1-24-2': 'For [MATH] and [MATH] we obtain [MATH], condition which usually holds for rotating astrophysical objects.', 'astro-ph-0006226-1-25-0': 'With these phenomenological assumptions Eqs. ([REF])-([REF]) lead to the following representation of the basic physical parameters of the rotating general relativistic star: [EQUATION]', 'astro-ph-0006226-1-25-1': 'We shall also suppose that [MATH] and [MATH] are universal constants and we shall choose [MATH] and [MATH].', 'astro-ph-0006226-1-26-0': 'The equatorial radius is defined only for values of the angular velocity satisfying the condition [MATH].', 'astro-ph-0006226-1-26-1': 'Therefore for the maximum admissible constant angular velocity of the maximally rotating star in uniform rotation we obtain the relation [EQUATION] where [EQUATION] for [MATH].', 'astro-ph-0006226-1-27-0': 'Equation ([REF]) is very similar to the "empirical" formula discussed in [1]-[7].', 'astro-ph-0006226-1-27-1': 'The coefficient of proportionality in ([REF]) is independent of the equation of state of the dense matter, but its numerical value does not fit the calculated value.On the other hand for [MATH] the radius of the star tends to infinity.', 'astro-ph-0006226-1-28-0': 'An alternative expression can be obtained by imposing the restriction [EQUATION] where [MATH] is the maxium allowable equatorial speed of the star.', 'astro-ph-0006226-1-28-1': 'Therefore we obtain [EQUATION]', 'astro-ph-0006226-1-28-2': 'By taking for twice of the ratio of the maximum static mass and radius a mean value of 0.58 and considering [MATH], we obtain [EQUATION] with [MATH],value that differs within [MATH] from the value [MATH] obtained in [16].', 'astro-ph-0006226-1-29-0': 'The maximum radius of the maximally rotating configuration can be obtained from [MATH], and is given by [EQUATION] where [EQUATION]', 'astro-ph-0006226-1-29-1': 'The mass [MATH] of the maximal radius rotating neutron star follows from ([REF]) and is given by [EQUATION] where [EQUATION]', 'astro-ph-0006226-1-29-2': 'The maximum mass [MATH] of the rotating star can be obtained from the equation [EQUATION] leading to [EQUATION] where [MATH] is a constant.', 'astro-ph-0006226-1-29-3': 'Therefore for the mass of the maximally rotating configuration we obtain [EQUATION]', 'astro-ph-0006226-1-29-4': 'Equations ([REF]) and ([REF]) show the existence of a proportionality relation between maximum mass and radius of the rotating and non-rotating configuration, respectively, as has already been suggested, on an empirical basis,in [12].', 'astro-ph-0006226-1-30-0': 'An investigation of 12 EOS performed in [12] for 12 realistic EOS of nuclear matter led to a (mean) value of [MATH] , while the same calculation performed by us with the use of data presented in [20] for other 14 different EOSs gives [MATH] , leading to a mean value of [MATH] for the 26 considered EOSs.Therefore we may conclude that the proportionality between the maximum mass of the rotating star and the maximum mass of the static configuration is exact and universal, being independent of the equation of state of dense matte.On the other hand the relation between the radius of the rotating and non-rotating configuration is EOS dependent,the coefficient of proportionality slightly decreasing with the increase of the mass-radius ratio of the static star.', 'astro-ph-0006226-1-31-0': '# APPLICATIONS TO NEUTRON STARS', 'astro-ph-0006226-1-32-0': "The correct mathematical and physical modelling of millisecond pulsars can be done only in the framework of general relativistic equilibrium models for rapidly rotating neutron stars.Such models are solutions of the Einstein's equations for the axisymmetric stationary gravitational field and they must be constructed numerically.", 'astro-ph-0006226-1-32-1': 'Recently several independent numerical codes have been developed by different groups of researchers and have been used to obtain rapidly rotating neutron star models based on a variety of realistic equations of state.Hence a large amount of numerical data is now available.', 'astro-ph-0006226-1-33-0': 'In the present Section we shall apply the results of the phenomenological formalism presented in the previous Section to the case of neutron stars described by realistic equations of state.', 'astro-ph-0006226-1-33-1': 'The data are selected from the paper by Cook, Shapiro and Teukolsky [20],who constructed general relativistic rotating star sequences for 14 nuclear matter equations of state.', 'astro-ph-0006226-1-33-2': 'Detailed data are presented only for 5 equations of state.', 'astro-ph-0006226-1-33-3': 'For the sake of comparison we have used the equations of state denoted A [21], AU [22], FPS [23] and L [24].', 'astro-ph-0006226-1-33-4': 'At low densities all these equations of state employ the Feynman,Metropolis and Teller [25] EOS and then join onto the Baym,Bethe and Pethick [26] EOS up to neutron drip.The equations of state are given in a tabular form and small changes in the way the tabulated equation of state is constructed do have a small effect on the resulting neutron star model.', 'astro-ph-0006226-1-34-0': 'In Tables 1-4 we present the comparison of the basic physical parameters of rotating stars obtained, for these four equations of state, with the help of Eqs. ([REF])-([REF]) and by numerically integrating the gravitational field equations.', 'astro-ph-0006226-1-35-0': 'For [MATH] sequences described by EOS A and FPS the maximum error of our prediction is around [MATH].', 'astro-ph-0006226-1-35-1': 'For the maximum mass normal sequences of EOS AU and L the maximum error in the predicted value of the mass does not exceed [MATH] but for large angular speeds it is around [MATH] for the radius of the rotating compact object.', 'astro-ph-0006226-1-36-0': '# APPLICATIONS TO STRANGE STARS', 'astro-ph-0006226-1-37-0': 'It is generally believed today that strange quark matter,consisting of u-,d- and s quarks is energetically the most favorable state of quark matter.', 'astro-ph-0006226-1-37-1': 'Witten [27] suggested that there are two ways of formation of the strange matter: the quark-hadron phase transition in the early universe and conversion of neutron stars into strange ones at ultrahigh densities.In the theories of strong interactions quark bag models suppose that the breaking of physical vacuum takes place inside hadrons.', 'astro-ph-0006226-1-37-2': 'As a result the vacuum energy densities inside and outside a hadron become essentially different and the vacuum pressure on a bag wall equilibrates the pressure of quarks thus stabilizing the system.', 'astro-ph-0006226-1-38-0': 'If the hypothesis of the quark matter is true, then some of neutrons stars could actually be strange stars, built entirely of strange matter [28],[29].', 'astro-ph-0006226-1-38-1': 'Caldwell and Friedman [30] have presented arguments against the existence of strange stars.', 'astro-ph-0006226-1-38-2': 'For a recent review of strange star properties see [31].', 'astro-ph-0006226-1-39-0': 'There are several proposed mechanisms for the formation of quark stars after galaxy formation.', 'astro-ph-0006226-1-39-1': 'Strange stars are expected to form during the collapse of the core of a massive star after the supernova explosion as a result of a first or second order phase transition, resulting in deconfined quark matter [32].', 'astro-ph-0006226-1-39-2': 'Another possibility for strange star formation is that some rapidly spinning neutron stars in low-mass X-ray binaries (LXMBs) can accrete sufficient mass to undergo a phase transition to become strange stars [33].', 'astro-ph-0006226-1-39-3': 'In this scenario it is supposed that at the beginning of accretion the mass of the neutron star is 1.4.', 'astro-ph-0006226-1-39-4': 'It has been shown [34] that the amounts of matter accreted by 18 millisecond pulsars in binary systems exceed [MATH].', 'astro-ph-0006226-1-39-5': 'Hence some of the millisecond pulsars may be strange stars.Strange stars have also been proposed as sources of unusual astrophysical phenomena, e.g. soft [MATH]-ray repeaters [35],pulsating X-ray burster [36], cosmological [MATH]-ray bursts [35], [37-38] etc.The mechanism of the phase transition from neutron to quark stars in low LXMBs also results in the excitation of stellar radial oscillations that can be damped by gravitational wave radiation instead of internal viscosity [39] .', 'astro-ph-0006226-1-39-6': 'The discovery of kHz quasi-periodic oscillation in LXMBs [40] implies that the compact stellar object must have very soft equation of state, which is consistent with that of strange stars [41-42].', 'astro-ph-0006226-1-40-0': 'Assuming that interactions of quarks and gluons are sufficiently small the energy density [MATH] and pressure [MATH] of a quark-gluon plasma at temperature [MATH] and chemical potential [MATH] (the subscript f denotes the various quark flavors u,d,s etc.) can be calculated by thermal theory.', 'astro-ph-0006226-1-40-1': 'Neglecting quark masses in first order perturbation theory and supposing that quarks are confined to the bag volume (in the case of a bare strange star, the boundary of the bag coincides with stellar surface), the equation of state is [EQUATION] where [MATH] is the difference between the energy density of the perturbative and non-perturbative QCD vacuum (the bag constant).', 'astro-ph-0006226-1-40-2': 'Equation ([REF]) is essentially the equation of state of a gas of massive particles with corrections due to the QCD trace anomaly and perturbative interactions.These are always negative , reducing the energy density at given temperature by about a factor two [43].', 'astro-ph-0006226-1-40-3': "In the limit [MATH] (at the star's surface) we have [MATH].", 'astro-ph-0006226-1-40-4': 'The equation of state ([REF]) does not depend upon quark flavor number,hence it will be correct either for strange quark matter ([MATH]) or for normal quark matter ([MATH]).', 'astro-ph-0006226-1-40-5': 'For any intermediate values of [MATH] the state equation ([REF]) gives the pressure with error less than 4% [31].', 'astro-ph-0006226-1-40-6': 'Thus the equation of state of strange matter is mainly determined by the vacuum energy density [MATH].', 'astro-ph-0006226-1-41-0': 'The bag model equation of state ([REF]) has been the basis for the study of most of the static relativistic models of strange stars [27],[29].', 'astro-ph-0006226-1-41-1': 'Based on the numerical integration of the mass continuity and hydrostatic equilibrium TOV (Tolman-Oppenheimer-Volkoff) equations for different values of the bag constant these authors obtained a complete description of static strange stars.', 'astro-ph-0006226-1-41-2': 'Using numerical methods Witten [27] and Haensel et al. [29] obtained the maximum gravitational mass [MATH], the maximum baryon mass [MATH]-the total baryon number of the stellar configuration) and the maximum radius [MATH] of the strange star ,as a function of the bag constant, in the form [27],[29],[9]: [EQUATION] where [MATH].', 'astro-ph-0006226-1-42-0': 'Colpi and Miller [44] and Glendenning and Weber [45] have investigated the rotational properties of strange stars in the slow rotation approximation.', 'astro-ph-0006226-1-42-1': 'As far as rotational deformations are concerned, there are a number of detailed differences between the strange star models and standard neutron stars.', 'astro-ph-0006226-1-42-2': 'Exact numerical calculations of rapidly rotating strange stars were done by Lattimer et al. [10], Gourgoulhon et al.[9] (by using a multi-domain spectral method that enable to treat exactly the density discontinuity at the surface of strange stars) and by Stergioulas,Kluzniak and Bulik [42].', 'astro-ph-0006226-1-42-3': 'Rotation increases maximum allowable mass of strange stars and the equatorial radius of the maximum mass configuration.', 'astro-ph-0006226-1-42-4': 'Gourgoulhon et al. [9] obtained for the maximum mass and radius of quark stars the following two exact formulae [EQUATION]', 'astro-ph-0006226-1-42-5': 'In the present section we shall derive, by using the formalism presented in Section I, analytic mass and radius formulae for general relativistic static and rotating equilibrium strange matter configurations described by the bag model equation of state ([REF]).', 'astro-ph-0006226-1-42-6': 'We shall begin with the study of the static strange star, but presenting and alternative and physically more involved discussion of this case.', 'astro-ph-0006226-1-43-0': 'The changes caused by the general theory of relativity in the conditions of thermal equilibrium,taking into account the gravitational field of the body,are of fundamental importance.In a constant gravitational field we must distinguish the conserved energy [MATH] of any small part of the stellar object from the energy [MATH] measured by an observer situated at a given point.', 'astro-ph-0006226-1-43-1': 'These two quantities are related by [MATH] [46], where [MATH] is the time component of the metric tensor.', 'astro-ph-0006226-1-43-2': 'A similar change occurs in the condition of the constancy of the chemical potential throughout the star.', 'astro-ph-0006226-1-43-3': 'The chemical potential is defined as the derivative of the energy with respect to the number of particles [MATH], [MATH].', 'astro-ph-0006226-1-43-4': 'Since this number is a constant for the stellar object, [MATH] ,for the chemical potential measured at any point inside the gravitating body we have the relation [46]: [EQUATION]', 'astro-ph-0006226-1-43-5': 'A similar relation also holds for the temperature [MATH], [MATH], since we suppose that the strange star is in thermal equilibrium [46].', 'astro-ph-0006226-1-43-6': 'Consequently, [MATH] inside the compact object.', 'astro-ph-0006226-1-43-7': 'Hence [MATH].', 'astro-ph-0006226-1-43-8': 'At constant volume (equal to unity) we have [MATH],where [MATH] and [MATH] are the entropy and number of particles in unit volume of the body,respectively.', 'astro-ph-0006226-1-43-9': 'With the use of [MATH] and taking into account that [MATH] we obtain the following equation relating the equilibrium chemical potential to the energy density and pressure of the star [46]: [EQUATION]', 'astro-ph-0006226-1-43-10': 'Consider now a static equilibrium quark matter configuration satisfying the bag model equation of state ([REF]).', 'astro-ph-0006226-1-43-11': 'Let us compare the values of the chemical potential [MATH] at two points:at the center of the star and at the vacuum boundary.', 'astro-ph-0006226-1-43-12': 'From equation ([REF]) we obtain [EQUATION] where the indices [MATH] and [MATH] refer to the center and to surface of quark star respectively.At the vacuum boundary the gravitational field of the strange star is described by the Schwarzschild solution, which gives 17: [EQUATION] where [MATH] and [MATH] are the total mass and radius of the static strange star,respectively.At the center of the star the time component of the metric tensor has a constant value (this also follows from the Bianchi identity [MATH] 17 and we denote [EQUATION]', 'astro-ph-0006226-1-43-13': 'From a physical point of view [MATH] can be related via the relation [MATH] to the redshift [MATH] of a photon emitted from the center of the quark star.', 'astro-ph-0006226-1-43-14': 'For a given static strange matter configuration the value of [MATH] depends only on the central density of the quark star [MATH] and on the bag constant.', 'astro-ph-0006226-1-43-15': 'Therefore from equation ([REF]) we obtain [EQUATION]', 'astro-ph-0006226-1-43-16': 'With the use of the bag model equation of state ([REF]) we can integrate equation ([REF]) to obtain [EQUATION]', 'astro-ph-0006226-1-43-17': 'The integration constant [MATH] can be determined by calculating the chemical potential at the center of the quark star.', 'astro-ph-0006226-1-43-18': 'Hence we obtain [EQUATION]', 'astro-ph-0006226-1-43-19': 'The variation of the chemical potential inside the quark star can be represented as [EQUATION]', 'astro-ph-0006226-1-43-20': 'At the surface of the star [MATH].', 'astro-ph-0006226-1-43-21': 'Therefore from equation ([REF]) it immediately follows that [EQUATION]', 'astro-ph-0006226-1-43-22': 'In order to simplify notation we shall introduce a dimensionless parameter [MATH], so that [MATH].', 'astro-ph-0006226-1-43-23': 'By eliminating [MATH] from equations ([REF]) and ([REF]) we obtain the following exact formula for the mass-radius ratio of a strange star: [EQUATION]', 'astro-ph-0006226-1-43-24': 'For a given equation of state the mass-radius ratio of the star depends on the values of the metric tensor component at the center of the star,[MATH] , only.A possible representation for the function giving the values of [MATH] at the center of the quark star is in the form of a power series [MATH],with [MATH] constants.', 'astro-ph-0006226-1-44-0': 'As applied on the star surface the mass continuity equation leads to a rough approximation of the quark star mass of the form [MATH].', 'astro-ph-0006226-1-44-1': 'A mass-radius relation of this form could also describe zero pressure quark matter,with [MATH],[MATH] and [MATH] .', 'astro-ph-0006226-1-44-2': 'But for densities greater than [MATH] the effects determined by the large central density become important.', 'astro-ph-0006226-1-44-3': 'Hence for strange quark stars we propose the following mass-radius relation: [EQUATION] with [MATH] a function describing the variation in the quark star mass due to the increase of the central density.', 'astro-ph-0006226-1-45-0': 'The exact form and the values of the functions [MATH] and [MATH] can be determined only by numerically integrating the gravitational field equations.By fitting the numerical data given in [47] for the mass and radius of the strange star with the expressions ([REF]) and ([REF]) we obtain the following representations for these functions (in the present paper we shall consider [MATH]): [EQUATION]', 'astro-ph-0006226-1-45-1': 'The numerical constants in equations ([REF]) and ([REF]) depend on [MATH] because the numerical data have been calculated at a given [MATH].', 'astro-ph-0006226-1-45-2': 'For the polynomial fittings ([REF]) and ([REF]) the correlation coefficient [MATH] and the probability [MATH].', 'astro-ph-0006226-1-45-3': 'Therefore for a given value of the bag constant [MATH] we obtain the following exact representations for the radius and mass of the static strange matter configuration obeying the MIT bag model equation of state: [EQUATION]', 'astro-ph-0006226-1-45-4': 'The variations of the radius and mass for a strange star ([MATH] ) as a function of the parameter [MATH] are represented in Figures 1 and 2.', 'astro-ph-0006226-1-45-5': 'For the sake of comparison we have also presented the data obtained by numerically integrating the TOV and hydrostatic equilibrium equations 47.U', 'astro-ph-0006226-1-45-6': 'Using ([REF])-([REF]) we can reproduce the values of the mass and radius of the quark star obtained by numerical integration with an error smaller than 1.', 'astro-ph-0006226-1-45-7': 'The maximum radius of the strange star is obtained from the condition [MATH].T', 'astro-ph-0006226-1-45-8': 'The corresponding algebraic equation has the solution [MATH] (this value depends of course on the value of [MATH]),giving the value of the ratio of the central pressure and bag constant for the maximum allowable radius [MATH] of the static strange star.This can be expressed as [EQUATION] and its numerical value for [MATH] is [MATH].F', 'astro-ph-0006226-1-45-9': 'From the condition [MATH] it follows that [MATH] and the maximum mass of the static quark star is given by [EQUATION]', 'astro-ph-0006226-1-45-10': 'From equation ([REF]) and for the chosen value of the bag constant,we obtain a value of [MATH].T', 'astro-ph-0006226-1-45-11': 'These results are in good agreement with the previously proposed (Witten [27],Haensel,Zdunik and Schaeffer [29]) maximum radius and mass values,given by equations ([REF]) (from equations ([REF]) and for [MATH] we obtain [MATH]).F', 'astro-ph-0006226-1-45-12': 'For values of [MATH] static quark star models would be unstable to radial perturbations.', 'astro-ph-0006226-1-46-0': 'As an application of the mass and radius formulae obtained for the static strange stars we shall derive an explicit expression for the total energy of the quark star.', 'astro-ph-0006226-1-46-1': 'The total energy (including the gravitational field contribution) inside an equipotential surface [MATH] can be defined, according to Lynden-Bell and Katz 48 and Gron and Johannesen 49 to be [EQUATION] where [MATH] is a Killing vector field of time translation, [MATH] its value at [MATH] and [MATH] is the jump across the shell of the trace of the extrinsic curvature of [MATH], considered as embedded in 2-space [MATH].', 'astro-ph-0006226-1-46-2': '[MATH] and [MATH] are the energy of the matter and of the gravitational field,respectively.', 'astro-ph-0006226-1-46-3': 'This definition is manifestly coordinate invariant.', 'astro-ph-0006226-1-46-4': 'In the case of the static strange star with the use of equation ([REF])and ([REF]) we obtain for the total energy (also including the gravitational contribution) the following exact expression: [EQUATION] where [MATH] is the total energy of the quark matter.', 'astro-ph-0006226-1-47-0': 'The variation of the total energy of the strange star as a function of the parameter [MATH] is represented in Figure 3.', 'astro-ph-0006226-1-48-0': 'The minimum value of the total-matter plus gravitational-energy of the strange matter configuration is obtained for [MATH].', 'astro-ph-0006226-1-48-1': 'The most stable static stellar configuration made of strange matter is given by quark stars with radius [MATH] and with mass [MATH], corresponding to values of the central density of the order of [MATH].', 'astro-ph-0006226-1-49-0': 'We shall consider now the study of the rotating strange star configurations.From equation ([REF]) and equations ([REF]) and ([REF]) it follows that the radius of the rotating strange star can be expressed, as a function of the central density and angular velocity only,in the following form: [EQUATION]', 'astro-ph-0006226-1-49-1': 'We shall compare our results obtained with the use of equations ([REF])-([REF]) and ([REF])-([REF]) with the results provided by Stergioulas,Kluzniak and Bulik 42 and obtained by numerically integrating the gravitational field equations for maximally rotating ("Keplerien") models of strange stars.The results of Stergioulas et al.42 are also in very good agreement with the results of the exact numerical models of rotating strange stars built of self bounded quark matter of Gourgoulhon et al. 9, the difference between these two works being smaller than [MATH].', 'astro-ph-0006226-1-49-2': 'In order to improve the accuracy of the expressions ([REF])-([REF]) we shall consider that the free parameters [MATH] and [MATH] are not constants, but some angular velocity dependent functions given by [EQUATION]', 'astro-ph-0006226-1-49-3': 'Equations ([REF])-([REF]) take into account the variation of the central density of the maximally rotating strange star due to the increase of the angular velocity.', 'astro-ph-0006226-1-50-0': 'In Fig.4 we have represented the variation of the radius of the strange stars,given by equation ([REF]) together with the angular velocity dependent functions [MATH] and [MATH] and the values given in Stergioulas et al.42, calculated for the same values of the central density and angular velocity.The mean of the difference between these two sets of values is smaller than [MATH].', 'astro-ph-0006226-1-50-1': 'The variation of the radius of the maximally rotating strange star as a function of both central density and angular velocity is represented in Fig.5.', 'astro-ph-0006226-1-50-2': 'Figs.6 and 7 present the variation of the mass of the rapidly rotating strange star as a function of [MATH] the angular velocity [MATH] and of [MATH] and central density, respectively.', 'astro-ph-0006226-1-51-0': 'Hence the basic rotational parameters of maximally rotating strange star can be represented in terms of the static configuration and of the angular velocity only.', 'astro-ph-0006226-1-52-0': '# DISCUSSIONS AND FINAL REMARKS', 'astro-ph-0006226-1-53-0': 'In the present paper we have suggested the possibility of the existence of a universal pattern expressing the basic properties of rotating compact object as simple functions of the parameters of the static object and of the angular velocity only.', 'astro-ph-0006226-1-53-1': 'We have obtained exact formulae which give the dependence of the radius and mass of the static and rotating stars on the central density of the stellar object and of its angular velocity.In the static case this is made possible due to the constancy of the chemical potential.', 'astro-ph-0006226-1-53-2': 'The two unknown functions involved in the model must be obtained by fitting the exact formulae with data obtained from the numerical integration of the structure equations of the neutron or quark star.', 'astro-ph-0006226-1-53-3': 'The resulting analytical expressions can reproduce the radius and mass of the strange star with an error smaller than [MATH] and they also provide a simple way to obtain the maximum mass and radius of the static configuration.', 'astro-ph-0006226-1-54-0': 'In the rotating case,with the use of the hydrostatic equilibrium condition,which is the consequence of the Bianchi identities, we have also obtained exact mass-radius relations,depending on three functions describing the effect of rotation on star structure.These relations are exact in the sense that they have been obtained without any special assumptions.By assuming some appropriate forms for the unknown functions we have obtained a general description of the mass and radius of the rotating neutron or strange stars, which generally and for a broad class of equations of state can reproduce the values obtained by numerical integration of the gravitational field equations with a mean error of around [MATH].', 'astro-ph-0006226-1-54-1': 'The expressions of the unknown parameters have been chosen following a close analogy with the static case,whose relevance for the study of rotating general relativistic configurations seems to be more important than previously believed.These functions also incorporate some other general relativistic effects not explicitly taken into account,like the variation of the moment of inertia of the star with the angular velocity.', 'astro-ph-0006226-1-54-2': 'Ravenhall and Pethick [19] have presented a formula valid for a broad range of realistic equations of state of dense matter expressing the moment of inertia in terms of stellar mass and radius.', 'astro-ph-0006226-1-54-3': 'We have not used these results,obtained in the slow rotation limit because,at least in the case of strange stars, the Newtonian expression of the moment of inertia also leads to a quite accurate physical description of the rotating objects.As an application of the obtained formulae we had given a derivation of the "empirical" formula relating the maximum angular velocity to the mass and radius of the static maximal stellar configurations.', 'astro-ph-0006226-1-55-0': 'The possibility of obtaining the basic parameters of general relativistic rotating objects in terms of static parameters could lead to a major computational simplification in the study of rotation.The relation between the presented formalism and the Einstein gravitational field equations will be the subject of a future publication.'}

{'astro-ph-0006226-2-0-0': 'We propose an approximate description of basic parameters (radius, mass and oblateness) of general relativistic compact rotating objects in terms of the parameters of the static configuration and of the angular velocity only.', 'astro-ph-0006226-2-0-1': 'The representation in terms of static properties is derived using the condition of stationary equilibrium together with some phenomenological assumptions.', 'astro-ph-0006226-2-0-2': 'The predicted radius and mass of rotating neutron star (described by some realistic equations of state) and strange star (described by the bag model equation of state) are compared with data obtained by numerical integration of gravitational field equation.The obtained formulae also allow a simple derivation of the "empirical" equation relating the maximum rotation frequency [MATH] of uniformly rotating star models to the mass and radius of the maximum allowable mass configuration of the non-rotating model.', 'astro-ph-0006226-2-1-0': 'Pacs Numbers:', 'astro-ph-0006226-2-2-0': '# INTRODUCTION', 'astro-ph-0006226-2-3-0': 'Rotation is a basic physical property of astrophysical objects.', 'astro-ph-0006226-2-3-1': 'Study of rotational properties of stars can lead to some restrictions imposed on the nuclear equation of state of dense matter at densities larger than nuclear density.Oscillations of rapidly rotating stars can become unstable hence producing detectable gravitational wave emissions.', 'astro-ph-0006226-2-4-0': 'In the past decades there were numerous attempts to construct analytic models for rotating perfect fluid bodies in general relativity.', 'astro-ph-0006226-2-4-1': 'But obtaining an exact interior solution for a rotating body proved to be a formidable task.The first step in this direction was the fundamental result of Kerr [1] who obtained the solution for the vacuum domain, outside the rotating star.', 'astro-ph-0006226-2-4-2': 'It took no less than three decades of investigations to obtain the first, highly idealized model of a general relativistic, thin rotating disk of dust, by Neugebauer and Meinel [2].', 'astro-ph-0006226-2-4-3': 'Various schemes have been developed for obtaining stationary and axisymmetric perfect fluid solutions of the gravitational equations like Petrov type D, local rotational symmetry, fluid kinematics, non-trivial Killing tensor, vanishing Simon tensor, electric magnetic Weyl curvature, lagrangian or static- stationary symmetry, geodesic eigenrays etc. (for a recent review of rotating perfect fluid models in general relativity see [3]).', 'astro-ph-0006226-2-5-0': 'If in the case of rapidly rotating stellar configurations there are still many unsolved problems, a remarkable progress has been made in the study of slowly and rigidly rotating perfect fluid configurations.', 'astro-ph-0006226-2-5-1': 'By casting the metric in the form [EQUATION]', 'astro-ph-0006226-2-5-2': 'Hartle [4] obtained a formalism that proved to be very useful in many investigations of the rotational properties of stars.', 'astro-ph-0006226-2-5-3': 'However, this model, which considers only first order corrections in [MATH], can not be used to compute models of rapidly rotating relativistic stars with sufficient accuracy.', 'astro-ph-0006226-2-6-0': 'On the other hand there has been recently a considerable advance in the numerical understanding of rotating stars.', 'astro-ph-0006226-2-6-1': 'Several high precision numerical codes are now avalaible and it has been shown that they agree with each other to remarkable accuracy (see [5] for a review of recent developments in numerical study of rotatation, nonaxisymmetric oscillations and instabilities of general relativistic stars).', 'astro-ph-0006226-2-7-0': 'The non-sphericity of rapidly rotating stationary stellar configurations and the complicated character of the interplay of the effects of rotation and of those of general relativity seem not to permit a simple universal description of rotating compact objects.', 'astro-ph-0006226-2-7-1': 'However, Haensel and Zdunik [6] and Friedman, Ipser and Parker [7] have found a simple relation connecting the maximum rotation frequency [MATH] with the maximum mass [MATH] and radius [MATH] of the static configuration: [EQUATION] with [MATH] a constant which does not depend on the equation of state of the dense matter.', 'astro-ph-0006226-2-7-2': 'The value of the constant [MATH] has been obtained by fitting equation ([REF]) with data obtained by numerically integrating the gravitational field equations.', 'astro-ph-0006226-2-7-3': 'It is given by [MATH] [7], [MATH] [8] or by [MATH] [9].', 'astro-ph-0006226-2-7-4': 'The empirical relation ([REF]) has been checked for many realistic equations of state for neutron stars [8], [10]-[12] and for the case of strange stars where the empirical formula also holds with a very good precision, the relative deviations do not exceeding 2% [9].', 'astro-ph-0006226-2-7-5': 'Equation ([REF]), obtained on the basis of analyzing numerical solutions of the gravitational field equations, provides an enormous simplification of the problem of dynamical effects of rotation because the solutions of the complicate general relativistic equations for a rotating star can be replaced with the solutions of the much simpler TOV equation.', 'astro-ph-0006226-2-7-6': 'Some attempts to explain this empirical relation were not concluded with satisfactory result.', 'astro-ph-0006226-2-7-7': 'Weber and Glendenning [13], [14] used numerical models of slowly rotating relativistic stars to show that the formula still hold, but with [MATH].', 'astro-ph-0006226-2-7-8': 'Also in the slow rotation limit Glendenning and Weber [15] derived a formula relating [MATH] to [MATH], in terms of the mass, equatorial radius, moment of inertia, angular momentum and quadrupole moment of the maximally rotating configuration only.', 'astro-ph-0006226-2-7-9': 'But it is not clear how the formula ([REF]) follows from their results.Up to now a clear physical understanding of this relation is still missing.', 'astro-ph-0006226-2-8-0': 'On the other hand a universal relation between the maximum mass and radius of non-rotating neutron star configuration and the mass [MATH] and radius [MATH] of the configuration rotating with [MATH] of the form [EQUATION] has also been found [12].', 'astro-ph-0006226-2-8-1': 'In ([REF]) [MATH] and [MATH] are specific equation of state dependent constants, whose values have been calculated, for a broad set of realistic EOS, in [12].', 'astro-ph-0006226-2-8-2': 'Their mean values are [MATH] and [MATH] [12].', 'astro-ph-0006226-2-8-3': 'The empirical constant [MATH] can be obtained, within an approximation better than [MATH] , from the formula [12] [EQUATION]', 'astro-ph-0006226-2-8-4': 'The general validity of ([REF]) and ([REF]) suggests the possibility that all basic physical parameters of general relativistic rotating stellar objects (like mass and radius) can be somehow related to the similar parameters of the static configuration.', 'astro-ph-0006226-2-8-5': 'It is the purpose of the present paper to propose a general description of basic physical parameters (mass, radius and oblateness) of compact general relativistic objects in terms of the physical properties of the static configuration and of the angular velocity only.', 'astro-ph-0006226-2-8-6': 'To obtain the representation in terms of static physical parameters we use only the general relativistic conditions of equilibrium for static and rotating stars and some phenomenological assumptions.The resulted approximate mass and radius formulae are compared with data obtained from numerical integration of gravitational field equations in case of neutron stars described by realistic equations of state and strange stars described by the bag model equation of state.', 'astro-ph-0006226-2-9-0': 'The present paper is organized as follows.The general formalism allowing to obtain mass and radius formulae for rotating general relativistic stars is presented in Section 2.', 'astro-ph-0006226-2-9-1': 'In Section 3 we apply our results to the case of neutron stars described by realistic equations of state.', 'astro-ph-0006226-2-9-2': 'In Section 4 we consider the case of strange stars.', 'astro-ph-0006226-2-9-3': 'We discuss and conclude our results in Section 5.', 'astro-ph-0006226-2-10-0': '# THE GENERAL FORMALISM', 'astro-ph-0006226-2-11-0': 'For a static equilibrium stellar type configuration, with interior described by the metric [EQUATION] the condition of the hydrostatic equilibrium, which follows from the Bianchi identities, can be written as [EQUATION] with [MATH] and [MATH] the energy density and the pressure of the matter respectively ( in the present paper we shall use units so that [MATH]).', 'astro-ph-0006226-2-12-0': 'We shall also make the assumption that the matter EOS is a one parameter dependent function, [EQUATION] with [MATH] the proper baryon density.', 'astro-ph-0006226-2-12-1': 'For such an equation of state the heat function defined by [EQUATION] is a regular function of [MATH].', 'astro-ph-0006226-2-12-2': 'It can always be written in the form [16] [EQUATION] where [EQUATION] is the enthalpy per baryon.', 'astro-ph-0006226-2-13-0': 'As applied at the center and at the surface of the star respectively, the hydrostatic equilibrium condition yields [EQUATION]', 'astro-ph-0006226-2-13-1': 'At the vacuum boundary of the static star the Schwarzschild exterior solution gives the metric and we have [17] [EQUATION] with [MATH] and [MATH] the mass and radius of the static stellar configuration.', 'astro-ph-0006226-2-13-2': 'We denote by [MATH] and [MATH] the baryon density at the center of the star and at the surface, respectively.', 'astro-ph-0006226-2-13-3': 'Therefore from ([REF]) and ([REF]) we obtain the following general and exact expression for the mass-radius ratio of the static star: [EQUATION] where we have denoted [EQUATION] and [EQUATION]', 'astro-ph-0006226-2-13-4': 'For a given equation of state [MATH] is a function of the central density only.', 'astro-ph-0006226-2-13-5': 'From a physical point of view [MATH] can be related via the relation [MATH] to the redshift [MATH] of a photon emitted from the center of the star.', 'astro-ph-0006226-2-14-0': 'For the mass of the star we can obtain another general representation by assuming [EQUATION] where [MATH] is a specific density that can be arbitrarily chosen (for example it is the central density of the minimum mass configuration) and [MATH] is a function describing the effects of the variation of the central density on the basic parameters of the stellar configuration.', 'astro-ph-0006226-2-15-0': 'The two unknown functions [MATH] and [MATH] can be determined by fitting equations ([REF]) and ([REF]) with the exact values of the mass and radius obtained by numerically integrating the TOV equation for a given equation of state.', 'astro-ph-0006226-2-15-1': 'Their knowledge allows us to construct exact mass and radius formulae for sequences of static general relativistic stars having different central densities.', 'astro-ph-0006226-2-15-2': 'From equations ([REF]) and ([REF]) we obtain the radius and the mass of the star in the form [EQUATION]', 'astro-ph-0006226-2-15-3': 'Solving the equations [MATH] and [MATH] for the value of the central density [MATH] will give, with the use of ([REF]) and ([REF]) the maximum values of the radius and mass of the static star, respectively.', 'astro-ph-0006226-2-16-0': 'To describe the interior of the rotating general relativistic star we shall adopt the formalism presented in [16] and [9].', 'astro-ph-0006226-2-16-1': 'Under the hypothesis of stationarity, axial symmetry and purely azimuthal motion a coordinate system [MATH] can be chosen so that inside the star the line element takes the form [EQUATION] where [MATH],[MATH],[MATH] and [MATH] are functions of [MATH] and [MATH] only.', 'astro-ph-0006226-2-16-2': 'As measured by the locally non-rotating observer the fluid 3-velocity is given by [MATH] , where [MATH] is the angular velocity of a fluid element moving in the [MATH]-direction (physically it is the angular velocity as measured by an observer at spatial infinity) [9],[16].', 'astro-ph-0006226-2-17-0': 'From the point of view of our phenomenological approach the most important result is the equation of the stationary motion, which results from the Bianchi identities and which for a rotating perfect fluid reduces to [16] [EQUATION] where [MATH] , [MATH] and [MATH].', 'astro-ph-0006226-2-17-1': 'If [MATH] (case called uniform or rigid rotation) equation ([REF]) can be integrated to give the following fundamental result describing the stationary equilibrium of a rotating general relativistic star: [EQUATION]', 'astro-ph-0006226-2-17-2': 'Equation ([REF]) is just the generalization to the case of rotation of the well-known static Bianchi identity [MATH] [17] we have already used to describe static stellar configurations.', 'astro-ph-0006226-2-18-0': 'We assume that the vacuum boundary of the rotating star is described by the Kerr metric [1] in the Boyer-Lindquist coordinates, which has the form [18]: [EQUATION]', 'astro-ph-0006226-2-18-1': 'In this form the Kerr metric is manifestly axially symmetric and closely resembles the Schwarzschild solution in its standard form.', 'astro-ph-0006226-2-18-2': '[MATH] is the mass of the source and the parameter [MATH] is the ratio between the angular momentum [MATH] and the mass[MATH] of the rotating star.', 'astro-ph-0006226-2-19-0': "Let's apply equation ([REF]) at two points: at the center of the dense core and at the pole of the rotating star.", 'astro-ph-0006226-2-19-1': 'We denote by [MATH] the value of the metric tensor component [MATH] at the center of the star.', 'astro-ph-0006226-2-19-2': 'At the polar point [MATH] and [MATH], where [MATH] is the polar radius of the star.', 'astro-ph-0006226-2-19-3': 'We also have [MATH] .', 'astro-ph-0006226-2-19-4': 'Therefore at this point the line element is given by [EQUATION]', 'astro-ph-0006226-2-19-5': 'Consequently from equations ([REF]) and ([REF]) we obtain the following exact mass-polar radius relation for the rotating general relativistic configuration: [EQUATION]', 'astro-ph-0006226-2-19-6': 'The function [MATH] is identical to that of the static case.', 'astro-ph-0006226-2-19-7': "Let's apply now equation ([REF]) for two points situated in the equatorial plan of the rotating star: at the center of the star and at the equator respectively.", 'astro-ph-0006226-2-19-8': 'At the equator [MATH] and [MATH] is the equatorial radius of the star).', 'astro-ph-0006226-2-19-9': 'For a uniform rotation the rotation angle of the source/observer at the equator is [MATH] .', 'astro-ph-0006226-2-19-10': 'Taking into account these results we obtain the Kerr metric at the equator of the rotating star in the form: [EQUATION]', 'astro-ph-0006226-2-19-11': 'With the use of ([REF]) and ([REF]) we obtain the following mass-equatorial radius relation: [EQUATION] where a new function [EQUATION] has also been defined.', 'astro-ph-0006226-2-20-0': 'From Eqs. ([REF]) and ([REF]) we obtain the ratio [MATH] of the polar and equatorial radius (the oblateness) of the star in the form: [EQUATION]', 'astro-ph-0006226-2-20-1': 'In the case of the static star we have also proposed the alternative Eq. ([REF]) for providing another mass-radius relation.', 'astro-ph-0006226-2-20-2': 'We shall generalize this equation to the rotating case by assuming that the following formula relates the mass of the rotating star to its equatorial radius: [EQUATION] with [MATH] a function describing the combined general relativistic effects of rotation and central density on the mass of the star and with [MATH] the same specific density as used in the static case.', 'astro-ph-0006226-2-20-3': 'For [MATH], [MATH] and Eq. ([REF]) must reduce to the static case equation ([REF]).', 'astro-ph-0006226-2-21-0': 'Equations ([REF]),([REF]) and ([REF]) give a complete and exact description of the mass and radius of the rotating general relativistic star.', 'astro-ph-0006226-2-21-1': 'But unfortunately the present approach, which is basically thermodynamic in its essence, can not predict the exact form and the values of the three unknown functions entering in the formalism.', 'astro-ph-0006226-2-21-2': 'The only thing we can do is to assume, also based on the static case, some empirical forms for the functions [MATH] and to check whether the resulting formulae can give a satisfactory description of rotating star configurations.', 'astro-ph-0006226-2-21-3': 'Therefore in the following we shall use the following five approximations:', 'astro-ph-0006226-2-22-0': 'i) As a first approximation we shall define the moment of inertia [MATH] of the rotating compact relativistic object via the Newtonian expression [EQUATION]', 'astro-ph-0006226-2-22-1': 'In fact over a wide range of [MATH] and [MATH] the corrections added by general relativistic effects to the moment of inertia can be approximated by [MATH] [15], but we shall not use this representation.', 'astro-ph-0006226-2-22-2': 'By adopting the Newtonian formula we obtain [EQUATION] ii) We assume that the function [MATH], describing the metric tensor component [MATH] at the center of the rotating star is given by [EQUATION] with [MATH] an EOS dependent function given by [EQUATION] and [MATH] a non-negative constant.', 'astro-ph-0006226-2-22-3': '[MATH] is also the function corresponding to the static case.', 'astro-ph-0006226-2-23-0': 'iii) We assume that inside the rotating star [MATH] is independent of the angular velocity [MATH] of the rotating compact object and can be represented by the function corresponding to the static case: [EQUATION] iv) We suppose that [EQUATION] [MATH] is again the function corresponding to the static case.', 'astro-ph-0006226-2-24-0': 'With these four phenomenological assumptions Eqs. ([REF])-([REF]) lead to the following representation of the basic physical parameters of the rotating general relativistic star: [EQUATION]', 'astro-ph-0006226-2-24-1': 'We shall also suppose that [MATH] is a universal constants and we shall choose [MATH].', 'astro-ph-0006226-2-24-2': 'In this formulation of the general relativistic problem of the rotation the oblateness parameter [MATH] of the star is given by the roots of the third order algebraic equation ([REF]).', 'astro-ph-0006226-2-25-0': 'The equatorial radius is defined only for values of the angular velocity satisfying the condition [MATH].', 'astro-ph-0006226-2-25-1': 'Therefore for the maximum admissible constant angular velocity of the maximally rotating star in uniform rotation we obtain the relation [EQUATION] where [EQUATION] for [MATH].', 'astro-ph-0006226-2-26-0': 'Equation ([REF]) is very similar to the "empirical" formula discussed in [1]-[7].', 'astro-ph-0006226-2-26-1': 'The coefficient of proportionality in ([REF]) is independent of the equation of state of the dense matter, but its numerical value does not fit the calculated value.On the other hand for [MATH] the radius of the star tends to infinity.', 'astro-ph-0006226-2-27-0': 'An alternative expression can be obtained by imposing the restriction [EQUATION] where [MATH] is the maxium allowable equatorial speed of the star and we have also used the Newtonian force balance equation between the gravitational and centrifugal force.', 'astro-ph-0006226-2-27-1': 'Therefore we obtain [EQUATION]', 'astro-ph-0006226-2-27-2': 'It is interesting to note that the values of [MATH] are in a narrow range of (0.467, 0.667) [8], [12].', 'astro-ph-0006226-2-27-3': 'Taking for [MATH] a mean value of 0.58 [12] it follows that [MATH] and we find [EQUATION] with [MATH], value which coincides with that proposed in [8] and differs only within [MATH] from the value [MATH] obtained in [9].', 'astro-ph-0006226-2-28-0': 'By taking for twice of the ratio of the maximum static mass and radius a mean value of 0.58 [12] and considering [MATH], we obtain [EQUATION] with [MATH],value wich coincides with the value obtained in that differs within [MATH] from the value [MATH] obtained in [20].', 'astro-ph-0006226-2-29-0': 'The maximum radius of the maximally rotating configuration can be obtained from [MATH], and is given by [EQUATION] where [EQUATION]', 'astro-ph-0006226-2-29-1': 'The mass [MATH] of the maximal radius rotating neutron star follows from ([REF]) and is given by [EQUATION] where [EQUATION]', 'astro-ph-0006226-2-29-2': 'The maximum mass [MATH] of the rotating star can be obtained from the equation [EQUATION] leading to [EQUATION] where [MATH] is a dimensionless EOS dependent function.', 'astro-ph-0006226-2-29-3': 'Therefore for the maximum mass of the maximally rotating configuration we obtain [EQUATION]', 'astro-ph-0006226-2-29-4': 'Equations ([REF]) and ([REF]) show the existence of a proportionality relation between maximum mass and radius of the rotating and non-rotating configuration, respectively, as has already been suggested, on an empirical basis,in [12].', 'astro-ph-0006226-2-30-0': 'An investigation of 12 EOS performed in [12] for 12 realistic EOS of nuclear matter led to a (mean) value of [MATH] , while the same calculation performed by us with the use of data presented in [20] for other 14 different EOSs gives [MATH] , leading to a mean value of [MATH] for the 26 considered EOSs.', 'astro-ph-0006226-2-31-0': 'Eqs. ([REF])-([REF]) lead, for [MATH] and [MATH] to values of [MATH] and [MATH].', 'astro-ph-0006226-2-32-0': 'Therefore we may conclude that the proportionality between the maximum mass of the rotating star and the maximum mass of the static configuration is universal, being with a very good approximation independent of the equation of state of dense matter.', 'astro-ph-0006226-2-32-1': 'On the other hand the relation between the radius of the rotating and non-rotating configuration is EOS dependent,the coefficient of proportionality slightly decreasing with the increase of the mass-radius ratio of the static star.', 'astro-ph-0006226-2-33-0': '# APPLICATIONS TO NEUTRON STARS', 'astro-ph-0006226-2-34-0': "The correct mathematical and physical modelling of millisecond pulsars can be done only in the framework of general relativistic equilibrium models for rapidly rotating neutron stars.Such models are solutions of the Einstein's equations for the axisymmetric stationary gravitational field and they must be constructed numerically.", 'astro-ph-0006226-2-34-1': 'Recently several independent numerical codes have been developed by different groups of researchers and have been used to obtain rapidly rotating neutron star models based on a variety of realistic equations of state.Hence a large amount of numerical data is now available.', 'astro-ph-0006226-2-35-0': 'In the present Section we shall apply the results of the phenomenological formalism presented in the previous Section to the case of neutron stars described by realistic equations of state.', 'astro-ph-0006226-2-35-1': 'The data are selected from the paper by Cook, Shapiro and Teukolsky [20],who constructed general relativistic rotating star sequences for 14 nuclear matter equations of state.', 'astro-ph-0006226-2-35-2': 'Detailed data are presented only for 5 equations of state.', 'astro-ph-0006226-2-35-3': 'For the sake of comparison we have used the equations of state denoted A [21], AU [22], FPS [23] and L [24].', 'astro-ph-0006226-2-35-4': 'At low densities all these equations of state employ the Feynman,Metropolis and Teller [25] EOS and then join onto the Baym,Bethe and Pethick [26] EOS up to neutron drip.The equations of state are given in a tabular form and small changes in the way the tabulated equation of state is constructed do have a small effect on the resulting neutron star model.', 'astro-ph-0006226-2-36-0': 'In Tables 1-4 we present the comparison of the basic physical parameters of rotating stars obtained, for these four equations of state, with the help of Eqs. ([REF])-([REF]) and by numerically integrating the gravitational field equations.', 'astro-ph-0006226-2-37-0': 'For [MATH] sequences described by EOS A and FPS the maximum error of our prediction is around [MATH].', 'astro-ph-0006226-2-37-1': 'For the maximum mass normal sequences of EOS AU and L the maximum error in the predicted value of the mass does not exceed [MATH] but for large angular speeds it is around [MATH] for the radius of the rotating compact object.', 'astro-ph-0006226-2-38-0': '# APPLICATIONS TO STRANGE STARS', 'astro-ph-0006226-2-39-0': 'It is generally believed today that strange quark matter,consisting of u-,d- and s quarks is energetically the most favorable state of quark matter.', 'astro-ph-0006226-2-39-1': 'Witten [27] suggested that there are two ways of formation of the strange matter: the quark-hadron phase transition in the early universe and conversion of neutron stars into strange ones at ultrahigh densities.In the theories of strong interactions quark bag models suppose that the breaking of physical vacuum takes place inside hadrons.', 'astro-ph-0006226-2-39-2': 'As a result the vacuum energy densities inside and outside a hadron become essentially different and the vacuum pressure on a bag wall equilibrates the pressure of quarks thus stabilizing the system.', 'astro-ph-0006226-2-40-0': 'If the hypothesis of the quark matter is true, then some of neutrons stars could actually be strange stars, built entirely of strange matter [28],[29].', 'astro-ph-0006226-2-40-1': 'Caldwell and Friedman [30] have presented arguments against the existence of strange stars.', 'astro-ph-0006226-2-40-2': 'For a recent review of strange star properties see [31].', 'astro-ph-0006226-2-41-0': 'There are several proposed mechanisms for the formation of quark stars after galaxy formation.', 'astro-ph-0006226-2-41-1': 'Strange stars are expected to form during the collapse of the core of a massive star after the supernova explosion as a result of a first or second order phase transition, resulting in deconfined quark matter [32].', 'astro-ph-0006226-2-41-2': 'Another possibility for strange star formation is that some rapidly spinning neutron stars in low-mass X-ray binaries (LXMBs) can accrete sufficient mass to undergo a phase transition to become strange stars [33].', 'astro-ph-0006226-2-41-3': 'In this scenario it is supposed that at the beginning of accretion the mass of the neutron star is 1.4.', 'astro-ph-0006226-2-41-4': 'It has been shown [34] that the amounts of matter accreted by 18 millisecond pulsars in binary systems exceed [MATH].', 'astro-ph-0006226-2-41-5': 'Hence some of the millisecond pulsars may be strange stars.Strange stars have also been proposed as sources of unusual astrophysical phenomena, e.g. soft [MATH]-ray repeaters [35],pulsating X-ray burster [36], cosmological [MATH]-ray bursts [35], [37-38] etc.The mechanism of the phase transition from neutron to quark stars in low LXMBs also results in the excitation of stellar radial oscillations that can be damped by gravitational wave radiation instead of internal viscosity [39] .', 'astro-ph-0006226-2-41-6': 'The discovery of kHz quasi-periodic oscillation in LXMBs [40] implies that the compact stellar object must have very soft equation of state, which is consistent with that of strange stars [41-42].', 'astro-ph-0006226-2-42-0': 'Assuming that interactions of quarks and gluons are sufficiently small the energy density [MATH] and pressure [MATH] of a quark-gluon plasma at temperature [MATH] and chemical potential [MATH] (the subscript f denotes the various quark flavors u,d,s etc.) can be calculated by thermal theory.', 'astro-ph-0006226-2-42-1': 'Neglecting quark masses in first order perturbation theory and supposing that quarks are confined to the bag volume (in the case of a bare strange star, the boundary of the bag coincides with stellar surface), the equation of state is [EQUATION] where [MATH] is the difference between the energy density of the perturbative and non-perturbative QCD vacuum (the bag constant).', 'astro-ph-0006226-2-42-2': 'Equation ([REF]) is essentially the equation of state of a gas of massive particles with corrections due to the QCD trace anomaly and perturbative interactions.These are always negative , reducing the energy density at given temperature by about a factor two [43].', 'astro-ph-0006226-2-42-3': "In the limit [MATH] (at the star's surface) we have [MATH].", 'astro-ph-0006226-2-42-4': 'The equation of state ([REF]) does not depend upon quark flavor number,hence it will be correct either for strange quark matter ([MATH]) or for normal quark matter ([MATH]).', 'astro-ph-0006226-2-42-5': 'For any intermediate values of [MATH] the state equation ([REF]) gives the pressure with error less than 4% [31].', 'astro-ph-0006226-2-42-6': 'Thus the equation of state of strange matter is mainly determined by the vacuum energy density [MATH].', 'astro-ph-0006226-2-43-0': 'The bag model equation of state ([REF]) has been the basis for the study of most of the static relativistic models of strange stars [27],[29].', 'astro-ph-0006226-2-43-1': 'Based on the numerical integration of the mass continuity and hydrostatic equilibrium TOV (Tolman-Oppenheimer-Volkoff) equations for different values of the bag constant these authors obtained a complete description of static strange stars.', 'astro-ph-0006226-2-43-2': 'Using numerical methods Witten [27] and Haensel et al. [29] obtained the maximum gravitational mass [MATH], the maximum baryon mass [MATH]-the total baryon number of the stellar configuration) and the maximum radius [MATH] of the strange star ,as a function of the bag constant, in the form [27],[29],[9]: [EQUATION] where [MATH].', 'astro-ph-0006226-2-44-0': 'Colpi and Miller [44] and Glendenning and Weber [45] have investigated the rotational properties of strange stars in the slow rotation approximation.', 'astro-ph-0006226-2-44-1': 'As far as rotational deformations are concerned, there are a number of detailed differences between the strange star models and standard neutron stars.', 'astro-ph-0006226-2-44-2': 'Exact numerical calculations of rapidly rotating strange stars were done by Lattimer et al. [10], Gourgoulhon et al.[9] (by using a multi-domain spectral method that enable to treat exactly the density discontinuity at the surface of strange stars) and by Stergioulas,Kluzniak and Bulik [42].', 'astro-ph-0006226-2-44-3': 'Rotation increases maximum allowable mass of strange stars and the equatorial radius of the maximum mass configuration.', 'astro-ph-0006226-2-44-4': 'Gourgoulhon et al. [9] obtained for the maximum mass and radius of quark stars the following two exact formulae [EQUATION]', 'astro-ph-0006226-2-44-5': 'In the present section we shall derive, by using the formalism presented in Section I, analytic mass and radius formulae for general relativistic static and rotating equilibrium strange matter configurations described by the bag model equation of state ([REF]).', 'astro-ph-0006226-2-44-6': 'We shall begin with the study of the static strange star, but presenting and alternative and physically more involved discussion of this case.', 'astro-ph-0006226-2-45-0': 'The changes caused by the general theory of relativity in the conditions of thermal equilibrium,taking into account the gravitational field of the body,are of fundamental importance.In a constant gravitational field we must distinguish the conserved energy [MATH] of any small part of the stellar object from the energy [MATH] measured by an observer situated at a given point.', 'astro-ph-0006226-2-45-1': 'These two quantities are related by [MATH] [46], where [MATH] is the time component of the metric tensor.', 'astro-ph-0006226-2-45-2': 'A similar change occurs in the condition of the constancy of the chemical potential throughout the star.', 'astro-ph-0006226-2-45-3': 'The chemical potential is defined as the derivative of the energy with respect to the number of particles [MATH], [MATH].', 'astro-ph-0006226-2-45-4': 'Since this number is a constant for the stellar object, [MATH] ,for the chemical potential measured at any point inside the gravitating body we have the relation [46]: [EQUATION]', 'astro-ph-0006226-2-45-5': 'A similar relation also holds for the temperature [MATH], [MATH], since we suppose that the strange star is in thermal equilibrium [46].', 'astro-ph-0006226-2-45-6': 'Consequently, [MATH] inside the compact object.', 'astro-ph-0006226-2-45-7': 'Hence [MATH].', 'astro-ph-0006226-2-45-8': 'At constant volume (equal to unity) we have [MATH],where [MATH] and [MATH] are the entropy and number of particles in unit volume of the body,respectively.', 'astro-ph-0006226-2-45-9': 'With the use of [MATH] and taking into account that [MATH] we obtain the following equation relating the equilibrium chemical potential to the energy density and pressure of the star [46]: [EQUATION]', 'astro-ph-0006226-2-45-10': 'Consider now a static equilibrium quark matter configuration satisfying the bag model equation of state ([REF]).', 'astro-ph-0006226-2-45-11': 'Let us compare the values of the chemical potential [MATH] at two points:at the center of the star and at the vacuum boundary.', 'astro-ph-0006226-2-45-12': 'From equation ([REF]) we obtain [EQUATION] where the indices [MATH] and [MATH] refer to the center and to surface of quark star respectively.At the vacuum boundary the gravitational field of the strange star is described by the Schwarzschild solution, which gives 17: [EQUATION] where [MATH] and [MATH] are the total mass and radius of the static strange star,respectively.At the center of the star the time component of the metric tensor has a constant value (this also follows from the Bianchi identity [MATH] 17 and we denote [EQUATION]', 'astro-ph-0006226-2-45-13': 'From a physical point of view [MATH] can be related via the relation [MATH] to the redshift [MATH] of a photon emitted from the center of the quark star.', 'astro-ph-0006226-2-45-14': 'For a given static strange matter configuration the value of [MATH] depends only on the central density of the quark star [MATH] and on the bag constant.', 'astro-ph-0006226-2-45-15': 'Therefore from equation ([REF]) we obtain [EQUATION]', 'astro-ph-0006226-2-45-16': 'With the use of the bag model equation of state ([REF]) we can integrate equation ([REF]) to obtain [EQUATION]', 'astro-ph-0006226-2-45-17': 'The integration constant [MATH] can be determined by calculating the chemical potential at the center of the quark star.', 'astro-ph-0006226-2-45-18': 'Hence we obtain [EQUATION]', 'astro-ph-0006226-2-45-19': 'The variation of the chemical potential inside the quark star can be represented as [EQUATION]', 'astro-ph-0006226-2-45-20': 'At the surface of the star [MATH].', 'astro-ph-0006226-2-45-21': 'Therefore from equation ([REF]) it immediately follows that [EQUATION]', 'astro-ph-0006226-2-45-22': 'In order to simplify notation we shall introduce a dimensionless parameter [MATH], so that [MATH].', 'astro-ph-0006226-2-45-23': 'By eliminating [MATH] from equations ([REF]) and ([REF]) we obtain the following exact formula for the mass-radius ratio of a strange star: [EQUATION]', 'astro-ph-0006226-2-45-24': 'For a given equation of state the mass-radius ratio of the star depends on the values of the metric tensor component at the center of the star,[MATH] , only.A possible representation for the function giving the values of [MATH] at the center of the quark star is in the form of a power series [MATH],with [MATH] constants.', 'astro-ph-0006226-2-46-0': 'As applied on the star surface the mass continuity equation leads to a rough approximation of the quark star mass of the form [MATH].', 'astro-ph-0006226-2-46-1': 'A mass-radius relation of this form could also describe zero pressure quark matter,with [MATH],[MATH] and [MATH] .', 'astro-ph-0006226-2-46-2': 'But for densities greater than [MATH] the effects determined by the large central density become important.', 'astro-ph-0006226-2-46-3': 'Hence for strange quark stars we propose the following mass-radius relation: [EQUATION] with [MATH] a function describing the variation in the quark star mass due to the increase of the central density.', 'astro-ph-0006226-2-47-0': 'The exact form and the values of the functions [MATH] and [MATH] can be determined only by numerically integrating the gravitational field equations.By fitting the numerical data given in [47] for the mass and radius of the strange star with the expressions ([REF]) and ([REF]) we obtain the following representations for these functions (in the present paper we shall consider [MATH]): [EQUATION]', 'astro-ph-0006226-2-47-1': 'The numerical constants in equations ([REF]) and ([REF]) depend on [MATH] because the numerical data have been calculated at a given [MATH].', 'astro-ph-0006226-2-47-2': 'For the polynomial fittings ([REF]) and ([REF]) the correlation coefficient [MATH] and the probability [MATH].', 'astro-ph-0006226-2-47-3': 'Therefore for a given value of the bag constant [MATH] we obtain the following exact representations for the radius and mass of the static strange matter configuration obeying the MIT bag model equation of state: [EQUATION]', 'astro-ph-0006226-2-47-4': 'The variations of the radius and mass for a strange star ([MATH] ) as a function of the parameter [MATH] are represented in Figures 1 and 2.', 'astro-ph-0006226-2-47-5': 'For the sake of comparison we have also presented the data obtained by numerically integrating the TOV and hydrostatic equilibrium equations 47.U', 'astro-ph-0006226-2-47-6': 'Using ([REF])-([REF]) we can reproduce the values of the mass and radius of the quark star obtained by numerical integration with an error smaller than 1.', 'astro-ph-0006226-2-47-7': 'The maximum radius of the strange star is obtained from the condition [MATH].T', 'astro-ph-0006226-2-47-8': 'The corresponding algebraic equation has the solution [MATH] (this value depends of course on the value of [MATH]),giving the value of the ratio of the central pressure and bag constant for the maximum allowable radius [MATH] of the static strange star.This can be expressed as [EQUATION] and its numerical value for [MATH] is [MATH].F', 'astro-ph-0006226-2-47-9': 'From the condition [MATH] it follows that [MATH] and the maximum mass of the static quark star is given by [EQUATION]', 'astro-ph-0006226-2-47-10': 'From equation ([REF]) and for the chosen value of the bag constant,we obtain a value of [MATH].T', 'astro-ph-0006226-2-47-11': 'These results are in good agreement with the previously proposed (Witten [27],Haensel,Zdunik and Schaeffer [29]) maximum radius and mass values,given by equations ([REF]) (from equations ([REF]) and for [MATH] we obtain [MATH]).F', 'astro-ph-0006226-2-47-12': 'For values of [MATH] static quark star models would be unstable to radial perturbations.', 'astro-ph-0006226-2-48-0': 'As an application of the mass and radius formulae obtained for the static strange stars we shall derive an explicit expression for the total energy of the quark star.', 'astro-ph-0006226-2-48-1': 'The total energy (including the gravitational field contribution) inside an equipotential surface [MATH] can be defined, according to Lynden-Bell and Katz 48 and Gron and Johannesen 49 to be [EQUATION] where [MATH] is a Killing vector field of time translation, [MATH] its value at [MATH] and [MATH] is the jump across the shell of the trace of the extrinsic curvature of [MATH], considered as embedded in 2-space [MATH].', 'astro-ph-0006226-2-48-2': '[MATH] and [MATH] are the energy of the matter and of the gravitational field,respectively.', 'astro-ph-0006226-2-48-3': 'This definition is manifestly coordinate invariant.', 'astro-ph-0006226-2-48-4': 'In the case of the static strange star with the use of equation ([REF])and ([REF]) we obtain for the total energy (also including the gravitational contribution) the following exact expression: [EQUATION] where [MATH] is the total energy of the quark matter.', 'astro-ph-0006226-2-49-0': 'The variation of the total energy of the strange star as a function of the parameter [MATH] is represented in Figure 3.', 'astro-ph-0006226-2-50-0': 'The minimum value of the total-matter plus gravitational-energy of the strange matter configuration is obtained for [MATH].', 'astro-ph-0006226-2-50-1': 'The most stable static stellar configuration made of strange matter is given by quark stars with radius [MATH] and with mass [MATH], corresponding to values of the central density of the order of [MATH].', 'astro-ph-0006226-2-51-0': 'We shall consider now the study of the rotating strange star configurations.', 'astro-ph-0006226-2-51-1': 'We shall compare our results obtained with the use of equations ([REF])-([REF]) and ([REF])-([REF]) with the results provided by Stergioulas,Kluzniak and Bulik 42 and obtained by numerically integrating the gravitational field equations for maximally rotating ("Keplerien") models of strange stars.The results of Stergioulas et al.42 are also in very good agreement with the results of the exact numerical models of rotating strange stars built of self bounded quark matter of Gourgoulhon et al. 9, the difference between these two works being smaller than [MATH].', 'astro-ph-0006226-2-51-2': 'In order to improve the accuracy of the expressions ([REF])-([REF]) we shall consider that the function [MATH] can be expressed in a more general form as [EQUATION] and we will assume that the parameters [MATH] and [MATH] are not constants, but some angular velocity dependent functions given by [EQUATION]', 'astro-ph-0006226-2-51-3': 'Equations ([REF])-([REF]) take into account the variation of the central density of the maximally rotating strange star due to the increase of the angular velocity.', 'astro-ph-0006226-2-52-0': 'In Fig.4 we have represented the variation of the radius of the strange stars,given by equation ([REF]) together with the angular velocity dependent functions [MATH] and [MATH] and the values given in Stergioulas et al.42, calculated for the same values of the central density and angular velocity.The mean of the difference between these two sets of values is smaller than [MATH].', 'astro-ph-0006226-2-52-1': 'The variation of the radius of the maximally rotating strange star as a function of both central density and angular velocity is represented in Fig.5.', 'astro-ph-0006226-2-52-2': 'Figs.6 and 7 present the variation of the mass of the rapidly rotating strange star as a function of [MATH] the angular velocity [MATH] and of [MATH] and central density, respectively.', 'astro-ph-0006226-2-53-0': 'From equation ([REF]) and equations ([REF]) and ([REF]) it follows that the radius of the rotating strange star can be expressed, as a function of the central density and angular velocity only,in the following form: [EQUATION]', 'astro-ph-0006226-2-53-1': 'Hence in this approximation the basic rotational parameters of maximally rotating strange star can be represented in terms of the static configuration and of the angular velocity only.', 'astro-ph-0006226-2-54-0': '# DISCUSSIONS AND FINAL REMARKS', 'astro-ph-0006226-2-55-0': 'In the present paper we have suggested the possibility of the existence of a universal pattern expressing the basic properties of rotating compact object as simple functions of the parameters of the static object and of the angular velocity only.', 'astro-ph-0006226-2-55-1': 'We have obtained exact formulae which give the dependence of the radius and mass of the static and rotating stars on the central density of the stellar object and of its angular velocity.In the static case this is made possible due to the constancy of the chemical potential.', 'astro-ph-0006226-2-55-2': 'The two unknown functions involved in the model must be obtained by fitting the exact formulae with data obtained from the numerical integration of the structure equations of the neutron or quark star.', 'astro-ph-0006226-2-55-3': 'The resulting analytical expressions can reproduce the radius and mass of the strange star with an error smaller than [MATH] and they also provide a simple way to obtain the maximum mass and radius of the static configuration.', 'astro-ph-0006226-2-56-0': 'In the rotating case,with the use of the hydrostatic equilibrium condition,which is the consequence of the Bianchi identities, we have also obtained exact mass-radius relations,depending on three functions describing the effect of rotation on star structure.These relations are exact in the sense that they have been obtained without any special assumptions.By assuming some appropriate forms for the unknown functions we have obtained a general description of the mass and radius of the rotating neutron or strange stars, which generally and for a broad class of equations of state can reproduce the values obtained by numerical integration of the gravitational field equations with a mean error of around [MATH].', 'astro-ph-0006226-2-56-1': 'The expressions of the unknown parameters have been chosen following a close analogy with the static case,whose relevance for the study of rotating general relativistic configurations seems to be more important than previously believed.These functions also incorporate some other general relativistic effects not explicitly taken into account,like the variation of the moment of inertia of the star with the angular velocity.', 'astro-ph-0006226-2-56-2': 'Ravenhall and Pethick [19] have presented a formula valid for a broad range of realistic equations of state of dense matter expressing the moment of inertia in terms of stellar mass and radius.', 'astro-ph-0006226-2-56-3': 'We have not used these results,obtained in the slow rotation limit because,at least in the case of strange stars, the Newtonian expression of the moment of inertia also leads to a quite accurate physical description of the rotating objects.As an application of the obtained formulae we had given a derivation of the "empirical" formula relating the maximum angular velocity to the mass and radius of the static maximal stellar configurations.', 'astro-ph-0006226-2-57-0': 'The possibility of obtaining the basic parameters of general relativistic rotating objects in terms of static parameters could lead to a major computational simplification in the study of rotation.The relation between the presented formalism and the Einstein gravitational field equations will be the subject of a future publication.'}

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'astro-ph-0006226-2-45-22'], ['astro-ph-0006226-1-43-23', 'astro-ph-0006226-2-45-23'], ['astro-ph-0006226-1-43-24', 'astro-ph-0006226-2-45-24'], ['astro-ph-0006226-1-26-0', 'astro-ph-0006226-2-25-0'], ['astro-ph-0006226-1-26-1', 'astro-ph-0006226-2-25-1'], ['astro-ph-0006226-1-12-1', 'astro-ph-0006226-2-12-1'], ['astro-ph-0006226-1-12-2', 'astro-ph-0006226-2-12-2'], ['astro-ph-0006226-1-35-0', 'astro-ph-0006226-2-37-0'], ['astro-ph-0006226-1-35-1', 'astro-ph-0006226-2-37-1'], ['astro-ph-0006226-1-28-1', 'astro-ph-0006226-2-27-1'], ['astro-ph-0006226-1-22-0', 'astro-ph-0006226-2-22-0'], ['astro-ph-0006226-1-22-1', 'astro-ph-0006226-2-22-1'], ['astro-ph-0006226-1-23-1', 'astro-ph-0006226-2-22-3']]

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[]

[['astro-ph-0006226-1-24-0', 'astro-ph-0006226-2-23-0'], ['astro-ph-0006226-1-49-0', 'astro-ph-0006226-2-51-0'], ['astro-ph-0006226-1-30-0', 'astro-ph-0006226-2-30-0'], ['astro-ph-0006226-1-28-0', 'astro-ph-0006226-2-27-0']]

[]

['astro-ph-0006226-1-1-0', 'astro-ph-0006226-1-7-3', 'astro-ph-0006226-1-21-3', 'astro-ph-0006226-1-43-7', 'astro-ph-0006226-2-1-0', 'astro-ph-0006226-2-7-3', 'astro-ph-0006226-2-21-3', 'astro-ph-0006226-2-31-0', 'astro-ph-0006226-2-45-7']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0006226

0812.4692

{'0812.4692-1-0-0': 'RNA polymerase (RNAP) is like a mobile molecular workshop that polymerizes a RNA molecule by adding monomeric subunits one by one, while moving step by step on the DNA template itself.', '0812.4692-1-0-1': 'Here we develop a theoretical model by incorporating their steric interactions and mechanochemical cycles which explicitly captures the cyclical shape changes of each motor.', '0812.4692-1-0-2': 'Using this model, we explain not only the dependence of the average velocity of a RNAP on the externally applied load force, but also predict a nonmotonic variation of the average velocity on external torque.', '0812.4692-1-0-3': 'We also show the effect of steric interactions of the motors on the total rate of RNA synthesis.', '0812.4692-1-0-4': 'In principle, our predictions can be tested by carrying out in-vitro experiments.', '0812.4692-1-1-0': '# Introduction', '0812.4692-1-2-0': 'Molecular motors [CITATION] in living cells are either proteins or macromolecular complexes made of proteins and ribonucleic acids (RNAs).', '0812.4692-1-2-1': 'Like their macroscpic counterparts, these motors perform mechanical work, while translocating on a filamentous track, by converting input energy which is often supplied as chemical energy [CITATION].', '0812.4692-1-2-2': 'In this paper we study a special class of motors, called RNA polymerase (RNAP) which play crucial roles in gene expression [CITATION].', '0812.4692-1-3-0': 'Transcription of a gene encoded in the sequence of nucleotides in a specific segment of a DNA is carried out by RNAP motors which treat the DNA as a template [CITATION].', '0812.4692-1-3-1': 'An RNAP is more like a mobile workshop that performs three functions simultaneously: (i) it decodes the genetic message encoded in the template DNA and selects the appropriate nucleotide, the monomeric subunit of RNA, as dictated by the template, (ii) it catalyzes the addition of the monomeric subunit thus selected to the growing RNA molecule, (iii) it steps forward by one nucleotide on its template without completely destabilizing the ternary complex consisting of the polymerase, the template DNA and the product RNA.', '0812.4692-1-3-2': 'The free energy released by the polymerization of the RNA molecule serves as the input energy for the driving the mechanical movements of the RNAP.', '0812.4692-1-3-3': 'Therefore, these enzymes are also regarded as molecular motors [CITATION].', '0812.4692-1-4-0': 'During the transcription of a gene, the collective movement of the RNAPs on the same DNA track is often referred to as RNAP traffic because of its superficial similarity with vehicular traffic [CITATION].', '0812.4692-1-4-1': 'The beginning and the end of the specific sequence corresponding to a gene are the analogs of the on-ramp and off-ramp of vehicular traffic on highways.', '0812.4692-1-4-2': 'The average number of RNAPs, which complete the synthesis of a RNA molecule per unit time interval can be identified as the flux in RNAP traffic.', '0812.4692-1-4-3': 'Note that flux is the product of the number density and average velocity of the motors.', '0812.4692-1-4-4': 'Thus, flux in RNAP traffic is identical to the average rate of synthesis of the corresponding RNA.', '0812.4692-1-4-5': "Using the terminology of traffic science [CITATION], we'll call the relation between the flux and the number density of the motors as the fundamental diagram.", '0812.4692-1-4-6': 'The fundamental diagram is an important quantitative characteristic of traffic flow.', '0812.4692-1-5-0': 'The dependence of the velocity of the motor on an externally imposed load (opposing) force is called the force-velocity relation which is one of the most important characteristics of a molecular motor.', '0812.4692-1-5-1': 'The force-velocity relation for RNAP motors have been measured by carrying out single molecule experiments [CITATION].', '0812.4692-1-5-2': 'However, to our knowledge, the response of an RNAP motor to an externally applied torque has not been investigated so far.', '0812.4692-1-5-3': 'The effects of steric interactions of the RNAP motors on their dynamics has been studied only in a few experiments [CITATION]; but, none of these addressed the question of the nature of the overall spatio-temporal organization of the RNAP motors in RNAP traffic.', '0812.4692-1-6-0': 'The traffic-like collective dynamics of cytoskeletal molecular motors [CITATION] and that of ribosomes on mRNA tracks [CITATION] have been investigated theoretically in the physics literature.', '0812.4692-1-6-1': 'However, so far, RNAP traffic has received far less attention [CITATION] In this paper we develop a model that captures not only the steric interactions between the RNAPs, but also separately the biochemical reactions catalyzed by an RNAP and the cyclic shape changes it undergoes during each mechano-chemical cycle.', '0812.4692-1-6-2': 'This model may be regarded as a "unified" description in the sense that the same model describes the single RNAP properties (e.g., the force-velocity and torque-velocity relations) as well as the collective spatio-temporal organization (and rate of RNA synthesis).', '0812.4692-1-7-0': '# Basic mechano-chemistry and the model', '0812.4692-1-8-0': 'The main stages in the polymerization of polynucleotides by the polymerase machines are common:', '0812.4692-1-9-0': '(a) initiation: Once the polymerase encounters a specific sequence on the template that acts as a chemically coded start signal, it initiates the synthesis of the product.', '0812.4692-1-9-1': 'The RNAP, together with the DNA bubble and the growing RNA transcript, forms a "transcription elongation complex" (TEC).', '0812.4692-1-9-2': 'This stage is completed when the nascent product RNA becomes long enough to stabilize the TEC against dissociation from the template.', '0812.4692-1-10-0': '(b) elongation: During this stage, the nascent product gets elongated by the addition of nucleotides; during elongation [CITATION], each successful addition of a nucleotide to the elongating mRNA leads to a forward stepping of the RNAP.', '0812.4692-1-11-0': '(c) termination: Normally, the process of synthesis is terminated, and the newly polymerized full length product molecule is released, when the polymerase encounters the terminator (or, stop) sequence on the template.', '0812.4692-1-12-0': 'In this paper we are interested mainly in the elongation of the mRNA transcripts.', '0812.4692-1-13-0': 'One of the key experimental observations is that an RNAP cycles between a "closed" and an "open" shape during each mechano-chemical cycle (see fig. [REF]a) [CITATION].', '0812.4692-1-13-1': 'Recognition of the correct incoming nucleotide (dictated by the template) and its binding with the catalytic site on the RNAP leads to its closing.', '0812.4692-1-13-2': 'It remains closed as long as the hydrolysis of the NTP takes place.', '0812.4692-1-13-3': 'The nascent RNA is elongated by one monomer on completion of this reaction which also produces pyrophosphate.', '0812.4692-1-13-4': 'Then, the shape of the RNAP switches back to its open shape from the closed one thereby facilitating release of the pyrophosphate.', '0812.4692-1-13-5': 'In the open shape, the RNAP is weakly bound to its DNA track and, in principle, can execute Brownian motion.', '0812.4692-1-14-0': 'To our knowledge, none of the models of RNAP traffic reported earlier, explicitly capture these cyclic change of shape of the polymerase motor.', '0812.4692-1-14-1': 'Liverpool et al.[CITATION] as well as Klumpp and Hwa [CITATION] treat each RNAP as a rigid rod and describe its translocation from one nucleotide to the next on its template in terms of a single parameter.', '0812.4692-1-14-2': 'Thus, both these models capture the effects of the mechano-chemistry of indivisual RNAPs by an effective "hopping" rate contstant.', '0812.4692-1-14-3': 'In an attempt to capture the most essential aspects of the mechano-chemical cycle of individual RNAP motors, we have asigned two possible "internal" states to an RNAP at each spatial position on its track [CITATION].', '0812.4692-1-14-4': 'However, each of these two states is, in reality, not a single bio-chemical state; a sequence of bio-chemical states which interconvert sufficiently rapidly are collectively represented by one "internal" state whereas the transition from one internal state to the other is the slowest of all the transitions and, therefore, rate-limiting.', '0812.4692-1-14-5': 'This minimal model neither distinguishes between purely chemical transitions pure shape changes, nor does it assign distinct internal states to the "open" and "closed" shapes of the RNAP motor.', '0812.4692-1-14-6': 'Therefore, in this paper, we extend our earlier 2-state model to a 4-state model.', '0812.4692-1-15-0': 'We shall use the terms RNAP and TEC interchangeably.', '0812.4692-1-15-1': 'We extend the 2-state model of ref.[CITATION] by assigning four possible distinct "internal" state to the RNAP to capture its transitions between open and closed shapes.', '0812.4692-1-15-2': 'The composition of these four states and the spatial position of the RNAP as well as the transitions between these states are shown in fig.[REF].', '0812.4692-1-16-0': '# RNAP TRAFFIC UNDER PERIODIC BOUNDARY CONDITION', '0812.4692-1-17-0': 'In our model a one dimensional lattice of [MATH] sites represents the DNA template.', '0812.4692-1-17-1': 'Each site of these lattice correspinds to a single base pair.', '0812.4692-1-17-2': 'The total number of RNAP motors which move simultaneously on this template (and transcribe the same gene sequentially) is [MATH]; the linear size of each RNAP is [MATH] in the units of base pairs.', '0812.4692-1-17-3': 'Therefore, the number density of the RNAP motors is [MATH] whereas the corresponding coverage density is [MATH].', '0812.4692-1-17-4': 'The instantaneous spatial position of a RNAP is denoted by the leftmost site of the lattice it covers at that instant of time; however, it also "covers" all the adjacent [MATH] on the right side of of the site which denotes its position.', '0812.4692-1-17-5': 'In order to capture the steric interaction (mutual exclusion) of the RNAPs, no site of the lattice is allowed to be be covered simultaneously by more than one RNAP.', '0812.4692-1-17-6': 'Thus, this model of RNAP traffic may be regarded as a totally asymmetric simple exclusion process (TASEP) [CITATION] for hard rods each of which can exist in one of its four possible "intenal" states at any arbitrary position on the lattice.', '0812.4692-1-18-0': 'Let [MATH] denote the probability that there is a RNAP at the spatial position [MATH] and in the chemical state [MATH] at time [MATH].', '0812.4692-1-18-1': 'At any arbitrary site [MATH] and time [MATH], these probabilities must satisfy the normalization condition [EQUATION]', '0812.4692-1-18-2': 'Under the mean-field approximation, the master equations for [MATH] are as follows: [EQUATION] where [MATH] is the conditional probability that, given an RNAP in site i, site j is empty.', '0812.4692-1-18-3': 'We calculate [MATH] following the same steps as we did in ref. [CITATION], getting [EQUATION]', '0812.4692-1-18-4': 'The 4-state model reduces to our earlier 2-state model [CITATION] in the limit [MATH],[MATH]', '0812.4692-1-19-0': "In the steady state, all the [MATH]'s become independent of time.", '0812.4692-1-19-1': "Moreover, because of the periodic boundary conditions, all the sites are equivalent so that the steady-state probabilities [MATH]'s are also independent of site index [MATH].", '0812.4692-1-19-2': 'Solving Eqn. ([REF],[REF],[REF],[REF]), together with normalization condition ([REF]), in the steady state under periodic boundary conditions, we get [EQUATION]', '0812.4692-1-19-3': 'Where; [EQUATION]', '0812.4692-1-19-4': 'The corresponding flux [MATH] is given by [EQUATION]', '0812.4692-1-19-5': 'Hence, the average velocity [MATH] of a single RNAP is given by [EQUATION]', '0812.4692-1-19-6': 'The average velocity [MATH] depends on the number density [MATH] through the [MATH]-dependence of [MATH].', '0812.4692-1-20-0': "In the following subsections we'll use the formula ([REF]) in the regime of sufficiently low coverage density [MATH] to predict the dependence of [MATH] of single RNAP motors on external force and torque.", '0812.4692-1-20-1': 'Such dependences of [MATH] on external force and torque can be probed by carrying out single-molecule experiments in-vitro.', '0812.4692-1-20-2': "In order to predict the total rate of RNA synthesis at coverage densities where steric interactions between the RNAPs is not negligibly small, we'll use the expression ([REF]).", '0812.4692-1-20-3': 'Thus, our model may be regarded as a "unified" description of transcription in the sense that it can account for properties of single RNAP motors as well as their collective behaviour.', '0812.4692-1-21-0': 'For numerical calculations, we use the rate constants extracted by Wang et al.[CITATION] from the empirical data.', '0812.4692-1-21-1': 'In the absence of external force and torque, these rate constants are as follows: [EQUATION]', '0812.4692-1-22-0': '## Effects of externally applied force and torque', '0812.4692-1-23-0': 'In this section we examine the effects of external force [MATH] and torque [MATH] on the rate of RNA polymerization.', '0812.4692-1-23-1': 'From the force-velocity relation, we extract the stall force [MATH], the load force which just stalls the a single RNAP motor.', '0812.4692-1-23-2': 'In order to extract these single RNAP properties from our model, we first use the formula ([REF]) in the regime of extremely low coverage density of the RNAP motors.', '0812.4692-1-24-0': '### Effects of external force', '0812.4692-1-25-0': 'We assume that the load force [MATH] significantly affects only those steps of the mechano-chemical cycle of an RNAP which involve its mechanical movement, i.e., forward or backward movement in real space.', '0812.4692-1-25-1': 'Therefore, we assume that force-dependence of these rate constants are given by [EQUATION] where [MATH] and [MATH] are the corresponding values of the rate constants in the absence of the load force.', '0812.4692-1-25-2': 'In equation ([REF]) the symbol [MATH] nm is the typical length of a single nucleotide.', '0812.4692-1-25-3': 'None of the rate constants other than the three listed in ([REF]) are affected by the load force [MATH].', '0812.4692-1-26-0': 'The force-velocity relation of individual RNAP motors in our model is shown in fig. [REF] for three different concentrations of the NTPs.', '0812.4692-1-26-1': 'The average velocity decreases monotonically with increasing load force; the convex shape of the load-velocity curves are very similar to those reported by Wang et al. [CITATION] in their single RNAP model.', '0812.4692-1-26-2': 'In this model, the higher is the NTP concentration the larger is the stall force [MATH].', '0812.4692-1-26-3': 'Moreover, for a given load force [MATH], the average velocity of the RNAP is larger at higher NTP concentration.', '0812.4692-1-27-0': '### Effects of external torque', '0812.4692-1-28-0': 'Now we consider the effects of an externally imposed torque which assists opening, but opposes closing of the palm-like shape of an isolated RNAP.', '0812.4692-1-28-1': 'These competing effects of the same torque on the two steps of each mechano-chemical cycle of individual RNAP motors has nontrivial effects on the rate of RNA synthesis.', '0812.4692-1-28-2': 'The above mentioned effects of the torque [MATH] on opening and closing of RNAP are captured by the following choice of the correspondig rate constants [EQUATION] where [MATH] and [MATH] on the right hand sides of the equations are the rate constants in the absence of external torque.', '0812.4692-1-29-0': 'The average velocity of the RNAP motors are plotted against the torque in fig. [REF] for different sets of values of the model parameters.', '0812.4692-1-29-1': 'The most dramatic effect of the torque is the nonmonotonic variation of [MATH] with [MATH] in several experimentally accessible regimes of parameter values.', '0812.4692-1-29-2': 'Increasing torque increases the opening rate and decreases the closing rate.', '0812.4692-1-29-3': 'As long as the opening rate is still lower (and, hence, rate limiting), the velocity increases with increasing torque.', '0812.4692-1-29-4': 'A peak appears where closing becomes the rate limiting step.', '0812.4692-1-29-5': 'In order to demonstrate the effect of this crossover from a regime dominated by opening to that dominated closing of RNAP, we have plotted one curve in fig. [REF](b) corresponding to [MATH] for which the maximum occurs at [MATH].', '0812.4692-1-30-0': '## Effects of steric interactions: flux-density relation', '0812.4692-1-31-0': 'We have plotted mean-field estimate ([REF]) of flux [MATH] against the coverage density [MATH] in fig. [REF] for three different valus of the ratio [MATH] at fixed NTP concentration of [MATH].', '0812.4692-1-31-1': 'In the traffic science literature [CITATION] such flux-density relations are usually referred to as the fundamental diagram.', '0812.4692-1-31-2': 'The qualitative features of the fundamental diagrams in fig. [REF] are similar to those observed earlier in the two state model of RNAP traffic [CITATION].', '0812.4692-1-31-3': 'The most notable feature of these fundamental diagrams is their asymmetric shape; this is in sharp contrast to the symmetry of fundamental diagram of a TASEP about [MATH].', '0812.4692-1-31-4': 'The physical reason for the asymmetric shape of the fundamental diagram in fig. [REF] is as follows: To get maximum current, system should have maximum number of particle-hole pairs.', '0812.4692-1-31-5': 'For [MATH],maximum number of particle hole pairs are when [MATH], [MATH].', '0812.4692-1-31-6': 'For [MATH], maximum number of particle-hole pairs can be found when', '0812.4692-1-32-0': '[MATH], i.e., [MATH], which corresponds to [EQUATION]', '0812.4692-1-33-0': '# RNAP TRAFFIC MODEL UNDER OPEN BOUNDARY CONDITION', '0812.4692-1-34-0': 'For modeling transcription, the open boundary condition is more realistic than the periodic boundary conditions.', '0812.4692-1-34-1': 'Under this conditon a RNAP can attach to start site(labled as [MATH]) with rate [MATH] provided none of the first [MATH] sites on the lattice is covered by any other RNAP.', '0812.4692-1-34-2': 'Similarly, after reaching the site [MATH], an RNAP leaves the DNA track with rate [MATH].', '0812.4692-1-34-3': 'We calculate the conditional probability [MATH] same way as we have done for the 2-state model of RNAP [CITATION].', '0812.4692-1-34-4': 'In this case, under mean-field approximation, the master equations for the probabilities [MATH] are given by [EQUATION]', '0812.4692-1-34-5': 'Where [MATH] is the Heaviside step function defined by [EQUATION] and [EQUATION]', '0812.4692-1-34-6': 'We have numerically solved the mean-field equations ([REF])-([REF]) in the steady-state to compute the corresponding flux of the RNAP motors.', '0812.4692-1-34-7': 'These mean-field theoretic estimates of flux are compared with the corresponding data obtained from direct computer simulations of the model.', '0812.4692-1-34-8': 'These results are plotted as functions of the rate constants [MATH] and [MATH], respectively, in figs. [REF] (a) and (b).', '0812.4692-1-34-9': 'The flux rises with increasing [MATH], but the rate of rise decreases with increasing [MATH].', '0812.4692-1-34-10': 'Eventually, the flux saturates because [MATH] is no longer rate-limiting.', '0812.4692-1-34-11': 'This trend of variation of the total rate of RNA synthesis with the concentration of NTP is similar to that observed earlier in our 2-state model of RNAP traffic [CITATION].', '0812.4692-1-34-12': 'Thus, a 2-state model of RNAP would be adequate to capture the dependence of the rate of RNA synthesis on the NTP concentration.', '0812.4692-1-35-0': '# Summary and conclusions', '0812.4692-1-36-0': 'In this paper we have extended our earlier 2-state model of RNAP traffic to a 4-state model so as to capture some important cyclic shape changes in the mechano-chemical cycle of each RNAP.', '0812.4692-1-36-1': 'The new model predicts the effects of these shape changes on the rate of RNA synthesis.', '0812.4692-1-36-2': 'Moreover, we have used the same model in the extremely low density limit to extract the force-velocity relation.', '0812.4692-1-36-3': 'Finally, we have demonstrated a novel nonmonotonic variation of the average speed of the RNAP motors (and, hence, a nonmonotonic variation of the rate of synthesis of RNA) with the increase of an externally imposed torque on the individual motors.', '0812.4692-1-36-4': 'In principle, it should be possible to test the new predictions of our model by carrying out laboratory experiments in-vitro.', '0812.4692-1-37-0': 'In spite of some crucial differences, most of the polynucleotide polymerases seem to share a common "cupped right hand" architecture [CITATION].', '0812.4692-1-37-1': 'Therefore, it should be possible to make minor modifications in our 4-state model of RNAP so as to develop similar models of other polynucleotide polymerases.'}

{'0812.4692-2-0-0': 'RNA polymerase (RNAP) is like a mobile molecular workshop that polymerizes a RNA molecule by adding monomeric subunits one by one, while moving step by step on the DNA template itself.', '0812.4692-2-0-1': 'Here we develop a theoretical model by incorporating their steric interactions and mechanochemical cycles which explicitly captures the cyclical shape changes of each motor.', '0812.4692-2-0-2': 'Using this model, we explain not only the dependence of the average velocity of a RNAP on the externally applied load force, but also predict a nonmotonic variation of the average velocity on external torque.', '0812.4692-2-0-3': 'We also show the effect of steric interactions of the motors on the total rate of RNA synthesis.', '0812.4692-2-0-4': 'In principle, our predictions can be tested by carrying out in-vitro experiments.', '0812.4692-2-1-0': '# Introduction', '0812.4692-2-2-0': 'Molecular motors [CITATION] in living cells are either proteins or macromolecular complexes made of proteins and ribonucleic acids (RNAs).', '0812.4692-2-2-1': 'Like their macroscpic counterparts, these motors perform mechanical work, while translocating on a filamentous track, by converting input energy which is often supplied as chemical energy [CITATION].', '0812.4692-2-2-2': 'In this paper we study a special class of motors, called RNA polymerase (RNAP) which play crucial roles in gene expression [CITATION].', '0812.4692-2-3-0': 'Transcription of a gene encoded in the sequence of nucleotides in a specific segment of a DNA is carried out by RNAP motors which treat the DNA as a template [CITATION].', '0812.4692-2-3-1': 'An RNAP is more like a mobile workshop that performs three functions simultaneously: (i) it decodes the genetic message encoded in the template DNA and selects the appropriate nucleotide, the monomeric subunit of RNA, as dictated by the template, (ii) it catalyzes the addition of the monomeric subunit thus selected to the growing RNA molecule, (iii) it steps forward by one nucleotide on its template without completely destabilizing the ternary complex consisting of the polymerase, the template DNA and the product RNA.', '0812.4692-2-3-2': 'The free energy released by the polymerization of the RNA molecule serves as the input energy for the driving the mechanical movements of the RNAP.', '0812.4692-2-3-3': 'Therefore, these enzymes are also regarded as molecular motors [CITATION].', '0812.4692-2-4-0': 'During the transcription of a gene, the collective movement of the RNAPs on the same DNA track is often referred to as RNAP traffic because of its superficial similarity with vehicular traffic [CITATION].', '0812.4692-2-4-1': 'The beginning and the end of the specific sequence corresponding to a gene are the analogs of the on-ramp and off-ramp of vehicular traffic on highways.', '0812.4692-2-4-2': 'The average number of RNAPs, which complete the synthesis of a RNA molecule per unit time interval can be identified as the flux in RNAP traffic.', '0812.4692-2-4-3': 'Note that flux is the product of the number density and average velocity of the motors.', '0812.4692-2-4-4': 'Thus, flux in RNAP traffic is identical to the average rate of synthesis of the corresponding RNA.', '0812.4692-2-4-5': "Using the terminology of traffic science [CITATION], we'll call the relation between the flux and the number density of the motors as the fundamental diagram.", '0812.4692-2-4-6': 'The fundamental diagram is an important quantitative characteristic of traffic flow.', '0812.4692-2-5-0': 'The dependence of the velocity of the motor on an externally imposed load (opposing) force is called the force-velocity relation which is one of the most important characteristics of a molecular motor.', '0812.4692-2-5-1': 'The force-velocity relation for RNAP motors have been measured by carrying out single molecule experiments [CITATION].', '0812.4692-2-5-2': 'However, to our knowledge, the response of an RNAP motor to an externally applied torque has not been investigated so far.', '0812.4692-2-5-3': 'The effects of steric interactions of the RNAP motors on their dynamics has been studied only in a few experiments [CITATION]; but, none of these addressed the question of the nature of the overall spatio-temporal organization of the RNAP motors in RNAP traffic.', '0812.4692-2-6-0': 'The traffic-like collective dynamics of cytoskeletal molecular motors [CITATION] and that of ribosomes on mRNA tracks [CITATION] have been investigated theoretically in the physics literature.', '0812.4692-2-6-1': 'However, so far, RNAP traffic has received far less attention [CITATION] In this paper we develop a model that captures not only the steric interactions between the RNAPs, but also separately the biochemical reactions catalyzed by an RNAP and the cyclic shape changes it undergoes during each mechano-chemical cycle.', '0812.4692-2-6-2': 'This model may be regarded as a "unified" description in the sense that the same model describes the single RNAP properties (e.g., the force-velocity and torque-velocity relations) as well as the collective spatio-temporal organization (and rate of RNA synthesis).', '0812.4692-2-7-0': '# Basic mechano-chemistry and the model', '0812.4692-2-8-0': 'The main stages in the polymerization of polynucleotides by the polymerase machines are common:', '0812.4692-2-9-0': '(a) initiation: Once the polymerase encounters a specific sequence on the template that acts as a chemically coded start signal, it initiates the synthesis of the product.', '0812.4692-2-9-1': 'The RNAP, together with the DNA bubble and the growing RNA transcript, forms a "transcription elongation complex" (TEC).', '0812.4692-2-9-2': 'This stage is completed when the nascent product RNA becomes long enough to stabilize the TEC against dissociation from the template.', '0812.4692-2-10-0': '(b) elongation: During this stage, the nascent product gets elongated by the addition of nucleotides; during elongation [CITATION], each successful addition of a nucleotide to the elongating mRNA leads to a forward stepping of the RNAP.', '0812.4692-2-11-0': '(c) termination: Normally, the process of synthesis is terminated, and the newly polymerized full length product molecule is released, when the polymerase encounters the terminator (or, stop) sequence on the template.', '0812.4692-2-12-0': 'In this paper we are interested mainly in the elongation of the mRNA transcripts.', '0812.4692-2-13-0': 'One of the key experimental observations is that an RNAP cycles between a "closed" and an "open" shape during each mechano-chemical cycle (see fig. [REF] and fig. [REF]a) [CITATION].', '0812.4692-2-13-1': 'Recognition of the correct incoming nucleotide (dictated by the template) and its binding with the catalytic site on the RNAP leads to its closing.', '0812.4692-2-13-2': 'It remains closed as long as the hydrolysis of the NTP takes place.', '0812.4692-2-13-3': 'The nascent RNA is elongated by one monomer on completion of this reaction which also produces pyrophosphate.', '0812.4692-2-13-4': 'Then, the shape of the RNAP switches back to its open shape from the closed one thereby facilitating release of the pyrophosphate.', '0812.4692-2-13-5': 'In the open shape, the RNAP is weakly bound to its DNA track and, in principle, can execute Brownian motion.', '0812.4692-2-14-0': 'To our knowledge, none of the models of RNAP traffic reported earlier, explicitly capture these cyclic change of shape of the polymerase motor.', '0812.4692-2-14-1': 'Liverpool et al.[CITATION] as well as Klumpp and Hwa [CITATION] treat each RNAP as a rigid rod and describe its translocation from one nucleotide to the next on its template in terms of a single parameter.', '0812.4692-2-14-2': 'Thus, both these models capture the effects of the mechano-chemistry of indivisual RNAPs by an effective "hopping" rate contstant.', '0812.4692-2-14-3': 'In an attempt to capture the most essential aspects of the mechano-chemical cycle of individual RNAP motors, we have asigned two possible "internal" states to an RNAP at each spatial position on its track [CITATION].', '0812.4692-2-14-4': 'However, each of these two states is, in reality, not a single bio-chemical state; a sequence of bio-chemical states which interconvert sufficiently rapidly are collectively represented by one "internal" state whereas the transition from one internal state to the other is the slowest of all the transitions and, therefore, rate-limiting.', '0812.4692-2-14-5': 'This minimal model neither distinguishes between purely chemical transitions pure shape changes, nor does it assign distinct internal states to the "open" and "closed" shapes of the RNAP motor.', '0812.4692-2-14-6': 'Therefore, in this paper, we extend our earlier 2-state model to a 4-state model.', '0812.4692-2-15-0': 'We shall use the terms RNAP and TEC interchangeably.', '0812.4692-2-15-1': 'We extend the 2-state model of ref.[CITATION] by assigning four possible distinct "internal" state to the RNAP to capture its transitions between open and closed shapes.', '0812.4692-2-15-2': 'The composition of these four states and the spatial position of the RNAP as well as the transitions between these states are shown in fig.[REF](b).', '0812.4692-2-16-0': '# RNAP TRAFFIC UNDER PERIODIC BOUNDARY CONDITION', '0812.4692-2-17-0': 'In our model a one dimensional lattice of [MATH] sites represents the DNA template.', '0812.4692-2-17-1': 'Each site of these lattice correspinds to a single base pair.', '0812.4692-2-17-2': 'The total number of RNAP motors which move simultaneously on this template (and transcribe the same gene sequentially) is [MATH]; the linear size of each RNAP is [MATH] in the units of base pairs.', '0812.4692-2-17-3': 'Therefore, the number density of the RNAP motors is [MATH] whereas the corresponding coverage density is [MATH].', '0812.4692-2-17-4': 'The instantaneous spatial position of a RNAP is denoted by the leftmost site of the lattice it covers at that instant of time; however, it also "covers" all the adjacent [MATH] on the right side of of the site which denotes its position.', '0812.4692-2-17-5': 'In order to capture the steric interaction (mutual exclusion) of the RNAPs, no site of the lattice is allowed to be be covered simultaneously by more than one RNAP.', '0812.4692-2-17-6': 'Thus, this model of RNAP traffic may be regarded as a totally asymmetric simple exclusion process (TASEP) [CITATION] for hard rods each of which can exist in one of its four possible "intenal" states at any arbitrary position on the lattice.', '0812.4692-2-18-0': 'Let [MATH] denote the probability that there is a RNAP at the spatial position [MATH] and in the chemical state [MATH] at time [MATH].', '0812.4692-2-18-1': 'At any arbitrary site [MATH] and time [MATH], these probabilities must satisfy the normalization condition [EQUATION]', '0812.4692-2-18-2': 'Under the mean-field approximation, the master equations for [MATH] are as follows: [EQUATION] where [MATH] is the conditional probability that, given an RNAP in site i, site j is empty.', '0812.4692-2-18-3': 'We calculate [MATH] following the same steps as we did in ref. [CITATION], getting [EQUATION]', '0812.4692-2-18-4': 'The 4-state model reduces to our earlier 2-state model [CITATION] in the limit [MATH],[MATH]', '0812.4692-2-19-0': "In the steady state, all the [MATH]'s become independent of time.", '0812.4692-2-19-1': "Moreover, because of the periodic boundary conditions, all the sites are equivalent so that the steady-state probabilities [MATH]'s are also independent of site index [MATH].", '0812.4692-2-19-2': 'Solving Eqn. ([REF],[REF],[REF],[REF]), together with normalization condition ([REF]), in the steady state under periodic boundary conditions, we get [EQUATION]', '0812.4692-2-19-3': 'Where; [EQUATION]', '0812.4692-2-19-4': 'The corresponding flux [MATH] is given by [EQUATION]', '0812.4692-2-19-5': 'Hence, the average velocity [MATH] of a single RNAP is given by [EQUATION]', '0812.4692-2-19-6': 'The average velocity [MATH] depends on the number density [MATH] through the [MATH]-dependence of [MATH].', '0812.4692-2-20-0': "In the following subsections we'll use the formula ([REF]) in the regime of sufficiently low coverage density [MATH] to predict the dependence of [MATH] of single RNAP motors on external force and torque.", '0812.4692-2-20-1': 'Such dependences of [MATH] on external force and torque can be probed by carrying out single-molecule experiments in-vitro.', '0812.4692-2-20-2': "In order to predict the total rate of RNA synthesis at coverage densities where steric interactions between the RNAPs is not negligibly small, we'll use the expression ([REF]).", '0812.4692-2-20-3': 'Thus, our model may be regarded as a "unified" description of transcription in the sense that it can account for properties of single RNAP motors as well as their collective behaviour.', '0812.4692-2-21-0': 'For numerical calculations, we use the rate constants extracted by Wang et al.[CITATION] from the empirical data.', '0812.4692-2-21-1': 'In the absence of external force and torque, these rate constants are as follows: [EQUATION]', '0812.4692-2-22-0': '## Effects of externally applied force and torque', '0812.4692-2-23-0': 'In this section we examine the effects of external force [MATH] and torque [MATH] on the rate of RNA polymerization.', '0812.4692-2-23-1': 'From the force-velocity relation, we extract the stall force [MATH], the load force which just stalls the a single RNAP motor.', '0812.4692-2-23-2': 'In order to extract these single RNAP properties from our model, we first use the formula ([REF]) in the regime of extremely low coverage density of the RNAP motors.', '0812.4692-2-24-0': '### Effects of external force', '0812.4692-2-25-0': 'We assume that the load force [MATH] significantly affects only those steps of the mechano-chemical cycle of an RNAP which involve its mechanical movement, i.e., forward or backward movement in real space.', '0812.4692-2-25-1': 'Therefore, we assume that force-dependence of these rate constants are given by [EQUATION] where [MATH] and [MATH] are the corresponding values of the rate constants in the absence of the load force.', '0812.4692-2-25-2': 'In equation ([REF]) the symbol [MATH] nm is the typical length of a single nucleotide.', '0812.4692-2-25-3': 'None of the rate constants other than the three listed in ([REF]) are affected by the load force [MATH].', '0812.4692-2-26-0': 'The force-velocity relation of individual RNAP motors in our model is shown in fig. [REF] for three different concentrations of the NTPs.', '0812.4692-2-26-1': 'The average velocity decreases monotonically with increasing load force; the convex shape of the load-velocity curves are very similar to those reported by Wang et al. [CITATION] in their single RNAP model.', '0812.4692-2-26-2': 'In this model, the higher is the NTP concentration the larger is the stall force [MATH].', '0812.4692-2-26-3': 'Moreover, for a given load force [MATH], the average velocity of the RNAP is larger at higher NTP concentration.', '0812.4692-2-27-0': '### Effects of external torque', '0812.4692-2-28-0': 'Now we consider the effects of an externally imposed torque which assists opening, but opposes closing of the palm-like shape of an isolated RNAP.', '0812.4692-2-28-1': 'These competing effects of the same torque on the two steps of each mechano-chemical cycle of individual RNAP motors has nontrivial effects on the rate of RNA synthesis.', '0812.4692-2-28-2': 'The above mentioned effects of the torque [MATH] on opening and closing of RNAP are captured by the following choice of the correspondig rate constants [EQUATION] where [MATH] and [MATH] on the right hand sides of the equations are the rate constants in the absence of external torque.', '0812.4692-2-29-0': 'The average velocity of the RNAP motors are plotted against the torque in fig. [REF] for different sets of values of the model parameters.', '0812.4692-2-29-1': 'The most dramatic effect of the torque is the nonmonotonic variation of [MATH] with [MATH] in several experimentally accessible regimes of parameter values.', '0812.4692-2-29-2': 'Increasing torque increases the opening rate and decreases the closing rate.', '0812.4692-2-29-3': 'As long as the opening rate is still lower (and, hence, rate limiting), the velocity increases with increasing torque.', '0812.4692-2-29-4': 'A peak appears where closing becomes the rate limiting step.', '0812.4692-2-29-5': 'In order to demonstrate the effect of this crossover from a regime dominated by opening to that dominated closing of RNAP, we have plotted one curve in fig. [REF](b) corresponding to [MATH] for which the maximum occurs at [MATH].', '0812.4692-2-30-0': '## Effects of steric interactions: flux-density relation', '0812.4692-2-31-0': 'We have plotted mean-field estimate ([REF]) of flux [MATH] against the coverage density [MATH] in fig. [REF] for three different valus of the ratio [MATH] at fixed NTP concentration of [MATH].', '0812.4692-2-31-1': 'In the traffic science literature [CITATION] such flux-density relations are usually referred to as the fundamental diagram.', '0812.4692-2-31-2': 'The qualitative features of the fundamental diagrams in fig. [REF] are similar to those observed earlier in the two state model of RNAP traffic [CITATION].', '0812.4692-2-31-3': 'The most notable feature of these fundamental diagrams is their asymmetric shape; this is in sharp contrast to the symmetry of fundamental diagram of a TASEP about [MATH].', '0812.4692-2-31-4': 'The physical reason for the asymmetric shape of the fundamental diagram in fig. [REF] is as follows: To get maximum current, system should have maximum number of particle-hole pairs.', '0812.4692-2-31-5': 'For [MATH],maximum number of particle hole pairs are when [MATH], [MATH].', '0812.4692-2-31-6': 'For [MATH], maximum number of particle-hole pairs can be found when', '0812.4692-2-32-0': '[MATH], i.e., [MATH], which corresponds to [EQUATION]', '0812.4692-2-33-0': '# RNAP TRAFFIC MODEL UNDER OPEN BOUNDARY CONDITION', '0812.4692-2-34-0': 'For modeling transcription, the open boundary condition is more realistic than the periodic boundary conditions.', '0812.4692-2-34-1': 'Under this conditon a RNAP can attach to start site(labled as [MATH]) with rate [MATH] provided none of the first [MATH] sites on the lattice is covered by any other RNAP.', '0812.4692-2-34-2': 'Similarly, after reaching the site [MATH], an RNAP leaves the DNA track with rate [MATH].', '0812.4692-2-34-3': 'We calculate the conditional probability [MATH] same way as we have done for the 2-state model of RNAP [CITATION].', '0812.4692-2-34-4': 'In this case, under mean-field approximation, the master equations for the probabilities [MATH] are given by [EQUATION]', '0812.4692-2-34-5': 'Where [MATH] is the Heaviside step function defined by [EQUATION] and [EQUATION]', '0812.4692-2-34-6': 'We have numerically solved the mean-field equations ([REF])-([REF]) in the steady-state to compute the corresponding flux of the RNAP motors.', '0812.4692-2-34-7': 'These mean-field theoretic estimates of flux are compared with the corresponding data obtained from direct computer simulations of the model.', '0812.4692-2-34-8': 'These results are plotted as functions of the rate constants [MATH] and [MATH], respectively, in figs. [REF] (a) and (b).', '0812.4692-2-34-9': 'The flux rises with increasing [MATH], but the rate of rise decreases with increasing [MATH].', '0812.4692-2-34-10': 'Eventually, the flux saturates because [MATH] is no longer rate-limiting.', '0812.4692-2-34-11': 'This trend of variation of the total rate of RNA synthesis with the concentration of NTP is similar to that observed earlier in our 2-state model of RNAP traffic [CITATION].', '0812.4692-2-34-12': 'Thus, a 2-state model of RNAP would be adequate to capture the dependence of the rate of RNA synthesis on the NTP concentration.', '0812.4692-2-35-0': '# Summary and conclusions', '0812.4692-2-36-0': 'In this paper we have extended our earlier 2-state model of RNAP traffic to a 4-state model so as to capture some important cyclic shape changes in the mechano-chemical cycle of each RNAP.', '0812.4692-2-36-1': 'The new model predicts the effects of these shape changes on the rate of RNA synthesis.', '0812.4692-2-36-2': 'Moreover, we have used the same model in the extremely low density limit to extract the force-velocity relation.', '0812.4692-2-36-3': 'Finally, we have demonstrated a novel nonmonotonic variation of the average speed of the RNAP motors (and, hence, a nonmonotonic variation of the rate of synthesis of RNA) with the increase of an externally imposed torque on the individual motors.', '0812.4692-2-36-4': 'In principle, it should be possible to test the new predictions of our model by carrying out laboratory experiments in-vitro.', '0812.4692-2-37-0': 'In spite of some crucial differences, most of the polynucleotide polymerases seem to share a common "cupped right hand" architecture [CITATION].', '0812.4692-2-37-1': 'Therefore, it should be possible to make minor modifications in our 4-state model of RNAP so as to develop similar models of other polynucleotide polymerases.'}

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['0812.4692-3-33-3', '0812.4692-4-33-3'], ['0812.4692-3-33-4', '0812.4692-4-33-4'], ['0812.4692-3-33-5', '0812.4692-4-33-5'], ['0812.4692-3-33-6', '0812.4692-4-33-6'], ['0812.4692-3-33-7', '0812.4692-4-33-7'], ['0812.4692-3-33-8', '0812.4692-4-33-8'], ['0812.4692-3-33-9', '0812.4692-4-33-9'], ['0812.4692-3-33-10', '0812.4692-4-33-10'], ['0812.4692-3-33-11', '0812.4692-4-33-11'], ['0812.4692-3-33-12', '0812.4692-4-33-12'], ['0812.4692-3-20-0', '0812.4692-4-20-0'], ['0812.4692-3-20-1', '0812.4692-4-20-1'], ['0812.4692-3-20-2', '0812.4692-4-20-2'], ['0812.4692-3-20-3', '0812.4692-4-20-3'], ['0812.4692-3-31-0', '0812.4692-4-31-0'], ['0812.4692-3-31-1', '0812.4692-4-31-1'], ['0812.4692-3-31-2', '0812.4692-4-31-2'], ['0812.4692-3-31-3', '0812.4692-4-31-3'], ['0812.4692-3-31-4', '0812.4692-4-31-4'], ['0812.4692-3-31-5', '0812.4692-4-31-5'], ['0812.4692-3-31-6', '0812.4692-4-31-6'], ['0812.4692-3-18-0', '0812.4692-4-19-0'], ['0812.4692-3-18-1', '0812.4692-4-19-1'], ['0812.4692-3-18-2', '0812.4692-4-19-2'], ['0812.4692-3-18-3', '0812.4692-4-19-3'], ['0812.4692-3-19-0', '0812.4692-4-19-4'], ['0812.4692-3-19-1', '0812.4692-4-19-5'], ['0812.4692-3-19-2', '0812.4692-4-19-6'], ['0812.4692-3-19-4', '0812.4692-4-19-8'], ['0812.4692-3-19-5', '0812.4692-4-19-9'], ['0812.4692-3-19-6', '0812.4692-4-19-10']]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0812.4692

{'0812.4692-3-0-0': 'RNA polymerase (RNAP) is like a mobile molecular workshop that polymerizes a RNA molecule by adding monomeric subunits one by one, while moving step by step on the DNA template itself.', '0812.4692-3-0-1': 'Here we develop a theoretical model by incorporating their steric interactions and mechanochemical cycles which explicitly captures the cyclical shape changes of each motor.', '0812.4692-3-0-2': 'Using this model, we explain not only the dependence of the average velocity of a RNAP on the externally applied load force, but also predict a nonmotonic variation of the average velocity on external torque.', '0812.4692-3-0-3': 'We also show the effect of steric interactions of the motors on the total rate of RNA synthesis.', '0812.4692-3-0-4': 'In principle, our predictions can be tested by carrying out in-vitro experiments.', '0812.4692-3-1-0': '# Introduction', '0812.4692-3-2-0': 'Molecular motors [CITATION] in living cells are either proteins or macromolecular complexes made of proteins and ribonucleic acids (RNAs).', '0812.4692-3-2-1': 'Like their macroscpic counterparts, these motors perform mechanical work, while translocating on a filamentous track, by converting input energy which is often supplied as chemical energy [CITATION].', '0812.4692-3-2-2': 'In this paper we study a special class of motors, called RNA polymerase (RNAP) which play crucial roles in gene expression [CITATION].', '0812.4692-3-3-0': 'Transcription of a gene encoded in the sequence of nucleotides in a specific segment of a DNA is carried out by RNAP motors which treat the DNA as a template [CITATION].', '0812.4692-3-3-1': 'An RNAP is more like a mobile workshop that performs three functions simultaneously: (i) it decodes the genetic message encoded in the template DNA and selects the appropriate nucleotide, the monomeric subunit of RNA, as dictated by the template, (ii) it catalyzes the addition of the monomeric subunit thus selected to the growing RNA molecule, (iii) it steps forward by one nucleotide on its template without completely destabilizing the ternary complex consisting of the polymerase, the template DNA and the product RNA.', '0812.4692-3-3-2': 'The free energy released by the polymerization of the RNA molecule serves as the input energy for the driving the mechanical movements of the RNAP.', '0812.4692-3-3-3': 'Therefore, these enzymes are also regarded as molecular motors [CITATION].', '0812.4692-3-4-0': 'During the transcription of a gene, the collective movement of the RNAPs on the same DNA track is often referred to as RNAP traffic because of its superficial similarity with vehicular traffic [CITATION].', '0812.4692-3-4-1': 'The beginning and the end of the specific sequence corresponding to a gene are the analogs of the on-ramp and off-ramp of vehicular traffic on highways.', '0812.4692-3-4-2': 'The average number of RNAPs, which complete the synthesis of a RNA molecule per unit time interval can be identified as the flux in RNAP traffic.', '0812.4692-3-4-3': 'Note that flux is the product of the number density and average velocity of the motors.', '0812.4692-3-4-4': 'Thus, flux in RNAP traffic is identical to the average rate of synthesis of the corresponding RNA.', '0812.4692-3-4-5': "Using the terminology of traffic science [CITATION], we'll call the relation between the flux and the number density of the motors as the fundamental diagram.", '0812.4692-3-4-6': 'The fundamental diagram is an important quantitative characteristic of traffic flow.', '0812.4692-3-5-0': 'The dependence of the velocity of the motor on an externally imposed load (opposing) force is called the force-velocity relation which is one of the most important characteristics of a molecular motor.', '0812.4692-3-5-1': 'The force-velocity relation for RNAP motors have been measured by carrying out single molecule experiments [CITATION].', '0812.4692-3-5-2': 'However, to our knowledge, the response of an RNAP motor to an externally applied torque has not been investigated so far.', '0812.4692-3-5-3': 'The effects of steric interactions of the RNAP motors on their dynamics has been studied only in a few experiments [CITATION]; but, none of these addressed the question of the nature of the overall spatio-temporal organization of the RNAP motors in RNAP traffic.', '0812.4692-3-6-0': 'The traffic-like collective dynamics of cytoskeletal molecular motors [CITATION] and that of ribosomes on mRNA tracks [CITATION] have been investigated theoretically in the physics literature.', '0812.4692-3-6-1': 'However, so far, RNAP traffic has received far less attention [CITATION] In this paper we develop a model that captures not only the steric interactions between the RNAPs, but also separately the biochemical reactions catalyzed by an RNAP and the cyclic shape changes it undergoes during each mechano-chemical cycle.', '0812.4692-3-6-2': 'This model may be regarded as a "unified" description in the sense that the same model describes the single RNAP properties (e.g., the force-velocity and torque-velocity relations) as well as the collective spatio-temporal organization (and rate of RNA synthesis).', '0812.4692-3-7-0': '# Basic mechano-chemistry and the model', '0812.4692-3-8-0': 'The main stages in the polymerization of polynucleotides by the polymerase machines are common:', '0812.4692-3-9-0': '(a) initiation: Once the polymerase encounters a specific sequence on the template that acts as a chemically coded start signal, it initiates the synthesis of the product.', '0812.4692-3-9-1': 'The RNAP, together with the DNA bubble and the growing RNA transcript, forms a "transcription elongation complex" (TEC).', '0812.4692-3-9-2': 'This stage is completed when the nascent product RNA becomes long enough to stabilize the TEC against dissociation from the template.', '0812.4692-3-10-0': '(b) elongation: During this stage, the nascent product gets elongated by the addition of nucleotides; during elongation [CITATION], each successful addition of a nucleotide to the elongating mRNA leads to a forward stepping of the RNAP.', '0812.4692-3-11-0': '(c) termination: Normally, the process of synthesis is terminated, and the newly polymerized full length product molecule is released, when the polymerase encounters the terminator (or, stop) sequence on the template.', '0812.4692-3-12-0': 'In this paper we are interested mainly in the elongation of the mRNA transcripts.', '0812.4692-3-13-0': 'One of the key experimental observations is that an RNAP cycles between a "closed" and an "open" shape during each mechano-chemical cycle (see fig. [REF] and fig. [REF]a) [CITATION].', '0812.4692-3-13-1': 'Recognition of the correct incoming nucleotide (dictated by the template) and its binding with the catalytic site on the RNAP leads to its closing.', '0812.4692-3-13-2': 'It remains closed as long as the hydrolysis of the NTP takes place.', '0812.4692-3-13-3': 'The nascent RNA is elongated by one monomer on completion of this reaction which also produces pyrophosphate.', '0812.4692-3-13-4': 'Then, the shape of the RNAP switches back to its open shape from the closed one thereby facilitating release of the pyrophosphate.', '0812.4692-3-13-5': 'In the open shape, the RNAP is weakly bound to its DNA track and, in principle, can execute Brownian motion.', '0812.4692-3-14-0': 'To our knowledge, none of the models of RNAP traffic reported earlier, explicitly capture these cyclic change of shape of the polymerase motor.', '0812.4692-3-14-1': 'Liverpool et al.[CITATION] as well as Klumpp and Hwa [CITATION] treat each RNAP as a rigid rod and describe its translocation from one nucleotide to the next on its template in terms of a single parameter.', '0812.4692-3-14-2': 'Thus, both these models capture the effects of the mechano-chemistry of indivisual RNAPs by an effective "hopping" rate contstant.', '0812.4692-3-14-3': 'In an attempt to capture the most essential aspects of the mechano-chemical cycle of individual RNAP motors, we have asigned two possible "internal" states to an RNAP at each spatial position on its track [CITATION].', '0812.4692-3-14-4': 'However, each of these two states is, in reality, not a single bio-chemical state; a sequence of bio-chemical states which interconvert sufficiently rapidly are collectively represented by one "internal" state whereas the transition from one internal state to the other is the slowest of all the transitions and, therefore, rate-limiting.', '0812.4692-3-14-5': 'This minimal model neither distinguishes between purely chemical transitions pure shape changes, nor does it assign distinct internal states to the "open" and "closed" shapes of the RNAP motor.', '0812.4692-3-14-6': 'Therefore, in this paper, we extend our earlier 2-state model to a 4-state model.', '0812.4692-3-15-0': 'We shall use the terms RNAP and TEC interchangeably.', '0812.4692-3-15-1': 'We extend the 2-state model of ref.[CITATION] by assigning four possible distinct "internal" state to the RNAP to capture its transitions between open and closed shapes.', '0812.4692-3-15-2': 'The composition of these four states and the spatial position of the RNAP as well as the transitions between these states are shown in fig.[REF](b).', '0812.4692-3-16-0': '# RNAP TRAFFIC UNDER PERIODIC BOUNDARY CONDITION', '0812.4692-3-17-0': 'In our model a one dimensional lattice of [MATH] sites represents the DNA template.', '0812.4692-3-17-1': 'Each site of these lattice corresponds to a single base pair.', '0812.4692-3-17-2': 'The total number of RNAP motors which move simultaneously on this template (and transcribe the same gene sequentially) is [MATH]; the linear size of each RNAP is [MATH] in the units of base pairs.', '0812.4692-3-17-3': 'Therefore, the number density of the RNAP motors is [MATH] whereas the corresponding coverage density is [MATH].', '0812.4692-3-17-4': 'The instantaneous spatial position of a RNAP is denoted by the leftmost site of the lattice it covers at that instant of time; however, it also "covers" all the adjacent [MATH] on the right side of of the site which denotes its position.', '0812.4692-3-17-5': 'In order to capture the steric interaction (mutual exclusion) of the RNAPs, no site of the lattice is allowed to be be covered simultaneously by more than one RNAP.', '0812.4692-3-17-6': 'Thus, this model of RNAP traffic may be regarded as a totally asymmetric simple exclusion process (TASEP) [CITATION] for hard rods each of which can exist in one of its four possible "intenal" states at any arbitrary position on the lattice.', '0812.4692-3-18-0': 'Let [MATH] denote the probability that there is a RNAP at the spatial position [MATH] and in the chemical state [MATH] at time [MATH].', '0812.4692-3-18-1': 'At any arbitrary site [MATH] and time [MATH], these probabilities must satisfy the normalization condition [EQUATION]', '0812.4692-3-18-2': 'Under the mean-field approximation, the master equations for [MATH] are as follows: [EQUATION] where [MATH] is the conditional probability that, given an RNAP in site i, site j is empty.', '0812.4692-3-18-3': 'We calculate [MATH] following the same steps as we did in ref. [CITATION], getting [EQUATION]', '0812.4692-3-18-4': 'The 4-state model reduces to our earlier 2-state model [CITATION] in the limit [MATH],[MATH]', '0812.4692-3-19-0': "In the steady state, all the [MATH]'s become independent of time.", '0812.4692-3-19-1': "Moreover, because of the periodic boundary conditions, all the sites are equivalent so that the steady-state probabilities [MATH]'s are also independent of site index [MATH].", '0812.4692-3-19-2': 'Solving Eqn. ([REF],[REF],[REF],[REF]), together with normalization condition ([REF]), in the steady state under periodic boundary conditions, we get [EQUATION]', '0812.4692-3-19-3': 'Where; [EQUATION]', '0812.4692-3-19-4': 'The corresponding flux [MATH] is given by [EQUATION]', '0812.4692-3-19-5': 'Hence, the average velocity [MATH] of a single RNAP is given by [EQUATION]', '0812.4692-3-19-6': 'The average velocity [MATH] depends on the number density [MATH] through the [MATH]-dependence of [MATH].', '0812.4692-3-20-0': "In the following subsections we'll use the formula ([REF]) in the regime of sufficiently low coverage density [MATH] to predict the dependence of [MATH] of single RNAP motors on external force and torque.", '0812.4692-3-20-1': 'Such dependences of [MATH] on external force and torque can be probed by carrying out single-molecule experiments in-vitro.', '0812.4692-3-20-2': "In order to predict the total rate of RNA synthesis at coverage densities where steric interactions between the RNAPs is not negligibly small, we'll use the expression ([REF]).", '0812.4692-3-20-3': 'Thus, our model may be regarded as a "unified" description of transcription in the sense that it can account for properties of single RNAP motors as well as their collective behaviour.', '0812.4692-3-21-0': 'For numerical calculations, we use the rate constants extracted by Wang et al.[CITATION] from the empirical data.', '0812.4692-3-21-1': 'In the absence of external force and torque, these rate constants are as follows: [EQUATION]', '0812.4692-3-22-0': '## Effects of externally applied force and torque', '0812.4692-3-23-0': 'In this section we examine the effects of external force [MATH] and torque [MATH] on the rate of RNA polymerization.', '0812.4692-3-23-1': 'From the force-velocity relation, we extract the stall force [MATH], the load force which just stalls the a single RNAP motor.', '0812.4692-3-23-2': 'In order to extract these single RNAP properties from our model, we first use the formula ([REF]) in the regime of extremely low coverage density of the RNAP motors.', '0812.4692-3-24-0': '### Effects of external force', '0812.4692-3-25-0': 'We assume that the load force [MATH] significantly affects only those steps of the mechano-chemical cycle of an RNAP which involve its mechanical movement, i.e., forward or backward movement in real space.', '0812.4692-3-25-1': 'Therefore, we assume that force-dependence of these rate constants are given by [EQUATION] where [MATH] and [MATH] are the corresponding values of the rate constants in the absence of the load force.', '0812.4692-3-25-2': 'In equation ([REF]) the symbol [MATH] nm is the typical length of a single nucleotide.', '0812.4692-3-25-3': 'None of the rate constants other than the three listed in ([REF]) are affected by the load force [MATH].', '0812.4692-3-26-0': 'The force-velocity relation of individual RNAP motors in our model is shown in fig. [REF] for three different concentrations of the NTPs.', '0812.4692-3-26-1': 'The average velocity decreases monotonically with increasing load force; the convex shape of the load-velocity curves are very similar to those reported by Wang et al. [CITATION] in their single RNAP model.', '0812.4692-3-26-2': 'In this model, the higher is the NTP concentration the larger is the stall force [MATH].', '0812.4692-3-26-3': 'Moreover, for a given load force [MATH], the average velocity of the RNAP is larger at higher NTP concentration.', '0812.4692-3-27-0': '### Effects of external torque', '0812.4692-3-28-0': 'Now we consider the effects of an externally imposed torque which assists opening, but opposes closing of the palm-like shape of an isolated RNAP.', '0812.4692-3-28-1': 'These competing effects of the same torque on the two steps of each mechano-chemical cycle of individual RNAP motors has nontrivial effects on the rate of RNA synthesis.', '0812.4692-3-28-2': 'The above mentioned effects of the torque [MATH] on opening and closing of RNAP are captured by the following choice of the correspondig rate constants [EQUATION] where [MATH] and [MATH] on the right hand sides of the equations are the rate constants in the absence of external torque.', '0812.4692-3-29-0': 'The average velocity of the RNAP motors are plotted against the torque in fig. [REF] for different sets of values of the model parameters.', '0812.4692-3-29-1': 'The most dramatic effect of the torque is the nonmonotonic variation of [MATH] with [MATH] in several experimentally accessible regimes of parameter values.', '0812.4692-3-29-2': 'Increasing torque increases the opening rate and decreases the closing rate.', '0812.4692-3-29-3': 'As long as the opening rate is still lower (and, hence, rate limiting), the velocity increases with increasing torque.', '0812.4692-3-29-4': 'A peak appears where closing becomes the rate limiting step.', '0812.4692-3-29-5': 'In order to demonstrate the effect of this crossover from a regime dominated by opening to that dominated closing of RNAP, we have plotted one curve in fig. [REF](b) corresponding to [MATH] for which the maximum occurs at [MATH].', '0812.4692-3-30-0': '## Effects of steric interactions: flux-density relation', '0812.4692-3-31-0': 'We have plotted mean-field estimate ([REF]) of flux [MATH] against the coverage density [MATH] in fig. [REF] for three different valus of the ratio [MATH] at fixed NTP concentration of [MATH].', '0812.4692-3-31-1': 'In the traffic science literature [CITATION] such flux-density relations are usually referred to as the fundamental diagram.', '0812.4692-3-31-2': 'The qualitative features of the fundamental diagrams in fig. [REF] are similar to those observed earlier in the two state model of RNAP traffic [CITATION].', '0812.4692-3-31-3': 'The most notable feature of these fundamental diagrams is their asymmetric shape; this is in sharp contrast to the symmetry of fundamental diagram of a TASEP about [MATH].', '0812.4692-3-31-4': 'The physical reason for the asymmetric shape of the fundamental diagram in fig. [REF] is as follows: to get maximum current, system should have maximum number of particle-hole pairs.', '0812.4692-3-31-5': 'For [MATH], maximum number of particle hole pairs occur when [MATH], i.e., [MATH].', '0812.4692-3-31-6': 'For [MATH], the magnitude of [MATH] can be found from the corresponding condition [MATH], i.e., [MATH], which corresponds to the coverage density [EQUATION]', '0812.4692-3-32-0': '# RNAP TRAFFIC MODEL UNDER OPEN BOUNDARY CONDITION', '0812.4692-3-33-0': 'For modeling transcription, the open boundary condition is more realistic than the periodic boundary conditions.', '0812.4692-3-33-1': 'Under this conditon a RNAP can attach to start site(labled as [MATH]) with rate [MATH] provided none of the first [MATH] sites on the lattice is covered by any other RNAP.', '0812.4692-3-33-2': 'Similarly, after reaching the site [MATH], an RNAP leaves the DNA track with rate [MATH].', '0812.4692-3-33-3': 'We calculate the conditional probability [MATH] same way as we have done for the 2-state model of RNAP [CITATION].', '0812.4692-3-33-4': 'In this case, under mean-field approximation, the master equations for the probabilities [MATH] are given by [EQUATION]', '0812.4692-3-33-5': 'Where [MATH] is the Heaviside step function defined by [EQUATION] and [EQUATION]', '0812.4692-3-33-6': 'We have numerically solved the mean-field equations ([REF])-([REF]) in the steady-state to compute the corresponding flux of the RNAP motors.', '0812.4692-3-33-7': 'These mean-field theoretic estimates of flux are compared with the corresponding data obtained from direct computer simulations of the model.', '0812.4692-3-33-8': 'These results are plotted as functions of the rate constants [MATH] and [MATH], respectively, in figs. [REF] (a) and (b).', '0812.4692-3-33-9': 'The flux rises with increasing [MATH], but the rate of rise decreases with increasing [MATH].', '0812.4692-3-33-10': 'Eventually, the flux saturates because [MATH] is no longer rate-limiting.', '0812.4692-3-33-11': 'This trend of variation of the total rate of RNA synthesis with the concentration of NTP is similar to that observed earlier in our 2-state model of RNAP traffic [CITATION].', '0812.4692-3-33-12': 'Thus, a 2-state model of RNAP would be adequate to capture the dependence of the rate of RNA synthesis on the NTP concentration.', '0812.4692-3-34-0': '# Summary and conclusions', '0812.4692-3-35-0': 'In this paper we have extended our earlier 2-state model of RNAP traffic to a 4-state model so as to capture some important cyclic shape changes in the mechano-chemical cycle of each RNAP.', '0812.4692-3-35-1': 'The new model predicts the effects of these shape changes on the rate of RNA synthesis.', '0812.4692-3-35-2': 'Moreover, we have used the same model in the extremely low density limit to extract the force-velocity relation.', '0812.4692-3-35-3': 'Finally, we have demonstrated a novel nonmonotonic variation of the average speed of the RNAP motors (and, hence, a nonmonotonic variation of the rate of synthesis of RNA) with the increase of an externally imposed torque on the individual motors.', '0812.4692-3-35-4': 'In principle, it should be possible to test the new predictions of our model by carrying out laboratory experiments in-vitro.', '0812.4692-3-36-0': 'In spite of some crucial differences, most of the polynucleotide polymerases seem to share a common "cupped right hand" architecture [CITATION].', '0812.4692-3-36-1': 'Therefore, it should be possible to make minor modifications in our 4-state model of RNAP so as to develop similar models of other polynucleotide polymerases.'}

{'0812.4692-4-0-0': 'RNA polymerase (RNAP) is a mobile molecular workshop that polymerizes a RNA molecule by adding monomeric subunits one by one, while moving step by step on the DNA template itself.', '0812.4692-4-0-1': 'Here we develop a theoretical model by incorporating the steric interactions of the RNAPs and their mechanochemical cycles which explicitly captures the cyclical shape changes of each motor.', '0812.4692-4-0-2': 'Using this model, we explain not only the dependence of the average velocity of a RNAP on the externally applied load force, but also predict a nonmotonic variation of the average velocity on external torque.', '0812.4692-4-0-3': 'We also show the effect of steric interactions of the motors on the total rate of RNA synthesis.', '0812.4692-4-0-4': 'In principle, our predictions can be tested by carrying out in-vitro experiments which we suggest here.', '0812.4692-4-1-0': '# Introduction', '0812.4692-4-2-0': 'Molecular motors [CITATION] in living cells are either proteins or macromolecular complexes made of proteins and ribonucleic acids (RNAs).', '0812.4692-4-2-1': 'Like their macroscopic counterparts, these motors perform mechanical work, while translocating on a filamentous track, by converting input energy which is often supplied as chemical energy [CITATION].', '0812.4692-4-2-2': 'In this paper we study a special class of motors, called RNA polymerase (RNAP) which play crucial roles in gene expression [CITATION].', '0812.4692-4-3-0': 'Transcription of a gene encoded in the sequence of nucleotides in a specific segment of a DNA is carried out by RNAP motors which treat the DNA as a template [CITATION].', '0812.4692-4-3-1': 'An RNAP is more like a mobile workshop that performs three functions simultaneously: (i) it decodes the genetic message encoded in the template DNA and selects the appropriate nucleotide, the monomeric subunit of RNA, as dictated by the template, (ii) it catalyzes the addition of the monomeric subunit thus selected to the growing RNA molecule, (iii) it steps forward by one nucleotide on its template without completely destabilizing the ternary complex consisting of the polymerase, the template DNA and the product RNA.', '0812.4692-4-3-2': 'The free energy released by the polymerization of the RNA molecule serves as the input energy for the driving the mechanical movements of the RNAP.', '0812.4692-4-3-3': 'Therefore, these enzymes are also regarded as molecular motors [CITATION].', '0812.4692-4-4-0': 'During the transcription of a gene, the collective movement of the RNAPs on the same DNA track is often referred to as RNAP traffic because of its superficial similarity with vehicular traffic [CITATION].', '0812.4692-4-4-1': 'The beginning and the end of the specific sequence corresponding to a gene are the analogs of the on-ramp and off-ramp of vehicular traffic on highways.', '0812.4692-4-4-2': 'The average number of RNAPs, which complete the synthesis of a RNA molecule per unit time interval can be identified as the flux in RNAP traffic.', '0812.4692-4-4-3': 'Note that flux is the product of the number density and average velocity of the motors.', '0812.4692-4-4-4': 'Thus, flux in RNAP traffic is identical to the average rate of synthesis of the corresponding RNA.', '0812.4692-4-4-5': "Using the terminology of traffic science [CITATION], we'll call the relation between the flux and the number density of the motors as the fundamental diagram.", '0812.4692-4-4-6': 'The fundamental diagram is an important quantitative characteristic of traffic flow.', '0812.4692-4-5-0': 'The dependence of the velocity of the motor on an externally imposed load (opposing) force is called the force-velocity relation which is one of the most important characteristics of a molecular motor.', '0812.4692-4-5-1': 'The force-velocity relation for RNAP motors have been measured by carrying out single molecule experiments [CITATION].', '0812.4692-4-5-2': 'However, to our knowledge, the response of an RNAP motor to an externally applied torque has not been investigated so far.', '0812.4692-4-5-3': 'The effects of steric interactions of the RNAP motors on their dynamics has been studied only in a few experiments [CITATION]; but, none of these addressed the question of the nature of the overall spatio-temporal organization of the RNAP motors in RNAP traffic.', '0812.4692-4-6-0': 'The traffic-like collective dynamics of cytoskeletal molecular motors [CITATION] and that of ribosomes on mRNA tracks [CITATION] have been investigated theoretically in the physics literature.', '0812.4692-4-6-1': 'However, so far, RNAP traffic has received far less attention [CITATION] In this paper we develop a model that captures not only the steric interactions between the RNAPs, but also separately the biochemical reactions catalyzed by an RNAP and the cyclic shape changes it undergoes during each mechano-chemical cycle.', '0812.4692-4-6-2': 'This model may be regarded as a "unified" description in the sense that the same model describes the single RNAP properties (e.g., the force-velocity and torque-velocity relations) as well as the collective spatio-temporal organization (and rate of RNA synthesis).', '0812.4692-4-7-0': '# Basic mechano-chemistry and the model', '0812.4692-4-8-0': 'The main stages in the polymerization of polynucleotides by the polymerase machines are common:', '0812.4692-4-9-0': '(a) initiation: Once the polymerase encounters a specific sequence on the template that acts as a chemically coded start signal, it initiates the synthesis of the product.', '0812.4692-4-9-1': 'The RNAP, together with the DNA bubble and the growing RNA transcript, forms a "transcription elongation complex" (TEC).', '0812.4692-4-9-2': 'This stage is completed when the nascent product RNA becomes long enough to stabilize the TEC against dissociation from the template.', '0812.4692-4-10-0': '(b) elongation: During this stage, the nascent product gets elongated by the addition of nucleotides; during elongation [CITATION], each successful addition of a nucleotide to the elongating mRNA leads to a forward stepping of the RNAP.', '0812.4692-4-11-0': '(c) termination: Normally, the process of synthesis is terminated, and the newly polymerized full length product molecule is released, when the polymerase encounters the terminator (or, stop) sequence on the template.', '0812.4692-4-12-0': 'In this paper we are interested mainly in the elongation of the mRNA transcripts.', '0812.4692-4-13-0': 'One of the key experimental observations is that an RNAP cycles between a "closed" and an "open" shape during each mechano-chemical cycle (see fig. [REF] and fig. [REF]a) [CITATION].', '0812.4692-4-13-1': 'Recognition of the correct incoming nucleotide (dictated by the template) and its binding with the catalytic site on the RNAP leads to its closing.', '0812.4692-4-13-2': 'It remains closed as long as the hydrolysis of the NTP takes place.', '0812.4692-4-13-3': 'The nascent RNA is elongated by one monomer on completion of this reaction which also produces pyrophosphate.', '0812.4692-4-13-4': 'Then, the shape of the RNAP switches back to its open shape from the closed one thereby facilitating release of the pyrophosphate.', '0812.4692-4-13-5': 'In the open shape, the RNAP is weakly bound to its DNA track and, in principle, can execute Brownian motion.', '0812.4692-4-14-0': 'To our knowledge, none of the models of RNAP traffic reported earlier, explicitly capture these cyclic change of shape of the polymerase motor.', '0812.4692-4-14-1': 'Liverpool et al.[CITATION] as well as Klumpp and Hwa [CITATION] treat each RNAP as a rigid rod and describe its translocation from one nucleotide to the next on its template in terms of a single parameter.', '0812.4692-4-14-2': 'Thus, both these models capture the effects of the mechano-chemistry of indivisual RNAPs by an effective "hopping" rate contstant.', '0812.4692-4-14-3': 'In an attempt to capture the most essential aspects of the mechano-chemical cycle of individual RNAP motors, we have assigned two possible "internal" states to an RNAP at each spatial position on its track [CITATION].', '0812.4692-4-14-4': 'However, each of these two states is, in reality, not a single bio-chemical state; a sequence of bio-chemical states which interconvert sufficiently rapidly are collectively represented by one "internal" state whereas the transition from one internal state to the other is the slowest of all the transitions and, therefore, rate-limiting.', '0812.4692-4-14-5': 'This minimal model neither distinguishes between purely chemical transitions pure shape changes, nor does it assign distinct internal states to the "open" and "closed" shapes of the RNAP motor.', '0812.4692-4-14-6': 'Therefore, in this paper, we extend our earlier 2-state model to a 4-state model.', '0812.4692-4-15-0': 'We shall use the terms RNAP and TEC interchangeably.', '0812.4692-4-15-1': 'We extend the 2-state model of ref.[CITATION] by assigning four possible distinct "internal" state to the RNAP to capture its transitions between open and closed shapes.', '0812.4692-4-15-2': 'The composition of these four states and the spatial position of the RNAP as well as the transitions between these states are shown in fig.[REF](b).', '0812.4692-4-16-0': 'The subscript [MATH] of the symbol "RNAP" denotes the position of the RNAP motor on its track, measured from a nucleotide that marks the starting point of transcription by the RNAP.', '0812.4692-4-16-1': 'The two superscripts [MATH] and [MATH] of "RNAP" refer to its "open" and "closed" shapes, restectively.', '0812.4692-4-16-2': 'The four distinct states in each cycle are labelled by the integers 1,2,3,4.', '0812.4692-4-16-3': 'Let us begin with the state 1, where the RNAP is open and is located at [MATH] so that the length of the corresponding elongating mRNA transcript is also [MATH].', '0812.4692-4-16-4': 'Arrival of the correct NTP subunit causes the transition to the state 2 where the open RNAP, now located at the new position [MATH], forms a complex with the mRNA transcript of length [MATH] and the newly arrived NTP subunit.', '0812.4692-4-16-5': 'Next, the shape of the RNAP changes from open to closed form and the new state of the system is labelled by the integer index 3.', '0812.4692-4-16-6': 'Then, the closed RNAP catalyzes the formation of a new covalent bond between the mRNA transcript and the newly arrived NTP subunit which leads to the elongation of the transcript from [MATH] to [MATH] and the corresponding transition from state 3 to state 4.', '0812.4692-4-16-7': 'Finally, the RNAP opens again and releases the pyrophosphate ([MATH]); this transition takes the open RNAP to the state 1 at the position [MATH].', '0812.4692-4-16-8': 'The 4-state model reduces to our earlier 2-state model [CITATION] in the limit [MATH],[MATH]', '0812.4692-4-17-0': '# RNAP TRAFFIC UNDER PERIODIC BOUNDARY CONDITION', '0812.4692-4-18-0': 'In our model a one dimensional lattice of [MATH] sites represents the DNA template.', '0812.4692-4-18-1': 'Each site of these lattice corresponds to a single base pair.', '0812.4692-4-18-2': 'The total number of RNAP motors which move simultaneously on this template (and transcribe the same gene sequentially) is [MATH]; the linear size of each RNAP is [MATH] in the units of base pairs.', '0812.4692-4-18-3': 'Therefore, the number density of the RNAP motors is [MATH] whereas the corresponding coverage density is [MATH].', '0812.4692-4-18-4': 'The instantaneous spatial position of a RNAP is denoted by the leftmost site of the lattice it covers at that instant of time; however, it also "covers" all the adjacent [MATH] on the right side of of the site which denotes its position.', '0812.4692-4-18-5': 'In order to capture the steric interaction (mutual exclusion) of the RNAPs, no site of the lattice is allowed to be be covered simultaneously by more than one RNAP.', '0812.4692-4-18-6': 'Thus, this model of RNAP traffic may be regarded as a totally asymmetric simple exclusion process (TASEP) [CITATION] for hard rods each of which can exist in one of its four possible "internal" states at any arbitrary position on the lattice.', '0812.4692-4-19-0': 'Let [MATH] denote the probability that there is a RNAP at the spatial position [MATH] and in the chemical state [MATH] at time [MATH].', '0812.4692-4-19-1': 'At any arbitrary site [MATH] and time [MATH], these probabilities must satisfy the normalization condition [EQUATION]', '0812.4692-4-19-2': 'Under the mean-field approximation, the master equations for [MATH] are as follows: [EQUATION] where [MATH] is the conditional probability that, given an RNAP in site i, site j is empty.', '0812.4692-4-19-3': 'We calculate [MATH] following the same steps as we did in ref. [CITATION], getting [EQUATION]', '0812.4692-4-19-4': "In the steady state, all the [MATH]'s become independent of time.", '0812.4692-4-19-5': "Moreover, because of the periodic boundary conditions, all the sites are equivalent so that the steady-state probabilities [MATH]'s are also independent of site index [MATH].", '0812.4692-4-19-6': 'Solving Eqn. ([REF],[REF],[REF],[REF]), together with normalization condition ([REF]), in the steady state under periodic boundary conditions, we get [EQUATION]', '0812.4692-4-19-7': 'Where; [EQUATION]', '0812.4692-4-19-8': 'The corresponding flux [MATH] is given by [EQUATION]', '0812.4692-4-19-9': 'Hence, the average velocity [MATH] of a single RNAP is given by [EQUATION]', '0812.4692-4-19-10': 'The average velocity [MATH] depends on the number density [MATH] through the [MATH]-dependence of [MATH].', '0812.4692-4-20-0': "In the following subsections we'll use the formula ([REF]) in the regime of sufficiently low coverage density [MATH] to predict the dependence of [MATH] of single RNAP motors on external force and torque.", '0812.4692-4-20-1': 'Such dependences of [MATH] on external force and torque can be probed by carrying out single-molecule experiments in-vitro.', '0812.4692-4-20-2': "In order to predict the total rate of RNA synthesis at coverage densities where steric interactions between the RNAPs is not negligibly small, we'll use the expression ([REF]).", '0812.4692-4-20-3': 'Thus, our model may be regarded as a "unified" description of transcription in the sense that it can account for properties of single RNAP motors as well as their collective behaviour.', '0812.4692-4-21-0': 'For numerical calculations, we use the rate constants extracted by Wang et al.[CITATION] from the empirical data.', '0812.4692-4-21-1': 'In the absence of external force and torque, the typical numerical values of these rate constants are as follows: [EQUATION]', '0812.4692-4-21-2': 'So far as the two new rate constants [MATH] and [MATH] are concerned, we investigate their effects on the rate of transcription by varying their magnitudes over a wide range.', '0812.4692-4-22-0': '## Effects of externally applied force and torque', '0812.4692-4-23-0': 'In this section we examine the effects of external force [MATH] and torque [MATH] on the rate of RNA polymerization.', '0812.4692-4-23-1': 'From the force-velocity relation, we extract the stall force [MATH], the load force which just stalls the a single RNAP motor.', '0812.4692-4-23-2': 'In order to extract these single RNAP properties from our model, we first use the formula ([REF]) in the regime of extremely low coverage density of the RNAP motors.', '0812.4692-4-24-0': '### Effects of external force', '0812.4692-4-25-0': 'Following the standard practice in the master equation approach to molecular motors [CITATION] for capturing the effects of external force on the mechano-chemistry [CITATION], we multiply the "mechanical" steps of the cycle by an appropriate Boltzmann-like factor.', '0812.4692-4-25-1': 'More precisely, we assume that the load force [MATH] significantly affects only those steps of the mechano-chemical cycle of an RNAP which involve its mechanical movement, i.e., forward or backward movement in real space.', '0812.4692-4-25-2': 'Therefore, we assume that force-dependence of these rate constants are given by [EQUATION] where [MATH] and [MATH] are the corresponding values of the rate constants in the absence of the load force.', '0812.4692-4-25-3': 'In equation ([REF]) the symbol [MATH] nm is the typical length of a single nucleotide.', '0812.4692-4-25-4': 'None of the rate constants other than the three listed in ([REF]) are affected by the load force [MATH].', '0812.4692-4-26-0': 'The force-velocity relation of individual RNAP motors in our model is shown in fig. [REF] for three different concentrations of the NTPs.', '0812.4692-4-26-1': 'The average velocity decreases monotonically with increasing load force; the convex shape of the load-velocity curves are very similar to those reported by Wang et al. [CITATION] in their single RNAP model.', '0812.4692-4-26-2': 'In this model, the higher is the NTP concentration the larger is the stall force [MATH].', '0812.4692-4-26-3': 'Moreover, for a given load force [MATH], the average velocity of the RNAP is larger at higher NTP concentration.', '0812.4692-4-27-0': '### Effects of external torque', '0812.4692-4-28-0': 'Now we consider the effects of an externally imposed torque which assists opening, but opposes closing of the palm-like shape of an isolated RNAP.', '0812.4692-4-28-1': 'These competing effects of the same torque on the two steps of each mechano-chemical cycle of individual RNAP motors has nontrivial effects on the rate of RNA synthesis.', '0812.4692-4-28-2': 'The above mentioned effects of the torque [MATH] on opening and closing of RNAP are captured by the following choice of the corresponding rate constants [EQUATION] where [MATH] and [MATH] on the right hand sides of the equations are the rate constants in the absence of external torque.', '0812.4692-4-29-0': 'The average velocity of the RNAP motors are plotted against the torque in fig. [REF] for different sets of values of the model parameters.', '0812.4692-4-29-1': 'The most dramatic effect of the torque is the nonmonotonic variation of [MATH] with [MATH] in several experimentally accessible regimes of parameter values.', '0812.4692-4-29-2': 'Increasing torque increases the opening rate and decreases the closing rate.', '0812.4692-4-29-3': 'As long as the opening rate is still lower (and, hence, rate limiting), the velocity increases with increasing torque.', '0812.4692-4-29-4': 'A peak appears where closing becomes the rate limiting step.', '0812.4692-4-29-5': 'In order to demonstrate the effect of this crossover from a regime dominated by opening to that dominated closing of RNAP, we have plotted one curve in fig. [REF](b) corresponding to [MATH] for which the maximum occurs at [MATH].', '0812.4692-4-30-0': '## Effects of steric interactions: flux-density relation', '0812.4692-4-31-0': 'We have plotted mean-field estimate ([REF]) of flux [MATH] against the coverage density [MATH] in fig. [REF] for three different valus of the ratio [MATH] at fixed NTP concentration of [MATH].', '0812.4692-4-31-1': 'In the traffic science literature [CITATION] such flux-density relations are usually referred to as the fundamental diagram.', '0812.4692-4-31-2': 'The qualitative features of the fundamental diagrams in fig. [REF] are similar to those observed earlier in the two state model of RNAP traffic [CITATION].', '0812.4692-4-31-3': 'The most notable feature of these fundamental diagrams is their asymmetric shape; this is in sharp contrast to the symmetry of fundamental diagram of a TASEP about [MATH].', '0812.4692-4-31-4': 'The physical reason for the asymmetric shape of the fundamental diagram in fig. [REF] is as follows: to get maximum current, system should have maximum number of particle-hole pairs.', '0812.4692-4-31-5': 'For [MATH], maximum number of particle hole pairs occur when [MATH], i.e., [MATH].', '0812.4692-4-31-6': 'For [MATH], the magnitude of [MATH] can be found from the corresponding condition [MATH], i.e., [MATH], which corresponds to the coverage density [EQUATION]', '0812.4692-4-32-0': '# RNAP TRAFFIC MODEL UNDER OPEN BOUNDARY CONDITION', '0812.4692-4-33-0': 'For modeling transcription, the open boundary condition is more realistic than the periodic boundary conditions.', '0812.4692-4-33-1': 'Under this conditon a RNAP can attach to start site(labled as [MATH]) with rate [MATH] provided none of the first [MATH] sites on the lattice is covered by any other RNAP.', '0812.4692-4-33-2': 'Similarly, after reaching the site [MATH], an RNAP leaves the DNA track with rate [MATH].', '0812.4692-4-33-3': 'We calculate the conditional probability [MATH] same way as we have done for the 2-state model of RNAP [CITATION].', '0812.4692-4-33-4': 'In this case, under mean-field approximation, the master equations for the probabilities [MATH] are given by [EQUATION]', '0812.4692-4-33-5': 'Where [MATH] is the Heaviside step function defined by [EQUATION] and [EQUATION]', '0812.4692-4-33-6': 'We have numerically solved the mean-field equations ([REF])-([REF]) in the steady-state to compute the corresponding flux of the RNAP motors.', '0812.4692-4-33-7': 'These mean-field theoretic estimates of flux are compared with the corresponding data obtained from direct computer simulations of the model.', '0812.4692-4-33-8': 'These results are plotted as functions of the rate constants [MATH] and [MATH], respectively, in figs. [REF] (a) and (b).', '0812.4692-4-33-9': 'The flux rises with increasing [MATH], but the rate of rise decreases with increasing [MATH].', '0812.4692-4-33-10': 'Eventually, the flux saturates because [MATH] is no longer rate-limiting.', '0812.4692-4-33-11': 'This trend of variation of the total rate of RNA synthesis with the concentration of NTP is similar to that observed earlier in our 2-state model of RNAP traffic [CITATION].', '0812.4692-4-33-12': 'Thus, a 2-state model of RNAP would be adequate to capture the dependence of the rate of RNA synthesis on the NTP concentration.', '0812.4692-4-34-0': '# Summary and conclusions', '0812.4692-4-35-0': 'In this paper we have extended our earlier 2-state model of RNAP traffic to a 4-state model so as to capture some important cyclic shape changes in the mechano-chemical cycle of each RNAP.', '0812.4692-4-35-1': 'The new model predicts the effects of these shape changes on the rate of RNA synthesis.', '0812.4692-4-35-2': 'Moreover, we have used the same model in the extremely low density limit to extract the force-velocity relation.', '0812.4692-4-35-3': 'Finally, we have demonstrated a novel nonmonotonic variation of the average speed of the RNAP motors (and, hence, a nonmonotonic variation of the rate of synthesis of RNA) with the increase of an externally imposed torque on the individual motors.', '0812.4692-4-35-4': 'Nonmonotonic variation of average speed with external load force (not external torque) has been observed earlier [CITATION] in the case of several DNA polymerases.', '0812.4692-4-35-5': 'But, the physical origin of this nonmonotonicity, as interpreted in ref. [CITATION], is quite different from that responsible for the nonmonotonic variation of average speed with external torque.', '0812.4692-4-35-6': 'Moreover, the present version of our model does not explicitly capture the effect of possible twisting of the DNA by the polymerase.', '0812.4692-4-35-7': 'In principle, it should be possible to test the new predictions of our model by carrying out laboratory experiments in-vitro.', '0812.4692-4-36-0': 'To our knowledge, it may not be possible, at present, to apply a torque that would directly oppose (or assist) the opening (or closing) of a RNAP.', '0812.4692-4-36-1': 'Therefore, we now suggest a possible alternative technique.', '0812.4692-4-36-2': 'The nonmonotonic variation of the velocity of the RNAP with external torque will result also in the case where the torque opposes, rather than assisting, the opening of the individual RNAP motors (i.e., opposite to the direction of the torque shown in fig. [REF]).', '0812.4692-4-36-3': 'Suppose, the finger-like domain of each RNAP is genetically cross-linked with the palm domain by a protein whose entropic elastic constant is significant.', '0812.4692-4-36-4': 'When such an RNAP tends to open up, the cross-linking protein opposes the opening.', '0812.4692-4-36-5': 'By repeating the experiment with different cross-linkers, whose entropic elastic constants are different, one can study the variation of the average velocity of the RNAP motors with the applied torque.', '0812.4692-4-37-0': 'In spite of some crucial differences, most of the polynucleotide polymerases seem to share a common "cupped right hand" architecture [CITATION].', '0812.4692-4-37-1': 'Therefore, it should be possible to make minor modifications in our 4-state model of RNAP so as to develop similar models of other polynucleotide polymerases.'}

1302.2461

{'1302.2461-1-0-0': 'Solid solutions of the magnetic insulators Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] (Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] ) have been prepared in polycrystalline form for the first time.', '1302.2461-1-0-1': 'Single crystalline material was obtained using a mirror image floating zone technique.', '1302.2461-1-0-2': 'X-ray diffraction data taken at room temperature indicate that the space group of Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] remains unchanged for all values of [MATH] , while the cell parameters depend on the chemical composition, as expected.', '1302.2461-1-0-3': 'Magnetization data, measured from 300K down to 2K, suggests that the interaction constant [MATH] within the Cr [MATH] dimers varies in a peculiar way as a function of [MATH] , starting at [MATH] K for [MATH] , then first slightly dropping to [MATH] K for [MATH] , before reaching [MATH] K for [MATH] .', '1302.2461-1-1-0': '# Introduction', '1302.2461-1-2-0': 'In the last years, many spin dimer systems have been shown to exhibit a field induced phase transition of the magnetic subsystem [CITATION] below a critical temperature and above a certain critical magnetic field [MATH] .', '1302.2461-1-2-1': 'This phenomenon has been interpreted in terms of a Bose-Einstein condensation of the magnetic quasiparticles (triplons) in connection with the collective dimer states[CITATION].', '1302.2461-1-3-0': 'The materials Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] are examples for such spin dimer systems, and they have been considered as possible candidates for a triplon BEC [CITATION].', '1302.2461-1-3-1': 'Recently, these two compounds have been proposed as candidates for a new experimental scheme to probe the macroscopic phase coherence that should be found in all Bose-Einstein condensates [CITATION].', '1302.2461-1-3-2': 'It has been argued that a solid solution of Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] might have a different critical field than the mother compounds, but with otherwise similar physical properties.', '1302.2461-1-3-3': 'The chemical potential for the triplon quasiparticles in such compounds is controlled by the external magnetic field through [MATH] [CITATION] with [MATH] the Bohr magneton and [MATH] the Landé factor.', '1302.2461-1-3-4': 'A change in the chemical potential stemming from, e.g., a changed stoichiometry leading to slightly differing values for [MATH] between the two compounds, might induce Josephson effects in a device of two of such crystals coupled together[CITATION].', '1302.2461-1-3-5': 'We therefore systematically studied the system Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] to obtain information about its physical properties, in particular about the magnetic interaction constants that determine the value of the critical fields [MATH] .', '1302.2461-1-4-0': '# Experimental details', '1302.2461-1-5-0': 'The samples were first synthesized as polycrystalline powders using standard solid-state reaction schemes.', '1302.2461-1-5-1': 'We used three different methods to prepare the polycrystalline material.', '1302.2461-1-5-2': 'First, Ba(NO [MATH] , Sr(NO [MATH] and Cr(NO [MATH] 9H [MATH] O were used as reactants.', '1302.2461-1-5-3': 'The powders were mixed according to [EQUATION] dissolved in water and heated afterwards while permanently stirred to keep the mixture homogeneous.', '1302.2461-1-5-4': 'After evaporating the water, the remaining powder was ground and heated under flowing argon at 915 [MATH] C for 24h to remove any excess water and NO [MATH] .', '1302.2461-1-5-5': 'The resulting oxide powder was ground again, mechanically pressed into pellets and sintered at 1100 [MATH] C for 48 h under flowing Ar.', '1302.2461-1-5-6': 'For [MATH] and [MATH] , rods were hydrostatically pressed and annealed under the same conditions.', '1302.2461-1-5-7': 'The thus obtained polycrystalline material was black.', '1302.2461-1-5-8': 'An optical floating zone method was used to grow crystals from those rods [CITATION].', '1302.2461-1-5-9': 'Two of the resulting polycrystalline rods were then mounted in a Crystal Systems Inc. image furnace using Pt wire, a short one (about 3cm) as a seed rod, and a longer one (about 10cm) as a feed rod.', '1302.2461-1-5-10': 'The crystal growth was performed in two stages, both in a high-purity environment of flowing Ar.', '1302.2461-1-5-11': 'The first stage was performed relatively fast (growth rates between 21mm/h and 25mm/h) to produce a premelted rod with a high density.', '1302.2461-1-5-12': 'For the second stage, the premelted rod was used as a feed rod with the remnants of the former feed rod as a seed.', '1302.2461-1-5-13': 'The growth rate was relatively low (between 2mm/h and 5mm/h), but was varied during the growth to maintain the zone stability.', '1302.2461-1-5-14': 'The resulting boule consisted of several, well oriented grains (see Fig. [REF] for the X-ray Laue-diffractogram).', '1302.2461-1-6-0': 'While we obtained single crystalline samples for [MATH] , attempts with [MATH] have not yet resulted in single crystalline material.', '1302.2461-1-7-0': 'Additionally, polycrystalline samples of several stoichiometries were prepared according to [EQUATION] and [EQUATION]', '1302.2461-1-7-1': 'The respective carbonates and chromium oxides were mixed, ground and heated under flowing argon at 1000°C for 24h.', '1302.2461-1-7-2': 'The resulting material was ground again and sealed in an alumina crucible inside a quartz tube under vacuum to prevent a reaction between sample and tube.', '1302.2461-1-7-3': 'The tube was heated up to 1200°C for 24h and quenched in water.', '1302.2461-1-7-4': 'This last step was repeated several times, with intermediate grindings.', '1302.2461-1-7-5': 'Each of the three preparation methods yielded homogeneous black polycrystalline material and we did not find relevant differences for any value of x between 0 and 3.', '1302.2461-1-7-6': 'However, we noted that grinding the material again thoroughly in air or dry He-atmosphere lead to a slightly green color of the ground powder for every value of x.', '1302.2461-1-8-0': 'The results shown below were obtained from the samples prepared by the first method mentioned above, and from single crystalline material (for [MATH] ).', '1302.2461-1-9-0': '# Crystallographic structure', '1302.2461-1-10-0': 'For the structural analysis, parts of the grown boules as well as corresponding polycrystalline samples with [MATH] were ground and examined using Cu [MATH] radiation.', '1302.2461-1-10-1': 'The resulting patterns were analyzed using the Rietveld method (FullProf suite).', '1302.2461-1-10-2': 'As expected, no difference was found between ground single crystals and polycrystalline material.', '1302.2461-1-10-3': 'All of the observed reflections of the mixed crystals are in good agreement with the space group R [MATH] m (see Fig. [REF]), which is the same as that of the mother compounds with [MATH] and [MATH] .', '1302.2461-1-10-4': 'As scattering coefficients for Cr [MATH] were not available, we used the known values for Cr [MATH] .', '1302.2461-1-10-5': 'We did not find any relevant systematic change in the relative atomic positions with varying x, although it should be noted that the sensitivity on the oxygen position is not high enough to exclude such a shift.', '1302.2461-1-10-6': "The absolue values of the cell parameters, however, depend linearly on the strontium content [MATH] , in agreement with Vegard's law (see Fig. [REF]).", '1302.2461-1-11-0': '# Magnetic properties', '1302.2461-1-12-0': 'The magnetic properties of the polycrystalline samples were analyzed using a commercial SQUID magnetometer (Quantum Design Inc.).', '1302.2461-1-12-1': 'The magnetization was measured for temperatures between 5K and 300K and in magnetic fields of [MATH] and [MATH] .', '1302.2461-1-12-2': 'For compositions with [MATH] and [MATH] , the magnetization shows a pronounced maximum at low temperatures.', '1302.2461-1-12-3': 'The experimental data can be well described with the Bleaney-Bowers formula for interacting dimers, [EQUATION] with [MATH] which is only slightly anisotropic in these systems[CITATION], [MATH] the density of the coupled ions and [MATH] and [MATH] the inter- and intradimer coupling constant, respectively.', '1302.2461-1-12-4': 'For intermediate values of [MATH] around 1.5, a paramagnetic background becomes more and more relevant so that a Brillouin term had to be included, [EQUATION] where [MATH] denotes the density of the corresponding uncoupled ions.', '1302.2461-1-12-5': 'The experimental data are reasonably well described by the sum of above terms (see Fig [REF]).', '1302.2461-1-12-6': 'However, as the fit is not very sensitive to the interdimer interaction constant [MATH] (especially in the presence of a strong paramagnetic background) we could not extract reliable values for it.', '1302.2461-1-13-0': 'The thus obtained intradimer interaction constant [MATH] strongly depends on the stoichiometry, as shown in Fig [REF].', '1302.2461-1-13-1': 'Surprisingly, it first decreases with increasing strontium content up to [MATH] , before increasing again up to the value for pure Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-1-13-2': 'As the relative atomic positions do not change with [MATH] , we attribute the overall increase of the interaction constant to a decrease of the absolute intradimer distance of the Cr [MATH] -ions from pure Ba [MATH] Cr [MATH] O [MATH] to pure Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-1-13-3': 'As a consequence, the variation of [MATH] with [MATH] should lead to an associated change in the critical field [MATH] , which we will investigate in a separate report.', '1302.2461-1-14-0': 'In Fig. [REF] we also show the fraction [MATH] of dimerized Cr [MATH] ions and we compare it in Fig. [REF] with the density [MATH] of uncoupled Cr [MATH] ions, both treated as independent fitting parameters.', '1302.2461-1-14-1': 'The paramagnetic background for intermediate [MATH] is substantial, but its microscopic origin is not clear up to now.', '1302.2461-1-14-2': 'The sum of the prefactors [MATH] and [MATH] is reasonably close to 1, which supports the validity of our fitting procedure.', '1302.2461-1-14-3': 'An increase of the preparation temperature to 1250°C as well as repeated grinding and annealing of the polycrystalline samples did not reduce this background contribution substantially, but our corresponding X-ray diffraction data do, most interestingly, not suggest the appearance of possible impurity phases.', '1302.2461-1-14-4': 'We therefore believe that this magnetic background is intrinsic to the polycrystalline system Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-1-15-0': '# Conclusion', '1302.2461-1-16-0': 'We have for the first time synthesized mixed crystals of the magnetic insulators Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-1-16-1': 'The structure of the solid solution series remains unchanged, but with linearly decreasing cell parameters for increasing Sr-content [MATH] .', '1302.2461-1-16-2': 'The magnetic behavior indicates the presence of a dimerized spin system for all values of [MATH] , with an interaction constant [MATH] that strongly depends on the composition and surprisingly falls below the value of pure Ba [MATH] Cr [MATH] O [MATH] in certain range of [MATH] .', '1302.2461-1-16-3': 'For intermediate values ( [MATH] ), a paramagnetic background appears that does not seem to be related to impurity phases and is therefore probably intrinsic to Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .'}

{'1302.2461-2-0-0': 'Solid solutions of the magnetic insulators Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] (Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] ) have been prepared in polycrystalline form for the first time.', '1302.2461-2-0-1': 'Single crystalline material was obtained using a mirror image floating zone technique.', '1302.2461-2-0-2': 'X-ray diffraction data taken at room temperature indicate that the space group of Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] remains unchanged for all values of [MATH] , while the cell parameters depend on the chemical composition, as expected.', '1302.2461-2-0-3': 'Magnetization data, measured from 300K down to 2K, suggests that the interaction constant [MATH] within the Cr [MATH] dimers varies in a peculiar way as a function of [MATH] , starting at [MATH] K for [MATH] , then first slightly dropping to [MATH] K for [MATH] , before reaching [MATH] K for [MATH] .', '1302.2461-2-1-0': '# Introduction', '1302.2461-2-2-0': 'In the last years, many spin dimer systems have been shown to exhibit a field induced phase transition of the magnetic subsystem [CITATION] below a critical temperature and above a certain critical magnetic field [MATH] .', '1302.2461-2-2-1': 'This phenomenon has been interpreted in terms of a Bose-Einstein condensation of the magnetic quasiparticles (triplons) in connection with the collective dimer states[CITATION].', '1302.2461-2-3-0': 'The materials Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] are examples for such spin dimer systems, and they have been considered as possible candidates for a triplon BEC [CITATION].', '1302.2461-2-3-1': 'Recently, these two compounds have been proposed as candidates for a new experimental scheme to probe the macroscopic phase coherence that should be found in all Bose-Einstein condensates [CITATION].', '1302.2461-2-3-2': 'It has been argued that a solid solution of Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] might have a different critical field than the mother compounds, but with otherwise similar physical properties.', '1302.2461-2-3-3': 'The chemical potential for the triplon quasiparticles in such compounds is controlled by the external magnetic field through [MATH] [CITATION] with [MATH] the Bohr magneton and [MATH] the Landé factor.', '1302.2461-2-3-4': 'A change in the chemical potential stemming from, e.g., a changed stoichiometry leading to slightly differing values for [MATH] between the two compounds, might induce Josephson effects in a device of two of such crystals coupled together[CITATION].', '1302.2461-2-3-5': 'We therefore systematically studied the system Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] to obtain information about its physical properties, in particular about the magnetic interaction constants that determine the value of the critical fields [MATH] .', '1302.2461-2-4-0': '# Experimental details', '1302.2461-2-5-0': 'The samples were first synthesized as polycrystalline powders using standard solid-state reaction schemes.', '1302.2461-2-5-1': 'We used three different methods to prepare the polycrystalline material.', '1302.2461-2-5-2': 'First, Ba(NO [MATH] , Sr(NO [MATH] and Cr(NO [MATH] 9H [MATH] O were used as reactants.', '1302.2461-2-5-3': 'The powders were mixed according to [EQUATION] dissolved in water and heated afterwards while continuously stirred to keep the mixture homogeneous.', '1302.2461-2-5-4': 'After evaporating the water, the remaining powder was ground and heated under flowing argon at 915 [MATH] C for 24h to remove any excess water and NO [MATH] .', '1302.2461-2-5-5': 'The resulting oxide powder was ground again, mechanically pressed into pellets and sintered at 1100 [MATH] C for 48 h under flowing Ar.', '1302.2461-2-5-6': 'For [MATH] and [MATH] , rods were hydrostatically pressed and annealed under the same conditions.', '1302.2461-2-5-7': 'The thus obtained polycrystalline material was black.', '1302.2461-2-5-8': 'An optical floating zone method was used to grow crystals from those rods [CITATION].', '1302.2461-2-5-9': 'Two of the resulting polycrystalline rods were then mounted in a Crystal Systems Inc. image furnace using Pt wire, a short one (about 3cm) as a seed rod, and a longer one (about 10cm) as a feed rod.', '1302.2461-2-5-10': 'The crystal growth was performed in two stages, both in a high-purity environment of flowing Ar.', '1302.2461-2-5-11': 'The first stage was performed relatively fast (growth rates between 21mm/h and 25mm/h) to produce a premelted rod with a high density.', '1302.2461-2-5-12': 'For the second stage, the premelted rod was used as a feed rod with the remnants of the former feed rod as a seed.', '1302.2461-2-5-13': 'The growth rate was relatively low (between 2mm/h and 5mm/h), but was varied during the growth to maintain the zone stability.', '1302.2461-2-5-14': 'The resulting boule consisted of several, well oriented grains (see Fig. [REF] for the X-ray Laue-diffractogram).', '1302.2461-2-6-0': 'While we obtained single crystalline samples for [MATH] , attempts with [MATH] have not yet resulted in single crystalline material.', '1302.2461-2-7-0': 'Additionally, polycrystalline samples of several stoichiometries were prepared according to [EQUATION] and [EQUATION]', '1302.2461-2-7-1': 'The respective carbonates and chromium oxides were mixed, ground and heated under flowing argon at 1000°C for 24h.', '1302.2461-2-7-2': 'The resulting material was ground again and sealed in an alumina crucible inside a quartz tube under vacuum to prevent a reaction between sample and tube.', '1302.2461-2-7-3': 'The tube was heated up to 1200°C for 24h and quenched in water.', '1302.2461-2-7-4': 'This last step was repeated several times, with intermediate grindings.', '1302.2461-2-7-5': 'Each of the three preparation methods yielded homogeneous black polycrystalline material and we did not find relevant differences for any value of x between 0 and 3.', '1302.2461-2-7-6': 'However, we noted that grinding the material again thoroughly in air or dry He-atmosphere lead to a slightly green color of the ground powder for every value of x.', '1302.2461-2-8-0': 'The results shown below were obtained from the samples prepared by the first method mentioned above, and from single crystalline material (for [MATH] ).', '1302.2461-2-9-0': '# Crystallographic structure', '1302.2461-2-10-0': 'In order to perform structural analysis, parts of the boule as well as corresponding polycrystalline samples with [MATH] were ground and examined using Cu [MATH] X-ray radiation.', '1302.2461-2-10-1': 'The resulting patterns were analyzed using the Rietveld method (FullProf suite).', '1302.2461-2-10-2': 'As expected, no difference was found between ground single crystals and polycrystalline material.', '1302.2461-2-10-3': 'All of the observed reflections of the mixed crystals are in good agreement with the space group R [MATH] m (see Fig. [REF]), which is the same as that of the mother compounds with [MATH] and [MATH] .', '1302.2461-2-10-4': 'As scattering coefficients for Cr [MATH] were not available, we used the known values for Cr [MATH] .', '1302.2461-2-10-5': 'We did not find any relevant systematic change in the relative atomic positions as a function of x, although it should be noted that the sensitivity on the oxygen position is not high enough to exclude such a shift.', '1302.2461-2-10-6': "The absolue values of the cell parameters, however, depend linearly on the strontium content [MATH] , in agreement with Vegard's law (see Fig. [REF]).", '1302.2461-2-11-0': '# Magnetic properties', '1302.2461-2-12-0': 'The magnetic properties of the polycrystalline samples were analyzed using a commercial SQUID magnetometer (Quantum Design Inc.).', '1302.2461-2-12-1': 'The magnetization was measured for temperatures between 5K and 300K and in magnetic fields of [MATH] and [MATH] .', '1302.2461-2-12-2': 'For compositions with [MATH] and [MATH] , the magnetization shows a pronounced maximum at low temperatures.', '1302.2461-2-12-3': 'The experimental data can be well described with the Bleaney-Bowers formula for interacting dimers, [EQUATION] with [MATH] which is only slightly anisotropic in these systems[CITATION], [MATH] the density of the coupled ions and [MATH] and [MATH] the inter- and intradimer coupling constant, respectively.', '1302.2461-2-12-4': 'For intermediate values of [MATH] around 1.5, a paramagnetic background becomes more and more relevant so that a Brillouin term had to be included, [EQUATION] where [MATH] denotes the density of the corresponding uncoupled ions.', '1302.2461-2-12-5': 'The experimental data are reasonably well described by the sum of above terms (see Fig [REF]).', '1302.2461-2-12-6': 'However, as the fit is not very sensitive to the interdimer interaction constant [MATH] (especially in the presence of a strong paramagnetic background) we could not extract reliable values for it.', '1302.2461-2-13-0': 'The thus obtained intradimer interaction constant [MATH] strongly depends on the stoichiometry, as shown in Fig [REF].', '1302.2461-2-13-1': 'Surprisingly, it first decreases with increasing strontium content up to [MATH] , before increasing again up to the value for pure Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-2-13-2': 'As the relative atomic positions do not change with [MATH] , we attribute the overall increase of the interaction constant to a decrease of the absolute intradimer distance of the Cr [MATH] -ions from pure Ba [MATH] Cr [MATH] O [MATH] to pure Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-2-13-3': 'As a consequence, the variation of [MATH] with [MATH] should lead to an associated change in the critical field [MATH] , which we will investigate in a separate report.', '1302.2461-2-14-0': 'In Fig. [REF] we also show the fraction [MATH] of dimerized Cr [MATH] ions and we compare it in Fig. [REF] with the density [MATH] of uncoupled Cr [MATH] ions, both treated as independent fitting parameters.', '1302.2461-2-14-1': 'The paramagnetic background for intermediate [MATH] is substantial, but its microscopic origin is not clear up to now.', '1302.2461-2-14-2': 'The sum of the prefactors [MATH] and [MATH] is reasonably close to 1, which supports the validity of our fitting procedure.', '1302.2461-2-14-3': 'An increase of the preparation temperature to 1250°C as well as repeated grinding and annealing of the polycrystalline samples did not reduce this background contribution substantially, but our corresponding X-ray diffraction data do, most interestingly, not suggest the appearance of possible impurity phases.', '1302.2461-2-14-4': 'We therefore believe that this magnetic background is intrinsic to the polycrystalline system Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-2-15-0': '# Conclusion', '1302.2461-2-16-0': 'We have for the first time synthesized mixed crystals of the magnetic insulators Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-2-16-1': 'The structure of the solid solution series remains unchanged, but with linearly decreasing cell parameters for increasing Sr-content [MATH] .', '1302.2461-2-16-2': 'The magnetic behavior indicates the presence of a dimerized spin system for all values of [MATH] , with an interaction constant [MATH] that strongly depends on the composition and surprisingly falls below the value of pure Ba [MATH] Cr [MATH] O [MATH] in certain range of [MATH] .', '1302.2461-2-16-3': 'For intermediate values ( [MATH] ), a paramagnetic background appears that does not seem to be related to impurity phases and is therefore probably intrinsic to Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .'}

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'1302.2461-2-0-0'], ['1302.2461-1-0-1', '1302.2461-2-0-1'], ['1302.2461-1-0-2', '1302.2461-2-0-2'], ['1302.2461-1-0-3', '1302.2461-2-0-3'], ['1302.2461-1-13-0', '1302.2461-2-13-0'], ['1302.2461-1-13-1', '1302.2461-2-13-1'], ['1302.2461-1-13-2', '1302.2461-2-13-2'], ['1302.2461-1-13-3', '1302.2461-2-13-3'], ['1302.2461-1-10-1', '1302.2461-2-10-1'], ['1302.2461-1-10-2', '1302.2461-2-10-2'], ['1302.2461-1-10-3', '1302.2461-2-10-3'], ['1302.2461-1-10-4', '1302.2461-2-10-4'], ['1302.2461-1-10-6', '1302.2461-2-10-6'], ['1302.2461-1-8-0', '1302.2461-2-8-0'], ['1302.2461-1-2-0', '1302.2461-2-2-0'], ['1302.2461-1-2-1', '1302.2461-2-2-1'], ['1302.2461-1-7-0', '1302.2461-2-7-0'], ['1302.2461-1-7-1', '1302.2461-2-7-1'], ['1302.2461-1-7-2', '1302.2461-2-7-2'], ['1302.2461-1-7-3', '1302.2461-2-7-3'], ['1302.2461-1-7-4', '1302.2461-2-7-4'], ['1302.2461-1-7-5', '1302.2461-2-7-5'], ['1302.2461-1-7-6', '1302.2461-2-7-6'], ['1302.2461-1-12-0', '1302.2461-2-12-0'], ['1302.2461-1-12-1', '1302.2461-2-12-1'], ['1302.2461-1-12-2', '1302.2461-2-12-2'], ['1302.2461-1-12-3', '1302.2461-2-12-3'], ['1302.2461-1-12-4', '1302.2461-2-12-4'], ['1302.2461-1-12-5', '1302.2461-2-12-5'], ['1302.2461-1-12-6', '1302.2461-2-12-6'], ['1302.2461-1-6-0', '1302.2461-2-6-0'], ['1302.2461-1-14-0', '1302.2461-2-14-0'], ['1302.2461-1-14-1', '1302.2461-2-14-1'], ['1302.2461-1-14-2', '1302.2461-2-14-2'], ['1302.2461-1-14-3', '1302.2461-2-14-3'], ['1302.2461-1-14-4', '1302.2461-2-14-4'], ['1302.2461-2-13-1', '1302.2461-3-13-1'], ['1302.2461-2-14-0', '1302.2461-3-14-0'], ['1302.2461-2-14-2', '1302.2461-3-14-2'], ['1302.2461-2-14-3', '1302.2461-3-14-3'], ['1302.2461-2-14-4', '1302.2461-3-14-6'], ['1302.2461-2-3-0', '1302.2461-3-3-0'], ['1302.2461-2-3-1', '1302.2461-3-3-1'], ['1302.2461-2-3-5', '1302.2461-3-3-5'], ['1302.2461-2-16-0', '1302.2461-3-16-0'], ['1302.2461-2-0-0', '1302.2461-3-0-0'], ['1302.2461-2-0-1', '1302.2461-3-0-1'], ['1302.2461-2-12-0', '1302.2461-3-12-0'], ['1302.2461-2-12-1', '1302.2461-3-12-1'], ['1302.2461-2-12-2', '1302.2461-3-12-2'], ['1302.2461-2-12-6', '1302.2461-3-12-6'], ['1302.2461-2-7-0', '1302.2461-3-7-0'], ['1302.2461-2-7-1', '1302.2461-3-7-1'], ['1302.2461-2-7-2', '1302.2461-3-7-2'], ['1302.2461-2-7-4', '1302.2461-3-7-4'], ['1302.2461-2-10-0', '1302.2461-3-10-0'], ['1302.2461-2-10-1', '1302.2461-3-10-1'], ['1302.2461-2-10-2', '1302.2461-3-10-2'], ['1302.2461-2-10-3', '1302.2461-3-10-3'], ['1302.2461-2-10-4', '1302.2461-3-10-4'], ['1302.2461-2-10-5', '1302.2461-3-10-8'], ['1302.2461-2-2-0', '1302.2461-3-2-0'], ['1302.2461-2-5-0', '1302.2461-3-5-0'], ['1302.2461-2-5-1', '1302.2461-3-5-1'], ['1302.2461-2-5-2', '1302.2461-3-5-2'], ['1302.2461-2-5-3', '1302.2461-3-5-3'], ['1302.2461-2-5-4', '1302.2461-3-5-4'], ['1302.2461-2-5-5', '1302.2461-3-5-5'], ['1302.2461-2-5-6', '1302.2461-3-5-6'], ['1302.2461-2-5-8', '1302.2461-3-5-8'], ['1302.2461-2-5-9', '1302.2461-3-5-9'], ['1302.2461-2-5-10', '1302.2461-3-5-10'], ['1302.2461-2-5-11', '1302.2461-3-5-11'], ['1302.2461-2-5-12', '1302.2461-3-5-12'], ['1302.2461-2-5-13', '1302.2461-3-5-13'], ['1302.2461-2-5-14', '1302.2461-3-5-14'], ['1302.2461-1-5-3', '1302.2461-2-5-3'], ['1302.2461-1-10-0', '1302.2461-2-10-0'], ['1302.2461-1-10-5', '1302.2461-2-10-5'], ['1302.2461-2-13-0', '1302.2461-3-13-0'], ['1302.2461-2-13-2', '1302.2461-3-13-2'], ['1302.2461-2-13-3', '1302.2461-3-13-3'], ['1302.2461-2-14-1', '1302.2461-3-14-1'], ['1302.2461-2-3-2', '1302.2461-3-3-2'], ['1302.2461-2-3-3', '1302.2461-3-3-3'], ['1302.2461-2-3-4', '1302.2461-3-3-4'], ['1302.2461-2-16-1', '1302.2461-3-16-1'], ['1302.2461-2-0-2', '1302.2461-3-0-2'], ['1302.2461-2-0-3', '1302.2461-3-0-3'], ['1302.2461-2-12-3', '1302.2461-3-12-3'], ['1302.2461-2-12-4', '1302.2461-3-12-4'], ['1302.2461-2-12-5', '1302.2461-3-12-5'], ['1302.2461-2-6-0', '1302.2461-3-6-0'], ['1302.2461-2-7-3', '1302.2461-3-7-3'], ['1302.2461-2-7-5', '1302.2461-3-7-5'], ['1302.2461-2-7-6', '1302.2461-3-7-6'], ['1302.2461-2-2-1', '1302.2461-3-2-1'], ['1302.2461-2-8-0', '1302.2461-3-8-0'], ['1302.2461-2-5-7', '1302.2461-3-5-7'], ['1302.2461-2-16-2', '1302.2461-3-16-2'], ['1302.2461-2-16-3', '1302.2461-3-16-2'], ['1302.2461-2-10-6', '1302.2461-3-10-5']]

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[]

[['1302.2461-2-16-2', '1302.2461-3-16-2'], ['1302.2461-2-16-3', '1302.2461-3-16-2'], ['1302.2461-2-10-6', '1302.2461-3-10-5']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1302.2461

{'1302.2461-3-0-0': 'Solid solutions of the magnetic insulators Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] (Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] ) have been prepared in polycrystalline form for the first time.', '1302.2461-3-0-1': 'Single crystalline material was obtained using a mirror image floating zone technique.', '1302.2461-3-0-2': 'X-ray diffraction data taken at room temperature indicate that the space group of Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] remains unchanged for all values of x, while the cell parameters depend on the chemical composition, as expected.', '1302.2461-3-0-3': 'Magnetization data, measured from 300K down to 2K, suggest that the interaction constant [MATH] within the Cr [MATH] dimers varies in a peculiar way as a function of x, starting at [MATH] K for [MATH] , then first slightly dropping to [MATH] K for [MATH] , before reaching [MATH] K for [MATH] .', '1302.2461-3-1-0': '# Introduction', '1302.2461-3-2-0': 'In the last years, many spin dimer systems have been shown to exhibit a field induced phase transition of the magnetic subsystem [CITATION] below a critical temperature and above a certain critical magnetic field [MATH] .', '1302.2461-3-2-1': 'This phenomenon has been interpreted in terms of a Bose-Einstein condensation (BEC) of the magnetic quasiparticles (triplons) in connection with the collective dimer states[CITATION].', '1302.2461-3-3-0': 'The materials Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] are examples for such spin dimer systems, and they have been considered as possible candidates for a triplon BEC [CITATION].', '1302.2461-3-3-1': 'Recently, these two compounds have been proposed as candidates for a new experimental scheme to probe the macroscopic phase coherence that should be found in all Bose-Einstein condensates [CITATION].', '1302.2461-3-3-2': 'It has been argued that compounds in a solid solution of Ba [MATH] Cr [MATH] O [MATH] and Sr [MATH] Cr [MATH] O [MATH] might have different critical fields than the mother compounds, but with otherwise similar physical properties.', '1302.2461-3-3-3': 'The chemical potential [MATH] for the triplon quasiparticles in such materials is controlled by the external magnetic field [MATH] through [MATH] [CITATION] with [MATH] the Bohr magneton and [MATH] the Landé factor.', '1302.2461-3-3-4': 'A change in the chemical potential stemming from, e.g., a changed stoichiometry leading to slightly differing values for [MATH] between two compounds, might induce Josephson effects in a device of two of such crystals coupled together[CITATION].', '1302.2461-3-3-5': 'We therefore systematically studied the system Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] to obtain information about its physical properties, in particular about the magnetic interaction constants that determine the value of the critical fields [MATH] .', '1302.2461-3-4-0': '# Experimental details', '1302.2461-3-5-0': 'The samples were first synthesized as polycrystalline powders using standard solid-state reaction schemes.', '1302.2461-3-5-1': 'We used three different methods to prepare the polycrystalline material.', '1302.2461-3-5-2': 'First, Ba(NO [MATH] , Sr(NO [MATH] and Cr(NO [MATH] 9H [MATH] O were used as reactants.', '1302.2461-3-5-3': 'The powders were mixed according to [EQUATION] dissolved in water and heated afterwards while continuously stirred to keep the mixture homogeneous.', '1302.2461-3-5-4': 'After evaporating the water, the remaining powder was ground and heated under flowing argon at 915 [MATH] C for 24h to remove any excess water and NO [MATH] .', '1302.2461-3-5-5': 'The resulting oxide powder was ground again, mechanically pressed into pellets and sintered at 1100 [MATH] C for 48 h under flowing Ar.', '1302.2461-3-5-6': 'For [MATH] and [MATH] , rods were hydrostatically pressed and annealed under the same conditions.', '1302.2461-3-5-7': 'The obtained polycrystalline material was black.', '1302.2461-3-5-8': 'An optical floating zone method was used to grow crystals from those rods [CITATION].', '1302.2461-3-5-9': 'Two of the resulting polycrystalline rods were then mounted in a Crystal Systems Inc. image furnace using Pt wire, a short one (about 3cm) as a seed rod, and a longer one (about 10cm) as a feed rod.', '1302.2461-3-5-10': 'The crystal growth was performed in two stages, both in a high-purity environment of flowing Ar.', '1302.2461-3-5-11': 'The first stage was performed relatively fast (growth rates between 21mm/h and 25mm/h) to produce a premelted rod with a high density.', '1302.2461-3-5-12': 'For the second stage, the premelted rod was used as a feed rod with the remnants of the former feed rod as a seed.', '1302.2461-3-5-13': 'The growth rate was relatively low (between 2mm/h and 5mm/h), but was varied during the growth to maintain the zone stability.', '1302.2461-3-5-14': 'The resulting boule consisted of several, well oriented grains (see Fig. [REF] for the X-ray Laue-diffractogram).', '1302.2461-3-6-0': 'While we obtained crystalline samples for [MATH] , attempts with [MATH] have not yet resulted in crystalline material.', '1302.2461-3-7-0': 'Additionally, polycrystalline samples of several stoichiometries were prepared according to [EQUATION] and [EQUATION]', '1302.2461-3-7-1': 'The respective carbonates and chromium oxides were mixed, ground and heated under flowing argon at 1000°C for 24h.', '1302.2461-3-7-2': 'The resulting material was ground again and sealed in an alumina crucible inside a quartz tube under vacuum to prevent a reaction between sample and tube.', '1302.2461-3-7-3': 'The tube was heated up to 1200°C for 24h and quenched into water.', '1302.2461-3-7-4': 'This last step was repeated several times, with intermediate grindings.', '1302.2461-3-7-5': 'Each of the three preparation methods yielded homogeneous black polycrystalline material and we did not find relevant differences between the respective products for any value of x between 0 and 3.', '1302.2461-3-7-6': 'However, we noted that grinding the material again thoroughly in air or dry He-atmosphere led to a slightly green color of the ground powder for every value of x.', '1302.2461-3-8-0': 'The results shown below were obtained from the samples prepared by the first method described above, and from single crystalline material (for [MATH] ).', '1302.2461-3-9-0': '# Crystallographic structure', '1302.2461-3-10-0': 'In order to perform structural analysis, parts of the boule as well as corresponding polycrystalline samples with [MATH] were ground and examined using Cu [MATH] X-ray radiation.', '1302.2461-3-10-1': 'The resulting patterns were analyzed using the Rietveld method (FullProf suite).', '1302.2461-3-10-2': 'As expected, no difference was found between ground single crystals and polycrystalline material.', '1302.2461-3-10-3': 'All of the observed reflections of the mixed crystals are in good agreement with the space group R [MATH] m (see Fig. [REF]), which is the same as that of the mother compounds with [MATH] and [MATH] .', '1302.2461-3-10-4': 'As scattering coefficients for Cr [MATH] were not available, we used the known values for Cr [MATH] .', '1302.2461-3-10-5': "The absolute values of the cell parameters depend linearly on the strontium content x, in agreement with Vegard's law (see Fig. [REF]).", '1302.2461-3-10-6': 'In Fig. [REF], we have plotted the distance between two Cr ions forming a single dimer.', '1302.2461-3-10-7': 'As expected, this intradimer distance also decreases smoothly as a function of x.', '1302.2461-3-10-8': 'We did not find any relevant systematic change in the relative atomic positions as a function of x, although it should be noted that the sensitivity on the oxygen position is not high enough to exclude such a shift.', '1302.2461-3-10-9': 'In Fig. [REF], we show the z-value of the atomic position for the Cr and Ba/Sr ions while the respective x and y coordinates were assumed to be zero.', '1302.2461-3-10-10': 'All other coordinates where kept zero for all x.', '1302.2461-3-11-0': '# Magnetic properties', '1302.2461-3-12-0': 'The magnetic properties of the polycrystalline samples were analyzed using a commercial SQUID magnetometer (Quantum Design Inc.).', '1302.2461-3-12-1': 'The magnetization was measured for temperatures between 5K and 300K and in magnetic fields of [MATH] and [MATH] .', '1302.2461-3-12-2': 'For compositions with [MATH] and [MATH] , the magnetization shows a pronounced maximum at low temperatures.', '1302.2461-3-12-3': 'The experimental data can be well described with the Bleaney-Bowers formula for interacting dimers, [EQUATION] with [MATH] which is only slightly anisotropic in these systems[CITATION], [MATH] the density of the coupled ions and [MATH] and [MATH] the inter- and intradimer coupling constants, respectively.', '1302.2461-3-12-4': 'For intermediate values of x around 1.5, a paramagnetic background becomes more and more relevant so that a Brillouin term had to be included, [EQUATION] where [MATH] denotes the density of the corresponding uncoupled ions.', '1302.2461-3-12-5': 'The experimental data are reasonably well described by the sum of the above terms (see Fig. [REF]).', '1302.2461-3-12-6': 'However, as the fit is not very sensitive to the interdimer interaction constant [MATH] (especially in the presence of a strong paramagnetic background) we could not extract reliable values for it.', '1302.2461-3-13-0': 'The thus obtained intradimer interaction constant [MATH] strongly depends on the stoichiometry, as shown in Fig. [REF].', '1302.2461-3-13-1': 'Surprisingly, it first decreases with increasing strontium content up to [MATH] , before increasing again up to the value for pure Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-3-13-2': 'As the relative atomic positions do not change with x (see Fig. [REF]), we attribute the overall increase of the interaction constant to a decrease of the absolute intradimer distance of the Cr [MATH] -ions from pure Ba [MATH] Cr [MATH] O [MATH] to pure Sr [MATH] Cr [MATH] O [MATH] (see Fig. [REF]).', '1302.2461-3-13-3': 'As a consequence, the variation of [MATH] with x should lead to an associated change in the critical field [MATH] , which we will investigate in a separate report.', '1302.2461-3-14-0': 'In Fig. [REF] we also show the fraction [MATH] of dimerized Cr [MATH] ions and we compare it in Fig. [REF] with the density [MATH] of uncoupled Cr [MATH] ions, both treated as independent fitting parameters.', '1302.2461-3-14-1': 'The paramagnetic background for intermediate x is substantial, but its microscopic origin is not clear up to now.', '1302.2461-3-14-2': 'The sum of the prefactors [MATH] and [MATH] is reasonably close to 1, which supports the validity of our fitting procedure.', '1302.2461-3-14-3': 'An increase of the preparation temperature to 1250°C as well as repeated grinding and annealing of the polycrystalline samples did not reduce this background contribution substantially, but our corresponding X-ray diffraction data do, most interestingly, not suggest the appearance of possible impurity phases.', '1302.2461-3-14-4': 'In Fig. [REF] we show the part of the total scattering intensity that is neither associated with the expected diffraction peaks of Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] ) nor with the diffuse background ( [MATH] ), normalized to the total integrated intensity without the diffuse background.', '1302.2461-3-14-5': 'Within the accuracy of this procedure, diffraction peaks stemming from unwanted impurity phases are virtually absent for all values of x.', '1302.2461-3-14-6': 'We therefore believe that this magnetic background is intrinsic to the polycrystalline system Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-3-15-0': '# Conclusion', '1302.2461-3-16-0': 'We have for the first time synthesized mixed crystals of the magnetic insulators Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .', '1302.2461-3-16-1': 'The structure of the solid solution series remains unchanged, but with linearly decreasing cell parameters for increasing Sr-content x.', '1302.2461-3-16-2': 'The magnetic behavior indicates the presence of a dimerized spin system for all values of x, with an interaction constant [MATH] that strongly depends on the composition and surprisingly falls below the value of pure Ba [MATH] Cr [MATH] O [MATH] in certain range of x. For intermediate values ( [MATH] ), a paramagnetic background appears that does not seem to be related to impurity phases and is therefore probably intrinsic to Ba [MATH] Sr [MATH] Cr [MATH] O [MATH] .'}

1204.0880

{'1204.0880-1-0-0': 'In 1978 E. De Giorgi fromulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation [MATH], which are monotone in some direction.', '1204.0880-1-0-1': 'In this paper we prove the analogous statement for the equation [MATH], where the Laplacian is replaced by the Ornstein-Uhlenbeck operator.', '1204.0880-1-0-2': 'Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain an similar result in infinite dimensions by a limit procedure.', '1204.0880-1-1-0': '# Introduction', '1204.0880-1-2-0': 'A celebrated conjecture by De Giorgi [CITATION] asks if bounded entire solutions to the equation [EQUATION] which are strictly increasing in some direction are one-dimensional, in the sense that the level sets [MATH] are hyperplanes, at least if [MATH].', '1204.0880-1-2-1': 'This conjecture has been proved by Ghoussoub and Gui [CITATION] in dimension [MATH], and by Ambrosio and Cabre [CITATION] in dimension [MATH], and a counterexample has been given by del Pino, Kowalczyk and Wei in [CITATION] for [MATH].', '1204.0880-1-2-2': 'While the conjecture is still open for [MATH], a very nice proof has been presented by O. Savin [CITATION] under the additional assumption that [MATH] connects [MATH] to [MATH] along the direction where it increases.', '1204.0880-1-2-3': 'See also [CITATION] for another proof in dimension [MATH] and [CITATION] for a review on the subject.', '1204.0880-1-3-0': 'In this paper, we are interested in a variant of [REF] where the Laplacian [MATH] is substituted by the Ornstein-Uhlenbeck operator [MATH].', '1204.0880-1-3-1': 'Namely, we consider the semilinear elliptic equation [EQUATION] and show the one-dimensional symmetry of bounded entire solutions which are monotone in some direction.', '1204.0880-1-4-0': 'Let us state our main result.', '1204.0880-1-5-0': 'Let [MATH], [MATH].', '1204.0880-1-5-1': 'Let [MATH] be a solution of [EQUATION] where [MATH] is a locally Lipschitz function.', '1204.0880-1-6-0': 'Assume that [EQUATION] for some [MATH].', '1204.0880-1-6-1': 'Then, [MATH] is one-dimensional, i.e. there exist [MATH] and [MATH] such that [EQUATION] for any [MATH].', '1204.0880-1-7-0': 'Notice that [REF] can be regarded as the analog of [REF] in the so-called Gauss space, that is, in [MATH] endowed with the Gaussian instead of the Lebesgue measure.', '1204.0880-1-7-1': 'Indeed, while the Pde in [REF] is the Euler-Lagrange equation of the Allen-Cahn Energy [EQUATION] the Pde in [REF] is the Euler-Lagrange equation of the functional [EQUATION] where [MATH] and [EQUATION] is the standard Gaussian probability measure.', '1204.0880-1-7-2': 'It is interesting to remark that Theorem [REF] holds for general type of nonlinearities, as it happens for the conjecture of De Giorgi when [MATH] (see [CITATION], and this is a major difference with respect to the techniques in [CITATION]).', '1204.0880-1-8-0': 'As in the case of the Laplacian, Theorem [REF] is closely related to the Bernstein problem in the Gauss space, which asks for flatness of entire minimal surfaces which are graphs in some direction.', '1204.0880-1-8-1': 'We point out that minimal surfaces in the Gauss space are interesting geometric objects, since they correspond to self-similar shrinkers of the mean curvature flow (see for instance [CITATION]), and satisfy the equation [EQUATION] where [MATH] is the mean curvature at [MATH] and [MATH] is the normal vector.', '1204.0880-1-8-2': 'In this context, the analog of the Bernstein Theorem has been proved by Ecker and Huisken [CITATION], under a polynomial growth assumption on the volume of the minimal surface, and more recently by Wang in [CITATION] without any further assumption.', '1204.0880-1-8-3': 'We point out that, differently from the Euclidean case, the result holds without any restriction on the dimension of the ambient space, and in fact there is no such restriction also in Theorem [REF].', '1204.0880-1-8-4': 'This is due to the exponential decay of the Gaussian measure associated to the Ornstein-Uhlenbeck operator which allows for better estimates than the corresponding Euclidean ones.', '1204.0880-1-9-0': 'Since Theorem [REF] holds in any dimension and the Gauss space [MATH] formally converges to a Wiener space [MATH] (see Section [REF] for a precise definition) as [MATH], one may expect that an analogous result holds in such infinite dimensional setting.', '1204.0880-1-9-1': 'Indeed, in this paper we confirm this expectation and show the infinite dimensional extension of Theorem [REF]:', '1204.0880-1-10-0': 'Let [MATH] satisfy [EQUATION] where [MATH] is a locally Lipschitz function.', '1204.0880-1-10-1': 'Assume that [EQUATION] and [EQUATION] for all [MATH], for all [MATH] and for some [MATH].', '1204.0880-1-10-2': 'Then, [MATH] is one-dimensional, in the sense that there exist [MATH] and [MATH] such that [EQUATION]', '1204.0880-1-10-3': 'Notice that Theorem [REF] can be recovered as a corollary of Theorem [REF], when the function [MATH] depends only on finitely many variables.', '1204.0880-1-10-4': 'As far as we know, Theorem [REF] is the first result of De Giorgi conjecture type in an infinite dimensional setting.', '1204.0880-1-10-5': 'The proof that we perform exploits and generalizes some geometric ideas of [CITATION].', '1204.0880-1-11-0': '# Notation', '1204.0880-1-12-0': 'We denote by [MATH] the [MATH]-dimensional Gauss space, where [MATH] is the standard Gaussian measure on [MATH] defined in [REF].', '1204.0880-1-13-0': '## The Wiener space', '1204.0880-1-14-0': 'An abstract Wiener space is defined as a triple [MATH] where [MATH] is a separable Banach space, endowed with the norm [MATH], [MATH] is a nondegenerate centered Gaussian measure, and [MATH] is the Cameron-Martin space associated to the measure [MATH], that is, [MATH] is a separable Hilbert space densely embedded in [MATH], endowed with the inner product [MATH] and with the norm [MATH].', '1204.0880-1-14-1': 'The requirement that [MATH] is a centered Gaussian measure means that for any [MATH], the measure [MATH] is a centered Gaussian measure on the real line [MATH], that is, the Fourier transform of [MATH] is given by [EQUATION] here the operator [MATH] is the covariance operator and it is uniquely determined by formula [EQUATION].', '1204.0880-1-14-2': "The nondegeneracy of [MATH] implies that [MATH] is positive definite: the boundedness of [MATH] follows by Fernique's Theorem (see for instance [CITATION]), asserting that there exists a positive number [MATH] such that [EQUATION].", '1204.0880-1-14-3': 'This implies also that the maps [MATH] belong to [MATH] for any [MATH] and [MATH], where [MATH] denotes the space of all functions [MATH] such that [EQUATION].', '1204.0880-1-14-4': 'In particular, any element [MATH] can be seen as a map [MATH], and we denote by [MATH] the identification map [MATH].', '1204.0880-1-14-5': 'The space [MATH] given by the closure of [MATH] in [MATH] is called reproducing kernel.', '1204.0880-1-14-6': 'By considering the map [MATH] defined as [EQUATION] we obtain that [MATH] is an injective [MATH]-Radonifying operator, which is Hilbert-Schmidt when [MATH] is Hilbert.', '1204.0880-1-14-7': 'We also have [MATH].', '1204.0880-1-14-8': 'The space [MATH], equipped with the inner product [MATH] and norm [MATH] induced by [MATH] via [MATH], is the Cameron-Martin space and is a dense subspace of [MATH].', '1204.0880-1-14-9': 'The continuity of [MATH] implies that the embedding of [MATH] in [MATH] is continuous, that is, there exists [MATH] such that [EQUATION].', '1204.0880-1-14-10': 'We have also that the measure [MATH] is absolutely continuous with respect to translation along Cameron-Martin directions; in fact, for [MATH], [MATH], the measure [MATH] is absolutely continuous with respect to [MATH] with density given by [EQUATION]', '1204.0880-1-15-0': '## Cylindrical functions and differential operators', '1204.0880-1-16-0': 'For [MATH] we choose [MATH] in such a way that [MATH], or equivalently [MATH], form an orthonormal basis of [MATH].', '1204.0880-1-16-1': 'We order the vectors [MATH] in such a way that the numbers [MATH] form a decreasing sequence.', '1204.0880-1-17-0': 'Given [MATH], we also let [MATH], and [MATH] be the closure of the orthogonal projection from [MATH] to [MATH] [EQUATION].', '1204.0880-1-18-0': 'The map [MATH] induces the decomposition [MATH], with [MATH], and [MATH], with [MATH] and [MATH] Gaussian measures on [MATH] and [MATH] respectively, having [MATH] and [MATH] as Cameron-Martin spaces.', '1204.0880-1-18-1': 'When no confusion is possible we identify [MATH] with [MATH]; with this identification the measure [MATH] is the standard Gaussian measure on [MATH] (see [CITATION]).', '1204.0880-1-18-2': 'Given [MATH], we denote by [MATH] the projection [MATH], and by [MATH] the infinite dimensional component of [MATH], so that [MATH].', '1204.0880-1-18-3': 'When we identify [MATH] with [MATH] we shall rather write [MATH].', '1204.0880-1-19-0': 'We say that [MATH] is a cylindrical function if [MATH] for some [MATH] and [MATH].', '1204.0880-1-20-0': 'We denote by [MATH], [MATH], the space of all [MATH] cylindrical functions, that is, functions of the form [MATH] with [MATH], with continuous and bounded derivatives up to the order [MATH].', '1204.0880-1-20-1': 'We denote by [MATH] the space generated by all functions of the form [MATH], with [MATH] and [MATH].', '1204.0880-1-21-0': 'We let [EQUATION] where [MATH] and [MATH] is the adjoint operator of [MATH].', '1204.0880-1-21-1': 'With this notation, the integration by parts formula holds: [EQUATION]', '1204.0880-1-21-2': 'In particular, thanks to [REF], the operator [MATH] is closable in [MATH], and we denote by [MATH] the domain of its closure.', '1204.0880-1-21-3': 'The Sobolev spaces [MATH], with [MATH] and [MATH], can be defined analogously [CITATION], and [MATH] is dense in [MATH], for all [MATH] and [MATH] with [MATH].', '1204.0880-1-22-0': 'Given a vector field [MATH], [MATH], using [REF] we can define [MATH] in the distributional sense, taking test functions [MATH] in [MATH] with [MATH].', '1204.0880-1-23-0': 'We say that [MATH] if this linear functional can be extended to all test functions [MATH].', '1204.0880-1-23-1': 'This is true in particular if [MATH].', '1204.0880-1-24-0': 'Let [MATH], [MATH] and [MATH].', '1204.0880-1-24-1': 'From [REF], with [MATH] and [MATH], we get [EQUATION]', '1204.0880-1-24-2': 'Let now [MATH].', '1204.0880-1-24-3': 'If we apply [REF] with [MATH], we obtain [EQUATION] which, summing up in [MATH], gives [EQUATION]', '1204.0880-1-25-0': 'The operator [MATH] is usually called the Ornstein-Uhlenbeck operator.', '1204.0880-1-26-0': 'Notice that, if [MATH] is a cylindrical function, that is [MATH] with [MATH] and [MATH], then [EQUATION]', '1204.0880-1-26-1': 'We write [MATH] if [MATH] is continuous and [MATH] if both [MATH] and [MATH] are continuous.', '1204.0880-1-27-0': 'For simplicity of notation, from now on we will omit the explicit dependence on [MATH] of operators and spaces.', '1204.0880-1-27-1': 'We also indicate by [MATH] and [MATH] respectively the scalar product and the norm in [MATH].', '1204.0880-1-27-2': 'When no confusion is possible, we shall also write [MATH] to indicate the derivative [MATH].', '1204.0880-1-28-0': '# Proof of Theorem [REF]', '1204.0880-1-29-0': 'Recalling the integration by parts formula [REF], equation [REF] can be written in a weak form as [EQUATION] which is meaningful for [MATH].', '1204.0880-1-29-1': 'Notice that, as [MATH] is dense in [MATH], it is enough to require [REF] for all [MATH].', '1204.0880-1-30-0': 'Since [MATH], by [CITATION] we have that a bounded weak solution of [REF] belongs to [MATH].', '1204.0880-1-31-0': '## The linearized equation', '1204.0880-1-32-0': 'We now consider the equation solved by the derivatives of the solution [MATH].', '1204.0880-1-33-0': 'Let [MATH] satisfy [REF].', '1204.0880-1-33-1': 'For any [MATH] let [MATH], then [EQUATION]', '1204.0880-1-33-2': 'Notice first that it is enough to prove [REF] for all [MATH].', '1204.0880-1-33-3': 'Letting [MATH], we multiply [REF] by [MATH] and recall [REF], to get [EQUATION] where the last inequality follows from [REF], with [MATH] replaced by [MATH].', '1204.0880-1-34-0': '## A variational inequality implied by the monotonicity', '1204.0880-1-35-0': 'The next result shows that monotone solutions of [REF] satisfy a variational inequality.', '1204.0880-1-35-1': 'In the Euclidean case, this fact boils down to the classical stability condition (namely, the second derivative of the energy functional being nonnegative).', '1204.0880-1-35-2': 'Differently from this, in our case, a negative eigenvalue appears in the inequality.', '1204.0880-1-36-0': 'Let [MATH] satisfy [REF] and [REF].', '1204.0880-1-36-1': 'Then, for any [MATH] it holds [EQUATION]', '1204.0880-1-36-2': 'The proof is a variation of a classical technique (see, e.g., [CITATION]).', '1204.0880-1-37-0': 'Without loss of generality we may assume [MATH], and we let [MATH] be such that [MATH].', '1204.0880-1-37-1': 'Notice that, thanks to [REF], the space of such functions is dense in [MATH].', '1204.0880-1-37-2': 'We use [REF], with [MATH] and test function [MATH], and we obtain [EQUATION]', '1204.0880-1-38-0': '## A geometric Poincare inequality', '1204.0880-1-39-0': 'We show that a sort of geometric Poincare inequality stems from solutions of [REF] satisfying [REF].', '1204.0880-1-40-0': 'In the Euclidean case, it boils down to the inequality discovered in [CITATION].', '1204.0880-1-41-0': 'Let [MATH] satisfy [REF] and [REF].', '1204.0880-1-41-1': 'For any [MATH] we have [EQUATION] where [EQUATION].', '1204.0880-1-42-0': 'We use [REF] with test function [MATH], and we see that [EQUATION]', '1204.0880-1-42-1': 'We now exploit [REF] with test function [MATH] and we get [EQUATION]', '1204.0880-1-42-2': 'Summing over [MATH], we conclude that [EQUATION]', '1204.0880-1-42-3': 'From [REF] and [REF], we conclude that [EQUATION] which gives [REF].', '1204.0880-1-43-0': 'Let [MATH] satisfying [REF], let [MATH] and [MATH].', '1204.0880-1-43-1': 'We consider the map [MATH] defined as [MATH], and let [EQUATION] be its noncritical set.', '1204.0880-1-43-2': 'By the Implicit Function Theorem, the level set of [MATH] in [MATH] are [MATH]-dimensional hypersurfaces of class [MATH].', '1204.0880-1-43-3': 'Thus we can consider the principal curvatures of these hypersurfaces, that we denote by [MATH], and the tangential gradient of [MATH], that we denote by [MATH].', '1204.0880-1-43-4': 'We also set [EQUATION] and [EQUATION].', '1204.0880-1-44-0': 'With this notation, we have the following', '1204.0880-1-45-0': 'Let [MATH] satisfy [REF], [REF] and [REF], and fix [MATH].', '1204.0880-1-45-1': 'For any [MATH] we have [EQUATION]', '1204.0880-1-45-2': 'Let [EQUATION]', '1204.0880-1-45-3': 'Since [MATH] and [EQUATION] for any [MATH], [MATH], it follows that [EQUATION] so that [MATH] is nondecreasing in [MATH].', '1204.0880-1-45-4': 'Accordingly, [EQUATION] for any [MATH].', '1204.0880-1-45-5': "Moreover, by Stampacchia's Theorem we have that [MATH] for almost any [MATH], and similarly [MATH] for almost any [MATH].", '1204.0880-1-46-0': 'Therefore [EQUATION]', '1204.0880-1-46-1': 'On the other hand, by [CITATION], [EQUATION]', '1204.0880-1-46-2': 'From this, [REF] and [REF], we obtain [EQUATION] which, recalling [REF], implies [REF].', '1204.0880-1-47-0': '## A symmetry result', '1204.0880-1-48-0': 'We now use the previous material to obtain a one-dimensional symmetry result for the [MATH]-dimensional projection of the solution.', '1204.0880-1-48-1': 'The idea of using geometric Poincare inequalities as the ones in [CITATION] in order to obtain symmetry properties goes back to [CITATION] and it was widely used in [CITATION] in the finite dimensional Euclidean setting.', '1204.0880-1-48-2': 'The result we present here is the following:', '1204.0880-1-49-0': 'Fix [MATH] and [MATH].', '1204.0880-1-49-1': 'Let [MATH] satisfy [REF], [REF] and [REF].', '1204.0880-1-49-2': 'Then, the map [MATH]', '1204.0880-1-50-0': 'is one-dimensional, i.e. there exists [MATH] and [MATH], with [MATH], such that [EQUATION] for any [MATH].', '1204.0880-1-51-0': 'We fix [MATH], to be taken arbitrarily large in what follows, and let [MATH].', '1204.0880-1-51-1': 'Let [MATH] be such that [MATH] if [MATH], [MATH] if [MATH] and [MATH] for any [MATH].', '1204.0880-1-51-2': 'We take [MATH].', '1204.0880-1-51-3': 'Then [MATH], and [REF] yields [EQUATION]', '1204.0880-1-51-4': 'Also, due to our assumptions on [MATH], [EQUATION].', '1204.0880-1-51-5': 'Therefore, by sending [MATH] in [REF], we conclude that [EQUATION] for any [MATH].', '1204.0880-1-51-6': 'From this and [CITATION] we get [REF].', '1204.0880-1-52-0': 'From the finite dimensional symmetry result of Proposition [REF], one can take the limit as [MATH] and obtain:', '1204.0880-1-53-0': 'Let [MATH] satisfy [REF], [REF] and [REF].', '1204.0880-1-53-1': 'Then, [MATH] is necessarily one-dimensional, i.e. there exists [MATH] and [MATH] such that [EQUATION] for any [MATH].', '1204.0880-1-54-0': 'We first show that there exists [MATH] such that [EQUATION]', '1204.0880-1-54-1': 'Let [MATH] be defined as [MATH].', '1204.0880-1-54-2': 'Since [MATH] is a dense subset of [MATH], it is enough to show that [REF] holds in [MATH], where [MATH].', '1204.0880-1-55-0': 'However, from Proposition [REF] we know that [EQUATION] which implies that [EQUATION]', '1204.0880-1-55-1': 'From [REF] it follows that there exists a function [MATH] such that [MATH] for all [MATH], and [EQUATION]', '1204.0880-1-55-2': 'Moreover, [MATH] is a bounded nondecreasing solution to the ODE [EQUATION].', '1204.0880-1-55-3': 'Being [MATH] continuous, if [MATH] is nonconstant (otherwise the thesis follows immediately) then the function [MATH] is also continuous, so that [MATH] and [MATH] for some [MATH], which implies the thesis.', '1204.0880-1-56-0': '## Proof of Theorem [REF]', '1204.0880-1-57-0': 'The proof of Theorem [REF] follows directly from Lemma [REF] and Corollary [REF].', '1204.0880-1-58-0': 'We observe that, in the infinite dimensional case, there may exist weak solutions to [REF], satisfying [REF], which are not continuous.', '1204.0880-1-58-1': 'Indeed, given [MATH] satisfying [REF] and [REF] below, the function [MATH] in [REF] is a solution to [REF], monotone in the direction given by [MATH], for any [MATH].', '1204.0880-1-58-2': 'However, such a solution is continuous only if [MATH].', '1204.0880-1-58-3': 'As a possible generalization of Theorem [REF], one could ask if any bounded weak solution to [REF], satisfying [MATH] for some [MATH], is of this form.', '1204.0880-1-59-0': '# Heteroclinic solutions', '1204.0880-1-60-0': 'The results in Theorems [REF] and [REF] may be seen either as classification results (when one knows explicitly the solutions of the associated one-dimensional problem) or as nonexistence result (when the associated one-dimensional problem does not admit any solution).', '1204.0880-1-60-1': 'For this, we now give some simple conditions on the nonlinearity [MATH] ensuring existence or nonexistence of bounded solutions to the ODE [EQUATION] satisfying [EQUATION]', '1204.0880-1-60-2': 'Notice that, from [REF] it follows that there exist [MATH], with [MATH], such that [EQUATION]', '1204.0880-1-60-3': 'Moreover, passing to the limit in [REF] we also get [EQUATION]', '1204.0880-1-60-4': 'We start with a nonexistence result.', '1204.0880-1-61-0': 'Assume that there exists [MATH] such that [EQUATION].', '1204.0880-1-61-1': 'Then, there are no solutions to [REF] satisfying [REF].', '1204.0880-1-62-0': 'Let us assume that [MATH] in [MATH], since the argument is analogous in the other case, and assume by contradiction that we are given a solution [MATH] of [REF], [REF].', '1204.0880-1-63-0': 'Letting [MATH] be such that [MATH], we have that [MATH] satisfies the differential inequality [EQUATION] which implies, by direct integration, [EQUATION] contradicting [REF].', '1204.0880-1-64-0': 'We consider the potential [MATH], defined as [EQUATION] where [MATH].', '1204.0880-1-64-1': 'Notice that, if [MATH] is convex or concave, from [REF] if follows that [MATH] in [MATH], so that by Proposition [REF] there are no solutions to [REF] satisfying [REF].', '1204.0880-1-65-0': 'Given [MATH], we let [EQUATION] where [MATH].', '1204.0880-1-65-1': 'Notice that [REF] is the Euler-Lagrange equation of [MATH].', '1204.0880-1-66-0': 'As a counterpart of the nonexistence result in Proposition [REF], we now give an existence result for monotone solutions to [REF].', '1204.0880-1-67-0': 'Assume that [MATH] satisfies the following properties: [EQUATION]', '1204.0880-1-67-1': 'Assume also that there exists [MATH] such that [MATH] and [EQUATION]', '1204.0880-1-67-2': 'Then, there exists a monotone solution to [REF], connecting [MATH] to [MATH].', '1204.0880-1-68-0': 'Let [MATH] be a solution to the minimum problem [EQUATION]', '1204.0880-1-68-1': 'Note that [REF] implies [EQUATION] so that we may assume [MATH] for all [MATH].', '1204.0880-1-69-0': 'Let now [MATH] be the Ehrhard rearrangement of [MATH] [CITATION], which is defined in such a way that [MATH] is nondecreasing on [MATH], and [EQUATION]', '1204.0880-1-70-0': 'Notice that [MATH] and [MATH] for all [MATH].', '1204.0880-1-70-1': 'By [CITATION] (see also [CITATION]), we have [MATH] and [EQUATION] so that [EQUATION].', '1204.0880-1-70-2': 'In particular, we may assume that [MATH], i.e. that [MATH] is nondecreasing on [MATH].', '1204.0880-1-71-0': 'As [MATH] and [MATH] are solutions to [REF], which is the Euler-Lagrange equation of [MATH], we get that either [MATH] or [MATH] or [EQUATION]', '1204.0880-1-71-1': 'On the other hand, thanks to [REF] and the fact that [MATH], we can exclude the first two possibilities, so that [REF] holds.', '1204.0880-1-71-2': 'Moreover, since [MATH] is nondecreasing and [MATH] for all [MATH], it follows that [MATH] for all [MATH] and [EQUATION].', '1204.0880-1-71-3': 'Since by [REF] the function [MATH] is a monotone solution to [REF] on [MATH], we get that the odd extension of [MATH] on [MATH] is a solution to [REF] on the whole of [MATH] which satisfies [REF] and connects [MATH] to [MATH].', '1204.0880-1-72-0': 'Notice that for all [MATH] we have [EQUATION].', '1204.0880-1-72-1': 'If we let [MATH] be the unique solution to [EQUATION] we have [EQUATION].', '1204.0880-1-72-2': 'In particular, condition [REF] is verified whenever [EQUATION] which is satisfied, for instance, by the standard double-well potential [MATH].'}

{'1204.0880-2-0-0': 'In 1978 E. De Giorgi fromulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation [MATH], which are monotone in some direction.', '1204.0880-2-0-1': 'In this paper we prove the analogous statement for the equation [MATH], where the Laplacian is replaced by the Ornstein-Uhlenbeck operator.', '1204.0880-2-0-2': 'Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain an similar result in infinite dimensions by a limit procedure.', '1204.0880-2-1-0': '# Introduction', '1204.0880-2-2-0': 'A celebrated conjecture by De Giorgi [CITATION] asks if bounded entire solutions to the equation [EQUATION] which are strictly increasing in some direction are one-dimensional, in the sense that the level sets [MATH] are hyperplanes, at least if [MATH].', '1204.0880-2-2-1': 'This conjecture has been proved by Ghoussoub and Gui [CITATION] in dimension [MATH], and by Ambrosio and Cabre [CITATION] in dimension [MATH], and a counterexample has been given by del Pino, Kowalczyk and Wei in [CITATION] for [MATH].', '1204.0880-2-2-2': 'While the conjecture is still open for [MATH], a very nice proof has been presented by O. Savin [CITATION] under the additional assumption that [MATH] connects [MATH] to [MATH] along the direction where it increases.', '1204.0880-2-2-3': 'See also [CITATION] for another proof in dimension [MATH] and [CITATION] for a review on the subject.', '1204.0880-2-3-0': 'In this paper, we are interested in a variant of [REF] where the Laplacian [MATH] is substituted by the Ornstein-Uhlenbeck operator [MATH].', '1204.0880-2-3-1': 'Namely, we consider the semilinear elliptic equation [EQUATION] and show the one-dimensional symmetry of bounded entire solutions which are monotone in some direction.', '1204.0880-2-4-0': 'Let us state our main result.', '1204.0880-2-5-0': 'Let [MATH], [MATH].', '1204.0880-2-5-1': 'Let [MATH] be a solution of [EQUATION] where [MATH] is a locally Lipschitz function.', '1204.0880-2-6-0': 'Assume that [EQUATION] for some [MATH].', '1204.0880-2-6-1': 'Then, [MATH] is one-dimensional, i.e. there exist [MATH] and [MATH] such that [EQUATION] for any [MATH].', '1204.0880-2-7-0': 'Notice that [REF] can be regarded as the analog of [REF] in the so-called Gauss space, that is, in [MATH] endowed with the Gaussian instead of the Lebesgue measure.', '1204.0880-2-7-1': 'Indeed, while the Pde in [REF] is the Euler-Lagrange equation of the Allen-Cahn Energy [EQUATION] the Pde in [REF] is the Euler-Lagrange equation of the functional [EQUATION] where [MATH] and [EQUATION] is the standard Gaussian probability measure.', '1204.0880-2-7-2': 'It is interesting to remark that Theorem [REF] holds for general type of nonlinearities, as it happens for the conjecture of De Giorgi when [MATH] (see [CITATION], and this is a major difference with respect to the techniques in [CITATION]).', '1204.0880-2-8-0': 'As in the case of the Laplacian, Theorem [REF] is closely related to the Bernstein problem in the Gauss space, which asks for flatness of entire minimal surfaces which are graphs in some direction.', '1204.0880-2-8-1': 'We point out that minimal surfaces in the Gauss space are interesting geometric objects, since they correspond to self-similar shrinkers of the mean curvature flow (see for instance [CITATION]), and satisfy the equation [EQUATION] where [MATH] is the mean curvature at [MATH] and [MATH] is the normal vector.', '1204.0880-2-8-2': 'In this context, the analog of the Bernstein Theorem has been proved by Ecker and Huisken [CITATION], under a polynomial growth assumption on the volume of the minimal surface, and more recently by Wang in [CITATION] without any further assumption.', '1204.0880-2-8-3': 'We point out that, differently from the Euclidean case, the result holds without any restriction on the dimension of the ambient space, and in fact there is no such restriction also in Theorem [REF].', '1204.0880-2-8-4': 'This is due to the exponential decay of the Gaussian measure associated to the Ornstein-Uhlenbeck operator which allows for better estimates than the corresponding Euclidean ones.', '1204.0880-2-9-0': 'Since Theorem [REF] holds in any dimension and the Gauss space [MATH] formally converges to a Wiener space [MATH] (see Section [REF] for a precise definition) as [MATH], one may expect that an analogous result holds in such infinite dimensional setting.', '1204.0880-2-9-1': 'Indeed, in this paper we confirm this expectation and show the infinite dimensional extension of Theorem [REF]:', '1204.0880-2-10-0': 'Let [MATH] satisfy [EQUATION] where [MATH] is a locally Lipschitz function.', '1204.0880-2-10-1': 'Assume that [EQUATION] and [EQUATION] for all [MATH], for all [MATH] and for some [MATH].', '1204.0880-2-10-2': 'Then, [MATH] is one-dimensional, in the sense that there exist [MATH] and [MATH] such that [EQUATION]', '1204.0880-2-10-3': 'Notice that Theorem [REF] can be recovered as a corollary of Theorem [REF], when the function [MATH] depends only on finitely many variables.', '1204.0880-2-10-4': 'As far as we know, Theorem [REF] is the first result of De Giorgi conjecture type in an infinite dimensional setting.', '1204.0880-2-10-5': 'The proof that we perform exploits and generalizes some geometric ideas of [CITATION].', '1204.0880-2-11-0': '# Notation', '1204.0880-2-12-0': 'We denote by [MATH] the [MATH]-dimensional Gauss space, where [MATH] is the standard Gaussian measure on [MATH] defined in [REF].', '1204.0880-2-13-0': '## The Wiener space', '1204.0880-2-14-0': 'An abstract Wiener space is defined as a triple [MATH] where [MATH] is a separable Banach space, endowed with the norm [MATH], [MATH] is a nondegenerate centered Gaussian measure, and [MATH] is the Cameron-Martin space associated to the measure [MATH], that is, [MATH] is a separable Hilbert space densely embedded in [MATH], endowed with the inner product [MATH] and with the norm [MATH].', '1204.0880-2-14-1': 'The requirement that [MATH] is a centered Gaussian measure means that for any [MATH], the measure [MATH] is a centered Gaussian measure on the real line [MATH], that is, the Fourier transform of [MATH] is given by [EQUATION] here the operator [MATH] is the covariance operator and it is uniquely determined by formula [EQUATION].', '1204.0880-2-14-2': "The nondegeneracy of [MATH] implies that [MATH] is positive definite: the boundedness of [MATH] follows by Fernique's Theorem (see for instance [CITATION]), asserting that there exists a positive number [MATH] such that [EQUATION].", '1204.0880-2-14-3': 'This implies also that the maps [MATH] belong to [MATH] for any [MATH] and [MATH], where [MATH] denotes the space of all functions [MATH] such that [EQUATION].', '1204.0880-2-14-4': 'In particular, any element [MATH] can be seen as a map [MATH], and we denote by [MATH] the identification map [MATH].', '1204.0880-2-14-5': 'The space [MATH] given by the closure of [MATH] in [MATH] is called reproducing kernel.', '1204.0880-2-14-6': 'By considering the map [MATH] defined as [EQUATION] we obtain that [MATH] is an injective [MATH]-Radonifying operator, which is Hilbert-Schmidt when [MATH] is Hilbert.', '1204.0880-2-14-7': 'We also have [MATH].', '1204.0880-2-14-8': 'The space [MATH], equipped with the inner product [MATH] and norm [MATH] induced by [MATH] via [MATH], is the Cameron-Martin space and is a dense subspace of [MATH].', '1204.0880-2-14-9': 'The continuity of [MATH] implies that the embedding of [MATH] in [MATH] is continuous, that is, there exists [MATH] such that [EQUATION].', '1204.0880-2-14-10': 'We have also that the measure [MATH] is absolutely continuous with respect to translation along Cameron-Martin directions; in fact, for [MATH], [MATH], the measure [MATH] is absolutely continuous with respect to [MATH] with density given by [EQUATION]', '1204.0880-2-15-0': '## Cylindrical functions and differential operators', '1204.0880-2-16-0': 'For [MATH] we choose [MATH] in such a way that [MATH], or equivalently [MATH], form an orthonormal basis of [MATH].', '1204.0880-2-16-1': 'We order the vectors [MATH] in such a way that the numbers [MATH] form a decreasing sequence.', '1204.0880-2-17-0': 'Given [MATH], we also let [MATH], and [MATH] be the closure of the orthogonal projection from [MATH] to [MATH] [EQUATION].', '1204.0880-2-18-0': 'The map [MATH] induces the decomposition [MATH], with [MATH], and [MATH], with [MATH] and [MATH] Gaussian measures on [MATH] and [MATH] respectively, having [MATH] and [MATH] as Cameron-Martin spaces.', '1204.0880-2-18-1': 'When no confusion is possible we identify [MATH] with [MATH]; with this identification the measure [MATH] is the standard Gaussian measure on [MATH] (see [CITATION]).', '1204.0880-2-18-2': 'Given [MATH], we denote by [MATH] the projection [MATH], and by [MATH] the infinite dimensional component of [MATH], so that [MATH].', '1204.0880-2-18-3': 'When we identify [MATH] with [MATH] we shall rather write [MATH].', '1204.0880-2-19-0': 'We say that [MATH] is a cylindrical function if [MATH] for some [MATH] and [MATH].', '1204.0880-2-20-0': 'We denote by [MATH], [MATH], the space of all [MATH] cylindrical functions, that is, functions of the form [MATH] with [MATH], with continuous and bounded derivatives up to the order [MATH].', '1204.0880-2-20-1': 'We denote by [MATH] the space generated by all functions of the form [MATH], with [MATH] and [MATH].', '1204.0880-2-21-0': 'We let [EQUATION] where [MATH] and [MATH] is the adjoint operator of [MATH].', '1204.0880-2-21-1': 'With this notation, the integration by parts formula holds: [EQUATION]', '1204.0880-2-21-2': 'In particular, thanks to [REF], the operator [MATH] is closable in [MATH], and we denote by [MATH] the domain of its closure.', '1204.0880-2-21-3': 'The Sobolev spaces [MATH], with [MATH] and [MATH], can be defined analogously [CITATION], and [MATH] is dense in [MATH], for all [MATH] and [MATH] with [MATH].', '1204.0880-2-22-0': 'Given a vector field [MATH], [MATH], using [REF] we can define [MATH] in the distributional sense, taking test functions [MATH] in [MATH] with [MATH].', '1204.0880-2-23-0': 'We say that [MATH] if this linear functional can be extended to all test functions [MATH].', '1204.0880-2-23-1': 'This is true in particular if [MATH].', '1204.0880-2-24-0': 'Let [MATH], [MATH] and [MATH].', '1204.0880-2-24-1': 'From [REF], with [MATH] and [MATH], we get [EQUATION]', '1204.0880-2-24-2': 'Let now [MATH].', '1204.0880-2-24-3': 'If we apply [REF] with [MATH], we obtain [EQUATION] which, summing up in [MATH], gives [EQUATION]', '1204.0880-2-25-0': 'The operator [MATH] is usually called the Ornstein-Uhlenbeck operator.', '1204.0880-2-26-0': 'Notice that, if [MATH] is a cylindrical function, that is [MATH] with [MATH] and [MATH], then [EQUATION]', '1204.0880-2-26-1': 'We write [MATH] if [MATH] is continuous and [MATH] if both [MATH] and [MATH] are continuous.', '1204.0880-2-27-0': 'For simplicity of notation, from now on we will omit the explicit dependence on [MATH] of operators and spaces.', '1204.0880-2-27-1': 'We also indicate by [MATH] and [MATH] respectively the scalar product and the norm in [MATH].', '1204.0880-2-27-2': 'When no confusion is possible, we shall also write [MATH] to indicate the derivative [MATH].', '1204.0880-2-28-0': '# Proof of Theorem [REF]', '1204.0880-2-29-0': 'Recalling the integration by parts formula [REF], equation [REF] can be written in a weak form as [EQUATION] which is meaningful for [MATH].', '1204.0880-2-29-1': 'Notice that, as [MATH] is dense in [MATH], it is enough to require [REF] for all [MATH].', '1204.0880-2-30-0': 'Since [MATH], by [CITATION] we have that a bounded weak solution of [REF] belongs to [MATH].', '1204.0880-2-31-0': '## The linearized equation', '1204.0880-2-32-0': 'We now consider the equation solved by the derivatives of the solution [MATH].', '1204.0880-2-33-0': 'Let [MATH] satisfy [REF].', '1204.0880-2-33-1': 'For any [MATH] let [MATH], then [EQUATION]', '1204.0880-2-33-2': 'Notice first that it is enough to prove [REF] for all [MATH].', '1204.0880-2-33-3': 'Letting [MATH], we multiply [REF] by [MATH] and recall [REF], to get [EQUATION] where the last inequality follows from [REF], with [MATH] replaced by [MATH].', '1204.0880-2-34-0': '## A variational inequality implied by the monotonicity', '1204.0880-2-35-0': 'The next result shows that monotone solutions of [REF] satisfy a variational inequality.', '1204.0880-2-35-1': 'In the Euclidean case, this fact boils down to the classical stability condition (namely, the second derivative of the energy functional being nonnegative).', '1204.0880-2-35-2': 'Differently from this, in our case, a negative eigenvalue appears in the inequality.', '1204.0880-2-36-0': 'Let [MATH] satisfy [REF] and [REF].', '1204.0880-2-36-1': 'Then, for any [MATH] it holds [EQUATION]', '1204.0880-2-36-2': 'The proof is a variation of a classical technique (see, e.g., [CITATION]).', '1204.0880-2-37-0': 'Without loss of generality we may assume [MATH], and we let [MATH] be such that [MATH].', '1204.0880-2-37-1': 'Notice that, thanks to [REF], the space of such functions is dense in [MATH].', '1204.0880-2-37-2': 'We use [REF], with [MATH] and test function [MATH], and we obtain [EQUATION]', '1204.0880-2-38-0': '## A geometric Poincare inequality', '1204.0880-2-39-0': 'We show that a sort of geometric Poincare inequality stems from solutions of [REF] satisfying [REF].', '1204.0880-2-40-0': 'In the Euclidean case, it boils down to the inequality discovered in [CITATION].', '1204.0880-2-41-0': 'Let [MATH] satisfy [REF] and [REF].', '1204.0880-2-41-1': 'For any [MATH] we have [EQUATION] where [EQUATION].', '1204.0880-2-42-0': 'We use [REF] with test function [MATH], and we see that [EQUATION]', '1204.0880-2-42-1': 'We now exploit [REF] with test function [MATH] and we get [EQUATION]', '1204.0880-2-42-2': 'Summing over [MATH], we conclude that [EQUATION]', '1204.0880-2-42-3': 'From [REF] and [REF], we conclude that [EQUATION] which gives [REF].', '1204.0880-2-43-0': 'Let [MATH] satisfying [REF], let [MATH] and [MATH].', '1204.0880-2-43-1': 'We consider the map [MATH] defined as [MATH], and let [EQUATION] be its noncritical set.', '1204.0880-2-43-2': 'By the Implicit Function Theorem, the level set of [MATH] in [MATH] are [MATH]-dimensional hypersurfaces of class [MATH].', '1204.0880-2-43-3': 'Thus we can consider the principal curvatures of these hypersurfaces, that we denote by [MATH], and the tangential gradient of [MATH], that we denote by [MATH].', '1204.0880-2-43-4': 'We also set [EQUATION] and [EQUATION].', '1204.0880-2-44-0': 'With this notation, we have the following', '1204.0880-2-45-0': 'Let [MATH] satisfy [REF], [REF] and [REF], and fix [MATH].', '1204.0880-2-45-1': 'For any [MATH] we have [EQUATION]', '1204.0880-2-45-2': 'Let [EQUATION]', '1204.0880-2-45-3': 'Since [MATH] and [EQUATION] for any [MATH], [MATH], it follows that [EQUATION] so that [MATH] is nondecreasing in [MATH].', '1204.0880-2-45-4': 'Accordingly, [EQUATION] for any [MATH].', '1204.0880-2-45-5': "Moreover, by Stampacchia's Theorem we have that [MATH] for almost any [MATH], and similarly [MATH] for almost any [MATH].", '1204.0880-2-46-0': 'Therefore [EQUATION]', '1204.0880-2-46-1': 'On the other hand, by [CITATION], [EQUATION]', '1204.0880-2-46-2': 'From this, [REF] and [REF], we obtain [EQUATION] which, recalling [REF], implies [REF].', '1204.0880-2-47-0': '## A symmetry result', '1204.0880-2-48-0': 'We now use the previous material to obtain a one-dimensional symmetry result for the [MATH]-dimensional projection of the solution.', '1204.0880-2-48-1': 'The idea of using geometric Poincare inequalities as the ones in [CITATION] in order to obtain symmetry properties goes back to [CITATION] and it was widely used in [CITATION] in the finite dimensional Euclidean setting.', '1204.0880-2-48-2': 'The result we present here is the following:', '1204.0880-2-49-0': 'Fix [MATH] and [MATH].', '1204.0880-2-49-1': 'Let [MATH] satisfy [REF], [REF] and [REF].', '1204.0880-2-49-2': 'Then, the map [MATH]', '1204.0880-2-50-0': 'is one-dimensional, i.e. there exists [MATH] and [MATH], with [MATH], such that [EQUATION] for any [MATH].', '1204.0880-2-51-0': 'We fix [MATH], to be taken arbitrarily large in what follows, and let [MATH].', '1204.0880-2-51-1': 'Let [MATH] be such that [MATH] if [MATH], [MATH] if [MATH] and [MATH] for any [MATH].', '1204.0880-2-51-2': 'We take [MATH].', '1204.0880-2-51-3': 'Then [MATH], and [REF] yields [EQUATION]', '1204.0880-2-51-4': 'Also, due to our assumptions on [MATH], [EQUATION].', '1204.0880-2-51-5': 'Therefore, by sending [MATH] in [REF], we conclude that [EQUATION] for any [MATH].', '1204.0880-2-51-6': 'From this and [CITATION] we get [REF].', '1204.0880-2-52-0': 'From the finite dimensional symmetry result of Proposition [REF], one can take the limit as [MATH] and obtain:', '1204.0880-2-53-0': 'Let [MATH] satisfy [REF], [REF] and [REF].', '1204.0880-2-53-1': 'Then, [MATH] is necessarily one-dimensional, i.e. there exists [MATH] and [MATH] such that [EQUATION] for any [MATH].', '1204.0880-2-54-0': 'We first show that there exists [MATH] such that [EQUATION]', '1204.0880-2-54-1': 'Let [MATH] be defined as [MATH].', '1204.0880-2-54-2': 'Since [MATH] is a dense subset of [MATH], it is enough to show that [REF] holds in [MATH], where [MATH].', '1204.0880-2-55-0': 'However, from Proposition [REF] we know that [EQUATION] which implies that [EQUATION]', '1204.0880-2-55-1': 'From [REF] it follows that there exists a function [MATH] such that [MATH] for all [MATH], and [EQUATION]', '1204.0880-2-55-2': 'Moreover, [MATH] is a bounded nondecreasing solution to the ODE [EQUATION].', '1204.0880-2-55-3': 'Being [MATH] continuous, if [MATH] is nonconstant (otherwise the thesis follows immediately) then the function [MATH] is also continuous, so that [MATH] and [MATH] for some [MATH], which implies the thesis.', '1204.0880-2-56-0': '## Proof of Theorem [REF]', '1204.0880-2-57-0': 'The proof of Theorem [REF] follows directly from Lemma [REF] and Corollary [REF].', '1204.0880-2-58-0': 'We observe that, in the infinite dimensional case, there may exist weak solutions to [REF], satisfying [REF], which are not continuous.', '1204.0880-2-58-1': 'Indeed, given [MATH] satisfying [REF] and [REF] below, the function [MATH] in [REF] is a solution to [REF], monotone in the direction given by [MATH], for any [MATH].', '1204.0880-2-58-2': 'However, such a solution is continuous only if [MATH].', '1204.0880-2-58-3': 'As a possible generalization of Theorem [REF], one could ask if any bounded weak solution to [REF], satisfying [MATH] for some [MATH], is of this form.', '1204.0880-2-59-0': '# Heteroclinic solutions', '1204.0880-2-60-0': 'The results in Theorems [REF] and [REF] may be seen either as classification results (when one knows explicitly the solutions of the associated one-dimensional problem) or as nonexistence result (when the associated one-dimensional problem does not admit any solution).', '1204.0880-2-60-1': 'For this, we now give some simple conditions on the nonlinearity [MATH] ensuring existence or nonexistence of bounded solutions to the ODE [EQUATION] satisfying [EQUATION]', '1204.0880-2-60-2': 'Notice that, from [REF] it follows that there exist [MATH], with [MATH], such that [EQUATION]', '1204.0880-2-60-3': 'Moreover, passing to the limit in [REF] we also get [EQUATION]', '1204.0880-2-60-4': 'We start with a nonexistence result.', '1204.0880-2-61-0': 'Assume that there exists [MATH] such that [EQUATION].', '1204.0880-2-61-1': 'Then, there are no solutions to [REF] satisfying [REF].', '1204.0880-2-62-0': 'Let us assume that [MATH] in [MATH], since the argument is analogous in the other case, and assume by contradiction that we are given a solution [MATH] of [REF], [REF].', '1204.0880-2-63-0': 'Letting [MATH] be such that [MATH], we have that [MATH] satisfies the differential inequality [EQUATION] which implies, by direct integration, [EQUATION] contradicting [REF].', '1204.0880-2-64-0': 'We consider the potential [MATH], defined as [EQUATION] where [MATH].', '1204.0880-2-64-1': 'Notice that, if [MATH] is convex or concave, from [REF] if follows that [MATH] in [MATH], so that by Proposition [REF] there are no solutions to [REF] satisfying [REF].', '1204.0880-2-65-0': 'Given [MATH], we let [EQUATION] where [MATH].', '1204.0880-2-65-1': 'Notice that [REF] is the Euler-Lagrange equation of [MATH].', '1204.0880-2-66-0': 'As a counterpart of the nonexistence result in Proposition [REF], we now give an existence result for monotone solutions to [REF].', '1204.0880-2-67-0': 'Assume that [MATH] satisfies the following properties: [EQUATION]', '1204.0880-2-67-1': 'Assume also that there exists [MATH] such that [MATH] and [EQUATION]', '1204.0880-2-67-2': 'Then, there exists a monotone solution to [REF], connecting [MATH] to [MATH].', '1204.0880-2-68-0': 'Let [MATH] be a solution to the minimum problem [EQUATION]', '1204.0880-2-68-1': 'Note that [REF] implies [EQUATION] so that we may assume [MATH] for all [MATH].', '1204.0880-2-69-0': 'Let now [MATH] be the Ehrhard rearrangement of [MATH] [CITATION], which is defined in such a way that [MATH] is nondecreasing on [MATH], and [EQUATION]', '1204.0880-2-70-0': 'Notice that [MATH] and [MATH] for all [MATH].', '1204.0880-2-70-1': 'By [CITATION] (see also [CITATION]), we have [MATH] and [EQUATION] so that [EQUATION].', '1204.0880-2-70-2': 'In particular, we may assume that [MATH], i.e. that [MATH] is nondecreasing on [MATH].', '1204.0880-2-71-0': 'As [MATH] and [MATH] are solutions to [REF], which is the Euler-Lagrange equation of [MATH], we get that either [MATH] or [MATH] or [EQUATION]', '1204.0880-2-71-1': 'On the other hand, thanks to [REF] and the fact that [MATH], we can exclude the first two possibilities, so that [REF] holds.', '1204.0880-2-71-2': 'Moreover, since [MATH] is nondecreasing and [MATH] for all [MATH], it follows that [MATH] for all [MATH] and [EQUATION].', '1204.0880-2-71-3': 'Since by [REF] the function [MATH] is a monotone solution to [REF] on [MATH], we get that the odd extension of [MATH] on [MATH] is a solution to [REF] on the whole of [MATH] which satisfies [REF] and connects [MATH] to [MATH].', '1204.0880-2-72-0': 'Notice that for all [MATH] we have [EQUATION].', '1204.0880-2-72-1': 'If we let [MATH] be the unique solution to [EQUATION] we have [EQUATION].', '1204.0880-2-72-2': 'In particular, condition [REF] is verified whenever [EQUATION] which is satisfied, for instance, by the standard double-well potential [MATH].'}

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'1204.0880-2-72-1'], ['1204.0880-1-72-2', '1204.0880-2-72-2'], ['1204.0880-1-17-0', '1204.0880-2-17-0'], ['1204.0880-1-30-0', '1204.0880-2-30-0'], ['1204.0880-1-54-0', '1204.0880-2-54-0'], ['1204.0880-1-54-1', '1204.0880-2-54-1'], ['1204.0880-1-54-2', '1204.0880-2-54-2'], ['1204.0880-1-63-0', '1204.0880-2-63-0'], ['1204.0880-1-37-0', '1204.0880-2-37-0'], ['1204.0880-1-37-1', '1204.0880-2-37-1'], ['1204.0880-1-37-2', '1204.0880-2-37-2'], ['1204.0880-1-40-0', '1204.0880-2-40-0'], ['1204.0880-1-7-0', '1204.0880-2-7-0'], ['1204.0880-1-7-1', '1204.0880-2-7-1'], ['1204.0880-1-7-2', '1204.0880-2-7-2'], ['1204.0880-1-32-0', '1204.0880-2-32-0'], ['1204.0880-1-35-0', '1204.0880-2-35-0'], ['1204.0880-1-35-1', '1204.0880-2-35-1'], ['1204.0880-1-35-2', '1204.0880-2-35-2'], ['1204.0880-1-60-0', '1204.0880-2-60-0'], ['1204.0880-1-60-1', '1204.0880-2-60-1'], ['1204.0880-1-60-2', '1204.0880-2-60-2'], ['1204.0880-1-60-3', '1204.0880-2-60-3'], ['1204.0880-1-60-4', '1204.0880-2-60-4'], ['1204.0880-1-67-0', '1204.0880-2-67-0'], ['1204.0880-1-67-1', '1204.0880-2-67-1'], ['1204.0880-1-67-2', '1204.0880-2-67-2'], ['1204.0880-1-23-0', '1204.0880-2-23-0'], ['1204.0880-1-23-1', '1204.0880-2-23-1'], ['1204.0880-1-39-0', '1204.0880-2-39-0'], ['1204.0880-1-3-0', '1204.0880-2-3-0'], ['1204.0880-1-3-1', '1204.0880-2-3-1'], ['1204.0880-1-55-0', '1204.0880-2-55-0'], ['1204.0880-1-55-1', '1204.0880-2-55-1'], ['1204.0880-1-55-2', '1204.0880-2-55-2'], ['1204.0880-1-55-3', '1204.0880-2-55-3'], ['1204.0880-1-12-0', '1204.0880-2-12-0'], ['1204.0880-1-53-1', '1204.0880-2-53-1'], ['1204.0880-1-48-0', '1204.0880-2-48-0'], ['1204.0880-1-48-1', '1204.0880-2-48-1'], ['1204.0880-1-2-0', '1204.0880-2-2-0'], ['1204.0880-1-2-1', '1204.0880-2-2-1'], ['1204.0880-1-2-2', '1204.0880-2-2-2'], ['1204.0880-1-2-3', '1204.0880-2-2-3'], ['1204.0880-1-27-0', '1204.0880-2-27-0'], ['1204.0880-1-27-1', '1204.0880-2-27-1'], ['1204.0880-1-27-2', '1204.0880-2-27-2'], ['1204.0880-1-16-0', '1204.0880-2-16-0'], ['1204.0880-1-16-1', '1204.0880-2-16-1'], ['1204.0880-1-57-0', '1204.0880-2-57-0'], ['1204.0880-1-0-0', '1204.0880-2-0-0'], ['1204.0880-1-0-1', '1204.0880-2-0-1'], ['1204.0880-1-0-2', '1204.0880-2-0-2'], ['1204.0880-1-64-0', '1204.0880-2-64-0'], ['1204.0880-1-64-1', '1204.0880-2-64-1'], ['1204.0880-1-18-0', '1204.0880-2-18-0'], ['1204.0880-1-18-1', '1204.0880-2-18-1'], ['1204.0880-1-18-2', '1204.0880-2-18-2'], ['1204.0880-1-18-3', '1204.0880-2-18-3'], ['1204.0880-1-5-1', '1204.0880-2-5-1'], ['1204.0880-1-21-0', '1204.0880-2-21-0'], ['1204.0880-1-21-1', '1204.0880-2-21-1'], ['1204.0880-1-21-2', '1204.0880-2-21-2'], ['1204.0880-1-21-3', '1204.0880-2-21-3'], ['1204.0880-1-8-0', '1204.0880-2-8-0'], ['1204.0880-1-8-1', '1204.0880-2-8-1'], ['1204.0880-1-8-2', '1204.0880-2-8-2'], ['1204.0880-1-8-3', '1204.0880-2-8-3'], ['1204.0880-1-8-4', '1204.0880-2-8-4'], ['1204.0880-1-14-0', '1204.0880-2-14-0'], ['1204.0880-1-14-1', '1204.0880-2-14-1'], ['1204.0880-1-14-2', '1204.0880-2-14-2'], ['1204.0880-1-14-3', '1204.0880-2-14-3'], ['1204.0880-1-14-4', '1204.0880-2-14-4'], ['1204.0880-1-14-5', '1204.0880-2-14-5'], ['1204.0880-1-14-6', '1204.0880-2-14-6'], ['1204.0880-1-14-7', '1204.0880-2-14-7'], ['1204.0880-1-14-8', '1204.0880-2-14-8'], ['1204.0880-1-14-9', '1204.0880-2-14-9'], ['1204.0880-1-14-10', '1204.0880-2-14-10'], ['1204.0880-1-58-0', '1204.0880-2-58-0'], ['1204.0880-1-58-1', '1204.0880-2-58-1'], ['1204.0880-1-58-2', '1204.0880-2-58-2'], ['1204.0880-1-58-3', '1204.0880-2-58-3'], ['1204.0880-1-36-0', '1204.0880-2-36-0'], ['1204.0880-1-36-1', '1204.0880-2-36-1'], ['1204.0880-1-36-2', '1204.0880-2-36-2'], ['1204.0880-1-20-0', '1204.0880-2-20-0'], ['1204.0880-1-20-1', '1204.0880-2-20-1'], ['1204.0880-1-10-0', '1204.0880-2-10-0'], ['1204.0880-1-10-1', '1204.0880-2-10-1'], ['1204.0880-1-10-2', '1204.0880-2-10-2'], ['1204.0880-1-10-3', '1204.0880-2-10-3'], ['1204.0880-1-10-4', '1204.0880-2-10-4'], ['1204.0880-1-10-5', '1204.0880-2-10-5'], ['1204.0880-1-22-0', '1204.0880-2-22-0'], ['1204.0880-1-29-0', '1204.0880-2-29-0'], ['1204.0880-1-29-1', '1204.0880-2-29-1']]

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'1204.0880-2-72-1'], ['1204.0880-1-72-2', '1204.0880-2-72-2'], ['1204.0880-1-17-0', '1204.0880-2-17-0'], ['1204.0880-1-30-0', '1204.0880-2-30-0'], ['1204.0880-1-54-0', '1204.0880-2-54-0'], ['1204.0880-1-54-1', '1204.0880-2-54-1'], ['1204.0880-1-54-2', '1204.0880-2-54-2'], ['1204.0880-1-63-0', '1204.0880-2-63-0'], ['1204.0880-1-37-0', '1204.0880-2-37-0'], ['1204.0880-1-37-1', '1204.0880-2-37-1'], ['1204.0880-1-37-2', '1204.0880-2-37-2'], ['1204.0880-1-40-0', '1204.0880-2-40-0'], ['1204.0880-1-7-0', '1204.0880-2-7-0'], ['1204.0880-1-7-1', '1204.0880-2-7-1'], ['1204.0880-1-7-2', '1204.0880-2-7-2'], ['1204.0880-1-32-0', '1204.0880-2-32-0'], ['1204.0880-1-35-0', '1204.0880-2-35-0'], ['1204.0880-1-35-1', '1204.0880-2-35-1'], ['1204.0880-1-35-2', '1204.0880-2-35-2'], ['1204.0880-1-60-0', '1204.0880-2-60-0'], ['1204.0880-1-60-1', '1204.0880-2-60-1'], ['1204.0880-1-60-2', '1204.0880-2-60-2'], ['1204.0880-1-60-3', '1204.0880-2-60-3'], ['1204.0880-1-60-4', '1204.0880-2-60-4'], ['1204.0880-1-67-0', '1204.0880-2-67-0'], ['1204.0880-1-67-1', '1204.0880-2-67-1'], ['1204.0880-1-67-2', '1204.0880-2-67-2'], ['1204.0880-1-23-0', '1204.0880-2-23-0'], ['1204.0880-1-23-1', '1204.0880-2-23-1'], ['1204.0880-1-39-0', '1204.0880-2-39-0'], ['1204.0880-1-3-0', '1204.0880-2-3-0'], ['1204.0880-1-3-1', '1204.0880-2-3-1'], ['1204.0880-1-55-0', '1204.0880-2-55-0'], ['1204.0880-1-55-1', '1204.0880-2-55-1'], ['1204.0880-1-55-2', '1204.0880-2-55-2'], ['1204.0880-1-55-3', '1204.0880-2-55-3'], ['1204.0880-1-12-0', '1204.0880-2-12-0'], ['1204.0880-1-53-1', '1204.0880-2-53-1'], ['1204.0880-1-48-0', '1204.0880-2-48-0'], ['1204.0880-1-48-1', '1204.0880-2-48-1'], ['1204.0880-1-2-0', '1204.0880-2-2-0'], ['1204.0880-1-2-1', '1204.0880-2-2-1'], ['1204.0880-1-2-2', '1204.0880-2-2-2'], ['1204.0880-1-2-3', '1204.0880-2-2-3'], ['1204.0880-1-27-0', '1204.0880-2-27-0'], ['1204.0880-1-27-1', '1204.0880-2-27-1'], ['1204.0880-1-27-2', '1204.0880-2-27-2'], ['1204.0880-1-16-0', '1204.0880-2-16-0'], ['1204.0880-1-16-1', '1204.0880-2-16-1'], ['1204.0880-1-57-0', '1204.0880-2-57-0'], ['1204.0880-1-0-0', '1204.0880-2-0-0'], ['1204.0880-1-0-1', '1204.0880-2-0-1'], ['1204.0880-1-0-2', '1204.0880-2-0-2'], ['1204.0880-1-64-0', '1204.0880-2-64-0'], ['1204.0880-1-64-1', '1204.0880-2-64-1'], ['1204.0880-1-18-0', '1204.0880-2-18-0'], ['1204.0880-1-18-1', '1204.0880-2-18-1'], ['1204.0880-1-18-2', '1204.0880-2-18-2'], ['1204.0880-1-18-3', '1204.0880-2-18-3'], ['1204.0880-1-5-1', '1204.0880-2-5-1'], ['1204.0880-1-21-0', '1204.0880-2-21-0'], ['1204.0880-1-21-1', '1204.0880-2-21-1'], ['1204.0880-1-21-2', '1204.0880-2-21-2'], ['1204.0880-1-21-3', '1204.0880-2-21-3'], ['1204.0880-1-8-0', '1204.0880-2-8-0'], ['1204.0880-1-8-1', '1204.0880-2-8-1'], ['1204.0880-1-8-2', '1204.0880-2-8-2'], ['1204.0880-1-8-3', '1204.0880-2-8-3'], ['1204.0880-1-8-4', '1204.0880-2-8-4'], ['1204.0880-1-14-0', '1204.0880-2-14-0'], ['1204.0880-1-14-1', '1204.0880-2-14-1'], ['1204.0880-1-14-2', '1204.0880-2-14-2'], ['1204.0880-1-14-3', '1204.0880-2-14-3'], ['1204.0880-1-14-4', '1204.0880-2-14-4'], ['1204.0880-1-14-5', '1204.0880-2-14-5'], ['1204.0880-1-14-6', '1204.0880-2-14-6'], ['1204.0880-1-14-7', '1204.0880-2-14-7'], ['1204.0880-1-14-8', '1204.0880-2-14-8'], ['1204.0880-1-14-9', '1204.0880-2-14-9'], ['1204.0880-1-14-10', '1204.0880-2-14-10'], ['1204.0880-1-58-0', '1204.0880-2-58-0'], ['1204.0880-1-58-1', '1204.0880-2-58-1'], ['1204.0880-1-58-2', '1204.0880-2-58-2'], ['1204.0880-1-58-3', '1204.0880-2-58-3'], ['1204.0880-1-36-0', '1204.0880-2-36-0'], ['1204.0880-1-36-1', '1204.0880-2-36-1'], ['1204.0880-1-36-2', '1204.0880-2-36-2'], ['1204.0880-1-20-0', '1204.0880-2-20-0'], ['1204.0880-1-20-1', '1204.0880-2-20-1'], ['1204.0880-1-10-0', '1204.0880-2-10-0'], ['1204.0880-1-10-1', '1204.0880-2-10-1'], ['1204.0880-1-10-2', '1204.0880-2-10-2'], ['1204.0880-1-10-3', '1204.0880-2-10-3'], ['1204.0880-1-10-4', '1204.0880-2-10-4'], ['1204.0880-1-10-5', '1204.0880-2-10-5'], ['1204.0880-1-22-0', '1204.0880-2-22-0'], ['1204.0880-1-29-0', '1204.0880-2-29-0'], ['1204.0880-1-29-1', '1204.0880-2-29-1']]

[]

[]

[]

[]

['1204.0880-1-4-0', '1204.0880-1-5-0', '1204.0880-1-6-0', '1204.0880-1-6-1', '1204.0880-1-9-1', '1204.0880-1-24-0', '1204.0880-1-24-1', '1204.0880-1-24-2', '1204.0880-1-24-3', '1204.0880-1-25-0', '1204.0880-1-26-0', '1204.0880-1-26-1', '1204.0880-1-33-0', '1204.0880-1-33-1', '1204.0880-1-33-2', '1204.0880-1-33-3', '1204.0880-1-41-0', '1204.0880-1-41-1', '1204.0880-1-44-0', '1204.0880-1-45-0', '1204.0880-1-45-1', '1204.0880-1-45-2', '1204.0880-1-45-3', '1204.0880-1-45-4', '1204.0880-1-45-5', '1204.0880-1-46-0', '1204.0880-1-46-1', '1204.0880-1-46-2', '1204.0880-1-48-2', '1204.0880-1-49-0', '1204.0880-1-49-1', '1204.0880-1-49-2', '1204.0880-1-50-0', '1204.0880-1-51-0', '1204.0880-1-51-1', '1204.0880-1-51-2', '1204.0880-1-51-3', '1204.0880-1-51-4', '1204.0880-1-51-5', '1204.0880-1-51-6', '1204.0880-1-52-0', '1204.0880-1-53-0', '1204.0880-1-65-0', '1204.0880-1-65-1', '1204.0880-1-70-0', '1204.0880-1-70-1', '1204.0880-1-70-2', '1204.0880-2-4-0', '1204.0880-2-5-0', '1204.0880-2-6-0', '1204.0880-2-6-1', '1204.0880-2-9-1', '1204.0880-2-24-0', '1204.0880-2-24-1', '1204.0880-2-24-2', '1204.0880-2-24-3', '1204.0880-2-25-0', '1204.0880-2-26-0', '1204.0880-2-26-1', '1204.0880-2-33-0', '1204.0880-2-33-1', '1204.0880-2-33-2', '1204.0880-2-33-3', '1204.0880-2-41-0', '1204.0880-2-41-1', '1204.0880-2-44-0', '1204.0880-2-45-0', '1204.0880-2-45-1', '1204.0880-2-45-2', '1204.0880-2-45-3', '1204.0880-2-45-4', '1204.0880-2-45-5', '1204.0880-2-46-0', '1204.0880-2-46-1', '1204.0880-2-46-2', '1204.0880-2-48-2', '1204.0880-2-49-0', '1204.0880-2-49-1', '1204.0880-2-49-2', '1204.0880-2-50-0', '1204.0880-2-51-0', '1204.0880-2-51-1', '1204.0880-2-51-2', '1204.0880-2-51-3', '1204.0880-2-51-4', '1204.0880-2-51-5', '1204.0880-2-51-6', '1204.0880-2-52-0', '1204.0880-2-53-0', '1204.0880-2-65-0', '1204.0880-2-65-1', '1204.0880-2-70-0', '1204.0880-2-70-1', '1204.0880-2-70-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1204.0880

1807.06782

{'1807.06782-1-0-0': 'We report the measurement of the branching fraction and [MATH] asymmetry for the [MATH] decay.', '1807.06782-1-0-1': 'The analysis is performed on a data sample of 711 [MATH] collected at the [MATH] resonance with the Belle detector at the KEKB asymmetric-energy [MATH] collider.', '1807.06782-1-0-2': 'We obtain a branching fraction of [MATH] and an [MATH] of [MATH], where the first uncertainties are statistical and the second are systematic.', '1807.06782-1-0-3': 'Hints of peaking structures are also observed in the differential branching fraction as functions of Dalitz variables.', '1807.06782-1-1-0': 'Three-body charmless hadronic [MATH] decays are suppressed in the Standard Model (SM) and are also sensitive to localized [MATH] violation in the phase space [CITATION].', '1807.06782-1-1-1': 'The [MATH] [CITATION] decays with even number of kaons have a smaller decay rate compared to the other cases with odd number of kaons.', '1807.06782-1-1-2': 'These proceed via the [MATH] tree-level, the [MATH]-exchange, and the [MATH] penguin process with a virtual loop, which provides an opportunity to search for physics beyond the SM since new bosons may cause deviations from SM predictions.', '1807.06782-1-1-3': 'Measurements by the BABAR experiment [CITATION] find hints of structures in the low [MATH] and [MATH] regions that have highly asymmetric helicity angular distributions.', '1807.06782-1-1-4': 'However, the yield is not enough to draw firm conclusions with a full Dalitz analysis.', '1807.06782-1-1-5': 'Similar studies on [MATH] were performed by Belle [CITATION], BaBar [CITATION], and LHCb [CITATION], in which strong evidence of localized [MATH] violation was found in the low [MATH] region.', '1807.06782-1-2-0': 'By using the full data of Belle, we expect to measure the branching fraction and [MATH] asymmetry of [MATH] decay more precisely.', '1807.06782-1-2-1': 'Towards that end, we use the charges of final-state particles to define the [MATH] asymmetry parameter ([MATH]) as [EQUATION] where [MATH] denotes the measured signal yield of the corresponding [MATH] final states.', '1807.06782-1-2-2': 'This is similar to the definition of the [MATH] measurement in the [MATH] decays [CITATION].', '1807.06782-1-2-3': 'In addition, we use the [MATH] [CITATION] method to obtain the background-subtracted yields for the Dalitz variables [MATH], [MATH], and [MATH], and hence determine their differential branching fractions.', '1807.06782-1-3-0': 'Our measurements are obtained from data sample of 711 [MATH] corresponding to [MATH] pairs collected with the Belle detector [CITATION] operating at the KEKB asymmetric-energy [MATH] collider [CITATION].', '1807.06782-1-3-1': 'The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF) and an electromagnetic calorimeter comprised of CsI(Tl) crystals located inside a superconducting solenoid that provides a 1.5 T magnetic field.', '1807.06782-1-3-2': 'An iron flux-return yoke located outside the solenoid is instrumented to detect [MATH] mesons and muons.', '1807.06782-1-3-3': 'The detector is described in detail elsewhere [CITATION].', '1807.06782-1-4-0': 'This analysis uses the data sets with two different inner-detector configurations.', '1807.06782-1-4-1': 'About 156 [MATH] were collected with a beam-pipe of radius 2 cm and with 3 layers of SVD, while the rest of the data set was collected with a beam-pipe of radius 1.5 cm and 4 layers of SVD [CITATION].', '1807.06782-1-4-2': 'Large Monte Carlo (MC) samples for signal and backgrounds are generated with EvtGen [CITATION] and subsequently simulated with GEANT3 [CITATION] with the configurations of the Belle detector.', '1807.06782-1-4-3': 'These samples are used to obtain expected distributions of various physical quantities for signal and backgrounds, to optimize selection criteria as well as to determine the signal selection efficiency.', '1807.06782-1-5-0': 'The selection criteria for the final-state charged particles in the [MATH] reconstruction are based on information obtained from the tracking systems (SVD and CDC) and the charged-hadron identification (PID) systems namely CDC, ACC, and TOF.', '1807.06782-1-5-1': 'The charged kaons and pions are required to have an impact parameter within [MATH] cm of the interaction point (IP) in the transverse plane, and within [MATH] cm along the [MATH] beam direction.', '1807.06782-1-5-2': 'The likelihood values of each track for different particle types ([MATH] and [MATH]) are determined from the information provided by the PID system.', '1807.06782-1-5-3': 'The track is identified as a kaon if [MATH] else as a pion.', '1807.06782-1-5-4': 'The [MATH] candidates are reconstructed via the [MATH] decay, and the identification is enhanced by using a neural network (NN) [CITATION] which combines seven kinematic variables of [MATH] [CITATION].', '1807.06782-1-5-5': 'The invariant mass of the [MATH] candidates are required to be within [MATH] MeV/[MATH] of the world average, which corresponds to about three times of the resolution of measurement.', '1807.06782-1-5-6': 'The vertex fit of [MATH] is required to succeed with the goodness-of-fit value [MATH] less than 20.', '1807.06782-1-6-0': '[MATH] mesons are identified among the reconstructed [MATH] candidates with two kinematic variables calculated in the center-of-mass frame: the beam-energy-constrained-mass [MATH], and the energy difference [MATH], where [MATH] is the beam energy, and [MATH]) is the momentum (energy) of the reconstructed [MATH] meson.', '1807.06782-1-6-1': 'The [MATH] candidates are required to have [MATH] 5.255 GeV/[MATH] and [MATH] 0.15 GeV, and the signal region is defined as 5.272 GeV/[MATH] 5.288 GeV/[MATH] and [MATH] 0.05 GeV.', '1807.06782-1-6-2': 'We require a vertex fit for [MATH] candidates with [MATH] 100.', '1807.06782-1-6-3': 'If two or more [MATH] candidates are found in an event, we choose the one with the smallest [MATH] value.', '1807.06782-1-6-4': 'We find that 9% of events have more than one [MATH] candidates and the best [MATH] selection chooses the correct candidate in 99.5% of cases.', '1807.06782-1-7-0': 'In this measurement, the dominant background is from the continuum [MATH] processes.', '1807.06782-1-7-1': 'To suppress this, we construct a Fisher discriminant [CITATION] from 17 modified Fox-Wolfram moments [CITATION].', '1807.06782-1-7-2': 'To further improve the distinguishing power, we combine the output of the Fisher discriminant with four more variables in a NN.', '1807.06782-1-7-3': "These are: the cosine of the angle between the reconstructed [MATH] flight direction and the beam direction in the CM frame, the offset between the reconstructed [MATH]'s vertex and the rest of the tracks' vertex along the [MATH] axis, the cosine of the angle between the thrust axis [CITATION] of the reconstructed [MATH] and that of the rest of the event, and a [MATH] meson flavor tagging quality variable.", '1807.06782-1-7-4': 'The NN is trained by using signal MC and continuum MC samples.', '1807.06782-1-7-5': 'The NN output ([MATH]) ranges from [MATH] to 1, and it is required to be greater than 0.7.', '1807.06782-1-7-6': 'This removes 93% of continuum background while 82% of the signal is retained.', '1807.06782-1-7-7': '[MATH] is then transformed to [MATH], where [MATH] is 0.7 and [MATH] is the max value of [MATH] among the generated MC samples.', '1807.06782-1-8-0': 'Background events from [MATH] decays through the [MATH] process (generic [MATH] decays) and charmless (rare) [MATH] decays exhibit peaking structures in the signal region.', '1807.06782-1-8-1': 'They are mainly due to the two-body decays of [MATH] mesons and [MATH], e.g. [MATH], [MATH], [MATH], [MATH], and [MATH].', '1807.06782-1-8-2': 'These decays can be identified by peaks at the nominal [MATH] and [MATH] mass in the distributions of the invariant masses of two of the final-state particles ([MATH], [MATH], [MATH], and the cases with changing the mass hypothesis of charged kaon or pion).', '1807.06782-1-8-3': 'We exclude the events within [MATH] of the peaking structures to suppress the contributions from [MATH] mesons and [MATH].', '1807.06782-1-9-0': 'The rare [MATH] background is studied with a large MC sample in which the branching fractions are much larger than the measured or expected value.', '1807.06782-1-9-1': 'Two modes are found to have peaks near the [MATH] signal region: [MATH] and [MATH], including their intermediate resonant modes, while the rest of the rare [MATH] events have a relatively flat [MATH] distribution.', '1807.06782-1-10-0': 'The signal yield and [MATH] are extracted from a three-dimensional extended unbinned maximum likelihood fit, with the likelihood defined as [EQUATION] where [EQUATION] [MATH] is the total number of candidate events, [MATH] is the number of events in category [MATH], [MATH] denotes event index, [MATH] is the charge of [MATH] of the [MATH]-th event, [MATH] is the value of [MATH] violation parameter of the [MATH]-th category, [MATH] represents the value of the corresponding three-dimensional probability density function (PDF) for the [MATH]-th category, and [MATH], [MATH], and [MATH] are the [MATH], [MATH], and [MATH] value of the [MATH]-th event, respectively.', '1807.06782-1-11-0': "With all the selection criteria applied, the signal MC sample contains 97.6% of the correctly-reconstructed signal [MATH] events ('true' signal) and 2.4% self-crossfeed (scf) events.", '1807.06782-1-11-1': 'In the fit, the ratio of scf events to true signal events is fixed, so the signal yield ([MATH]) is defined as combined yield of the two PDF ([MATH]).', '1807.06782-1-11-2': 'In addition to the signal part, five more categories are included in the fit: continuum background, generic [MATH] background, [MATH], [MATH], and the rest of the rare [MATH] background.', '1807.06782-1-11-3': 'The true signal PDF is described by a product of a double Gaussian function in [MATH], a triple Gaussian function in [MATH], and a bifurcated Gaussian function in [MATH].', '1807.06782-1-11-4': 'These signal PDF shapes are calibrated by data-MC differences obtained from study of the control mode: [MATH] with [MATH].', '1807.06782-1-11-5': 'The continuum background PDF is described by a product of an ARGUS function [CITATION] in [MATH], a second-order polynomial in [MATH], and a combination of a single Gaussian and a bifurcated Gaussian function in [MATH].', '1807.06782-1-11-6': 'The shape of the continuum background PDF is free in the data fit.', '1807.06782-1-11-7': 'For the others (self-crossfeed, generic [MATH], [MATH], [MATH], and rare [MATH]), their PDFs are described by a smoothed histogram in [MATH] and [MATH], and a bifurcated Gaussian function in [MATH] whose shape is based on MC.', '1807.06782-1-11-8': 'The yield of each category is floated.', '1807.06782-1-11-9': 'Except for the signal, [MATH] is fixed to zero for the other categories.', '1807.06782-1-12-0': 'The projections of the fit are shown in Fig. [REF].', '1807.06782-1-12-1': 'We obtain a signal yield of [MATH] with a statistical significance of 13.0, and an [MATH] of [MATH].', '1807.06782-1-12-2': 'The statistical significance of fitted yield is defined as [MATH], where [MATH] and [MATH] are the likelihood values obtained by the fit with and without the signal yield fixed as zero, respectively.', '1807.06782-1-13-0': 'The branching fraction is calculated using [EQUATION] where [MATH], [MATH], [MATH], and [MATH] are the fitted signal yield, the number of [MATH] pairs ([MATH]), the reconstruction efficiency of signal, and the efficiency calibration factor, respectively.', '1807.06782-1-13-1': 'The efficiency calibration factor ([MATH]) contains calibrations due to various systematic effects: [MATH], where [MATH]) and [MATH]) are the corrections due to the [MATH] and [MATH] identification with requirement on [MATH] and [MATH], and are obtained by the control sample study of [MATH] with [MATH], [MATH] is due to the requirement on [MATH] and is obtained by the [MATH] with [MATH] control sample study, and [MATH] is due to fit bias and is obtained by ensemble test on the fitter.', '1807.06782-1-13-2': 'The reconstruction efficiency for the signal ([MATH]) is [MATH] with all the selection criteria applied.', '1807.06782-1-14-0': 'After separating the three Dalitz variables into five bins, we obtain their background-subtracted histograms by the [MATH] method, and then calculate the differential branching fraction as function of the three Dalitz variables with the yield and reconstruction efficiency within each bin.', '1807.06782-1-14-1': 'The background-subtracted Dalitz plot is shown in Fig. [REF].', '1807.06782-1-14-2': 'Fig. [REF] shows the differential branching fraction as functions of the three Dalitz variables including comparison to the MC with a three-body phase space decay model.', '1807.06782-1-14-3': 'At the second bin (about 1.2 GeV/[MATH]) of the [MATH] spectrum, a large deviation with 6.6[MATH] local excess from the phase space MC is observed.', '1807.06782-1-14-4': 'At the fifth bin (about 4.2 GeV/[MATH]) of the [MATH] spectrum, we also observe deviation with 3.4[MATH] local excess from the phase space MC.', '1807.06782-1-15-0': 'To check the localized [MATH] asymmetry, differential branching fraction for the [MATH] and [MATH] final states are shown in Fig. [REF].', '1807.06782-1-15-1': 'Within each bin of the Dalitz variables, we see null asymmetry since there is no significant difference in the branching fractions between the two final states cases.', '1807.06782-1-15-2': 'The details of differential branching fraction calculation in each bin are summarized in Table [REF].', '1807.06782-1-16-0': 'Sources of various systematic uncertainties on the branching fraction calculation are shown in Table [REF].', '1807.06782-1-16-1': 'The uncertainty due to the total number of [MATH] pairs is 1.4.', '1807.06782-1-16-2': 'The uncertainty due to the charged-track reconstruction efficiency is estimated to be 0.35 per track by using the partially reconstructed [MATH] with [MATH] events.', '1807.06782-1-16-3': 'The uncertainty due to the [MATH] branching fraction is based on the world average value [MATH] [CITATION].', '1807.06782-1-16-4': 'The uncertainty due to [MATH] identification is estimated to be 1.6% based on a control sample of [MATH] with [MATH].', '1807.06782-1-16-5': 'The uncertainty of the reconstruction efficiency is estimated based on MC statistics.', '1807.06782-1-16-6': 'The uncertainty due to the shape-fixed signal and background PDFs is estimated by the deviation of fitted signal yield with varying the conditions of those PDFs in different cases.', '1807.06782-1-17-0': 'Sources of various systematic uncertainties on [MATH] are listed in Table [REF].', '1807.06782-1-17-1': 'The uncertainty due to [MATH] and [MATH] detection bias are obtained by control sample studies of [MATH] and [MATH] [CITATION], and [MATH] [CITATION], respectively.', '1807.06782-1-17-2': 'The uncertainty due to the shape-fixed signal and background PDFs is estimated by the deviation of fitted [MATH] with varying the conditions of those shape-fixed PDFs in different cases.', '1807.06782-1-18-0': 'In conclusion, we have performed a measurement of branching fraction and [MATH] of the [MATH] decay based on a data sample of 711 fb[MATH] collected by Belle.', '1807.06782-1-18-1': 'We obtain a branching fraction of [MATH] and an [MATH] of [MATH], where their first uncertainty is statistical and the second is systematic.', '1807.06782-1-18-2': 'The measured [MATH] is consistent with null asymmetry.', '1807.06782-1-18-3': 'In addition, we also utilize the background-subtracted distributions to calculate the differential branching fraction as functions of [MATH], [MATH], and [MATH].', '1807.06782-1-18-4': 'Compared with the phase space MC, hints of peaking resonances are seen at about 1.2 GeV/[MATH] of [MATH] with 6.6[MATH] local excess and about 4.2 GeV/[MATH] of [MATH] with 3.4[MATH].', '1807.06782-1-18-5': 'No localized [MATH] asymmetry is observed.', '1807.06782-1-18-6': 'In the near future, the experiments with large data sets such as Belle II and LHCb can provide more detailed analysis employing a full Dalitz analysis to search for the intermediate resonances and localized [MATH] asymmetry.', '1807.06782-1-19-0': 'We thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics group for the efficient operation of the solenoid; and the KEK computer group, the National Institute of Informatics, and the PNNL/EMSL computing group for valuable computing and SINET4 network support.', '1807.06782-1-19-1': 'We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council; Austrian Science Fund under Grant No. P 22742-N16 and P 26794-N20; the National Natural Science Foundation of China under Contracts No. 10575109, No. 10775142, No. 10875115, No. 11175187, No. 11475187 and No. 11575017; the Chinese Academy of Science Center for Excellence in Particle Physics; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. LG14034; the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Universe, and the VolkswagenStiftung; the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; the WCU program of the Ministry of Education, National Research Foundation (NRF) of Korea Grants No. 2011-0029457, No. 2012-0008143, No. 2012R1A1A2008330, No. 2013R1A1A3007772, No. 2014R1A2A2A01005286, No. 2014R1A2A2A01002734, No. 2015R1A2A2A01003280 , No. 2015H1A2A1033649; the Basic Research Lab program under NRF Grant No. KRF-2011-0020333, Center for Korean J-PARC Users, No. NRF-2013K1A3A7A06056592; the Brain Korea 21-Plus program and Radiation Science Research Institute; the Polish Ministry of Science and Higher Education and the National Science Center; the Ministry of Education and Science of the Russian Federation and the Russian Foundation for Basic Research; the Slovenian Research Agency; Ikerbasque, Basque Foundation for Science and the Euskal Herriko Unibertsitatea (UPV/EHU) under program UFI 11/55 (Spain); the Swiss National Science Foundation; the Ministry of Education and the Ministry of Science and Technology of Taiwan; and the U.S. Department of Energy and the National Science Foundation.', '1807.06782-1-19-2': 'This work is supported by a Grant-in-Aid from MEXT for Science Research in a Priority Area ("New Development of Flavor Physics") and from JSPS for Creative Scientific Research ("Evolution of Tau-lepton Physics").'}

{'1807.06782-2-0-0': 'We report the measurement of the branching fraction and final-state asymmetry for the [MATH] decays.', '1807.06782-2-0-1': 'The analysis is based on a data sample of 711 [MATH] collected at the [MATH] resonance with the Belle detector at the KEKB asymmetric-energy [MATH] collider.', '1807.06782-2-0-2': 'We obtain a branching fraction of [MATH] and a final-state asymmetry of [MATH], where the first uncertainties are statistical and the second are systematic.', '1807.06782-2-0-3': 'Hints of peaking structures are observed in the differential branching fraction plotted as functions of Dalitz variables.', '1807.06782-2-1-0': 'Three-body charmless hadronic [MATH] decays are suppressed in the standard model (SM) and are also sensitive to localized [MATH] violation in the phase space [CITATION].', '1807.06782-2-1-1': 'The [MATH] [CITATION] decays with even number of kaons have a smaller decay rate compared to the cases with odd number of kaons.', '1807.06782-2-1-2': 'These proceed via the [MATH] tree-level, the [MATH]-exchange, and the [MATH] penguin process with a virtual loop, which provides an opportunity to search for physics beyond the SM since new heavy particles may cause deviations from SM predictions.', '1807.06782-2-2-0': 'Previous measurements by the BaBar [CITATION] and LHCb [CITATION] experiments find hints of structures at the low [MATH] and [MATH] regions that have highly asymmetric helicity angular distributions.', '1807.06782-2-2-1': 'However, the yield is not enough to draw firm conclusions with a full Dalitz analysis.', '1807.06782-2-2-2': 'Similar studies on [MATH] were performed by Belle [CITATION], BaBar [CITATION], and LHCb [CITATION], in which strong evidence of localized [MATH] violation was found in the low [MATH] region.', '1807.06782-2-3-0': 'By using the full data set of Belle, we expect to measure the branching fraction and final-state asymmetry of [MATH] decays more precisely.', '1807.06782-2-3-1': 'Using charges of final-state particles, the latter is defined as [EQUATION] where [MATH] denotes the measured signal yield of the corresponding [MATH] final states.', '1807.06782-2-3-2': '[MATH] is distinct from the direct [MATH] asymmetry; rather it is an asymmetry between the decay final states of [MATH] and [MATH] where [MATH] leads to a [MATH].', '1807.06782-2-3-3': "We measure [MATH] as the measurement of direct [MATH] asymmetry based on flavor tagging won't be so precise.", '1807.06782-2-3-4': 'Only about 30% of events can be effectively flavor-tagged, which would be further affected by [MATH]-[MATH] mixing.', '1807.06782-2-3-5': 'In addition, we use the [MATH] [CITATION] method to obtain the background-subtracted yields for the Dalitz variables [MATH], [MATH], and [MATH], and hence determine their differential branching fractions.', '1807.06782-2-3-6': 'The total branching fraction is extracted by integrating the differential branching fraction.', '1807.06782-2-4-0': 'Our measurements are obtained from a data sample of 711 [MATH], corresponding to [MATH] pairs, collected with the Belle detector [CITATION] operating at the KEKB asymmetric-energy [MATH] collider [CITATION].', '1807.06782-2-4-1': 'The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF) and an electromagnetic calorimeter comprised of CsI(Tl) crystals, all located inside a superconducting solenoid that provides a 1.5 T magnetic field.', '1807.06782-2-4-2': 'An iron flux-return yoke located outside the solenoid is instrumented to detect [MATH] mesons and muons.', '1807.06782-2-4-3': 'The detector is described in detail elsewhere [CITATION].', '1807.06782-2-5-0': 'This analysis uses the data sets with two different inner-detector configurations.', '1807.06782-2-5-1': 'About 140 [MATH] were collected with a beam-pipe of radius 2.0 cm and with 3 layers of SVD, while the rest of the data set was recorded with a beam-pipe of radius 1.5 cm and 4 layers of SVD [CITATION].', '1807.06782-2-5-2': 'Large samples Monte Carlo (MC) events for signal and backgrounds are generated with EvtGen [CITATION] and subsequently simulated with GEANT3 [CITATION] with the configurations of the Belle detector.', '1807.06782-2-5-3': 'These samples are used to obtain expected distributions of various physical quantities for signal and backgrounds, to optimize selection criteria as well as to determine the signal detection efficiency.', '1807.06782-2-6-0': 'The selection criteria for the final-state charged particles in the [MATH] reconstruction are based on information obtained from the tracking systems (SVD and CDC) and the charged-hadron identification (PID) systems namely CDC, ACC, and TOF.', '1807.06782-2-6-1': 'The charged kaons and pions are required to have an impact parameter within [MATH] cm of the interaction point (IP) in the transverse plane, and within [MATH] cm along the [MATH] beam direction.', '1807.06782-2-6-2': 'The likelihood values of each track for kaon and pion hypotheses ([MATH] and [MATH]) are determined from the information provided by the PID system.', '1807.06782-2-6-3': 'The track is identified as a kaon if [MATH] else as a pion.', '1807.06782-2-6-4': 'The efficiency for identifying a pion (kaon) is about 88% (86%), which depends on the momentum of the track, while the probability for a pion or a kaon to be misidentified as the other one is less than 10%.', '1807.06782-2-6-5': 'The efficiency and misidentification probability are averaged over the momentum of the final-state particles.', '1807.06782-2-6-6': 'The [MATH] candidates are reconstructed via the [MATH] decay, and the identification is enhanced by using a neural network (NN) [CITATION] which combines seven kinematic variables of [MATH] [CITATION].', '1807.06782-2-6-7': 'The invariant mass of the [MATH] candidates are required to be within [MATH] MeV/[MATH] of the world average, which corresponds to about three times of the resolution.', '1807.06782-2-6-8': 'The vertex fit of [MATH] is required to succeed with the goodness-of-fit value [MATH] less than 20.', '1807.06782-2-7-0': '[MATH] mesons are identified with two kinematic variables calculated in the center-of-mass frame: the beam-energy-constrained-mass [MATH], and the energy difference [MATH], where [MATH] is the beam energy, and [MATH]) is the momentum (energy) of the reconstructed [MATH] meson.', '1807.06782-2-7-1': 'The [MATH] candidates are required to have [MATH] 5.255 GeV/[MATH] and [MATH] 0.15 GeV, and the signal region is defined as 5.272 GeV/[MATH] 5.288 GeV/[MATH] and [MATH] 0.05 GeV.', '1807.06782-2-7-2': 'We require a vertex fit for [MATH] candidates with [MATH] 100.', '1807.06782-2-7-3': 'We find that 9% of events have more than one [MATH] candidates.', '1807.06782-2-7-4': 'In those cases, we choose the one with the smallest [MATH] value.', '1807.06782-2-7-5': 'Our best [MATH] selection method chooses the correct candidate in 99% of cases.', '1807.06782-2-8-0': 'The dominant background arises from the continuum [MATH] process.', '1807.06782-2-8-1': 'To suppress this, we construct a Fisher discriminant [CITATION] from 17 modified Fox-Wolfram moments [CITATION].', '1807.06782-2-8-2': 'To further improve the distinguishing power, we combine the output of the Fisher discriminant with four more variables in a NN.', '1807.06782-2-8-3': "These are: the cosine of the angle between the reconstructed [MATH] flight direction and the beam direction in the CM frame, the offset between the vertex of the reconstructed [MATH] and that of the rest of the tracks' vertex along the [MATH] axis, the cosine of the angle between the thrust axis [CITATION] of the reconstructed [MATH] and that of the rest of the event in the CM frame, and a [MATH] meson flavor tagging quality variable.", '1807.06782-2-8-4': 'The NN is trained with signal and continuum MC samples.', '1807.06782-2-8-5': 'The NN output ([MATH]) ranges from [MATH] to 1, and it is required to be greater than 0.7.', '1807.06782-2-8-6': 'This removes 93% of continuum background while 82% of the signal is retained.', '1807.06782-2-8-7': 'We transform [MATH] to [MATH], where [MATH] is 0.7 and [MATH] is the maximum value of [MATH].', '1807.06782-2-9-0': 'Background events from [MATH] decays mediated via the [MATH] transition (generic [MATH] decays) exhibit peaking structures in the signal region.', '1807.06782-2-9-1': 'They are mainly due to the two-body decays of [MATH] mesons and [MATH], e.g., [MATH], [MATH], [MATH], [MATH], and [MATH].', '1807.06782-2-9-2': 'These decays can be identified by peaks at the nominal [MATH] and [MATH] mass in the distributions of the invariant masses of two of the final-state particles ([MATH], [MATH], [MATH], and the cases with changing the masses hypothesis of charged kaon or pion).', '1807.06782-2-9-3': 'We exclude the events within [MATH] of the peaking structures to suppress the contributions from [MATH] mesons and [MATH].', '1807.06782-2-10-0': 'The rare [MATH] background coming from [MATH] transitions is studied with a large MC sample in which the branching fractions are much larger than the measured or expected value.', '1807.06782-2-10-1': 'Two modes are found to have peaks near the [MATH] signal region: [MATH] and [MATH], including their intermediate resonant modes.', '1807.06782-2-10-2': 'Rest of the rare [MATH] events have a relatively flat [MATH] distribution.', '1807.06782-2-11-0': 'The signal yield and [MATH] are extracted from a three-dimensional extended unbinned maximum likelihood fit, with the likelihood defined as [EQUATION] where, [EQUATION] [MATH] is the total number of candidate events, [MATH] is the number of events in category [MATH], [MATH] denotes event index, [MATH] is the charge of [MATH] in the [MATH]-th event, [MATH] is the value of final-state asymmetry of the [MATH]-th category, [MATH] represents the value of the corresponding three-dimensional probability density function (PDF), and [MATH], [MATH], and [MATH] are the [MATH], [MATH], and [MATH] value of the [MATH]-th event, respectively.', '1807.06782-2-12-0': "With all the selection criteria applied, the signal MC sample contains 98% of the correctly-reconstructed signal [MATH] events ('true' signal) and 2% self-crossfeed (scf) events.", '1807.06782-2-12-1': 'In the fit, the ratio of scf to true signal events is fixed.', '1807.06782-2-12-2': 'The signal yield ([MATH]) is the combined yield of the two PDFs.', '1807.06782-2-12-3': 'In addition to the signal part, five more categories are included in the fit: continuum background, generic [MATH] background, [MATH], [MATH], and the rest of the rare [MATH] background.', '1807.06782-2-12-4': 'The true signal PDF is described by a product of a sum of two Gaussian functions in [MATH], a sum of three Gaussian functions in [MATH], and an asymmetric Gaussian function in [MATH].', '1807.06782-2-12-5': 'These signal PDF shapes are calibrated including possible data-MC differences obtained from study of the control mode: [MATH] with [MATH].', '1807.06782-2-12-6': 'The continuum background PDF is described by a product of an ARGUS function [CITATION] in [MATH], a second-order polynomial in [MATH], and a combination of a Gaussian and an asymmetric Gaussian function in [MATH].', '1807.06782-2-12-7': 'The shape parameters of the continuum background PDF are free in the data fit, except for the ARGUS end-point which is fixed to 5.2892 GeV/[MATH].', '1807.06782-2-12-8': 'For the others (scf, generic [MATH], [MATH], [MATH], and rare [MATH]), their PDFs are described by a smoothed histogram in [MATH] and [MATH], and an asymmetric Gaussian function in [MATH] whose shape is based on MC.', '1807.06782-2-12-9': 'The yield of each category is floated.', '1807.06782-2-12-10': 'Except for the signal, [MATH] is fixed to zero for the other categories.', '1807.06782-2-13-0': 'The projections of the fit are shown in Fig. [REF].', '1807.06782-2-13-1': 'We obtain a signal yield of [MATH] with a statistical significance of 13 standard deviations, and an [MATH] of [MATH].', '1807.06782-2-13-2': 'The significance is defined as [MATH], where [MATH] and [MATH] are the likelihood values obtained by the fit with and without the signal yield fixed to zero, respectively.', '1807.06782-2-14-0': 'The branching fraction is calculated using [EQUATION] where [MATH], [MATH], [MATH], and [MATH] are the fitted signal yield, the number of [MATH] pairs ([MATH]), the reconstruction efficiency of signal, and the efficiency calibration factor, respectively.', '1807.06782-2-14-1': 'The last quantity contains calibrations due to various systematic effects: [MATH], where [MATH]) and [MATH]) are the corrections due to the [MATH] and [MATH] identification with requirement on [MATH] and [MATH], and are obtained by the control sample study of [MATH] with [MATH], [MATH] is due to the requirement on [MATH] and is obtained by the [MATH] with [MATH] control sample study, and [MATH] is due to fit bias and is obtained by ensemble test on the fitter.', '1807.06782-2-14-2': 'The reconstruction efficiency for the signal ([MATH]) is [MATH] with all the selection criteria applied.', '1807.06782-2-15-0': 'Figure [REF] shows the background-subtracted Dalitz plot obtained with the [MATH] method.', '1807.06782-2-15-1': 'There seem to be some structures around the region of 2 GeV[MATH]/[MATH] and 7 GeV[MATH]/[MATH] 23 GeV[MATH]/[MATH].', '1807.06782-2-15-2': 'To check the projections on the Dalitz variables, we also obtain their background-subtracted distributions after separating them into five bins, and then calculate the differential branching fraction as function of the three Dalitz variables with the yield and reconstruction efficiency within each bin.', '1807.06782-2-15-3': 'Figure [REF] shows the differential branching fraction as functions of the three Dalitz variables including comparison to the MC with a three-body phase space decay model.', '1807.06782-2-15-4': 'Large deviation from the phase space expectation is found at the second bin (around 1.2 GeV/[MATH]) of the [MATH] spectrum and at the fifth bin (around 4.2 GeV/[MATH]) of the [MATH] spectrum.', '1807.06782-2-15-5': "In addition, no obvious structure is seen at both the low [MATH] and [MATH] regions, which are also consistent with the previous two-body decays' measurements of [MATH] [CITATION] and [MATH] [CITATION].", '1807.06782-2-16-0': 'To check the localized final-state asymmetry, differential branching fraction for the [MATH] and [MATH] final states are shown in Fig. [REF].', '1807.06782-2-16-1': 'Within each bin of the Dalitz variables, we see null asymmetry since there is no significant difference in the branching fractions between the two final states.', '1807.06782-2-16-2': 'The details of differential branching fraction calculation in each bin are summarized in Table [REF].', '1807.06782-2-17-0': 'Sources of various systematic uncertainties on the branching fraction calculation are shown in Table [REF].', '1807.06782-2-17-1': 'The uncertainty due to the total number of [MATH] pairs is 1.4.', '1807.06782-2-17-2': 'The uncertainty due to the charged-track reconstruction efficiency is estimated to be 0.35 per track by using the partially reconstructed [MATH] with [MATH] events.', '1807.06782-2-17-3': 'The uncertainty due to the [MATH] and [MATH] identification are obtained by the control sample study of [MATH] with [MATH].', '1807.06782-2-17-4': 'The uncertainty due to the [MATH] branching fraction is based on the world average value [MATH] [CITATION].', '1807.06782-2-17-5': 'The uncertainty due to [MATH] identification is estimated to be 1.6% based on a control sample of [MATH] with [MATH].', '1807.06782-2-17-6': 'The uncertainty due to continuum suppression with the requirement on [MATH] and is obtained by the [MATH] with [MATH] control sample study.', '1807.06782-2-17-7': 'The uncertainty of the reconstruction efficiency is estimated due to limited MC statistics.', '1807.06782-2-17-8': 'The uncertainty due to the fixed signal and background PDF shapes is estimated by the deviation of fitted signal yield with varying the conditions of those PDFs in different cases.', '1807.06782-2-17-9': 'For all the smoothed histograms, we vary the binning conditions of those histograms.', '1807.06782-2-17-10': 'For the other PDFs with fixed parameterization, the fixed parameters are randomized by using Gaussian random number to repeat data fits with various parameter sets, and the uncertainty of the yield distribution is quoted.', '1807.06782-2-17-11': 'The uncertainty due to fit bias is obtained by ensemble test on the fitter.', '1807.06782-2-18-0': 'Sources of various systematic uncertainties on [MATH] are listed in Table [REF].', '1807.06782-2-18-1': 'The uncertainty due to [MATH] and [MATH] detection bias are obtained by control sample studies of [MATH] and [MATH] [CITATION], and [MATH] [CITATION], respectively.', '1807.06782-2-18-2': 'The uncertainty due to the fixed signal and background PDF shapes is using the same way as the one for the uncertainty on branching fraction.', '1807.06782-2-18-3': 'It is also estimated by the deviation of fitted [MATH] with varying the conditions of those PDFs in different cases.', '1807.06782-2-19-0': 'In conclusion, we have performed a measurement of branching fraction and [MATH] of the [MATH] decay based on a data sample of 711 fb[MATH] collected by Belle.', '1807.06782-2-19-1': 'We obtain a branching fraction of [MATH] and an [MATH] of [MATH], where their first uncertainty is statistical and the second is systematic.', '1807.06782-2-19-2': 'The measured [MATH] is consistent with null asymmetry.', '1807.06782-2-19-3': 'Hints of peaking structures are seen around a region of 2 GeV[MATH]/[MATH] and 7 GeV[MATH]/[MATH] 23 GeV[MATH]/[MATH] in the Dalitz plot.', '1807.06782-2-19-4': 'A cross-check is done by the differential branching fraction with projecting on each Dalitz variable, and hints of peaking resonances are seen at around 1.2 GeV/[MATH] of [MATH] and around 4.2 GeV/[MATH] of [MATH] when compared to the phase space MC.', '1807.06782-2-19-5': 'No obvious [MATH] structure is seen at both the low [MATH] and [MATH] spectrum, which is also consistent with the BaBar and LHCb results [CITATION].', '1807.06782-2-19-6': 'No localized final-state asymmetry is observed.', '1807.06782-2-19-7': 'In the near future, the experiments with large data sets such as Belle II and LHCb can provide more detailed analysis employing a full Dalitz analysis to search for the intermediate resonances and localized final-state asymmetry.', '1807.06782-2-20-0': 'We thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics group for the efficient operation of the solenoid; and the KEK computer group, the National Institute of Informatics, and the PNNL/EMSL computing group for valuable computing and SINET4 network support.', '1807.06782-2-20-1': 'We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council; Austrian Science Fund under Grant No. P 22742-N16 and P 26794-N20; the National Natural Science Foundation of China under Contracts No. 10575109, No. 10775142, No. 10875115, No. 11175187, No. 11475187 and No. 11575017; the Chinese Academy of Science Center for Excellence in Particle Physics; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. LG14034; the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Universe, and the VolkswagenStiftung; the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; the WCU program of the Ministry of Education, National Research Foundation (NRF) of Korea Grants No. 2011-0029457, No. 2012-0008143, No. 2012R1A1A2008330, No. 2013R1A1A3007772, No. 2014R1A2A2A01005286, No. 2014R1A2A2A01002734, No. 2015R1A2A2A01003280 , No. 2015H1A2A1033649; the Basic Research Lab program under NRF Grant No. KRF-2011-0020333, Center for Korean J-PARC Users, No. NRF-2013K1A3A7A06056592; the Brain Korea 21-Plus program and Radiation Science Research Institute; the Polish Ministry of Science and Higher Education and the National Science Center; the Ministry of Education and Science of the Russian Federation and the Russian Foundation for Basic Research; the Slovenian Research Agency; Ikerbasque, Basque Foundation for Science and the Euskal Herriko Unibertsitatea (UPV/EHU) under program UFI 11/55 (Spain); the Swiss National Science Foundation; the Ministry of Education and the Ministry of Science and Technology of Taiwan; and the U.S. Department of Energy and the National Science Foundation.', '1807.06782-2-20-2': 'This work is supported by a Grant-in-Aid from MEXT for Science Research in a Priority Area ("New Development of Flavor Physics") and from JSPS for Creative Scientific Research ("Evolution of Tau-lepton Physics").'}

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[]

[['1807.06782-1-17-2', '1807.06782-2-18-2'], ['1807.06782-1-17-2', '1807.06782-2-18-3'], ['1807.06782-1-14-1', '1807.06782-2-15-0'], ['1807.06782-1-8-0', '1807.06782-2-9-0'], ['1807.06782-1-11-1', '1807.06782-2-12-1'], ['1807.06782-1-11-1', '1807.06782-2-12-2'], ['1807.06782-1-11-3', '1807.06782-2-12-4'], ['1807.06782-1-11-6', '1807.06782-2-12-7'], ['1807.06782-1-2-1', '1807.06782-2-3-1'], ['1807.06782-1-18-4', '1807.06782-2-19-4'], ['1807.06782-1-18-5', '1807.06782-2-19-6'], ['1807.06782-1-16-5', '1807.06782-2-17-7'], ['1807.06782-1-7-0', '1807.06782-2-8-0'], ['1807.06782-1-7-7', '1807.06782-2-8-7'], ['1807.06782-1-9-1', '1807.06782-2-10-1'], ['1807.06782-1-9-1', '1807.06782-2-10-2'], ['1807.06782-1-6-3', '1807.06782-2-7-4'], ['1807.06782-1-6-4', '1807.06782-2-7-3'], ['1807.06782-1-6-4', '1807.06782-2-7-5']]

[]

['1807.06782-1-6-1', '1807.06782-1-19-1', '1807.06782-2-7-1', '1807.06782-2-20-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1807.06782

1211.1915

{'1211.1915-1-0-0': 'The activated dynamics exponent [MATH] which relates the correlation length in glasses to the alpha relaxation time is argued to equal [MATH], the dimensionality of the system.', '1211.1915-1-0-1': 'This result is obtained via a detailed study of relaxation processes in small systems of disks confined in a square and extended to large systems by employing the mosaic picture of glasses.', '1211.1915-1-0-2': 'It is in excellent agreement with the simulational studies on hard spheres of Berthier and Witten.', '1211.1915-1-1-0': 'As the temperature [MATH] of a fragile glass is reduced, the alpha relaxation time [MATH], the fundamental glass time scale, rapidly increases.', '1211.1915-1-1-1': 'In the old theory of Adam and Gibbs [CITATION], and the more recent random first-order transition (RFOT) theory of glasses [CITATION], [CITATION] and its elaboration as a mosaic picture (for a review see [CITATION]), this increase is attributed to cooperatively rearranging regions (CRR) of size [MATH].', '1211.1915-1-1-2': 'This length scale increases as the temperature decreases and has actually been determined numerically from a study of "point-to-set" correlations [CITATION].', '1211.1915-1-1-3': 'It is also expected to be related to the linear extent of the dynamical heterogeneities [CITATION].', '1211.1915-1-1-4': '[MATH] and [MATH] are thought to be related by the activated dynamics equation [EQUATION] where [MATH] is a non-universal constant, [MATH] is a length of the order of a molecular diameter and [MATH] is a microscopic time scale.', '1211.1915-1-1-5': 'Much interest attaches to the value of the exponent [MATH].', '1211.1915-1-1-6': 'In this paper we shall argue that [EQUATION] where [MATH] is the dimensionality of the system.', '1211.1915-1-1-7': 'This result should hold both for hard spheres and disks, and might also extend to soft potentials.', '1211.1915-1-1-8': 'For hard spheres ([MATH]) or hard disks ([MATH]), the time scales and length scales are increasing functions of the packing fraction [MATH] rather than the temperature [CITATION].', '1211.1915-1-1-9': 'In general, one expects that the correlation length grows as [EQUATION] where in the approach of Fullerton and Moore [CITATION] (see also [CITATION]), [EQUATION] [MATH] is the spin glass domain wall energy exponent, and for [MATH] is approximately [MATH] [CITATION].', '1211.1915-1-1-10': '[MATH] in Eq. ([REF]) is close to (or even identical to) the random close packing fraction [MATH].', '1211.1915-1-1-11': 'Then, if [MATH], we predict that the alpha relaxation time varies as [EQUATION] where the exponent [MATH] in three dimensions.', '1211.1915-1-1-12': 'Berthier and Witten [CITATION] found that [MATH] in their study of hard spheres, when they used a value of [MATH] for their fitting.', '1211.1915-1-1-13': 'Note that Eq. ([REF]) applies to correlations which exist in the supercooled liquid up to alpha relaxation times.', '1211.1915-1-1-14': 'It does not apply to behavior on time scales very much longer than the alpha relaxation time, when real supercooled liquids may phase separate (if they are mixtures of species) or simply crystallize.', '1211.1915-1-2-0': 'The basic idea of Adam and Gibbs [CITATION] and the mosaic picture [CITATION] is that regions of size less than [MATH] are glassy and rarely relax.', '1211.1915-1-2-1': '(In the simulations of Ref. [CITATION], for cavities of a size larger than [MATH] the particles could escape from their initial configurations; but for cavities of size [MATH] or smaller, the particles were often trapped near their initial configurations when realistic, non-swap Monte-Carlo dynamics were used.)', '1211.1915-1-2-2': 'It is by a careful examination of just how the atoms within a region of linear extent [MATH] have to move in order to allow the atoms to escape from their initial configurations that we are able to argue that [MATH].', '1211.1915-1-2-3': 'On the other hand, RFOT with wetting predicts that [MATH] and [MATH] [CITATION] so that [MATH], which is the Vogel-Fulcher-Tamman law [CITATION].', '1211.1915-1-2-4': 'If one uses the RFOT form [MATH], with our value of [MATH], then [MATH], which is also the value obtained by the very different arguments of Schweizer [CITATION].', '1211.1915-1-2-5': 'The simulations in Ref. [CITATION] were claimed to give [MATH], a value which is hard to reconcile with the value of [MATH] of Berthier and Witten [CITATION].', '1211.1915-1-3-0': 'We shall illustrate the process of escape from the initial configuration for two simple systems, consisting of either two or five disks confined in a square.', '1211.1915-1-3-1': 'At high density, escape involves squeezing through a narrow neck in configuration space, which for these simple systems we can make explicit.', '1211.1915-1-3-2': 'We shall show that transition state theory provides a quantitative account of the slow relaxational processes in these small systems.', '1211.1915-1-3-3': 'The transition state theory is then generalized to large systems of hard disks or spheres, when the length scale of the dynamical heterogeneities is of order [MATH], and we derive Eq. ([REF]).', '1211.1915-1-3-4': '(We shall assume throughout that the dynamical and static length scales are proportional when they are both large [CITATION].)', '1211.1915-1-3-5': 'Our generalization from the explicit calculations where the number of disks, [MATH], is small ([MATH] or [MATH]) to the case of large [MATH] involves assumptions about the nature and validity of the mosaic picture [CITATION], but the good agreement for the value [MATH] with the results of Ref. [CITATION] suggests that the assumptions made are actually correct.', '1211.1915-1-4-0': 'We begin with some definitions for our study of two and five disks confined in a square: [MATH] is defined as the length of one side of the square; [MATH] is the radius of the disks; and the packing fraction [MATH] is defined as the fraction of the area of the square that is covered by the disks, [MATH].', '1211.1915-1-4-1': 'A configuration of the disks is defined by the [MATH] coordinates of the disk centers (for disks [MATH]); every configuration belongs to a state, which is the set of configurations that can be reached from it without violating the no-overlap constraint.', '1211.1915-1-5-0': 'Two-disk system.', '1211.1915-1-5-1': 'The system of two disks confined to a square has been considered previously by Speedy [CITATION].', '1211.1915-1-5-2': 'For [MATH] (or [MATH]) the disks can pass each other, though this becomes more difficult as [MATH].', '1211.1915-1-5-3': 'For larger values of [MATH], the configuration space (disregarding the identity of the disks) is broken into two states in which (for [MATH], or [MATH]) the disk centers lie on the same diagonal of the square (see Fig. [REF]).', '1211.1915-1-5-4': 'These limiting configurations are the inherent structures introduced in Ref. [CITATION].', '1211.1915-1-6-0': 'Speedy [CITATION] has considered the thermodynamics of this system and, in particular, finds weakly non-analytic contributions to the thermodynamic quantities, such as the pressure.', '1211.1915-1-6-1': 'Bowles et al. [CITATION] have studied the dynamics of a similar system of two disks confined to a channel, and find that the mean first-passage times of the disks past each other varies as [MATH], where [MATH] for inertial motion of the disks.', '1211.1915-1-6-2': 'Our event-driven molecular dynamics results (Fig. [REF]) confirm the value of the exponent, [MATH], for the case of two disks in a square box.', '1211.1915-1-7-0': 'The origin of this behavior can be obtained from transition state theory [CITATION].', '1211.1915-1-7-1': 'In this well-studied approximation, which works best when the transition rate over a barrier is small, the transition rate [MATH] between two states varies as [EQUATION] where [MATH] is the partition function evaluated at the top of the barrier; see Ref. [CITATION] for a full description of the transition state formalism and the definition of [MATH].', '1211.1915-1-7-2': 'In the case of two disks passing this means that instead of the full partition function integral over [MATH] and [MATH], there is a constraint that [MATH], so it is effectively a three-dimensional integral.', '1211.1915-1-7-3': 'The integral over [MATH] gives a trivial factor [MATH] and the remaining two integrals give a factor [MATH] in the limit [MATH].', '1211.1915-1-7-4': '[MATH] itself is essentially just a constant: it has a very mild singularity, [MATH], when the packing fraction [MATH] approaches [MATH] from above [CITATION].', '1211.1915-1-7-5': 'Transition state theory thus predicts a slope of [MATH] in Fig. [REF] for the dependence of the relaxation rate on packing fraction.', '1211.1915-1-8-0': 'Configurations of the five-disk system.', '1211.1915-1-8-1': 'The configuration space of five disks confined to a square has been analyzed previously: Bowles and Speedy [CITATION] have discussed the thermodynamics and dynamics; Hinow [CITATION] has studied the jammed states of this system; and Carlsson et al. [CITATION] have given a detailed analysis of how the topology of the configuration space depends on [MATH].', '1211.1915-1-8-2': 'We refer to Fig. [REF] for configurations of the disks at two critical values of the radius.', '1211.1915-1-9-0': 'Below the liquid-glass critical point, i.e., for [MATH], the system is fluid (any pair of disks can exchange position), but for slightly greater values of [MATH] the configuration space is fractured into two states: a "crystalline" state in which one disk is surrounded by the four others, confined near the corners of the box; and an "amorphous" state in which all five disks lie close to the walls of the box and are unable to change their order.', '1211.1915-1-9-1': 'Above [MATH], the state fractures further into four "frozen" amorphous states of the kind illustrated in Fig. [REF] (b), in which one disk is confined near a corner of the box.', '1211.1915-1-9-2': 'Above [MATH], the system can exist only in the crystalline state.', '1211.1915-1-10-0': 'It may be noticed that [MATH] differs significantly from the value 0.1871 stated by Carlsson et al. [CITATION].', '1211.1915-1-10-1': 'We have been unable to find a path between glass-like and crystal-like metastable states that passes via the configuration proposed in their paper.', '1211.1915-1-10-2': 'We find, moreover, that their proposed state with [MATH] is not a stationary point of the softened potential energy function [MATH] introduced in Ref. [CITATION]: instead, it is a minimum of [MATH] at which [MATH].', '1211.1915-1-10-3': 'On the other hand, we can show that our own configuration at [MATH] lies on a path between crystal-like and glass-like states and also that this configuration corresponds very precisely to an ordinary saddle point of [MATH].', '1211.1915-1-10-4': 'Such a reaction path is illustrated by an animation provided in the supplement to this paper [CITATION].', '1211.1915-1-11-0': 'Dynamics of the five-disk system.', '1211.1915-1-11-1': 'As for the case of two disks, an event-driven molecular dynamics algorithm [CITATION] was used to simulate the motion of the five-disk system and calculate the mean time of passage between metastable states.', '1211.1915-1-11-2': 'The initial velocities of the disks were chosen to be of fixed magnitude, but with random directions.', '1211.1915-1-12-0': 'A very simple method was used in our work to identify when a transition had taken place.', '1211.1915-1-12-1': 'For the transition from crystal to amorphous states, the simulation is started in a typical "crystal" configuration with one disk [Fig. [REF] (a), shaded] near to the center of the box.', '1211.1915-1-12-2': "The shaded disk's first collision with any wall is an unambiguous sign that the transition to the amorphous state has occurred.", '1211.1915-1-12-3': 'Transitions between metastable amorphous states can be identified in a similar way, using the fact that every disk remains near the perimeter of the box.', '1211.1915-1-12-4': 'From Fig. [REF] (b), we can see that a transition has occurred if any disk makes a collision with a wall other than the one (or two) it was close to in the initial configuration.', '1211.1915-1-12-5': 'For each kind of transition, the time of first occurrence of the diagnostic event is recorded and the simulation restarted with random initial velocities.', '1211.1915-1-13-0': 'Results from our simulations are plotted in Fig. [REF].', '1211.1915-1-14-0': 'They show that, as expected, the mean time of passage increases rapidly as [MATH] approaches a critical value [MATH] or [MATH].', '1211.1915-1-14-1': 'Because of the slowing-down of the dynamics, our simulations were unable to reach a regime where a power-law dependence of [MATH] on [MATH] could be demonstrated unambiguously: the steepest gradients in the graphs of Fig. [REF] (a) and (b) are approximately [MATH] and [MATH], respectively.', '1211.1915-1-14-2': 'The first of these results (for the transition between metastable amorphous states) is easy to understand in the transition-state theory.', '1211.1915-1-14-3': 'If the reaction coordinate is fixed at the value it takes in the transition state, the hard-disk constraints define a configuration space with nine spatial dimensions: accordingly, the constrained partition function (and hence [MATH]) should be expected to vary as [MATH]; we have verified this dependence by a separate Monte Carlo sampling of the partition function.', '1211.1915-1-14-4': 'A similar argument could, of course, be made for the transition between metastable crystal and amorphous states, if [MATH] was sufficiently close to [MATH]; our results from molecular dynamics suggest that the anticipated power-law behaviour is restricted to a very narrow range of [MATH] and that this asymptotic regime was not reached in our simulations.', '1211.1915-1-15-0': 'Our conclusion from these studies is that a transition between states in a region containing [MATH] particles generally requires coordinated motion of all the [MATH] particles in order to squeeze through the neck in the phase space.', '1211.1915-1-15-1': 'The rate at which this will occur is, according to transition state theory, [EQUATION] where [MATH] is the packing fraction of the jammed state closest to the initial configuration of the particles and [MATH] is the highest packing fraction below [MATH] at which a neck first opens to allow escape from the jammed state.', '1211.1915-1-15-2': 'In studies of jammed states in large systems, Torquato and Jiao [CITATION] found an algorithm which generated jammed states over a wide range of packing fractions.', '1211.1915-1-15-3': '(The commonly used Lubachevsky-Stillinger algorithm [CITATION] mainly produces jammed states with [MATH].', '1211.1915-1-15-4': 'When the number of particles is small, the value of [MATH] is reduced and the distribution of the range of [MATH] broadens [CITATION].)', '1211.1915-1-15-5': 'We attribute the slow dynamics in the system to the squeezing through narrow necks in the configuration space which is required to escape from the vicinity of the initial conditions.', '1211.1915-1-16-0': 'The key to the derivation of our main result, [MATH], is the following picture of the allowed configuration space of [MATH] hard spheres or disks in a box.', '1211.1915-1-16-1': 'Consider spheres.', '1211.1915-1-16-2': 'As the packing fraction increases beyond [MATH], there is marked slowing down of the dynamics of the system; [MATH] grows rapidly as the packing fraction increases past this density: the spheres are caged and can only escape from the cages on the alpha relaxation time scale.', '1211.1915-1-16-3': 'We are imagining that typical initial conditions place the system close to one of the jammed states but that there is a way of escaping from this state through a neck in configuration space.', '1211.1915-1-16-4': 'Suppose that the packing fraction [MATH] is close to but below [MATH].', '1211.1915-1-16-5': 'Let [EQUATION] then from Eq. ([REF]), provided [MATH] is large, [EQUATION]', '1211.1915-1-16-6': 'Note that the form of the equation (i.e. its dependence on [MATH]) would be hardly altered if a finite fraction of the [MATH] particles were rattlers.', '1211.1915-1-17-0': 'The next step in the argument is to apply Eq. ([REF]) in the context of the mosaic picture of dynamic heterogeneity [CITATION].', '1211.1915-1-17-1': 'The scale of the tiles within the mosaic is [MATH].', '1211.1915-1-17-2': 'It is pictured that the dynamics of glasses proceeds by a series of rearrangements in each tile.', '1211.1915-1-17-3': 'Once in one tile the atoms have reached a different state; this will sometimes trigger state changes in neigboring tiles.', '1211.1915-1-17-4': 'Such avalanches of movement are commonly observed [CITATION].', '1211.1915-1-17-5': 'To rearrange the spheres in a region of size [MATH] a neck will have to be passed through in phase space and the time scale for doing this will be as given in Eq. ([REF]) but with [MATH] replaced by the number of spheres in the region of size [MATH] which is [MATH].', '1211.1915-1-17-6': 'Different regions will have different values for [MATH], which is in accord with the fact that there is a spread of relaxation times associated with the dynamic heterogeneities [CITATION].', '1211.1915-1-17-7': 'The alpha relaxation time sets the time scale for these processes, so in Eq. ([REF]) it follows that [MATH].', '1211.1915-1-17-8': 'Note that the mosaic size [MATH] is the size of the region on which one has to make re-arrangements in order to allow an atom to escape from its surrounding cage, so that the timescale [MATH] is naturally identified in this picture with the alpha relaxation time.', '1211.1915-1-18-0': 'We should like to thank Chris Fullerton for many useful discussions.'}

{'1211.1915-2-0-0': 'The dynamics of two and five disk systems confined in a square has been studied using molecular dynamics simulations and compared with the predictions of transition state theory.', '1211.1915-2-0-1': 'We determine the partition functions [MATH] and [MATH] of transition state theory using a procedure first used by Salsburg and Wood for the pressure.', '1211.1915-2-0-2': 'Our simulations show this procedure and transition state theory are in excellent agreement with the simulations.', '1211.1915-2-0-3': 'A generalization of the transition state theory to the case of a large number of disks [MATH] is made and shown to be in full agreement with simulations of disks moving in a narrow channel.', '1211.1915-2-0-4': 'The same procedure for hard spheres in three dimensions leads to the Vogel-Fulcher-Tammann formula for their alpha relaxation time.', '1211.1915-2-1-0': '# Introduction', '1211.1915-2-2-0': 'The long relaxation times seen in supercooled liquids have long been a challenge to understand [CITATION].', '1211.1915-2-2-1': 'Glassy behavior has been extensively modelled by studying hard spheres in three dimensions and hard disks in two dimensions.', '1211.1915-2-2-2': 'Some of this work is reviewed in Ref. [CITATION].', '1211.1915-2-2-3': 'In this paper we study small systems of disks, in particular two disks and five disks confined in a square, first using event driven molecular dynamics and then by means of transition state theory [CITATION].', '1211.1915-2-2-4': 'In the final section of the paper we use the insights gained from studying small systems to speculate about the behavior of large numbers of hard spheres or disks.', '1211.1915-2-3-0': 'It is convenient from the outset to introduce the following terminology.', '1211.1915-2-3-1': 'The transition state is the neck in configuration space through which the system has to pass to escape its initial state.', '1211.1915-2-3-2': 'A configuration of the [MATH] disks is defined by the [MATH] coordinates of the disk centers (for disks, [MATH]); every configuration belongs to a state, which is the set of configurations that can be reached from it without violating the no-overlap constraint appropriate for hard disks and spheres.', '1211.1915-2-3-3': 'The transition state theory will be found to work well when the neck is narrow, that is, when there are long relaxation times in the system.', '1211.1915-2-4-0': 'We shall illustrate the process of escape from the initial configuration for two simple systems, consisting of either two or five disks confined in a square.', '1211.1915-2-4-1': 'For these simple systems we can make explicit the narrow necks in configuration space through which the system can escape from its initial configuration near an inherent state [CITATION].', '1211.1915-2-4-2': 'We shall show that transition state theory provides a quantitative account of the slow relaxational processes in these small systems.', '1211.1915-2-4-3': 'The theory requires that one evaluates a variant of the partition function of the system at the neck ([MATH]) and to do this we adopt the procedure first used by Salsburg and Wood [CITATION] to calculate the pressure of hard spheres near their largest packing density.', '1211.1915-2-4-4': 'We have checked its accuracy for these small systems by comparing its predictions for the pressure of the system and the relaxation times with results obtained directly from event driven molecular dynamics.', '1211.1915-2-5-0': 'Of course, one is only interested in small systems of disks because of the light their study might shine on large systems of hard disks or spheres.', '1211.1915-2-5-1': 'We shall show that as [MATH], the number of disks or spheres, becomes large, then, under certain circumstances, our transition state formula for the relaxation time in the system goes over to the well-known Vogel-Fulcher-Tammann (VFT) equation.', '1211.1915-2-5-2': 'These circumstances are evidently realized in at least one case, that of disks moving in a narrow channel [CITATION].', '1211.1915-2-5-3': 'The agreement is quantitative in this case [CITATION].', '1211.1915-2-5-4': 'For hard spheres in three dimensions their alpha relaxation time can be fitted by the VFT form [CITATION], but with the divergence occurring at a density below that of random close packing.', '1211.1915-2-5-5': 'This matter is discussed in Sec. [REF], where we then go on to give a speculative extension to our procedure which leads to a generalized VFT formula where the divergence takes place at a density similar to that of random close packing.', '1211.1915-2-6-0': 'Throughout our study of two and five disks confined in a square, [MATH] denotes the length of one side of the square and [MATH] denotes the radius of the disks.', '1211.1915-2-6-1': 'The packing fraction [MATH] is then defined as the fraction of the area of the square that is covered by the disks, [MATH].', '1211.1915-2-6-2': 'The two disk system is studied in Sec. [REF].', '1211.1915-2-6-3': 'In Sec. [REF] the inherent states and the necks in configuration space which separate them are discussed for the five disk system.', '1211.1915-2-6-4': 'In Sec. [REF] we compare the results of our event driven molecular dynamics simulations for five disks with the predictions of transition state theory.', '1211.1915-2-7-0': '# The two-disk system', '1211.1915-2-8-0': 'Two disks confined to a square have been considered previously by Speedy [CITATION].', '1211.1915-2-8-1': 'For [MATH] (or [MATH]) the disks can pass each other, though this becomes more difficult as [MATH] (see Fig. [REF]).', '1211.1915-2-8-2': 'Awazu [CITATION] studied an autocorrelation function which developed a plateau in this limit, and which he argued showed similarities with the [MATH] and [MATH] relaxation processes found in glasses.', '1211.1915-2-8-3': 'For larger values of [MATH], the configuration space (disregarding the identity of the disks) is broken into two states.', '1211.1915-2-8-4': 'For [MATH], or [MATH], the maximum density possible, the disk centers lie on the same diagonal of the square.', '1211.1915-2-8-5': 'These (two) limiting configurations are the inherent structures introduced in Ref. [CITATION].', '1211.1915-2-9-0': 'Speedy [CITATION] has considered the thermodynamics of this system and, in particular, finds weakly non-analytic contributions to the thermodynamic quantities, such as the pressure, at [MATH].', '1211.1915-2-9-1': 'Speedy used transition state theory to determine the alpha relaxation time of the system, which according to Awazu, is the time [MATH] to flip between the two configurations.', '1211.1915-2-9-2': 'The origin of this behavior can be obtained from transition state theory [CITATION].', '1211.1915-2-9-3': 'In this well-studied approximation, which works best when the transition rate over a barrier is small, the transition rate [MATH] between two states varies as [EQUATION] where [MATH] is a typical particle speed and [MATH] is the partition function evaluated at the top of the barrier along the trajectory which separates the states; see Ref. [CITATION] for a full description of the transition state formalism and the definition of [MATH].', '1211.1915-2-9-4': 'In the case of two disks passing this means that instead of the full partition function integral over [MATH] and [MATH], there is a constraint that [MATH], (say) so it is effectively a three-dimensional integral.', '1211.1915-2-9-5': 'The integral over [MATH] gives a trivial factor [MATH] and the remaining two integrals give a factor [MATH] in the limit [MATH] by the argument used by Salsburg and Wood [CITATION].', '1211.1915-2-9-6': '[MATH] itself is essentially just a constant: it has a very mild singularity, [MATH], when the packing fraction [MATH] approaches [MATH] from above [CITATION].', '1211.1915-2-9-7': 'Transition state theory thus predicts a slope of [MATH] in Fig. [REF] for the dependence of the relaxation time on packing fraction as [MATH].', '1211.1915-2-9-8': '(The full integrals for [MATH] and [MATH] were explicitly evaluated by Speedy [CITATION].)', '1211.1915-2-9-9': 'Our event-driven molecular dynamics results (Fig. [REF]) are consistent with the transition state theory prediction that [MATH], with [MATH], for the case of two disks in a square box, in the limit [MATH].', '1211.1915-2-10-0': '# Configurations of the five-disk system.', '1211.1915-2-11-0': 'The configuration space of five disks confined to a square has been analyzed previously: Bowles and Speedy [CITATION] have discussed the thermodynamics and dynamics; Hinow [CITATION] has studied the jammed states of this system; and Carlsson et al. [CITATION] have given a detailed analysis of how the topology of the configuration space depends on [MATH].', '1211.1915-2-11-1': 'We refer to Fig. [REF] for configurations of the disks at two critical values of the radius.', '1211.1915-2-12-0': 'Below the fluid-crystal critical point, i.e., for [MATH], the system is fluid (any pair of disks can exchange position), but for slightly greater values of [MATH] the configuration space is fractured into two states: a "crystal" state in which one disk is surrounded by the four others, confined near the corners of the box; and a "glass" state in which all five disks lie close to the walls of the box and are unable to change their order.', '1211.1915-2-12-1': 'Above [MATH], the glass state fractures further into four "frozen" glass states of the kind illustrated in Fig. [REF] (b), in which one disk is confined near a corner of the box.', '1211.1915-2-12-2': 'Above [MATH], the system can exist only in the crystalline state.', '1211.1915-2-13-0': 'It may be noticed that [MATH] differs significantly from the value 0.1871 stated by Carlsson et al. [CITATION].', '1211.1915-2-13-1': 'We have been unable to find a path between glass-like and crystal-like metastable states that passes via the configuration proposed in their paper.', '1211.1915-2-13-2': 'We find, moreover, that their proposed state with [MATH] is not a stationary point of the softened potential energy function [MATH] introduced in Ref. [CITATION]: instead, it is a minimum of [MATH] at which [MATH].', '1211.1915-2-13-3': 'It is a dead-end configuration, illustrated in Fig. [REF]: it can be reached from the crystal by the steps in the first two panels of Fig. [REF](a), but progress to the glass state of the third panel is not possible as the central disk cannot escape to the edge of the square.', '1211.1915-2-13-4': 'On the other hand, we can show that our own configuration at [MATH] lies on a path between crystal-like and glass-like states and also that this configuration corresponds very precisely to an ordinary saddle point of [MATH].', '1211.1915-2-13-5': 'Such a reaction path is illustrated by an animation provided in the supplement to this paper [CITATION].', '1211.1915-2-14-0': '# Dynamics of the five-disk system', '1211.1915-2-15-0': 'As for the case of two disks, an event-driven molecular dynamics algorithm [CITATION] was used to simulate the motion of the five-disk system and calculate the mean time of passage between metastable states.', '1211.1915-2-15-1': 'The initial velocities of the disks were drawn from the Maxwell-Boltzmann distribution.', '1211.1915-2-16-0': 'A very simple method was used in our work to identify when a transition had taken place.', '1211.1915-2-16-1': 'For the transition from crystal to glass states, the simulation is started in a typical "crystal" configuration with one disk [Fig. [REF] (a), shaded] near to the center of the box.', '1211.1915-2-16-2': "The shaded disk's first collision with any wall is an unambiguous sign that the transition to the glass state has occurred.", '1211.1915-2-16-3': 'Transitions between metastable glass states can be identified in a similar way.', '1211.1915-2-16-4': 'From Fig. [REF] (b), we can see that a transition has occurred if a disk [e.g. the shaded disk in Fig. [REF] (b)] makes a collision with a wall other than the one it was close to in the initial configuration.', '1211.1915-2-16-5': 'For each kind of transition, the time of first occurrence of the diagnostic event is recorded and the simulation restarted with random initial velocities.', '1211.1915-2-17-0': 'Transition state theory requires us to evaluate [MATH] and [MATH].', '1211.1915-2-17-1': 'We shall use the procedure introduced by Salsburg and Wood [CITATION] to determine these as it becomes essentially exact as the density approaches its maximum value (called [MATH]) appropriate for a given inherent state.', '1211.1915-2-17-2': 'Thus for the crystal state [MATH], so [MATH].', '1211.1915-2-17-3': 'Let [MATH] denote the average spacing between the centers of the particles, where [MATH].', '1211.1915-2-17-4': 'The Salsburg-Wood approximation is that as [MATH], [MATH], where here [MATH], [MATH].', '1211.1915-2-17-5': 'Similarly, as [MATH] from below, [MATH] where [MATH].', '1211.1915-2-17-6': 'Hence, according to transition state theory, the transition rate [MATH] from the crystal to the glass state should vary as [EQUATION]', '1211.1915-2-17-7': 'If the reaction coordinate is fixed at the value it takes in the transition state, the hard-disk constraints define a configuration space with nine spatial dimensions in our two-dimensional hard disk system.', '1211.1915-2-17-8': 'Accordingly, the constrained partition function should be expected to vary as [MATH] using the procedure of Salsburg and Wood [CITATION].', '1211.1915-2-18-0': 'Our simulations to test this for both the glass glass transition and the crystal glass transition are plotted in Fig. [REF].', '1211.1915-2-18-1': 'The agreement is excellent as [MATH].', '1211.1915-2-19-0': 'To further examine the accuracy of the Salsburg-Wood procedure for calculating [MATH] and [MATH], we have determined from our molecular dynamics simulations the pressure of the system in the glass states.', '1211.1915-2-19-1': 'The temperature was obtained from the average kinetic energy, using [MATH].', '1211.1915-2-19-2': 'The results are shown in Fig. [REF].', '1211.1915-2-20-0': 'The Salsburg-Wood approximation for [MATH] predicts that the pressure [EQUATION] on using the relation [MATH].', '1211.1915-2-20-1': 'The straight line in Fig. [REF] represents the prediction of the Salsburg-Wood calculation for the pressure, i.e. Eq. ([REF]), and is in perfect agreement with the data as [MATH].', '1211.1915-2-20-2': 'Notice that at the neck, [MATH], which is indicated by the vertical dashed line in Fig. [REF], the singularities are so mild as to be invisible, which means it is adequate to use in Eq. ([REF]) the form of [MATH] valid near [MATH], even for [MATH] close to [MATH].', '1211.1915-2-20-3': '(Also, the expression for [MATH] is dominated by the form of [MATH], which is rapidly approaching zero as [MATH], while [MATH] is there only slowly varying.)', '1211.1915-2-21-0': '# Large numbers of spheres or disks', '1211.1915-2-22-0': 'In our studies of two and five disks we found that a transition between states in a region containing [MATH] particles generally requires coordinated motion of all the [MATH] particles in order to squeeze through the neck in the phase space.', '1211.1915-2-22-1': 'The rate at which this will occur was given by the transition state formula of Eq. ([REF]).', '1211.1915-2-22-2': 'In this section, we examine the consequences of assuming that the formula can be extended to systems containing a large number [MATH] of spheres or disks.', '1211.1915-2-23-0': 'We shall first suppose that one is at a packing fraction below that of the neck out of an inherent state whose largest density is at a packing fraction [MATH] and that [MATH] is the highest packing fraction below [MATH] at which a neck first opens to allow escape from the inherent state, and that one is in a configuration close to that of the inherent state.', '1211.1915-2-23-1': 'Furthermore we shall assume that when [MATH] is large, [EQUATION] where [MATH] is a positive constant of O(1).', '1211.1915-2-23-2': 'The assumption behind Eq. ([REF]) is that escape from a jammed state will become possible if the volume of the system is increased by an amount of the order of the volume of a single sphere.', '1211.1915-2-23-3': 'With this assumption, and taking [MATH] to be large, Eq. ([REF]) reduces to [EQUATION] where [MATH] denotes the typical time between collisions of the disks.', '1211.1915-2-23-4': 'Then as [MATH], [EQUATION] which is the Vogel-Fulcher-Tammann formula.', '1211.1915-2-24-0': 'Given a particular configuration of the [MATH] particles with a packing fraction [MATH] we need to know the packing fraction [MATH] of the nearby inherent state close to the initial configuration.', '1211.1915-2-24-1': 'In other words, we need the Stillinger map to the jammed inherent states [CITATION].', '1211.1915-2-24-2': 'For the problem of disks moving in a long narrow channel such a map was explicitly constructed in [CITATION] and the function [MATH] exhibited.', '1211.1915-2-24-3': 'Except for quite small values of [MATH], [MATH] is essentially a constant independent of [MATH] and close to the largest packing fraction possible in the system.', '1211.1915-2-24-4': 'The map is similar to what would have been obtained in an extremely rapid compression.', '1211.1915-2-24-5': 'The relaxation times [MATH] in this narrow channel system are consistent with Eq. ([REF]) [CITATION].', '1211.1915-2-24-6': 'It has proved possible to identify the inherent states and the necks which have to be squeezed through to escape from the vicinity of the inherent states and as a consequence the value of the coefficient [MATH] can be explicitly determined for this system [CITATION].', '1211.1915-2-25-0': 'In dimensions [MATH] much less can be said with certainty.', '1211.1915-2-25-1': 'Fits of the alpha relaxation time to the VFT formula for three dimensional hard spheres were made by Brambilla et al. [CITATION] and a fit was achieved with a value of [MATH].', '1211.1915-2-25-2': 'One might have expected that the appropriate value of [MATH] if the map from [MATH] to the inherent state is essentially a rapid compression would be that of random close packing, [MATH].', '1211.1915-2-25-3': 'The result that [MATH] was obtained for studies of [MATH] at [MATH] and it might require data at larger values of [MATH] to produce [MATH] values closer to [MATH].', '1211.1915-2-26-0': 'We have been assuming that the Stillinger map in two and three dimensions, [MATH], is essentially a constant independent of [MATH].', '1211.1915-2-26-1': 'This lack of any [MATH] dependence of [MATH] seems unlikely according to the studies in [CITATION].', '1211.1915-2-26-2': 'Suppose that instead the Stillinger map in two and three dimensions takes the form, for [MATH] close to [MATH], [EQUATION] with [MATH].', '1211.1915-2-26-3': 'To test this supposition, one would need to start the rapid compression from the well-equilibrated fluid system at a packing fraction near [MATH].', '1211.1915-2-26-4': 'Producing this initial state would be difficult.', '1211.1915-2-26-5': 'If Eq. ([REF]) is valid, it would lead to the following expression for the alpha relaxation time [EQUATION]', '1211.1915-2-26-6': 'In Ref. [CITATION], a good fit was obtained with [MATH] and a value for [MATH] - a commonly quoted value.', '1211.1915-2-27-0': 'In words, Eq. ([REF]) states that if one starts from the equilibrated system at a packing fraction [MATH] close to [MATH], then the rapid compression (or the Stillinger map) finds a jammed state whose packing fraction [MATH] only differs from [MATH] by a quantity of order [MATH], which is small when [MATH].', '1211.1915-2-27-1': 'The physical implication is that equilibrated systems at such high densities are always close to a jammed state.', '1211.1915-2-27-2': 'However, Eq. ([REF]) also assumes that [MATH] is a well-defined density and this is contentious [CITATION].', '1211.1915-2-27-3': 'Notice that our difficulties in using Eq. ([REF]) stem from just not knowing the form of the Stillinger map [MATH] for two and three dimensional systems.', '1211.1915-2-27-4': 'It is possible that it takes a form that would leave [MATH] finite for all [MATH] less than that of the maximum density.', '1211.1915-2-27-5': 'In this situation it could be that for [MATH] well below [MATH] might appear to be diverging as [MATH], but if studies could be performed nearer [MATH] the relaxation times would be very long but finite.', '1211.1915-2-28-0': 'In conclusion we have shown that the long relaxation times seen in small systems of two and five disks confined in a square are due to squeezing through necks in configuration space, and can be understood quantitatively with the aid of transition state theory.', '1211.1915-2-28-1': 'We have suggested that a similar mechanism might be relevant to hard spheres in higher dimensions and could lead either to the VFT formula or possibly a generalization of it.', '1211.1915-2-29-0': 'We should like to thank Chris Fullerton for many useful discussions, and Steve Teitel for supplying useful references.'}

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[]

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[['1211.1915-1-14-3', '1211.1915-2-17-7']]

['1211.1915-1-5-0', '1211.1915-1-13-0', '1211.1915-1-16-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1211.1915

cond-mat-0303338

{'cond-mat-0303338-1-0-0': 'We present a fully nonequilibrium calculation of the low-temperature transport properties of a single molecular quantum dot coupled to local phonon mode when an ac field is applied to the gate.', 'cond-mat-0303338-1-0-1': 'The resonant behavior is shown in the time-averaged differential conductance as the ac frequency matches the frequency of the local phonon mode, which is a direct consequence of the satellite-phonon-peak structure in the dot electron spectral function.', 'cond-mat-0303338-1-0-2': 'The different step structure with and without the external irradiation is found in the I-V curves, and the oscillation behavior is found in the step height as a function of the irradiation intensity.', 'cond-mat-0303338-1-1-0': 'Recent advances in nanotechnology have allowed the fabrication of very small molecular quantum dots weakly coupled to the macroscopic charge reserviors (leads)[CITATION].', 'cond-mat-0303338-1-1-1': 'In contrast to the semiconductor dot, which is quite rigid in space, the molecules involved in the electron tunneling process naturally possess the vibrational degrees of freedom which will inevitably react to the transform of electrons through the molecular quantum dots[CITATION].', 'cond-mat-0303338-1-1-2': 'Theoretically, a lot of effort has been focused on the quantum conductance of molecular systems based on the kinetic equation approach[CITATION], the rate equation approach[CITATION] , the correlation effects[CITATION], the nonequilibrium quantum theory[CITATION], and considering the strong coupling to enviornment[CITATION].', 'cond-mat-0303338-1-1-3': 'So far, the stationary quantum transport through the molecular dots has been considered, while the influence of a time-dependent ac field on the current has not been well addressed.', 'cond-mat-0303338-1-1-4': 'Irradiation of a quantum dot with an ac field offers a new way of affacting its dynamics, which enables one to study the effect of electron-phonon interaction on the transport phenomenon of molecular dots in essentially nonequilibrium condition.', 'cond-mat-0303338-1-2-0': 'In this Letter, we use the Keldysh nonequilibrium-Green-function technique to derive the time-dependent current of a single-molecular quantum dot coupled to a local phonon mode with the external irradiation applied to the gate, for the first time.', 'cond-mat-0303338-1-2-1': 'After a canonical transformation, we obtain a formula for the time-dependent current in general terms of bias, temperature, the intensity and frequency of the external ac field, and the electron-phonon coupling.', 'cond-mat-0303338-1-2-2': 'We show that the satellite-peak structure due to the electron-phonon interaction can be probed by imposing on top of gate bias an ac bias voltage.', 'cond-mat-0303338-1-2-3': 'The satellite-peak structure in the dot electron spectral function gives rise to resonant behaviors in the time-averaged current as the ac frequency matches the frequency of local phonon mode, which can be observed directly in experiments.', 'cond-mat-0303338-1-2-4': 'The calculated I-V curves also show the different step structures with and without the external irradiation, and the step height shows the Bessel-type oscillation behavior as a function of irradiation intensity.', 'cond-mat-0303338-1-3-0': 'In this work we consider a simplest Holstein-type model with a single phonon mode is employed to address the vibrational degrees of freedom in the molecular dot.', 'cond-mat-0303338-1-3-1': 'All other complexity of real molecular devices, apart from interaction with phonon mode, is ignored.', 'cond-mat-0303338-1-3-2': 'The system Hamiltonian is [EQUATION] where [EQUATION] [MATH] are creation (annihilation) operators for the noninteracting electrons with momentum [MATH] and spin index [MATH] in the left [MATH] or right [MATH] metallic leads.', 'cond-mat-0303338-1-3-3': '[MATH] is the frequency of the single phonon mode, and [MATH] is the phonon creation (annihilation) operator.', 'cond-mat-0303338-1-3-4': '[MATH] describes the electron in the quantum dot coupled to the local phonon mode with the coupling constant [MATH], where [MATH] is the dot-electron creation (annihilation) operator, and [MATH] is the single energy level of the dot which can be tunned by the external irradiation, [MATH] for harmonic bias.', 'cond-mat-0303338-1-3-5': 'Here we assume that the metallic leads are dc biased, neglecting the possible leakage of the irradiating ac field to the leads.', 'cond-mat-0303338-1-3-6': 'The generalization onto the case of nonzero ac bias is straightforward.', 'cond-mat-0303338-1-3-7': '[MATH] describes the tunneling coupling between the dot and the leads, where the tunneling matrix elements [MATH] transfer electrons through an insulating barrier out of the dot.', 'cond-mat-0303338-1-4-0': 'Based on the Keldysh nonequilibrium Green function formalism[CITATION] , the time-dependent current from [MATH] lead to the dot is given by[CITATION] [EQUATION] where [MATH] are the Fermi distribution function of the left (right) leads, which has different chemical potential upon a dc biase voltage [MATH].', 'cond-mat-0303338-1-4-1': '[MATH] characterizes the coupling between the dot and the leads, and [MATH] is the spin-[MATH] band density of states in the leads.', 'cond-mat-0303338-1-4-2': "[MATH] is the retarded (lesser) Green's function of the dot.", 'cond-mat-0303338-1-5-0': "In order to compute the time-dependent current, one has to compute the dot electron Green's functions in the presence of both the electron-phonon interaction and the tunneling coupling between dot and leads.", 'cond-mat-0303338-1-5-1': "Here we show that the Green's function can be calculated by performing the canonical transformation [MATH][CITATION].", 'cond-mat-0303338-1-5-2': 'Then the single energy level of dot is renormalized to [MATH], where [MATH], and the tunneling coupling between dot and leads is also renormalized as [MATH], where [MATH].', 'cond-mat-0303338-1-5-3': "The dot-electron retarded Green's function can be decoupled as [MATH], where [MATH], [MATH], [MATH], and [MATH].", 'cond-mat-0303338-1-5-4': 'The renormalization factor due to the electron-phonon interaction is[CITATION] [MATH], where [MATH], and [MATH].', 'cond-mat-0303338-1-5-5': "The retarded Green's function can be easily obtained by the standard Dyson equation approach[CITATION], and the result is [EQUATION] where [MATH] is the total tunneling coupling to the leads.", 'cond-mat-0303338-1-5-6': "Following the operational rules[CITATION] to the Dyson equation for the contour-ordered Green's function, the decoupled Keldysh less Green's function is [EQUATION] with the less self-energy [EQUATION] and then the full Green's function is [MATH].", 'cond-mat-0303338-1-6-0': "Substitution of the Green's functions into Eq. ([REF]) gives rise to [EQUATION] with [MATH].", 'cond-mat-0303338-1-6-1': "Obviously, in the time-independent case, [MATH] is just the Fourier transform of the retarded Green's function [MATH].", 'cond-mat-0303338-1-6-2': 'After some algebra, we find that for this model, [EQUATION] where [MATH] is the [MATH]-th ordered Bessel function, [MATH] is the Bessel function of complex argument, and [MATH], [MATH], [MATH], [MATH], [MATH].', 'cond-mat-0303338-1-6-3': 'Eq. ([REF]) together with the current expression Eq. ([REF]) provides the complete solution to the time-dependent transport of molecular quantum dot coupled to local phonon mode in the presence of external irradiation.', 'cond-mat-0303338-1-7-0': 'Now we discuss the time-averaged current [MATH] and the assiciated conductance, which could be directly relevant to experiment.', 'cond-mat-0303338-1-7-1': 'For this model, we then obtain [EQUATION]', 'cond-mat-0303338-1-7-2': 'In the absence of external irradiation, i.e., [MATH], Eq. ([REF]) fully agrees with the result of dc current in Ref. [CITATION].', 'cond-mat-0303338-1-7-3': 'The contribution of the external irradiation to the transport becomes significant when the argument of the Bessel function, [MATH], is of the order of unity.', 'cond-mat-0303338-1-7-4': 'Another important consequence is that the time-averaged current under irradiation is proportional to [MATH].', 'cond-mat-0303338-1-7-5': 'As a result, one should expect the change in current due to the external irradiation to depend on the intensity of irradiation.', 'cond-mat-0303338-1-8-0': 'For simplicity, we consider the tunneling coupling between the molecular dot and the two leads to be symmetric and independent of spin index, i.e., [MATH], and assume that the leads have broad and flat density of states (the wide-band limit).', 'cond-mat-0303338-1-8-1': 'In Fig. 1, we plot the zero-temperature differential conductance as a function of the Fermi energy measured relative to the single level [MATH] of the dot in the external ac field with the frequency [MATH] and different values of irradiation intensities.', 'cond-mat-0303338-1-8-2': 'For comparision, we also plot the differential conductance in the absence of ac field.', 'cond-mat-0303338-1-8-3': 'At zero ac biase voltage, our results agrees well with that in Ref. [CITATION], where the electron-phonon coupling can lead to the satellite resonant peaks and the shift of overall spectrum.', 'cond-mat-0303338-1-8-4': 'The ac field with the frequency [MATH] can lead to the enhancement of the satellite resonant peaks at positive energy region and the appearance of the new resonant peaks at negative energy region.', 'cond-mat-0303338-1-8-5': 'One can also see that the main peak is suppressed by the irradiation when increasing the intensity.', 'cond-mat-0303338-1-8-6': 'In Fig. 2, we plot the zero-temperature differential conductance as a function of the frequency of ac field for different values of irradiation intensities by fixing the Fermi energy of the leads [MATH] as [MATH] to avoid unnecessary complications.', 'cond-mat-0303338-1-8-7': 'As the intensity of ac field increases, resonant signals are clearly shown when the frequency of ac field satisfies [MATH].', 'cond-mat-0303338-1-8-8': 'The satellite-phonon-peak structure in the dot electron spectral function, as shown in Ref. [CITATION] and also Fig. 1 in this Letter, gives rise to resonant behaviors in the time-averaged differential conductance as the irradiating ac frequency matches the frequency of local phonon mode, and can be observed directly in experiments.', 'cond-mat-0303338-1-8-9': 'Fig. 2 also shows that more resonant signals can be observed when increasing the irradiation intensity.', 'cond-mat-0303338-1-9-0': 'Fig. 3 shows the calculated zero-temperature current-voltage curves with and without the external ac irradiation.', 'cond-mat-0303338-1-9-1': 'Here we assume the leads to be symmetrically voltage biased, i.e., [MATH] on the left lead and [MATH] on the right one, to avoid unnecessary complications.', 'cond-mat-0303338-1-9-2': 'In the absence of external ac field, clear steps appear at roughly [MATH] intervals in the weak tunneling coupling limit, corresponding to [MATH], with [MATH], [MATH], [MATH], and the height of the [MATH]-th step decreases with [MATH], which can be easily understood from Eq. ([REF]) [see below] by taking [MATH] for small electron-phonon interaction.', 'cond-mat-0303338-1-9-3': 'In the presence of ac field with frequency [MATH], steps appear at the same intervals [MATH] as those in the zero ac field, while the step height is modulated by Bessel function due to the irradiation [see Eq. ([REF]) in below].', 'cond-mat-0303338-1-9-4': 'Fig. 3 also shows that more steps appear at the intervals [MATH] in the case of [MATH], corresponding to [MATH] with [MATH], [MATH], [MATH].', 'cond-mat-0303338-1-9-5': "The zero-temperature [MATH]-th step's height in an external ac field with frequency [MATH] can be analytically obtained from Eq. ([REF]) at the fixed Fermi energy [MATH] as [MATH], [EQUATION]", 'cond-mat-0303338-1-9-6': 'In Fig. 4, we plot the height of the [MATH]-th step, where [MATH] and [MATH], as a function of the irradiation intensity for different values of the electron-phonon coupling constant.', 'cond-mat-0303338-1-9-7': 'The oscillation behavior of the step height is clearly observed for the small electron-phonon interaction due to the external irradiation [see Eq. ([REF])], while this oscillation smears out for large electron-phonon interaction.', 'cond-mat-0303338-1-9-8': 'The large electron-phonon interaction enhances the processes of absorption and emission of many [MATH] phonons when electron tunnels through the dot, and then the summation of Bessel function with large values of indices [MATH] results in the smearing of the oscillation behavior [see Eq. ([REF])].', 'cond-mat-0303338-1-10-0': "Summarizing, using the Keldysh nonequilibrium Green's function technique, we have studied in this work, to the best of our knowledge, for the first time the time-dependent transport through a single molecular quantum dot coupled to a local phonon mode in the presence of an external ac field.", 'cond-mat-0303338-1-10-1': 'We show that the external irradiation provides another important experimental tool where both the equilibrium and out of equilibrium transport phenomenon can be probed.', 'cond-mat-0303338-1-10-2': 'In particular, resonant behavior as the ac frequency matches the frequency of local phonon mode is shown to exist as a result of the satellite-phonon-peak structure in the dot electron density of states.', 'cond-mat-0303338-1-10-3': 'The nonlinear I-V curves exhibit new structure caused by the external irradiation, which can be investigated experimentally.', 'cond-mat-0303338-1-11-0': 'We thank Jian-Xin Zhu for many stimulating discussions which lead to this work and useful discussions with Zhi-Rong Liu.'}

{'cond-mat-0303338-2-0-0': 'We present a fully nonequilibrium calculation of the low-temperature transport properties of a single molecular quantum dot coupled to local phonon mode when an ac field is applied to the gate.', 'cond-mat-0303338-2-0-1': 'The resonant behavior is shown in the time-averaged differential conductance as the ac frequency matches the frequency of the local phonon mode, which is a direct consequence of the satellite-phonon-peak structure in the dot electron spectral function.', 'cond-mat-0303338-2-0-2': 'The different step structure with and without the external irradiation is found in the I-V curves, and the oscillation behavior is found in the step height as a function of the irradiation intensity.', 'cond-mat-0303338-2-1-0': 'Recent advances in nanotechnology have allowed the fabrication of very small molecular quantum dots weakly coupled to the macroscopic charge reservoirs (leads)[CITATION].', 'cond-mat-0303338-2-1-1': 'In contrast to the semiconductor dot, which is quite rigid in space, the molecules involved in the electron tunneling process naturally possess the vibrational degrees of freedom which will inevitably react to the transform of electrons through the molecular quantum dots[CITATION].', 'cond-mat-0303338-2-1-2': 'In addition to the importance in molecular-scale electronics from the application point of view, these artificial, tunable devices are potentically important for understanding the basic physics including the many-body effect.', 'cond-mat-0303338-2-1-3': 'Theoretically, a lot of effort has been focused on the quantum conductance of molecular systems based on the kinetic equation approach[CITATION], the rate equation approach[CITATION] , the correlation effects[CITATION], the nonequilibrium quantum theory[CITATION], and the strong coupling to environment[CITATION].', 'cond-mat-0303338-2-1-4': 'So far, the stationary quantum transport through the molecular dots has been considered, while the influence of a time-dependent ac field on the current has not been well addressed.', 'cond-mat-0303338-2-1-5': 'Irradiation of a quantum dot with an ac field[CITATION] offers a new way of affecting its dynamics, which enables one to study the effect of electron-phonon interaction on the transport phenomenon of molecular dots in essentially nonequilibrium condition.', 'cond-mat-0303338-2-2-0': "In this Letter, we use the Keldysh nonequilibrium Green's function technique to study the nonlinear ac transport through a single-molecular quantum dot coupled to a local phonon mode with the external irradiation applied to the gate, for the first time.", 'cond-mat-0303338-2-2-1': 'After a canonical transformation, we obtain a formula for the time-dependent current in general terms of bias, temperature, the intensity and frequency of the external ac field, and the electron-phonon coupling.', 'cond-mat-0303338-2-2-2': 'We show that the satellite-peak structure due to the electron-phonon interaction can be probed by imposing on top of gate bias an ac bias voltage.', 'cond-mat-0303338-2-2-3': 'The satellite-peak structure in the dot electron spectral function gives rise to resonant behavior in the time-averaged current as the ac frequency matches the frequency of local phonon mode, which can be observed directly in experiments.', 'cond-mat-0303338-2-2-4': 'The calculated I-V curves also show the different step structures with and without the external irradiation, and the step height shows the Bessel-type oscillation behavior as a function of irradiation intensity.', 'cond-mat-0303338-2-3-0': 'In this work we consider a simplest Holstein-type model with a single phonon mode is employed to address the vibrational degrees of freedom in the molecular dot.', 'cond-mat-0303338-2-3-1': 'All other complexity of real molecular devices, apart from interaction with phonon mode, is ignored.', 'cond-mat-0303338-2-3-2': 'Then the system Hamiltonian can be written as [EQUATION] where [EQUATION] [MATH] are creation (annihilation) operators for the noninteracting electrons with momentum [MATH] and spin index [MATH] in the left [MATH] or right [MATH] metallic leads.', 'cond-mat-0303338-2-3-3': '[MATH] is the frequency of the single phonon mode, and [MATH] is the phonon creation (annihilation) operator.', 'cond-mat-0303338-2-3-4': '[MATH] describes the electron in the quantum dot coupled to the local phonon mode with the coupling constant [MATH], where [MATH] is the dot-electron creation (annihilation) operator, and [MATH] is the single energy level of the dot which can be tuned by the external irradiation, [MATH] for harmonic bias.', 'cond-mat-0303338-2-3-5': 'Here we assume that the metallic leads are dc biased, neglecting the possible leakage of the irradiating ac field to the leads.', 'cond-mat-0303338-2-3-6': 'The generalization onto the case of nonzero ac bias is straightforward.', 'cond-mat-0303338-2-3-7': '[MATH] describes the tunneling coupling between the dot and the leads, where the tunneling matrix elements [MATH] transfer electrons through an insulating barrier out of the dot.', 'cond-mat-0303338-2-4-0': "Based on the Keldysh nonequilibrium Green's function formalism[CITATION] , the time-dependent current from the [MATH] lead to the dot is given by [CITATION] [EQUATION] where [MATH] are the Fermi distribution function of the left (right) leads, which have different chemical potentials upon a dc bias voltage [MATH].", 'cond-mat-0303338-2-4-1': '[MATH] characterizes the coupling between the dot and the leads, and [MATH] is the spin-[MATH] band density of states in the leads.', 'cond-mat-0303338-2-4-2': 'Here we assume that the leads give rise to a flat, energy independent, density of states (i. e., the wide-band limit).', 'cond-mat-0303338-2-4-3': "[MATH] is the retarded (lesser) Green's function of the dot.", 'cond-mat-0303338-2-5-0': "In order to compute the time-dependent current, one has to compute the dot electron Green's functions in the presence of both the electron-phonon interaction and the tunneling coupling between dot and leads.", 'cond-mat-0303338-2-5-1': "The Green's function can be calculated by performing the canonical transformation [MATH][CITATION], and then the dot level is renormalized to [MATH], where [MATH], and the tunneling coupling term is also renormalized as [MATH], where [MATH].", 'cond-mat-0303338-2-5-2': "Ignoring the effects of narrowing the bands of leads due to the phonons[CITATION], the dot-electron Green's function can be decoupled as [MATH], where [MATH], [MATH], [MATH], and [MATH].", 'cond-mat-0303338-2-5-3': 'The renormalization factor due to the electron-phonon interaction is[CITATION] [MATH], where [MATH], and [MATH].', 'cond-mat-0303338-2-5-4': "The retarded Green's function can be easily obtained by the standard Dyson equation approach[CITATION], and the result is [EQUATION] where [MATH] is the total tunneling coupling to the leads.", 'cond-mat-0303338-2-5-5': "Following the operational rules[CITATION] to the Dyson equation for the contour-ordered Green's function, the Keldysh Green's function is found to be [EQUATION] with the less self-energy [EQUATION]", 'cond-mat-0303338-2-5-6': 'Without electron-phonon interaction, the above result fully agrees with that for time-dependent transport through noninteracting quantum dot[CITATION] .', 'cond-mat-0303338-2-6-0': "Substitution of the Green's functions into Eq. ([REF]) gives [EQUATION] with [MATH].", 'cond-mat-0303338-2-6-1': "Obviously, in the time-independent case, [MATH] is just the Fourier transform of the retarded Green's function [MATH].", 'cond-mat-0303338-2-6-2': 'After some algebra, we find that for this model, [EQUATION] where [MATH] is the Bessel function of the [MATH]-th order, [MATH] is the Bessel function of complex argument, and [MATH], [MATH], [MATH], [MATH], [MATH].', 'cond-mat-0303338-2-6-3': 'Eq. ([REF]) together with the current expression Eq. ([REF]) provides the complete solution to the time-dependent transport of molecular quantum dot coupled to local phonon mode in the ac field.', 'cond-mat-0303338-2-7-0': 'Experimentally what is interest is the current on a time scale long compared to [MATH].', 'cond-mat-0303338-2-7-1': 'Here we discuss the time-averaged current [MATH], which could be directly relevant to experiment.', 'cond-mat-0303338-2-7-2': 'For this model, we then obtain [EQUATION]', 'cond-mat-0303338-2-7-3': 'In the absence of external irradiation, i.e., [MATH], Eq. ([REF]) fully agrees with the result of dc current in Refs. [CITATION].', 'cond-mat-0303338-2-7-4': 'The contribution of the external irradiation to the transport becomes significant when the argument of the Bessel function, [MATH], is of the order of unity.', 'cond-mat-0303338-2-7-5': 'Another important consequence is that the time-averaged current under irradiation is proportional to [MATH].', 'cond-mat-0303338-2-7-6': 'As a result, one should expect the change in current due to the external irradiation to depend on the intensity of irradiation.', 'cond-mat-0303338-2-8-0': 'For simplicity, we consider the tunneling coupling between the molecular dot and the two leads to be symmetric and independent of the spin index, i.e., [MATH].', 'cond-mat-0303338-2-8-1': 'In Fig. 1, we plot the zero-temperature differential conductance as a function of the Fermi energy measured relative to the single level [MATH] of the dot in the external ac field with frequency [MATH] and different values of irradiation intensities.', 'cond-mat-0303338-2-8-2': 'For comparison, we also plot the differential conductance in the absence of ac field.', 'cond-mat-0303338-2-8-3': 'At zero ac bias voltage, our results agrees well with that in Refs. [CITATION], where the electron-phonon coupling can lead to the satellite resonant peaks.', 'cond-mat-0303338-2-8-4': 'Fig. 1 shows that the ac field with frequency [MATH] can lead to the enhancement of satellite resonant peaks in the positive energy region and the appearance of new peaks in the negative energy region.', 'cond-mat-0303338-2-8-5': 'One can also see that the main peak is suppressed by the irradiation while increasing the intensity.', 'cond-mat-0303338-2-8-6': 'In Fig. 2, we plot the zero-temperature differential conductance as a function of the frequency of ac field for different values of irradiation intensities by fixing the Fermi energy of the leads [MATH] as [MATH] to avoid unnecessary complications.', 'cond-mat-0303338-2-8-7': 'As the intensity of ac field increases, resonant signals are clearly shown when the frequency of ac field satisfies [MATH].', 'cond-mat-0303338-2-8-8': 'The satellite-phonon-peak structure in the dot electron spectral function, as shown in Refs. [CITATION] and also in Fig. 1 of this Letter, gives rise to resonant behavior in the conductance as the irradiating ac frequency matches the frequency of local phonon mode, and can be observed directly in experiments.', 'cond-mat-0303338-2-8-9': 'Fig. 2 also shows that more resonant signals can be observed while increasing the irradiation intensity.', 'cond-mat-0303338-2-9-0': 'Fig. 3 shows the calculated zero-temperature current-voltage curves with and without the external ac irradiation.', 'cond-mat-0303338-2-9-1': 'Here we assume the leads to be symmetrically voltage biased, i.e., [MATH] on the left lead and [MATH] on the right one, to avoid unnecessary complications.', 'cond-mat-0303338-2-9-2': 'In the absence of external ac field, clear steps appear at roughly [MATH] intervals in the weak tunneling coupling limit, corresponding to [MATH], with [MATH], [MATH], [MATH], and the height of the [MATH]-th step decreases with [MATH], which can be easily understood from Eq. ([REF]) [see below] by taking [MATH] for small electron-phonon interaction.', 'cond-mat-0303338-2-9-3': 'In the presence of ac field with frequency [MATH], steps appear at the same intervals [MATH] as those in the zero ac field, while the step height is modulated by Bessel function due to the irradiation [see Eq. ([REF]) in below].', 'cond-mat-0303338-2-9-4': 'Fig. 3 also shows that more steps appear at the intervals [MATH] in the case of [MATH], corresponding to [MATH] with [MATH], [MATH], [MATH].', 'cond-mat-0303338-2-9-5': "The zero-temperature [MATH]-th step's height in an external ac field with frequency [MATH] can be analytically obtained from Eq. ([REF]) at the fixed Fermi energy [MATH] as [MATH], [EQUATION]", 'cond-mat-0303338-2-9-6': 'In Fig. 4, we plot the height of the [MATH]-th step, where [MATH] and [MATH], as a function of the irradiation intensity for different values of the electron-phonon coupling constant.', 'cond-mat-0303338-2-9-7': 'The oscillation behavior of the step height is clearly observed for the small electron-phonon interaction due to the external irradiation [see Eq. ([REF])], while this oscillation smears out for large electron-phonon interaction.', 'cond-mat-0303338-2-9-8': 'The large electron-phonon interaction enhances the processes of absorption and emission of many [MATH] phonons when electron tunnels through the dot, and then the summation of Bessel function with large values of indices [MATH] results in the smearing of the oscillation behavior [see Eq. ([REF])].', 'cond-mat-0303338-2-10-0': "Summarizing, using the Keldysh nonequilibrium Green's function technique, we have studied in this work, to the best of our knowledge, for the first time the time-dependent transport through a single molecular quantum dot coupled to a local phonon mode in the presence of an external ac field.", 'cond-mat-0303338-2-10-1': 'We show that the external irradiation provides another important experimental tool where both the equilibrium and out of equilibrium transport phenomenon can be probed.', 'cond-mat-0303338-2-10-2': 'In particular, resonant behavior as the ac frequency matches the frequency of local phonon mode is shown to exist as a result of the satellite-phonon-peak structure in the dot electron density of states.', 'cond-mat-0303338-2-10-3': 'The nonlinear I-V curves exhibit new structure caused by the external irradiation, which can be investigated experimentally.', 'cond-mat-0303338-2-11-0': 'We thank J.-X. Zhu for stimulating discussions which lead to this work and Z.-R. Liu for useful discussions.'}

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[]

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[]

['cond-mat-0303338-2-11-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0303338

hep-th-0312318

{'hep-th-0312318-1-0-0': 'In this paper we work in perturbative Quantum Gravity and we introduce a new effective model for gravity.', 'hep-th-0312318-1-0-1': 'Expanding the Einstein-Hilbert Lagrangian in graviton field powers we have an infinite number of terms.', 'hep-th-0312318-1-0-2': 'In this paper we study the possibility of an interpretation of more than three graviton interacting vertices as effective vertices of a most fundamental theory that contain tensor fields .', 'hep-th-0312318-1-0-3': 'Here we introduce a Lagrangian model named I.T.B. (Intermediate-Tensor-Boson) where four gravitational "pseudo-currents" that contain two gravitons couple to three massive tensorial fields of ranks one, three and five respectively.', 'hep-th-0312318-1-0-4': 'We show that the exchange of those massive particles reproduces, at low energy, the interacting vertices for four or more gravitons.', 'hep-th-0312318-1-0-5': 'In a particolar version, the model contains a dimensionless coupling constant "[MATH]" and the mass [MATH] of the intermediate bosons as free parameters.', 'hep-th-0312318-1-0-6': 'The universal gravitational constant [MATH] is shown to be proportional to the inverse of mass squared of mediator fields, particularly [MATH].', 'hep-th-0312318-1-0-7': 'A foresighting choice of the dimensionless coupling constant could lower the energy scale where quantum gravity aspects show up.', 'hep-th-0312318-1-1-0': '# Introduction to the Model', 'hep-th-0312318-1-2-0': 'In this paper we consider the perturbative expansion of the Einstein-Hilbert action in graviton ([MATH]) field powers.', 'hep-th-0312318-1-2-1': 'After the quadratic order that is the free Graviton Lagrangian, the first not trivial order is the three gravitons interaction.', 'hep-th-0312318-1-2-2': 'The subsequent is the four gravitons interaction and so on.', 'hep-th-0312318-1-2-3': 'The interaction vertex that we study with particular attention is the local four graviton interaction vertex; we show that we can see this term as a "current-current" interaction in analogy with Fermi Theory of Weak Interactions.', 'hep-th-0312318-1-2-4': 'In a second step we introduce the I.T.B. (Intermediate Tensor Boson) theory for Quantum Gravity in analogy with the I.V.B. (Intermediate Vector Boson) theory that is the non local extension of the Fermi theory.', 'hep-th-0312318-1-2-5': 'In this Model we introduce an opportune number of Lagrangian terms describing the coupling between currents, made only of gravitons, and tensorial bosons of rank 1, 3, 5.', 'hep-th-0312318-1-2-6': 'The bosons are the analog of the [MATH] and [MATH] vectors for I.V.B. theory.', 'hep-th-0312318-1-3-0': 'Moreover we show that using only another Lagrangian term we can reproduce the tensor structure of all interactions with [MATH], [MATH], [MATH], [MATH] gravitons.', 'hep-th-0312318-1-3-1': 'Finally we compare the I.T.B model with string theory and we propose a new picture where the two theories are two different phases of a more fundamental theory.', 'hep-th-0312318-1-3-2': 'This can be a theory of massless Tensor Fields [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], or a Yang-Mills theory coupled with Sub-Quark [CITATION], [CITATION], [CITATION].', 'hep-th-0312318-1-3-3': 'In the second case String Theory and the I.T.B model could be two different hadronic phases of a more microscopic sub-quarks theory.', 'hep-th-0312318-1-4-0': '# [MATH]-Action Gravitons Expansion', 'hep-th-0312318-1-5-0': 'The [MATH] action for Gravity is [EQUATION] where [EQUATION]', 'hep-th-0312318-1-5-1': 'We consider now the fluctuation of the [MATH] metric around the flat background [MATH] [EQUATION]', 'hep-th-0312318-1-5-2': 'This expansion around the flat background is correct only locally.', 'hep-th-0312318-1-5-3': 'In fact, for Manifold with non trivial topology, the metric can non be globally expanded in [MATH] plus a fluctuation.', 'hep-th-0312318-1-5-4': 'This expansion [REF] is relevant for the description of nature because, in the portion of the Universe where our solar system is placed, the intensity of the fluctuation is about [MATH].', 'hep-th-0312318-1-6-0': 'If we observe the Lagrangian [REF] we see that at any order we have only two derivatives.', 'hep-th-0312318-1-6-1': 'This is a consequence of the definition of the Cristoffel symbols, in fact such symbols have only low indices and so the form in terms of [MATH] is exact [EQUATION]', 'hep-th-0312318-1-6-2': 'Now using the relation [MATH] we can expand the determinant and the inverse of the metric [MATH] in power of [MATH] : [EQUATION]', 'hep-th-0312318-1-6-3': 'This second relation can be obtained from the Taylor expansion of [MATH].', 'hep-th-0312318-1-6-4': 'Using those relations the Lorentz gauge condition [MATH] and, the mass-shell condition ([MATH], [MATH]) the amplitudes can be written [EQUATION]', 'hep-th-0312318-1-6-5': 'Since we are studying the tree level gravitons interaction it is sufficient to consider only the on shell Lagrangian terms.', 'hep-th-0312318-1-6-6': 'In fact all the interactions that contain the tensors [MATH], [MATH] and [MATH], produce amplitudes proportional to [MATH], [MATH] and [MATH] that are identically zero on external gravitons states.', 'hep-th-0312318-1-7-0': 'Now we introduce tensors with tree and five indexes, that we call "tensor currents" and we rewrite the Lagrangian ([REF]) as a "current-current" interaction in analogy with Fermi theory of the weak interactions.', 'hep-th-0312318-1-7-1': 'The currents need to do it are [EQUATION]', 'hep-th-0312318-1-7-2': 'Using those currents in the Lagrangian ([REF]) we obtain this compact form for the four gravitons interaction [EQUATION]', 'hep-th-0312318-1-8-0': '# [MATH] Constant Expansion', 'hep-th-0312318-1-9-0': 'The fundamental constant in gravity is the Newton constant [MATH] that in natural units [MATH] and [MATH] is [EQUATION]', 'hep-th-0312318-1-9-1': 'In front of the Lagrangian we have the factor [MATH], and we can reabsorb it in the definition of the graviton field : [EQUATION]', 'hep-th-0312318-1-9-2': 'In this way we obtain mass dimension for the graviton [MATH] while the metric is dimensionless.', 'hep-th-0312318-1-9-3': 'With this definition we obtain a factor [MATH] in front of the [MATH]-oder interaction in the gravitons expansion of the Lagrangian.', 'hep-th-0312318-1-9-4': 'The scheme of the Lagrangian with correct powers of [MATH] is [EQUATION]', 'hep-th-0312318-1-9-5': 'Where [MATH] factors are understood.', 'hep-th-0312318-1-10-0': 'At this point we can rewrite the Lagrangian [REF] in the following way : [EQUATION]', 'hep-th-0312318-1-11-0': '# The I.T.B Theory for Quantum Gravity', 'hep-th-0312318-1-12-0': 'As said in the first section the I.V.B theory replaces the Fermi Theory of the weak interactions.', 'hep-th-0312318-1-12-1': 'In this new model the fermionic local interaction becomes a non local interaction with the exchange of a massive boson ([MATH] or [MATH]).', 'hep-th-0312318-1-12-2': 'The mass of the bosons is exactly the inverse of the [MATH] Fermi constant ([MATH]).', 'hep-th-0312318-1-12-3': 'Now we introduce for Gravity an analog of the I.V.B model that we call I.T.B (Intermediate Tensor Boson Model).', 'hep-th-0312318-1-13-0': 'In the I.T.B Model the analog of the [MATH] bosons are tensor fields of rang [MATH], [MATH] and [MATH] : [EQUATION]', 'hep-th-0312318-1-13-1': 'Such fields couple to what we call "gravitational currents", in analogy with the Fermi model, through the following Lagrangian terms with a dimensionless coupling constant, as in the I.V.B model.', 'hep-th-0312318-1-13-2': '[EQUATION]', 'hep-th-0312318-1-13-3': 'Those Lagrangian terms reproduce all the local four graviton interactions as a low energy limit ([MATH]) of amplitudes in which gravitons interact through the exchange of massive fields [MATH], [MATH] or [MATH].', 'hep-th-0312318-1-14-0': 'We introduce now the following tensor factors for the propagetors of the higher spin fields [EQUATION]', 'hep-th-0312318-1-14-1': 'We start with the field of rank [MATH].', 'hep-th-0312318-1-14-2': 'In this case the Lagrangian is [MATH] and we calculate the scattering amplitudes for [MATH] gravitons using the [MATH]-matrix.', 'hep-th-0312318-1-15-0': 'We recall that the [MATH]-matrix connects the final state [MATH] to the initial state [MATH] defined by [EQUATION]', 'hep-th-0312318-1-15-1': 'The probability to obtain the system in the general state [MATH] after the scattering, is [EQUATION] and the unitarity of the [MATH]-matrix can be written as [EQUATION]', 'hep-th-0312318-1-15-2': 'The general form of the [MATH]-matrix that has those property is [EQUATION]', 'hep-th-0312318-1-15-3': 'Now we consider the second order in the [MATH]-matrix for the Lagrangian [MATH], with [MATH] and [MATH] defined in [REF] : [EQUATION]', 'hep-th-0312318-1-15-4': 'Introducing the propagator for the field [MATH] in [MATH] at the second order and taking [MATH] we obtain the local amplitude for [MATH] gravitons.', 'hep-th-0312318-1-15-5': 'This amplitude is exactly what we obtain from Einstein action ([REF]) at first order in the [MATH]-matrix.', 'hep-th-0312318-1-15-6': "In this section we don't give calculations detail report the calculus but only the idea of the mechanism (see the Appendix).", 'hep-th-0312318-1-15-7': 'Using Feyman diagrams we can depict the interaction at high ([MATH]) and low ([MATH]) energy, and this is very similar to what we find when we pass from I.V.B model to Fermi theory provided the following identifications [EQUATION]', 'hep-th-0312318-1-15-8': 'Qualitatively the [MATH]-matrix at the second order is : [EQUATION]', 'hep-th-0312318-1-15-9': 'At this point we analyse the result from the I.T.B model.', 'hep-th-0312318-1-16-0': 'In this model we can interpret the Newton constant [MATH] as proportional to the mass of the fields of rank [MATH], [MATH], and [MATH], [MATH], [MATH] and [MATH] that are the mediators of the gravitational interaction.', 'hep-th-0312318-1-16-1': '[EQUATION]', 'hep-th-0312318-1-16-2': 'The other important fact is that we can have a lower energy scale where gravity takes a quantum nature.', 'hep-th-0312318-1-16-3': 'In the low energy limit the non-local amplitude becomes local and comparing with the Einstein theory we obtain the following identification [EQUATION] or : [EQUATION]', 'hep-th-0312318-1-16-4': 'If the adimentional coupling constant "[MATH]" is very small ([MATH]), we can apply the perturbative theory and for [MATH] we obtain [MATH], that is the Grand Unification scale.', 'hep-th-0312318-1-16-5': 'If [MATH] than [MATH] and we cauld see Quantum Gravitational effects at accelerators.', 'hep-th-0312318-1-17-0': '## Tensor Fields as Reducible Representation of the Poincare Group', 'hep-th-0312318-1-18-0': 'The tensor factors [REF], [REF] e [REF] inside the propagators have no definite symmetry.', 'hep-th-0312318-1-18-1': 'Now we rewrite the tensor factors [REF], [REF] as a sum of propagators for other fields, with always rank [MATH] and [MATH], that are reducible representations of the Poincare group with definite tensor symmetry.', 'hep-th-0312318-1-18-2': 'We can write the tensors [MATH] as following.', 'hep-th-0312318-1-19-0': 'For [MATH] we have [EQUATION]', 'hep-th-0312318-1-19-1': 'Any line in the tensor ([REF]) represents the propagator of a field that can be decomposed in irreducible representations of the Poncare group.', 'hep-th-0312318-1-20-0': 'For [MATH] we have the following decomposition [EQUATION]', 'hep-th-0312318-1-21-0': '# Interaction with [MATH] Gravitons (Tensor Structure)', 'hep-th-0312318-1-22-0': 'The interaction vertexes that contain [MATH] gravitons have a tensor structure such that we can obtain those interactions introducing one or plus gravitons into the four graviton vertex.', 'hep-th-0312318-1-22-1': 'To carry out this idea we must introduce another Lagrangian term to the I.T.B model; such term couples two massive fields with a graviton.', 'hep-th-0312318-1-22-2': 'This Interaction term in the low energy limit reproduces the tensor behaviour of the local interactions of the Einstein action.', 'hep-th-0312318-1-23-0': 'For example we consider a typical [MATH]-gravitons interaction that we can obtain from the expansion of the Einstein-Hilbert action [EQUATION]', 'hep-th-0312318-1-23-1': 'The Lagrangian term that we must introduce in the I.T.B model to reproduce ([REF]) is [EQUATION]', 'hep-th-0312318-1-23-2': 'Omitting the indexes, the total lagrangian is [EQUATION]', 'hep-th-0312318-1-23-3': 'To calculate the [MATH] gravitons amplitude in the I.T.B model, we consider the third order in the [MATH]-matrix at tree level : [EQUATION]', 'hep-th-0312318-1-23-4': 'Comparing the result with the Einstein action we obtain [MATH] and so [MATH].', 'hep-th-0312318-1-24-0': '# "[MATH]" Graviton Interaction (Tensor Structure)', 'hep-th-0312318-1-25-0': 'In the previous section we have introduced another Lagrangian term that reproduces all the interactions for [MATH] or in general "[MATH]" gravitons ([MATH]) at least in the tensor structure.', 'hep-th-0312318-1-25-1': 'Now we omit the tensor indexes and we explain the mechanism.', 'hep-th-0312318-1-25-2': 'A typical Lagrangian term for [MATH] gravitons that we obtain from the Einstein action is : [EQUATION]', 'hep-th-0312318-1-25-3': 'This local interaction can be reproduced with the I.T.B effective Lagrangian [EQUATION] where [MATH] were defined before.', 'hep-th-0312318-1-26-0': 'At fourth order we obtain the [MATH] graviton interaction [EQUATION]', 'hep-th-0312318-1-26-1': 'For the "[MATH]" graviton interaction we must take the [MATH]-matrix at the "[MATH]" order : [EQUATION]', 'hep-th-0312318-1-26-2': 'From the amplitude we obtain [MATH].', 'hep-th-0312318-1-27-0': '# I.T.B. Model Off-Shell', 'hep-th-0312318-1-28-0': 'In the previous sections we have taken on-shell gravitons that satisfy the following relations : [EQUATION]', 'hep-th-0312318-1-28-1': 'We can build the I.T.B model for off-shell gravitons too.', 'hep-th-0312318-1-28-2': 'In this case the rank [MATH] fields couple to the following general currents : [EQUATION]', 'hep-th-0312318-1-29-0': '# String Theory and ITB Model Toward a More Fundamental Theory', 'hep-th-0312318-1-30-0': 'In this section we analyse the possible connections between string theory, in the critical dimension [MATH] for the Bosonic String or [MATH] for the Superstring, and the ITB Model introduced in the previous sections.', 'hep-th-0312318-1-30-1': 'In particular in the string theory contest it is simple to verify (at least in the bosonic case) that in the low energy limit [MATH] the effective vertexes for gravitons massive fields coupling contain only the tensor fields with rank [MATH] as in the ITB Model.', 'hep-th-0312318-1-31-0': 'The fondamental idea is the following:', 'hep-th-0312318-1-32-0': 'at very high energy ([MATH]) we have a Gauge Theory of Massless Tensor Fields (G.T.H.S) that can lives in any dimension.', 'hep-th-0312318-1-32-1': 'In fact String Theory is Weyl anomaly free in this regime as we can see from the Virasoro algebra ([REF]).', 'hep-th-0312318-1-32-2': "This fundamental theory could have many different Higgs' s phases related with many different relative minimum of complicated potential.", 'hep-th-0312318-1-32-3': 'The conjecture is that two of those are the String Theory and the I.T.B Model vacuums.', 'hep-th-0312318-1-32-4': 'In this picture we see that there is not a prefered dimension as in field theory, but the dimension is a consequence of the symmetry breaking.', 'hep-th-0312318-1-32-5': 'There is a vacuum where the massless fields of the microscopic theory become all massive with the exception of the gravitational multiplet and the spectrum is that of the String Theory so the dimension is D=10, and there is another vacuum where the theory reproduces the gravitational theory in analogy with the Fermi theory of the weak interactions.', 'hep-th-0312318-1-32-6': 'The first vacuum is a finite theory (String Theory) and the other is a renormalizable gauge theory of Higher Spin in [MATH] dimension with spontaneous symmetry breaking that contains only the graviton as massless state and a residual gauge symmetry that is the infinitesimal version of the [MATH]-invariance of General Relativity.', 'hep-th-0312318-1-32-7': '[EQUATION]', 'hep-th-0312318-1-32-8': 'Now we summarize the analogies between the Fermi Theory of weak interactions (F.T.W.I), General Relativity (G.R) and String Theory.', 'hep-th-0312318-1-32-9': 'At low energy the F.T.W.I model is a good approsimation of the weak interactions, it consists of a local vertex with four fermions and a fondamental constant [MATH].', 'hep-th-0312318-1-32-10': 'At higher energies we see that the theory contains a vertex that consists of a fermionic current and a massive vector "W" (Intermediate Vector Boson Model, I.V.B).', 'hep-th-0312318-1-32-11': 'When we take the limit [MATH] we obtain the F.T.W.I with [MATH].', 'hep-th-0312318-1-32-12': 'In analogy we have introduced the Intermediate Tensor Boson Model (I.T.B) where now the fermionic current become a "pseudocurrent" with two gravitons and the W-bosons become fields with ramk [MATH].', 'hep-th-0312318-1-32-13': 'Finally we have a theory very similar to string theory in the [MATH], in fact in this limit we obtain that the dominant Lagrangian terms obtained from scattering amplitudes in string theory are equal to the terms introduced in the I.T.B model.', 'hep-th-0312318-1-33-0': 'It is likely there could be a microscopic theory of gravity similar to the standard model and we call this model :', 'hep-th-0312318-1-34-0': '[MATH] (G.S.M),', 'hep-th-0312318-1-35-0': 'where all the fields are massless and the big gauge invariance that includes all the fields produce a renormalizable theory in [MATH]-d.', 'hep-th-0312318-1-36-0': 'On the other hand in the [MATH] limit, String Theory can live in any dimension and it is likely that it is a theory for massless higher spin fields [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], that we call :', 'hep-th-0312318-1-37-0': '[MATH] (H.S.G.T).', 'hep-th-0312318-1-38-0': 'In this pictures String Theory and the I.T.B theory are two different phases of a more microscopic theory of Higher Spin Fields but it is possible there is another description of the microscopic Universe.', 'hep-th-0312318-1-39-0': 'It is likely that String Theory and the I.T.B model are two different Hadronic phases of a Sub-Quark theory [CITATION], [CITATION], [CITATION].', 'hep-th-0312318-1-39-1': 'At high energy we have an [MATH] Yang-Mills Gauge Theory that contains [MATH] Sub-Quarks in a free phase.', 'hep-th-0312318-1-39-2': 'On the other side at low energy in the confinement phase we obtain an Hadronic spectrum that reproduce all the string spectrum near a vacuum or the I.T.B model near another vacuum.', 'hep-th-0312318-1-40-0': '# Outlook and Conclusions', 'hep-th-0312318-1-41-0': 'In this paper we introduced a simple model calling I.T.B (Intermediate Tensor Boson) for the description of interactions with four or more gravitons.', 'hep-th-0312318-1-41-1': 'We took Fermi theory of weak interactions as a starting point.', 'hep-th-0312318-1-41-2': 'In particular we focused on the evolution of the Fermi theory in the I.V.B theory (Intermediate Vector Boson).', 'hep-th-0312318-1-41-3': 'In our model the square root of [MATH] constant is proportional to the inverse mediator mass [MATH], in analogy with the Fermi theory of weak interactions in which the square root of Fermi constant is proportional to the inverse proton mass[MATH].', 'hep-th-0312318-1-42-0': 'In the four gravitons interaction case we introduced four Lagrangian terms that couple four "Gravitational Currents" with tensor fields of spin [MATH].', 'hep-th-0312318-1-42-1': 'This is in analogy with the Fermi theory of weak interactions which couple the fermionic currents, containg two fermions, with the Bosons [MATH] and [MATH].', 'hep-th-0312318-1-42-2': 'We can summary the analogy between the I.V.B model and I.T.B model in the following identifications [EQUATION]', 'hep-th-0312318-1-42-3': 'In order to obtain the local interactions of Einstein action we introduced tensor field propagators that are reducible representations of the Poincare group (see section [REF]).', 'hep-th-0312318-1-43-0': 'We write the complete interaction Lagrangian for four gravitons interactions [EQUATION]', 'hep-th-0312318-1-43-1': 'We also studied the interactions with [MATH] gravitons and we showed that the tensor structure can be reproduced in the I.T.B model using another Lagrangian term that contain a tensor field and two graviton without derivative.', 'hep-th-0312318-1-43-2': 'We write the complete interaction Lagrangian reproducing the tensor structure of all the interactions with [MATH] gravitons [EQUATION]', 'hep-th-0312318-1-43-3': 'In this simple model the coupling constants are all dimensionless constants and the Newton constant is connected with the mediators mass [MATH].', 'hep-th-0312318-1-43-4': 'Although the theory is not renormalizable because the mediators are higher spin fields.', 'hep-th-0312318-1-44-0': 'As we said in the section [REF] it is likely there is a theory of gravity similar to the standard model for weak and electro-magnetic interactions.', 'hep-th-0312318-1-44-1': 'This more fundamental theory could be a gauge theory of higher spin fields and so the Quantum Gravity would be a possible broken phase of this theory.', 'hep-th-0312318-1-44-2': 'Another possible phase of our new theory could be a string theory as we said in [REF].', 'hep-th-0312318-1-45-0': 'Another possibility is that there is an alternative more fundamental theory ([REF]).', 'hep-th-0312318-1-45-1': 'It is likely that at high energy level we have an [MATH] Yang-Mills Gauge Theory that contain [MATH] Sub-Quark in a free phase.', 'hep-th-0312318-1-45-2': 'On the other side at low energy in the confinement phase we obtain an hadronic spectrum that reproduce the I.T.B model near a vacuum or all the string spectrum near another vacuum.'}

{'hep-th-0312318-2-0-0': 'In this paper we work in perturbative Quantum Gravity and we introduce a new effective model for gravity.', 'hep-th-0312318-2-0-1': 'Expanding the Einstein-Hilbert Lagrangian in graviton field powers we have an infinite number of terms.', 'hep-th-0312318-2-0-2': 'In this paper we study the possibility of an interpretation of more than three graviton interacting vertices as effective vertices of a most fundamental theory that contain tensor fields .', 'hep-th-0312318-2-0-3': 'Here we introduce a Lagrangian model named I.T.B. (Intermediate-Tensor-Boson) where four gravitational "pseudo-currents" that contain two gravitons couple to three massive tensorial fields of ranks one, three and five respectively.', 'hep-th-0312318-2-0-4': 'We show that the exchange of those massive particles reproduces, at low energy, the interacting vertices for four or more gravitons.', 'hep-th-0312318-2-0-5': 'In a particolar version, the model contains a dimensionless coupling constant "[MATH]" and the mass [MATH] of the intermediate bosons as free parameters.', 'hep-th-0312318-2-0-6': 'The universal gravitational constant [MATH] is shown to be proportional to the inverse of mass squared of mediator fields, particularly [MATH].', 'hep-th-0312318-2-0-7': 'A foresighting choice of the dimensionless coupling constant could lower the energy scale where quantum gravity aspects show up.', 'hep-th-0312318-2-1-0': '# Introduction to the Model', 'hep-th-0312318-2-2-0': 'In this paper we consider the perturbative expansion of the Einstein-Hilbert action in graviton ([MATH]) field powers.', 'hep-th-0312318-2-2-1': 'After the quadratic order that is the free Graviton Lagrangian, the first not trivial order is the three gravitons interaction.', 'hep-th-0312318-2-2-2': 'The subsequent is the four gravitons interaction and so on.', 'hep-th-0312318-2-2-3': 'The interaction vertex that we study with particular attention is the local four graviton interaction vertex; we show that we can see this term as a "current-current" interaction in analogy with Fermi Theory of Weak Interactions.', 'hep-th-0312318-2-2-4': 'In a second step we introduce the I.T.B. (Intermediate Tensor Boson) theory for Quantum Gravity in analogy with the I.V.B. (Intermediate Vector Boson) theory that is the non local extension of the Fermi theory.', 'hep-th-0312318-2-2-5': 'In this Model we introduce an opportune number of Lagrangian terms describing the coupling between currents, made only of gravitons, and tensorial bosons of rank 1, 3, 5.', 'hep-th-0312318-2-2-6': 'The bosons are the analog of the [MATH] and [MATH] vectors for I.V.B. theory.', 'hep-th-0312318-2-3-0': 'Moreover we show that using only another Lagrangian term we can reproduce the tensor structure of all interactions with [MATH], [MATH], [MATH] , [MATH] gravitons.', 'hep-th-0312318-2-3-1': 'Finally we compare the I.T.B model with string theory and we propose a new picture where the two theories are two different phases of a more fundamental theory.', 'hep-th-0312318-2-3-2': 'This can be a theory of massless Tensor Fields [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION] or a Yang-Mills theory coupled with Sub-Quark [CITATION], [CITATION], [CITATION].', 'hep-th-0312318-2-3-3': 'In the second case String Theory and the I.T.B model could be two different hadronic phases of a more microscopic sub-quarks theory.', 'hep-th-0312318-2-4-0': '# [MATH]-Action Gravitons Expansion', 'hep-th-0312318-2-5-0': 'The [MATH] action for Gravity is [EQUATION] where [EQUATION]', 'hep-th-0312318-2-5-1': 'We consider now the fluctuation of the [MATH] metric around the flat background [MATH] [EQUATION]', 'hep-th-0312318-2-5-2': 'This expansion around the flat background is correct only locally.', 'hep-th-0312318-2-5-3': 'In fact, for Manifold with non trivial topology, the metric can non be globally expanded in [MATH] plus a fluctuation.', 'hep-th-0312318-2-5-4': 'This expansion [REF] is relevant for the description of nature because, in the portion of the Universe where our solar system is placed, the intensity of the fluctuation is about [MATH].', 'hep-th-0312318-2-6-0': 'If we observe the Lagrangian [REF] we see that at any order we have only two derivatives.', 'hep-th-0312318-2-6-1': 'This is a consequence of the definition of the Cristoffel symbols, in fact such symbols have only low indices and so the form in terms of [MATH] is exact [EQUATION]', 'hep-th-0312318-2-6-2': 'Now using the relation [MATH] we can expand the determinant and the inverse of the metric [MATH] in power of [MATH] : [EQUATION]', 'hep-th-0312318-2-6-3': 'This second relation can be obtained from the Taylor expansion of [MATH].', 'hep-th-0312318-2-6-4': 'Using those relations the Lorentz gauge condition [MATH] and, the mass-shell condition ([MATH], [MATH]) the amplitudes can be written [EQUATION]', 'hep-th-0312318-2-6-5': 'Since we are studying the tree level gravitons interaction it is sufficient to consider only the on shell Lagrangian terms.', 'hep-th-0312318-2-6-6': 'In fact all the interactions that contain the tensors [MATH], [MATH] and [MATH], produce amplitudes proportional to [MATH], [MATH] and [MATH] that are identically zero on external gravitons states.', 'hep-th-0312318-2-7-0': 'Now we introduce tensors with tree and five indexes, that we call "tensor currents" and we rewrite the Lagrangian ([REF]) as a "current-current" interaction in analogy with Fermi theory of the weak interactions.', 'hep-th-0312318-2-7-1': 'The currents need to do it are [EQUATION]', 'hep-th-0312318-2-7-2': 'Using those currents in the Lagrangian ([REF]) we obtain this compact form for the four gravitons interaction [EQUATION]', 'hep-th-0312318-2-8-0': '# [MATH] Constant Expansion', 'hep-th-0312318-2-9-0': 'The fundamental constant in gravity is the Newton constant [MATH] that in natural units [MATH] and [MATH] is [EQUATION]', 'hep-th-0312318-2-9-1': 'In front of the Lagrangian we have the factor [MATH], and we can reabsorb it in the definition of the graviton field : [EQUATION]', 'hep-th-0312318-2-9-2': 'In this way we obtain mass dimension for the graviton [MATH] while the metric is dimensionless.', 'hep-th-0312318-2-9-3': 'With this definition we obtain a factor [MATH] in front of the [MATH]-oder interaction in the gravitons expansion of the Lagrangian.', 'hep-th-0312318-2-9-4': 'The scheme of the Lagrangian with correct powers of [MATH] is [EQUATION]', 'hep-th-0312318-2-9-5': 'Where [MATH] factors are understood.', 'hep-th-0312318-2-10-0': 'At this point we can rewrite the Lagrangian [REF] in the following way : [EQUATION]', 'hep-th-0312318-2-11-0': '# The I.T.B Theory for Quantum Gravity', 'hep-th-0312318-2-12-0': 'As said in the first section the I.V.B theory replaces the Fermi Theory of the weak interactions.', 'hep-th-0312318-2-12-1': 'In this new model the fermionic local interaction becomes a non local interaction with the exchange of a massive boson ([MATH] or [MATH]).', 'hep-th-0312318-2-12-2': 'The mass of the bosons is exactly the inverse of the [MATH] Fermi constant ([MATH]).', 'hep-th-0312318-2-12-3': 'Now we introduce for Gravity an analog of the I.V.B model that we call I.T.B (Intermediate Tensor Boson Model).', 'hep-th-0312318-2-13-0': 'In the I.T.B Model the analog of the [MATH] bosons are tensor fields of rang [MATH], [MATH] and [MATH] : [EQUATION]', 'hep-th-0312318-2-13-1': 'Such fields couple to what we call "gravitational currents", in analogy with the Fermi model, through the following Lagrangian terms with a dimensionless coupling constant, as in the I.V.B model.', 'hep-th-0312318-2-13-2': '[EQUATION]', 'hep-th-0312318-2-13-3': 'Those Lagrangian terms reproduce all the local four graviton interactions as a low energy limit ([MATH]) of amplitudes in which gravitons interact through the exchange of massive fields [MATH], [MATH] or [MATH].', 'hep-th-0312318-2-14-0': 'We introduce now the following tensor factors for the propagetors of the higher spin fields [EQUATION]', 'hep-th-0312318-2-14-1': 'We start with the field of rank [MATH].', 'hep-th-0312318-2-14-2': 'In this case the Lagrangian is [MATH] and we calculate the scattering amplitudes for [MATH] gravitons using the [MATH]-matrix.', 'hep-th-0312318-2-15-0': 'We recall that the [MATH]-matrix connects the final state [MATH] to the initial state [MATH] defined by [EQUATION]', 'hep-th-0312318-2-15-1': 'The probability to obtain the system in the general state [MATH] after the scattering, is [EQUATION] and the unitarity of the [MATH]-matrix can be written as [EQUATION]', 'hep-th-0312318-2-15-2': 'The general form of the [MATH]-matrix that has those property is [EQUATION]', 'hep-th-0312318-2-15-3': 'Now we consider the second order in the [MATH]-matrix for the Lagrangian [MATH], with [MATH] and [MATH] defined in [REF] : [EQUATION]', 'hep-th-0312318-2-15-4': 'Introducing the propagator for the field [MATH] in [MATH] at the second order and taking [MATH] we obtain the local amplitude for [MATH] gravitons.', 'hep-th-0312318-2-15-5': 'This amplitude is exactly what we obtain from Einstein action ([REF]) at first order in the [MATH]-matrix.', 'hep-th-0312318-2-15-6': "In this section we don't give calculations detail report the calculus but only the idea of the mechanism (see the Appendix).", 'hep-th-0312318-2-15-7': 'Using Feyman diagrams we can depict the interaction at high ([MATH]) and low ([MATH]) energy, and this is very similar to what we find when we pass from I.V.B model to Fermi theory provided the following identifications [EQUATION]', 'hep-th-0312318-2-15-8': 'Qualitatively the [MATH]-matrix at the second order is : [EQUATION]', 'hep-th-0312318-2-15-9': 'At this point we analyse the result from the I.T.B model.', 'hep-th-0312318-2-16-0': 'In this model we can interpret the Newton constant [MATH] as proportional to the mass of the fields of rank [MATH], [MATH], and [MATH], [MATH], [MATH] and [MATH] that are the mediators of the gravitational interaction.', 'hep-th-0312318-2-16-1': '[EQUATION]', 'hep-th-0312318-2-16-2': 'The other important fact is that we can have a lower energy scale where gravity takes a quantum nature.', 'hep-th-0312318-2-16-3': 'In the low energy limit the non-local amplitude becomes local and comparing with the Einstein theory we obtain the following identification [EQUATION] or : [EQUATION]', 'hep-th-0312318-2-16-4': 'If the adimentional coupling constant "[MATH]" is very small ([MATH]), we can apply the perturbative theory and for [MATH] we obtain [MATH], that is the Grand Unification scale.', 'hep-th-0312318-2-16-5': 'If [MATH] than [MATH] and we cauld see Quantum Gravitational effects at accelerators.', 'hep-th-0312318-2-17-0': '## Tensor Fields as Reducible Representation of the Poincare Group', 'hep-th-0312318-2-18-0': 'The tensor factors [REF], [REF] e [REF] inside the propagators have no definite symmetry.', 'hep-th-0312318-2-18-1': 'Now we rewrite the tensor factors [REF], [REF] as a sum of propagators for other fields, with always rank [MATH] and [MATH], that are reducible representations of the Poincare group with definite tensor symmetry.', 'hep-th-0312318-2-18-2': 'We can write the tensors [MATH] as following.', 'hep-th-0312318-2-19-0': 'For [MATH] we have [EQUATION]', 'hep-th-0312318-2-19-1': 'Any line in the tensor ([REF]) represents the propagator of a field that can be decomposed in irreducible representations of the Poncare group.', 'hep-th-0312318-2-20-0': 'For [MATH] we have the following decomposition [EQUATION]', 'hep-th-0312318-2-21-0': '# Interaction with [MATH] Gravitons (Tensor Structure)', 'hep-th-0312318-2-22-0': 'The interaction vertexes that contain [MATH] gravitons have a tensor structure such that we can obtain those interactions introducing one or plus gravitons into the four graviton vertex.', 'hep-th-0312318-2-22-1': 'To carry out this idea we must introduce another Lagrangian term to the I.T.B model; such term couples two massive fields with a graviton.', 'hep-th-0312318-2-22-2': 'This Interaction term in the low energy limit reproduces the tensor behaviour of the local interactions of the Einstein action.', 'hep-th-0312318-2-23-0': 'For example we consider a typical [MATH]-gravitons interaction that we can obtain from the expansion of the Einstein-Hilbert action [EQUATION]', 'hep-th-0312318-2-23-1': 'The Lagrangian term that we must introduce in the I.T.B model to reproduce ([REF]) is [EQUATION]', 'hep-th-0312318-2-23-2': 'Omitting the indexes, the total lagrangian is [EQUATION]', 'hep-th-0312318-2-23-3': 'To calculate the [MATH] gravitons amplitude in the I.T.B model, we consider the third order in the [MATH]-matrix at tree level : [EQUATION]', 'hep-th-0312318-2-23-4': 'Comparing the result with the Einstein action we obtain [MATH] and so [MATH].', 'hep-th-0312318-2-24-0': '# "[MATH]" Graviton Interaction (Tensor Structure)', 'hep-th-0312318-2-25-0': 'In the previous section we have introduced another Lagrangian term that reproduces all the interactions for [MATH] or in general "[MATH]" gravitons ([MATH]) at least in the tensor structure.', 'hep-th-0312318-2-25-1': 'Now we omit the tensor indexes and we explain the mechanism.', 'hep-th-0312318-2-25-2': 'A typical Lagrangian term for [MATH] gravitons that we obtain from the Einstein action is : [EQUATION]', 'hep-th-0312318-2-25-3': 'This local interaction can be reproduced with the I.T.B effective Lagrangian [EQUATION] where [MATH] were defined before.', 'hep-th-0312318-2-26-0': 'At fourth order we obtain the [MATH] graviton interaction [EQUATION]', 'hep-th-0312318-2-26-1': 'For the "[MATH]" graviton interaction we must take the [MATH]-matrix at the "[MATH]" order : [EQUATION]', 'hep-th-0312318-2-26-2': 'From the amplitude we obtain [MATH].', 'hep-th-0312318-2-27-0': '# I.T.B. Model Off-Shell', 'hep-th-0312318-2-28-0': 'In the previous sections we have taken on-shell gravitons that satisfy the following relations : [EQUATION]', 'hep-th-0312318-2-28-1': 'We can build the I.T.B model for off-shell gravitons too.', 'hep-th-0312318-2-28-2': 'In this case the rank [MATH] fields couple to the following general currents : [EQUATION]', 'hep-th-0312318-2-29-0': '# String Theory and ITB Model Toward a More Fundamental Theory', 'hep-th-0312318-2-30-0': 'In this section we analyse the possible connections between string theory, in the critical dimension [MATH] for the Bosonic String or [MATH] for the Superstring, and the ITB Model introduced in the previous sections [CITATION].', 'hep-th-0312318-2-30-1': 'In particular in the string theory contest it is simple to verify (at least in the bosonic case) that in the low energy limit [MATH] the effective vertexes for gravitons massive fields coupling contain only the tensor fields with rank [MATH] as in the ITB Model.', 'hep-th-0312318-2-31-0': 'The fondamental idea is the following:', 'hep-th-0312318-2-32-0': 'at very high energy ([MATH]) we have a Gauge Theory of Massless Tensor Fields (G.T.H.S) that can lives in any dimension.', 'hep-th-0312318-2-32-1': 'In fact String Theory is Weyl anomaly free in this regime as we can see from the Virasoro algebra ([REF]).', 'hep-th-0312318-2-32-2': "This fundamental theory could have many different Higgs' s phases related with many different relative minimum of complicated potential.", 'hep-th-0312318-2-32-3': 'The conjecture is that two of those are the String Theory and the I.T.B Model vacuums.', 'hep-th-0312318-2-32-4': 'In this picture we see that there is not a prefered dimension as in field theory, but the dimension is a consequence of the symmetry breaking.', 'hep-th-0312318-2-32-5': 'There is a vacuum where the massless fields of the microscopic theory become all massive with the exception of the gravitational multiplet and the spectrum is that of the String Theory so the dimension is D=10, and there is another vacuum where the theory reproduces the gravitational theory in analogy with the Fermi theory of the weak interactions.', 'hep-th-0312318-2-32-6': 'The first vacuum is a finite theory (String Theory) and the other is a renormalizable gauge theory of Higher Spin in [MATH] dimension with spontaneous symmetry breaking that contains only the graviton as massless state and a residual gauge symmetry that is the infinitesimal version of the [MATH]-invariance of General Relativity.', 'hep-th-0312318-2-32-7': '[EQUATION]', 'hep-th-0312318-2-32-8': 'Now we summarize the analogies between the Fermi Theory of weak interactions (F.T.W.I), General Relativity (G.R) and String Theory.', 'hep-th-0312318-2-32-9': 'At low energy the F.T.W.I model is a good approsimation of the weak interactions, it consists of a local vertex with four fermions and a fondamental constant [MATH].', 'hep-th-0312318-2-32-10': 'At higher energies we see that the theory contains a vertex that consists of a fermionic current and a massive vector "W" (Intermediate Vector Boson Model, I.V.B).', 'hep-th-0312318-2-32-11': 'When we take the limit [MATH] we obtain the F.T.W.I with [MATH].', 'hep-th-0312318-2-32-12': 'In analogy we have introduced the Intermediate Tensor Boson Model (I.T.B) where now the fermionic current become a "pseudocurrent" with two gravitons and the W-bosons become fields with ramk [MATH].', 'hep-th-0312318-2-32-13': 'Finally we have a theory very similar to string theory in the [MATH], in fact in this limit we obtain that the dominant Lagrangian terms obtained from scattering amplitudes in string theory are equal to the terms introduced in the I.T.B model.', 'hep-th-0312318-2-33-0': 'It is likely there could be a microscopic theory of gravity similar to the standard model and we call this model :', 'hep-th-0312318-2-34-0': '[MATH] (G.S.M),', 'hep-th-0312318-2-35-0': 'where all the fields are massless and the big gauge invariance that includes all the fields produce a renormalizable theory in [MATH]-d.', 'hep-th-0312318-2-36-0': 'On the other hand in the [MATH] limit, String Theory can live in any dimension and it is likely that it is a theory for massless higher spin fields [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION] that we call :', 'hep-th-0312318-2-37-0': '[MATH] (H.S.G.T).', 'hep-th-0312318-2-38-0': 'In this pictures String Theory and the I.T.B theory are two different phases of a more microscopic theory of Higher Spin Fields but it is possible there is another description of the microscopic Universe.', 'hep-th-0312318-2-39-0': 'It is likely that String Theory and the I.T.B model are two different Hadronic phases of a Sub-Quark theory [CITATION], [CITATION], [CITATION].', 'hep-th-0312318-2-39-1': 'At high energy we have an [MATH] Yang-Mills Gauge Theory that contains [MATH] Sub-Quarks in a free phase.', 'hep-th-0312318-2-39-2': 'On the other side at low energy in the confinement phase we obtain an Hadronic spectrum that reproduce all the string spectrum near a vacuum or the I.T.B model near another vacuum.', 'hep-th-0312318-2-40-0': '# Outlook and Conclusions', 'hep-th-0312318-2-41-0': 'In this paper we introduced a simple model calling I.T.B (Intermediate Tensor Boson) for the description of interactions with four or more gravitons.', 'hep-th-0312318-2-41-1': 'We took Fermi theory of weak interactions as a starting point.', 'hep-th-0312318-2-41-2': 'In particular we focused on the evolution of the Fermi theory in the I.V.B theory (Intermediate Vector Boson).', 'hep-th-0312318-2-41-3': 'In our model the square root of [MATH] constant is proportional to the inverse mediator mass [MATH], in analogy with the Fermi theory of weak interactions in which the square root of Fermi constant is proportional to the inverse proton mass[MATH].', 'hep-th-0312318-2-42-0': 'In the four gravitons interaction case we introduced four Lagrangian terms that couple four "Gravitational Currents" with tensor fields of spin [MATH].', 'hep-th-0312318-2-42-1': 'This is in analogy with the Fermi theory of weak interactions which couple the fermionic currents, containg two fermions, with the Bosons [MATH] and [MATH].', 'hep-th-0312318-2-42-2': 'We can summary the analogy between the I.V.B model and I.T.B model in the following identifications [EQUATION]', 'hep-th-0312318-2-42-3': 'In order to obtain the local interactions of Einstein action we introduced tensor field propagators that are reducible representations of the Poincare group (see section [REF]).', 'hep-th-0312318-2-43-0': 'We write the complete interaction Lagrangian for four gravitons interactions [EQUATION]', 'hep-th-0312318-2-43-1': 'We also studied the interactions with [MATH] gravitons and we showed that the tensor structure can be reproduced in the I.T.B model using another Lagrangian term that contain a tensor field and two graviton without derivative.', 'hep-th-0312318-2-43-2': 'We write the complete interaction Lagrangian reproducing the tensor structure of all the interactions with [MATH] gravitons [EQUATION]', 'hep-th-0312318-2-43-3': 'In this simple model the coupling constants are all dimensionless constants and the Newton constant is connected with the mediators mass [MATH].', 'hep-th-0312318-2-43-4': 'Although the theory is not renormalizable because the mediators are higher spin fields.', 'hep-th-0312318-2-44-0': 'As we said in the section [REF] it is likely there is a theory of gravity similar to the standard model for weak and electro-magnetic interactions.', 'hep-th-0312318-2-44-1': 'This more fundamental theory could be a gauge theory of higher spin fields and so the Quantum Gravity would be a possible broken phase of this theory.', 'hep-th-0312318-2-44-2': 'Another possible phase of our new theory could be a string theory as we said in [REF].', 'hep-th-0312318-2-45-0': 'Another possibility is that there is an alternative more fundamental theory ([REF]).', 'hep-th-0312318-2-45-1': 'It is likely that at high energy level we have an [MATH] Yang-Mills Gauge Theory that contain [MATH] Sub-Quark in a free phase.', 'hep-th-0312318-2-45-2': 'On the other side at low energy in the confinement phase we obtain an hadronic spectrum that reproduce the I.T.B model near a vacuum or all the string spectrum near another vacuum.'}

[['hep-th-0312318-1-13-0', 'hep-th-0312318-2-13-0'], ['hep-th-0312318-1-13-1', 'hep-th-0312318-2-13-1'], ['hep-th-0312318-1-13-3', 'hep-th-0312318-2-13-3'], ['hep-th-0312318-1-35-0', 'hep-th-0312318-2-35-0'], ['hep-th-0312318-1-2-0', 'hep-th-0312318-2-2-0'], ['hep-th-0312318-1-2-1', 'hep-th-0312318-2-2-1'], ['hep-th-0312318-1-2-2', 'hep-th-0312318-2-2-2'], ['hep-th-0312318-1-2-3', 'hep-th-0312318-2-2-3'], ['hep-th-0312318-1-2-4', 'hep-th-0312318-2-2-4'], ['hep-th-0312318-1-2-5', 'hep-th-0312318-2-2-5'], ['hep-th-0312318-1-2-6', 'hep-th-0312318-2-2-6'], ['hep-th-0312318-1-30-1', 'hep-th-0312318-2-30-1'], ['hep-th-0312318-1-6-0', 'hep-th-0312318-2-6-0'], ['hep-th-0312318-1-6-1', 'hep-th-0312318-2-6-1'], ['hep-th-0312318-1-6-2', 'hep-th-0312318-2-6-2'], ['hep-th-0312318-1-6-3', 'hep-th-0312318-2-6-3'], ['hep-th-0312318-1-6-4', 'hep-th-0312318-2-6-4'], ['hep-th-0312318-1-6-5', 'hep-th-0312318-2-6-5'], ['hep-th-0312318-1-6-6', 'hep-th-0312318-2-6-6'], ['hep-th-0312318-1-15-0', 'hep-th-0312318-2-15-0'], ['hep-th-0312318-1-15-1', 'hep-th-0312318-2-15-1'], ['hep-th-0312318-1-15-2', 'hep-th-0312318-2-15-2'], ['hep-th-0312318-1-15-3', 'hep-th-0312318-2-15-3'], ['hep-th-0312318-1-15-4', 'hep-th-0312318-2-15-4'], ['hep-th-0312318-1-15-5', 'hep-th-0312318-2-15-5'], ['hep-th-0312318-1-15-6', 'hep-th-0312318-2-15-6'], ['hep-th-0312318-1-15-7', 'hep-th-0312318-2-15-7'], ['hep-th-0312318-1-15-8', 'hep-th-0312318-2-15-8'], ['hep-th-0312318-1-15-9', 'hep-th-0312318-2-15-9'], ['hep-th-0312318-1-25-0', 'hep-th-0312318-2-25-0'], ['hep-th-0312318-1-25-1', 'hep-th-0312318-2-25-1'], ['hep-th-0312318-1-25-2', 'hep-th-0312318-2-25-2'], ['hep-th-0312318-1-25-3', 'hep-th-0312318-2-25-3'], ['hep-th-0312318-1-18-0', 'hep-th-0312318-2-18-0'], ['hep-th-0312318-1-18-1', 'hep-th-0312318-2-18-1'], ['hep-th-0312318-1-18-2', 'hep-th-0312318-2-18-2'], ['hep-th-0312318-1-45-0', 'hep-th-0312318-2-45-0'], ['hep-th-0312318-1-45-1', 'hep-th-0312318-2-45-1'], ['hep-th-0312318-1-45-2', 'hep-th-0312318-2-45-2'], ['hep-th-0312318-1-41-0', 'hep-th-0312318-2-41-0'], ['hep-th-0312318-1-41-1', 'hep-th-0312318-2-41-1'], ['hep-th-0312318-1-41-2', 'hep-th-0312318-2-41-2'], ['hep-th-0312318-1-41-3', 'hep-th-0312318-2-41-3'], ['hep-th-0312318-1-7-0', 'hep-th-0312318-2-7-0'], ['hep-th-0312318-1-7-1', 'hep-th-0312318-2-7-1'], ['hep-th-0312318-1-7-2', 'hep-th-0312318-2-7-2'], ['hep-th-0312318-1-19-0', 'hep-th-0312318-2-19-0'], ['hep-th-0312318-1-19-1', 'hep-th-0312318-2-19-1'], ['hep-th-0312318-1-12-0', 'hep-th-0312318-2-12-0'], ['hep-th-0312318-1-12-1', 'hep-th-0312318-2-12-1'], ['hep-th-0312318-1-12-2', 'hep-th-0312318-2-12-2'], ['hep-th-0312318-1-12-3', 'hep-th-0312318-2-12-3'], ['hep-th-0312318-1-23-0', 'hep-th-0312318-2-23-0'], ['hep-th-0312318-1-23-1', 'hep-th-0312318-2-23-1'], ['hep-th-0312318-1-23-2', 'hep-th-0312318-2-23-2'], ['hep-th-0312318-1-23-3', 'hep-th-0312318-2-23-3'], ['hep-th-0312318-1-23-4', 'hep-th-0312318-2-23-4'], ['hep-th-0312318-1-9-0', 'hep-th-0312318-2-9-0'], ['hep-th-0312318-1-9-1', 'hep-th-0312318-2-9-1'], ['hep-th-0312318-1-9-2', 'hep-th-0312318-2-9-2'], ['hep-th-0312318-1-9-3', 'hep-th-0312318-2-9-3'], ['hep-th-0312318-1-9-4', 'hep-th-0312318-2-9-4'], ['hep-th-0312318-1-9-5', 'hep-th-0312318-2-9-5'], ['hep-th-0312318-1-32-0', 'hep-th-0312318-2-32-0'], ['hep-th-0312318-1-32-1', 'hep-th-0312318-2-32-1'], ['hep-th-0312318-1-32-2', 'hep-th-0312318-2-32-2'], ['hep-th-0312318-1-32-3', 'hep-th-0312318-2-32-3'], ['hep-th-0312318-1-32-4', 'hep-th-0312318-2-32-4'], ['hep-th-0312318-1-32-5', 'hep-th-0312318-2-32-5'], ['hep-th-0312318-1-32-6', 'hep-th-0312318-2-32-6'], ['hep-th-0312318-1-32-8', 'hep-th-0312318-2-32-8'], ['hep-th-0312318-1-32-9', 'hep-th-0312318-2-32-9'], ['hep-th-0312318-1-32-10', 'hep-th-0312318-2-32-10'], ['hep-th-0312318-1-32-11', 'hep-th-0312318-2-32-11'], ['hep-th-0312318-1-32-12', 'hep-th-0312318-2-32-12'], ['hep-th-0312318-1-32-13', 'hep-th-0312318-2-32-13'], ['hep-th-0312318-1-5-0', 'hep-th-0312318-2-5-0'], ['hep-th-0312318-1-5-1', 'hep-th-0312318-2-5-1'], ['hep-th-0312318-1-5-2', 'hep-th-0312318-2-5-2'], ['hep-th-0312318-1-5-3', 'hep-th-0312318-2-5-3'], ['hep-th-0312318-1-5-4', 'hep-th-0312318-2-5-4'], ['hep-th-0312318-1-0-0', 'hep-th-0312318-2-0-0'], ['hep-th-0312318-1-0-1', 'hep-th-0312318-2-0-1'], ['hep-th-0312318-1-0-2', 'hep-th-0312318-2-0-2'], ['hep-th-0312318-1-0-3', 'hep-th-0312318-2-0-3'], ['hep-th-0312318-1-0-4', 'hep-th-0312318-2-0-4'], ['hep-th-0312318-1-0-5', 'hep-th-0312318-2-0-5'], ['hep-th-0312318-1-0-6', 'hep-th-0312318-2-0-6'], ['hep-th-0312318-1-0-7', 'hep-th-0312318-2-0-7'], ['hep-th-0312318-1-39-0', 'hep-th-0312318-2-39-0'], ['hep-th-0312318-1-39-1', 'hep-th-0312318-2-39-1'], ['hep-th-0312318-1-39-2', 'hep-th-0312318-2-39-2'], ['hep-th-0312318-1-16-0', 'hep-th-0312318-2-16-0'], ['hep-th-0312318-1-16-2', 'hep-th-0312318-2-16-2'], ['hep-th-0312318-1-16-3', 'hep-th-0312318-2-16-3'], ['hep-th-0312318-1-16-4', 'hep-th-0312318-2-16-4'], ['hep-th-0312318-1-16-5', 'hep-th-0312318-2-16-5'], ['hep-th-0312318-1-28-0', 'hep-th-0312318-2-28-0'], ['hep-th-0312318-1-28-1', 'hep-th-0312318-2-28-1'], ['hep-th-0312318-1-28-2', 'hep-th-0312318-2-28-2'], ['hep-th-0312318-1-26-0', 'hep-th-0312318-2-26-0'], ['hep-th-0312318-1-26-1', 'hep-th-0312318-2-26-1'], ['hep-th-0312318-1-26-2', 'hep-th-0312318-2-26-2'], ['hep-th-0312318-1-42-0', 'hep-th-0312318-2-42-0'], ['hep-th-0312318-1-42-1', 'hep-th-0312318-2-42-1'], ['hep-th-0312318-1-42-2', 'hep-th-0312318-2-42-2'], ['hep-th-0312318-1-42-3', 'hep-th-0312318-2-42-3'], ['hep-th-0312318-1-43-0', 'hep-th-0312318-2-43-0'], ['hep-th-0312318-1-43-1', 'hep-th-0312318-2-43-1'], ['hep-th-0312318-1-43-2', 'hep-th-0312318-2-43-2'], ['hep-th-0312318-1-43-3', 'hep-th-0312318-2-43-3'], ['hep-th-0312318-1-43-4', 'hep-th-0312318-2-43-4'], ['hep-th-0312318-1-3-1', 'hep-th-0312318-2-3-1'], ['hep-th-0312318-1-3-3', 'hep-th-0312318-2-3-3'], ['hep-th-0312318-1-38-0', 'hep-th-0312318-2-38-0'], ['hep-th-0312318-1-44-0', 'hep-th-0312318-2-44-0'], ['hep-th-0312318-1-44-1', 'hep-th-0312318-2-44-1'], ['hep-th-0312318-1-44-2', 'hep-th-0312318-2-44-2'], ['hep-th-0312318-1-22-0', 'hep-th-0312318-2-22-0'], ['hep-th-0312318-1-22-1', 'hep-th-0312318-2-22-1'], ['hep-th-0312318-1-22-2', 'hep-th-0312318-2-22-2'], ['hep-th-0312318-1-14-0', 'hep-th-0312318-2-14-0'], ['hep-th-0312318-1-14-1', 'hep-th-0312318-2-14-1'], ['hep-th-0312318-1-14-2', 'hep-th-0312318-2-14-2'], ['hep-th-0312318-1-10-0', 'hep-th-0312318-2-10-0'], ['hep-th-0312318-1-30-0', 'hep-th-0312318-2-30-0'], ['hep-th-0312318-1-3-0', 'hep-th-0312318-2-3-0'], ['hep-th-0312318-1-3-2', 'hep-th-0312318-2-3-2']]

[['hep-th-0312318-1-13-0', 'hep-th-0312318-2-13-0'], ['hep-th-0312318-1-13-1', 'hep-th-0312318-2-13-1'], ['hep-th-0312318-1-13-3', 'hep-th-0312318-2-13-3'], ['hep-th-0312318-1-35-0', 'hep-th-0312318-2-35-0'], ['hep-th-0312318-1-2-0', 'hep-th-0312318-2-2-0'], ['hep-th-0312318-1-2-1', 'hep-th-0312318-2-2-1'], ['hep-th-0312318-1-2-2', 'hep-th-0312318-2-2-2'], ['hep-th-0312318-1-2-3', 'hep-th-0312318-2-2-3'], ['hep-th-0312318-1-2-4', 'hep-th-0312318-2-2-4'], ['hep-th-0312318-1-2-5', 'hep-th-0312318-2-2-5'], ['hep-th-0312318-1-2-6', 'hep-th-0312318-2-2-6'], ['hep-th-0312318-1-30-1', 'hep-th-0312318-2-30-1'], ['hep-th-0312318-1-6-0', 'hep-th-0312318-2-6-0'], ['hep-th-0312318-1-6-1', 'hep-th-0312318-2-6-1'], ['hep-th-0312318-1-6-2', 'hep-th-0312318-2-6-2'], ['hep-th-0312318-1-6-3', 'hep-th-0312318-2-6-3'], ['hep-th-0312318-1-6-4', 'hep-th-0312318-2-6-4'], ['hep-th-0312318-1-6-5', 'hep-th-0312318-2-6-5'], ['hep-th-0312318-1-6-6', 'hep-th-0312318-2-6-6'], ['hep-th-0312318-1-15-0', 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'hep-th-0312318-2-9-0'], ['hep-th-0312318-1-9-1', 'hep-th-0312318-2-9-1'], ['hep-th-0312318-1-9-2', 'hep-th-0312318-2-9-2'], ['hep-th-0312318-1-9-3', 'hep-th-0312318-2-9-3'], ['hep-th-0312318-1-9-4', 'hep-th-0312318-2-9-4'], ['hep-th-0312318-1-9-5', 'hep-th-0312318-2-9-5'], ['hep-th-0312318-1-32-0', 'hep-th-0312318-2-32-0'], ['hep-th-0312318-1-32-1', 'hep-th-0312318-2-32-1'], ['hep-th-0312318-1-32-2', 'hep-th-0312318-2-32-2'], ['hep-th-0312318-1-32-3', 'hep-th-0312318-2-32-3'], ['hep-th-0312318-1-32-4', 'hep-th-0312318-2-32-4'], ['hep-th-0312318-1-32-5', 'hep-th-0312318-2-32-5'], ['hep-th-0312318-1-32-6', 'hep-th-0312318-2-32-6'], ['hep-th-0312318-1-32-8', 'hep-th-0312318-2-32-8'], ['hep-th-0312318-1-32-9', 'hep-th-0312318-2-32-9'], ['hep-th-0312318-1-32-10', 'hep-th-0312318-2-32-10'], ['hep-th-0312318-1-32-11', 'hep-th-0312318-2-32-11'], ['hep-th-0312318-1-32-12', 'hep-th-0312318-2-32-12'], ['hep-th-0312318-1-32-13', 'hep-th-0312318-2-32-13'], ['hep-th-0312318-1-5-0', 'hep-th-0312318-2-5-0'], ['hep-th-0312318-1-5-1', 'hep-th-0312318-2-5-1'], ['hep-th-0312318-1-5-2', 'hep-th-0312318-2-5-2'], ['hep-th-0312318-1-5-3', 'hep-th-0312318-2-5-3'], ['hep-th-0312318-1-5-4', 'hep-th-0312318-2-5-4'], ['hep-th-0312318-1-0-0', 'hep-th-0312318-2-0-0'], ['hep-th-0312318-1-0-1', 'hep-th-0312318-2-0-1'], ['hep-th-0312318-1-0-2', 'hep-th-0312318-2-0-2'], ['hep-th-0312318-1-0-3', 'hep-th-0312318-2-0-3'], ['hep-th-0312318-1-0-4', 'hep-th-0312318-2-0-4'], ['hep-th-0312318-1-0-5', 'hep-th-0312318-2-0-5'], ['hep-th-0312318-1-0-6', 'hep-th-0312318-2-0-6'], ['hep-th-0312318-1-0-7', 'hep-th-0312318-2-0-7'], ['hep-th-0312318-1-39-0', 'hep-th-0312318-2-39-0'], ['hep-th-0312318-1-39-1', 'hep-th-0312318-2-39-1'], ['hep-th-0312318-1-39-2', 'hep-th-0312318-2-39-2'], ['hep-th-0312318-1-16-0', 'hep-th-0312318-2-16-0'], ['hep-th-0312318-1-16-2', 'hep-th-0312318-2-16-2'], ['hep-th-0312318-1-16-3', 'hep-th-0312318-2-16-3'], ['hep-th-0312318-1-16-4', 'hep-th-0312318-2-16-4'], ['hep-th-0312318-1-16-5', 'hep-th-0312318-2-16-5'], ['hep-th-0312318-1-28-0', 'hep-th-0312318-2-28-0'], ['hep-th-0312318-1-28-1', 'hep-th-0312318-2-28-1'], ['hep-th-0312318-1-28-2', 'hep-th-0312318-2-28-2'], ['hep-th-0312318-1-26-0', 'hep-th-0312318-2-26-0'], ['hep-th-0312318-1-26-1', 'hep-th-0312318-2-26-1'], ['hep-th-0312318-1-26-2', 'hep-th-0312318-2-26-2'], ['hep-th-0312318-1-42-0', 'hep-th-0312318-2-42-0'], ['hep-th-0312318-1-42-1', 'hep-th-0312318-2-42-1'], ['hep-th-0312318-1-42-2', 'hep-th-0312318-2-42-2'], ['hep-th-0312318-1-42-3', 'hep-th-0312318-2-42-3'], ['hep-th-0312318-1-43-0', 'hep-th-0312318-2-43-0'], ['hep-th-0312318-1-43-1', 'hep-th-0312318-2-43-1'], ['hep-th-0312318-1-43-2', 'hep-th-0312318-2-43-2'], ['hep-th-0312318-1-43-3', 'hep-th-0312318-2-43-3'], ['hep-th-0312318-1-43-4', 'hep-th-0312318-2-43-4'], ['hep-th-0312318-1-3-1', 'hep-th-0312318-2-3-1'], ['hep-th-0312318-1-3-3', 'hep-th-0312318-2-3-3'], ['hep-th-0312318-1-38-0', 'hep-th-0312318-2-38-0'], ['hep-th-0312318-1-44-0', 'hep-th-0312318-2-44-0'], ['hep-th-0312318-1-44-1', 'hep-th-0312318-2-44-1'], ['hep-th-0312318-1-44-2', 'hep-th-0312318-2-44-2'], ['hep-th-0312318-1-22-0', 'hep-th-0312318-2-22-0'], ['hep-th-0312318-1-22-1', 'hep-th-0312318-2-22-1'], ['hep-th-0312318-1-22-2', 'hep-th-0312318-2-22-2'], ['hep-th-0312318-1-14-0', 'hep-th-0312318-2-14-0'], ['hep-th-0312318-1-14-1', 'hep-th-0312318-2-14-1'], ['hep-th-0312318-1-14-2', 'hep-th-0312318-2-14-2'], ['hep-th-0312318-1-10-0', 'hep-th-0312318-2-10-0']]

[['hep-th-0312318-1-30-0', 'hep-th-0312318-2-30-0'], ['hep-th-0312318-1-3-0', 'hep-th-0312318-2-3-0'], ['hep-th-0312318-1-3-2', 'hep-th-0312318-2-3-2']]

[]

[]

[]

['hep-th-0312318-1-13-2', 'hep-th-0312318-1-16-1', 'hep-th-0312318-1-20-0', 'hep-th-0312318-1-31-0', 'hep-th-0312318-1-32-7', 'hep-th-0312318-1-33-0', 'hep-th-0312318-1-34-0', 'hep-th-0312318-1-36-0', 'hep-th-0312318-1-37-0', 'hep-th-0312318-2-13-2', 'hep-th-0312318-2-16-1', 'hep-th-0312318-2-20-0', 'hep-th-0312318-2-31-0', 'hep-th-0312318-2-32-7', 'hep-th-0312318-2-33-0', 'hep-th-0312318-2-34-0', 'hep-th-0312318-2-36-0', 'hep-th-0312318-2-37-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/0312318

1908.06426

{'1908.06426-1-0-0': 'In this paper we show some Orlicz-Jensen-Hermite-Hadamard inequality and a reverse to that inequality.', '1908.06426-1-0-1': 'This establishes, in particular, one of the first multidimensional Hermite-Hadamard inequality in this generality.', '1908.06426-1-0-2': 'We then show several consequences of those inequalities.', '1908.06426-1-1-0': 'Using the direct version of the inequality above, we derive a Milman-Pajor-Spingarn type inequality, which in some sense, improves the previous versions.', '1908.06426-1-1-1': 'Moreover, as a particular case we answer a question posed by Francisco Santos on upper bounds of the volume of a convex body in terms of the volume of one of its central sections.', '1908.06426-1-1-2': 'Using the reverse version, we derive a generalization of the Rogers-Shephard inequality for the volume of sections and projections of convex bodies.', '1908.06426-1-2-0': 'Finally, we also derive some new tight Grunbaum type inequalities, choosing a different center than in the corresponding inequalities of Makai-Martini, Fradelizi, and Stephen-Yaskin.', '1908.06426-1-2-1': 'These new inequalities will help in quantifying the improvement of the new volumetric inequalities derived before.', '1908.06426-1-3-0': "The classical Jensen's inequality [CITATION] states that if [MATH] is a probability space, then for any concave [MATH] and any [MATH]-integrable function [MATH], we have that [EQUATION] and moreover, equality holds if and only if either [MATH] is affine or [MATH] is independent of [MATH].", '1908.06426-1-4-0': 'Let [MATH] (resp. [MATH]) be the set of n-dimensional compact, convex (resp. 0-symmetric) sets.', '1908.06426-1-4-1': 'For any set [MATH], we denote by [MATH] the volume (or Lebesgue measure) of [MATH] computed in its affine hull [MATH], i.e., the smallest affine subspace containing [MATH].', '1908.06426-1-4-2': 'Let [MATH] be the Euclidean norm of [MATH], let [MATH] be the Euclidean unit ball of [MATH], and let [MATH] be its volume.', '1908.06426-1-4-3': 'The center of mass of [MATH] is the point [EQUATION].', '1908.06426-1-4-4': "A well-known consequence of Jensen's inequality is the following Hermite-Hadamard inequality: for any [MATH] and [MATH] concave, then [EQUATION] with equality sign if and only if [MATH] is affine.", '1908.06426-1-4-5': 'It was named after Hermite 1881 and Hadamard 1893 proved independently [REF] in the 1-dimensional case.', '1908.06426-1-4-6': 'See [CITATION] (and [CITATION] or [CITATION]) and the references on it for other historical considerations and a comprehensive and complete view of this type of inequalities.', '1908.06426-1-5-0': 'Let [MATH] be the set of i-dimensional linear subspaces in [MATH].', '1908.06426-1-5-1': 'For [MATH] and [MATH], let [MATH] be the orthogonal projection of [MATH] onto [MATH].', '1908.06426-1-5-2': 'Moreover, let [MATH] be the vectors of the canonical basis of [MATH].', '1908.06426-1-5-3': 'For every [MATH], let [MATH] be the linear hull of [MATH], and let [MATH] be the orthogonal subspace to [MATH], and let [MATH] be the boundary of [MATH].', '1908.06426-1-6-0': 'In 2017 during the conference Convex, Discrete and Integral Geometry Francisco Santos asked the following question: for every [MATH] such that [MATH], what is the smallest [MATH] such that [EQUATION].', '1908.06426-1-6-1': 'One of our aims is to compute this constant [MATH].', '1908.06426-1-6-2': 'The same inequality is derived in [CITATION] where it is used to bound the volume of empty lattice 4-simplices in terms of volumes of 3-lattice polytopes.', '1908.06426-1-6-3': "Notice that for every [MATH] and [MATH], Fubini's theorem implies that [EQUATION].", '1908.06426-1-6-4': 'There exist special subspaces for which the inequality above strengthens.', '1908.06426-1-6-5': 'In this regard Spingarn [CITATION] and later Milman and Pajor [CITATION] proved that if [MATH] and [MATH], then [EQUATION]', '1908.06426-1-6-6': 'It is even known the worst deviation between the maximal volume section and the one passing through the centroid of [MATH] (cf. [CITATION],[CITATION], and further extensions in [CITATION]).', '1908.06426-1-6-7': "Surprisingly enough, a consequence of Jensen's inequality [REF] shows that (cf. Theorem [REF] below) for every [MATH] and [MATH] then [EQUATION] and this choice can be sometimes better than [REF] (cf. Remarks [REF] and [REF]).", '1908.06426-1-6-8': 'In this regard, we prove the following result, extending the inequality above when [MATH] and answering to the question posed at the beginning.', '1908.06426-1-6-9': 'The result below can be sometimes better than [REF] (cf. Remark [REF]) up to a linear factor in the dimension of the subspace.', '1908.06426-1-7-0': 'Let [MATH] and [MATH] be such that [MATH].', '1908.06426-1-7-1': 'If we assume w.l.o.g. that [MATH], there is equality above if and only if there exist [MATH], and [MATH] such that [MATH], where [EQUATION] for every [MATH].', '1908.06426-1-7-2': 'If in addition [MATH], then there exist [MATH] and [MATH] such that [MATH] and that [EQUATION] for every [MATH].', '1908.06426-1-8-0': 'At the core of the proof of Theorem [REF] rests a generalization of [REF], which is the main result of the paper.', '1908.06426-1-8-1': 'We call it an Orlicz-Jensen-Hermite-Hadamard inequality, due to the similarities with the Orlicz-Brunn-Minkowski inequality [CITATION].', '1908.06426-1-8-2': 'In a sense, in this paper we highlight the intimate relation between Hermite-Hadamard inequalities and some Rogers-Shephard and Milman-Pajor type inequalities.', '1908.06426-1-9-0': 'Let [MATH], let [MATH] be concave, and let [MATH] be convex, non-decreasing be such that [MATH] and [MATH].', '1908.06426-1-9-1': 'If [MATH] is strictly convex and strictly monotonic, equality holds if and only after applying a suitable rotation then there exist [MATH] and [MATH] with [MATH] such that [EQUATION] and such that [MATH] is an affine function with [MATH], for every [MATH].', '1908.06426-1-10-0': 'Notice that the constant [MATH] in Theorem [REF] is typically larger than 1, as in the case [MATH], [MATH] (cf. Corollary [REF]).', '1908.06426-1-10-1': 'One can generally obtain a similar inequality with constant 1 at the cost of choosing a worse center.', '1908.06426-1-10-2': 'Milman and Pajor (see [CITATION]) proved that if [MATH] is an integrable log-concave function (i.e. [MATH] is concave), and [MATH] is a probability measure, then [EQUATION] and equality holds if and only if [MATH] is independent of [MATH].', '1908.06426-1-10-3': 'A direct consequence of this result is the following Hermite-Hadamard inequality: for any [MATH], [MATH] concave, and [MATH], then [EQUATION] where [MATH] is log-concave if [MATH] is concave).', '1908.06426-1-10-4': 'Notice that the point [MATH] depends on [MATH] and does not coincide in general with [MATH] (cf. Corollary [REF]).', '1908.06426-1-11-0': 'We find reverse inequalities to [REF] in the literature (see for instance [CITATION]) even integrated with respect to more general measures (cf. [CITATION]).', '1908.06426-1-11-1': 'Here we give a reverse inequality in the sense of Theorem [REF].', '1908.06426-1-11-2': 'Notice that for every [MATH], we let [MATH] be the convex hull of [MATH].', '1908.06426-1-11-3': 'Moreover, for every function [MATH], [MATH], we let [MATH] be the graph of [MATH].', '1908.06426-1-12-0': 'Let [MATH] with [MATH], let [MATH] be a concave function and let [MATH] be convex, strictly increasing function with [MATH] and [MATH].', '1908.06426-1-12-1': 'Moreover, equality holds if and only if [MATH].', '1908.06426-1-13-0': 'As a consequence of Theorem [REF] we will derive (cf. Section [REF]) the well-known reverse inequality to [REF] due to Rogers and Shephard [CITATION].', '1908.06426-1-13-1': 'It states that for every [MATH] with [MATH], and [MATH], then [EQUATION]', '1908.06426-1-13-2': 'One of the consequences of Rogers and Shephard inequality [REF] states that for any [MATH], then [EQUATION] cf. [CITATION], and see [CITATION] for its equality cases.', '1908.06426-1-13-3': 'This inequality is considered as the reverse inequality to the so-called Brunn-Minkowski inequality, which states that for any [MATH] and [MATH], then [EQUATION] and equality holds if and only if [MATH] and [MATH] are dilates, or if they are lower dimensional, then they must be contained in parallel hyperplanes.', '1908.06426-1-14-0': 'We split the proofs of the results into three sections.', '1908.06426-1-14-1': 'In Section [REF] we proof all the functional inequalities, i.e. Theorems [REF] and [REF].', '1908.06426-1-14-2': 'Afterwards in Section [REF] we show some applications, in particular, some volumetric inequalities solving in particular the question posed above.', '1908.06426-1-14-3': 'In Section [REF] we prove some Grunbaum type inequalities for a compact convex set [MATH] and a subspace passing through the centroid of an orthogonal projection of [MATH].', '1908.06426-1-14-4': 'With these results, we will quantify the improvements of the results in Section [REF] compared to previously known results.', '1908.06426-1-15-0': '# Proof of the Orlicz-Jensen-Hermite-Hadamard type inequalities', '1908.06426-1-16-0': 'Let us start this section by remembering that the Schwarz symmetrization of [MATH] with respect to [MATH], [MATH], is the set [EQUATION] where [MATH] is such that [MATH].', '1908.06426-1-16-1': 'It is well-known that [MATH] and that [MATH] (cf. [CITATION] or [CITATION] for more details).', '1908.06426-1-16-2': 'For every [MATH] and [MATH], the support function of [MATH] at [MATH] is defined by [MATH].', '1908.06426-1-17-0': '[Proof of Theorem [REF]] Since [MATH] is concave and non-negative in [MATH], then let [MATH] be an affine function such that [EQUATION].', '1908.06426-1-17-1': 'Since [MATH] is non-decreasing, then [EQUATION]', '1908.06426-1-17-2': 'Now let [MATH], where [MATH] is the graph of [MATH], and observe that [MATH] is an affine hyperplane in [MATH].', '1908.06426-1-17-3': 'Let us furthermore observe that since [MATH] then [MATH], and thus we let [MATH] be such that [EQUATION].', '1908.06426-1-17-4': 'After a suitable rotation, we can assume that [MATH], that [MATH], and that [EQUATION] for some [MATH] and some [MATH].', '1908.06426-1-17-5': 'Since [MATH] is 0-symmetric and [MATH] is affine, thus [MATH] too.', '1908.06426-1-17-6': 'Observe that [MATH], i.e., [MATH].', '1908.06426-1-18-0': 'Observe also that [MATH] gets constant value on each affine subspace [MATH], [MATH].', '1908.06426-1-18-1': 'Hence, if [MATH], let [EQUATION].', '1908.06426-1-18-2': "Using Fubini's formula we have that [EQUATION].", '1908.06426-1-19-0': 'Let us consider now [MATH].', '1908.06426-1-19-1': 'If we denote by [MATH] for every [MATH], then [EQUATION] and in particular [MATH].', '1908.06426-1-19-2': 'Moreover, we also have that [MATH] for every [MATH].', '1908.06426-1-19-3': 'Therefore [EQUATION].', '1908.06426-1-19-4': 'We now define the cylinders [EQUATION].', '1908.06426-1-19-5': 'Since [MATH] is 0-symmetric and convex, then [MATH].', '1908.06426-1-19-6': 'Moreover, [MATH] is a continuously decreasing family, and thus there exists [MATH] such that [MATH].', '1908.06426-1-19-7': 'Let [MATH] and let [MATH] for [MATH].', '1908.06426-1-19-8': 'Let us observe that since [MATH] and [MATH] are 0-symmetric and [MATH] and [MATH] are [MATH]-Euclidean balls centered at [MATH] then [EQUATION].', '1908.06426-1-19-9': 'We also observe that [MATH] implies that [MATH].', '1908.06426-1-19-10': 'Let us furthermore denote by [EQUATION].', '1908.06426-1-19-11': 'Then [EQUATION]', '1908.06426-1-19-12': 'We start bounding from above the simpler left integral in [REF], whose domain of integration is [MATH].', '1908.06426-1-19-13': 'Since [MATH] is a convex function, then [MATH] is convex too, from which we see that [EQUATION].', '1908.06426-1-20-0': 'Now we focus in bounding from above the right integral in [REF], whose domain is [MATH], partially using ideas from above.', '1908.06426-1-20-1': 'Using again that [MATH] is convex, then [MATH] is convex too, and thus [EQUATION]', '1908.06426-1-20-2': 'Once more since [MATH] is convex then [MATH] is convex too, and thus [EQUATION]', '1908.06426-1-20-3': 'These two upper bounds prove from [REF] that [EQUATION] where [MATH] is an affine function with [MATH] and [MATH] for every [MATH].', '1908.06426-1-20-4': 'Again by Fubini we now get that [EQUATION] concluding the proof of the inequality.', '1908.06426-1-21-0': 'For the equality case, let us suppose that [MATH] is strictly convex and strictly increasing.', '1908.06426-1-21-1': 'we must have equality in all inequalities above.', '1908.06426-1-21-2': 'Equality in [REF] together with the strict monotonicity of [MATH] implies that [MATH] must be an affine function.', '1908.06426-1-21-3': 'Equalities in [REF] together with the strict convexity of [MATH] force that [MATH] for every [MATH], i.e., [MATH] has to fulfill [EQUATION] for every [MATH] and some constant [MATH].', '1908.06426-1-21-4': 'Since [MATH], hence we also have that [EQUATION] and thus the equality case of Brunn-Minkowski inequality [REF] implies that [MATH] is a translation of the same [MATH]-dimensional set for every [MATH].', '1908.06426-1-21-5': 'This is equivalent to the fact that [EQUATION] where [MATH] and [MATH].', '1908.06426-1-21-6': 'Finally, equality in [REF] forces that [MATH], i.e., that [MATH] for every [MATH], which concludes the equality case.', '1908.06426-1-22-0': 'Notice that for any [MATH] then [EQUATION] attain equality in Theorem [REF] for every convex, non-decreasing function [MATH] with [MATH] and [MATH].', '1908.06426-1-23-0': 'Our first corollary follows from applying Theorem [REF] to [MATH].', '1908.06426-1-23-1': 'In the next section we will use it to give new estimates of the volume of a convex body in terms of the volumes of some of its sections and projections.', '1908.06426-1-24-0': 'Let [MATH], let [MATH] be concave, and let [MATH].', '1908.06426-1-24-1': 'Then [EQUATION].', '1908.06426-1-24-2': 'Equality holds if [MATH] is affine and if moreover [MATH], if and only if [MATH] is a generalized cylinder [MATH], for some [MATH] with [MATH], and such that [MATH] for every [MATH] with [MATH].', '1908.06426-1-25-0': 'Yet another corollary to Theorem [REF] is when we apply it to [MATH].', '1908.06426-1-26-0': 'Let [MATH] and let [MATH] be concave.', '1908.06426-1-26-1': 'Then [EQUATION].', '1908.06426-1-26-2': 'Equality holds if and only if [MATH] is a generalized cylinder [MATH], for some [MATH] with [MATH], and [MATH] is an affine function such that [MATH] for every [MATH] with [MATH].', '1908.06426-1-27-0': '[Proof of Theorem [REF]] Since [MATH] is a non-negative concave function, then the function [MATH] whose graph is the truncated cone with base at [MATH] and apex at [MATH] fulfills [MATH] for every [MATH] and [MATH].', '1908.06426-1-27-1': 'Since [MATH] is increasing, then [MATH] for every [MATH] and thus [EQUATION].', '1908.06426-1-27-2': 'Let us observe that if [MATH], for any [MATH] then [EQUATION].', '1908.06426-1-28-0': 'This means that for every [MATH], then [EQUATION].', '1908.06426-1-28-1': 'Observe that [MATH] is well-defined and continuous as [MATH] is strictly increasing and continuous on its interior as it is convex too.', '1908.06426-1-28-2': 'Thus [EQUATION] which concludes the proof.', '1908.06426-1-29-0': 'Since [MATH] is strictly increasing, there is equality above if and only if [MATH] coincides with [MATH], therefore concluding the equality case.', '1908.06426-1-30-0': 'A direct consequence of Theorem [REF] is the following result.', '1908.06426-1-31-0': 'Let [MATH] with [MATH], let [MATH] be a concave function, and let [MATH].', '1908.06426-1-31-1': 'Then [EQUATION].', '1908.06426-1-31-2': 'Moreover, equality holds if and only if [MATH]', '1908.06426-1-32-0': 'Applying Theorem [REF] to [MATH], we see that [EQUATION] concluding the proof.', '1908.06426-1-33-0': 'The equality holds the same way as in Theorem [REF].', '1908.06426-1-34-0': '# Estimating sizes of convex sets by their marginals', '1908.06426-1-35-0': 'We start this section by proving Theorem [REF] as a consequence of Corollary [REF].', '1908.06426-1-35-1': "[Proof of Theorem [REF]] By Fubini's formula, we have that [EQUATION].", '1908.06426-1-35-2': 'By Brunn-Minkowski inequality [REF] then [EQUATION] is a concave function.', '1908.06426-1-35-3': 'Assuming that [MATH], then Corollary [REF] implies that [EQUATION] concluding the result.', '1908.06426-1-36-0': 'For the equality case, we must have equality in Corollary [REF] where [MATH], [MATH], and [MATH].', '1908.06426-1-36-1': 'Hence, first of all, [MATH] must be an affine function.', '1908.06426-1-36-2': 'We hence can write [EQUATION] for some [MATH].', '1908.06426-1-36-3': 'This means in particular that [EQUATION].', '1908.06426-1-36-4': 'Hence, using Brunn-Minkowski equality case [REF], we have that [MATH] are dilates, of volume [EQUATION].', '1908.06426-1-36-5': 'Since [MATH] is convex, then there exists a matrix [MATH] of the form [EQUATION] where [MATH] and [MATH] is the identity matrix of [MATH], such that [EQUATION] with [EQUATION].', '1908.06426-1-36-6': 'Second, if [MATH], i.e. [MATH], then we moreover have that there exist [MATH] and [MATH] such that [MATH].', '1908.06426-1-36-7': 'Moreover, we must have also that [EQUATION] i.e., that [MATH].', '1908.06426-1-36-8': 'Once more since [MATH] is affine, this means that [EQUATION] i.e., that [EQUATION] thus concluding the equality case.', '1908.06426-1-37-0': 'For any [MATH], [MATH], the set [EQUATION] together with the subspace [MATH] achieves equality in Theorem [REF].', '1908.06426-1-38-0': 'We now properly state [REF] along with the characterization of its equality cases.', '1908.06426-1-38-1': 'Notice that here we do not require [MATH] to be 0-symmetric.', '1908.06426-1-39-0': 'If we assume w.l.o.g. that [MATH] and that [MATH], there is equality above if and only if there exist [MATH] and [MATH] such that [MATH], where [EQUATION] for every [MATH].', '1908.06426-1-40-0': 'Let us consider the function [EQUATION] which by Brunn-Minkowski inequality [REF], is a concave function.', '1908.06426-1-40-1': "Hence, using Fubini's formula and [REF] we directly obtain that [EQUATION] as desired.", '1908.06426-1-41-0': 'The equality case follows as the equality case of Theorem [REF].', '1908.06426-1-42-0': 'We now give two interesting observations out of Theorem [REF].', '1908.06426-1-43-0': 'Let us observe that Theorem [REF] sometimes gives a tighter inequality than [REF].', '1908.06426-1-43-1': 'Indeed, if we consider the cone [MATH] with apex at [MATH] and basis [MATH], and consider [MATH], it is straightforward to check that [EQUATION] and thus, that [EQUATION].', '1908.06426-1-44-0': 'One can combine two of those inequalities to show that any point in the line segment determined by two good choices of points (as in [REF]), is again a good choice.', '1908.06426-1-45-0': 'If for some [MATH] and [MATH] there exist points [MATH] such that [EQUATION] then, for every [MATH], the Brunn-Minkowski inequality [REF] gives that [EQUATION] i.e., all points [MATH] also fulfills the inequality [REF], [MATH].', '1908.06426-1-45-1': 'In particular, from Theorem [REF] and [REF] with [MATH], we obtain that [MATH] gives also an inequality of the same type.', '1908.06426-1-46-0': 'We finish this section by proving [REF] as a direct consequence of Corollary [REF].', '1908.06426-1-46-1': '[Proof of [REF]] Let us observe that Brunn-Minkowski inequality [REF] implies that [EQUATION] is a concave function.', '1908.06426-1-46-2': "Therefore Fubini's formula and Corollary [REF] imply that [EQUATION] proving the inequality.", '1908.06426-1-47-0': 'The equality case follows immediately as in the equality case of Theorem [REF].', '1908.06426-1-48-0': '# Grunbaum type inequalities through centroid of projections', '1908.06426-1-49-0': 'As observed above, replacing in [REF] [MATH] by [MATH] gives sometimes a better choice.', '1908.06426-1-49-1': 'To see this, remember that if [MATH] with [MATH] and [MATH], then Fradelizi [CITATION] (earlier Makai and Martini [CITATION] in the case [MATH]) proved that [EQUATION]', '1908.06426-1-49-2': 'However, if we translate [MATH] such that [MATH], then we obtain a much tighter estimate (cf. [REF] and Theorem [REF]).', '1908.06426-1-49-3': 'For any [MATH] we denote by [MATH] the relative boundary of [MATH].', '1908.06426-1-50-0': 'Let [MATH] and let [MATH], for some [MATH].', '1908.06426-1-50-1': 'Observe that [MATH].', '1908.06426-1-50-2': 'The convexity of [MATH] implies that [EQUATION] and Brunn-Minkowski inequality [REF] then implies that [EQUATION] as desired.', '1908.06426-1-51-0': 'If we have equality above, then we have equality in all inequalities.', '1908.06426-1-51-1': 'This means that there exists some [MATH] for which [MATH], [MATH] and [MATH].', '1908.06426-1-51-2': 'These three values together with Brunn-Minkowski inequality imply that [MATH] is an affine function on [MATH], ranging from value 0 at [MATH] to the maximum value at [MATH], and thus [EQUATION] for every [MATH].', '1908.06426-1-51-3': 'By the Brunn-Minkowski equality cases, we hence have that [MATH] are dilates for every [MATH], and thus there must exist [MATH] and another affine function [MATH], [MATH], such that [EQUATION] for every [MATH], as desired.', '1908.06426-1-52-0': 'We can construct very general examples of convex sets sharp in Theorem [REF].', '1908.06426-1-52-1': 'For instance, let [MATH], [MATH], and let [MATH].', '1908.06426-1-52-2': 'Then the set [EQUATION] attains equality in Theorem [REF] for the subspace [MATH].', '1908.06426-1-53-0': 'Using Theorem [REF] we see that if [MATH] is near the extreme cases (w.r.t. Hausdorff or Banach-Mazur distance), [EQUATION] which applied in Theorem [REF] gives us [EQUATION].', '1908.06426-1-53-1': 'If instead we consider [REF] we have that [EQUATION] for some [MATH], which applied in [REF] gives [EQUATION].', '1908.06426-1-53-2': 'Hence implying that Theorem [REF] essentially improves the choice of [REF] by the linear factor [MATH].', '1908.06426-1-54-0': 'For the sake of completeness, we also show the general inequality in the regard of Theorem [REF].', '1908.06426-1-54-1': 'In order to establish it, let us remember that Hammer [CITATION] proved that if [MATH] has [MATH] and [MATH] and [MATH], then [EQUATION]', '1908.06426-1-54-2': 'Let [MATH] and let [MATH].', '1908.06426-1-54-3': 'Moreover, equality holds if and only if after a suitable rigid motion, there exist [MATH], [MATH], [MATH], [MATH], and [MATH], such that [MATH] with [MATH], and such that [MATH] with [EQUATION] for every [MATH].', '1908.06426-1-55-0': 'Let us suppose after a translation of [MATH] that [MATH].', '1908.06426-1-55-1': 'If [MATH] and if [MATH], by [REF] we have that [MATH].', '1908.06426-1-56-0': 'Observe that [MATH].', '1908.06426-1-56-1': 'The convexity of [MATH] implies that [EQUATION] and Brunn-Minkowski inequality [REF] then implies that [EQUATION] as desired.', '1908.06426-1-57-0': 'The equality case follows proceeding as in the equality case of Theorem [REF].', '1908.06426-1-58-0': 'Using Theorem [REF] for some [MATH] we see that if [MATH] is near the extreme cases (in Hausdorff or Banach-Mazur distance), [EQUATION] which applied in Theorem [REF] gives us [EQUATION].', '1908.06426-1-58-1': 'If instead we consider [REF] we have that [EQUATION] for some [MATH], which applied in [REF] gives [EQUATION].', '1908.06426-1-58-2': 'Thus showing that Theorem [REF] improves upon the choice of [REF] by the linear factor [MATH].'}

{'1908.06426-2-0-0': 'In this paper we show an Orlicz-Jensen-Hermite-Hadamard inequality and a reverse to that inequality.', '1908.06426-2-0-1': 'This establishes, in particular, one of the first multidimensional Hermite-Hadamard inequality in this generality.', '1908.06426-2-0-2': 'We then show several consequences of those results.', '1908.06426-2-1-0': 'Using the direct version of the inequality above, we derive a Milman-Pajor-Spingarn type inequality, which in some sense, improves the previous versions.', '1908.06426-2-1-1': 'Moreover, as a particular case we answer a question posed by Francisco Santos on upper bounds of the volume of a convex body in terms of the volume of one of its central sections.', '1908.06426-2-1-2': 'Using the reverse version, we derive a generalization of a result by Rogers and Shephard for the volume of sections and projections of convex bodies.', '1908.06426-2-2-0': 'Finally, we also derive some new tight Grunbaum type inequalities, choosing a different center than in the corresponding inequalities of Makai-Martini, Fradelizi, and Stephen-Yaskin.', '1908.06426-2-2-1': 'These new inequalities will help in quantifying the improvement of the new volumetric inequalities derived before.', '1908.06426-2-3-0': "The classical Jensen's inequality [CITATION] states that if [MATH] is a probability space, then for any concave [MATH] and any [MATH]-integrable function [MATH], we have that [EQUATION] and moreover, equality holds if and only if either [MATH] is affine or [MATH] is independent of [MATH].", '1908.06426-2-4-0': 'Let [MATH] (resp. [MATH]) be the set of n-dimensional compact, convex (resp. 0-symmetric) sets.', '1908.06426-2-4-1': 'For any set [MATH], we denote by [MATH] the volume (or Lebesgue measure) of [MATH] computed in its affine hull [MATH], i.e., the smallest affine subspace containing [MATH].', '1908.06426-2-4-2': 'Let [MATH] be the Euclidean norm of [MATH], let [MATH] be the Euclidean unit ball of [MATH], and let [MATH] be its volume.', '1908.06426-2-4-3': 'The center of mass of [MATH] is the point [EQUATION].', '1908.06426-2-4-4': "A well-known consequence of Jensen's inequality is the following Hermite-Hadamard inequality: for any [MATH] and [MATH] concave, then [EQUATION] with equality sign if and only if [MATH] is affine.", '1908.06426-2-4-5': 'It was named after Hermite 1881 and Hadamard 1893, who proved independently [REF] in the 1-dimensional case.', '1908.06426-2-4-6': 'See [CITATION] (and [CITATION] or [CITATION]) and the references on it for other historical considerations and a comprehensive and complete view of this type of inequalities.', '1908.06426-2-4-7': 'The mean value of [MATH] measured in [MATH] (the left-term in [REF]) has repeatedly appeared during the development of different topics of Analysis and Geometry (cf. [CITATION]), for instance, in the Hardy-Littlewood Maximal Function (cf. [CITATION]).', '1908.06426-2-4-8': 'Notice that the Mean Value Theorem ensures the existence of a point [MATH] (which depends on [MATH]) such that [MATH] whenever [MATH] is continuous.', '1908.06426-2-5-0': 'Let [MATH] be the set of i-dimensional linear subspaces in [MATH].', '1908.06426-2-5-1': 'For [MATH] and [MATH], let [MATH] be the orthogonal projection of [MATH] onto [MATH].', '1908.06426-2-5-2': 'Moreover, let [MATH] be the vectors of the canonical basis of [MATH].', '1908.06426-2-5-3': 'For every [MATH], let [MATH] be the linear hull of [MATH], and let [MATH] be the orthogonal subspace to [MATH], and let [MATH] be the boundary of [MATH].', '1908.06426-2-6-0': 'In 2017 during the conference Convex, Discrete and Integral Geometry Francisco Santos asked the following question: for every [MATH] such that [MATH], what is the smallest [MATH] such that [EQUATION].', '1908.06426-2-6-1': 'One of our aims is to compute this constant [MATH].', '1908.06426-2-6-2': 'A similar inequality is derived in [CITATION] where it is used to bound the volume of empty lattice 4-simplices in terms of volumes of 3-lattice polytopes.', '1908.06426-2-6-3': "Notice that for every [MATH] and [MATH], Fubini's theorem implies that [EQUATION].", '1908.06426-2-6-4': 'There exist special subspaces for which the inequality above strengthens.', '1908.06426-2-6-5': 'In this regard Spingarn [CITATION] and later Milman and Pajor [CITATION] proved that if [MATH] and [MATH], then [EQUATION]', '1908.06426-2-6-6': 'It is even known the worst deviation between the maximal volume section and the one passing through the centroid of [MATH] (cf. [CITATION],[CITATION], and further extensions in [CITATION]).', '1908.06426-2-6-7': "Surprisingly enough, a consequence of Jensen's inequality [REF] shows that (cf. Theorem [REF] below) for every [MATH] and [MATH] then [EQUATION] and this choice can be sometimes better than [REF] (cf. Remarks [REF] and [REF]).", '1908.06426-2-6-8': 'In this regard, we prove the following result, extending the inequality above when [MATH] and answering to the question posed at the beginning.', '1908.06426-2-6-9': 'The result below can be sometimes better than [REF] (cf. Remark [REF]) up to a linear factor in the dimension of the subspace.', '1908.06426-2-7-0': 'Let [MATH] and [MATH] be such that [MATH].', '1908.06426-2-7-1': 'If we assume w.l.o.g. that [MATH], there is equality above if and only if there exist [MATH], and [MATH] such that [MATH], where [EQUATION] for every [MATH].', '1908.06426-2-7-2': 'If in addition [MATH], then there exist [MATH] and [MATH] such that [MATH] and that [EQUATION] for every [MATH].', '1908.06426-2-8-0': 'At the core of the proof of Theorem [REF] rests a generalization of [REF], which is the main result of the paper.', '1908.06426-2-8-1': 'We call it an Orlicz-Jensen-Hermite-Hadamard inequality, due to the similarities with the Orlicz-Brunn-Minkowski inequality [CITATION].', '1908.06426-2-8-2': 'In a sense, in this paper we highlight the intimate relation between Hermite-Hadamard inequalities and some Rogers-Shephard and Milman-Pajor type inequalities.', '1908.06426-2-9-0': 'Let [MATH], let [MATH] be concave, and let [MATH] be convex, non-decreasing such that [MATH] and [MATH].', '1908.06426-2-9-1': 'If [MATH] is strictly convex and strictly monotonic, equality holds if and only after applying a suitable rotation then there exist [MATH] and [MATH] with [MATH] such that [EQUATION] and such that [MATH] is an affine function with [MATH], for every [MATH].', '1908.06426-2-10-0': 'Notice that, due to the convexity of [MATH], the term [MATH] in Theorem [REF] is larger than [MATH], for some constant [MATH], as in the case [MATH], [MATH] (cf. Corollary [REF]).', '1908.06426-2-10-1': 'One can generally obtain a similar inequality with constant 1 at the cost of choosing a worse center.', '1908.06426-2-10-2': 'Milman and Pajor (see [CITATION]) proved that if [MATH] is an integrable log-concave function (i.e. [MATH] is concave), and [MATH] is a probability measure, then [EQUATION] and equality holds if and only if [MATH] is independent of [MATH].', '1908.06426-2-10-3': 'A direct consequence of this result is the following Hermite-Hadamard inequality: for any [MATH], [MATH] concave, and [MATH], then [EQUATION] where [MATH] is log-concave if [MATH] is concave).', '1908.06426-2-10-4': 'Notice that the point [MATH] depends on [MATH] and does not coincide in general with [MATH] (cf. Corollary [REF] and compare this to [REF]).', '1908.06426-2-11-0': 'Using Theorem [REF] we also derive a Hermite-Hadamard inequality as in [REF] evaluated at the center of mass of the domain.', '1908.06426-2-11-1': 'Notice that if [MATH], since [MATH], the right-term below becomes [MATH].', '1908.06426-2-12-0': 'Let [MATH] and let [MATH] be log-concave.', '1908.06426-2-12-1': 'Equality holds if and only after applying a suitable rotation there exist [MATH] and [MATH] with [MATH] such that [EQUATION] and if moreover [MATH], with [MATH] concave function, then [MATH] is an affine function with [MATH], for every [MATH].', '1908.06426-2-13-0': 'We find reverse inequalities to [REF] in the literature (see for instance [CITATION]) even integrated with respect to more general measures (cf. [CITATION]).', '1908.06426-2-13-1': 'Here we give a reverse inequality in the sense of Theorem [REF].', '1908.06426-2-13-2': 'Notice that for every [MATH], we let [MATH] be the convex hull of [MATH].', '1908.06426-2-13-3': 'Moreover, for every function [MATH], [MATH], we let [MATH] be the graph of [MATH].', '1908.06426-2-14-0': 'Let [MATH] with [MATH], let [MATH] be a concave function and let [MATH] be convex, strictly increasing function with [MATH] and [MATH].', '1908.06426-2-14-1': 'Moreover, equality holds if and only if [MATH].', '1908.06426-2-15-0': 'As a consequence of Theorem [REF] we will derive (cf. Section [REF]) the well-known reverse inequality to [REF] due to Rogers and Shephard [CITATION].', '1908.06426-2-15-1': 'It states that for every [MATH] with [MATH], and [MATH], then [EQUATION]', '1908.06426-2-15-2': 'One of the consequences of Rogers and Shephard inequality [REF] states that for any [MATH], then [EQUATION] cf. [CITATION], and see [CITATION] for its equality cases.', '1908.06426-2-15-3': 'This inequality is considered as the reverse inequality to the so-called Brunn-Minkowski inequality, which states that for any [MATH] and [MATH], then [EQUATION] and equality holds if and only if [MATH] and [MATH] are dilates, or if they are lower dimensional, then they must be contained in parallel hyperplanes (see [CITATION] and the references therein for an insightful and complete study of this inequality).', '1908.06426-2-16-0': 'As another consequence of Theorem [REF] we derive a reverse inequality to the one proven in Theorem [REF].', '1908.06426-2-17-0': 'Let [MATH] with [MATH], and let [MATH] be a log-concave function.', '1908.06426-2-17-1': 'If [MATH] with [MATH], equality holds if and only if [MATH].', '1908.06426-2-18-0': 'We split the proofs of the results into three sections.', '1908.06426-2-18-1': 'In Section [REF] we prove all the functional inequalities, i.e. Theorems [REF] and [REF].', '1908.06426-2-18-2': 'From those results we derive some consequences, such as Theorems [REF] and [REF].', '1908.06426-2-18-3': 'Afterwards in Section [REF] we show further applications, in particular, some volumetric inequalities solving in particular the question posed above.', '1908.06426-2-18-4': 'In Section [REF] we prove some Grunbaum type inequalities for a compact convex set [MATH] and a subspace passing through the centroid of an orthogonal projection of [MATH].', '1908.06426-2-18-5': 'With these results, we will quantify the improvements of the results in Section [REF] compared to previously known results.', '1908.06426-2-19-0': '# Proof of the Orlicz-Jensen-Hermite-Hadamard type inequalities', '1908.06426-2-20-0': 'Let us start this section by remembering that the Schwarz symmetrization of [MATH] with respect to [MATH], [MATH], is the set [EQUATION] where [MATH] is such that [MATH].', '1908.06426-2-20-1': 'It is well-known that [MATH] and that [MATH] (cf. [CITATION] or [CITATION] for more details).', '1908.06426-2-20-2': 'For every [MATH] and [MATH], the support function of [MATH] at [MATH] is defined by [MATH].', '1908.06426-2-21-0': '[Proof of Theorem [REF]] Since [MATH] is concave and non-negative in [MATH], then let [MATH] be an affine function such that [EQUATION].', '1908.06426-2-21-1': 'Since [MATH] is non-decreasing, then [EQUATION]', '1908.06426-2-21-2': 'Now let [MATH], where [MATH] is the graph of [MATH], and observe that [MATH] is an affine hyperplane in [MATH].', '1908.06426-2-21-3': 'Let us furthermore observe that since [MATH] then [MATH], and thus we let [MATH] be such that [EQUATION].', '1908.06426-2-21-4': 'After a suitable rotation, we can assume that [MATH], that [MATH], and that [EQUATION] for some [MATH] and some [MATH].', '1908.06426-2-21-5': 'Since [MATH] is 0-symmetric and [MATH] is affine, thus [MATH] too.', '1908.06426-2-21-6': 'Observe that [MATH], i.e., [MATH].', '1908.06426-2-22-0': 'Observe also that [MATH] gets constant value on each affine subspace [MATH], [MATH].', '1908.06426-2-22-1': 'Hence, if [MATH], let [EQUATION].', '1908.06426-2-22-2': "Using Fubini's formula we have that [EQUATION].", '1908.06426-2-23-0': 'Let us consider now [MATH].', '1908.06426-2-23-1': 'If we denote by [MATH] for every [MATH], then [EQUATION] and in particular [MATH].', '1908.06426-2-23-2': 'Moreover, we also have that [MATH] for every [MATH].', '1908.06426-2-23-3': 'Therefore [EQUATION].', '1908.06426-2-23-4': 'We now define the cylinders [EQUATION].', '1908.06426-2-23-5': 'Since [MATH] is 0-symmetric and convex, then [MATH].', '1908.06426-2-23-6': 'Moreover, [MATH] is a continuously decreasing family, and thus there exists [MATH] such that [MATH].', '1908.06426-2-23-7': 'Let [MATH] and let [MATH] for [MATH].', '1908.06426-2-23-8': 'Let us observe that since [MATH] and [MATH] are 0-symmetric and [MATH] and [MATH] are [MATH]-Euclidean balls centered at [MATH] then [EQUATION].', '1908.06426-2-23-9': 'We also observe that [MATH] implies that [MATH].', '1908.06426-2-23-10': 'Let us furthermore denote by [EQUATION].', '1908.06426-2-23-11': 'Then [EQUATION]', '1908.06426-2-23-12': 'We start bounding from above the simpler left integral in [REF], whose domain of integration is [MATH].', '1908.06426-2-23-13': 'Since [MATH] is a convex function, then [MATH] is convex too, from which we see that [EQUATION].', '1908.06426-2-24-0': 'Now we focus in bounding from above the right integral in [REF], whose domain is [MATH], partially using ideas from above.', '1908.06426-2-24-1': 'Using again that [MATH] is convex, then [MATH] is convex too, and thus [EQUATION]', '1908.06426-2-24-2': 'Once more since [MATH] is convex then [MATH] is convex too, and thus [EQUATION]', '1908.06426-2-24-3': 'These two upper bounds prove from [REF] that [EQUATION] where [MATH] is an affine function with [MATH] and [MATH] for every [MATH].', '1908.06426-2-24-4': 'Again by Fubini we now get that [EQUATION] concluding the proof of the inequality.', '1908.06426-2-25-0': 'For the equality case, let us suppose that [MATH] is strictly convex and strictly increasing.', '1908.06426-2-25-1': 'we must have equality in all inequalities above.', '1908.06426-2-25-2': 'Equality in [REF] together with the strict monotonicity of [MATH] implies that [MATH] must be an affine function.', '1908.06426-2-25-3': 'Equalities in [REF] together with the strict convexity of [MATH] force that [MATH] for every [MATH], i.e., [MATH] has to fulfill [EQUATION] for every [MATH] and some constant [MATH].', '1908.06426-2-25-4': 'Since [MATH], hence we also have that [EQUATION] and thus the equality case of Brunn-Minkowski inequality [REF] implies that [MATH] is a translation of the same [MATH]-dimensional set for every [MATH].', '1908.06426-2-25-5': 'This is equivalent to the fact that [EQUATION] where [MATH] and [MATH].', '1908.06426-2-25-6': 'Finally, equality in [REF] forces that [MATH], i.e., that [MATH] for every [MATH], which concludes the equality case.', '1908.06426-2-26-0': 'Notice that for any [MATH] then [EQUATION] attain equality in Theorem [REF] for every convex, non-decreasing function [MATH] with [MATH] and [MATH].', '1908.06426-2-27-0': 'Our first corollary follows from applying Theorem [REF] to [MATH].', '1908.06426-2-27-1': 'In the next section we will use it to give new estimates of the volume of a convex body in terms of the volumes of some of its sections and projections.', '1908.06426-2-28-0': 'Let [MATH], let [MATH] be concave, and let [MATH].', '1908.06426-2-28-1': 'Then [EQUATION].', '1908.06426-2-28-2': 'Equality holds if [MATH] is affine and if moreover [MATH], if and only if [MATH] is a generalized cylinder [MATH], for some [MATH] with [MATH], and such that [MATH] for every [MATH] with [MATH].', '1908.06426-2-29-0': 'Yet another corollary to Theorem [REF] is when we apply it to [MATH].', '1908.06426-2-29-1': '[Proof of Theorem [REF]] Let [MATH], with [MATH] a concave function.', '1908.06426-2-29-2': 'Applying Theorem [REF] to the function [MATH] in [MATH] and to [MATH] we obtain that [EQUATION] which shows the result.', '1908.06426-2-30-0': 'Since [MATH] is strictly convex and strictly increasing, the equality case follows immediately from the equality case of Theorem [REF].', '1908.06426-2-31-0': '[Proof of Theorem [REF]] Since [MATH] is a non-negative concave function, then the function [MATH] whose graph is the truncated cone with base at [MATH] and apex at [MATH] fulfills [MATH] for every [MATH] and [MATH].', '1908.06426-2-31-1': 'Since [MATH] is increasing, then [MATH] for every [MATH] and thus [EQUATION].', '1908.06426-2-31-2': 'Let us observe that if [MATH], for any [MATH] then [EQUATION].', '1908.06426-2-32-0': 'This means that for every [MATH], then [EQUATION].', '1908.06426-2-32-1': 'Observe that [MATH] is well-defined and continuous as [MATH] is strictly increasing and continuous on its interior as it is convex too.', '1908.06426-2-32-2': 'Thus [EQUATION] which concludes the proof.', '1908.06426-2-33-0': 'Since [MATH] is strictly increasing, there is equality above if and only if [MATH] coincides with [MATH], therefore concluding the equality case.', '1908.06426-2-34-0': 'A direct consequence of Theorem [REF] is the following result.', '1908.06426-2-35-0': 'Let [MATH] with [MATH], let [MATH] be a concave function, and let [MATH].', '1908.06426-2-35-1': 'Then [EQUATION].', '1908.06426-2-35-2': 'Moreover, equality holds if and only if [MATH]', '1908.06426-2-36-0': 'Applying Theorem [REF] to [MATH], we see that [EQUATION] concluding the proof.', '1908.06426-2-37-0': 'The equality holds the same way as in Theorem [REF].', '1908.06426-2-38-0': 'A second consequence of Theorem [REF] is Theorem [REF].', '1908.06426-2-38-1': 'In order to prove it, we use the following lemma.', '1908.06426-2-39-0': 'For every [MATH] and [MATH], we have that [EQUATION].', '1908.06426-2-40-0': 'Since [EQUATION] thus [EQUATION].', '1908.06426-2-40-1': 'This implies recursively that [EQUATION] i.e., in general that [EQUATION] as desired.', '1908.06426-2-41-0': '[Proof of Theorem [REF]] Let [MATH] be such that [MATH].', '1908.06426-2-41-1': 'Taking [MATH], Theorem [REF] states that [EQUATION].', '1908.06426-2-41-2': 'Using Lemma [REF] we thus get that [EQUATION] as desired.', '1908.06426-2-42-0': 'Since [MATH] is convex and strictly increasing, the equality case follows immediately from the equality case of Theorem [REF].', '1908.06426-2-43-0': '# Estimating sizes of convex sets by their marginals', '1908.06426-2-44-0': 'We start this section by proving Theorem [REF] as a consequence of Corollary [REF].', '1908.06426-2-44-1': "[Proof of Theorem [REF]] By Fubini's formula, we have that [EQUATION].", '1908.06426-2-44-2': 'By Brunn-Minkowski inequality [REF] then [EQUATION] is a concave function.', '1908.06426-2-44-3': 'After a suitable rigid motion, we assume that [MATH].', '1908.06426-2-44-4': 'Corollary [REF] then implies that [EQUATION] concluding the result.', '1908.06426-2-45-0': 'For the equality case, we must have equality in Corollary [REF] where [MATH], [MATH], and [MATH].', '1908.06426-2-45-1': 'Hence, first of all, [MATH] must be an affine function.', '1908.06426-2-45-2': 'We hence can write [EQUATION] for some [MATH].', '1908.06426-2-45-3': 'This means in particular that [EQUATION].', '1908.06426-2-45-4': 'Hence, using Brunn-Minkowski equality case [REF], we have that [MATH] are dilates, of volume [EQUATION].', '1908.06426-2-45-5': 'Since [MATH] is convex, then there exists a matrix [MATH] of the form [EQUATION] where [MATH] and [MATH] is the identity matrix of [MATH], such that [EQUATION] with [EQUATION].', '1908.06426-2-45-6': 'Second, if [MATH], i.e. [MATH], then we moreover have that there exist [MATH] and [MATH] such that [MATH].', '1908.06426-2-45-7': 'Moreover, we must have also that [EQUATION] i.e., that [MATH].', '1908.06426-2-45-8': 'Once more since [MATH] is affine, this means that [EQUATION] i.e., that [EQUATION] thus concluding the equality case.', '1908.06426-2-46-0': 'For any [MATH], [MATH], the set [EQUATION] together with the subspace [MATH] achieves equality in Theorem [REF].', '1908.06426-2-47-0': 'We now properly state [REF] along with the characterization of its equality cases.', '1908.06426-2-47-1': 'Notice that here we do not require [MATH] to be 0-symmetric.', '1908.06426-2-48-0': 'If we assume w.l.o.g. that [MATH] and that [MATH], there is equality above if and only if there exist [MATH] and [MATH] such that [MATH], where [EQUATION] for every [MATH].', '1908.06426-2-49-0': 'Let us consider the function [EQUATION] which by Brunn-Minkowski inequality [REF], is a concave function.', '1908.06426-2-49-1': "Hence, using Fubini's formula and [REF] we directly obtain that [EQUATION] as desired.", '1908.06426-2-50-0': 'The equality case follows as the equality case of Theorem [REF].', '1908.06426-2-51-0': 'We now give two interesting observations out of Theorem [REF].', '1908.06426-2-52-0': 'Let us observe that Theorem [REF] sometimes gives a tighter inequality than [REF].', '1908.06426-2-52-1': 'Indeed, if we consider the cone [MATH] with apex at [MATH] and basis [MATH], and consider [MATH], it is straightforward to check that [MATH], that [EQUATION] and thus, since [MATH], then [EQUATION].', '1908.06426-2-53-0': 'One can combine two of those inequalities to show that any point in the line segment determined by two good choices of points (as in [REF]), is again a good choice.', '1908.06426-2-54-0': 'If for some [MATH] and [MATH] there exist points [MATH] such that [EQUATION] then, for every [MATH], the Brunn-Minkowski inequality [REF] gives that [EQUATION] i.e., all points [MATH] also fulfills the inequality [REF], [MATH].', '1908.06426-2-54-1': 'In particular, from Theorem [REF] and [REF] with [MATH], we obtain that [MATH] gives also an inequality of the same type.', '1908.06426-2-55-0': 'We finish this section by proving [REF] as a direct consequence of Corollary [REF].', '1908.06426-2-55-1': '[Proof of [REF]] Let us observe that Brunn-Minkowski inequality [REF] implies that [EQUATION] is a concave function.', '1908.06426-2-55-2': "Therefore Fubini's formula and Corollary [REF] imply that [EQUATION] proving the inequality.", '1908.06426-2-56-0': 'The equality case follows immediately as in the equality case of Theorem [REF].', '1908.06426-2-57-0': '# Grunbaum type inequalities through centroid of projections', '1908.06426-2-58-0': 'As observed above, replacing in [REF] [MATH] by [MATH] gives sometimes a better choice.', '1908.06426-2-58-1': 'To see this, remember that if [MATH] with [MATH] and [MATH], then Fradelizi [CITATION] (earlier Makai and Martini [CITATION] in the case [MATH]) proved that [EQUATION]', '1908.06426-2-58-2': 'However, if we translate [MATH] such that [MATH], then we obtain a much tighter estimate (cf. [REF] and Theorem [REF]).', '1908.06426-2-58-3': 'For any [MATH] we denote by [MATH] the relative boundary of [MATH].', '1908.06426-2-59-0': 'Let [MATH] and let [MATH], for some [MATH].', '1908.06426-2-59-1': 'Observe that [MATH].', '1908.06426-2-59-2': 'The convexity of [MATH] implies that [EQUATION] and Brunn-Minkowski inequality [REF] then implies that [EQUATION] as desired.', '1908.06426-2-60-0': 'If we have equality above, then we have equality in all inequalities.', '1908.06426-2-60-1': 'This means that there exists some [MATH] for which [MATH], [MATH] and [MATH].', '1908.06426-2-60-2': 'These three values together with Brunn-Minkowski inequality imply that [MATH] is an affine function on [MATH], ranging from value 0 at [MATH] to the maximum value at [MATH], and thus [EQUATION] for every [MATH].', '1908.06426-2-60-3': 'By the Brunn-Minkowski equality cases, we hence have that [MATH] are dilates for every [MATH], and thus there must exist [MATH] and another affine function [MATH], [MATH], such that [EQUATION] for every [MATH], as desired.', '1908.06426-2-61-0': 'We can construct very general examples of convex sets sharp in Theorem [REF].', '1908.06426-2-61-1': 'For instance, let [MATH], [MATH], and let [MATH].', '1908.06426-2-61-2': 'Then the set [EQUATION] attains equality in Theorem [REF] for the subspace [MATH].', '1908.06426-2-62-0': 'Using Theorem [REF] we see that if [MATH] is near the extreme cases (w.r.t. Hausdorff or Banach-Mazur distance), [EQUATION] which applied in Theorem [REF] gives us [EQUATION].', '1908.06426-2-62-1': 'If instead we consider [REF] we have that [EQUATION] for some [MATH], which applied in [REF] gives [EQUATION].', '1908.06426-2-62-2': 'Hence implying that Theorem [REF] essentially improves the choice of [REF] by the linear factor [MATH].', '1908.06426-2-63-0': 'For the sake of completeness, we also show the general inequality in the regard of Theorem [REF].', '1908.06426-2-63-1': 'In order to establish it, let us remember that Hammer [CITATION] proved that if [MATH] has [MATH] and [MATH] and [MATH], then [EQUATION]', '1908.06426-2-63-2': 'Let [MATH] and let [MATH].', '1908.06426-2-63-3': 'Moreover, equality holds if and only if after a suitable rigid motion, there exist [MATH], [MATH], [MATH], [MATH], and [MATH], such that [MATH] with [MATH], and such that [MATH] with [EQUATION] for every [MATH].', '1908.06426-2-64-0': 'Let us suppose after a translation of [MATH] that [MATH].', '1908.06426-2-64-1': 'If [MATH] and if [MATH], by [REF] we have that [MATH].', '1908.06426-2-65-0': 'Observe that [MATH].', '1908.06426-2-65-1': 'The convexity of [MATH] implies that [EQUATION] and Brunn-Minkowski inequality [REF] then implies that [EQUATION] as desired.', '1908.06426-2-66-0': 'The equality case follows proceeding as in the equality case of Theorem [REF].', '1908.06426-2-67-0': 'Using Theorem [REF] for some [MATH] we see that if [MATH] is near the extreme cases (in Hausdorff or Banach-Mazur distance), [EQUATION] which applied in Theorem [REF] gives us [EQUATION].', '1908.06426-2-67-1': 'If instead we consider [REF] we have that [EQUATION] for some [MATH], which applied in [REF] gives [EQUATION].', '1908.06426-2-67-2': 'Thus showing that Theorem [REF] improves upon the choice of [REF] by the linear factor [MATH].'}

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[]

[['1908.06426-1-10-0', '1908.06426-2-10-0']]

[['1908.06426-1-26-0', '1908.06426-2-12-0']]

['1908.06426-1-17-6', '1908.06426-1-18-1', '1908.06426-1-19-3', '1908.06426-1-19-11', '1908.06426-1-24-1', '1908.06426-1-26-1', '1908.06426-1-30-0', '1908.06426-1-31-1', '1908.06426-1-33-0', '1908.06426-1-42-0', '1908.06426-1-50-1', '1908.06426-1-56-0', '1908.06426-2-21-6', '1908.06426-2-22-1', '1908.06426-2-23-3', '1908.06426-2-23-11', '1908.06426-2-28-1', '1908.06426-2-34-0', '1908.06426-2-35-1', '1908.06426-2-37-0', '1908.06426-2-39-0', '1908.06426-2-51-0', '1908.06426-2-59-1', '1908.06426-2-65-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1908.06426

1703.02280

{'1703.02280-1-0-0': 'We employ a weak measurement approach to demonstrate the very existence of the photoexcited interband superposition states in intrinsic graphene.', '1703.02280-1-0-1': 'We propose an optical two-beam setup where such measurements are possible and derive an explicit formula for the differential optical absorption that contains a signature of such states.', '1703.02280-1-0-2': 'We provide an interpretation of our results in terms of a non-Markovian weak measurement formalism applied to the pseudospin degree of freedom coupled with an electromagnetic wave.', '1703.02280-1-1-0': 'Introduction.', '1703.02280-1-1-1': '- Quantum coherence is among the most subtle concepts in quantum mechanics: It cannot be directly observed because any superposition state collapses immediately to an eigenstate of the operator corresponding to the observable one is trying to measure [CITATION].', '1703.02280-1-1-2': 'In attempts to circumvent the projection postulate, the concept of weak measurements has been introduced nearly 30 years ago [CITATION].', '1703.02280-1-1-3': 'To perform a weak measurement [CITATION] of a quantum state [MATH] the detector must also represent a quantum mechanical object described by a wave function [MATH].', '1703.02280-1-1-4': "The measured system interacts with the detector, and it is the detector's wave function [MATH] that is collapsed by a strong measurement.", '1703.02280-1-1-5': 'Due to the interaction between these two states the outcome of the measurement on [MATH] contains some information about coherence of the original quantum state [MATH].', '1703.02280-1-1-6': 'In the past decade several experimental attempts have been made to measure coherent superpositions with photons [CITATION] and, more recently, with nuclear spins [CITATION].', '1703.02280-1-1-7': 'Despite theoretical efforts [CITATION] the interpretation of weak measurements remains controversial [CITATION].', '1703.02280-1-2-0': 'In this Letter, we shift the focus of this problem to photocarriers in semiconductors, namely, we propose an experiment to measure the photoexcited interband (pseudospin) superposition states in intrinsic graphene.', '1703.02280-1-2-1': 'Our quantum steady state is created by linearly polarized monochromatic radiation and described by a 2[MATH]2 photocarrier density matrix [MATH] including non-zero coherences representing interband superpositions, see Fig. [REF].', '1703.02280-1-2-2': 'The probe radiation in turn creates a density matrix [MATH] that depends on [MATH] but is collapsed by a strong measurement of the photocarrier occupation.', '1703.02280-1-2-3': 'We show that the differential relative absorption of the probe radiation, see Fig. [REF], contains a certain quantum information about initial superposition states encoded in [MATH].', '1703.02280-1-3-0': 'Model in depth.', '1703.02280-1-3-1': '- The effective two-level system shown in Fig. [REF] is described by the two-dimensional massless Dirac-like Hamiltonian [CITATION] [MATH], where [MATH] is the two-component momentum operator, [MATH] is the pseudospin operator constructed from the Pauli matrices, and [MATH] ms[MATH] is the carrier velocity in graphene.', '1703.02280-1-3-2': 'The corresponding eigenvalues and eigenstates are [MATH] and [MATH] with [MATH].', '1703.02280-1-3-3': 'The index [MATH]) stands for the conduction (valence) band respectively, see Fig. [REF].', '1703.02280-1-3-4': 'The pseudospin is parallel (antiparallel) to momentum while being in a conduction (valence) band eigenstate.', '1703.02280-1-3-5': 'Hence, a certain interband coherent state [MATH] we are trying to measure represents, at the same time, a pseudospin superposition state.', '1703.02280-1-4-0': 'In order to create and measure such a superposition state we employ an electromagnetic wave [MATH] with the frequency [MATH] and amplitude [MATH] linearly-polarized in the direction [MATH].', '1703.02280-1-4-1': 'The equilibrium carriers are then perturbed by the field-pseudospin interaction [CITATION] [MATH], where [MATH] with [MATH] being the electron charge.', '1703.02280-1-4-2': 'We assume normal incidence without momentum transfer from photons to electrons so that we always superimpose the states [MATH] with the same momentum.', '1703.02280-1-4-3': 'It is convenient to define the coupling parameter [MATH] that entangles the electric field and the pseudospin [MATH].', '1703.02280-1-4-4': 'In contrast to conventional semiconductors, where a similar coupling constant can also be defined, graphene allows tuning [MATH] in a very broad range and in a very predictable manner.', '1703.02280-1-4-5': 'Indeed, the excitation frequency [MATH] can be shifted from the ultraviolet to far infrared regions while keeping the photocarrier velocity constant, hence, changing [MATH] by orders of magnitude.', '1703.02280-1-4-6': 'In conventional semiconductors the lowest possible [MATH] is limited by the band gap, and the photocarrier velocity is dependent on the excitation frequency reflecting their complicated band structure.', '1703.02280-1-4-7': 'Thus, graphene offers an unprecedented opportunity to study the pseudospin (or interband) coherence by optical means.', '1703.02280-1-5-0': 'Creating a pseudospin superposition state.', '1703.02280-1-5-1': '- The operator [MATH] written in the eigenbasis of [MATH] is given by the matrix [EQUATION] where [MATH] describes the polarization plane orientation.', '1703.02280-1-5-2': 'The pseudospin-momentum coupling in graphene results in both off- and diagonal terms in Eq. ([REF]) responsible for the generation of coherences in the electron density matrix.', '1703.02280-1-5-3': 'This is the crucial ingredient that makes the optical generation and detection of superposition states possible.', '1703.02280-1-5-4': 'It is natural to describe the pseudospin coherence by a [MATH] density matrix [MATH] whose evolution is governed by the Liouville-von Neumann equation [MATH], where the generation rate is given in the lowest non-zero order in [MATH] by [CITATION] [EQUATION]', '1703.02280-1-5-5': 'Here, the adiabaticity parameter [MATH] has been introduced in order to take into account the finite Lorentzian spectral width of the radiation flow.', '1703.02280-1-5-6': 'We assume that [MATH] at the end of the day.', '1703.02280-1-5-7': 'This approximation corresponds to the continuous wave (CW) operation of the laser.', '1703.02280-1-5-8': 'We now write the density matrix [MATH] and the corresponding Liouville-von Neumann equation in the eigenstate basis of [MATH] (helicity basis), and, at the same time, add relaxation and dephasing terms.', '1703.02280-1-5-9': 'The equation then reads [EQUATION] where the functions [MATH] represent the change in the density matrix [MATH] written in basis of the unperturbed eigenstates of [MATH], [MATH] is the equilibrium distribution matrix diagonal in the same basis, [MATH] is the relaxation time, [MATH] is the dephasing time.', '1703.02280-1-5-10': 'The generation rate is transformed within the rotating wave approximation [CITATION] and reads [EQUATION] and [MATH], [MATH].', '1703.02280-1-5-11': 'Here, [MATH].', '1703.02280-1-5-12': 'Note that [MATH] is proportional to a [MATH]-function at [MATH], whereas [MATH] is not.', '1703.02280-1-5-13': 'Since the interaction matrix [MATH] contains both diagonal and non-diagonal terms, the generation rate [MATH] does the same.', '1703.02280-1-5-14': 'This is the reason why the light-carrier interaction creates the interband coherent states.', '1703.02280-1-5-15': 'If the generation [MATH] were turned off at [MATH], then the coherences in [MATH] would be given by [MATH], [MATH] representing a rapidly oscillating function with the amplitude determined by the initial condition at [MATH].', '1703.02280-1-5-16': 'Measuring such an oscillating function could be a challenging task and, therefore, we focus on a steady-state limit, when [MATH].', '1703.02280-1-5-17': 'In this limit, the solution of ([REF]) is given by [MATH], [MATH], [MATH], [MATH].', '1703.02280-1-5-18': 'At [MATH], the diagonal terms [MATH] represent occupations of the corresponding bands given by the delta function [MATH] whereas the coherences [MATH] describe the pseudospin superposition states and remain finite even at [MATH].', '1703.02280-1-6-0': 'Measuring the superposition state.', '1703.02280-1-6-1': '- The interband coherences schematically depicted in Fig. [REF] are generated by [MATH] which couples the electric field and pseudospin - the quantity we are trying to measure.', '1703.02280-1-6-2': 'The weak measurement is performed by means of the interaction between the probe electromagnetic wave and pseudospin: [MATH] with [MATH], where [MATH].', '1703.02280-1-6-3': 'The orthogonality between [MATH] and [MATH] allows us to filter out the absorption of any of these two electromagnetic waves easily.', '1703.02280-1-6-4': 'Similar to [MATH], the electric field amplitude [MATH] and pseudospin [MATH] are coupled by the coupling parameter [MATH].', '1703.02280-1-6-5': 'The probe radiation is assumed to have the intensity [MATH] much lower than [MATH] so that the probe almost does not change the coherent state created initially by [MATH].', '1703.02280-1-6-6': 'It is also crucial for the probe interaction [MATH] to be weak enough to justify the weak measurement criterion within the whole frequency interval we are interested in, i.e. the probe intensity should be tuned while changing the probe frequency [MATH] so that [MATH] remains constant.', '1703.02280-1-6-7': 'The complete quantum mechanical weak measurement procedure also requires averaging over many single weak measurements [CITATION].', '1703.02280-1-6-8': 'Since we employ the density matrix (not a wave function) for the quantum state description, the statistics is taken into account automatically within our model.', '1703.02280-1-6-9': 'In the experiment, the CW operating probe serves for the desired statistics.', '1703.02280-1-7-0': 'The probe generation rate [MATH] is formally given by the equation similar to Eq. ([REF]), but the resulting [MATH] depends not only on the band occupations [MATH], [MATH] but also on the coherences [MATH], [MATH] introduced above.', '1703.02280-1-7-1': 'The coherences vanish in equilibrium, however, once the steady-state non-equilibrium is created, the coherences can be observed by measuring the optical absorption calculated from [MATH].', '1703.02280-1-7-2': 'The relative optical absorption is defined as [MATH], where [MATH] is the absorbed intensity per spin/valley channel given by [MATH].', '1703.02280-1-7-3': 'Here, we need only one component of the generation matrix [MATH] representing the valence-to-conduction band transition rate given by [EQUATION]', '1703.02280-1-7-4': 'The first term formally looks similar to Eq. ([REF]) but the equilibrium occupations [MATH] are substituted by their steady-state counterparts [MATH] determined by the relaxation time [MATH].', '1703.02280-1-7-5': 'This term dominates in the absorption at [MATH] and corresponds to the conventional signal in the optical pump-probe measurements [CITATION].', '1703.02280-1-7-6': 'The second term contains quantum information about the initial steady-state via coherences [MATH] determined by the dephasing time [MATH].', '1703.02280-1-7-7': 'At [MATH], [MATH] and [MATH] are weighted in the relative absorption by the delta-function [MATH] and the principle value [MATH] respectively.', '1703.02280-1-7-8': '[MATH] dominates in the absorption at [MATH].', '1703.02280-1-8-0': 'The description in terms of a non-Markovian weak measurement formalism [CITATION] can provide insight into the interaction of the probe beam and the pseudospin.', '1703.02280-1-8-1': 'Starting from a microscopic model, the field-field correlator of the outgoing electric field after the interaction with the graphene is given by [EQUATION] where [MATH] is the response function and [MATH] is the noise function of the probe beam.', '1703.02280-1-8-2': 'Note, that the dimensionless electric field [MATH] is quantized in this formalism and the probe intensity is determined by the photon number [MATH] to which we normalize here.', '1703.02280-1-8-3': 'The parameter [MATH] takes into account a Lorentzian broadening of the laser beam.', '1703.02280-1-8-4': 'The square and curly brackets denote commutator and anticommutator, respectively.', '1703.02280-1-8-5': 'The first term in Eq. ([REF]) does not contribute to the differential absorption because the trace is taken over the steady-state distribution function [MATH] determined by Eq. ([REF]) resulting in [MATH].', '1703.02280-1-8-6': 'Hence, we focus on the second term.', '1703.02280-1-8-7': 'It describes the self-interaction of the electric field mediated by the graphene and therefore contains the pseudospin response function [MATH] which is sensitive to the presence of coherence.', '1703.02280-1-8-8': 'Therefore it adds a signature of the quantum state of the graphene to the outgoing electric field.', '1703.02280-1-8-9': 'The steady-state approximation is realized as [MATH].', '1703.02280-1-8-10': 'The pseudospin is considered in the eigen state basis of [MATH] and its dynamics is governed by the equation [MATH].', '1703.02280-1-8-11': 'The statistical average is performed as [MATH].', '1703.02280-1-8-12': 'In particular, [MATH], [MATH], and [MATH].', '1703.02280-1-8-13': 'The resulting correlator reads [EQUATION]', '1703.02280-1-8-14': 'We divide the correlator by the laser pulse duration [MATH].', '1703.02280-1-8-15': 'Then, we employ the rotating wave approximation when integrating Eq. ([REF]).', '1703.02280-1-8-16': 'The result for small [MATH] reads [EQUATION] cf. Eq. ([REF]).', '1703.02280-1-8-17': 'In the simplest case of a zero-temperature equilibrium state, we have [MATH], [MATH], hence, [MATH], [MATH].', '1703.02280-1-8-18': 'To calculate the total absorbed energy per unit time and unit area from Eq. ([REF]) we integrate over all momenta and multiply by [MATH].', '1703.02280-1-8-19': 'To obtain the relative absorption we divide the result by the incident intensity [MATH] and multiply by [MATH] for the valley and spin degeneracy.', '1703.02280-1-8-20': 'The equilibrium absorption then reads [EQUATION] which is a known result [CITATION].', '1703.02280-1-8-21': 'Most saliently, either of Eqs. ([REF]) and ([REF]) contains non-equilibrium quantum information encoded in the coherences [MATH] and [MATH] whose manifestation can also be seen in the absorption.', '1703.02280-1-9-0': 'Discussion.', '1703.02280-1-9-1': '- We focus on the relative absorption difference [MATH] at [MATH] and [MATH].', '1703.02280-1-9-2': 'This difference is negative due to the so-called Pauli bleaching and reads [EQUATION] where [MATH] is the equilibrium absorption, and [EQUATION] is a constant that controls the overall signal strength via the coupling parameters [MATH], [MATH].', '1703.02280-1-9-3': 'We estimate [MATH] to be of the order of [MATH] for typical CW lasers with the power of about 1 mW focused on a [MATH]m-size spot and the frequencies [MATH], [MATH] being of the order of 0.1 and 1 eV respectively.', '1703.02280-1-9-4': 'Measuring such a low signal is a challenging task, however, the relative differential transmission of the order of [MATH] has already been measured recently on graphene using an optical pump-probe setup [CITATION].', '1703.02280-1-9-5': 'Note that [MATH] can be changed in a very broad range by playing with [MATH] and [MATH], as there is no theoretical limit for excitation frequencies from below (no band gap).', '1703.02280-1-10-0': 'Eq. ([REF]) is among the main results of our work.', '1703.02280-1-10-1': 'This signal is a response to the pseudospin superposition states created by [MATH] and measured by [MATH].', '1703.02280-1-10-2': 'The signal should vanish if such states were not there.', '1703.02280-1-10-3': 'The relative absorption ([REF]) is solely due to the pseudospin coherence because the conventional occupation-related dip shrinks to zero at [MATH].', '1703.02280-1-10-4': 'For this reason Eq. ([REF]) does not depend on [MATH] and vanishes at [MATH] (very fast dephasing).', '1703.02280-1-10-5': 'On the other hand, the signal saturates at [MATH] and equals to [EQUATION]', '1703.02280-1-10-6': 'The value of the dephasing time [MATH] is unknown but it enters Eq. ([REF]) only in products with [MATH] and [MATH] resulting in the universal behavior shown in Fig. [REF].', '1703.02280-1-10-7': 'One can see that reducing [MATH] affects the signal in the same way as faster dephasing does.', '1703.02280-1-10-8': 'On the other hand, faster dephasing is equivalent to higher [MATH].', '1703.02280-1-10-9': 'This effect can also be seen in Eq. ([REF]) vanishing at [MATH] even though dephasing is absent there.', '1703.02280-1-11-0': 'Eq. ([REF]) can be further simplified at [MATH] as [EQUATION]', '1703.02280-1-11-1': 'As it is expected from Eq. ([REF]), the signal ([REF]) saturates at [MATH] as [MATH].', '1703.02280-1-11-2': 'Let us remind that in order to keep [MATH] constant in the formal limit [MATH] we should increase [MATH] to infinity as well making this regime experimentally irrelevant.', '1703.02280-1-11-3': 'The regime [MATH] is most convenient for the pseudospin coherence observation, where the differential absorption is approaching its maximum and, most importantly, the conventional occupation dip at [MATH] is not present even if it is broadened by relaxation.', '1703.02280-1-12-0': 'Conclusion and Outlook.', '1703.02280-1-12-1': '- To conclude, we have demonstrated an analytical framework for a weak measurement description, where the quantum mechanical coherence and carrier statistics are treated using the Liouville-von Neumann equation and, alternatively, the field-field correlator.', '1703.02280-1-12-2': 'We apply this approach to intrinsic graphene whose peculiar band structure makes it possible to generate and observe the quantum mechanical pseudospin coherence.', '1703.02280-1-12-3': 'The weak measurement of pseudospin coherence is provided by coupling the pseudospin to the electric field of an electromagnetic wave.', '1703.02280-1-12-4': 'This coupling is what makes graphene special and, in combination with the results presented above, provides a reliable solid-state platform for fundamental quantum research.'}

{'1703.02280-2-0-0': 'We employ a weak measurement approach to demonstrate the very existence of the photoexcited interband superposition states in intrinsic graphene.', '1703.02280-2-0-1': 'We propose an optical two-beam setup where such measurements are possible and derive an explicit formula for the differential optical absorption that contains a signature of such states.', '1703.02280-2-0-2': 'We provide an interpretation of our results in terms of a non-Markovian weak measurement formalism applied to the pseudospin degree of freedom coupled with an electromagnetic wave.', '1703.02280-2-1-0': '# Introduction', '1703.02280-2-2-0': 'The past decade has witnessed many experiments addressing coherent quantum superpositions in single quantum systems directly.', '1703.02280-2-2-1': 'Among these are systems with photons [CITATION], with nuclear spins [CITATION], and, more recently, in electronic setups [CITATION].', '1703.02280-2-2-2': 'Since the quantum system should be left unchanged during the coherent evolution the concept of weak measurements was introduced nearly 30 years ago [CITATION] and still remains a highly active topic [CITATION].', '1703.02280-2-2-3': 'In this paper, we would like to add another possible manifestation of this problem by considering an optical measurement of the photocarriers in semiconductors, which can be readily implemented.', '1703.02280-2-2-4': 'To this end, we propose an experiment to weakly measure the photoexcited interband (electron-hole) superposition states in intrinsic graphene.', '1703.02280-2-2-5': 'To the best of our knowledge, these electron-hole pairs have never been considered from the weak measurement point of view, despite the very extensive literature on semiconductor optics.', '1703.02280-2-2-6': 'By establishing a detailed correspondence between the optical excitation-absorption process and the recently developed non-Markovian weak measurement formalism [CITATION] we find the signature of the coherent oscillation in the optical absorption signal.', '1703.02280-2-3-0': 'To perform a weak measurement [CITATION] of a quantum state [MATH] the detector must also represent a quantum-mechanical object described by a wave function [MATH].', '1703.02280-2-3-1': "The measured system interacts with the detector, and it is the detector's wave function [MATH] that is collapsed by a strong measurement.", '1703.02280-2-3-2': 'Due to the interaction between these two states the outcome of the measurement on [MATH] contains some information about coherence of the original quantum state [MATH].', '1703.02280-2-3-3': 'In our setup, the role of the quantum state [MATH] is played by a 2[MATH]2 photocarrier density matrix written in the eigenstate basis of the massless Dirac Hamiltonian for charge carriers in graphene.', '1703.02280-2-3-4': 'Once an external radiation source is applied, the matrix represents a sum [MATH], where [MATH] is the equilibrium distribution, and [MATH] is a non-equilibrium addition on top of [MATH].', '1703.02280-2-3-5': 'The matrix [MATH] includes non-zero coherences representing interband superpositions (see Fig. [REF]).', '1703.02280-2-3-6': 'The probe radiation in turn changes the photocarrier density matrix to [MATH], where [MATH] depends on [MATH].', '1703.02280-2-3-7': 'Due to the weak interaction with the probe radiation the photocarrier density matrix is left almost unaffected [MATH].', '1703.02280-2-3-8': 'That is what is known as a linear-response regime.', '1703.02280-2-3-9': 'In contrast to the classical linear-response absorption measurement, we take into account coherences and show that they could be detected in such settings.', '1703.02280-2-3-10': 'We explicitly show that the differential relative absorption of the probe radiation given by Eq. ([REF]) and shown in Fig. [REF] contains a response from the initial superposition states encoded in [MATH], [MATH].', '1703.02280-2-3-11': 'We confirm this result using a non-Markovian weak measurement formalism developed recently, hence, providing an alternative method for weak measurement description.', '1703.02280-2-4-0': '# Model in depth', '1703.02280-2-5-0': 'The effective two-level system shown in Fig. [REF] is described by the two-dimensional massless Dirac-like Hamiltonian [CITATION] [MATH], where [MATH] is the two-component momentum operator, [MATH] is the pseudospin operator constructed from the Pauli matrices, and [MATH] ms[MATH] is the carrier velocity in graphene.', '1703.02280-2-5-1': 'The corresponding eigenvalues and eigenstates are [MATH] and [MATH] with [MATH].', '1703.02280-2-5-2': 'The index [MATH]) stands for the conduction (valence) band (see Fig. [REF]).', '1703.02280-2-5-3': 'The pseudospin is parallel (antiparallel) to momentum while being in a conduction-band (valence-band) eigenstate.', '1703.02280-2-5-4': 'Hence, a certain interband coherent state [MATH] we are trying to measure represents, at the same time, a pseudospin superposition state.', '1703.02280-2-6-0': 'In order to create and measure such a superposition state we employ an electromagnetic wave [MATH] with the frequency [MATH] and amplitude [MATH] linearly polarized in the direction [MATH].', '1703.02280-2-6-1': 'The equilibrium carriers are then perturbed by the field-pseudospin interaction [CITATION] [MATH], where [MATH] with [MATH] being the electron charge.', '1703.02280-2-6-2': 'We assume normal incidence without momentum transfer from photons to electrons so that we always superimpose the states [MATH] with the same momentum.', '1703.02280-2-6-3': 'It is convenient to define the coupling parameter [MATH] that entangles the electric field and the pseudospin [MATH].', '1703.02280-2-6-4': 'In contrast to conventional semiconductors, where a similar coupling constant can also be defined, graphene allows tuning [MATH] in a very broad range and in a very predictable manner.', '1703.02280-2-6-5': 'Indeed, the excitation frequency [MATH] can be shifted from the ultraviolet to far infrared regions while keeping the photocarrier velocity constant, hence, changing [MATH] by orders of magnitude.', '1703.02280-2-6-6': 'In conventional semiconductors the lowest possible [MATH] is limited by the band gap, and the photocarrier velocity is dependent on the excitation frequency reflecting their complicated band structure.', '1703.02280-2-6-7': 'Thus, graphene offers an unprecedented opportunity to study the pseudospin (or interband) coherence by optical means.', '1703.02280-2-7-0': '# Creating a pseudospin superposition state', '1703.02280-2-8-0': 'The operator [MATH] written in the eigenbasis of [MATH] is given by the matrix [EQUATION] where [MATH] describes the polarization plane orientation.', '1703.02280-2-8-1': 'The pseudospin-momentum coupling in graphene results in both off-diagonal and diagonal terms in Eq. ([REF]) responsible for the generation of coherences in the electron density matrix.', '1703.02280-2-8-2': 'This is the crucial ingredient that makes the optical generation and detection of superposition states possible.', '1703.02280-2-8-3': 'It is natural to describe the pseudospin coherence by a [MATH] density matrix [MATH] the evolution of which is governed by the Liouville-von Neumann equation [MATH], where the generation rate is given in the lowest non-zero order in [MATH] by [CITATION] [EQUATION]', '1703.02280-2-8-4': 'Here, the adiabaticity parameter [MATH] has been introduced in order to take into account the finite Lorentzian spectral width of the radiation flow.', '1703.02280-2-8-5': 'We assume that [MATH] at the end of the day.', '1703.02280-2-8-6': 'This approximation corresponds to the cw operation of the laser.', '1703.02280-2-8-7': 'We now write the density matrix [MATH] and the corresponding Liouville-von Neumann equation in the eigenstate basis of [MATH] (helicity basis), and, at the same time, add relaxation and dephasing terms.', '1703.02280-2-8-8': 'The equation then reads [EQUATION] where the functions [MATH] represent the change in the density matrix [MATH] written in the basis of the unperturbed eigenstates of [MATH], [MATH] is the equilibrium distribution matrix diagonal in the same basis, [MATH] is the relaxation time, and [MATH] is the dephasing time.', '1703.02280-2-8-9': 'The generation rate is transformed within the rotating wave approximation [CITATION] and reads [EQUATION] and [MATH], [MATH].', '1703.02280-2-8-10': 'Here, [MATH].', '1703.02280-2-8-11': 'Note that [MATH] is proportional to a [MATH]-function at [MATH], whereas [MATH] is not.', '1703.02280-2-8-12': 'Since the interaction matrix [MATH] contains both diagonal and non-diagonal terms, the generation rate [MATH] does the same.', '1703.02280-2-8-13': 'This is the reason why the light-carrier interaction creates the interband coherent states.', '1703.02280-2-8-14': 'If the generation [MATH] were turned off at [MATH], then the coherences in [MATH] would be given by [MATH], [MATH] representing a rapidly oscillating function with the amplitude determined by the initial condition at [MATH].', '1703.02280-2-8-15': 'Measuring such an oscillating function could be a challenging task and, therefore, we focus on a steady-state limit, when [MATH].', '1703.02280-2-8-16': 'In this limit, the solution of Eq. ([REF]) is given by [MATH], [MATH], [MATH], [MATH] (see Appendix [REF]).', '1703.02280-2-8-17': 'At [MATH], the diagonal terms [MATH] represent occupations of the corresponding bands given by the delta function [MATH] whereas the coherences [MATH] describe the pseudospin superposition states and remain finite even at [MATH].', '1703.02280-2-8-18': 'The explicit expressions for [MATH] as derived in Appendix [REF] are given by [EQUATION] and [EQUATION]', '1703.02280-2-9-0': '# Measuring the superposition state', '1703.02280-2-10-0': 'The interband coherences schematically depicted in Fig. [REF] are generated by [MATH], which couples the electric field and pseudospin - the quantity we are trying to measure.', '1703.02280-2-10-1': 'The weak measurement is performed by means of the interaction between the probe electromagnetic wave and pseudospin: [MATH] with [MATH], where [MATH].', '1703.02280-2-10-2': 'The orthogonality between [MATH] and [MATH] allows us to filter out the absorption of any of these two electromagnetic waves easily.', '1703.02280-2-10-3': 'Similar to [MATH], the electric-field amplitude [MATH] and pseudospin [MATH] are coupled by the coupling parameter [MATH].', '1703.02280-2-10-4': 'The probe radiation is assumed to have the intensity [MATH] much lower than [MATH] so that the probe almost does not change the coherent state created initially by [MATH].', '1703.02280-2-10-5': 'It is also crucial for the probe interaction [MATH] to be weak enough to justify the weak measurement criterion within the whole frequency interval we are interested in, i.e. the probe intensity should be tuned while changing the probe frequency [MATH] so that [MATH] remains constant.', '1703.02280-2-10-6': 'The complete quantum-mechanical weak measurement procedure also requires averaging over many single weak measurements [CITATION].', '1703.02280-2-10-7': 'Since we employ the density matrix (not a wave function) for the quantum state description, the statistics is taken into account automatically within our model.', '1703.02280-2-10-8': 'In the experiment, the cw operating probe serves for the desired statistics.', '1703.02280-2-11-0': 'The probe generation rate [MATH] is formally given by the equation similar to Eq. ([REF]), but the resulting [MATH] depends not only on the band occupations [MATH], [MATH] but also on the coherences [MATH], [MATH] introduced above.', '1703.02280-2-11-1': 'The coherences vanish in equilibrium; however, once the steady-state non-equilibrium is created, the coherences can be observed by measuring the optical absorption calculated from [MATH].', '1703.02280-2-11-2': 'The relative optical absorption is defined as [MATH], where [MATH] is the absorbed intensity per spin/valley channel given by [MATH].', '1703.02280-2-11-3': 'Here, we need only one component of the generation matrix [MATH] representing the valence-to-conduction-band transition rate given by [EQUATION]', '1703.02280-2-11-4': 'The first term formally looks similar to Eq. ([REF]) but the equilibrium occupations [MATH] are substituted by their steady-state counterparts [MATH] determined by the relaxation time [MATH].', '1703.02280-2-11-5': 'This term dominates in the absorption at [MATH] and corresponds to the conventional signal in the optical pump-probe measurements [CITATION].', '1703.02280-2-11-6': 'The second term contains quantum information about the initial steady state via coherences [MATH] determined by the dephasing time [MATH].', '1703.02280-2-11-7': 'At [MATH], [MATH] and [MATH] are weighted in the relative absorption by the [MATH]-function [MATH] and the principle value [MATH], respectively.', '1703.02280-2-11-8': '[MATH] dominates in the absorption at [MATH].', '1703.02280-2-12-0': 'The description in terms of a non-Markovian weak measurement formalism [CITATION] can provide insight into the interaction of the probe beam and the pseudospin.', '1703.02280-2-12-1': 'Starting from a microscopic model [CITATION] (see Appendix [REF]), the field-field correlator of the outgoing electric field after the interaction with the graphene is given by [EQUATION] where [MATH] is the response function and [MATH] is the noise function of the probe beam.', '1703.02280-2-12-2': 'Note, that the dimensionless electric field [MATH] is quantized in this formalism and the probe intensity is determined by the photon number [MATH] to which we normalize here.', '1703.02280-2-12-3': 'The parameter [MATH] takes into account a Lorentzian broadening of the laser beam.', '1703.02280-2-12-4': 'The square and curly brackets denote commutator and anticommutator, respectively.', '1703.02280-2-12-5': 'The first term in Eq. ([REF]) does not contribute to the differential absorption because the trace is taken over the steady-state distribution function [MATH] determined by Eq. ([REF]) resulting in [MATH].', '1703.02280-2-12-6': 'Hence, we focus on the second term.', '1703.02280-2-12-7': 'It describes the self-interaction of the electric field mediated by the graphene and therefore contains the pseudospin response function [MATH] which is sensitive to the presence of coherence.', '1703.02280-2-12-8': 'Therefore it adds a signature of the quantum state of the graphene to the outgoing electric field.', '1703.02280-2-12-9': 'The steady-state approximation is realized as [MATH].', '1703.02280-2-12-10': 'The pseudospin is considered in the eigen state basis of [MATH] and its dynamics is governed by the equation [MATH].', '1703.02280-2-12-11': 'The statistical average is performed as [MATH].', '1703.02280-2-12-12': 'In particular, [MATH], [MATH], and [MATH].', '1703.02280-2-12-13': 'The resulting correlator reads [EQUATION]', '1703.02280-2-12-14': 'We divide the correlator by the laser pulse duration [MATH].', '1703.02280-2-12-15': 'Then, we employ the rotating wave approximation when integrating Eq. ([REF]).', '1703.02280-2-12-16': 'The result for small [MATH] reads [EQUATION] cf. Eq. ([REF]).', '1703.02280-2-12-17': 'In the simplest case of a zero-temperature equilibrium state, we have [MATH], [MATH], hence, [MATH], [MATH].', '1703.02280-2-12-18': 'To calculate the total absorbed energy per unit time and unit area from Eq. ([REF]) we integrate over all momenta and multiply by [MATH].', '1703.02280-2-13-0': 'For isotropic [MATH] or [MATH] the corresponding parts average out by this momenta integration.', '1703.02280-2-13-1': 'However, our excitations depend on the direction [c.f. [MATH]] and the occurring integrals in Eq. ([REF]), [MATH] and [MATH], do not vanish for orthogonal pump and probe beams [MATH].', '1703.02280-2-14-0': 'To obtain the relative absorption we divide the result by the incident intensity [MATH] and multiply by [MATH] for the valley and spin degeneracy.', '1703.02280-2-14-1': 'The equilibrium absorption then reads [EQUATION] which is a known result [CITATION].', '1703.02280-2-14-2': 'Most saliently, either of Eqs. ([REF]) and ([REF]) contains non-equilibrium quantum information encoded in the coherences [MATH] and [MATH] the manifestation of which can also be seen in the absorption.', '1703.02280-2-15-0': 'We focus on the relative absorption difference [MATH] at [MATH] and [MATH] when the first term in Eq. ([REF]) vanishes so that the occupations do not contribute.', '1703.02280-2-15-1': 'To neglect the influence of occupations at [MATH] the probe frequency [MATH] must be fixed further away from [MATH].', '1703.02280-2-15-2': 'The differential absorption is negative due to the so-called Pauli bleaching and reads [EQUATION] where [MATH] is the equilibrium absorption, and [EQUATION] is a constant that controls the overall signal strength via the coupling parameters [MATH], [MATH].', '1703.02280-2-15-3': 'We estimate [MATH] to be of the order of [MATH] for typical cw lasers with the power of about 1 mW focused on a [MATH]m-size spot and the frequencies [MATH], [MATH] being of the order of 0.1 and 1 eV, respectively.', '1703.02280-2-15-4': 'Measuring such a low signal is a challenging task; however, the relative differential transmission of the order of [MATH] has already been measured recently on graphene using an optical pump-probe setup [CITATION].', '1703.02280-2-15-5': 'Note that [MATH] can be changed in a very broad range by playing with [MATH] and [MATH], as there is no theoretical limit for excitation frequencies from below (no band gap).', '1703.02280-2-16-0': '# Discussion', '1703.02280-2-17-0': 'Equation ([REF]) is among the main results of our paper.', '1703.02280-2-17-1': 'This signal is a response to the pseudospin superposition states created by [MATH] and measured by [MATH].', '1703.02280-2-17-2': 'The signal should vanish if such states were not there.', '1703.02280-2-17-3': 'The relative absorption ([REF]) is solely due to the pseudospin coherence because the conventional occupation-related dip shrinks to zero at [MATH].', '1703.02280-2-17-4': 'For this reason Eq. ([REF]) does not depend on [MATH] and vanishes at [MATH] (very fast dephasing).', '1703.02280-2-17-5': 'On the other hand, the signal saturates at [MATH] and equals to [EQUATION]', '1703.02280-2-17-6': 'The value of the dephasing time [MATH] is unknown but it enters Eq. ([REF]) only in products with [MATH] and [MATH] resulting in the universal behavior shown in Fig. [REF].', '1703.02280-2-17-7': 'One can see that reducing [MATH] affects the signal in the same way as faster dephasing does.', '1703.02280-2-17-8': 'On the other hand, faster dephasing is equivalent to higher [MATH].', '1703.02280-2-17-9': 'This effect can also be seen in Eq. ([REF]) vanishing at [MATH] even though dephasing is absent there.', '1703.02280-2-18-0': 'Equation ([REF]) can be further simplified at [MATH] as [EQUATION]', '1703.02280-2-18-1': 'As it is expected from Eq. ([REF]), the signal ([REF]) saturates at [MATH] as [MATH].', '1703.02280-2-18-2': 'Let us recall that in order to keep [MATH] constant in the formal limit [MATH] we should increase [MATH] to infinity as well making this regime experimentally irrelevant.', '1703.02280-2-18-3': 'The regime [MATH] is most convenient for the pseudospin coherence observation, where the differential absorption is approaching its maximum and, most importantly, the conventional occupation dip at [MATH] is not present even if it is broadened by relaxation.', '1703.02280-2-19-0': '# Conclusion To conclude, we have demonstrated a framework for a weak measurement description, where the quantum-mechanical coherence and carrier statistics are treated using the Liouville-von Neumann equation and, alternatively, the field-field correlator.', '1703.02280-2-19-1': 'We apply this approach to intrinsic graphene the peculiar band structure of which makes it possible to generate and observe the quantum-mechanical pseudospin coherence.', '1703.02280-2-19-2': 'The weak measurement of pseudospin coherence is provided by coupling the pseudospin to the electric field of an electromagnetic wave.', '1703.02280-2-19-3': 'This coupling is what makes graphene special and, in combination with the results presented above, provides a reliable solid-state platform for fundamental quantum research.', '1703.02280-2-20-0': 'We thank Daniele Brida for multiple discussions of these results.', '1703.02280-2-20-1': 'This paper was supported financially by the Center of Applied Photonics, the European Research Council Advanced Grant UltraPhase of Alfred Leitenstorfer, and the Deutsche Forschungsgemeinschaft through SFB 767.', '1703.02280-2-21-0': '# Optically excited coherences', '1703.02280-2-22-0': 'Equation (2) in the main text and the Liouville-von Neumann equation itself are the operator equations, and to get the corresponding equation for the distribution function we rewrite the both in the eigenbasis of [MATH].', '1703.02280-2-22-1': 'The density matrix [MATH] is then transformed as [EQUATION].', '1703.02280-2-22-2': 'The terms in the left-hand side of the Liouville-von Neumann equation are transformed as [EQUATION].', '1703.02280-2-22-3': 'Here, [MATH] are conduction- and valence-band dispersions.', '1703.02280-2-22-4': 'Note, that the integrand in the generation rate consists of two similar terms [MATH].', '1703.02280-2-22-5': 'Each of these two terms consists in turn of four terms transformed as [EQUATION]', '1703.02280-2-22-6': 'Here, the matrix [MATH] is given by Eq. (1).', '1703.02280-2-23-0': 'Once we are in equilibrium [MATH], [MATH], [MATH], where [MATH], [MATH] are the Fermi-Dirac distributions.', '1703.02280-2-23-1': 'Thus, the [MATH] dependence remains only in the exponents, and the excitation rate reads [EQUATION]', '1703.02280-2-23-2': 'Here, [EQUATION] where [MATH].', '1703.02280-2-23-3': 'Note, that [MATH] and since [MATH] some [MATH]-functions do not contribute at [MATH].', '1703.02280-2-23-4': 'Further on, the coherence generation rates read [EQUATION]', '1703.02280-2-23-5': 'Note, that [MATH].', '1703.02280-2-23-6': 'The distributions [MATH] can be found from [EQUATION] where [MATH] is the pseudospin relaxation time.', '1703.02280-2-23-7': 'Thus, [EQUATION] where the general relations [MATH], [MATH] hold.', '1703.02280-2-24-0': '# Non-Markovian weak measurement', '1703.02280-2-25-0': 'We derive the cross-correlation of a non-Markovian weak measurement from a microscopic model.', '1703.02280-2-25-1': 'In our paper the detector is given by the electric field of a laser beam [MATH] which is coupled to the pseudospin [MATH] of a graphene system and subsequently measured by a photodetector.', '1703.02280-2-25-2': 'In order to model the cross-correlation we label two electric fields (two detectors) with [MATH] and [MATH] so that the interaction Hamiltonian can be written as [EQUATION] where the [MATH] denote the coupling strength.', '1703.02280-2-25-3': 'The probability to find the outcomes [MATH] and [MATH] is given by [EQUATION] with [MATH], where [MATH] is the unitary time evolution of the undisturbed systems and [MATH] labels the interaction with the time-ordering [MATH].', '1703.02280-2-25-4': 'The readout of the electric field after the interaction with the pseudospin is denoted by the projection operators [MATH].', '1703.02280-2-25-5': 'Following the weak coupling assumption we expand the time evolution in the interaction picture to second order in [MATH] [EQUATION] where the expectation values are taken in the undisturbed subsystems.', '1703.02280-2-25-6': 'We already omitted terms which will be proportional to [MATH].', '1703.02280-2-25-7': 'The expectation value of the weak measurement output is given as [EQUATION]', '1703.02280-2-25-8': 'With [MATH] and by introducing the response function [MATH] and the noise function [MATH] the field-field correlator can be expressed as [EQUATION] where we set [MATH].', '1703.02280-2-25-9': 'In this case the cross-correlation differs from the measured quantity - the intensity of the laser beam which corresponds to the self-correlation - only by the intrinsic noise [MATH].', '1703.02280-2-25-10': 'This is due to the scalar commutator of the electric field [MATH].', '1703.02280-2-25-11': 'Since the intrinsic electric-field noise is independent of the measured system, it does not contribute to the absorption difference [MATH] in which we are interested.', '1703.02280-2-26-0': 'With the operator [MATH] we have the correlators [EQUATION] with [MATH].', '1703.02280-2-26-1': 'Note that the second term in Eq. ([REF]) is proportional to the photon number [MATH] whereas the first is not.', '1703.02280-2-26-2': 'Therefore the system operators commutator [MATH] dominates the result for [MATH].'}

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[['1703.02280-1-9-2', '1703.02280-2-15-2'], ['1703.02280-1-9-3', '1703.02280-2-15-3'], ['1703.02280-1-9-4', '1703.02280-2-15-4'], ['1703.02280-1-12-2', '1703.02280-2-19-1'], ['1703.02280-1-4-0', '1703.02280-2-6-0'], ['1703.02280-1-3-1', '1703.02280-2-5-0'], ['1703.02280-1-3-4', '1703.02280-2-5-3'], ['1703.02280-1-5-1', '1703.02280-2-8-0'], ['1703.02280-1-5-2', '1703.02280-2-8-1'], ['1703.02280-1-5-4', '1703.02280-2-8-3'], ['1703.02280-1-5-9', '1703.02280-2-8-8'], ['1703.02280-1-6-1', '1703.02280-2-10-0'], ['1703.02280-1-6-4', '1703.02280-2-10-3'], ['1703.02280-1-6-7', '1703.02280-2-10-6'], ['1703.02280-1-6-9', '1703.02280-2-10-8'], ['1703.02280-1-7-1', '1703.02280-2-11-1'], ['1703.02280-1-7-3', '1703.02280-2-11-3'], ['1703.02280-1-7-6', '1703.02280-2-11-6'], ['1703.02280-1-7-7', '1703.02280-2-11-7'], ['1703.02280-1-11-0', '1703.02280-2-18-0'], ['1703.02280-1-11-2', '1703.02280-2-18-2'], ['1703.02280-1-8-1', '1703.02280-2-12-1'], ['1703.02280-1-8-21', '1703.02280-2-14-2'], ['1703.02280-1-1-3', '1703.02280-2-3-0']]

[]

[['1703.02280-1-9-1', '1703.02280-2-15-0'], ['1703.02280-1-3-3', '1703.02280-2-5-2'], ['1703.02280-1-5-7', '1703.02280-2-8-6'], ['1703.02280-1-5-17', '1703.02280-2-8-16'], ['1703.02280-1-10-0', '1703.02280-2-17-0'], ['1703.02280-1-2-1', '1703.02280-2-3-3'], ['1703.02280-1-2-1', '1703.02280-2-3-5'], ['1703.02280-1-2-2', '1703.02280-2-3-6'], ['1703.02280-1-2-3', '1703.02280-2-3-10']]

[]

['1703.02280-1-1-0', '1703.02280-1-3-0', '1703.02280-1-5-11', '1703.02280-1-8-12', '1703.02280-1-9-0', '1703.02280-1-12-0', '1703.02280-2-8-10', '1703.02280-2-12-12', '1703.02280-2-23-2', '1703.02280-2-23-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1703.02280

1105.3840

{'1105.3840-1-0-0': 'The discontinuities of electromagnetic test fields generated by general layers of electric and magnetic monopoles and dipoles are investigated in general curved spacetimes.', '1105.3840-1-0-1': 'The equivalence of electric currents and magnetic dipoles is discussed.', '1105.3840-1-0-2': 'The results are used to describe exact "Schwarzschild disk" solutions endowed with such sources.', '1105.3840-1-0-3': 'The resulting distributions of charge and dipole densities on the disks are corroborated using the membrane paradigm.', '1105.3840-1-1-0': '# Introduction', '1105.3840-1-2-0': 'The investigation of electromagnetic fields coupled to strong gravitational fields have an interest from both theoretical perspectives and from a variety of applications in astrophysics.', '1105.3840-1-2-1': 'Examples on the theory side include studies of gravitational collapse of charged configurations (see, e.g., [CITATION]), of the validity of the cosmic censorship conjecture [CITATION], of the existence and properties of quasi black holes and wormholes (for recent accounts, see, e.g., [CITATION] and references therein), membranes producing repulsive gravity [CITATION], and of many other issues.', '1105.3840-1-2-2': 'Very often analytical works employ, as tractable physical models, 2-dimensional thin shells sweeping out 3-dimensional timelike hypersurfaces.', '1105.3840-1-2-3': 'Recently, we used this idealization to construct "spherical gravitating condensers" - two concentric charged shells made of perfect fluids (satisfying energy conditions) under the condition that the electric field is non-vanishing only between the shells (see [CITATION] and further references on charged shells therein).', '1105.3840-1-3-0': 'The literature on electromagnetic fields in relativistic astrophysics is vast.', '1105.3840-1-3-1': 'Here we restrict ourselves to referring to several monographs dealing in detail with black-hole electrodynamics, e.g., [CITATION], and we mention the relatively recent work [CITATION] on electromagnetic fields around compact objects in which various papers are also summarized.', '1105.3840-1-3-2': 'In [CITATION] solutions to the Maxwell equations are presented both in the interior and outside a rotating neutron star and the matching conditions of the electromagnetic field at the stellar surface are analyzed in detail.', '1105.3840-1-3-3': 'The fields are not continuous across the stellar surface which gives rise to charges and currents.', '1105.3840-1-4-0': 'In the present paper we study electromagnetic sources distributed on shells in curved spacetimes in general, considering in particular possible discontinuities of the electromagnetic field across the shells.', '1105.3840-1-4-1': 'The sources discussed are layers with monopole or dipole currents.', '1105.3840-1-4-2': 'As far as we are aware electric or magnetic dipole layers and the matching conditions for their fields were not studied before in the context of general relativity.', '1105.3840-1-5-0': 'In general, in case of dipoles the currents and the electromagnetic field tensor will be distribution valued.', '1105.3840-1-5-1': 'This implies products of distributions in the stress-energy tensor.', '1105.3840-1-5-2': 'In order to avoid this, one can treat the electromagnetic field as a test field and solve the Maxwell equations in a given background metric.', '1105.3840-1-5-3': 'In many astrophysical situations this approach is well justified since typically the averaged energy density of the electromagnetic field is much smaller than that of the gravitational field.', '1105.3840-1-5-4': 'This approach is followed here and thus only the standard theory of generalized functions is used.', '1105.3840-1-6-0': 'In another work [CITATION] we discuss spherical thin shells endowed with arbitrary, not necessarily spherical distributions of charge or dipole densities in a Schwarzschild spacetime.', '1105.3840-1-6-1': 'There it was possible to employ the results of [CITATION] to calculate the fields directly and read off their discontinuities across the shell.', '1105.3840-1-6-2': 'In the present paper we generalize the jump conditions to general backgrounds and general hypersurfaces.', '1105.3840-1-6-3': 'As a by-product of those jump conditions the equivalence of the external fields of magnetic dipoles and certain electric charge currents is proven in general.', '1105.3840-1-6-4': 'For elementary dipoles this was already known in special backgrounds like the Kerr spacetime [CITATION].', '1105.3840-1-7-0': 'The jump conditions can be used to obtain massive disks endowed with charge and dipole densities using the Israel-Darboux formalism.', '1105.3840-1-7-1': 'In the examples studied here, we use the Schwarzschild disk spacetime as a background, cf. [CITATION].', '1105.3840-1-7-2': 'Therefore, we generate massive thin disks (Schwarzschild disks) endowed with either electric/magnetic test charges or electric/magnetic test dipoles.', '1105.3840-1-7-3': 'The surface currents are depicted and explained using the membrane paradigm.', '1105.3840-1-8-0': 'We use throughout the article the metric signature [MATH] and units in which [MATH].', '1105.3840-1-9-0': '# Monopole and dipole layers in general', '1105.3840-1-10-0': 'Although in the examples analyzed in section [REF] we use the Schwarzschild disk spacetimes as backgrounds, the results in the next section, i.e., the source terms and the jump conditions hold in a more general backgrounds.', '1105.3840-1-10-1': 'Of course, it has to admit a hypersurface, where the sources are situated, and the derivation of a dipole current requires, that a family of "parallel" hypersurfaces as defined below exists.', '1105.3840-1-10-2': 'In [CITATION] the case of charged massive shells were already discussed in full Einstein-Maxwell theory.', '1105.3840-1-10-3': 'Nonetheless, we consider test charges in our work, mainly to show in which cases the field generated by an arbitrary dipole distribution can be seen outside of the source as generated by moving charges.', '1105.3840-1-11-0': '## The 4-currents for charges or dipoles distributed on a shell', '1105.3840-1-12-0': 'Denoting by [MATH] the Maxwell tensor and by [MATH] its dual, the Maxwell equations in a complex form read as follows: [EQUATION] where [MATH] is a self-dual 2-form.', '1105.3840-1-12-1': 'The 4-current [MATH] consists of an electric part [MATH] and a magnetic part [MATH].', '1105.3840-1-12-2': 'If [MATH] is vanishing the imaginary part of the Maxwell equations [REF] allows us to introduce an electric 4-potential [MATH] such that [MATH].', '1105.3840-1-12-3': 'In case there are no electric sources present, we can analogously introduce a magnetic 4-potential [MATH] such that [MATH].', '1105.3840-1-12-4': 'In the vacuum region both 4-potentials can be defined and we denote the complex linear combination by [MATH].', '1105.3840-1-13-0': 'Timelike hypersurfaces [MATH] representing the history of charged 2-surfaces (shells) are discussed widely in the literature, see, e.g., [CITATION].', '1105.3840-1-13-1': 'We recall their main properties, in particular the form of the 4-current which will help us in formulating the expressions for the dipole current.', '1105.3840-1-13-2': 'Suppose the hypersurface [MATH] is described by [MATH], where [MATH] are coordinates in the two parts of the spacetime on the two sides of [MATH] and the index [MATH] denotes from which side a quantity is seen.', '1105.3840-1-13-3': 'The unit normal of [MATH] is given by [MATH], where [MATH] and [MATH] is chosen such that the normal points from [MATH] to [MATH].', '1105.3840-1-13-4': 'To shorten the notation we drop the index [MATH] in the following wherever no confusion is to be expected.', '1105.3840-1-13-5': 'If the intrinsic coordinates of [MATH] are called [MATH], where [MATH] runs from [MATH] to [MATH] and [MATH] is a timelike coordinate, then the tangential vectors are [MATH].', '1105.3840-1-13-6': 'A tensor field [MATH] can be projected onto these directions at [MATH] and we denote this by [EQUATION]', '1105.3840-1-13-7': 'The 4-current of an electrically charged monopole layer is given by [EQUATION] where [MATH] is the surface current of the electrical charges, [MATH] is the rest electric surface charge density and [MATH] the 4-velocity of the charged particles projected onto [MATH].', '1105.3840-1-14-0': 'Let us consider, at least locally, a Gaussian normal coordinate system generated by the geodesics [MATH] orthogonal to [MATH] and going out from points [MATH].', '1105.3840-1-14-1': 'Then the metric is block diagonal [EQUATION] and [MATH].', '1105.3840-1-14-2': 'The family of hypersurfaces [MATH], i.e., [MATH], are still orthogonal to the family of geodesics [MATH] and are at a proper distance [MATH] measured along [MATH] from [MATH].', '1105.3840-1-14-3': 'The coordinates [MATH] can be used as intrinsic coordinates, thus [MATH] with the Kronecker delta [MATH], and [MATH] denote the intrinsic metrics of the hypersurfaces [MATH] and [MATH] their determinant.', '1105.3840-1-14-4': 'A slightly more general form of the metric arises when the coordinate [MATH] along geodesics [MATH] is not measuring anymore the proper distance, implying [MATH].', '1105.3840-1-14-5': 'These "generalized" Gaussian normal coordinates are used in the examples in section [REF].', '1105.3840-1-14-6': 'The 4-current of a charge distribution on the surface in the generalized Gaussian normal coordinates is given by [EQUATION]', '1105.3840-1-14-7': 'To avoid confusion, we recall the definition of the 1-dimensional [MATH]-distribution: For any sufficiently smooth test function [MATH] the following integral over a spacetime region [MATH] in the generalized Gaussian normal coordinates reduces to the integral over [MATH] as follows: [EQUATION] where [MATH] is a spacetime volume element, [MATH] is a volume element of the hypersurface [MATH].', '1105.3840-1-15-0': 'We construct dipole layers from two oppositely charged monopole layers which are separated by a proper distance [MATH].', '1105.3840-1-15-1': 'For simplicity we derive the 4-current in the Gaussian normal coordinates [REF] and make a coordinate transformation to the generalized Gaussian normal coordinates subsequently.', '1105.3840-1-15-2': 'Dipole layers arise in the limit of vanishing [MATH] with a simultaneous limit to infinite (and opposite) rest surface charge densities of the two shells.', '1105.3840-1-15-3': 'This means we consider two hypersurfaces [MATH] and [MATH] endowed with surface rest charge densities of opposite sign [MATH] and [MATH] and with velocity fields [MATH] and [MATH], so giving rise to two 4-currents.', '1105.3840-1-15-4': 'Note that the change of the charge densities in the limit is such that it does not effect the velocity fields.', '1105.3840-1-15-5': 'Dipole layers result only in the limiting process [MATH] if certain properties hold true in the limit which, for simplicity, we assume to hold throughout the entire limiting procedure.', '1105.3840-1-15-6': 'The family of geodesics [MATH] gives locally rise to an equivalence relation of points similarly to [CITATION], i.e., [MATH] if there exist a point [MATH] such that [MATH], cf. Fig. [REF].', '1105.3840-1-16-0': 'Since the intrinsic coordinates are carried along the geodesics, equivalent points are characterized by the same intrinsic coordinates.', '1105.3840-1-16-1': 'Let us assume that two charge elements initially placed at two equivalent points [MATH] and [MATH] stay in course of there motion in equivalent points at every moment of time, e.g., the intrinsic time coordinate [MATH], cf. Fig. [REF].', '1105.3840-1-16-2': 'Then the coordinate velocities of the charge elements are the same, so we have [EQUATION]', '1105.3840-1-16-3': 'Analogously to the the equivalence of points, we also can, with each area element [MATH] in [MATH], associate an area element [MATH] in [MATH] which is cut out by the geodesics emanating from the boundary of [MATH], cf. Fig. [REF].', '1105.3840-1-16-4': 'Since the total charge of the dipole shell has to vanish, we suppose that the charge [MATH] enclosed in any area [MATH] (as seen by observers at rest with respect to the intrinsic coordinates) is the opposite of that enclosed in [MATH].', '1105.3840-1-16-5': 'This condition yields [EQUATION]', '1105.3840-1-16-6': 'As in the classical case, the charge density [MATH] as [MATH].', '1105.3840-1-16-7': 'The electrical rest dipole moment surface density is then naturally defined as [MATH].', '1105.3840-1-17-0': 'Therefore, the limiting procedure based on [REF], [REF] and [REF] yields the resulting dipole 4-current in the form [EQUATION]', '1105.3840-1-17-1': 'Of course, the total charge contained in any proper volume enclosing a part of the electric dipole layer is vanishing.', '1105.3840-1-17-2': 'Rewriting this in the generalized Gaussian normal coordinate system we find the 4-current to read [EQUATION] where [MATH] is the surface current of the electrical dipoles and [MATH] is the 4-velocity of the dipoles projected onto [MATH].', '1105.3840-1-17-3': 'Let us repeat the definition of the normal derivative of a [MATH]-distribution in a curved background.', '1105.3840-1-17-4': 'For an arbitrary, sufficiently smooth test function [MATH] the following holds: [EQUATION]', '1105.3840-1-17-5': 'Note that even though a derivative of the delta function appears, no metric functions have to be differentiated because of the integral definition of distributions where [MATH] appears and cancels with the only metric term in the 4-current depending on [MATH].', '1105.3840-1-17-6': 'Thus also metrics which are not [MATH] as they arise in the Israel formalism are allowed.', '1105.3840-1-17-7': 'It is also clear by construction and a short calculation, that the continuity equation for [MATH] implies that the surface currents [MATH] and [MATH] satisfy the continuity equation on [MATH].', '1105.3840-1-17-8': 'The currents for shells endowed with a magnetic charge or a magnetic dipole density are analogously defined, i.e., we just have to replace the index [MATH] by the index [MATH].', '1105.3840-1-18-0': '## Discontinuities in the potential and the fields', '1105.3840-1-19-0': 'As is well known from flat space, the jumps of various components of the fields or potentials across a surface are related to electromagnetic sources distributed on that surface.', '1105.3840-1-19-1': 'However, even in special relativity magnetic charges are usually not discussed.', '1105.3840-1-19-2': 'The jumps resulting from a dipole layer were, to the best of our knowledge, not discussed in curved spacetimes.', '1105.3840-1-19-3': 'We denote the jumps of a function [MATH] by [MATH].', '1105.3840-1-19-4': 'We study the four cases of electric/magnetic charged shells and electric/magnetic dipole shells separately.', '1105.3840-1-19-5': 'All of them can be obtained using the equivalence principle and Maxwell theory.', '1105.3840-1-20-0': 'In the case of an electrically charged surface Kuchar showed in [CITATION] (see also [CITATION]) that [EQUATION]', '1105.3840-1-20-1': 'Note that these equations are covariant with respect to a change of intrinsic coordinates [MATH] and scalars with respect to the coordinates [MATH].', '1105.3840-1-20-2': 'For the electric 4-potential in an appropriate Lorenz gauge it follows [EQUATION]', '1105.3840-1-20-3': 'The magnetic 4-potential [MATH] will in general not be continuous across [MATH] owing to the fact that it can only be introduced in the absence of electrical currents and, therefore, different potentials will occur in the lower and the upper half of the spacetime.', '1105.3840-1-20-4': 'Furthermore, introducing the potential [MATH] on both sides of [MATH] in different gauges will not change the external field, however, jumps in the potential are, as seen below, related to dipole densities and therefore describe a different physical system; in particular, the field in [MATH] is changed.', '1105.3840-1-21-0': 'In case of a shell endowed with magnetic charges the same equations as [REF] and [REF] hold for the dual of the Maxwell tensor and for the magnetic 4-potential in a Lorenz gauge: [EQUATION]', '1105.3840-1-21-1': 'For the Maxwell tensor it follows that the tangential components jump and the normal components are continuous: [EQUATION] where [MATH] is the volume form of [MATH] related to the induced metric [MATH] of [MATH] whereas [MATH] is the volume form of the spacetime.', '1105.3840-1-21-2': 'Tangential indices are raised and lowered with the induced metric and its inverse.', '1105.3840-1-22-0': 'Analogously, from the equivalence principle the discontinuities of the Maxwell tensor for electric and magnetic dipole densities follow: [EQUATION]', '1105.3840-1-22-1': 'Here the antisymmetrization in the derivatives of [MATH] is defined as [MATH].', '1105.3840-1-22-2': 'Note that a layer with a curl-free [MATH] will not produce a jump in the external field and thus the source can only be detected by observing the trajectories of particles crossing that layer, i.e., by measuring the internal field in [MATH].', '1105.3840-1-23-0': 'The 4-potentials satisfy in these cases the following jump conditions: [EQUATION]', '1105.3840-1-23-1': 'Additionally, the normal components of the Maxwell tensor have a [MATH]-like contribution [MATH], the field "between the two layers".', '1105.3840-1-23-2': 'In order to see this contribution, we insert the aforementioned jumps into the Maxwell equations and calculate the source.', '1105.3840-1-23-3': 'Using again Gaussian normal coordinates we start with an electric 4-potential which is discontinuous across [MATH] and calculate the sources.', '1105.3840-1-23-4': 'Hence, we write [EQUATION] with [MATH], which implies the Maxwell tensor to be [EQUATION]', '1105.3840-1-23-5': 'Inserting this into the Maxwell equations and using the jump conditions above yields [EQUATION] where the first two terms are the source terms for a charged layer and for a dipole layer.', '1105.3840-1-23-6': 'The last two terms are sources outside of [MATH], for instance a volume charge density.', '1105.3840-1-23-7': 'In the remainder we will assume that outside of the shell there are no magnetic or electric sources.', '1105.3840-1-24-0': '## The equivalence of electric charges and magnetic dipoles', '1105.3840-1-25-0': 'In flat spacetimes and also in certain cases of electromagnetism in curved backgrounds, e.g., in the Schwarzschild and the Kerr spacetimes [CITATION], the equivalence of the external field of a magnetic point dipole and of an infinitesimal electric charge current loop is known and often used.', '1105.3840-1-25-1': 'Naturally, it can also be easily shown that the external field of an electric point dipole is indistinguishable from that of an infinitesimal magnetic charge current loop.', '1105.3840-1-25-2': 'A similar result can be shown to hold in the case of layers of dipoles.', '1105.3840-1-25-3': 'In our Gaussian normal coordinates the dual of the Maxwell tensor for a shell endowed with magnetic dipoles reads as follows, cf. [REF] and [REF]: [EQUATION]', '1105.3840-1-25-4': 'Of course, the internal field must be changed to transform locally from sources in the form of magnetic dipoles to electric currents.', '1105.3840-1-25-5': 'However, if we remove the last term in [REF] from the field the external field remains unchanged.', '1105.3840-1-25-6': 'An observer outside can detect the difference only by examining trajectories of charged test particles crossing the shell.', '1105.3840-1-25-7': 'Furthermore, the jumps of the Maxwell tensor remain the same: [EQUATION]', '1105.3840-1-25-8': 'Using equation [REF], these jumps are produced by an electric current [MATH] if [EQUATION]', '1105.3840-1-25-9': 'The electric charge current defined in such a way can also be seen as a source.', '1105.3840-1-25-10': 'The continuity equation for [MATH] is satisfied trivially.', '1105.3840-1-25-11': 'However, since the charge density [MATH] does not need to vanish, electrical charges are introduced in general.', '1105.3840-1-25-12': 'The total charge is in principle detectable at infinity in the asymptotics of the field assuming it falls off sufficiently fast.', '1105.3840-1-25-13': 'Nonetheless, the total charge for a field generated by magnetic dipoles is vanishing.', '1105.3840-1-25-14': 'How is this to be resolved?', '1105.3840-1-25-15': 'The total electric charge [MATH] of [MATH] as seen for observers at rest with respect to the intrinsic coordinates is given by [EQUATION]', '1105.3840-1-25-16': "Together with equation [REF] and Stokes' theorem we obtain [EQUATION]", '1105.3840-1-25-17': 'The asymptotic behavior of the field implies a vanishing current at infinity.', '1105.3840-1-25-18': 'Thus, no total electric charge [MATH] will be present though "local" volumes can contain a net charge.', '1105.3840-1-25-19': 'This is also in correspondence with the known results for point dipoles.', '1105.3840-1-25-20': 'In a rest frame of a point dipole the external field can be seen as caused by an infinitesimal charge current loop with a vanishing time component.', '1105.3840-1-25-21': 'This is usually interpreted as two currents of positive and negative charges such that the charge densities in the rest frame of the dipole cancel each other and - for example the positive charges are at rest (ions of the conductor) and the negative charges (electrons) contribute to the current.', '1105.3840-1-25-22': 'However, in a general frame as used here the charge densities do not necessarily cancel anymore.', '1105.3840-1-25-23': 'To generalize this to layers these point dipoles have to be superposed and so the current loops.', '1105.3840-1-25-24': 'The net current can have a charge density because one is not in a comoving frame of the dipoles.', '1105.3840-1-26-0': 'If the fields do not fall off sufficiently fast, then the total charge of the shell need not vanish or be definable.', '1105.3840-1-26-1': 'In such a case charges can also be "placed at infinity" which is reflected by a corresponding boundary condition.', '1105.3840-1-26-2': 'An example is given in section [REF].', '1105.3840-1-27-0': 'The argument given above can be reversed and used to show that the external field of every electric charge surface current can also be produced by a charge density at rest in a given frame of reference and a magnetic dipole surface current.', '1105.3840-1-27-1': 'The integrability condition of equation [REF] for [MATH] is then equivalent to the continuity equation of the electric charge surface current.', '1105.3840-1-27-2': 'It is obvious that an analogous equivalence between electric dipoles and magnetic charges can be established.', '1105.3840-1-27-3': 'Except for this kind of non-uniqueness, the field and its sources are completely determined by the jump conditions [REF]-[REF].', '1105.3840-1-28-0': '# Schwarzschild disks with electric/magnetic charge and dipole density', '1105.3840-1-29-0': 'The Schwarzschild metric in Schwarzschild coordinates [MATH] reads [EQUATION]', '1105.3840-1-29-1': 'In [CITATION] massive disks of counterrotating matter, the "Schwarzschild disks", were constructed from this spacetime using the Israel-Darboux formalism and Weyl coordinates [MATH] [EQUATION]', '1105.3840-1-29-2': 'This was done by identifying the surfaces [MATH] and [MATH].', '1105.3840-1-29-3': 'From the jumps of the extrinsic curvature of the resulting surface an energy-momentum density of the disk was obtained.', '1105.3840-1-29-4': 'The disks are infinite but their mass is finite and the mass density decreases rapidly at large radii.', '1105.3840-1-29-5': 'We show here how to endow such disks with an electric/magnetic charge densities or electric/magnetic dipole densities in a test field approach.', '1105.3840-1-29-6': 'We demonstrate this with two examples using the asymptotic homogeneous field and the field generated by a point charge.', '1105.3840-1-29-7': 'The same can be done to model more general distributions using the general solutions of the Maxwell equations for test fields on a Schwarzschild background given in [CITATION].', '1105.3840-1-30-0': 'In [MATH] defined by [MATH], we introduce intrinsic coordinates [MATH] which coincide with the Schwarzschild coordinates [MATH] in the disk but are capitalized to prevent confusion.', '1105.3840-1-30-1': 'The components of the normal vector in Schwarzschild coordinates are given by [EQUATION] where again "[MATH]" denotes the quantities as seen from [MATH] and "[MATH]" as seen from [MATH].', '1105.3840-1-30-2': 'Note that [MATH].', '1105.3840-1-31-0': '## Asymptotically homogeneous electric and magnetic field', '1105.3840-1-32-0': 'The first test field to be discussed is the asymptotically homogeneous electric and magnetic field, for which the complex 4-potential and Maxwell tensor in Schwarzschild coordinates read as follows (see, e.g., [CITATION]) [EQUATION]', '1105.3840-1-32-1': 'The 4-potential is in fact not given in [CITATION] but can be calculated easily.', '1105.3840-1-32-2': 'Assume the field in the upper/lower half is parametrized by [MATH] and [MATH].', '1105.3840-1-32-3': 'The jumps of the potential across [MATH] are given by [EQUATION]', '1105.3840-1-32-4': 'As it should be according to the equations [REF], [REF] and [REF], the orthogonal component of the potential is continuous.', '1105.3840-1-32-5': 'Furthermore, the radial component is continuous as well, i.e., the dipole currents (electric or magnetic) in the radial direction are vanishing.', '1105.3840-1-32-6': 'The dipole density approaches a constant value, so does the current in the [MATH] direction, as one can expect from the analogous result obtained in Maxwell theory in flat space or after setting the mass [MATH] to zero in the equations above.', '1105.3840-1-32-7': 'The jumps in the fields read [EQUATION]', '1105.3840-1-32-8': 'Using equations [REF] and [REF]-[REF] we observe again that for electric/magnetic charges the radial current is vanishing and that the electric and magnetic charges do rotate around the axis.', '1105.3840-1-32-9': 'The current is vanishing for [MATH].', '1105.3840-1-32-10': 'The total electric or magnetic charge of such a system will be infinite.', '1105.3840-1-32-11': 'This will be different for the case of the field discussed in the next subsection.', '1105.3840-1-33-0': 'We will now treat the case of electric monopoles and magnetic dipoles independently of the case of magnetic monopoles and electric dipoles.', '1105.3840-1-33-1': 'Afterwards the results can be superposed.', '1105.3840-1-34-0': 'Electric monopoles or magnetic dipoles', '1105.3840-1-35-0': 'This case is obtained for [MATH] and [MATH], together with [MATH] and [MATH].', '1105.3840-1-35-1': 'This leads to a surface current [EQUATION]', '1105.3840-1-35-2': 'In the classical case [MATH] the charges are at rest with a charge density equal to the first factor in the first equation.', '1105.3840-1-35-3': 'The discontinuities in the magnetic potential and the tangential components of the dual of the Maxwell tensor are in this case understood as being caused by the discontinuities of the orthogonal components of the Maxwell tensor and the presence of the electric monopole layer and, hence, the impossibility to introduce a magnetic potential globally.', '1105.3840-1-35-4': 'Looking at the classical case [MATH], the principal problem mentioned after equation [REF] becomes apparent when dealing with fields which are not falling off sufficiently fast at infinity.', '1105.3840-1-35-5': 'The axial current vanishes in this limit and thus cannot cause the magnetic field.', '1105.3840-1-35-6': 'The existing magnetic field can be explained by "magnetic charges or electric currents at infinity".', '1105.3840-1-35-7': 'Therefore, the disk is not the only source of the external field.', '1105.3840-1-35-8': 'This problem does not occur for fields which are falling off sufficiently fast.', '1105.3840-1-35-9': 'Such are discussed in the next example.', '1105.3840-1-35-10': 'However, for completeness we give here the 4-current provided that the discontinuities are interpreted as the result of a magnetic dipole layer according to equation [REF]: [EQUATION]', '1105.3840-1-35-11': 'Here the constants [MATH] and [MATH] are chosen such that the current is not singular at the axis and the dipole density vanishes at infinity.', '1105.3840-1-36-0': 'Analogously, we can study disks endowed with a magnetic charge density or electric dipole density by setting [MATH] and [MATH].', '1105.3840-1-36-1': 'The results are very similar to [REF] and [REF]; they can be obtained by a substitution [MATH] and [MATH] into [REF] and [REF].', '1105.3840-1-37-0': '## Disks generated by point charges', '1105.3840-1-38-0': 'The question whether a field is generated solely by disks or also by sources at infinity is circumvented if a solution is chosen such that it falls off sufficiently fast at infinity.', '1105.3840-1-38-1': 'We now consider the electromagnetic field produced by a point charge [MATH] situated in an arbitrary position [MATH].', '1105.3840-1-38-2': 'The electric 4-potential for such a point charge was given in [CITATION], and in closed form by Linet in [CITATION].', '1105.3840-1-38-3': 'It reads: [EQUATION]', '1105.3840-1-38-4': 'We consider two different test fields in the Schwarzschild spacetimes: the field produced by a point charge at [MATH] and the field produced by a point charge at [MATH].', '1105.3840-1-38-5': 'In the spacetime with the first test field we make a cut at such [MATH] that the black hole and the point charge are below the cut.', '1105.3840-1-38-6': 'For the second test field the cut is made at [MATH] such that the charge and the black hole are above the cut.', '1105.3840-1-38-7': 'After identifying the two hypersurfaces [MATH] there is no black hole or point charge in the spacetime, rather a massive disk with electromagnetic sources.', '1105.3840-1-38-8': 'However, the electromagnetic field outside the disk and thus the sources can be understood using the field lines in the "original" spacetime for the "original" test field, i.e., the Schwarzschild black hole spacetime with a point charge.', '1105.3840-1-38-9': 'This point of view is employed several times in the following; e.g., the charge density of the disk is explained by referring to the "original" black hole and its polarization.', '1105.3840-1-39-0': 'The fields of the two point charges can be obtained from the 4-potential [REF] in a straightforward way and so also the jumps.', '1105.3840-1-39-1': 'In order to obtain a layer endowed with either only charges or dipoles we have to require that the point charges have to be located symmetrically in the original spacetime, i.e., [MATH], as well as that the charges are either equal, [MATH], or opposite, [MATH].', '1105.3840-1-39-2': 'Because of the axially symmetry of the spacetime we can set [MATH].', '1105.3840-1-39-3': 'The jumps evaluate to [EQUATION]', '1105.3840-1-39-4': 'Note that functions [MATH] and [MATH] have to be evaluated at the respective [MATH] with the respective point charge.', '1105.3840-1-39-5': 'However, it holds that [MATH], so the same holds for [MATH].', '1105.3840-1-39-6': 'Therefore, functions [MATH], [MATH] and [MATH] should be read as functions with the argument [MATH].', '1105.3840-1-39-7': 'The jumps of the tangential components of the Maxwell tensor can be inferred from the jumps of the 4-potential.', '1105.3840-1-39-8': 'We can now discuss two cases - a monopole layer and a dipole layer.', '1105.3840-1-40-0': 'Electric monopoles or magnetic dipoles', '1105.3840-1-41-0': 'In order to obtain continuous tangential components of the 4-potential we have to set [MATH].', '1105.3840-1-41-1': 'Then the surface 3-current can be read off [REF] and [REF].', '1105.3840-1-41-2': 'The only non-vanishing component is [MATH].', '1105.3840-1-41-3': 'However, it is possible to consider two counterrotating streams with an equal charge, cf. with the underlying matter currents in the Schwarzschild disk [CITATION].', '1105.3840-1-41-4': 'This would of course change the charge density seen by a comoving observer.', '1105.3840-1-41-5': 'There are several parameters governing the behavior of the solution: the cut parameter [MATH], the charge [MATH] which acts as scaling, and the position of the two charges [MATH].', '1105.3840-1-41-6': 'In general, there is one maximum associated with the position of the charge [MATH] as in classical electrodynamics, and there is also the second maximum due to the influence of the black hole, as depicted in Fig. [REF].', '1105.3840-1-41-7': 'Although for [MATH] an axially symmetric distribution is obtained, so, only one maximum is present in this case.', '1105.3840-1-42-0': 'In the general case the first maximum lies at [MATH] and the second at [MATH], i.e., on opposite the side of the black hole in the "original" spacetime.', '1105.3840-1-42-1': 'The second maximum can be understood using the membrane paradigm [CITATION] (alternatively by discussing the boundary conditions at the horizon [CITATION]).', '1105.3840-1-42-2': 'Interpreting the horizon as a conducting sphere, a polarization is to be expected due to the field of the test charge.', '1105.3840-1-42-3': 'This will lead to a fictitious charge density at the horizon, cf. [CITATION], as follows: [EQUATION] where the upper sign denotes the induced charge density for the charge [MATH] at [MATH] and the lower for the charge [MATH] at [MATH].', '1105.3840-1-42-4': 'In the following we discuss only the [MATH] case, the other one follows from the reflection symmetry.', '1105.3840-1-42-5': 'Assuming [MATH], the area of the conducting sphere characterized by [EQUATION] is negatively charged.', '1105.3840-1-42-6': 'The opening angle [MATH] as seen from the test charge [MATH] for this area was described in [CITATION].', '1105.3840-1-42-7': 'There it was also discussed, that the field lines emanating from [MATH] with an angle [MATH] are bent towards the horizon and cross it eventually.', '1105.3840-1-42-8': 'Field lines starting at [MATH] are first bent towards the horizon due to the opposite sign of its charge density and then bent away because of the change of sign in the polarization density.', '1105.3840-1-42-9': 'This leads to an increase/decrease of the tangential/normal components of the electric field in the disk close to the axis of the black hole facing [MATH].', '1105.3840-1-42-10': 'On the other side of the black hole the normal/tangential components of the electric field in the disk are increased/decreased.', '1105.3840-1-42-11': 'Thus in general, two maxima for the charge density are obtained on opposite sides of the axis.', '1105.3840-1-42-12': 'For the dipole density also two maxima are to be expected but both are lying on the side of the black hole facing [MATH].', '1105.3840-1-43-0': 'The surface charge current in [MATH] behaves for [MATH] like [EQUATION]', '1105.3840-1-43-1': 'The fall off is sufficiently fast to permit the definition of the total charge which can of course be read off from the unchanged asymptotic behavior of the field and thus is still [MATH].', '1105.3840-1-43-2': 'Having fixed [MATH], the parameter [MATH] can be used to slow down the decrease of the charge density as can be seen from [REF], but since the total charge must remain the same, the charge gets only "smeared out".', '1105.3840-1-44-0': 'Dipole disk', '1105.3840-1-45-0': 'To obtain continuous normal components of the Maxwell tensor one has to choose [MATH]; the surface current is given by [REF] and [REF].', '1105.3840-1-45-1': 'Again, the surface current allows two interpretations: the distribution is static or it consists of two counterrotating streams.', '1105.3840-1-45-2': 'The same parameters arise here as in the last case and the generic behavior for some specific values is depicted in Fig. [REF].', '1105.3840-1-45-3': 'The two maxima can again be understood on the grounds of the membrane paradigm as described above.', '1105.3840-1-45-4': 'The asymptotic behavior of the dipole density is [EQUATION]', '1105.3840-1-45-5': 'The relation between a monopole distribution and a dipole distribution is illustrated in the following.', '1105.3840-1-45-6': 'Let us consider the electric 4-potential and the jumps in the tangential components of the Maxwell tensor as produced from the jumps in the normal components of the dual of the Maxwell tensor, i.e., of a magnetic charge density.', '1105.3840-1-45-7': 'If we remove the [MATH]distribution terms of the field, we obtain a field which is generated by a magnetic current which satisfies [EQUATION]', '1105.3840-1-45-8': 'As stated in section [REF] for the general case, it is obvious here that the continuity equation is also satisfied for the magnetic surface current.', '1105.3840-1-45-9': 'The magnetic charge density of this current is vanishing which can be interpreted as two currents with opposite charges, one of them at rest for example.', '1105.3840-1-45-10': 'Since the field falls off sufficiently fast and no total charge is present this is the sole source of the field.', '1105.3840-1-46-0': 'It is again clear from the symmetry of the Maxwell equations that the calculations of this section can be repeated for a magnetic point charge in order to obtain a magnetic charge density or a magnetic dipole density.', '1105.3840-1-47-0': 'From our analysis it follows that similarly we could endow disks with test charges and dipoles which produce Kerr spacetimes [CITATION].', '1105.3840-1-48-0': 'We thank Tomas Ledvinka for helpful discussions.', '1105.3840-1-48-1': 'JB acknowledges the partial support from Grant No. GACR 202/09/0772 of the Czech Republic, of Grants No. LC06014 and No. MSM0021620860 of the Ministry of Education.', '1105.3840-1-48-2': 'NG was financially supported by the Grants No. GAUK.', '1105.3840-1-48-3': '22708 and No. GACR 205/09/H033.', '1105.3840-1-48-4': 'JB and NG are also grateful to the Albert Einstein Institute in Golm for the kind hospitality.', '1105.3840-1-48-5': 'ACG-P acknowledges the hospitality of the Institute of Theoretical Physics, Charles University (Prague) and the financial support from COLCIENCIAS, Colombia.'}

{'1105.3840-2-0-0': 'The discontinuities of electromagnetic test fields generated by general layers of electric and magnetic monopoles and dipoles are investigated in general curved spacetimes.', '1105.3840-2-0-1': 'The equivalence of electric currents and magnetic dipoles is discussed.', '1105.3840-2-0-2': 'The results are used to describe exact "Schwarzschild disk" solutions endowed with such sources.', '1105.3840-2-0-3': 'The resulting distributions of charge and dipole densities on the disks are corroborated using the membrane paradigm.', '1105.3840-2-1-0': '# Introduction', '1105.3840-2-2-0': 'The investigation of electromagnetic fields coupled to strong gravitational fields have an interest from both theoretical perspectives and from a variety of applications in astrophysics.', '1105.3840-2-2-1': 'Examples on the theory side include studies of gravitational collapse of charged configurations (see, e.g., [CITATION]), of the validity of the cosmic censorship conjecture [CITATION], of the existence and properties of quasi black holes and wormholes (for recent accounts, see, e.g., [CITATION] and references therein), membranes producing repulsive gravity [CITATION], and of many other issues.', '1105.3840-2-2-2': 'Very often analytical works employ, as tractable physical models, 2-dimensional thin shells sweeping out 3-dimensional timelike hypersurfaces.', '1105.3840-2-2-3': 'Recently, we used this idealization to construct "spherical gravitating condensers" - two concentric charged shells made of perfect fluids (satisfying energy conditions) under the condition that the electric field is non-vanishing only between the shells (see [CITATION] and further references on charged shells therein).', '1105.3840-2-3-0': 'The literature on electromagnetic fields in relativistic astrophysics is vast.', '1105.3840-2-3-1': 'Here we restrict ourselves to referring to several monographs dealing in detail with black-hole electrodynamics, e.g., [CITATION], and we mention the relatively recent work [CITATION] on electromagnetic fields around compact objects in which various papers are also summarized.', '1105.3840-2-3-2': 'In [CITATION] solutions to the Maxwell equations are presented both in the interior and outside a rotating neutron star and the matching conditions of the electromagnetic field at the stellar surface are analyzed in detail.', '1105.3840-2-3-3': 'The fields are not continuous across the stellar surface which gives rise to charges and currents.', '1105.3840-2-4-0': 'In the present paper we study electromagnetic sources distributed on shells in curved spacetimes in general, considering in particular possible discontinuities of the electromagnetic field across the shells.', '1105.3840-2-4-1': 'The sources discussed are layers with monopole or dipole currents.', '1105.3840-2-4-2': 'As far as we are aware electric or magnetic dipole layers and the matching conditions for their fields were not studied before in the context of general relativity.', '1105.3840-2-5-0': 'In general, in case of dipoles the currents and the electromagnetic field tensor will be distribution valued.', '1105.3840-2-5-1': 'This implies products of distributions in the stress-energy tensor.', '1105.3840-2-5-2': 'In order to avoid this, one can treat the electromagnetic field as a test field and solve the Maxwell equations in a given background metric.', '1105.3840-2-5-3': 'In many astrophysical situations this approach is well justified since typically the averaged energy density of the electromagnetic field is much smaller than that of the gravitational field.', '1105.3840-2-5-4': 'This approach is followed here and thus only the standard theory of generalized functions is used.', '1105.3840-2-6-0': 'In another work [CITATION] we discuss spherical thin shells endowed with arbitrary, not necessarily spherical distributions of charge or dipole densities in a Schwarzschild spacetime.', '1105.3840-2-6-1': 'There it was possible to employ the results of [CITATION] to calculate the fields directly and read off their discontinuities across the shell.', '1105.3840-2-6-2': 'In the present paper we generalize the jump conditions to general backgrounds and general hypersurfaces.', '1105.3840-2-6-3': 'As a by-product of those jump conditions the equivalence of the external fields of magnetic dipoles and certain electric charge currents is proven in general.', '1105.3840-2-6-4': 'For elementary dipoles this was already known in special backgrounds like the Kerr spacetime [CITATION].', '1105.3840-2-7-0': 'The jump conditions can be used to obtain massive disks endowed with charge and dipole densities using the Israel-Darboux formalism.', '1105.3840-2-7-1': 'In the examples studied here, we use the Schwarzschild disk spacetime as a background, cf. [CITATION].', '1105.3840-2-7-2': 'Therefore, we generate massive thin disks (Schwarzschild disks) endowed with either electric/magnetic test charges or electric/magnetic test dipoles.', '1105.3840-2-7-3': 'The surface currents are depicted and explained using the membrane paradigm.', '1105.3840-2-8-0': 'We use throughout the article the metric signature [MATH] and units in which [MATH].', '1105.3840-2-9-0': '# Monopole and dipole layers in general', '1105.3840-2-10-0': 'Although in the examples analyzed in section [REF] we use the Schwarzschild disk spacetimes as backgrounds, the results in the next section, i.e., the source terms and the jump conditions hold in a more general backgrounds.', '1105.3840-2-10-1': 'Of course, it has to admit a hypersurface, where the sources are situated, and the derivation of a dipole current requires, that a family of "parallel" hypersurfaces as defined below exists.', '1105.3840-2-10-2': 'In [CITATION] the case of charged massive shells were already discussed in full Einstein-Maxwell theory.', '1105.3840-2-10-3': 'Nonetheless, we consider test charges in our work, mainly to show in which cases the field generated by an arbitrary dipole distribution can be seen outside of the source as generated by moving charges.', '1105.3840-2-11-0': '## The 4-currents for charges or dipoles distributed on a shell', '1105.3840-2-12-0': 'Denoting by [MATH] the Maxwell tensor and by [MATH] its dual, the Maxwell equations in a complex form read as follows: [EQUATION] where [MATH] is a self-dual 2-form.', '1105.3840-2-12-1': 'The 4-current [MATH] consists of an electric part [MATH] and a magnetic part [MATH].', '1105.3840-2-12-2': 'If [MATH] is vanishing the imaginary part of the Maxwell equations [REF] allows us to introduce an electric 4-potential [MATH] such that [MATH].', '1105.3840-2-12-3': 'In case there are no electric sources present, we can analogously introduce a magnetic 4-potential [MATH] such that [MATH].', '1105.3840-2-12-4': 'In the vacuum region both 4-potentials can be defined and we denote the complex linear combination by [MATH].', '1105.3840-2-13-0': 'Timelike hypersurfaces [MATH] representing the history of charged 2-surfaces (shells) are discussed widely in the literature, see, e.g., [CITATION].', '1105.3840-2-13-1': 'We recall their main properties, in particular the form of the 4-current which will help us in formulating the expressions for the dipole current.', '1105.3840-2-13-2': 'Suppose the hypersurface [MATH] is described by [MATH], where [MATH] are coordinates in the two parts of the spacetime on the two sides of [MATH] and the index [MATH] denotes from which side a quantity is seen.', '1105.3840-2-13-3': 'The unit normal of [MATH] is given by [MATH], where [MATH] and [MATH] is chosen such that the normal points from [MATH] to [MATH].', '1105.3840-2-13-4': 'To shorten the notation we drop the index [MATH] in the following wherever no confusion is to be expected.', '1105.3840-2-13-5': 'If the intrinsic coordinates of [MATH] are called [MATH], where [MATH] runs from [MATH] to [MATH] and [MATH] is a timelike coordinate, then the tangential vectors are [MATH].', '1105.3840-2-13-6': 'A tensor field [MATH] can be projected onto these directions at [MATH] and we denote this by [EQUATION]', '1105.3840-2-13-7': 'The 4-current of an electrically charged monopole layer is given by [EQUATION] where [MATH] is the surface current of the electrical charges, [MATH] is the rest electric surface charge density and [MATH] the 4-velocity of the charged particles projected onto [MATH].', '1105.3840-2-14-0': 'Let us consider, at least locally, a Gaussian normal coordinate system generated by the geodesics [MATH] orthogonal to [MATH] and going out from points [MATH].', '1105.3840-2-14-1': 'Then the metric is block diagonal [EQUATION] and [MATH].', '1105.3840-2-14-2': 'The family of hypersurfaces [MATH], i.e., [MATH], are still orthogonal to the family of geodesics [MATH] and are at a proper distance [MATH] measured along [MATH] from [MATH].', '1105.3840-2-14-3': 'The coordinates [MATH] can be used as intrinsic coordinates, thus [MATH] with the Kronecker delta [MATH], and [MATH] denote the intrinsic metrics of the hypersurfaces [MATH] and [MATH] their determinant.', '1105.3840-2-14-4': 'A slightly more general form of the metric arises when the coordinate [MATH] along geodesics [MATH] is not measuring anymore the proper distance, implying [MATH].', '1105.3840-2-14-5': 'These "generalized" Gaussian normal coordinates are used in the examples in section [REF].', '1105.3840-2-14-6': 'The 4-current of a charge distribution on the surface in the generalized Gaussian normal coordinates is given by [EQUATION]', '1105.3840-2-14-7': 'To avoid confusion, we recall the definition of the 1-dimensional [MATH]-distribution: For any sufficiently smooth test function [MATH] the following integral over a spacetime region [MATH] in the generalized Gaussian normal coordinates reduces to the integral over [MATH] as follows: [EQUATION] where [MATH] is a spacetime volume element, [MATH] is a volume element of the hypersurface [MATH].', '1105.3840-2-15-0': 'We construct dipole layers from two oppositely charged monopole layers which are separated by a proper distance [MATH].', '1105.3840-2-15-1': 'For simplicity we derive the 4-current in the Gaussian normal coordinates [REF] and make a coordinate transformation to the generalized Gaussian normal coordinates subsequently.', '1105.3840-2-15-2': 'Dipole layers arise in the limit of vanishing [MATH] with a simultaneous limit to infinite (and opposite) rest surface charge densities of the two shells.', '1105.3840-2-15-3': 'This means we consider two hypersurfaces [MATH] and [MATH] endowed with surface rest charge densities of opposite sign [MATH] and [MATH] and with velocity fields [MATH] and [MATH], so giving rise to two 4-currents.', '1105.3840-2-15-4': 'Note that the change of the charge densities in the limit is such that it does not effect the velocity fields.', '1105.3840-2-15-5': 'Dipole layers result only in the limiting process [MATH] if certain properties hold true in the limit which, for simplicity, we assume to hold throughout the entire limiting procedure.', '1105.3840-2-15-6': 'The family of geodesics [MATH] gives locally rise to an equivalence relation of points similarly to [CITATION], i.e., [MATH] if there exist a point [MATH] such that [MATH], cf. Fig. [REF].', '1105.3840-2-16-0': 'Since the intrinsic coordinates are carried along the geodesics, equivalent points are characterized by the same intrinsic coordinates.', '1105.3840-2-16-1': 'Let us assume that two charge elements initially placed at two equivalent points [MATH] and [MATH] stay in course of there motion in equivalent points at every moment of time, e.g., the intrinsic time coordinate [MATH], cf. Fig. [REF].', '1105.3840-2-16-2': 'Then the coordinate velocities of the charge elements are the same, so we have [EQUATION]', '1105.3840-2-16-3': 'Analogously to the the equivalence of points, we also can, with each area element [MATH] in [MATH], associate an area element [MATH] in [MATH] which is cut out by the geodesics emanating from the boundary of [MATH], cf. Fig. [REF].', '1105.3840-2-16-4': 'Since the total charge of the dipole shell has to vanish, we suppose that the charge [MATH] enclosed in any area [MATH] (as seen by observers at rest with respect to the intrinsic coordinates) is the opposite of that enclosed in [MATH].', '1105.3840-2-16-5': 'This condition yields [EQUATION]', '1105.3840-2-16-6': 'As in the classical case, the charge density [MATH] as [MATH].', '1105.3840-2-16-7': 'The electrical rest dipole moment surface density is then naturally defined as [MATH].', '1105.3840-2-17-0': 'Therefore, the limiting procedure based on [REF], [REF] and [REF] yields the resulting dipole 4-current in the form [EQUATION]', '1105.3840-2-17-1': 'Of course, the total charge contained in any proper volume enclosing a part of the electric dipole layer is vanishing.', '1105.3840-2-17-2': 'Rewriting this in the generalized Gaussian normal coordinate system we find the 4-current to read [EQUATION] where [MATH] is the surface current of the electrical dipoles and [MATH] is the 4-velocity of the dipoles projected onto [MATH].', '1105.3840-2-17-3': 'Let us repeat the definition of the normal derivative of a [MATH]-distribution in a curved background.', '1105.3840-2-17-4': 'For an arbitrary, sufficiently smooth test function [MATH] the following holds: [EQUATION]', '1105.3840-2-17-5': 'Note that even though a derivative of the delta function appears, no metric functions have to be differentiated because of the integral definition of distributions where [MATH] appears and cancels with the only metric term in the 4-current depending on [MATH].', '1105.3840-2-17-6': 'Thus also metrics which are not [MATH] as they arise in the Israel formalism are allowed.', '1105.3840-2-17-7': 'It is also clear by construction and a short calculation, that the continuity equation for [MATH] implies that the surface currents [MATH] and [MATH] satisfy the continuity equation on [MATH].', '1105.3840-2-17-8': 'The currents for shells endowed with a magnetic charge or a magnetic dipole density are analogously defined, i.e., we just have to replace the index [MATH] by the index [MATH].', '1105.3840-2-18-0': '## Discontinuities in the potential and the fields', '1105.3840-2-19-0': 'As is well known from flat space, the jumps of various components of the fields or potentials across a surface are related to electromagnetic sources distributed on that surface.', '1105.3840-2-19-1': 'However, even in special relativity magnetic charges are usually not discussed.', '1105.3840-2-19-2': 'The jumps resulting from a dipole layer were, to the best of our knowledge, not discussed in curved spacetimes.', '1105.3840-2-19-3': 'We denote the jumps of a function [MATH] by [MATH].', '1105.3840-2-19-4': 'We study the four cases of electric/magnetic charged shells and electric/magnetic dipole shells separately.', '1105.3840-2-19-5': 'All of them can be obtained using the equivalence principle and Maxwell theory.', '1105.3840-2-20-0': 'In the case of an electrically charged surface Kuchar showed in [CITATION] (see also [CITATION]) that [EQUATION]', '1105.3840-2-20-1': 'Note that these equations are covariant with respect to a change of intrinsic coordinates [MATH] and scalars with respect to the coordinates [MATH].', '1105.3840-2-20-2': 'For the electric 4-potential in an appropriate Lorenz gauge it follows [EQUATION]', '1105.3840-2-20-3': 'The magnetic 4-potential [MATH] will in general not be continuous across [MATH] owing to the fact that it can only be introduced in the absence of electrical currents and, therefore, different potentials will occur in the lower and the upper half of the spacetime.', '1105.3840-2-20-4': 'Furthermore, introducing the potential [MATH] on both sides of [MATH] in different gauges will not change the external field, however, jumps in the potential are, as seen below, related to dipole densities and therefore describe a different physical system; in particular, the field in [MATH] is changed.', '1105.3840-2-21-0': 'In case of a shell endowed with magnetic charges the same equations as [REF] and [REF] hold for the dual of the Maxwell tensor and for the magnetic 4-potential in a Lorenz gauge: [EQUATION]', '1105.3840-2-21-1': 'For the Maxwell tensor it follows that the tangential components jump and the normal components are continuous: [EQUATION] where [MATH] is the volume form of [MATH] related to the induced metric [MATH] of [MATH] whereas [MATH] is the volume form of the spacetime.', '1105.3840-2-21-2': 'Tangential indices are raised and lowered with the induced metric and its inverse.', '1105.3840-2-22-0': 'Analogously, from the equivalence principle the discontinuities of the Maxwell tensor for electric and magnetic dipole densities follow: [EQUATION]', '1105.3840-2-22-1': 'Here the antisymmetrization in the derivatives of [MATH] is defined as [MATH].', '1105.3840-2-22-2': 'Note that a layer with a curl-free [MATH] will not produce a jump in the external field and thus the source can only be detected by observing the trajectories of particles crossing that layer, i.e., by measuring the internal field in [MATH].', '1105.3840-2-23-0': 'The 4-potentials satisfy in these cases the following jump conditions: [EQUATION]', '1105.3840-2-23-1': 'Additionally, the normal components of the Maxwell tensor have a [MATH]-like contribution [MATH], the field "between the two layers".', '1105.3840-2-23-2': 'In order to see this contribution, we insert the aforementioned jumps into the Maxwell equations and calculate the source.', '1105.3840-2-23-3': 'Using again Gaussian normal coordinates we start with an electric 4-potential which is discontinuous across [MATH] and calculate the sources.', '1105.3840-2-23-4': 'Hence, we write [EQUATION] with [MATH], which implies the Maxwell tensor to be [EQUATION]', '1105.3840-2-23-5': 'Inserting this into the Maxwell equations and using the jump conditions above yields [EQUATION] where the first two terms are the source terms for a charged layer and for a dipole layer.', '1105.3840-2-23-6': 'The last two terms are sources outside of [MATH], for instance a volume charge density.', '1105.3840-2-23-7': 'In the remainder we will assume that outside of the shell there are no magnetic or electric sources.', '1105.3840-2-24-0': '## The equivalence of electric charges and magnetic dipoles', '1105.3840-2-25-0': 'In flat spacetimes and also in certain cases of electromagnetism in curved backgrounds, e.g., in the Schwarzschild and the Kerr spacetimes [CITATION], the equivalence of the external field of a magnetic point dipole and of an infinitesimal electric charge current loop is known and often used.', '1105.3840-2-25-1': 'Naturally, it can also be easily shown that the external field of an electric point dipole is indistinguishable from that of an infinitesimal magnetic charge current loop.', '1105.3840-2-25-2': 'A similar result can be shown to hold in the case of layers of dipoles.', '1105.3840-2-25-3': 'In our Gaussian normal coordinates the dual of the Maxwell tensor for a shell endowed with magnetic dipoles reads as follows, cf. [REF] and [REF]: [EQUATION]', '1105.3840-2-25-4': 'Of course, the internal field must be changed to transform locally from sources in the form of magnetic dipoles to electric currents.', '1105.3840-2-25-5': 'However, if we remove the last term in [REF] from the field the external field remains unchanged.', '1105.3840-2-25-6': 'An observer outside can detect the difference only by examining trajectories of charged test particles crossing the shell.', '1105.3840-2-25-7': 'Furthermore, the jumps of the Maxwell tensor remain the same: [EQUATION]', '1105.3840-2-25-8': 'Using equation [REF], these jumps are produced by an electric current [MATH] if [EQUATION]', '1105.3840-2-25-9': 'The electric charge current defined in such a way can also be seen as a source.', '1105.3840-2-25-10': 'The continuity equation for [MATH] is satisfied trivially.', '1105.3840-2-25-11': 'However, since the charge density [MATH] does not need to vanish, electrical charges are introduced in general.', '1105.3840-2-25-12': 'The total charge is in principle detectable at infinity in the asymptotics of the field assuming it falls off sufficiently fast.', '1105.3840-2-25-13': 'Nonetheless, the total charge for a field generated by magnetic dipoles is vanishing.', '1105.3840-2-25-14': 'How is this to be resolved?', '1105.3840-2-25-15': 'The total electric charge [MATH] of [MATH] as seen for observers at rest with respect to the intrinsic coordinates is given by [EQUATION]', '1105.3840-2-25-16': "Together with equation [REF] and Stokes' theorem we obtain [EQUATION]", '1105.3840-2-25-17': 'The asymptotic behavior of the field implies a vanishing current at infinity.', '1105.3840-2-25-18': 'Thus, no total electric charge [MATH] will be present though "local" volumes can contain a net charge.', '1105.3840-2-25-19': 'This is also in correspondence with the known results for point dipoles.', '1105.3840-2-25-20': 'In a rest frame of a point dipole the external field can be seen as caused by an infinitesimal charge current loop with a vanishing time component.', '1105.3840-2-25-21': 'This is usually interpreted as two currents of positive and negative charges such that the charge densities in the rest frame of the dipole cancel each other and - for example the positive charges are at rest (ions of the conductor) and the negative charges (electrons) contribute to the current.', '1105.3840-2-25-22': 'However, in a general frame as used here the charge densities do not necessarily cancel anymore.', '1105.3840-2-25-23': 'To generalize this to layers these point dipoles have to be superposed and so the current loops.', '1105.3840-2-25-24': 'The net current can have a charge density because one is not in a comoving frame of the dipoles.', '1105.3840-2-26-0': 'If the fields do not fall off sufficiently fast, then the total charge of the shell need not vanish or be definable.', '1105.3840-2-26-1': 'In such a case charges can also be "placed at infinity" which is reflected by a corresponding boundary condition.', '1105.3840-2-26-2': 'An example is given in section [REF].', '1105.3840-2-27-0': 'The argument given above can be reversed and used to show that the external field of every electric charge surface current can also be produced by a charge density at rest in a given frame of reference and a magnetic dipole surface current.', '1105.3840-2-27-1': 'The integrability condition of equation [REF] for [MATH] is then equivalent to the continuity equation of the electric charge surface current.', '1105.3840-2-27-2': 'It is obvious that an analogous equivalence between electric dipoles and magnetic charges can be established.', '1105.3840-2-27-3': 'Except for this kind of non-uniqueness, the field and its sources are completely determined by the jump conditions [REF]-[REF].', '1105.3840-2-28-0': '# Schwarzschild disks with electric/magnetic charge and dipole density', '1105.3840-2-29-0': 'The Schwarzschild metric in Schwarzschild coordinates [MATH] reads [EQUATION]', '1105.3840-2-29-1': 'In [CITATION] massive disks of counterrotating matter, the "Schwarzschild disks", were constructed from this spacetime using the Israel-Darboux formalism and Weyl coordinates [MATH] [EQUATION]', '1105.3840-2-29-2': 'This was done by identifying the surfaces [MATH] and [MATH].', '1105.3840-2-29-3': 'From the jumps of the extrinsic curvature of the resulting surface an energy-momentum density of the disk was obtained.', '1105.3840-2-29-4': 'The disks are infinite but their mass is finite and the mass density decreases rapidly at large radii.', '1105.3840-2-29-5': 'We show here how to endow such disks with an electric/magnetic charge densities or electric/magnetic dipole densities in a test field approach.', '1105.3840-2-29-6': 'We demonstrate this with two examples using the asymptotic homogeneous field and the field generated by a point charge.', '1105.3840-2-29-7': 'The same can be done to model more general distributions using the general solutions of the Maxwell equations for test fields on a Schwarzschild background given in [CITATION].', '1105.3840-2-30-0': 'In [MATH] defined by [MATH], we introduce intrinsic coordinates [MATH] which coincide with the Schwarzschild coordinates [MATH] in the disk but are capitalized to prevent confusion.', '1105.3840-2-30-1': 'The components of the normal vector in Schwarzschild coordinates are given by [EQUATION] where again "[MATH]" denotes the quantities as seen from [MATH] and "[MATH]" as seen from [MATH].', '1105.3840-2-30-2': 'Note that [MATH].', '1105.3840-2-31-0': '## Asymptotically homogeneous electric and magnetic field', '1105.3840-2-32-0': 'The first test field to be discussed is the asymptotically homogeneous electric and magnetic field, for which the complex 4-potential and Maxwell tensor in Schwarzschild coordinates read as follows (see, e.g., [CITATION]) [EQUATION]', '1105.3840-2-32-1': 'The 4-potential is in fact not given in [CITATION] but can be calculated easily.', '1105.3840-2-32-2': 'Assume the field in the upper/lower half is parametrized by [MATH] and [MATH].', '1105.3840-2-32-3': 'The jumps of the potential across [MATH] are given by [EQUATION]', '1105.3840-2-32-4': 'As it should be according to the equations [REF], [REF] and [REF], the orthogonal component of the potential is continuous.', '1105.3840-2-32-5': 'Furthermore, the radial component is continuous as well, i.e., the dipole currents (electric or magnetic) in the radial direction are vanishing.', '1105.3840-2-32-6': 'The dipole density approaches a constant value, so does the current in the [MATH] direction, as one can expect from the analogous result obtained in Maxwell theory in flat space or after setting the mass [MATH] to zero in the equations above.', '1105.3840-2-32-7': 'The jumps in the fields read [EQUATION]', '1105.3840-2-32-8': 'Using equations [REF] and [REF]-[REF] we observe again that for electric/magnetic charges the radial current is vanishing and that the electric and magnetic charges do rotate around the axis.', '1105.3840-2-32-9': 'The current is vanishing for [MATH].', '1105.3840-2-32-10': 'The total electric or magnetic charge of such a system will be infinite.', '1105.3840-2-32-11': 'This will be different for the case of the field discussed in the next subsection.', '1105.3840-2-33-0': 'We will now treat the case of electric monopoles and magnetic dipoles independently of the case of magnetic monopoles and electric dipoles.', '1105.3840-2-33-1': 'Afterwards the results can be superposed.', '1105.3840-2-34-0': 'Electric monopoles or magnetic dipoles', '1105.3840-2-35-0': 'This case is obtained for [MATH] and [MATH], together with [MATH] and [MATH].', '1105.3840-2-35-1': 'This leads to a surface current [EQUATION]', '1105.3840-2-35-2': 'In the classical case [MATH] the charges are at rest with a charge density equal to the first factor in the first equation.', '1105.3840-2-35-3': 'The discontinuities in the magnetic potential and the tangential components of the dual of the Maxwell tensor are in this case understood as being caused by the discontinuities of the orthogonal components of the Maxwell tensor and the presence of the electric monopole layer and, hence, the impossibility to introduce a magnetic potential globally.', '1105.3840-2-35-4': 'Looking at the classical case [MATH], the principal problem mentioned after equation [REF] becomes apparent when dealing with fields which are not falling off sufficiently fast at infinity.', '1105.3840-2-35-5': 'The axial current vanishes in this limit and thus cannot cause the magnetic field.', '1105.3840-2-35-6': 'The existing magnetic field can be explained by "magnetic charges or electric currents at infinity".', '1105.3840-2-35-7': 'Therefore, the disk is not the only source of the external field.', '1105.3840-2-35-8': 'This problem does not occur for fields which are falling off sufficiently fast.', '1105.3840-2-35-9': 'Such are discussed in the next example.', '1105.3840-2-35-10': 'However, for completeness we give here the 4-current provided that the discontinuities are interpreted as the result of a magnetic dipole layer according to equation [REF]: [EQUATION]', '1105.3840-2-35-11': 'Here the constants [MATH] and [MATH] are chosen such that the current is not singular at the axis and the dipole density vanishes at infinity.', '1105.3840-2-36-0': 'Analogously, we can study disks endowed with a magnetic charge density or electric dipole density by setting [MATH] and [MATH].', '1105.3840-2-36-1': 'The results are very similar to [REF] and [REF]; they can be obtained by a substitution [MATH] and [MATH] into [REF] and [REF].', '1105.3840-2-37-0': '## Disks generated by point charges', '1105.3840-2-38-0': 'The question whether a field is generated solely by disks or also by sources at infinity is circumvented if a solution is chosen such that it falls off sufficiently fast at infinity.', '1105.3840-2-38-1': 'We now consider the electromagnetic field produced by a point charge [MATH] situated in an arbitrary position [MATH].', '1105.3840-2-38-2': 'The electric 4-potential for such a point charge was given in [CITATION], and in closed form by Linet in [CITATION].', '1105.3840-2-38-3': 'It reads: [EQUATION]', '1105.3840-2-38-4': 'We consider two different test fields in the Schwarzschild spacetimes: the field produced by a point charge at [MATH] and the field produced by a point charge at [MATH].', '1105.3840-2-38-5': 'In the spacetime with the first test field we make a cut at such [MATH] that the black hole and the point charge are below the cut.', '1105.3840-2-38-6': 'For the second test field the cut is made at [MATH] such that the charge and the black hole are above the cut.', '1105.3840-2-38-7': 'After identifying the two hypersurfaces [MATH] there is no black hole or point charge in the spacetime, rather a massive disk with electromagnetic sources.', '1105.3840-2-38-8': 'However, the electromagnetic field outside the disk and thus the sources can be understood using the field lines in the "original" spacetime for the "original" test field, i.e., the Schwarzschild black hole spacetime with a point charge.', '1105.3840-2-38-9': 'This point of view is employed several times in the following; e.g., the charge density of the disk is explained by referring to the "original" black hole and its polarization.', '1105.3840-2-39-0': 'The fields of the two point charges can be obtained from the 4-potential [REF] in a straightforward way and so also the jumps.', '1105.3840-2-39-1': 'In order to obtain a layer endowed with either only charges or dipoles we have to require that the point charges have to be located symmetrically in the original spacetime, i.e., [MATH], as well as that the charges are either equal, [MATH], or opposite, [MATH].', '1105.3840-2-39-2': 'Because of the axially symmetry of the spacetime we can set [MATH].', '1105.3840-2-39-3': 'The jumps evaluate to [EQUATION]', '1105.3840-2-39-4': 'Note that functions [MATH] and [MATH] have to be evaluated at the respective [MATH] with the respective point charge.', '1105.3840-2-39-5': 'However, it holds that [MATH], so the same holds for [MATH].', '1105.3840-2-39-6': 'Therefore, functions [MATH], [MATH] and [MATH] should be read as functions with the argument [MATH].', '1105.3840-2-39-7': 'The jumps of the tangential components of the Maxwell tensor can be inferred from the jumps of the 4-potential.', '1105.3840-2-39-8': 'We can now discuss two cases - a monopole layer and a dipole layer.', '1105.3840-2-40-0': 'Electric monopoles or magnetic dipoles', '1105.3840-2-41-0': 'In order to obtain continuous tangential components of the 4-potential we have to set [MATH].', '1105.3840-2-41-1': 'Then the surface 3-current can be read off [REF] and [REF].', '1105.3840-2-41-2': 'The only non-vanishing component is [MATH].', '1105.3840-2-41-3': 'However, it is possible to consider two counterrotating streams with an equal charge, cf. with the underlying matter currents in the Schwarzschild disk [CITATION].', '1105.3840-2-41-4': 'This would of course change the charge density seen by a comoving observer.', '1105.3840-2-41-5': 'There are several parameters governing the behavior of the solution: the cut parameter [MATH], the charge [MATH] which acts as scaling, and the position of the two charges [MATH].', '1105.3840-2-41-6': 'In general, there is one maximum associated with the position of the charge [MATH] as in classical electrodynamics, and there is also the second maximum due to the influence of the black hole, as depicted in Fig. [REF].', '1105.3840-2-41-7': 'Although for [MATH] an axially symmetric distribution is obtained, so, only one maximum is present in this case.', '1105.3840-2-42-0': 'In the general case the first maximum lies at [MATH] and the second at [MATH], i.e., on opposite the side of the black hole in the "original" spacetime.', '1105.3840-2-42-1': 'The second maximum can be understood using the membrane paradigm [CITATION] (alternatively by discussing the boundary conditions at the horizon [CITATION]).', '1105.3840-2-42-2': 'Interpreting the horizon as a conducting sphere, a polarization is to be expected due to the field of the test charge.', '1105.3840-2-42-3': 'This will lead to a fictitious charge density at the horizon, cf. [CITATION], as follows: [EQUATION] where the upper sign denotes the induced charge density for the charge [MATH] at [MATH] and the lower for the charge [MATH] at [MATH].', '1105.3840-2-42-4': 'In the following we discuss only the [MATH] case, the other one follows from the reflection symmetry.', '1105.3840-2-42-5': 'Assuming [MATH], the area of the conducting sphere characterized by [EQUATION] is negatively charged.', '1105.3840-2-42-6': 'The opening angle [MATH] as seen from the test charge [MATH] for this area was described in [CITATION].', '1105.3840-2-42-7': 'There it was also discussed, that the field lines emanating from [MATH] with an angle [MATH] are bent towards the horizon and cross it eventually.', '1105.3840-2-42-8': 'Field lines starting at [MATH] are first bent towards the horizon due to the opposite sign of its charge density and then bent away because of the change of sign in the polarization density.', '1105.3840-2-42-9': 'This leads to an increase/decrease of the tangential/normal components of the electric field in the disk close to the axis of the black hole facing [MATH].', '1105.3840-2-42-10': 'On the other side of the black hole the normal/tangential components of the electric field in the disk are increased/decreased.', '1105.3840-2-42-11': 'Thus in general, two maxima for the charge density are obtained on opposite sides of the axis.', '1105.3840-2-42-12': 'For the dipole density also two maxima are to be expected but both are lying on the side of the black hole facing [MATH].', '1105.3840-2-43-0': 'The surface charge current in [MATH] behaves for [MATH] like [EQUATION]', '1105.3840-2-43-1': 'The fall off is sufficiently fast to permit the definition of the total charge which can of course be read off from the unchanged asymptotic behavior of the field and thus is still [MATH].', '1105.3840-2-43-2': 'Having fixed [MATH], the parameter [MATH] can be used to slow down the decrease of the charge density as can be seen from [REF], but since the total charge must remain the same, the charge gets only "smeared out".', '1105.3840-2-44-0': 'Dipole disk', '1105.3840-2-45-0': 'To obtain continuous normal components of the Maxwell tensor one has to choose [MATH]; the surface current is given by [REF] and [REF].', '1105.3840-2-45-1': 'Again, the surface current allows two interpretations: the distribution is static or it consists of two counterrotating streams.', '1105.3840-2-45-2': 'The same parameters arise here as in the last case and the generic behavior for some specific values is depicted in Fig. [REF].', '1105.3840-2-45-3': 'The two maxima can again be understood on the grounds of the membrane paradigm as described above.', '1105.3840-2-45-4': 'The asymptotic behavior of the dipole density is [EQUATION]', '1105.3840-2-45-5': 'The relation between a monopole distribution and a dipole distribution is illustrated in the following.', '1105.3840-2-45-6': 'Let us consider the electric 4-potential and the jumps in the tangential components of the Maxwell tensor as produced from the jumps in the normal components of the dual of the Maxwell tensor, i.e., of a magnetic charge density.', '1105.3840-2-45-7': 'If we remove the [MATH]distribution terms of the field, we obtain a field which is generated by a magnetic current which satisfies [EQUATION]', '1105.3840-2-45-8': 'As stated in section [REF] for the general case, it is obvious here that the continuity equation is also satisfied for the magnetic surface current.', '1105.3840-2-45-9': 'The magnetic charge density of this current is vanishing which can be interpreted as two currents with opposite charges, one of them at rest for example.', '1105.3840-2-45-10': 'Since the field falls off sufficiently fast and no total charge is present this is the sole source of the field.', '1105.3840-2-46-0': 'It is again clear from the symmetry of the Maxwell equations that the calculations of this section can be repeated for a magnetic point charge in order to obtain a magnetic charge density or a magnetic dipole density.', '1105.3840-2-47-0': 'From our analysis it follows that similarly we could endow disks with test charges and dipoles which produce Kerr spacetimes [CITATION].', '1105.3840-2-48-0': 'We thank Tomas Ledvinka for helpful discussions.', '1105.3840-2-48-1': 'JB acknowledges the partial support from Grant No. GACR 202/09/0772 of the Czech Republic, of Grants No. LC06014 and No. MSM0021620860 of the Ministry of Education.', '1105.3840-2-48-2': 'NG was financially supported by the Grants No. GAUK.', '1105.3840-2-48-3': '22708 and No. GACR 205/09/H033.', '1105.3840-2-48-4': 'JB and NG are also grateful to the Albert Einstein Institute in Golm for the kind hospitality.', '1105.3840-2-48-5': 'ACG-P acknowledges the hospitality of the Institute of Theoretical Physics, Charles University (Prague) and the financial support from COLCIENCIAS, Colombia.'}

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'1105.3840-2-5-1'], ['1105.3840-1-5-2', '1105.3840-2-5-2'], ['1105.3840-1-5-3', '1105.3840-2-5-3'], ['1105.3840-1-5-4', '1105.3840-2-5-4'], ['1105.3840-1-30-0', '1105.3840-2-30-0'], ['1105.3840-1-30-1', '1105.3840-2-30-1'], ['1105.3840-1-22-0', '1105.3840-2-22-0'], ['1105.3840-1-22-1', '1105.3840-2-22-1'], ['1105.3840-1-22-2', '1105.3840-2-22-2'], ['1105.3840-1-38-0', '1105.3840-2-38-0'], ['1105.3840-1-38-1', '1105.3840-2-38-1'], ['1105.3840-1-38-2', '1105.3840-2-38-2'], ['1105.3840-1-38-4', '1105.3840-2-38-4'], ['1105.3840-1-38-5', '1105.3840-2-38-5'], ['1105.3840-1-38-6', '1105.3840-2-38-6'], ['1105.3840-1-38-7', '1105.3840-2-38-7'], ['1105.3840-1-38-8', '1105.3840-2-38-8'], ['1105.3840-1-38-9', '1105.3840-2-38-9'], ['1105.3840-1-26-0', '1105.3840-2-26-0'], ['1105.3840-1-26-1', '1105.3840-2-26-1'], ['1105.3840-1-26-2', '1105.3840-2-26-2'], ['1105.3840-1-27-0', '1105.3840-2-27-0'], ['1105.3840-1-27-1', '1105.3840-2-27-1'], ['1105.3840-1-27-2', '1105.3840-2-27-2'], ['1105.3840-1-27-3', '1105.3840-2-27-3'], ['1105.3840-1-4-0', '1105.3840-2-4-0'], ['1105.3840-1-4-1', '1105.3840-2-4-1'], ['1105.3840-1-4-2', '1105.3840-2-4-2'], ['1105.3840-1-12-0', '1105.3840-2-12-0'], ['1105.3840-1-12-1', '1105.3840-2-12-1'], ['1105.3840-1-12-2', '1105.3840-2-12-2'], ['1105.3840-1-12-3', '1105.3840-2-12-3'], ['1105.3840-1-12-4', '1105.3840-2-12-4'], ['1105.3840-1-29-0', '1105.3840-2-29-0'], ['1105.3840-1-29-1', '1105.3840-2-29-1'], ['1105.3840-1-29-2', '1105.3840-2-29-2'], ['1105.3840-1-29-3', '1105.3840-2-29-3'], ['1105.3840-1-29-4', '1105.3840-2-29-4'], ['1105.3840-1-29-5', '1105.3840-2-29-5'], ['1105.3840-1-29-6', '1105.3840-2-29-6'], ['1105.3840-1-29-7', '1105.3840-2-29-7'], ['1105.3840-1-47-0', '1105.3840-2-47-0'], ['1105.3840-1-42-0', '1105.3840-2-42-0'], ['1105.3840-1-42-1', '1105.3840-2-42-1'], ['1105.3840-1-42-2', '1105.3840-2-42-2'], ['1105.3840-1-42-3', '1105.3840-2-42-3'], ['1105.3840-1-42-4', '1105.3840-2-42-4'], ['1105.3840-1-42-5', '1105.3840-2-42-5'], ['1105.3840-1-42-6', '1105.3840-2-42-6'], ['1105.3840-1-42-7', '1105.3840-2-42-7'], ['1105.3840-1-42-8', '1105.3840-2-42-8'], ['1105.3840-1-42-9', '1105.3840-2-42-9'], ['1105.3840-1-42-10', '1105.3840-2-42-10'], ['1105.3840-1-42-11', '1105.3840-2-42-11'], ['1105.3840-1-42-12', '1105.3840-2-42-12'], ['1105.3840-1-0-0', '1105.3840-2-0-0'], ['1105.3840-1-0-1', '1105.3840-2-0-1'], ['1105.3840-1-0-2', '1105.3840-2-0-2'], ['1105.3840-1-0-3', '1105.3840-2-0-3'], ['1105.3840-1-19-0', '1105.3840-2-19-0'], ['1105.3840-1-19-1', '1105.3840-2-19-1'], ['1105.3840-1-19-2', '1105.3840-2-19-2'], ['1105.3840-1-19-3', '1105.3840-2-19-3'], ['1105.3840-1-19-4', '1105.3840-2-19-4'], ['1105.3840-1-19-5', '1105.3840-2-19-5'], ['1105.3840-1-21-0', '1105.3840-2-21-0'], ['1105.3840-1-21-1', '1105.3840-2-21-1'], ['1105.3840-1-21-2', '1105.3840-2-21-2'], ['1105.3840-1-46-0', '1105.3840-2-46-0'], ['1105.3840-1-3-0', '1105.3840-2-3-0'], ['1105.3840-1-3-1', '1105.3840-2-3-1'], ['1105.3840-1-3-2', '1105.3840-2-3-2'], ['1105.3840-1-3-3', '1105.3840-2-3-3'], ['1105.3840-1-32-0', '1105.3840-2-32-0'], ['1105.3840-1-32-1', '1105.3840-2-32-1'], ['1105.3840-1-32-2', '1105.3840-2-32-2'], ['1105.3840-1-32-3', '1105.3840-2-32-3'], ['1105.3840-1-32-4', '1105.3840-2-32-4'], ['1105.3840-1-32-5', '1105.3840-2-32-5'], ['1105.3840-1-32-6', '1105.3840-2-32-6'], ['1105.3840-1-32-7', '1105.3840-2-32-7'], ['1105.3840-1-32-8', '1105.3840-2-32-8'], ['1105.3840-1-32-9', '1105.3840-2-32-9'], ['1105.3840-1-32-10', '1105.3840-2-32-10'], ['1105.3840-1-32-11', '1105.3840-2-32-11'], ['1105.3840-1-15-0', '1105.3840-2-15-0'], ['1105.3840-1-15-1', '1105.3840-2-15-1'], ['1105.3840-1-15-2', '1105.3840-2-15-2'], ['1105.3840-1-15-3', '1105.3840-2-15-3'], ['1105.3840-1-15-4', '1105.3840-2-15-4'], ['1105.3840-1-15-5', '1105.3840-2-15-5'], ['1105.3840-1-15-6', '1105.3840-2-15-6'], ['1105.3840-1-13-0', '1105.3840-2-13-0'], ['1105.3840-1-13-1', '1105.3840-2-13-1'], ['1105.3840-1-13-2', '1105.3840-2-13-2'], ['1105.3840-1-13-3', '1105.3840-2-13-3'], ['1105.3840-1-13-4', '1105.3840-2-13-4'], ['1105.3840-1-13-5', '1105.3840-2-13-5'], ['1105.3840-1-13-6', '1105.3840-2-13-6'], ['1105.3840-1-13-7', '1105.3840-2-13-7'], ['1105.3840-1-43-0', '1105.3840-2-43-0'], ['1105.3840-1-43-1', '1105.3840-2-43-1'], ['1105.3840-1-43-2', '1105.3840-2-43-2'], ['1105.3840-1-35-0', '1105.3840-2-35-0'], ['1105.3840-1-35-1', '1105.3840-2-35-1'], ['1105.3840-1-35-2', '1105.3840-2-35-2'], ['1105.3840-1-35-3', '1105.3840-2-35-3'], ['1105.3840-1-35-4', '1105.3840-2-35-4'], ['1105.3840-1-35-5', '1105.3840-2-35-5'], ['1105.3840-1-35-6', '1105.3840-2-35-6'], ['1105.3840-1-35-7', '1105.3840-2-35-7'], ['1105.3840-1-35-8', '1105.3840-2-35-8'], ['1105.3840-1-35-9', '1105.3840-2-35-9'], ['1105.3840-1-35-10', '1105.3840-2-35-10'], ['1105.3840-1-35-11', '1105.3840-2-35-11'], ['1105.3840-1-14-0', '1105.3840-2-14-0'], ['1105.3840-1-14-1', '1105.3840-2-14-1'], ['1105.3840-1-14-2', '1105.3840-2-14-2'], ['1105.3840-1-14-3', '1105.3840-2-14-3'], ['1105.3840-1-14-4', '1105.3840-2-14-4'], ['1105.3840-1-14-5', '1105.3840-2-14-5'], ['1105.3840-1-14-6', '1105.3840-2-14-6'], ['1105.3840-1-14-7', '1105.3840-2-14-7'], ['1105.3840-1-45-0', '1105.3840-2-45-0'], ['1105.3840-1-45-1', '1105.3840-2-45-1'], ['1105.3840-1-45-2', '1105.3840-2-45-2'], ['1105.3840-1-45-3', '1105.3840-2-45-3'], ['1105.3840-1-45-4', '1105.3840-2-45-4'], ['1105.3840-1-45-5', '1105.3840-2-45-5'], ['1105.3840-1-45-6', '1105.3840-2-45-6'], ['1105.3840-1-45-7', '1105.3840-2-45-7'], ['1105.3840-1-45-8', '1105.3840-2-45-8'], ['1105.3840-1-45-9', '1105.3840-2-45-9'], ['1105.3840-1-45-10', '1105.3840-2-45-10'], ['1105.3840-1-10-0', '1105.3840-2-10-0'], ['1105.3840-1-10-1', '1105.3840-2-10-1'], ['1105.3840-1-10-2', '1105.3840-2-10-2'], ['1105.3840-1-10-3', '1105.3840-2-10-3']]

[]

[]

[]

[]

['1105.3840-1-30-2', '1105.3840-1-34-0', '1105.3840-1-38-3', '1105.3840-1-40-0', '1105.3840-1-44-0', '1105.3840-1-48-1', '1105.3840-1-48-2', '1105.3840-1-48-3', '1105.3840-2-30-2', '1105.3840-2-34-0', '1105.3840-2-38-3', '1105.3840-2-40-0', '1105.3840-2-44-0', '1105.3840-2-48-1', '1105.3840-2-48-2', '1105.3840-2-48-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1105.3840

1805.06415

{'1805.06415-1-0-0': 'We consider the nonlinear Schrodinger equation [EQUATION] for [MATH]-subcritical or critical nonlinearities: [MATH].', '1805.06415-1-0-1': 'Under the additional technical assumptions [MATH] (and thus [MATH]), we construct [MATH] solutions that blow up in finite time with explicit blow-up profiles and blow-up rates.', '1805.06415-1-0-2': 'In particular, blowup can occur at any given finite set of points of [MATH].', '1805.06415-1-1-0': 'The construction involves explicit functions [MATH], solutions of the ordinary differential equation [MATH].', '1805.06415-1-1-1': 'In the simplest case, [MATH] for [MATH], [MATH].', '1805.06415-1-1-2': 'For [MATH] sufficiently large, [MATH] satisfies [MATH] close to the blow-up point [MATH], so that it is a suitable approximate solution of the problem.', '1805.06415-1-1-3': 'To construct an actual solution [MATH] close to [MATH], we use energy estimates and a compactness argument.', '1805.06415-1-2-0': '# Introduction', '1805.06415-1-3-0': 'We consider the nonlinear Schrodinger equation [EQUATION] on [MATH], where [MATH] satisfies [EQUATION]', '1805.06415-1-3-1': 'Under assumption [REF], equation [REF] is [MATH]-subcritical or critical, so that the corresponding Cauchy problem is locally well posed in [MATH] - see e.g. [CITATION].', '1805.06415-1-4-0': 'Equation [REF] is a member of the more general family of complex Ginzburg-Landau equations [EQUATION]', '1805.06415-1-4-1': 'It is proved in [CITATION] that (for [MATH]-subcritical [MATH]) equation [REF] has no global in time [MATH] solution that remains bounded in [MATH].', '1805.06415-1-4-2': 'In other words, every [MATH] solution blows up, in finite or infinite time.', '1805.06415-1-4-3': 'The question of finite-time blowup is left open in [CITATION].', '1805.06415-1-4-4': "It seems that no standard argument based on obstruction to global existence (Levine's method, variance argument) is applicable.", '1805.06415-1-5-0': 'The purpose of this article is to construct solutions of [REF] that blow up in finite time.', '1805.06415-1-5-1': 'For technical reasons we require [EQUATION] (The condition [MATH] is used in the proof of estimates [REF]-[REF], the stronger condition [MATH] is used in formula [REF].)', '1805.06415-1-5-2': 'Conditions [REF]-[REF] impose [MATH].', '1805.06415-1-5-3': 'More precisely, the allowed range of powers is [EQUATION]', '1805.06415-1-5-4': 'Our first blow-up result, related to single point blowup with a simple asymptotic profile is the following.', '1805.06415-1-6-0': 'Theorem [REF] is a particular case of a more general result (Theorem [REF] below), where asymptotic profiles more general than [REF] are allowed.', '1805.06415-1-6-1': 'Before stating precisely this more general result, we need to make precise our assumptions on the asymptotic profile.', '1805.06415-1-6-2': 'We consider an integer [MATH], real numbers [MATH], [MATH], [MATH] where [MATH] is a multi-index with [MATH], and points [MATH] such that [EQUATION]', '1805.06415-1-6-3': 'Let [MATH] satisfy [EQUATION]', '1805.06415-1-6-4': 'Our main result is the following [EQUATION]', '1805.06415-1-7-0': 'Here are some comments on Theorems [REF] and [REF].', '1805.06415-1-7-1': '[(i)]', '1805.06415-1-8-0': 'Theorem [REF] presents the simplest result with the choice of only two parameters [MATH] and [MATH].', '1805.06415-1-8-1': 'The parameter [MATH] has to be taken sufficiently large so that the ansatz [MATH] satisfies [MATH] (a similar strategy is used in [CITATION], see the comments below).', '1805.06415-1-8-2': 'Note that the choice of [MATH] determines the blow-up rates in [REF]-[REF].', '1805.06415-1-8-3': 'More parameters can chosen in Theorem [REF], which allows arbitrary locations for the blow-up points and flexibility on the blow-up rates.', '1805.06415-1-8-4': 'It is easy to construct explicitly functions [MATH] satisfying [REF].', '1805.06415-1-9-0': 'It follows from [REF] and [REF] that both [MATH] and [MATH] blow up as [MATH].', '1805.06415-1-10-0': 'The asymptotic profile [MATH] of [MATH] as [MATH], given by [REF], has the following properties: [MATH], [MATH] is bounded as [MATH] and [MATH] is [MATH] except at the points [MATH], where [MATH] has a singularity like [MATH].', '1805.06415-1-11-0': 'The solutions constructed in Theorems [REF] and [REF] are global for [MATH].', '1805.06415-1-11-1': 'Actually, this is a general fact : we show in Proposition [REF] that equation [REF] is globally well-posed in [MATH] in the negative sense of time.', '1805.06415-1-12-0': 'We prove Theorem [REF] by using the strategy of [CITATION].', '1805.06415-1-12-1': 'More precisely, we consider the sequence [MATH] of solutions of [REF] defined by [MATH], where [MATH] is defined by [REF].', '1805.06415-1-12-2': 'It follows that [MATH] is defined on [MATH].', '1805.06415-1-12-3': 'Since [MATH], and [MATH] is small compared to [MATH], [MATH] is almost a solution of [REF].', '1805.06415-1-12-4': 'Following the idea of [CITATION] (see also [CITATION] in the blow-up context) we estimate the solutions [MATH] by energy arguments.', '1805.06415-1-12-5': 'Note that the nice behavior of equation [REF] backwards in time, already discussed in Remark [REF] [REF], is important in this step.', '1805.06415-1-12-6': 'Finally, passing to the limit as [MATH] yields the solution [MATH] of Theorem [REF].', '1805.06415-1-13-0': 'The solution [MATH] given by Theorem [REF] blows up at [MATH] like the function [MATH] defined by [REF].', '1805.06415-1-13-1': 'Since [MATH], we see that the solution [MATH] displays an ODE-type blowup.', '1805.06415-1-14-0': 'We recall that ODE-type blowup has been intensively studied for several other nonlinear equations.', '1805.06415-1-14-1': 'For the nonlinear heat equation, the type of blowup obtained in Theorems [REF] and [REF] is called flat blowup in the literature and was recently investigated independently of our work in [CITATION] (see also previous references there) with applications to the Burgers equation.', '1805.06415-1-14-2': 'Even if the blow-up profile is identical (see 4.1 of [CITATION]), the strategy to proceed with the construction of an actual solution of the equation is different in this paper.', '1805.06415-1-15-0': 'Apart from such unstable forms of blowup, ODE-type blowup was also much studied as a stable form of blowup.', '1805.06415-1-15-1': 'We refer to [CITATION] and to references there for results in the parabolic context.', '1805.06415-1-15-2': 'For the semilinear wave and quasilinear wave equations, we refer e.g. to [CITATION] and to the references there.', '1805.06415-1-16-0': 'The rest of the paper is organized as follows.', '1805.06415-1-16-1': 'In Section [REF], we study the Cauchy problem for the backwards version of equation [REF], and in Section [REF], we derive various estimates on the function [MATH].', '1805.06415-1-16-2': 'Section [REF] is devoted to the construction and estimates of the approximate blow-up solutions, and the proof of Theorem [REF] is completed in Section [REF] by passing to the limit in the approximate solutions.', '1805.06415-1-17-0': '# The backwards equation', '1805.06415-1-18-0': 'In this section, we prove that the Cauchy problem for the backwards equation obtained by changing [MATH] to [MATH] in [REF] is globally well-posed in [MATH], under assumption [REF].', '1805.06415-1-19-0': 'Assumption [REF] ensures that equation [REF] is [MATH]-subcritical or critical, so that local well-posedness in [MATH] follows from standard arguments.', '1805.06415-1-19-1': '(See e.g. [CITATION].)', '1805.06415-1-20-0': 'We now prove that the Cauchy problem [REF] is globally well posed in [MATH].', '1805.06415-1-20-1': 'Indeed, let [MATH], and let [MATH] be the maximal existence time of the corresponding solution [MATH] of [REF].', '1805.06415-1-20-2': 'Multiplying the equation by [MATH] and taking the real part yields [EQUATION] so that [MATH] for all [MATH].', '1805.06415-1-20-3': 'Next, we multiply the equation by [MATH] and take the real part.', '1805.06415-1-20-4': 'We obtain (see [REF]-[REF] below) [EQUATION] so that [MATH] for all [MATH].', '1805.06415-1-20-5': 'These formal calculations can be justified by standard arguments, see e.g. [CITATION].', '1805.06415-1-20-6': 'This proves global existence in the subcritical case [MATH].', '1805.06415-1-20-7': 'In the critical case [MATH] and [MATH], we use the right-hand side of [REF] to obtain [EQUATION]', '1805.06415-1-20-8': "Since [EQUATION] we deduce by Sobolev's inequality that [EQUATION]", '1805.06415-1-20-9': 'We define [MATH] by [EQUATION] so that [EQUATION]', '1805.06415-1-20-10': 'Since by [REF] [EQUATION] we deduce from [REF] that [MATH].', '1805.06415-1-20-11': '(See [CITATION].)', '1805.06415-1-21-0': 'Finally, we prove the stronger uniqueness property, so we consider [MATH] and two solutions [MATH] of [REF] on [MATH].', '1805.06415-1-21-1': 'Setting [EQUATION] we see that [MATH] satisfies [EQUATION] in [MATH].', '1805.06415-1-21-2': 'Moreover, the map [MATH] is in [MATH] and [EQUATION] for a.a. [MATH].', '1805.06415-1-21-3': 'Taking the [MATH] duality product of equation [REF] with [MATH] and applying [REF], we deduce that for a.a. [MATH] [EQUATION]', '1805.06415-1-21-4': 'We recall that [EQUATION] for all [MATH].', '1805.06415-1-21-5': 'Indeed, [EQUATION]', '1805.06415-1-21-6': 'Since [MATH], we deduce from [REF] and [REF] that [MATH] on [MATH].', '1805.06415-1-22-0': '# Estimates of [MATH]', '1805.06415-1-23-0': 'In this section, we establish various estimates on the function [MATH] defined by [REF].', '1805.06415-1-24-0': 'Let [MATH] satisfy [REF]-[REF].', '1805.06415-1-24-1': 'If [MATH] is given by [REF], then [MATH] and [EQUATION] as [MATH], and [EQUATION] where [MATH] is given by [REF].', '1805.06415-1-25-0': 'We proceed in three steps.', '1805.06415-1-26-0': '1 Proof of estimates [REF]-[REF].', '1805.06415-1-26-1': 'We have [MATH], which implies [REF].', '1805.06415-1-26-2': 'Moreover, [EQUATION]', '1805.06415-1-26-3': 'We note that by [REF] and [REF], there exists [MATH] such that [EQUATION] for [MATH].', '1805.06415-1-26-4': 'Applying [REF], we deduce that [MATH] is bounded independently of [MATH] for [MATH].', '1805.06415-1-26-5': 'Given [MATH], let [EQUATION]', '1805.06415-1-26-6': 'It follows that [MATH], so that [MATH] is bounded independently of [MATH] and [MATH].', '1805.06415-1-26-7': 'For [MATH], assumption [REF] and formula [REF] imply [EQUATION] which proves [REF].', '1805.06415-1-27-0': '2 Proof of estimates [REF]-[REF].', '1805.06415-1-27-1': 'We have [EQUATION] and [EQUATION]', '1805.06415-1-27-2': 'We observe that by [REF], [REF], and [REF], there exists [MATH] such that [EQUATION] for all [MATH] and [MATH], so that [EQUATION]', '1805.06415-1-27-3': 'Moreover, [MATH] is bounded on [MATH], so that by [REF] and [REF], [EQUATION]', '1805.06415-1-27-4': 'On [MATH], we deduce from [REF] and formulas [REF] and [REF] that [EQUATION] and estimates [REF]-[REF] easily follow.', '1805.06415-1-28-0': '3 Proof of [REF]-[REF].', '1805.06415-1-28-1': 'It follows from [REF], [REF], [REF] and [REF] that [MATH] and [MATH] are bounded independently of [MATH].', '1805.06415-1-28-2': 'Furthermore, it is clear that [MATH] and [MATH] are also bounded independently of [MATH], for every [MATH].', '1805.06415-1-28-3': 'Therefore, we need only calculate [MATH] and [MATH] for [MATH] small.', '1805.06415-1-28-4': 'Assumption [REF] implies that [MATH] for [MATH] small, so that for small [MATH] [EQUATION]', '1805.06415-1-28-5': 'The limit [REF] easily follows.', '1805.06415-1-28-6': 'To prove [REF], we observe that by [REF] and [REF] [EQUATION] and we conclude as above.', '1805.06415-1-29-0': '4 [MATH].', '1805.06415-1-29-1': 'Given [MATH], we deduce from [REF], [REF], [REF] and [REF] that [EQUATION] for all [MATH] and [MATH].', '1805.06415-1-29-2': 'Since [MATH], the conclusion easily follows by dominated convergence.', '1805.06415-1-30-0': '# Construction and estimates of the approximate solutions', '1805.06415-1-31-0': 'We observe that by [REF], [MATH] satisfies [EQUATION]', '1805.06415-1-31-1': 'We now construct approximate solutions that behave like [MATH].', '1805.06415-1-31-2': 'More precisely, we set [EQUATION] for [MATH], and we consider the solution [MATH] of equation [REF] with the initial condition [EQUATION]', '1805.06415-1-31-3': 'It follows from Proposition [REF] that [MATH] is well defined, [MATH].', '1805.06415-1-31-4': 'We now establish estimates that are uniform in [MATH].', '1805.06415-1-31-5': 'For this we set [EQUATION] so that [MATH].', '1805.06415-1-32-0': 'If [MATH] is as above, then there exist [MATH] such that [EQUATION] for all [MATH].', '1805.06415-1-33-0': 'In the calculations that follow, we drop the index [MATH].', '1805.06415-1-33-1': 'Moreover, we let [MATH], where [MATH] is given by [REF].', '1805.06415-1-33-2': 'In addition, we make formal calculations, which can be justified for instance by the method of [CITATION].', '1805.06415-1-33-3': 'It is convenient to set [EQUATION] for all [MATH].', '1805.06415-1-33-4': 'We note that [EQUATION] for all [MATH].', '1805.06415-1-33-5': 'We will also use the estimates', '1805.06415-1-34-0': '_ z g( u+v ) - _z g(u) - _z g( v ) ( u^-1 v + v ^-1 u)', '1805.06415-1-35-0': '_ z g( u+v ) - _ z g(u) - _ z g( v ) ( u^-1 v + v ^-1 u)', '1805.06415-1-36-0': 'for all [MATH].', '1805.06415-1-36-1': 'We establish [REF], the proof of [REF] being similar.', '1805.06415-1-36-2': 'To prove [REF], we consider the three cases: [MATH].', '1805.06415-1-36-3': 'The second case is immediate, because then [MATH] and [MATH] are equivalent.', '1805.06415-1-36-4': 'Next, the first and third cases are equivalent, because the expressions on the right-hand side of [REF] are symmetric in [MATH].', '1805.06415-1-36-5': 'Therefore, we consider only the first case and, assuming without loss of generality [MATH], we have [EQUATION]', '1805.06415-1-36-6': 'Since [MATH] (recall that [MATH]), we need only estimate [MATH].', '1805.06415-1-36-7': 'We have [EQUATION] where [EQUATION]', '1805.06415-1-36-8': 'The function [MATH] is [MATH] on [MATH] and [MATH], so that there exists a constant [MATH] such that [MATH] for [MATH]; and so [EQUATION] which proves the desired estimate.', '1805.06415-1-37-0': 'The equation for [MATH] is, with the notation [REF] [EQUATION]', '1805.06415-1-37-1': 'Multiplying by [MATH] and taking the real part, we obtain after integration by parts [EQUATION]', '1805.06415-1-37-2': 'By [REF], the first term on the right-hand side is nonnegative, so that [EQUATION] hence, using [REF], [EQUATION] for [MATH].', '1805.06415-1-37-3': 'We observe that by [REF] and [REF] [EQUATION] so that integrating [REF] on [MATH] yields [EQUATION]', '1805.06415-1-37-4': 'We now multiply [REF] by [MATH], take the real part and integrate by parts.', '1805.06415-1-37-5': 'Since [EQUATION] we obtain [EQUATION]', '1805.06415-1-37-6': 'It follows from [REF]-[REF] that [EQUATION] so that [REF] yields [EQUATION]', '1805.06415-1-37-7': 'Applying [REF], we have [EQUATION]', '1805.06415-1-37-8': 'In view of [REF]-[REF], we obtain [EQUATION]', '1805.06415-1-37-9': "By Gagliardo-Nirenberg's inequality [EQUATION] so that [REF], [REF], [REF] and [REF] yield [EQUATION] where [EQUATION]", '1805.06415-1-37-10': 'Next, [EQUATION]', '1805.06415-1-37-11': 'The first term in the right-hand side of [REF] appears in [REF] and is estimated by the right-hand side of [REF], so we estimate the last term in [REF].', '1805.06415-1-37-12': 'By Cauchy-Schwarz [EQUATION] so that [EQUATION] for every [MATH].', '1805.06415-1-37-13': 'Since [MATH], we have [MATH], hence [EQUATION]', '1805.06415-1-37-14': "By Gagliardo-Nirenberg's inequality [EQUATION] thus we deduce from [REF], [REF], [REF] and [REF] that [EQUATION] where [EQUATION]", '1805.06415-1-37-15': 'We deduce from [REF], [REF], [REF] and [REF] that [EQUATION] hence [REF] and [REF] yield [EQUATION]', '1805.06415-1-37-16': 'We note that [REF] and [REF] imply [EQUATION] and one deduces easily that [EQUATION]', '1805.06415-1-37-17': 'Moreover, [EQUATION] and it follows from [REF] that [EQUATION] where [EQUATION]', '1805.06415-1-37-18': 'We now set [EQUATION]', '1805.06415-1-37-19': 'Since [MATH], we have [MATH], and it follows from [REF] that there exists a constant [MATH] independent of [MATH] such that [EQUATION] for all [MATH].', '1805.06415-1-37-20': 'This implies that there exists [MATH] such that [MATH].', '1805.06415-1-37-21': 'This completes the proof.', '1805.06415-1-38-0': '# Proof of Theorem [MATH]', '1805.06415-1-39-0': 'We consider the solution [MATH] of equation [REF] defined by [REF] and [REF], [MATH] defined by[REF], and we set', '1805.06415-1-40-0': 'V_n (t)= U( T_n -t )', '1805.06415-1-41-0': '_n (t)= _n ( T_n -t )', '1805.06415-1-42-0': 'for [MATH].', '1805.06415-1-42-1': 'It follows from [REF] that there exist [MATH] such that [EQUATION]', '1805.06415-1-42-2': 'Moreover, it follows from [REF] that [EQUATION]', '1805.06415-1-42-3': 'Using the estimate [MATH] and the embeddings [MATH], [MATH], we deduce that [EQUATION] so that, applying [REF], [REF], [REF] and [REF], there exists [MATH] such that [EQUATION]', '1805.06415-1-42-4': 'Given [MATH], it follows from [REF] and [REF] that the sequence [MATH] is bounded in [MATH].', '1805.06415-1-42-5': 'Therefore, after possibly extracting a subsequence, there exists [MATH] such that', '1805.06415-1-43-0': '_n _ n in L^((, ), H^1 (R^N ) ) weak[MATH]', '1805.06415-1-44-0': '_t _n _ n _t in L^((,), H^-1 (R^N ) ) weak[MATH]', '1805.06415-1-45-0': '_n (t) _ n (t) weakly in [MATH] and a.e. on [MATH], for all [MATH]', '1805.06415-1-46-0': 'Since [MATH] is arbitrary, a standard argument of diagonal extraction shows that there exists [MATH] such that (after extraction of a subsequence) [REF], [REF] and [REF] hold for all [MATH].', '1805.06415-1-46-1': 'Moreover, [REF] and [REF] imply that [EQUATION] and [REF] and [REF] imply that [EQUATION] for all [MATH].', '1805.06415-1-46-2': 'In addition, it follows easily from [REF] and the convergence properties [REF]-[REF] that [EQUATION] in [MATH], where [MATH].', '1805.06415-1-46-3': 'Therefore, setting [EQUATION] we see that [MATH] and that [EQUATION] in [MATH].', '1805.06415-1-46-4': 'We now claim that [EQUATION]', '1805.06415-1-46-5': 'Indeed, let [MATH], and we consider the solution [MATH] of [REF] such that [EQUATION] (See Proposition [REF].)', '1805.06415-1-46-6': 'The uniqueness property of Proposition [REF] implies that [MATH] on [MATH].', '1805.06415-1-46-7': 'In particular, [MATH], hence [REF] follows, since [MATH] is arbitrary.', '1805.06415-1-46-8': 'We may now extend [MATH] for [MATH] (see Proposition [REF]) to a solution [MATH] of [REF].', '1805.06415-1-46-9': 'Estimates [REF], [REF] and [REF] now follow from [REF], [REF] and [REF], respectively, and the convergence property [REF] follows from [REF].'}

{'1805.06415-2-0-0': 'We consider the nonlinear Schrodinger equation [EQUATION] for [MATH]-subcritical or critical nonlinearities: [MATH].', '1805.06415-2-0-1': 'Under the additional technical assumptions [MATH] (and thus [MATH]), we construct [MATH] solutions that blow up in finite time with explicit blow-up profiles and blow-up rates.', '1805.06415-2-0-2': 'In particular, blowup can occur at any given finite set of points of [MATH].', '1805.06415-2-1-0': 'The construction involves explicit functions [MATH], solutions of the ordinary differential equation [MATH].', '1805.06415-2-1-1': 'In the simplest case, [MATH] for [MATH], [MATH].', '1805.06415-2-1-2': 'For [MATH] sufficiently large, [MATH] satisfies [MATH] close to the blow-up point [MATH], so that it is a suitable approximate solution of the problem.', '1805.06415-2-1-3': 'To construct an actual solution [MATH] close to [MATH], we use energy estimates and a compactness argument.', '1805.06415-2-2-0': '# Introduction', '1805.06415-2-3-0': 'We consider the nonlinear Schrodinger equation [EQUATION] on [MATH], where [MATH] satisfies [EQUATION]', '1805.06415-2-3-1': 'Under assumption [REF], equation [REF] is [MATH]-subcritical or critical, so that the corresponding Cauchy problem is locally well posed in [MATH] - see e.g. [CITATION].', '1805.06415-2-4-0': 'Equation [REF] is a member of the more general family of complex Ginzburg-Landau equations [EQUATION]', '1805.06415-2-4-1': 'It is proved in [CITATION] that (for [MATH]) equation [REF] has no global in time [MATH] solution that remains bounded in [MATH].', '1805.06415-2-4-2': 'In other words, every [MATH] solution blows up, in finite or infinite time.', '1805.06415-2-4-3': 'The question of finite-time blowup is left open in [CITATION].', '1805.06415-2-4-4': "It seems that no standard argument based on obstruction to global existence (Levine's method, variance argument) is applicable.", '1805.06415-2-5-0': 'The purpose of this article is to construct solutions of [REF] that blow up in finite time.', '1805.06415-2-5-1': 'For technical reasons we require [EQUATION] (The condition [MATH] is used in the proof of estimates [REF]-[REF], the stronger condition [MATH] is used in formula [REF].)', '1805.06415-2-5-2': 'Conditions [REF]-[REF] impose [MATH].', '1805.06415-2-5-3': 'More precisely, the allowed range of powers is [EQUATION]', '1805.06415-2-5-4': 'Our first blow-up result, related to single point blowup with a simple asymptotic profile is the following.', '1805.06415-2-6-0': 'Theorem [REF] is a particular case of a more general result (Theorem [REF] below), where asymptotic profiles more general than [REF] are allowed.', '1805.06415-2-6-1': 'Before stating precisely this more general result, we need to make precise our assumptions on the asymptotic profile.', '1805.06415-2-6-2': 'We consider an integer [MATH], real numbers [MATH], [MATH], [MATH] where [MATH] is a multi-index with [MATH], and points [MATH] such that [EQUATION]', '1805.06415-2-6-3': 'Let [MATH] satisfy [EQUATION]', '1805.06415-2-6-4': 'Our main result is the following [EQUATION]', '1805.06415-2-7-0': 'Here are some comments on Theorems [REF] and [REF].', '1805.06415-2-7-1': '[(i)]', '1805.06415-2-8-0': 'Theorem [REF] presents the simplest result with the choice of only two parameters [MATH] and [MATH].', '1805.06415-2-8-1': 'The parameter [MATH] has to be taken sufficiently large so that the ansatz [MATH] satisfies [MATH] (a similar strategy is used in [CITATION], see the comments below).', '1805.06415-2-8-2': 'Note that the choice of [MATH] determines the blow-up rates in [REF]-[REF].', '1805.06415-2-8-3': 'More parameters can chosen in Theorem [REF], which allows arbitrary locations for the blow-up points and flexibility on the blow-up rates.', '1805.06415-2-8-4': 'It is easy to construct explicitly functions [MATH] satisfying [REF].', '1805.06415-2-9-0': 'It follows from [REF] and [REF] that both [MATH] and [MATH] blow up as [MATH].', '1805.06415-2-10-0': 'The asymptotic profile [MATH] of [MATH] as [MATH], given by [REF], has the following properties: [MATH], [MATH] is bounded as [MATH] and [MATH] is [MATH] except at the points [MATH], where [MATH] has a singularity like [MATH].', '1805.06415-2-11-0': 'The solutions constructed in Theorems [REF] and [REF] are global for [MATH].', '1805.06415-2-11-1': 'Actually, this is a general fact : we show in Proposition [REF] that equation [REF] is globally well-posed in [MATH] in the negative sense of time.', '1805.06415-2-12-0': 'We prove Theorem [REF] by using the strategy of [CITATION].', '1805.06415-2-12-1': 'More precisely, we consider the sequence [MATH] of solutions of [REF] defined by [MATH], where [MATH] is defined by [REF].', '1805.06415-2-12-2': 'It follows that [MATH] is defined on [MATH].', '1805.06415-2-12-3': 'Since [MATH], and [MATH] is small compared to [MATH], [MATH] is almost a solution of [REF].', '1805.06415-2-12-4': 'Following the idea of [CITATION] (see also [CITATION] in the blow-up context) we estimate the solutions [MATH] by energy arguments.', '1805.06415-2-12-5': 'Note that the nice behavior of equation [REF] backwards in time, already discussed in Remark [REF] [REF], is important in this step.', '1805.06415-2-12-6': 'Finally, passing to the limit as [MATH] yields the solution [MATH] of Theorem [REF].', '1805.06415-2-13-0': 'The solution [MATH] given by Theorem [REF] blows up at [MATH] like the function [MATH] defined by [REF].', '1805.06415-2-13-1': 'Since [MATH], we see that the solution [MATH] displays an ODE-type blowup.', '1805.06415-2-14-0': 'We recall that ODE-type blowup has been intensively studied for several other nonlinear equations.', '1805.06415-2-14-1': 'For the nonlinear heat equation, the type of blowup obtained in Theorems [REF] and [REF] is called flat blowup in the literature and was recently investigated independently of our work in [CITATION] (see also previous references there) with applications to the Burgers equation.', '1805.06415-2-14-2': 'Even if the blow-up profile is identical (see 4.1 of [CITATION]), the strategy to proceed with the construction of an actual solution of the equation is different in this paper.', '1805.06415-2-15-0': 'Apart from such unstable forms of blowup, ODE-type blowup was also much studied as a stable form of blowup.', '1805.06415-2-15-1': 'We refer to [CITATION] and to references there for results in the parabolic context.', '1805.06415-2-15-2': 'For the semilinear wave and quasilinear wave equations, we refer e.g. to [CITATION] and to the references there.', '1805.06415-2-16-0': 'The rest of the paper is organized as follows.', '1805.06415-2-16-1': 'In Section [REF], we study the Cauchy problem for the backwards version of equation [REF], and in Section [REF], we derive various estimates on the function [MATH].', '1805.06415-2-16-2': 'Section [REF] is devoted to the construction and estimates of the approximate blow-up solutions, and the proof of Theorem [REF] is completed in Section [REF] by passing to the limit in the approximate solutions.', '1805.06415-2-17-0': '# The backwards equation', '1805.06415-2-18-0': 'In this section, we prove that the Cauchy problem for the backwards equation obtained by changing [MATH] to [MATH] in [REF] is globally well-posed in [MATH], under assumption [REF].', '1805.06415-2-19-0': 'Assumption [REF] ensures that equation [REF] is [MATH]-subcritical or critical, so that local well-posedness in [MATH] follows from standard arguments.', '1805.06415-2-19-1': '(See e.g. [CITATION].)', '1805.06415-2-20-0': 'We now prove that the Cauchy problem [REF] is globally well posed in [MATH].', '1805.06415-2-20-1': 'Indeed, let [MATH], and let [MATH] be the maximal existence time of the corresponding solution [MATH] of [REF].', '1805.06415-2-20-2': 'Multiplying the equation by [MATH] and taking the real part yields [EQUATION] so that [MATH] for all [MATH].', '1805.06415-2-20-3': 'Next, we multiply the equation by [MATH] and take the real part.', '1805.06415-2-20-4': 'We obtain (see [REF]-[REF] below) [EQUATION] so that [MATH] for all [MATH].', '1805.06415-2-20-5': 'These formal calculations can be justified by standard arguments, see e.g. [CITATION].', '1805.06415-2-20-6': 'This proves global existence in the subcritical case [MATH].', '1805.06415-2-20-7': 'In the critical case [MATH] and [MATH], we use the right-hand side of [REF] to obtain [EQUATION]', '1805.06415-2-20-8': "Since [EQUATION] we deduce by Sobolev's inequality that [EQUATION]", '1805.06415-2-20-9': 'We define [MATH] by [EQUATION] so that [EQUATION]', '1805.06415-2-20-10': 'Since by [REF] [EQUATION] we deduce from [REF] that [MATH].', '1805.06415-2-20-11': '(See [CITATION].)', '1805.06415-2-21-0': 'Finally, we prove the stronger uniqueness property, so we consider [MATH] and two solutions [MATH] of [REF] on [MATH].', '1805.06415-2-21-1': 'Setting [EQUATION] we see that [MATH] satisfies [EQUATION] in [MATH].', '1805.06415-2-21-2': 'Moreover, the map [MATH] is in [MATH] and [EQUATION] for a.a. [MATH].', '1805.06415-2-21-3': 'Taking the [MATH] duality product of equation [REF] with [MATH] and applying [REF], we deduce that for a.a. [MATH] [EQUATION]', '1805.06415-2-21-4': 'We recall that [EQUATION] for all [MATH].', '1805.06415-2-21-5': 'Indeed, [EQUATION]', '1805.06415-2-21-6': 'Since [MATH], we deduce from [REF] and [REF] that [MATH] on [MATH].', '1805.06415-2-22-0': '# Estimates of [MATH]', '1805.06415-2-23-0': 'In this section, we establish various estimates on the function [MATH] defined by [REF].', '1805.06415-2-24-0': 'Let [MATH] satisfy [REF]-[REF].', '1805.06415-2-24-1': 'If [MATH] is given by [REF], then [MATH] and [EQUATION] as [MATH], and [EQUATION] where [MATH] is given by [REF].', '1805.06415-2-25-0': 'We proceed in three steps.', '1805.06415-2-26-0': '1 Proof of estimates [REF]-[REF].', '1805.06415-2-26-1': 'We have [MATH], which implies [REF].', '1805.06415-2-26-2': 'Moreover, [EQUATION]', '1805.06415-2-26-3': 'We note that by [REF] and [REF], there exists [MATH] such that [EQUATION] for [MATH].', '1805.06415-2-26-4': 'Applying [REF], we deduce that [MATH] is bounded independently of [MATH] for [MATH].', '1805.06415-2-26-5': 'Given [MATH], let [EQUATION]', '1805.06415-2-26-6': 'It follows that [MATH], so that [MATH] is bounded independently of [MATH] and [MATH].', '1805.06415-2-26-7': 'For [MATH], assumption [REF] and formula [REF] imply [EQUATION] which proves [REF].', '1805.06415-2-27-0': '2 Proof of estimates [REF]-[REF].', '1805.06415-2-27-1': 'We have [EQUATION] and [EQUATION]', '1805.06415-2-27-2': 'We observe that by [REF], [REF], and [REF], there exists [MATH] such that [EQUATION] for all [MATH] and [MATH], so that [EQUATION]', '1805.06415-2-27-3': 'Moreover, [MATH] is bounded on [MATH], so that by [REF] and [REF], [EQUATION]', '1805.06415-2-27-4': 'On [MATH], we deduce from [REF] and formulas [REF] and [REF] that [EQUATION] and estimates [REF]-[REF] easily follow.', '1805.06415-2-28-0': '3 Proof of [REF]-[REF].', '1805.06415-2-28-1': 'It follows from [REF], [REF], [REF] and [REF] that [MATH] and [MATH] are bounded independently of [MATH].', '1805.06415-2-28-2': 'Furthermore, it is clear that [MATH] and [MATH] are also bounded independently of [MATH], for every [MATH].', '1805.06415-2-28-3': 'Therefore, we need only calculate [MATH] and [MATH] for [MATH] small.', '1805.06415-2-28-4': 'Assumption [REF] implies that [MATH] for [MATH] small, so that for small [MATH] [EQUATION]', '1805.06415-2-28-5': 'The limit [REF] easily follows.', '1805.06415-2-28-6': 'To prove [REF], we observe that by [REF] and [REF] [EQUATION] and we conclude as above.', '1805.06415-2-29-0': '4 [MATH].', '1805.06415-2-29-1': 'Given [MATH], we deduce from [REF], [REF], [REF] and [REF] that [EQUATION] for all [MATH] and [MATH].', '1805.06415-2-29-2': 'Since [MATH], the conclusion easily follows by dominated convergence.', '1805.06415-2-30-0': '# Construction and estimates of the approximate solutions', '1805.06415-2-31-0': 'We observe that by [REF], [MATH] satisfies [EQUATION]', '1805.06415-2-31-1': 'We now construct approximate solutions that behave like [MATH].', '1805.06415-2-31-2': 'More precisely, we set [EQUATION] for [MATH], and we consider the solution [MATH] of equation [REF] with the initial condition [EQUATION]', '1805.06415-2-31-3': 'It follows from Proposition [REF] that [MATH] is well defined, [MATH].', '1805.06415-2-31-4': 'We now establish estimates that are uniform in [MATH].', '1805.06415-2-31-5': 'For this we set [EQUATION] so that [MATH].', '1805.06415-2-32-0': 'If [MATH] is as above, then there exist [MATH] such that [EQUATION] for all [MATH].', '1805.06415-2-33-0': 'In the calculations that follow, we drop the index [MATH].', '1805.06415-2-33-1': 'Moreover, we let [MATH], where [MATH] is given by [REF].', '1805.06415-2-33-2': 'In addition, we make formal calculations, which can be justified for instance by the method of [CITATION].', '1805.06415-2-33-3': 'It is convenient to set [EQUATION] for all [MATH].', '1805.06415-2-33-4': 'We note that [EQUATION] for all [MATH].', '1805.06415-2-33-5': 'We will also use the estimates', '1805.06415-2-34-0': '_ z g( u+v ) - _z g(u) - _z g( v ) ( u^-1 v + v ^-1 u)', '1805.06415-2-35-0': '_ z g( u+v ) - _ z g(u) - _ z g( v ) ( u^-1 v + v ^-1 u)', '1805.06415-2-36-0': 'for all [MATH].', '1805.06415-2-36-1': 'We establish [REF], the proof of [REF] being similar.', '1805.06415-2-36-2': 'To prove [REF], we consider the three cases: [MATH].', '1805.06415-2-36-3': 'The second case is immediate, because then [MATH] and [MATH] are equivalent.', '1805.06415-2-36-4': 'Next, the first and third cases are equivalent, because the expressions on the right-hand side of [REF] are symmetric in [MATH].', '1805.06415-2-36-5': 'Therefore, we consider only the first case and, assuming without loss of generality [MATH], we have [EQUATION]', '1805.06415-2-36-6': 'Since [MATH] (recall that [MATH]), we need only estimate [MATH].', '1805.06415-2-36-7': 'We have [EQUATION] where [EQUATION]', '1805.06415-2-36-8': 'The function [MATH] is [MATH] on [MATH] and [MATH], so that there exists a constant [MATH] such that [MATH] for [MATH]; and so [EQUATION] which proves the desired estimate.', '1805.06415-2-37-0': 'The equation for [MATH] is, with the notation [REF] [EQUATION]', '1805.06415-2-37-1': 'Multiplying by [MATH] and taking the real part, we obtain after integration by parts [EQUATION]', '1805.06415-2-37-2': 'By [REF], the first term on the right-hand side is nonnegative, so that [EQUATION] hence, using [REF], [EQUATION] for [MATH].', '1805.06415-2-37-3': 'We observe that by [REF] and [REF] [EQUATION] so that integrating [REF] on [MATH] yields [EQUATION]', '1805.06415-2-37-4': 'We now multiply [REF] by [MATH], take the real part and integrate by parts.', '1805.06415-2-37-5': 'Since [EQUATION] we obtain [EQUATION]', '1805.06415-2-37-6': 'It follows from [REF]-[REF] that [EQUATION] so that [REF] yields [EQUATION]', '1805.06415-2-37-7': 'Applying [REF], we have [EQUATION]', '1805.06415-2-37-8': 'In view of [REF]-[REF], we obtain [EQUATION]', '1805.06415-2-37-9': "By Gagliardo-Nirenberg's inequality [EQUATION] so that [REF], [REF], [REF] and [REF] yield [EQUATION] where [EQUATION]", '1805.06415-2-37-10': 'Next, [EQUATION]', '1805.06415-2-37-11': 'The first term in the right-hand side of [REF] appears in [REF] and is estimated by the right-hand side of [REF], so we estimate the last term in [REF].', '1805.06415-2-37-12': 'By Cauchy-Schwarz [EQUATION] so that [EQUATION] for every [MATH].', '1805.06415-2-37-13': 'Since [MATH], we have [MATH], hence [EQUATION]', '1805.06415-2-37-14': "By Gagliardo-Nirenberg's inequality [EQUATION] thus we deduce from [REF], [REF], [REF] and [REF] that [EQUATION] where [EQUATION]", '1805.06415-2-37-15': 'We deduce from [REF], [REF], [REF] and [REF] that [EQUATION] hence [REF] and [REF] yield [EQUATION]', '1805.06415-2-37-16': 'We note that [REF] and [REF] imply [EQUATION] and one deduces easily that [EQUATION]', '1805.06415-2-37-17': 'Moreover, [EQUATION] and it follows from [REF] that [EQUATION] where [EQUATION]', '1805.06415-2-37-18': 'We now set [EQUATION]', '1805.06415-2-37-19': 'Since [MATH], we have [MATH], and it follows from [REF] that there exists a constant [MATH] independent of [MATH] such that [EQUATION] for all [MATH].', '1805.06415-2-37-20': 'This implies that there exists [MATH] such that [MATH].', '1805.06415-2-37-21': 'This completes the proof.', '1805.06415-2-38-0': '# Proof of Theorem [MATH]', '1805.06415-2-39-0': 'We consider the solution [MATH] of equation [REF] defined by [REF] and [REF], [MATH] defined by[REF], and we set', '1805.06415-2-40-0': 'V_n (t)= U( T_n -t )', '1805.06415-2-41-0': '_n (t)= _n ( T_n -t )', '1805.06415-2-42-0': 'for [MATH].', '1805.06415-2-42-1': 'It follows from [REF] that there exist [MATH] such that [EQUATION]', '1805.06415-2-42-2': 'Moreover, it follows from [REF] that [EQUATION]', '1805.06415-2-42-3': 'Using the estimate [MATH] and the embeddings [MATH], [MATH], we deduce that [EQUATION] so that, applying [REF], [REF], [REF] and [REF], there exists [MATH] such that [EQUATION]', '1805.06415-2-42-4': 'Given [MATH], it follows from [REF] and [REF] that the sequence [MATH] is bounded in [MATH].', '1805.06415-2-42-5': 'Therefore, after possibly extracting a subsequence, there exists [MATH] such that', '1805.06415-2-43-0': '_n _ n in L^((, ), H^1 (R^N ) ) weak[MATH]', '1805.06415-2-44-0': '_t _n _ n _t in L^((,), H^-1 (R^N ) ) weak[MATH]', '1805.06415-2-45-0': '_n (t) _ n (t) weakly in [MATH] and a.e. on [MATH], for all [MATH]', '1805.06415-2-46-0': 'Since [MATH] is arbitrary, a standard argument of diagonal extraction shows that there exists [MATH] such that (after extraction of a subsequence) [REF], [REF] and [REF] hold for all [MATH].', '1805.06415-2-46-1': 'Moreover, [REF] and [REF] imply that [EQUATION] and [REF] and [REF] imply that [EQUATION] for all [MATH].', '1805.06415-2-46-2': 'In addition, it follows easily from [REF] and the convergence properties [REF]-[REF] that [EQUATION] in [MATH], where [MATH].', '1805.06415-2-46-3': 'Therefore, setting [EQUATION] we see that [MATH] and that [EQUATION] in [MATH].', '1805.06415-2-46-4': 'We now claim that [EQUATION]', '1805.06415-2-46-5': 'Indeed, let [MATH], and we consider the solution [MATH] of [REF] such that [EQUATION] (See Proposition [REF].)', '1805.06415-2-46-6': 'The uniqueness property of Proposition [REF] implies that [MATH] on [MATH].', '1805.06415-2-46-7': 'In particular, [MATH], hence [REF] follows, since [MATH] is arbitrary.', '1805.06415-2-46-8': 'We may now extend [MATH] for [MATH] (see Proposition [REF]) to a solution [MATH] of [REF].', '1805.06415-2-46-9': 'Estimates [REF], [REF] and [REF] now follow from [REF], [REF] and [REF], respectively, and the convergence property [REF] follows from [REF].'}

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[['1805.06415-1-4-1', '1805.06415-2-4-1']]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1805.06415

gr-qc-0508093

{'gr-qc-0508093-1-0-0': 'This paper examines the classical dynamics of false vacuum regions embedded in surrounding regions of true vacuum, in the thin-wall limit.', 'gr-qc-0508093-1-0-1': 'The dynamics of all generally relativistically allowed solutions - most but not all of which have been previously studied - are derived, enumerated, and interpreted.', 'gr-qc-0508093-1-0-2': 'We comment on the relation of these solutions to possible mechanisms whereby inflating regions may be spawned from non-inflating ones.', 'gr-qc-0508093-1-0-3': 'We then calculate the dynamics of first order deviations from spherical symmetry, finding that many solutions are unstable to such aspherical perturbations.', 'gr-qc-0508093-1-0-4': 'The parameter space in which the perturbations on bound solutions inevitably become nonlinear is mapped.', 'gr-qc-0508093-1-0-5': 'This instability has consequences for the Farhi-Guth-Guven mechanism for baby universe production via quantum tunneling.', 'gr-qc-0508093-1-1-0': '# Introduction', 'gr-qc-0508093-1-2-0': 'Nearly two decades ago, a series of papers [CITATION] began to investigate the possibility of creating an inflationary universe "in a laboratory" - that is, inside a surrounding region of much lower vacuum energy.', 'gr-qc-0508093-1-2-1': 'The spacetime of such a "bubble universe" was modeled as a spherically symmetric de Sitter region (the false vacuum) joined to a surrounding Schwarzschild geometry (the true vacuum) by an infinitesimally thin domain wall.', 'gr-qc-0508093-1-3-0': 'These studies found that the inflating (false vacuum) region could not avoid collapse unless either the null energy condition or cosmic censorship were violated in the full spacetime [CITATION], but that the creation of an enduring inflating region might be possible via quantum tunneling [CITATION].', 'gr-qc-0508093-1-3-1': 'In this picture, henceforth denoted the Farhi-Guth-Guven (FGG) mechanism, a classically constructable expanding bubble, which would classically re-collapse (both from the inside and outside perspective [CITATION]), instead tunnels to a new solution in which the inflating interior expands forever, while an outside observer sees a black hole .', 'gr-qc-0508093-1-3-2': 'The probability of this occurring can be calculated using the techniques of semi-classical quantum gravity, and is extraordinarily small.', 'gr-qc-0508093-1-4-0': 'Though nearly-miraculous, this process has garnered new interest recently, primarily because of evidence that our universe may have a fundamental positive cosmological constant.', 'gr-qc-0508093-1-4-1': 'If so, it will asymptotically approach everlasting equilibrium as de Sitter spacetime.', 'gr-qc-0508093-1-4-2': 'Given eternity, even the most unlikely process - such as the creation of inflating bubble universes - will eventually occur.', 'gr-qc-0508093-1-4-3': 'Taking this one step further, our observable universe could in fact be such a bubble universe, which arose from equilibrium de Sitter space and is currently returning to it.', 'gr-qc-0508093-1-4-4': 'This would realize the old idea of Boltzmann that the universe is fundamentally in equilibrium, but that extremely rare downward fluctuations in entropy periodically occur and allow the transitory existence of non-equilibrium regions that see entropy steadily increasing.', 'gr-qc-0508093-1-5-0': 'The classic problem with this idea was pointed out in its new context by Dyson, Susskind, Kleban [CITATION]: if our observable universe resulted from a downward entropy fluctuation that evolved with increasing entropy to the present time, then it is vastly more likely for this to have occurred by a fluctuation to our observable universe ten minutes ago (replete with incoming photons and memories in our brains to convince us that it is older) than by a fluctuation all the way to the much lower entropy corresponding to inflation .', 'gr-qc-0508093-1-5-1': 'Albrecht and Sorbo [CITATION], however, argue that inflation might avoid this problem by turning a tiny region of low entropy density into a very large one.', 'gr-qc-0508093-1-5-2': 'Their calculation shows that the FGG mechanism requires a much smaller entropy fluctuation than directly creating the large post-reheating region that will result from it, thus resolving the paradox.', 'gr-qc-0508093-1-6-0': 'There may, however, be reasons to doubt that the creation of small regions of false vacuum and subsequent tunneling are plausible events.', 'gr-qc-0508093-1-6-1': 'First, there is no regular instanton describing either the nucleation of small regions of false vacuum or the gravitational tunneling event; there is such an instanton describing tunneling to false-vacuum, but over a huge region, larger than the true-vacuum horizon (see, e.g., [CITATION]).', 'gr-qc-0508093-1-6-2': 'Second, Banks [CITATION] has pointed out that it is hard to understand the tunneling process holographically - it appears that the inflating region inside the bubble has many more states than the black hole it is "contained in" (see also [CITATION]).', 'gr-qc-0508093-1-6-3': 'Third, Dutta Vachaspati [CITATION] have recently (and previously in [CITATION]) argued that general causality considerations preclude the formation of a small false-vacuum region inside a large true-vacuum one.', 'gr-qc-0508093-1-7-0': 'Both in terms of the initial conditions for inflation, and also the more general issue of what processes can lead to transitions between vacuum states (as is important in understanding the string theory "landscape"), it is crucial to understand bubble universes, whether they can form, and with what probabilities.', 'gr-qc-0508093-1-7-1': 'To this question the present paper makes the following contributions: first, we organize and interpret all of the thin-walled, spherically symmetric one bubble spacetimes.', 'gr-qc-0508093-1-7-2': 'We then show that expanding false vacuum bubbles are unstable to non-spherical perturbations; if these bubbles start at a sufficiently small initial radius, then they inevitably become nonlinearly aspherical before tunneling might occur in the FGG mechanism.', 'gr-qc-0508093-1-7-3': 'It is unclear in this case whether tunneling to an inflationary universe inside the bubble can occur at all, or with what probability.', 'gr-qc-0508093-1-8-0': 'The plan of the paper is as follows.', 'gr-qc-0508093-1-8-1': 'In Sec. [REF], the allowed solutions to the junction conditions are enumerated, putting the previous work into context.', 'gr-qc-0508093-1-8-2': 'We provide a concise reference for all of the possible spacetimes with one false-vacuum bubble and arbitrary positive cosmological constant, then discuss the existing, and some new, interpretations of these solutions.', 'gr-qc-0508093-1-8-3': 'In Sec. [REF] we derive the first order perturbation equations, and demonstrate the existence of an instability in bound solutions.', 'gr-qc-0508093-1-8-4': 'In Sec. [REF] we integrate the equations to investigate the parameter space for which non-linear perturbations are unavoidable, and we conclude in Sec. [REF].', 'gr-qc-0508093-1-9-0': '# Junction Conditions', 'gr-qc-0508093-1-10-0': 'The dynamics of inflating regions has been discussed by a number of authors [CITATION] using the junction condition formalism.', 'gr-qc-0508093-1-10-1': 'These works study a spherically symmetric region of false vacuum (high energy density) in a surrounding region of true vacuum (lower energy density).', 'gr-qc-0508093-1-10-2': 'The wall separating the regions is assumed to be very thin compared to the radius of the region of false vacuum.', 'gr-qc-0508093-1-10-3': "One can obtain the dynamics of the wall by requiring metric continuity across the wall and then solving Einstein's equations.", 'gr-qc-0508093-1-10-4': "Under spherical symmetry, the dynamics of the problem then reduce to those of the bubble wall's radius.", 'gr-qc-0508093-1-10-5': 'This radius is a gauge invariant quantity because it simply quantifies the curvature of the bubble wall worldsheet, and any observer can measure it by comparing the normal to the wall at two nearby points.', 'gr-qc-0508093-1-11-0': '## Interior and Exterior Spacetimes', 'gr-qc-0508093-1-12-0': 'Let [MATH] be the cosmological constant in the true vacuum region.', 'gr-qc-0508093-1-12-1': 'Then if [MATH], the region is Schwarzschild-de Sitter spacetime with metric [EQUATION] in the static foliation.', 'gr-qc-0508093-1-12-2': 'Fixing [MATH], there are then three qualitatively different casual structures characterized by the value of [MATH] (see [CITATION]), due to the nature of the three roots of [MATH] [EQUATION] and [MATH].', 'gr-qc-0508093-1-12-3': 'We can label them as [EQUATION]', 'gr-qc-0508093-1-12-4': 'And the range of [MATH] means that they lie in the ranges [MATH].', 'gr-qc-0508093-1-13-0': 'The two positive roots correspond to the black hole and cosmological horizons.', 'gr-qc-0508093-1-13-1': 'The conformal diagram for this spacetime is shown in Fig. [REF].', 'gr-qc-0508093-1-13-2': '(See [CITATION] for a demonstration of the explicit form of the metric in global coordinates).', 'gr-qc-0508093-1-13-3': 'Surfaces of constant coordinate time [MATH] are drawn, with the circulating arrows denoting the direction of increasing [MATH].', 'gr-qc-0508093-1-13-4': 'We will consider region I to be the "causal patch" of a hypothetical observer (i.e., the region lying in both the causal past and causal future of the observer\'s world line) in what follows.', 'gr-qc-0508093-1-14-0': 'For [MATH], there are also three real roots: a double positive root [MATH] and a negative [MATH], given by: [EQUATION]', 'gr-qc-0508093-1-14-1': 'This mass is known as the Nariai mass, and in this spacetime there is only one horizon at the positive root.', 'gr-qc-0508093-1-14-2': 'The conformal diagram for this spacetime is shown in Fig. [REF] [CITATION].', 'gr-qc-0508093-1-14-3': 'There is also a time-reverse solution, starting at past null infinity and ending at [MATH].', 'gr-qc-0508093-1-14-4': 'For [MATH], there is one real negative root, and therefore no horizons in the spacetime.', 'gr-qc-0508093-1-14-5': 'The conformal diagram for this case is like figure [REF], but with the horizon lines excised.', 'gr-qc-0508093-1-15-0': 'Inside the false-vacuum region the spacetime is de Sitter space, with metric [EQUATION] in the static foliation.', 'gr-qc-0508093-1-15-1': 'Fig. [REF] shows the conformal diagram for the de Sitter region.', 'gr-qc-0508093-1-15-2': 'Again, surfaces of constant coordinate time [MATH] are shown, with the arrows denoting the direction of increasing [MATH].', 'gr-qc-0508093-1-15-3': 'We consider region III to be the causal patch in which our hypothetical false vacuum observer resides.', 'gr-qc-0508093-1-16-0': '## Equation of Motion', 'gr-qc-0508093-1-17-0': 'The bubble wall worldsheet has metric: [EQUATION] where [MATH] is the proper time in the frame of the wall, and ([MATH]) are the usual angular variables.', 'gr-qc-0508093-1-18-0': 'The coordinates in the full [MATH] spacetime are chosen to be Gaussian normal coordinates constructed in the neighborhood of the bubble wall worldsheet.', 'gr-qc-0508093-1-18-1': 'Three of the coordinates are [MATH] on the worldsheet, and the fourth, [MATH], is defined as the proper distance along a geodesic normal to the bubble worldsheet, with [MATH] increasing in the direction of SdS (true vacuum).', 'gr-qc-0508093-1-19-0': 'The transformation from the static coordinate systems of SdS and dS to the Gaussian normal system can be constructed in closed form using the methods of [CITATION], and the full metric takes the form: [EQUATION] where [MATH] defines the wall and therefore [MATH] and [MATH].', 'gr-qc-0508093-1-20-0': 'The energy momentum tensor on the wall is: [EQUATION] where [MATH] is the metric on the worldsheet of the wall for [MATH] and zero otherwise, and [MATH] is the energy density of the wall.', 'gr-qc-0508093-1-21-0': "Using the metric [REF] and the energy-momentum tensor [REF] together with the contributions from the dS interior and SdS exterior in Einstein's equations yields an equation of motion for the bubble wall of: [EQUATION] where [MATH] is the extrinsic curvature tensor in SdS and dS respectively.", 'gr-qc-0508093-1-21-1': 'In the Gaussian normal coordinates, this takes the form: [EQUATION]', 'gr-qc-0508093-1-21-2': 'Evaluating this in metric [REF], the [MATH] and [MATH] components of Eq. [REF] reduce to: [EQUATION] with the definitions [EQUATION]', 'gr-qc-0508093-1-21-3': 'Here, [MATH] is the metric coefficient in dS or SdS.', 'gr-qc-0508093-1-21-4': 'The sign of [MATH] is fixed by the trajectory because [MATH] could potentially be positive or negative (motion can be with or against the direction of increasing coordinate time indicated in Fig. [REF] and [REF]).', 'gr-qc-0508093-1-22-0': 'A set of dimensionless coordinates can be defined, in which Eq. [REF] can be written as the equation of motion of a particle of unit mass in a one dimensional potential [CITATION].', 'gr-qc-0508093-1-22-1': 'Let: [EQUATION] where M is the mass appearing in the SdS metric coefficient, and [EQUATION]', 'gr-qc-0508093-1-22-2': 'With these definitions, Eq. [REF] becomes [EQUATION] where the potential [MATH] and energy [MATH] are [EQUATION] with [EQUATION] and [EQUATION]', 'gr-qc-0508093-1-22-3': 'Note that a small negative [MATH] corresponds to a large mass, so that even between [MATH] the mass will vary by many orders of magnitude.', 'gr-qc-0508093-1-23-0': 'The parameter space allowed by the junction conditions is characterized by the value of the cosmological constant inside and outside the wall.', 'gr-qc-0508093-1-23-1': 'For [MATH], we have [MATH].', 'gr-qc-0508093-1-23-2': 'The maximum [MATH] of the potential [MATH] then satisfies [MATH].', 'gr-qc-0508093-1-23-3': 'The potential curves over the entire range of [MATH] are shown in Fig. [REF].', 'gr-qc-0508093-1-24-0': 'The interior and exterior cosmological constants can be expressed in terms of [MATH] as [MATH] and [MATH].', 'gr-qc-0508093-1-24-1': 'With these choices, the dynamics of the bubble wall are entirely determined by [MATH], [MATH], and [MATH].', 'gr-qc-0508093-1-25-0': 'Let us now discuss some realistic values for the parameters in this theory.', 'gr-qc-0508093-1-25-1': 'The interior cosmological constant ([MATH]) and the bubble wall surface energy density ([MATH]) will be set by the scale of inflation ([MATH]).', 'gr-qc-0508093-1-25-2': 'The exterior cosmological constant ([MATH]) will be set by a scale [MATH].', 'gr-qc-0508093-1-25-3': 'These yield [EQUATION]', 'gr-qc-0508093-1-25-4': 'We will consider three representative energy scales, covering the interesting range of energy scales for inflation.', 'gr-qc-0508093-1-25-5': 'For weak scale inflation ([MATH] GeV), [MATH], [MATH], and [MATH].', 'gr-qc-0508093-1-25-6': 'For an inflation scale near the GUT scale ([MATH] GeV [MATH]), we have [MATH], [MATH], and [MATH].', 'gr-qc-0508093-1-25-7': 'Near-Planck scale inflation ([MATH] GeV) yields [MATH], [MATH], and [MATH].', 'gr-qc-0508093-1-25-8': 'The most massive bound solution (that which just reaches the top of the potential) is given by converting from [MATH] to [MATH] using Eq. [REF].', 'gr-qc-0508093-1-25-9': 'This maximal mass is very different in each case, ranging from an ant-mass of [MATH] grams for [MATH] GeV to an Earth-mass of [MATH] grams, for [MATH] GeV.', 'gr-qc-0508093-1-26-0': '## Allowed Solutions', 'gr-qc-0508093-1-27-0': 'A bubble wall trajectory is characterized by [MATH]const., and there are three general types:', 'gr-qc-0508093-1-28-0': 'The qualitative features of a generic potential can be shown by considering the two illustrative (but unrealistic; see above) cases of ([MATH], [MATH]) shown in Fig. [REF] and ([MATH], [MATH]) shown in Fig. [REF].', 'gr-qc-0508093-1-28-1': 'The important features are:', 'gr-qc-0508093-1-29-0': 'The potential diagram contains all of the information needed to determine the conformal structure of the allowed one-bubble spacetimes.', 'gr-qc-0508093-1-29-1': 'A complete set of the qualitatively different trajectories for arbitrary interior and exterior positive definite cosmological constants (with [MATH]) and [MATH] are denoted in Fig. [REF] and [REF].', 'gr-qc-0508093-1-29-2': 'Figure [REF] and Fig. [REF] displays the conformal structure of these solutions .', 'gr-qc-0508093-1-29-3': 'The conformal diagrams in each row are matched along the bubble wall (solid line with an arrow) and the physical regions are shaded.', 'gr-qc-0508093-1-29-4': 'The naming scheme in Fig. [REF] and [REF] is chosen to reflect the structure of the conformal diagrams.', 'gr-qc-0508093-1-29-5': 'The first numbers are the regions of the SdS conformal diagram that the bubble wall passes through.', 'gr-qc-0508093-1-29-6': 'The numbers following the backslash are the regions of the dS conformal diagram that the bubble wall passes through.', 'gr-qc-0508093-1-30-0': 'For example, consider the IV-I-III/III solutions (solution 1 of Fig. [REF]).', 'gr-qc-0508093-1-30-1': 'These start at at [MATH], corresponding to the singularity in region IV of the SdS conformal diagram and the [MATH] surface in region III of the dS conformal diagram.', 'gr-qc-0508093-1-30-2': '[MATH] and [MATH] are both greater than zero over the entire trajectory.', 'gr-qc-0508093-1-30-3': 'Therefore, the wall on the SdS side must follow a path of increasing [MATH], pushing it into region I (crossing the black hole horizon).', 'gr-qc-0508093-1-30-4': 'The wall on the dS half must follow a path of decreasing coordinate time, and thus remain in region III.', 'gr-qc-0508093-1-30-5': 'The wall then reaches a turning point, falls through the black hole event horizon, and ends up back at [MATH] (the singularity in region II of the SdS diagram, and the [MATH] surface in region III of the dS diagram).', 'gr-qc-0508093-1-30-6': 'The construction of the other diagrams in Fig. [REF] and Fig. [REF] proceeds similarly.', 'gr-qc-0508093-1-31-0': 'Based on the inward pressure gradient, the IV-I-III/III solution is what one might expect the junction conditions to yield: a sphere of false vacuum which expands and then contracts.', 'gr-qc-0508093-1-31-1': 'Relativistic effects, however, lead to the qualitatively different behavior exhibited by the other solutions in Fig. [REF] and Fig. [REF].', 'gr-qc-0508093-1-31-2': 'For instance, the evolution of the IV-III-II/III (solution 2) solution is qualitatively similar to the IV-I-III/III solution, but is so massive that its evolution is always hidden behind the black hole event horizon.', 'gr-qc-0508093-1-31-3': "The IV-I-II'/III-II (solution 3) solution is a bubble which has enough kinetic energy to escape collapse by expanding through the cosmological horizon; observers inside (or who travel inside from region I) the false vacuum region will find themselves in an inflationary universe at late times.", 'gr-qc-0508093-1-31-4': 'In the time-reverse of this solution, a bubble implodes from infinity into the black hole horizon, and the interior undergoes collapse.', 'gr-qc-0508093-1-32-0': 'The remaining solutions, shown in Fig. [REF], have interiors which approach an inflationary universe at late times, but lie on the opposite side of the wormhole in the exterior SdS spacetime.', 'gr-qc-0508093-1-32-1': 'To an observer in region III of the SdS diagram, the IV"-III-II"/IV-I-II (solution 5) and IV"-III-II"/IV-III-II solutions (solution 6) would appear as a "sky" of false vacuum that encroaches from infinity, reaches a minimum radius and then expands back out.', 'gr-qc-0508093-1-32-2': 'It has been pointed out by Bousso [CITATION] that at late times (in an asymptotically flat spacetime) the wall trajectory according to the observer in region III of the S(dS) diagram approaches that of a true vacuum bubble [CITATION].', 'gr-qc-0508093-1-32-3': 'Amusingly, the SdS observer will think he is in a true vacuum bubble surrounded by a large region of false vacuum, while the dS observer will think she is in a false vacuum bubble surrounded by a large region of true vacuum.', 'gr-qc-0508093-1-33-0': 'This symmetry between true and false vacuum bubbles is made manifest in the analysis of the Coleman-De Luccia [CITATION] instanton, which describes the production of both true and false vacuum bubbles [CITATION].', 'gr-qc-0508093-1-33-1': 'These are zero energy solutions, and so we should look for an [MATH] unbound solution; this corresponds (via Eq. [REF]) to [MATH], and we see from Fig. [REF] and [REF] that the IV"-III-II"/IV-I-II solution (solution 5) can be identified as the false vacuum instanton.', 'gr-qc-0508093-1-33-2': 'The radius of the bubble wall at the turning point (relevant for construction of the euclidean instanton) is found by considering the limit as the potential (Eq. [REF]) goes to [MATH], where on the right (unbound) side of the potential hump the [MATH] term dominates.', 'gr-qc-0508093-1-33-3': 'Solving for [MATH] using Eq. [REF], we find the radius at turnaround to be [EQUATION] which agrees with the previous literature [CITATION] (see also [CITATION]).', 'gr-qc-0508093-1-33-4': 'Since the Schwarzschild mass is zero, we are now matching two pure de Sitter spacetimes across the bubble wall.', 'gr-qc-0508093-1-33-5': 'The conformal diagram for the exterior dS region (right) only contains the area between the vertical dashed lines (which are now identified as [MATH] surfaces) in solution 5 of Fig. [REF].', 'gr-qc-0508093-1-33-6': 'The interior half (left) of the diagram remains unchanged.', 'gr-qc-0508093-1-33-7': 'It can be seen that at turnaround, the bubble will be larger than both the interior horizon size and the (exterior) horizon size of the region it has replaced.', 'gr-qc-0508093-1-34-0': 'In the IV-III-II"/III-II solution (solution 4), the region of false vacuum surrounding the observer in region III of the SdS diagram would begin very small and then expand out of the cosmological horizon.', 'gr-qc-0508093-1-34-1': 'This solution will also have a time-reversed version in which the surrounding region of false vacuum implodes.', 'gr-qc-0508093-1-34-2': 'For the IV"-III"-II"/IV-III-II solution (solution 7), the "sky" of false vacuum would forever reside outside of the horizon of a region III observer.', 'gr-qc-0508093-1-34-3': 'Finally, the N/III-II solution (solution 8), which we will define as solutions with mass greater than or equal to the Nariai mass, will be an exploding (or imploding in the time-reversed solution) bubble of false vacuum centered on [MATH].', 'gr-qc-0508093-1-35-0': 'In a series of papers, Farhi et.', 'gr-qc-0508093-1-35-1': 'al. [CITATION] discussed the application of the Penrose theorem [CITATION] to the one-bubble spacetimes discussed above.', 'gr-qc-0508093-1-35-2': 'If the null energy condition (NEC) holds (as it does for the postulated energy momentum tensor) and there exists a non-compact Cauchy surface (as in the full SdS spacetime), then the existence of a closed anti-trapped surface in the spacetime implies the presence of an initial singularity.', 'gr-qc-0508093-1-35-3': 'Since each point on the conformal diagrams in Fig. [REF] and Fig. [REF] represents a two-sphere, such an anti-trapped surface exists if the ingoing and outgoing future-directed null rays both diverge.', 'gr-qc-0508093-1-35-4': 'For example, the 2-sphere represented by point [MATH] shown in Fig. [REF], solution 1, is a closed anti-trapped surface.', 'gr-qc-0508093-1-35-5': 'This can be seen by following the null ray (null rays are denoted by the dashed lines in Fig. [REF]) from [MATH] in region III of the dS diagram to [MATH] and noting that [MATH] increases monotonically as [MATH] is approached (the null rays are diverging).', 'gr-qc-0508093-1-35-6': 'But following the future-directed null rays in the opposite direction from [MATH] in region IV of the SdS diagram, across the bubble wall, and into the false vacuum region, shows that they also diverge.', 'gr-qc-0508093-1-35-7': 'Thus, an initial singularity is necessary for this solution to exist at and near [MATH].', 'gr-qc-0508093-1-35-8': 'This spacetime also, however, contains regions without anti-trapped surfaces.', 'gr-qc-0508093-1-35-9': 'Following the future-directed null rays to point [MATH], for example, we see that the ingoing rays diverge, but the outgoing rays converge.', 'gr-qc-0508093-1-35-10': 'For examples, see solutions 1-3 in Fig. [REF], where regions which contain anti-trapped surfaces are shaded blue (dark) and the regions which do not are shaded green (light).', 'gr-qc-0508093-1-36-0': 'If we cut the IV-I-II/III solution (solution 1) in the expanding phase on a spacelike hypersurface at a time where the radius of the bubble wall satisfies [MATH], then the spacetime would not necessarily contain an initial singularity .', 'gr-qc-0508093-1-36-1': "We can remove the initial singularity from the IV-I-II'/III-II (solution 3) solution as well by cutting on the same surface.", 'gr-qc-0508093-1-36-2': "Both the IV-I-II/III (solution 1) and IV-I-II'/III-II (solution 3) solutions are therefore classically buildable.", 'gr-qc-0508093-1-36-3': "The IV-I-II'/III-II solution (solution 3) is the only example where it is possible to form an inflationary universe from classically buildable initial conditions, but only exists when the interior and exterior cosmological constant are almost equal ([MATH]).", 'gr-qc-0508093-1-36-4': 'This solution might be of interest in understanding transitions between nearly degenerate vacua, for example in the context of eternal inflation.', 'gr-qc-0508093-1-37-0': 'Given the existence of a classically forbidden region in Fig. [REF] and [REF], one might ask if tunneling is allowed from one of the recollapsing solutions (solutions 1 and 2) to one of the expanding solutions (solutions 5-7 ) on the other side of the potential hump.', 'gr-qc-0508093-1-37-1': 'This event, shown in Fig. [REF], would constitute a violation of the NEC, and so the Penrose theorem would no longer apply to the antitrapped surfaces that exist after the tunneling event.', 'gr-qc-0508093-1-37-2': 'Such a process is indeed apparently allowed [CITATION], and would describe the quantum creation of an inflationary universe from classically buildable initial conditions (if the initial condition is the IV-I-II/III solution in solution 1).', 'gr-qc-0508093-1-38-0': 'This is a rather strange transition, however, as the unbound solution is behind the wormhole in SdS (see Fig. [REF]).', 'gr-qc-0508093-1-38-1': 'An observer in region I would see the bubble expand, reach its turning point, and then disappear, only to be replaced by a black hole.', 'gr-qc-0508093-1-38-2': 'An observer inside the bubble would see the wall expanding away and - just as it is about to turn around and start collapsing - instead disappear behind the cosmological horizon.', 'gr-qc-0508093-1-38-3': 'This observer will be inside an inflationary universe, but forever disconnected from region I.', 'gr-qc-0508093-1-38-4': 'If the black hole in the SdS spacetime then evaporates, the baby universe will become completely topologically disconnected.', 'gr-qc-0508093-1-39-0': 'Having considered one circumstance in which an NEC violation precipitates the creation of an inflationary universe, one might ask if there are others.', 'gr-qc-0508093-1-39-1': 'Quantum fluctuations of a scalar field in de Sitter space can violate the NEC, and so one might imagine that any of the solutions that we have discussed which allow inflation inside the bubble could be spontaneously created somewhere along their trajectories.', 'gr-qc-0508093-1-39-2': 'One example of this direct production of baby universes is a fluctuation into one of the unbound solutions (solutions 5-7).', 'gr-qc-0508093-1-39-3': 'Such scenarios have been considered in the context of the stochastic approach to baby universe production by Linde [CITATION] and in reference to eternal inflation by Carroll and Chen [CITATION].', 'gr-qc-0508093-1-39-4': 'The thermal decay of de Sitter vacua [CITATION] should also correspond to fluctuation into one of the solutions in our catalog.', 'gr-qc-0508093-1-40-0': 'With these considerations, there are at least three competing channels for the formation of baby universes: direct production, the FGG mechanism, and the zero-mass false vacuum instanton.', 'gr-qc-0508093-1-40-1': 'It would be desirable to develop a scheme to directly compare the relative probabilities of each of these processes, and this will be explored in future work.', 'gr-qc-0508093-1-41-0': 'Our full catalog of solutions is also interesting in regards to a recent proposal [CITATION] that false vacuum regions, assumed to be larger than the interior horizon, must at all times be larger than the exterior, true vacuum, horizon.', 'gr-qc-0508093-1-41-1': 'The basis of this conjecture is the condition that the divergence of a congruence of future directed null geodesics (defined as [MATH]) must satisfy [EQUATION] where [MATH] is an affine parameter, if the NEC holds for all [MATH].', 'gr-qc-0508093-1-41-2': 'Null rays in the dS and SdS spacetimes satisfy this inequality (in dS, the inequality is exactly zero), but we should check that the junction conditions do not violate it.', 'gr-qc-0508093-1-41-3': 'One requirement imposed by Eq. [REF] is that the divergence of the null rays does not increase at the position of the wall as they go from a true vacuum region into a false vacuum one.', 'gr-qc-0508093-1-41-4': 'Along any given null geodesic in the bubble interior or exterior, the value of [MATH] is either increasing or decreasing monotonically as a function of [MATH].', 'gr-qc-0508093-1-41-5': 'We can therefore state the condition Eq. [REF] as: one cannot have a null ray along which [MATH] outside the bubble and [MATH] inside the bubble.', 'gr-qc-0508093-1-41-6': 'Surveying the solutions in Fig. [REF] and Fig. [REF], we see that this is indeed always true.', 'gr-qc-0508093-1-42-0': 'The authors of Ref. [CITATION] intended to demonstrate that if one requires the false vacuum region to be larger than the interior horizon size at all times (so that inflation is unstoppable), it is necessarily larger than the exterior horizon size.', 'gr-qc-0508093-1-42-1': 'Although all of the allowed one-bubble spacetimes satisfy the condition Eq. [REF], there are only two examples in which all observers agree that this requirement is met: the false vacuum instanton (IV"-III-II"/IV-I-II solution (solution 5), with [MATH]) and the IV-I-II\'/III-II solution (solution 3, after turn-around).', 'gr-qc-0508093-1-42-2': 'In every other case (including the FGG spacetime in Fig. [REF]), the observers in region I of the SdS conformal diagram will see only a black hole horizon sized volume replaced by the false vacuum bubble.', 'gr-qc-0508093-1-42-3': 'We therefore conjecture that if one requires the false vacuum region to be larger than the interior horizon size at all times, then it will replace a volume larger than the exterior horizon size according to only some observers.', 'gr-qc-0508093-1-42-4': 'If one relaxes this requirement, then there are a diverse range of solutions which might describe the spawning of an inflationary universe.', 'gr-qc-0508093-1-42-5': 'For example, the bubbles in solutions 3, 4 and 8 of Fig. [REF] and Fig. [REF] grow from an arbitrarily small size.', 'gr-qc-0508093-1-43-0': 'Having discussed the character of the various solutions, we turn now to a potentially dangerous detail, which is particularly important for the FGG mechanism: there exists a classical instability against aspherical perturbations in the spherically symmetric solutions to the junction conditions.', 'gr-qc-0508093-1-44-0': '# Perturbations', 'gr-qc-0508093-1-45-0': 'The solutions described in Sec. [REF] assume that the region of false vacuum is spherically symmetric.', 'gr-qc-0508093-1-45-1': 'The stability of these solutions against aspherical perturbations has important consequences, especially if one hopes to build plausible cosmologies.', 'gr-qc-0508093-1-45-2': 'That there might be an instability in domain walls was first discussed by Adams, Freese and Widrow [CITATION].', 'gr-qc-0508093-1-45-3': 'The bubble wall can trade volume energy for surface energy and wall kinetic energy locally as well as globally, and so the bubble wall will become distorted if different sections of the wall have different kinetic energies.', 'gr-qc-0508093-1-45-4': "As long as the local distortions of the wall remain small compared to the size of the background solution's radius, this process can be formulated quantitatively as perturbation theory around a background spherically symmetric solution.", 'gr-qc-0508093-1-46-0': 'Previous authors [CITATION] have considered perturbations on expanding bubbles of true-vacuum [CITATION], which have zero total energy (surface, volume, and kinetic energies canceling), and so can expand asymptotically.', 'gr-qc-0508093-1-46-1': 'As was first pointed out by Garriga and Vilenkin [CITATION], even though local observers on the bubble wall see perturbations grow, external observers see them freeze out because they do not grow faster than bubble radius.', 'gr-qc-0508093-1-47-0': "The story is different for the bound (solutions 1 and 2) and unbound (solutions 5-7) false vacuum bubbles: since they reach a turning point, the perturbations have a chance to catch up to the bubble's expansion and become nonlinear.", 'gr-qc-0508093-1-47-1': 'The remainder of this work will focus on the instability of the bound IV-I-II/III solution (solution 1), since physically plausible initial conditions may be clearly formulated.', 'gr-qc-0508093-1-47-2': 'There is no obvious set of initial conditions for the perturbations on the unbound solutions, and so we simply observe that the results we will obtain for the bound solutions apply qualitatively here as well.', 'gr-qc-0508093-1-48-0': 'To simplify the problem, we assume that the full gravitational problem described in the previous sections can be treated as motion of the bubble wall in a fixed SdS background.', 'gr-qc-0508093-1-48-1': 'This assumption must be validated (as we do below), but we are mainly interested in the low-mass bound solutions for which we might expect the gravitational contributions to be small.', 'gr-qc-0508093-1-48-2': 'Assuming that a thin spherically symmetric bubble wall separates an internal dS from an external SdS spacetime, we can employ the action [CITATION]: [EQUATION] where [MATH] is the surface energy density on the bubble wall, [MATH]) is the metric on the worldsheet of the bubble wall, [MATH] is the difference in volume energy density on either side of the bubble wall: [EQUATION] and [MATH] is the metric of the background spacetime.', 'gr-qc-0508093-1-49-0': '## Wall Equation of Motion', 'gr-qc-0508093-1-50-0': 'The equation of motion resulting from Eq. [REF] is [CITATION]: [EQUATION] where [MATH] is the extrinsic curvature tensor of the worldsheet of the bubble wall, [EQUATION] and [MATH] is the covariant derivative and [MATH] is the unit normal to the bubble wall worldsheet.', 'gr-qc-0508093-1-51-0': 'We will use the static foliation of the SdS spacetime (see Eq. [REF]) as the coordinates [MATH] for the background spacetime.', 'gr-qc-0508093-1-51-1': 'The world sheet is given coordinates [MATH] as in Eq. [REF], and has metric: [EQUATION] with the gauge freedom in choosing [MATH] fixed by [EQUATION] so that [MATH].', 'gr-qc-0508093-1-51-2': 'Here and henceforth primes will denote derivatives with respect to [MATH].', 'gr-qc-0508093-1-51-3': 'The other non-zero components of [MATH] are [MATH] and [MATH].', 'gr-qc-0508093-1-52-0': "The first task at hand is to find the worldsheet's unit normal, which by spherical symmetry has only [MATH] and [MATH] components.", 'gr-qc-0508093-1-52-1': 'Requiring orthogonality to the worldsheet ([MATH]) and unit norm ([MATH]) yields its components: [EQUATION]', 'gr-qc-0508093-1-52-2': 'The components of [MATH] are given by [EQUATION]', 'gr-qc-0508093-1-52-3': 'Substituting Eq. [REF] into Eq. [REF] gives the equation of motion for the bubble wall: [EQUATION]', 'gr-qc-0508093-1-52-4': 'Eq. [REF] supplies the velocity of the bubble at some position along its trajectory [EQUATION]', 'gr-qc-0508093-1-52-5': 'Choosing this boundary condition is effectively restricting ourselves to the IV-I-II/III (solution 1) or IV-III-II/III (solution 2) solutions.', 'gr-qc-0508093-1-52-6': 'Since the solutions to Eq. [REF] approximate the dynamics of the junction condition problem, we should parametrize by [MATH], [MATH], and [MATH].', 'gr-qc-0508093-1-52-7': 'This can be done by using the conversions defined in Sec. [REF], and gives: [EQUATION] where [MATH] is written in terms of [MATH] as [EQUATION] and [EQUATION]', 'gr-qc-0508093-1-52-8': 'To justify the use of the simplified dynamics described above, Eq. [REF] was numerically integrated, and the position of the turning point compared to the corresponding point on the full junction condition potential.', 'gr-qc-0508093-1-52-9': 'Over the range of [MATH] corresponding to the bound solutions, we find excellent quantitative agreement (well within 1%) between the turning points of the solutions to Eq. [REF] and the junction condition potential.', 'gr-qc-0508093-1-52-10': 'This was repeated with equally good results for the weak, GUT, and Planck scale potentials and also for various initial positions between the black hole radius and the potential wall (turning point).', 'gr-qc-0508093-1-52-11': 'This shows that to zeroth order, dynamics as motion in a background is valid, and strongly suggests that it will be at higher orders as well.', 'gr-qc-0508093-1-53-0': '## Perturbations', 'gr-qc-0508093-1-54-0': 'We are now in a position to discuss the first-order perturbations on the spherically-symmetric background solutions discussed in Sec. [REF].', 'gr-qc-0508093-1-54-1': 'Physical perturbations are normal to the worldsheet of the (background) bubble wall, and can be described by scalar field [MATH] by taking the position of the perturbed worldsheet to be [EQUATION] where [MATH] is the spherically symmetric solution, and [MATH] is the unit normal to the worldsheet.', 'gr-qc-0508093-1-54-2': 'It is assumed that [MATH] is much smaller than the bubble wall radius, so that a perturbative analysis can be made.', 'gr-qc-0508093-1-55-0': 'The equation of motion for the perturbation field [MATH] in a curved spacetime background can be derived from the action Eq. [REF] after expanding to second order in [MATH] [CITATION] [EQUATION] where [EQUATION] and [MATH] is [EQUATION]', 'gr-qc-0508093-1-55-1': 'To solve the equation of motion, we can decompose [MATH] into spherical harmonics [EQUATION] and separate variables to get an equation for [MATH].', 'gr-qc-0508093-1-55-2': 'The geometrical factors in Eq. [REF] become dependent on [MATH] or [MATH] only at second order, so we will always be able to make this decomposition.', 'gr-qc-0508093-1-56-0': '[MATH] is then given by: [EQUATION]', 'gr-qc-0508093-1-56-1': 'The components of [MATH] are: [EQUATION]', 'gr-qc-0508093-1-56-2': 'The components of the Ricci tensor are given by: [EQUATION]', 'gr-qc-0508093-1-56-3': 'Contracting equations [REF] and [REF] gives: [EQUATION]', 'gr-qc-0508093-1-56-4': 'The Ricci scalar on the world sheet is [EQUATION] where [MATH] is given by Eq. [REF].', 'gr-qc-0508093-1-57-0': 'After substituting Eq. [REF], Eq. [REF], and Eq. [REF] into Eq. [REF], the equation of motion for [MATH] is [EQUATION]', 'gr-qc-0508093-1-57-1': 'In terms of the dimensionless variables of the junction condition problem this reads: [EQUATION] where [MATH] is the dimensionless perturbation field defined similarly to [MATH] (see Eq. [REF]).', 'gr-qc-0508093-1-57-2': 'The first term acts as a (anti)drag on (shrinking) growing perturbations.', 'gr-qc-0508093-1-57-3': 'The last term in this equation is always negative, acting as a restoring force.', 'gr-qc-0508093-1-57-4': 'Perturbations will grow when the other terms (which are positive over most of the trajectory in the expanding phase) in this equation dominate.', 'gr-qc-0508093-1-57-5': 'Further, the last term indicates that lower [MATH] modes will experience the largest growth.', 'gr-qc-0508093-1-57-6': 'The full details of the solutions, however, require a numerical approach, to which we now turn.', 'gr-qc-0508093-1-58-0': '# Application to the Farhi-Guth-Guven mechanism', 'gr-qc-0508093-1-59-0': 'The possibility of creating an inflating false-vacuum region via quantum tunneling (described in Sec. [REF]) has been investigated only under the assumption of spherical symmetry, and this would be grossly violated if perturbations on the bubble wall become nonlinear.', 'gr-qc-0508093-1-59-1': 'In this section, we investigate the circumstances under which this is the case.', 'gr-qc-0508093-1-59-2': 'The two basic questions at issue are: first, when do perturbations go nonlinear for some given set of initial perturbations, and second, what initial perturbations can be expected.', 'gr-qc-0508093-1-60-0': '## Dynamics of the Perturbation Field', 'gr-qc-0508093-1-61-0': 'Let us begin with the first issue.', 'gr-qc-0508093-1-61-1': 'Since Eq. [REF] is a second order ODE, it can be decomposed into the sum of two linearly independent solutions [EQUATION]', 'gr-qc-0508093-1-61-2': 'The functions [MATH] and [MATH] can be found numerically by alternately setting [MATH] and [MATH] to zero, then evolving the coupled Eq. [REF] and Eq. [REF] numerically for a time [MATH] with initial conditions for [MATH], [MATH], and [MATH].', 'gr-qc-0508093-1-61-3': 'If the bubble is to tunnel, it will do so at the time [MATH], when the bubble reaches its maximum radius and begins to re-collapse.', 'gr-qc-0508093-1-61-4': 'Given [MATH] and [MATH] at time [MATH], the size of the perturbations at the turning point for any [MATH], [MATH], [MATH], [MATH], and [MATH] can be determined.', 'gr-qc-0508093-1-61-5': 'An RK4 algorithm with adaptive step size was used to solve for [MATH] and [MATH], with numerical errors well within the [MATH] level.', 'gr-qc-0508093-1-62-0': 'The results of this analysis for [MATH] and for the low (weak) and intermediate (GUT) inflation scales discussed below Eq. [REF] are shown in Fig. [REF].', 'gr-qc-0508093-1-62-1': 'The solid lines show contours of constant (log) amplification factor [MATH] (left) and [MATH] (right) versus the bubble starting radius [MATH] and mass parameter [MATH], with bubble mass increasing toward the top.', 'gr-qc-0508093-1-62-2': 'The shaded regions indicate regions which we have disallowed as bubble starting radii because the bubble would not be classically buildable for [MATH] (marked as [MATH]), or the bubble is in the forbidden region [MATH] of the effective 1D equation of motion Eq. [REF], or the bubble would be too small to be treated classically.', 'gr-qc-0508093-1-62-3': 'We choose the latter radius as fifty times the Compton wavelength [MATH] of a piece of the bubble wall .', 'gr-qc-0508093-1-62-4': 'The choice of fifty Compton wavelengths is rather arbitrary; the effect of a larger bound would be to exclude more of the parameter space in Fig. [REF].', 'gr-qc-0508093-1-62-5': 'This (unshaded) parameter space includes all classical initial conditions which could be set up by the observer in region I of the SdS conformal diagram.', 'gr-qc-0508093-1-63-0': 'It can be seen in Fig. [REF] that the growth of the perturbations is in general larger for higher-mass bubbles (smaller [MATH], larger [MATH]).', 'gr-qc-0508093-1-63-1': 'The lower the inflation scale, the closer to zero the peak in the potential function becomes, and the smaller [MATH] (higher mass) bubbles are allowed, so at low inflation scales [MATH] and [MATH] can be very large.', 'gr-qc-0508093-1-63-2': 'Growth for the Planck-scale inflation bubbles is very small, with [MATH] of order 10 and [MATH] of order 1, and is not plotted.', 'gr-qc-0508093-1-64-0': 'The enhanced growth at small [MATH] is due to the suppression of the term in Eq. [REF] proportional to [MATH], which always acts to stabilize the perturbations.', 'gr-qc-0508093-1-64-1': 'Another consequence of this suppression is that the range in [MATH] over which solutions are unstable depends on [MATH]; as a general rule of thumb, approximately a few times [MATH]-modes are unstable (note that this is unlike the case true vacuum bubbles, for which only the [MATH] modes are unstable).', 'gr-qc-0508093-1-64-2': 'An example of the [MATH] function for [MATH] with the intermediate (GUT) inflation scale parameters is shown in Fig. [REF].', 'gr-qc-0508093-1-64-3': 'The [MATH] functions for very large [MATH] modes are stable and approach sinusoids with amplitudes less than one (see the inset of Fig. [REF]), meaning that the perturbations are never larger than their initial size.', 'gr-qc-0508093-1-65-0': '## Initial Conditions and Evolution to the Turning Point', 'gr-qc-0508093-1-66-0': 'Having fully characterized the growth of the perturbations, we now require an estimate for their initial values when the bubble is formed.', 'gr-qc-0508093-1-66-1': 'There is no reason to expect that a region of false vacuum will fluctuate into existence with anything near spherical symmetry, nor is it likely to have thin walls (there is no instanton or other mechanism to enforce these symmetries).', 'gr-qc-0508093-1-66-2': 'Since low-[MATH] (relative to [MATH]) modes are unstable, an initially aspherical bubble will only become more aspherical; this is in marked contrast to true vacuum bubbles, which both start spherical, and tend to become more spherical as they expand.', 'gr-qc-0508093-1-67-0': 'Suppose, however, that we consider the best-case scenario in which a bubble is, by chance or design, spherically symmetric.', 'gr-qc-0508093-1-67-1': 'It will nevertheless inevitably be dressed with zero-point quantum fluctuations of the perturbation field.', 'gr-qc-0508093-1-67-2': 'We may then check whether these fluctuations alone, considered as initial values for the perturbations of a bubble starting with a given [MATH] and [MATH], suffice to make the bubble nonlinearly aspherical by turnaround.', 'gr-qc-0508093-1-68-0': 'We assume that the ensemble average of the quantum fluctuations at the time of nucleation is zero; but the ensemble average of the square of the field (the space-like two-point function [MATH]) will not generally vanish.', 'gr-qc-0508093-1-68-1': 'We can write the mode functions (Eq. [REF]) in terms of it as: [EQUATION]', 'gr-qc-0508093-1-68-2': 'By spherical symmetry, the two-point function can be written as a function of the angular separation [MATH] between [MATH] and [MATH], and decomposed into Legendre polynomials: [EQUATION]', 'gr-qc-0508093-1-68-3': 'Using the addition theorem for spherical harmonics, we can write this as [EQUATION]', 'gr-qc-0508093-1-68-4': 'Substituting this into Eq. [REF] and using the orthogonality of the spherical harmonics yields the relation: [EQUATION]', 'gr-qc-0508093-1-68-5': 'Given some space-like two point function at the time the bubble is nucleated, we can obtain the [MATH] from [EQUATION] and therefore set the typical initial amplitudes of the mode functions as the r.m.s. value [MATH] from Eq. [REF].', 'gr-qc-0508093-1-68-6': 'The velocity field can be decomposed into spherical harmonics just as [MATH] was, and the analysis performed above carries over exactly.', 'gr-qc-0508093-1-68-7': 'The typical initial size of the velocity mode functions is then given by [EQUATION] with [EQUATION]', 'gr-qc-0508093-1-68-8': 'The initial amplitudes in Eq. [REF] and Eq. [REF] can now be evolved to the turning point, and the mode functions re-summed.', 'gr-qc-0508093-1-68-9': 'The ensemble average of the r.m.s. fluctuations in [MATH] at any time at a given point will then be: [EQUATION] which can be evaluated at [MATH].', 'gr-qc-0508093-1-69-0': 'A full model of the two-point functions [MATH] and [MATH] would involve quantizing the mode functions on the curved spacetime of the bubble wall worldsheet, which has a metric depending on [MATH].', 'gr-qc-0508093-1-69-1': 'Further, to treat large fluctuations, we would need to include non-linear terms in the equation of motion.', 'gr-qc-0508093-1-69-2': 'The exact two-point function is therefore a rather formidable object to compute.', 'gr-qc-0508093-1-69-3': 'As a simplified model, we will employ the two-point functions of a massless scalar field in flat spacetime, and replace the spatial distance [MATH] with the distance along the bubble wall [MATH].', 'gr-qc-0508093-1-69-4': 'This massless scalar corresponds to the perturbations on a flat wall separating domains of equal energy density in Minkowski space [CITATION].', 'gr-qc-0508093-1-69-5': 'Corrections to this picture in the presence of curvature should be small over small regions of the bubble wall.', 'gr-qc-0508093-1-69-6': 'We are also neglecting the large difference in energy densities across the bubble wall, which will give the field a (negative) mass to first order.', 'gr-qc-0508093-1-69-7': 'The apparent divergence of the correlator due to this negative mass will be rendered finite by the non-linear terms which must be introduced to discuss large fluctuations.', 'gr-qc-0508093-1-69-8': 'In light of all these difficulties, and several more approximations we will make below, this should be considered as a first, rough estimate of the amplitude of the quantum fluctuations on the bubble at the time of nucleation.', 'gr-qc-0508093-1-70-0': 'The space-like two-point function in Minkowski space at large separations is given by [EQUATION] where [MATH].', 'gr-qc-0508093-1-70-1': 'As in the work of Garriga and Vilenkin [CITATION], we introduce a smeared field operator to obtain a well-defined answer at close separations [EQUATION] where [MATH] is a smearing length, chosen to be the Compton wavelength [MATH] of a piece of wall, which is a physically reasonable lower bound on the size of a measurable region.', 'gr-qc-0508093-1-70-2': "The smeared correlator will then be given by [EQUATION] which evaluates at [MATH] to [EQUATION] where [MATH] is Catalin's constant.", 'gr-qc-0508093-1-70-3': 'We have been unable to integrate Eq. [REF] to obtain the exact form of the smeared two-point function for all [MATH].', 'gr-qc-0508093-1-70-4': 'However, it must smoothly interpolate between the value at [MATH] in Eq. [REF] to the functional form at [MATH] given by Eq. [REF].', 'gr-qc-0508093-1-70-5': "We therefore employ the following 'toy model' smeared field correlator [EQUATION] which has the correct asymptotics.", 'gr-qc-0508093-1-70-6': 'With [MATH], the dimensionless form of the toy correlator is given by: [EQUATION] where [MATH].', 'gr-qc-0508093-1-71-0': 'The velocity-velocity correlator can be calculated using the Hamiltonian approach.', 'gr-qc-0508093-1-71-1': 'For spacelike separations, this is given by [EQUATION] where we have introduced a hard momentum cut-off [MATH] to obtain a finite answer.', 'gr-qc-0508093-1-71-2': 'This cutoff will correspond to the inverse smearing length, with higher momentum scales higher accounted for in the smeared operator.', 'gr-qc-0508093-1-71-3': 'The integral can be evaluated in terms of generalized hypergeometric functions as [EQUATION]', 'gr-qc-0508093-1-71-4': 'At close separations, we construct a smeared operator [MATH].', 'gr-qc-0508093-1-71-5': 'The expectation value of this operator at zero separation is [EQUATION] where [MATH] is the Euler-Mascheroni constant.', 'gr-qc-0508093-1-71-6': 'Smoothly connecting the small [MATH] (Eq. [REF]) and large [MATH] (Eq. [REF]) behavior as in Eq. [REF], the [MATH] (defined similarly to [MATH]) correlator on the bubble at the time of nucleation is: [EQUATION]', 'gr-qc-0508093-1-71-7': 'The integrals Eq. [REF] and Eq. [REF] for the correlators Eq. [REF] and Eq. [REF] must be evaluated numerically.', 'gr-qc-0508093-1-71-8': 'Calculation of the coefficients for every [MATH] and [MATH] is unfortunately unfeasible because of the highly oscillatory behavior of the integrands and sheer number of mode functions that must be considered.', 'gr-qc-0508093-1-71-9': 'However, we have been able to deduce sufficiently good approximate fits for [MATH] and [MATH] as a function of both [MATH] and [MATH].', 'gr-qc-0508093-1-71-10': 'In both cases, the power is dominated by a peak at [MATH].', 'gr-qc-0508093-1-72-0': 'The [MATH] are nicely fit by the function [EQUATION]', 'gr-qc-0508093-1-72-1': 'The proposed fit for the [MATH] consists of two power laws matched at the [MATH] peak.', 'gr-qc-0508093-1-72-2': 'For [MATH], the best fit is [MATH] and for [MATH], the fit is [MATH].', 'gr-qc-0508093-1-72-3': 'Because these power law indices are slightly uncertain, we only count the modes with [MATH] in the [MATH].', 'gr-qc-0508093-1-72-4': 'This is conservative, and also justified because these modes will not contribute significantly to the sum in Eq. [REF].', 'gr-qc-0508093-1-73-0': 'With these initial conditions, we can now evolve each mode function using Eq. [REF] and then re-sum in Eq. [REF] to find the average size of the fluctuations at the turning point.', 'gr-qc-0508093-1-73-1': 'We have calculated [MATH] and [MATH] up to the [MATH] corresponding to the last unstable mode of the smallest [MATH], for all three inflation scales ([MATH] GeV, [MATH] GeV, and [MATH] GeV).', 'gr-qc-0508093-1-73-2': 'The results for weak- and GUT-scale inflation are shown in Fig. [REF], where the dotted line indicates the boundary of the region over which the perturbations become non-linear (non-linear to the left of the line).', 'gr-qc-0508093-1-73-3': 'It can be seen that in this model, even just quantum perturbations of the bubble wall grow nonlinear in bubbles that start at radii less than about one-tenth of the turnaround radius; this grossly violates the assumption of spherical symmetry used in tunneling calculations.', 'gr-qc-0508093-1-73-4': 'On the other hand, none of the parameter space in the high inflation scale case went non-linear, and at all scales there is always a region of initial bubble radii near the turnaround radius, for which nonlinearity never occurs.', 'gr-qc-0508093-1-74-0': '## Thick Walls and Radiation', 'gr-qc-0508093-1-75-0': 'Just as we have no reason to expect a fluctuated region to be spherically symmetric, we have no reason to assume that it will have thin walls.', 'gr-qc-0508093-1-75-1': 'An analysis of thick-walled true vacuum bubbles was undertaken in Ref. [CITATION], where it was found that the instabilities found in thin-walled case are still present in the form of deformations normal to the bubble profile.', 'gr-qc-0508093-1-75-2': 'In the case of small false vacuum bubbles, there is no obvious consideration (such as a corresponding instanton) to supply the profile of the bubble wall, and so we can merely conjecture by precedent that the instability would be retained in the thick-walled case as well.', 'gr-qc-0508093-1-76-0': 'Another consideration, applying to bubbles smaller than the false vacuum horizon size, is whether inflation is spoiled by non-vacuum contributions to the energy density.', 'gr-qc-0508093-1-76-1': 'The perturbations on the bubble wall translate into gravitational waves [CITATION], and since the bound bubble solutions remain relatively close to their gravitational radius and become distorted over many different length scales on a relatively short time scale (see the quasi-exponential growth in Fig. [REF]), they will be emitters of copious gravitational radiation.', 'gr-qc-0508093-1-76-2': 'Another problem arises if the kinetic and gradient energy of the field becomes appreciable in the bubble interior, either from intrusion of the wall (for example, imagine a bubble being pinched in half by some non-linear perturbation), or from particle production or other scalar modes propagating in from the wall.', 'gr-qc-0508093-1-76-3': 'If the emission of energy into the interior of the bubble from any combination of these modes makes a significant contribution to the equation of state, then inflation will not occur.', 'gr-qc-0508093-1-77-0': '# Summary Discussion', 'gr-qc-0508093-1-78-0': 'In this paper we have examined the feasibility of producing inflationary universes from a spacetime with a small cosmological constant.', 'gr-qc-0508093-1-78-1': 'Reviewing the solutions allowed by the junction conditions, there are several distinct possibilities if one allows for violations of the null energy condition.', 'gr-qc-0508093-1-78-2': 'The first is the Farhi, Guth Guven (FGG) mechanism, in which (referring to Fig. [REF] and Fig. [REF]) the IV-I-II/III (solution 1) or IV-III-II/III (solution 2) solution is fluctuated in the expanding phase and then tunnels to an unbound solution, as shown in Fig. [REF].', 'gr-qc-0508093-1-78-3': 'To the outside observer in region I of the SdS conformal diagram it appears that the bubble has disappeared behind the horizon, but on the other side of the wormhole, both an inflationary universe and a non-inflating universe (an asymptotically true-vacuum de Sitter region) have come into existence.', 'gr-qc-0508093-1-78-4': 'In one case, shown in solution 5 of Fig. [REF], we can imagine an observer in region III, then take the Schwarzschild mass [MATH] limit (thus removing regions I, II and IV entirely) and interpret the event as a transition from pure true-vacuum de Sitter space to the false vacuum.', 'gr-qc-0508093-1-78-5': 'This corresponds to tunneling to the false vacuum via a regular (Coleman-De Luccia) instanton as described by, e.g., [CITATION].', 'gr-qc-0508093-1-78-6': 'Since the very same spacetime takes part in both the FGG mechanism and the false vacuum instanton, there may be some way to smoothly interpolate between these two processes; we intend to explore this idea further in future work.', 'gr-qc-0508093-1-79-0': 'In addition to the FGG and regular instanton mechanisms, there are two more possibilities.', 'gr-qc-0508093-1-79-1': 'First, an inflationary universe might be directly produced by some null energy condition (NEC) violating fluctuation into one of the unbound or monotonic solutions shown in Solutions 4-8 of Fig. [REF].', 'gr-qc-0508093-1-79-2': 'To the outside observer in region I of the SdS conformal diagram, these solutions would be indistinguishable from the fluctuation of a black hole, but they would secretly entail the creation of everything on the other side of the wormhole, as in the FGG tunneling.', 'gr-qc-0508093-1-79-3': "A second method of direct production is the fluctuation (which does not require an NEC violation) of the IV-I-II'/III-II solution (solution 2).", 'gr-qc-0508093-1-79-4': 'These solutions only exist in the case where the interior and exterior cosmological constants are comparable, and so would not correspond to the nucleation of inflation from a universe like ours, but they might be of interest in understanding transitions between nearly degenerate vacua.', 'gr-qc-0508093-1-80-0': 'Examining these classical solutions to first order, we have shown that an instability to aspherical perturbations exists in those solutions which possess a turning point.', 'gr-qc-0508093-1-80-1': 'This includes the bound (solutions 1 and 2) and unbound solutions (solutions 5, 6, and 7).', 'gr-qc-0508093-1-80-2': 'In the latter case there is no clear way to set an initial radius or initial perturbation amplitude, so we can say only that collapsing bubbles are violently unstable.', 'gr-qc-0508093-1-80-3': 'The bound solutions are amenable to quantitative investigation, and we have focused on the growth of perturbations in the expanding phase that precedes tunneling in the FGG mechanism.', 'gr-qc-0508093-1-80-4': 'For bound expanding bubbles formed by the fluctuations of a scalar field in de Sitter space, there is no instanton to enforce spherical symmetry, so we would expect initial aspherical perturbations to be relatively large.', 'gr-qc-0508093-1-80-5': 'Since there is no detailed model for the fluctuating scalar field to see how large, we have instead calculated an estimate of the minimal deviations from spherical symmetry in light of quantum fluctuations, and present this as an extremely rare, best case scenario for spherical symmetry.', 'gr-qc-0508093-1-80-6': 'These minimal fluctuations were then evolved to the turning point of the bound solutions, which is the point in the FGG mechanism where there is a chance for tunneling to an inflationary universe to occur.', 'gr-qc-0508093-1-80-7': 'Of the three representative energy scales for inflation (false vacuum energy densities) we have studied, the evolved minimal perturbations on a Plank scale bubble remain small over most of the allowed parameter space, while the perturbations on GUT and weak scale bubbles can grow nonlinear if they start at a sufficiently small ([MATH]) fraction of the turnaround radius.', 'gr-qc-0508093-1-80-8': 'Thus even in the best-case scenario some bubbles become nonlinear, but on the other hand there will always in principle be some that do not.', 'gr-qc-0508093-1-81-0': 'The instability introduces complications into the use of the FGG mechanism as a means of baby universe production.', 'gr-qc-0508093-1-81-1': 'The existing calculations of the tunneling rate rely heavily on the assumption of spherical symmetry.', 'gr-qc-0508093-1-81-2': 'It is unclear how to perform a similar calculation for a (possible non-linearly) perturbed bubble, as the number of degrees of freedom has drastically increased and the assumption of a minisuperspace of spherically symmetric metrics is no longer good.', 'gr-qc-0508093-1-81-3': 'Further, the bubble interior will become filled with scalar gradient and kinetic energy and gravity waves, possibly upsetting the interior sufficiently to prevent inflation.', 'gr-qc-0508093-1-81-4': 'One might argue that in an eternal universe there is plenty of time to wait around for a fluctuation which is sufficiently spherical.', 'gr-qc-0508093-1-81-5': 'However, to fully understand the importance of the FGG mechanism, one must both have a model of the scalar field fluctuations, which would predict the distribution of bubble shapes and masses, and also a model for tunneling in the presence of asphericities.', 'gr-qc-0508093-1-81-6': 'In addition, one must understand the competition among the various processes which result in the formation of inflating false vacuum regions, and thus show that the FGG mechanism is not a relative rarity.', 'gr-qc-0508093-1-81-7': 'These problems represent significant calculations in a theory of quantum gravity which we do not yet possess, but progress in these areas would undoubtedly improve our understanding of the initial conditions for inflation.', 'gr-qc-0508093-1-82-0': 'The authors wish to thank A. Albrecht, T. Banks, P.J. Fox, S. Gratton, and T. Mai for their assistance in the development of this work.'}

{'gr-qc-0508093-2-0-0': 'This paper examines the classical dynamics of false vacuum regions embedded in surrounding regions of true vacuum, in the thin-wall limit.', 'gr-qc-0508093-2-0-1': 'The dynamics of all generally relativistically allowed solutions - most but not all of which have been previously studied - are derived, enumerated, and interpreted.', 'gr-qc-0508093-2-0-2': 'We comment on the relation of these solutions to possible mechanisms whereby inflating regions may be spawned from non-inflating ones.', 'gr-qc-0508093-2-0-3': 'We then calculate the dynamics of first order deviations from spherical symmetry, finding that many solutions are unstable to such aspherical perturbations.', 'gr-qc-0508093-2-0-4': 'The parameter space in which the perturbations on bound solutions inevitably become nonlinear is mapped.', 'gr-qc-0508093-2-0-5': 'This instability has consequences for the Farhi-Guth-Guven mechanism for baby universe production via quantum tunneling.', 'gr-qc-0508093-2-1-0': '# Introduction', 'gr-qc-0508093-2-2-0': 'Nearly two decades ago, a series of papers [CITATION] began to investigate the possibility of creating an inflationary universe "in a laboratory" - that is, inside a surrounding region of much lower vacuum energy.', 'gr-qc-0508093-2-2-1': 'The spacetime of such a "bubble universe" was modeled as a spherically symmetric de Sitter region (the false vacuum) joined to a surrounding Schwarzschild geometry (the true vacuum) by an infinitesimally thin domain wall.', 'gr-qc-0508093-2-3-0': 'These studies found that the inflating (false vacuum) region could not avoid collapse unless either the null energy condition or cosmic censorship were violated in the full spacetime [CITATION], but that the creation of an enduring inflating region might be possible via quantum tunneling [CITATION].', 'gr-qc-0508093-2-3-1': 'In this picture, henceforth denoted the Farhi-Guth-Guven (FGG) mechanism, a classically constructable expanding bubble, which would classically re-collapse (both from the inside and outside perspective [CITATION]), instead tunnels to a new solution in which the inflating interior expands forever, while an outside observer sees a black hole .', 'gr-qc-0508093-2-3-2': 'The probability of this occurring can be calculated using the techniques of semi-classical quantum gravity, and is extraordinarily small.', 'gr-qc-0508093-2-4-0': 'Though nearly-miraculous, this process has garnered new interest recently, primarily because of evidence that our universe may have a fundamental positive cosmological constant.', 'gr-qc-0508093-2-4-1': 'If so, it will asymptotically approach everlasting equilibrium as de Sitter spacetime.', 'gr-qc-0508093-2-4-2': 'Given eternity, even the most unlikely process - such as the creation of inflating bubble universes - will eventually occur.', 'gr-qc-0508093-2-4-3': 'Taking this one step further, our observable universe could in fact be such a bubble universe, which arose from equilibrium de Sitter space and is currently returning to it.', 'gr-qc-0508093-2-4-4': 'This would realize the old idea of Boltzmann that the universe is fundamentally in equilibrium, but that extremely rare downward fluctuations in entropy periodically occur and allow the transitory existence of non-equilibrium regions that see entropy steadily increasing.', 'gr-qc-0508093-2-5-0': 'The classic problem with this idea was pointed out in its new context by Dyson, Susskind, Kleban [CITATION]: if our observable universe resulted from a downward entropy fluctuation that evolved with increasing entropy to the present time, then it is vastly more likely for this to have occurred by a fluctuation to our observable universe ten minutes ago (replete with incoming photons and memories in our brains to convince us that it is older) than by a fluctuation all the way to the much lower entropy corresponding to inflation .', 'gr-qc-0508093-2-5-1': 'Albrecht and Sorbo [CITATION], however, argue that inflation might avoid this problem by turning a tiny region of low entropy density into a very large one.', 'gr-qc-0508093-2-5-2': 'Their calculation shows that the FGG mechanism requires a much smaller entropy fluctuation than directly creating the large post-reheating region that will result from it, thus resolving the paradox.', 'gr-qc-0508093-2-6-0': 'There may, however, be reasons to doubt that the creation of small regions of false vacuum and subsequent tunneling are plausible events.', 'gr-qc-0508093-2-6-1': 'First, there is no regular instanton describing either the nucleation of small regions of false vacuum or the gravitational tunneling event; there is such an instanton describing tunneling to false-vacuum, but over a huge region, larger than the true-vacuum horizon (see, e.g., [CITATION]).', 'gr-qc-0508093-2-6-2': 'Second, Banks [CITATION] has pointed out that it is hard to understand the tunneling process holographically - it appears that the inflating region inside the bubble has many more states than the black hole it is "contained in" (see also [CITATION]).', 'gr-qc-0508093-2-6-3': 'Third, Dutta Vachaspati [CITATION] have recently (and previously in [CITATION]) argued that general causality considerations preclude the formation of a small false-vacuum region inside a large true-vacuum one.', 'gr-qc-0508093-2-7-0': 'Both in terms of the initial conditions for inflation, and also the more general issue of what processes can lead to transitions between vacuum states (as is important in understanding the string theory "landscape"), it is crucial to understand bubble universes, whether they can form, and with what probabilities.', 'gr-qc-0508093-2-7-1': 'To this question the present paper makes the following contributions: first, we organize and interpret all of the thin-walled, spherically symmetric one bubble spacetimes.', 'gr-qc-0508093-2-7-2': 'We then show that expanding false vacuum bubbles are unstable to non-spherical perturbations; if these bubbles start at a sufficiently small initial radius, then they inevitably become nonlinearly aspherical before tunneling might occur in the FGG mechanism.', 'gr-qc-0508093-2-7-3': 'It is unclear in this case whether tunneling to an inflationary universe inside the bubble can occur at all, or with what probability.', 'gr-qc-0508093-2-8-0': 'The plan of the paper is as follows.', 'gr-qc-0508093-2-8-1': 'In Sec. [REF], the allowed solutions to the junction conditions are enumerated, putting the previous work into context.', 'gr-qc-0508093-2-8-2': 'We provide a concise reference for all of the possible spacetimes with one false-vacuum bubble and arbitrary positive cosmological constant, then discuss the existing, and some new, interpretations of these solutions.', 'gr-qc-0508093-2-8-3': 'In Sec. [REF] we derive the first order perturbation equations, and demonstrate the existence of an instability in bound solutions.', 'gr-qc-0508093-2-8-4': 'In Sec. [REF] we integrate the equations to investigate the parameter space for which non-linear perturbations are unavoidable, and we conclude in Sec. [REF].', 'gr-qc-0508093-2-9-0': '# Junction Conditions', 'gr-qc-0508093-2-10-0': 'The dynamics of inflating regions has been discussed by a number of authors [CITATION] using the junction condition formalism.', 'gr-qc-0508093-2-10-1': 'These works study a spherically symmetric region of false vacuum (high energy density) in a surrounding region of true vacuum (lower energy density).', 'gr-qc-0508093-2-10-2': 'The wall separating the regions is assumed to be very thin compared to the radius of the region of false vacuum.', 'gr-qc-0508093-2-10-3': "One can obtain the dynamics of the wall by requiring metric continuity across the wall and then solving Einstein's equations.", 'gr-qc-0508093-2-10-4': "Under spherical symmetry, the dynamics of the problem then reduce to those of the bubble wall's radius.", 'gr-qc-0508093-2-10-5': 'This radius is a gauge invariant quantity because it simply quantifies the curvature of the bubble wall worldsheet, and any observer can measure it by comparing the normal to the wall at two nearby points.', 'gr-qc-0508093-2-11-0': '## Interior and Exterior Spacetimes', 'gr-qc-0508093-2-12-0': 'Let [MATH] be the cosmological constant in the true vacuum region.', 'gr-qc-0508093-2-12-1': 'Then if [MATH], the region is Schwarzschild-de Sitter spacetime with metric [EQUATION] in the static foliation.', 'gr-qc-0508093-2-12-2': 'Fixing [MATH], there are then three qualitatively different casual structures characterized by the value of [MATH] (see [CITATION]), due to the nature of the three roots of [MATH] [EQUATION] and [MATH].', 'gr-qc-0508093-2-12-3': 'We can label them as [EQUATION]', 'gr-qc-0508093-2-12-4': 'And the range of [MATH] means that they lie in the ranges [MATH].', 'gr-qc-0508093-2-13-0': 'The two positive roots correspond to the black hole and cosmological horizons.', 'gr-qc-0508093-2-13-1': 'The conformal diagram for this spacetime is shown in Fig. [REF].', 'gr-qc-0508093-2-13-2': '(See [CITATION] for a demonstration of the explicit form of the metric in global coordinates).', 'gr-qc-0508093-2-13-3': 'Surfaces of constant coordinate time [MATH] are drawn, with the circulating arrows denoting the direction of increasing [MATH].', 'gr-qc-0508093-2-13-4': 'We will consider region I to be the "causal patch" of a hypothetical observer (i.e., the region lying in both the causal past and causal future of the observer\'s world line) in what follows.', 'gr-qc-0508093-2-14-0': 'For [MATH], there are also three real roots: a double positive root [MATH] and a negative [MATH], given by: [EQUATION]', 'gr-qc-0508093-2-14-1': 'This mass is known as the Nariai mass, and in this spacetime there is only one horizon at the positive root.', 'gr-qc-0508093-2-14-2': 'The conformal diagram for this spacetime is shown in Fig. [REF] [CITATION].', 'gr-qc-0508093-2-14-3': 'There is also a time-reverse solution, starting at past null infinity and ending at [MATH].', 'gr-qc-0508093-2-14-4': 'For [MATH], there is one real negative root, and therefore no horizons in the spacetime.', 'gr-qc-0508093-2-14-5': 'The conformal diagram for this case is like figure [REF], but with the horizon lines excised.', 'gr-qc-0508093-2-15-0': 'Inside the false-vacuum region the spacetime is de Sitter space, with metric [EQUATION] in the static foliation.', 'gr-qc-0508093-2-15-1': 'Fig. [REF] shows the conformal diagram for the de Sitter region.', 'gr-qc-0508093-2-15-2': 'Again, surfaces of constant coordinate time [MATH] are shown, with the arrows denoting the direction of increasing [MATH].', 'gr-qc-0508093-2-15-3': 'We consider region III to be the causal patch in which our hypothetical false vacuum observer resides.', 'gr-qc-0508093-2-16-0': '## Equation of Motion', 'gr-qc-0508093-2-17-0': 'The bubble wall worldsheet has metric: [EQUATION] where [MATH] is the proper time in the frame of the wall, and ([MATH]) are the usual angular variables.', 'gr-qc-0508093-2-18-0': 'The coordinates in the full [MATH] spacetime are chosen to be Gaussian normal coordinates constructed in the neighborhood of the bubble wall worldsheet.', 'gr-qc-0508093-2-18-1': 'Three of the coordinates are [MATH] on the worldsheet, and the fourth, [MATH], is defined as the proper distance along a geodesic normal to the bubble worldsheet, with [MATH] increasing in the direction of SdS (true vacuum).', 'gr-qc-0508093-2-19-0': 'The transformation from the static coordinate systems of SdS and dS to the Gaussian normal system can be constructed in closed form using the methods of [CITATION], and the full metric takes the form: [EQUATION] where [MATH] defines the wall and therefore [MATH] and [MATH].', 'gr-qc-0508093-2-20-0': 'The energy momentum tensor on the wall is: [EQUATION] where [MATH] is the metric on the worldsheet of the wall for [MATH] and zero otherwise, and [MATH] is the energy density of the wall.', 'gr-qc-0508093-2-21-0': "Using the metric [REF] and the energy-momentum tensor [REF] together with the contributions from the dS interior and SdS exterior in Einstein's equations yields an equation of motion for the bubble wall of [CITATION] : [EQUATION] where [MATH] is the extrinsic curvature tensor in SdS and dS respectively.", 'gr-qc-0508093-2-21-1': 'In the Gaussian normal coordinates, this takes the form: [EQUATION]', 'gr-qc-0508093-2-21-2': 'Evaluating this in metric [REF], the [MATH] and [MATH] components of Eq. [REF] reduce to: [EQUATION] with the definitions [EQUATION]', 'gr-qc-0508093-2-21-3': 'Here, [MATH] is the metric coefficient in dS or SdS.', 'gr-qc-0508093-2-21-4': 'The sign of [MATH] is fixed by the trajectory because [MATH] could potentially be positive or negative (motion can be with or against the direction of increasing coordinate time indicated in Fig. [REF] and [REF]).', 'gr-qc-0508093-2-22-0': 'A set of dimensionless coordinates can be defined, in which Eq. [REF] can be written as the equation of motion of a particle of unit mass in a one dimensional potential [CITATION].', 'gr-qc-0508093-2-22-1': 'Let: [EQUATION] where M is the mass appearing in the SdS metric coefficient, and [EQUATION]', 'gr-qc-0508093-2-22-2': 'With these definitions, Eq. [REF] becomes [EQUATION] where the potential [MATH] and energy [MATH] are [EQUATION] with [EQUATION] and [EQUATION]', 'gr-qc-0508093-2-22-3': 'Note that a small negative [MATH] corresponds to a large mass, so that even between [MATH] the mass will vary by many orders of magnitude.', 'gr-qc-0508093-2-23-0': 'The parameter space allowed by the junction conditions is characterized by the value of the cosmological constant inside and outside the wall.', 'gr-qc-0508093-2-23-1': 'For [MATH], we have [MATH].', 'gr-qc-0508093-2-23-2': 'The maximum [MATH] of the potential [MATH] then satisfies [MATH].', 'gr-qc-0508093-2-23-3': 'The potential curves over the entire range of [MATH] are shown in Fig. [REF].', 'gr-qc-0508093-2-24-0': 'The interior and exterior cosmological constants can be expressed in terms of [MATH] as [MATH] and [MATH].', 'gr-qc-0508093-2-24-1': 'With these choices, the dynamics of the bubble wall are entirely determined by [MATH], [MATH], and [MATH].', 'gr-qc-0508093-2-25-0': 'Let us now discuss some realistic values for the parameters in this theory.', 'gr-qc-0508093-2-25-1': 'The interior cosmological constant ([MATH]) and the bubble wall surface energy density ([MATH]) will be set by the scale of inflation ([MATH]).', 'gr-qc-0508093-2-25-2': 'The exterior cosmological constant ([MATH]) will be set by a scale [MATH].', 'gr-qc-0508093-2-25-3': 'These yield [EQUATION]', 'gr-qc-0508093-2-25-4': 'We will consider three representative energy scales, covering the interesting range of energy scales for inflation.', 'gr-qc-0508093-2-25-5': 'For weak scale inflation ([MATH] GeV), [MATH], [MATH], and [MATH].', 'gr-qc-0508093-2-25-6': 'For an inflation scale near the GUT scale ([MATH] GeV [MATH]), we have [MATH], [MATH], and [MATH].', 'gr-qc-0508093-2-25-7': 'Near-Planck scale inflation ([MATH] GeV) yields [MATH], [MATH], and [MATH].', 'gr-qc-0508093-2-25-8': 'The most massive bound solution (that which just reaches the top of the potential) is given by converting from [MATH] to [MATH] using Eq. [REF].', 'gr-qc-0508093-2-25-9': 'This maximal mass is very different in each case, ranging from an ant-mass of [MATH] grams for [MATH] GeV to an Earth-mass of [MATH] grams, for [MATH] GeV.', 'gr-qc-0508093-2-26-0': '## Allowed Solutions', 'gr-qc-0508093-2-27-0': 'A bubble wall trajectory is characterized by [MATH]const., and there are three general types:', 'gr-qc-0508093-2-28-0': 'The qualitative features of a generic potential can be shown by considering the four illustrative (but unrealistic; see above) cases of ([MATH], [MATH]) shown in Fig. [REF], ([MATH], [MATH]) shown in Fig. [REF], ([MATH], [MATH]) shown in Fig. [REF], and ([MATH], [MATH]) shown in Fig. [REF].', 'gr-qc-0508093-2-28-1': 'The important features are:', 'gr-qc-0508093-2-29-0': 'The potential diagram contains all of the information needed to determine the conformal structure of the allowed one-bubble spacetimes.', 'gr-qc-0508093-2-29-1': 'A complete set of the qualitatively different trajectories for arbitrary interior and exterior positive definite cosmological constants (with [MATH]) and [MATH] are denoted in Fig. [REF], [REF], [REF], and [REF].', 'gr-qc-0508093-2-29-2': 'Figures [REF] and [REF] display the conformal structure of these solutions .', 'gr-qc-0508093-2-29-3': 'The conformal diagrams in each row are matched along the bubble wall (solid line with an arrow) and the physical regions are shaded.', 'gr-qc-0508093-2-29-4': 'For solutions with qualitatively similar SdS diagrams, the various options for the dS interior are listed.', 'gr-qc-0508093-2-29-5': 'The naming scheme in Fig. [REF], [REF], [REF], and [REF] is chosen to reflect the structure of the conformal diagrams.', 'gr-qc-0508093-2-29-6': 'The first numbers are the regions of the SdS conformal diagram that the bubble wall passes through.', 'gr-qc-0508093-2-29-7': 'The numbers following the backslash are the regions of the dS conformal diagram that the bubble wall passes through.', 'gr-qc-0508093-2-30-0': 'For example, consider the IV-I-II/III solution (solution 1 of Fig. [REF]).', 'gr-qc-0508093-2-30-1': 'These start at at [MATH], corresponding to the singularity in region IV of the SdS conformal diagram and the [MATH] surface in region III of the dS conformal diagram.', 'gr-qc-0508093-2-30-2': '[MATH] and [MATH] are both greater than zero over the entire trajectory.', 'gr-qc-0508093-2-30-3': 'Therefore, the wall on the SdS side must follow a path of increasing [MATH], pushing it into region I (crossing the black hole horizon).', 'gr-qc-0508093-2-30-4': 'The wall on the dS half must follow a path of decreasing coordinate time, and thus remain in region III.', 'gr-qc-0508093-2-30-5': 'The wall then reaches a turning point, falls through the black hole event horizon, and ends up back at [MATH] (the singularity in region II of the SdS diagram, and the [MATH] surface in region III of the dS diagram).', 'gr-qc-0508093-2-30-6': 'The construction of the other diagrams in Fig. [REF] and Fig. [REF] proceeds similarly.', 'gr-qc-0508093-2-31-0': 'Based on the inward pressure gradient, the IV-I-II/III solution is what one might expect the junction conditions to yield: a sphere of false vacuum which expands and then contracts.', 'gr-qc-0508093-2-31-1': 'Relativistic effects, however, lead to the qualitatively different behavior exhibited by the other solutions in Fig. [REF] and Fig. [REF].', 'gr-qc-0508093-2-31-2': 'For instance, the evolution of the IV-III-II/III (solution 2) solution is qualitatively similar to the IV-I-II/III solution, but is so massive that its evolution is always hidden behind the black hole event horizon.', 'gr-qc-0508093-2-31-3': "The IV-I-II'/III-II (solutions 3 and 4) solution is a bubble which has enough kinetic energy to escape collapse by expanding through the cosmological horizon; observers inside (or who travel inside from region I) the false vacuum region will find themselves in an inflationary universe at late times.", 'gr-qc-0508093-2-31-4': 'In the time-reverse of this solution, a bubble implodes from infinity into the black hole horizon, and the interior undergoes collapse.', 'gr-qc-0508093-2-32-0': "Note that there are two options for the IV-I-II'/III-II solution.", 'gr-qc-0508093-2-32-1': 'Solution 3 corresponds to the situation [MATH], which contains both [MATH] and [MATH] sign changes (see Fig. [REF]).', 'gr-qc-0508093-2-32-2': 'This causes the bubble wall to accelerate in the direction of the true vacuum from both the interior and exterior perspectives.', 'gr-qc-0508093-2-32-3': 'Solution 4 corresponds to the situation [MATH], which does not contain a [MATH] sign change (see Fig. [REF]).', 'gr-qc-0508093-2-32-4': 'This causes the bubble wall to accelerate in the direction of the false vacuum from the interior perspective.', 'gr-qc-0508093-2-32-5': 'The reason for the disparity is that solution 4 exists only when the interior and exterior cosmological constants are similar in magnitude, and so the wall tension can have greater influence on the dynamics.', 'gr-qc-0508093-2-33-0': 'The remaining solutions, shown in Fig. [REF], have interiors which approach an inflationary universe at late times, but lie on the opposite side of the wormhole in the exterior SdS spacetime.', 'gr-qc-0508093-2-33-1': 'To an observer in region III of the SdS diagram, the IV"-III-II"/IV-I-II (solutions 5) and IV"-III-II"/IV-III-II solutions (solutions 6 and 7) would appear as a "sky" of false vacuum that encroaches from infinity, reaches a minimum radius and then expands back out.', 'gr-qc-0508093-2-33-2': 'It has been pointed out by Bousso [CITATION] that at late times (in an asymptotically flat spacetime) the wall trajectory according to the observer in region III of the S(dS) diagram approaches that of a true vacuum bubble [CITATION].', 'gr-qc-0508093-2-33-3': 'Amusingly, the SdS observer will think he is in a true vacuum bubble surrounded by a large region of false vacuum, while the dS observer will think she is in a false vacuum bubble surrounded by a large region of true vacuum.', 'gr-qc-0508093-2-34-0': 'This symmetry between true and false vacuum bubbles is made manifest in the analysis of the Coleman-De Luccia [CITATION] instanton, which describes the production of both true and false vacuum bubbles [CITATION].', 'gr-qc-0508093-2-34-1': 'These are zero energy solutions, and so we should look for an [MATH] unbound solution; this corresponds (via Eq. [REF]) to [MATH], and we see from Fig. [REF], [REF], [REF], and [REF] that the IV"-III-II"/IV-I-II solution (solution 5) or the IV"-III-II"/IV-III-II solution (solution 6), depending on the values of [MATH] and [MATH], can be identified as the analytically continued false vacuum instanton.', 'gr-qc-0508093-2-34-2': 'The radius of the bubble wall at the turning point is found by considering the limit as the potential (Eq. [REF]) goes to [MATH], where on the right (unbound) side of the potential hump the [MATH] term dominates.', 'gr-qc-0508093-2-34-3': 'Solving for [MATH] using Eq. [REF], we find the radius at turnaround to be [EQUATION] which agrees with the previous literature [CITATION] (see also [CITATION]).', 'gr-qc-0508093-2-34-4': 'Since the Schwarzschild mass is zero, we are now matching two pure de Sitter spacetimes across the bubble wall.', 'gr-qc-0508093-2-34-5': 'The conformal diagram for the exterior dS region (right) only contains the area between the vertical dashed lines (which are now identified as [MATH] surfaces) in solution 5 and 6 of Fig. [REF].', 'gr-qc-0508093-2-34-6': 'The interior half (left) of the diagram remains unchanged.', 'gr-qc-0508093-2-34-7': 'It can be seen that at turnaround, the bubble will be larger than both the interior horizon size and the (exterior) horizon size of the region it has replaced.', 'gr-qc-0508093-2-35-0': 'In the IV-III-II"/III-II solution (solutions 12 and 13), the region of false vacuum surrounding the observer in region III of the SdS diagram would begin very small and then expand out of the cosmological horizon.', 'gr-qc-0508093-2-35-1': 'This solution will also have a time-reversed version in which the surrounding region of false vacuum implodes.', 'gr-qc-0508093-2-35-2': 'For the IV"-III"-II"/IV-III-II solution (solutions 8 and 9), the "sky" of false vacuum would forever reside outside of the horizon of a region III observer.', 'gr-qc-0508093-2-35-3': 'Finally, the N/III-II solution (solutions 10 and 11), which we will define as solutions with mass greater than or equal to the Nariai mass, will be an exploding (or imploding in the time-reversed solution) bubble of false vacuum centered on [MATH].', 'gr-qc-0508093-2-36-0': 'In a series of papers, Farhi et.', 'gr-qc-0508093-2-36-1': 'al. [CITATION] discussed the application of the Penrose theorem [CITATION] to the one-bubble spacetimes discussed above.', 'gr-qc-0508093-2-36-2': 'If the null energy condition (NEC) holds (as it does for the postulated energy momentum tensor) and there exists a non-compact Cauchy surface (as in the full SdS spacetime), then the existence of a closed anti-trapped surface in the spacetime implies the presence of an initial singularity.', 'gr-qc-0508093-2-36-3': 'Since each point on the conformal diagrams in Fig. [REF] and Fig. [REF] represents a two-sphere, such an anti-trapped surface exists if the ingoing and outgoing future-directed null rays both diverge.', 'gr-qc-0508093-2-36-4': 'For example, the 2-sphere represented by point [MATH] shown in Fig. [REF], solution 1, is a closed anti-trapped surface.', 'gr-qc-0508093-2-36-5': 'This can be seen by following the null ray (null rays are denoted by the dashed lines in Fig. [REF]) from [MATH] in region III of the dS diagram to [MATH] and noting that [MATH] increases monotonically as [MATH] is approached (the null rays are diverging).', 'gr-qc-0508093-2-36-6': 'But following the future-directed null rays in the opposite direction from [MATH] in region IV of the SdS diagram, across the bubble wall, and into the false vacuum region, shows that they also diverge.', 'gr-qc-0508093-2-36-7': 'Thus, an initial singularity is necessary for this solution to exist at and near [MATH].', 'gr-qc-0508093-2-36-8': 'This spacetime also, however, contains regions without anti-trapped surfaces.', 'gr-qc-0508093-2-36-9': 'Following the future-directed null rays to point [MATH], for example, we see that the ingoing rays diverge, but the outgoing rays converge.', 'gr-qc-0508093-2-36-10': 'For examples, see solutions 1-4 in Fig. [REF], where regions which contain anti-trapped surfaces are shaded blue (dark) and the regions which do not are shaded green (light).', 'gr-qc-0508093-2-37-0': 'If we cut the IV-I-II/III solution (solution 1) in the expanding phase on a spacelike hypersurface at a time where the radius of the bubble wall satisfies [MATH], then the spacetime would not necessarily contain an initial singularity .', 'gr-qc-0508093-2-37-1': "We can remove the initial singularity from the IV-I-II'/III-II (solution 3 and 4) solution as well by cutting on the same surface.", 'gr-qc-0508093-2-37-2': "Both the IV-I-II/III (solution 1) and IV-I-II'/III-II (solution 3 and 4) solutions are therefore classically buildable.", 'gr-qc-0508093-2-37-3': "The IV-I-II'/III-II solution (solutions 3 and 4) is the only example where it is possible to form an inflationary universe from classically buildable initial conditions, but only exists when the interior and exterior cosmological constant are almost equal ([MATH]).", 'gr-qc-0508093-2-37-4': 'This solution might be of interest in understanding transitions between nearly degenerate vacua, for example in the context of eternal inflation.', 'gr-qc-0508093-2-38-0': 'Given the existence of a classically forbidden region in Fig. [REF], [REF], [REF], and [REF], one might ask if tunneling is allowed from one of the recollapsing solutions (solutions 1 and 2) to one of the expanding solutions (solutions 5-9 ) on the other side of the potential hump.', 'gr-qc-0508093-2-38-1': 'This event, shown in Fig. [REF], would constitute a violation of the NEC, and so the Penrose theorem would no longer apply to the antitrapped surfaces that exist after the tunneling event.', 'gr-qc-0508093-2-38-2': 'Such a process is indeed apparently allowed [CITATION], and would describe the quantum creation of an inflationary universe from classically buildable initial conditions (if the initial condition is the IV-I-II/III solution in solution 1).', 'gr-qc-0508093-2-39-0': 'This is a rather strange transition, however, as the unbound solution is behind the wormhole in SdS (see Fig. [REF]).', 'gr-qc-0508093-2-39-1': 'An observer in region I would see the bubble expand, reach its turning point, and then disappear, only to be replaced by a black hole.', 'gr-qc-0508093-2-39-2': 'An observer inside the bubble would see the wall expanding away and - just as it is about to turn around and start collapsing - instead disappear behind the cosmological horizon.', 'gr-qc-0508093-2-39-3': 'This observer will be inside an inflationary universe, but forever disconnected from region I.', 'gr-qc-0508093-2-39-4': 'If the black hole in the SdS spacetime then evaporates, the baby universe will become completely topologically disconnected.', 'gr-qc-0508093-2-40-0': 'Having considered one circumstance in which an NEC violation precipitates the creation of an inflationary universe, one might ask if there are others.', 'gr-qc-0508093-2-40-1': 'Quantum fluctuations of a scalar field in de Sitter space can violate the NEC, and so one might imagine that any of the solutions that we have discussed which allow inflation inside the bubble could be spontaneously created somewhere along their trajectories.', 'gr-qc-0508093-2-40-2': 'One example of this direct production of baby universes is a fluctuation into one of the unbound solutions (solutions 5-9).', 'gr-qc-0508093-2-40-3': 'Such scenarios have been considered in the context of the stochastic approach to baby universe production by Linde [CITATION] and in reference to eternal inflation by Carroll and Chen [CITATION].', 'gr-qc-0508093-2-41-0': 'Another example of the direct fluctuation into an inflationary universe is the thermal decay of de Sitter vacua discussed by Garriga and Megevand [CITATION] (Ref. [CITATION] discusses a related mechanism).', 'gr-qc-0508093-2-41-1': 'This process consists of the fluctuation from empty de Sitter of a bubble in unstable equilibrium between expansion and collapse.', 'gr-qc-0508093-2-41-2': 'This static solution is identified as the set of spacetimes which sit on top of the potentials in Fig. [REF], [REF], [REF], and [REF].', 'gr-qc-0508093-2-41-3': 'The two possible configurations are shown in Fig. [REF], where it can be seen that the bubble wall can lie on either side of the worm hole depending on the sign of [MATH] at the top of the potential.', 'gr-qc-0508093-2-41-4': 'It can be shown that the "Nariai limit" in Ref. [CITATION] corresponds to [MATH], where the [MATH] sign change occurs at the max of the potential.', 'gr-qc-0508093-2-42-0': 'With these considerations, there are at least three competing channels for the formation of baby universes: direct production, the FGG mechanism, and the zero-mass false vacuum instanton.', 'gr-qc-0508093-2-42-1': 'It would be desirable to develop a scheme to directly compare the relative probabilities of each of these processes, and this is currently being explored.', 'gr-qc-0508093-2-43-0': 'Our full catalog of solutions is also interesting in regards to a recent proposal [CITATION] that false vacuum regions, assumed to be larger than the interior horizon, must at all times be larger than the exterior, true vacuum, horizon.', 'gr-qc-0508093-2-43-1': 'The basis of this conjecture is the condition that the divergence of a congruence of future directed null geodesics (defined as [MATH]) must satisfy [EQUATION] where [MATH] is an affine parameter, if the NEC holds for all [MATH].', 'gr-qc-0508093-2-43-2': 'Null rays in the dS and SdS spacetimes satisfy this inequality (in dS, the inequality is exactly zero), but we should check that the junction conditions do not violate it.', 'gr-qc-0508093-2-43-3': 'One requirement imposed by Eq. [REF] is that the divergence of the null rays does not increase at the position of the wall as they go from a true vacuum region into a false vacuum one.', 'gr-qc-0508093-2-43-4': 'Along any given null geodesic in the bubble interior or exterior, the value of [MATH] is either increasing or decreasing monotonically as a function of [MATH].', 'gr-qc-0508093-2-43-5': 'We can therefore state the condition Eq. [REF] as: one cannot have a null ray along which [MATH] outside the bubble and [MATH] inside the bubble.', 'gr-qc-0508093-2-43-6': 'Surveying the solutions in Fig. [REF] and Fig. [REF], we see that this is indeed always true.', 'gr-qc-0508093-2-44-0': 'The authors of Ref. [CITATION] intended to demonstrate that if one requires the false vacuum region to be larger than the interior horizon size at all times (so that inflation is unstoppable), it is necessarily larger than the exterior horizon size.', 'gr-qc-0508093-2-44-1': 'Although all of the allowed one-bubble spacetimes satisfy the condition Eq. [REF], there are only two examples in which all observers agree that this requirement is met: the false vacuum instanton (IV"-III-II"/IV-I-II solution (solution 5), with [MATH]) and the IV-I-II\'/III-II solution (solution 3 and 4, after turn-around).', 'gr-qc-0508093-2-44-2': 'In every other case (including the FGG spacetime in Fig. [REF]), the observers in region I of the SdS conformal diagram will see only a black hole horizon sized volume replaced by the false vacuum bubble.', 'gr-qc-0508093-2-44-3': 'We therefore conjecture that if one requires the false vacuum region to be larger than the interior horizon size at all times, then it will replace a volume larger than the exterior horizon size according to only some observers.', 'gr-qc-0508093-2-44-4': 'If one relaxes this requirement, then there are a diverse range of solutions which might describe the spawning of an inflationary universe.', 'gr-qc-0508093-2-44-5': 'For example, the bubbles in solutions 3, 4 and 10-13 of Fig. [REF] and [REF] grow from an arbitrarily small size.', 'gr-qc-0508093-2-45-0': 'Having discussed the character of the various solutions, we turn now to a potentially dangerous detail, which is particularly important for the FGG mechanism: there exists a classical instability against aspherical perturbations in the spherically symmetric solutions to the junction conditions.', 'gr-qc-0508093-2-46-0': '# Perturbations', 'gr-qc-0508093-2-47-0': 'The solutions described in Sec. [REF] assume that the region of false vacuum is spherically symmetric.', 'gr-qc-0508093-2-47-1': 'The stability of these solutions against aspherical perturbations has important consequences, especially if one hopes to build plausible cosmologies.', 'gr-qc-0508093-2-47-2': 'That there might be an instability in domain walls was first discussed by Adams, Freese and Widrow [CITATION].', 'gr-qc-0508093-2-47-3': 'The bubble wall can trade volume energy for surface energy and wall kinetic energy locally as well as globally, and so the bubble wall will become distorted if different sections of the wall have different kinetic energies.', 'gr-qc-0508093-2-47-4': "As long as the local distortions of the wall remain small compared to the size of the background solution's radius, this process can be formulated quantitatively as perturbation theory around a background spherically symmetric solution.", 'gr-qc-0508093-2-48-0': 'Previous authors [CITATION] have considered perturbations on expanding bubbles of true-vacuum [CITATION], which have zero total energy (surface, volume, and kinetic energies canceling), and so can expand asymptotically.', 'gr-qc-0508093-2-48-1': 'As was first pointed out by Garriga and Vilenkin [CITATION], even though local observers on the bubble wall see perturbations grow, external observers see them freeze out because they do not grow faster than bubble radius.', 'gr-qc-0508093-2-49-0': "The story is different for the bound (solutions 1 and 2) and unbound (solutions 5-9) false vacuum bubbles: since they reach a turning point, the perturbations have a chance to catch up to the bubble's expansion and become nonlinear.", 'gr-qc-0508093-2-49-1': 'This also presumably has implications for the thermal decay mechanism of Garriga and Megevand [CITATION], depending on the duration of time the bubble wall sits in unstable equilibrium between expansion and collapse (see discussion in Sec. [REF]).', 'gr-qc-0508093-2-49-2': 'The remainder of this work will focus on the instability of the bound IV-I-II/III solution (solution 1), since physically plausible initial conditions may be clearly formulated.', 'gr-qc-0508093-2-49-3': 'There is no obvious set of initial conditions for the perturbations on the unbound solutions, and so we simply observe that the results we will obtain for the bound solutions apply qualitatively here as well.', 'gr-qc-0508093-2-50-0': 'To simplify the problem, we assume that the full gravitational problem described in the previous sections can be treated as motion of the bubble wall in a fixed SdS background.', 'gr-qc-0508093-2-50-1': 'This assumption must be validated (as we do below), but we are mainly interested in the low-mass bound solutions for which we might expect the gravitational contributions to be small.', 'gr-qc-0508093-2-50-2': 'Assuming that a thin spherically symmetric bubble wall separates an internal dS from an external SdS spacetime, we can employ the action [CITATION]: [EQUATION] where [MATH] is the surface energy density on the bubble wall, [MATH]) is the metric on the worldsheet of the bubble wall, [MATH] is the difference in volume energy density on either side of the bubble wall: [EQUATION] and [MATH] is the metric of the background spacetime.', 'gr-qc-0508093-2-51-0': '## Wall Equation of Motion', 'gr-qc-0508093-2-52-0': 'The equation of motion resulting from Eq. [REF] is [CITATION]: [EQUATION] where [MATH] is the extrinsic curvature tensor of the worldsheet of the bubble wall, [EQUATION] and [MATH] is the covariant derivative and [MATH] is the unit normal to the bubble wall worldsheet.', 'gr-qc-0508093-2-53-0': 'We will use the static foliation of the SdS spacetime (see Eq. [REF]) as the coordinates [MATH] for the background spacetime.', 'gr-qc-0508093-2-53-1': 'The world sheet is given coordinates [MATH] as in Eq. [REF], and has metric: [EQUATION] with the gauge freedom in choosing [MATH] fixed by [EQUATION] so that [MATH].', 'gr-qc-0508093-2-53-2': 'Here and henceforth primes will denote derivatives with respect to [MATH].', 'gr-qc-0508093-2-53-3': 'The other non-zero components of [MATH] are [MATH] and [MATH].', 'gr-qc-0508093-2-54-0': "The first task at hand is to find the worldsheet's unit normal, which by spherical symmetry has only [MATH] and [MATH] components.", 'gr-qc-0508093-2-54-1': 'Requiring orthogonality to the worldsheet ([MATH]) and unit norm ([MATH]) yields its components: [EQUATION]', 'gr-qc-0508093-2-54-2': 'The components of [MATH] are given by [EQUATION]', 'gr-qc-0508093-2-54-3': 'Substituting Eq. [REF] into Eq. [REF] gives the equation of motion for the bubble wall: [EQUATION]', 'gr-qc-0508093-2-54-4': 'Eq. [REF] supplies the velocity of the bubble at some position along its trajectory [EQUATION]', 'gr-qc-0508093-2-54-5': 'Choosing this boundary condition is effectively restricting ourselves to the IV-I-II/III (solution 1) or IV-III-II/III (solution 2) solutions.', 'gr-qc-0508093-2-54-6': 'Since the solutions to Eq. [REF] approximate the dynamics of the junction condition problem, we should parametrize by [MATH], [MATH], and [MATH].', 'gr-qc-0508093-2-54-7': 'This can be done by using the conversions defined in Sec. [REF], and gives: [EQUATION] where [MATH] is written in terms of [MATH] as [EQUATION] and [EQUATION]', 'gr-qc-0508093-2-54-8': 'To justify the use of the simplified dynamics described above, Eq. [REF] was numerically integrated, and the position of the turning point compared to the corresponding point on the full junction condition potential.', 'gr-qc-0508093-2-54-9': 'Over the range of [MATH] corresponding to the bound solutions, we find excellent quantitative agreement (well within 1%) between the turning points of the solutions to Eq. [REF] and the junction condition potential.', 'gr-qc-0508093-2-54-10': 'This was repeated with equally good results for the weak, GUT, and Planck scale potentials and also for various initial positions between the black hole radius and the potential wall (turning point).', 'gr-qc-0508093-2-54-11': 'This shows that to zeroth order, dynamics as motion in a background is valid, and strongly suggests that it will be at higher orders as well.', 'gr-qc-0508093-2-55-0': '## Perturbations', 'gr-qc-0508093-2-56-0': 'We are now in a position to discuss the first-order perturbations on the spherically-symmetric background solutions discussed in Sec. [REF].', 'gr-qc-0508093-2-56-1': 'Physical perturbations are normal to the worldsheet of the (background) bubble wall, and can be described by scalar field [MATH] by taking the position of the perturbed worldsheet to be [EQUATION] where [MATH] is the spherically symmetric solution, and [MATH] is the unit normal to the worldsheet.', 'gr-qc-0508093-2-56-2': 'It is assumed that [MATH] is much smaller than the bubble wall radius, so that a perturbative analysis can be made.', 'gr-qc-0508093-2-57-0': 'The equation of motion for the perturbation field [MATH] in a curved spacetime background can be derived from the action Eq. [REF] after expanding to second order in [MATH] [CITATION] [EQUATION] where [EQUATION] and [MATH] is [EQUATION]', 'gr-qc-0508093-2-57-1': 'To solve the equation of motion, we can decompose [MATH] into spherical harmonics [EQUATION] and separate variables to get an equation for [MATH].', 'gr-qc-0508093-2-57-2': 'The geometrical factors in Eq. [REF] become dependent on [MATH] or [MATH] only at second order, so we will always be able to make this decomposition.', 'gr-qc-0508093-2-58-0': '[MATH] is then given by: [EQUATION]', 'gr-qc-0508093-2-58-1': 'The components of [MATH] are: [EQUATION]', 'gr-qc-0508093-2-58-2': 'The components of the Ricci tensor are given by: [EQUATION]', 'gr-qc-0508093-2-58-3': 'Contracting equations [REF] and [REF] gives: [EQUATION]', 'gr-qc-0508093-2-58-4': 'The Ricci scalar on the world sheet is [EQUATION] where [MATH] is given by Eq. [REF].', 'gr-qc-0508093-2-59-0': 'After substituting Eq. [REF], Eq. [REF], and Eq. [REF] into Eq. [REF], the equation of motion for [MATH] is [EQUATION]', 'gr-qc-0508093-2-59-1': 'In terms of the dimensionless variables of the junction condition problem this reads: [EQUATION] where [MATH] is the dimensionless perturbation field defined similarly to [MATH] (see Eq. [REF]).', 'gr-qc-0508093-2-59-2': 'The first term acts as a (anti)drag on (shrinking) growing perturbations.', 'gr-qc-0508093-2-59-3': 'The last term in this equation is always negative, acting as a restoring force.', 'gr-qc-0508093-2-59-4': 'Perturbations will grow when the other terms (which are positive over most of the trajectory in the expanding phase) in this equation dominate.', 'gr-qc-0508093-2-59-5': 'Further, the last term indicates that lower [MATH] modes will experience the largest growth.', 'gr-qc-0508093-2-59-6': 'The full details of the solutions, however, require a numerical approach, to which we now turn.', 'gr-qc-0508093-2-60-0': '# Application to the Farhi-Guth-Guven mechanism', 'gr-qc-0508093-2-61-0': 'The possibility of creating an inflating false-vacuum region via quantum tunneling (described in Sec. [REF]) has been investigated only under the assumption of spherical symmetry, and this would be grossly violated if perturbations on the bubble wall become nonlinear.', 'gr-qc-0508093-2-61-1': 'In this section, we investigate the circumstances under which this is the case.', 'gr-qc-0508093-2-61-2': 'The two basic questions at issue are: first, when do perturbations go nonlinear for some given set of initial perturbations, and second, what initial perturbations can be expected.', 'gr-qc-0508093-2-62-0': '## Dynamics of the Perturbation Field', 'gr-qc-0508093-2-63-0': 'Let us begin with the first issue.', 'gr-qc-0508093-2-63-1': 'Since Eq. [REF] is a second order ODE, it can be decomposed into the sum of two linearly independent solutions [EQUATION]', 'gr-qc-0508093-2-63-2': 'The functions [MATH] and [MATH] can be found numerically by alternately setting [MATH] and [MATH] to zero, then evolving the coupled Eq. [REF] and Eq. [REF] numerically for a time [MATH] with initial conditions for [MATH], [MATH], and [MATH].', 'gr-qc-0508093-2-63-3': 'If the bubble is to tunnel, it will do so at the time [MATH], when the bubble reaches its maximum radius and begins to re-collapse.', 'gr-qc-0508093-2-63-4': 'Given [MATH] and [MATH] at time [MATH], the size of the perturbations at the turning point for any [MATH], [MATH], [MATH], [MATH], and [MATH] can be determined.', 'gr-qc-0508093-2-63-5': 'An RK4 algorithm with adaptive step size was used to solve for [MATH] and [MATH], with numerical errors well within the [MATH] level.', 'gr-qc-0508093-2-64-0': 'The results of this analysis for [MATH] and for the low (weak) and intermediate (GUT) inflation scales discussed below Eq. [REF] are shown in Fig. [REF].', 'gr-qc-0508093-2-64-1': 'The solid lines show contours of constant (log) amplification factor [MATH] (left) and [MATH] (right) versus the bubble starting radius [MATH] and mass parameter [MATH], with bubble mass increasing toward the top.', 'gr-qc-0508093-2-64-2': 'The shaded regions indicate regions which we have disallowed as bubble starting radii because the bubble would not be classically buildable for [MATH] (marked as [MATH]), or the bubble is in the forbidden region [MATH] of the effective 1D equation of motion Eq. [REF], or the bubble would be too small to be treated classically.', 'gr-qc-0508093-2-64-3': 'We choose the latter radius as fifty times the Compton wavelength [MATH] of a piece of the bubble wall .', 'gr-qc-0508093-2-64-4': 'The choice of fifty Compton wavelengths is rather arbitrary; the effect of a larger bound would be to exclude more of the parameter space in Fig. [REF].', 'gr-qc-0508093-2-64-5': 'This (unshaded) parameter space includes all classical initial conditions which could be set up by the observer in region I of the SdS conformal diagram.', 'gr-qc-0508093-2-65-0': 'It can be seen in Fig. [REF] that the growth of the perturbations is in general larger for higher-mass bubbles (smaller [MATH], larger [MATH]).', 'gr-qc-0508093-2-65-1': 'The lower the inflation scale, the closer to zero the peak in the potential function becomes, and the smaller [MATH] (higher mass) bubbles are allowed, so at low inflation scales [MATH] and [MATH] can be very large.', 'gr-qc-0508093-2-65-2': 'Growth for the Planck-scale inflation bubbles is very small, with [MATH] of order 10 and [MATH] of order 1, and is not plotted.', 'gr-qc-0508093-2-66-0': 'The enhanced growth at small [MATH] is due to the suppression of the term in Eq. [REF] proportional to [MATH], which always acts to stabilize the perturbations.', 'gr-qc-0508093-2-66-1': 'Another consequence of this suppression is that the range in [MATH] over which solutions are unstable depends on [MATH]; as a general rule of thumb, approximately a few times [MATH]-modes are unstable (note that this is unlike the case true vacuum bubbles, for which only the [MATH] modes are unstable).', 'gr-qc-0508093-2-66-2': 'An example of the [MATH] function for [MATH] with the intermediate (GUT) inflation scale parameters is shown in Fig. [REF].', 'gr-qc-0508093-2-66-3': 'The [MATH] functions for very large [MATH] modes are stable and approach sinusoids with amplitudes less than one (see the inset of Fig. [REF]), meaning that the perturbations are never larger than their initial size.', 'gr-qc-0508093-2-67-0': '## Initial Conditions and Evolution to the Turning Point', 'gr-qc-0508093-2-68-0': 'Having fully characterized the growth of the perturbations, we now require an estimate for their initial values when the bubble is formed.', 'gr-qc-0508093-2-68-1': 'There is no reason to expect that a region of false vacuum will fluctuate into existence with anything near spherical symmetry, nor is it likely to have thin walls (there is no instanton or other mechanism to enforce these symmetries).', 'gr-qc-0508093-2-68-2': 'Since low-[MATH] (relative to [MATH]) modes are unstable, an initially aspherical bubble will only become more aspherical; this is in marked contrast to true vacuum bubbles, which both start spherical, and tend to become more spherical as they expand.', 'gr-qc-0508093-2-69-0': 'Suppose, however, that we consider the best-case scenario in which a bubble is, by chance or design, spherically symmetric.', 'gr-qc-0508093-2-69-1': 'It will nevertheless inevitably be dressed with zero-point quantum fluctuations of the perturbation field.', 'gr-qc-0508093-2-69-2': 'We may then check whether these fluctuations alone, considered as initial values for the perturbations of a bubble starting with a given [MATH] and [MATH], suffice to make the bubble nonlinearly aspherical by turnaround.', 'gr-qc-0508093-2-70-0': 'We assume that the ensemble average of the quantum fluctuations at the time of nucleation is zero; but the ensemble average of the square of the field (the space-like two-point function [MATH]) will not generally vanish.', 'gr-qc-0508093-2-70-1': 'We can write the mode functions (Eq. [REF]) in terms of it as: [EQUATION]', 'gr-qc-0508093-2-70-2': 'By spherical symmetry, the two-point function can be written as a function of the angular separation [MATH] between [MATH] and [MATH], and decomposed into Legendre polynomials: [EQUATION]', 'gr-qc-0508093-2-70-3': 'Using the addition theorem for spherical harmonics, we can write this as [EQUATION]', 'gr-qc-0508093-2-70-4': 'Substituting this into Eq. [REF] and using the orthogonality of the spherical harmonics yields the relation: [EQUATION]', 'gr-qc-0508093-2-70-5': 'Given some space-like two point function at the time the bubble is nucleated, we can obtain the [MATH] from [EQUATION] and therefore set the typical initial amplitudes of the mode functions as the r.m.s. value [MATH] from Eq. [REF].', 'gr-qc-0508093-2-70-6': 'The velocity field can be decomposed into spherical harmonics just as [MATH] was, and the analysis performed above carries over exactly.', 'gr-qc-0508093-2-70-7': 'The typical initial size of the velocity mode functions is then given by [EQUATION] with [EQUATION]', 'gr-qc-0508093-2-70-8': 'The initial amplitudes in Eq. [REF] and Eq. [REF] can now be evolved to the turning point, and the mode functions re-summed.', 'gr-qc-0508093-2-70-9': 'The ensemble average of the r.m.s. fluctuations in [MATH] at any time at a given point will then be: [EQUATION] which can be evaluated at [MATH].', 'gr-qc-0508093-2-71-0': 'A full model of the two-point functions [MATH] and [MATH] would involve quantizing the mode functions on the curved spacetime of the bubble wall worldsheet, which has a metric depending on [MATH].', 'gr-qc-0508093-2-71-1': 'Further, to treat large fluctuations, we would need to include non-linear terms in the equation of motion.', 'gr-qc-0508093-2-71-2': 'The exact two-point function is therefore a rather formidable object to compute.', 'gr-qc-0508093-2-71-3': 'As a simplified model, we will employ the two-point functions of a massless scalar field in flat spacetime, and replace the spatial distance [MATH] with the distance along the bubble wall [MATH].', 'gr-qc-0508093-2-71-4': 'This massless scalar corresponds to the perturbations on a flat wall separating domains of equal energy density in Minkowski space [CITATION].', 'gr-qc-0508093-2-71-5': 'Corrections to this picture in the presence of curvature should be small over small regions of the bubble wall.', 'gr-qc-0508093-2-71-6': 'We are also neglecting the large difference in energy densities across the bubble wall, which will give the field a (negative) mass to first order.', 'gr-qc-0508093-2-71-7': 'The apparent divergence of the correlator due to this negative mass will be rendered finite by the non-linear terms which must be introduced to discuss large fluctuations.', 'gr-qc-0508093-2-71-8': 'In light of all these difficulties, and several more approximations we will make below, this should be considered as a first, rough estimate of the amplitude of the quantum fluctuations on the bubble at the time of nucleation.', 'gr-qc-0508093-2-72-0': 'The space-like two-point function in Minkowski space at large separations is given by [EQUATION] where [MATH].', 'gr-qc-0508093-2-72-1': 'As in the work of Garriga and Vilenkin [CITATION], we introduce a smeared field operator to obtain a well-defined answer at close separations [EQUATION] where [MATH] is a smearing length, chosen to be the Compton wavelength [MATH] of a piece of wall, which is a physically reasonable lower bound on the size of a measurable region.', 'gr-qc-0508093-2-72-2': "The smeared correlator will then be given by [EQUATION] which evaluates at [MATH] to [EQUATION] where [MATH] is Catalin's constant.", 'gr-qc-0508093-2-72-3': 'We have been unable to integrate Eq. [REF] to obtain the exact form of the smeared two-point function for all [MATH].', 'gr-qc-0508093-2-72-4': 'However, it must smoothly interpolate between the value at [MATH] in Eq. [REF] to the functional form at [MATH] given by Eq. [REF].', 'gr-qc-0508093-2-72-5': "We therefore employ the following 'toy model' smeared field correlator [EQUATION] which has the correct asymptotics.", 'gr-qc-0508093-2-72-6': 'With [MATH], the dimensionless form of the toy correlator is given by: [EQUATION] where [MATH].', 'gr-qc-0508093-2-73-0': 'The velocity-velocity correlator can be calculated using the Hamiltonian approach.', 'gr-qc-0508093-2-73-1': 'For spacelike separations, this is given by [EQUATION] where we have introduced a hard momentum cut-off [MATH] to obtain a finite answer.', 'gr-qc-0508093-2-73-2': 'This cutoff will correspond to the inverse smearing length, with higher momentum scales higher accounted for in the smeared operator.', 'gr-qc-0508093-2-73-3': 'The integral can be evaluated in terms of generalized hypergeometric functions as [EQUATION]', 'gr-qc-0508093-2-73-4': 'At close separations, we construct a smeared operator [MATH].', 'gr-qc-0508093-2-73-5': 'The expectation value of this operator at zero separation is [EQUATION] where [MATH] is the Euler-Mascheroni constant.', 'gr-qc-0508093-2-73-6': 'Smoothly connecting the small [MATH] (Eq. [REF]) and large [MATH] (Eq. [REF]) behavior as in Eq. [REF], the [MATH] (defined similarly to [MATH]) correlator on the bubble at the time of nucleation is: [EQUATION]', 'gr-qc-0508093-2-73-7': 'The integrals Eq. [REF] and Eq. [REF] for the correlators Eq. [REF] and Eq. [REF] must be evaluated numerically.', 'gr-qc-0508093-2-73-8': 'Calculation of the coefficients for every [MATH] and [MATH] is unfortunately unfeasible because of the highly oscillatory behavior of the integrands and sheer number of mode functions that must be considered.', 'gr-qc-0508093-2-73-9': 'However, we have been able to deduce sufficiently good approximate fits for [MATH] and [MATH] as a function of both [MATH] and [MATH].', 'gr-qc-0508093-2-73-10': 'In both cases, the power is dominated by a peak at [MATH].', 'gr-qc-0508093-2-74-0': 'The [MATH] are nicely fit by the function [EQUATION]', 'gr-qc-0508093-2-74-1': 'The proposed fit for the [MATH] consists of two power laws matched at the [MATH] peak.', 'gr-qc-0508093-2-74-2': 'For [MATH], the best fit is [MATH] and for [MATH], the fit is [MATH].', 'gr-qc-0508093-2-74-3': 'Because these power law indices are slightly uncertain, we only count the modes with [MATH] in the [MATH].', 'gr-qc-0508093-2-74-4': 'This is conservative, and also justified because these modes will not contribute significantly to the sum in Eq. [REF].', 'gr-qc-0508093-2-75-0': 'With these initial conditions, we can now evolve each mode function using Eq. [REF] and then re-sum in Eq. [REF] to find the average size of the fluctuations at the turning point.', 'gr-qc-0508093-2-75-1': 'We have calculated [MATH] and [MATH] up to the [MATH] corresponding to the last unstable mode of the smallest [MATH], for all three inflation scales ([MATH] GeV, [MATH] GeV, and [MATH] GeV).', 'gr-qc-0508093-2-75-2': 'The results for weak- and GUT-scale inflation are shown in Fig. [REF], where the dotted line indicates the boundary of the region over which the perturbations become non-linear (non-linear to the left of the line).', 'gr-qc-0508093-2-75-3': 'It can be seen that in this model, even just quantum perturbations of the bubble wall grow nonlinear in bubbles that start at radii less than about one-tenth of the turnaround radius; this grossly violates the assumption of spherical symmetry used in tunneling calculations.', 'gr-qc-0508093-2-75-4': 'On the other hand, none of the parameter space in the high inflation scale case went non-linear, and at all scales there is always a region of initial bubble radii near the turnaround radius, for which nonlinearity never occurs.', 'gr-qc-0508093-2-76-0': '## Thick Walls and Radiation', 'gr-qc-0508093-2-77-0': 'Just as we have no reason to expect a fluctuated region to be spherically symmetric, we have no reason to assume that it will have thin walls.', 'gr-qc-0508093-2-77-1': 'An analysis of thick-walled true vacuum bubbles was undertaken in Ref. [CITATION], where it was found that the instabilities found in thin-walled case are still present in the form of deformations normal to the bubble profile.', 'gr-qc-0508093-2-77-2': 'In the case of small false vacuum bubbles, there is no obvious consideration (such as a corresponding instanton) to supply the profile of the bubble wall, and so we can merely conjecture by precedent that the instability would be retained in the thick-walled case as well.', 'gr-qc-0508093-2-78-0': 'Another consideration, applying to bubbles smaller than the false vacuum horizon size, is whether inflation is spoiled by non-vacuum contributions to the energy density.', 'gr-qc-0508093-2-78-1': 'The perturbations on the bubble wall translate into gravitational waves [CITATION], and since the bound bubble solutions remain relatively close to their gravitational radius and become distorted over many different length scales on a relatively short time scale (see the quasi-exponential growth in Fig. [REF]), they will be emitters of copious gravitational radiation.', 'gr-qc-0508093-2-78-2': 'Another problem arises if the kinetic and gradient energy of the field becomes appreciable in the bubble interior, either from intrusion of the wall (for example, imagine a bubble being pinched in half by some non-linear perturbation), or from particle production or other scalar modes propagating in from the wall.', 'gr-qc-0508093-2-78-3': 'If the emission of energy into the interior of the bubble from any combination of these modes makes a significant contribution to the equation of state, then inflation will not occur.', 'gr-qc-0508093-2-79-0': '# Summary Discussion', 'gr-qc-0508093-2-80-0': 'In this paper we have examined the feasibility of producing inflationary universes from a spacetime with a small cosmological constant.', 'gr-qc-0508093-2-80-1': 'Reviewing the solutions allowed by the junction conditions, there are several distinct possibilities if one allows for violations of the null energy condition.', 'gr-qc-0508093-2-80-2': 'The first is the Farhi, Guth Guven (FGG) mechanism, in which (referring to Fig. [REF] and Fig. [REF]) the IV-I-II/III (solution 1) or IV-III-II/III (solution 2) solution is fluctuated in the expanding phase and then tunnels to an unbound solution, as shown in Fig. [REF].', 'gr-qc-0508093-2-80-3': 'To the outside observer in region I of the SdS conformal diagram it appears that the bubble has disappeared behind the horizon, but on the other side of the wormhole, both an inflationary universe and a non-inflating universe (an asymptotically true-vacuum de Sitter region) have come into existence.', 'gr-qc-0508093-2-80-4': 'In one case, shown in solution 5 of Fig. [REF], we can imagine an observer in region III, then take the Schwarzschild mass [MATH] limit (thus removing regions I, II and IV entirely) and interpret the spacetime as that of the analytically continued Coleman-De Luccia instanton.', 'gr-qc-0508093-2-80-5': 'Since the very same spacetime takes part in both the FGG mechanism and the Coleman-De Luccia false vacuum instanton, there may be some way to smoothly interpolate between these two processes; we intend to explore this idea further in future work.', 'gr-qc-0508093-2-81-0': 'In addition to the FGG and regular instanton mechanisms, there are two more possibilities.', 'gr-qc-0508093-2-81-1': 'First, an inflationary universe might be directly produced by some null energy condition (NEC) violating fluctuation into one of the unbound or monotonic solutions shown in Solutions 5-13 of Fig. [REF].', 'gr-qc-0508093-2-81-2': 'To the outside observer in region I of the SdS conformal diagram, these solutions would be indistinguishable from the fluctuation of a black hole, but they would secretly entail the creation of everything on the other side of the wormhole, as in the FGG tunneling.', 'gr-qc-0508093-2-81-3': "A second method of direct production is the fluctuation (which does not require an NEC violation) of the IV-I-II'/III-II solution (solution 3 and 4).", 'gr-qc-0508093-2-81-4': 'These solutions only exist in the case where the interior and exterior cosmological constants are comparable, and so would not correspond to the nucleation of inflation from a universe like ours, but they might be of interest in understanding transitions between nearly degenerate vacua.', 'gr-qc-0508093-2-82-0': 'Examining these classical solutions to first order, we have shown that an instability to aspherical perturbations exists in those solutions which possess a turning point.', 'gr-qc-0508093-2-82-1': 'This includes the bound (solutions 1 and 2) and unbound solutions (solutions 5-9).', 'gr-qc-0508093-2-82-2': 'In the latter case there is no clear way to set an initial radius or initial perturbation amplitude, so we can say only that collapsing bubbles are violently unstable.', 'gr-qc-0508093-2-82-3': 'The bound solutions are amenable to quantitative investigation, and we have focused on the growth of perturbations in the expanding phase that precedes tunneling in the FGG mechanism.', 'gr-qc-0508093-2-82-4': 'For bound expanding bubbles formed by the fluctuations of a scalar field in de Sitter space, there is no instanton to enforce spherical symmetry, so we would expect initial aspherical perturbations to be relatively large.', 'gr-qc-0508093-2-82-5': 'Since there is no detailed model for the fluctuating scalar field to see how large, we have instead calculated an estimate of the minimal deviations from spherical symmetry in light of quantum fluctuations, and present this as an extremely rare, best case scenario for spherical symmetry.', 'gr-qc-0508093-2-82-6': 'These minimal fluctuations were then evolved to the turning point of the bound solutions, which is the point in the FGG mechanism where there is a chance for tunneling to an inflationary universe to occur.', 'gr-qc-0508093-2-82-7': 'Of the three representative energy scales for inflation (false vacuum energy densities) we have studied, the evolved minimal perturbations on a Plank scale bubble remain small over most of the allowed parameter space, while the perturbations on GUT and weak scale bubbles can grow nonlinear if they start at a sufficiently small ([MATH]) fraction of the turnaround radius.', 'gr-qc-0508093-2-82-8': 'Thus even in the best-case scenario some bubbles become nonlinear, but on the other hand there will always in principle be some that do not.', 'gr-qc-0508093-2-83-0': 'The instability introduces complications into the use of the FGG mechanism as a means of baby universe production.', 'gr-qc-0508093-2-83-1': 'The existing calculations of the tunneling rate rely heavily on the assumption of spherical symmetry.', 'gr-qc-0508093-2-83-2': 'It is unclear how to perform a similar calculation for a (possible non-linearly) perturbed bubble, as the number of degrees of freedom has drastically increased and the assumption of a minisuperspace of spherically symmetric metrics is no longer good.', 'gr-qc-0508093-2-83-3': 'Further, the bubble interior will become filled with scalar gradient and kinetic energy and gravity waves, possibly upsetting the interior sufficiently to prevent inflation.', 'gr-qc-0508093-2-83-4': 'One might argue that in an eternal universe there is plenty of time to wait around for a fluctuation which is sufficiently spherical.', 'gr-qc-0508093-2-83-5': 'However, to fully understand the importance of the FGG mechanism, one must both have a model of the scalar field fluctuations, which would predict the distribution of bubble shapes and masses, and also a model for tunneling in the presence of asphericities.', 'gr-qc-0508093-2-83-6': 'In addition, one must understand the competition among the various processes which result in the formation of inflating false vacuum regions, and thus show that the FGG mechanism is not a relative rarity.', 'gr-qc-0508093-2-83-7': 'These problems represent significant calculations in a theory of quantum gravity which we do not yet possess, but progress in these areas would undoubtedly improve our understanding of the initial conditions for inflation.', 'gr-qc-0508093-2-84-0': 'The authors wish to thank A. Albrecht, T. Banks, P.J. Fox, S. Gratton, and T. Mai for their assistance in the development of this work.'}

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[]

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[]

['gr-qc-0508093-1-23-1', 'gr-qc-0508093-1-25-3', 'gr-qc-0508093-1-27-0', 'gr-qc-0508093-1-28-1', 'gr-qc-0508093-2-23-1', 'gr-qc-0508093-2-25-3', 'gr-qc-0508093-2-27-0', 'gr-qc-0508093-2-28-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/gr-qc/0508093

hep-th-9904188

{'hep-th-9904188-1-0-0': '0,2cm We construct a new class of two-dimensional field theories with target spaces that are finite multiparameter deformations of the usual coset [MATH]-spaces.', 'hep-th-9904188-1-0-1': 'They arise naturally, when certain models, related by Poisson-Lie T-duality, develop a local gauge invariance at specific points of their classical moduli space.', 'hep-th-9904188-1-0-2': 'We show that canonical equivalences in this context can be formulated in loop space in terms of parafermionic-type algebras with a central extension.', 'hep-th-9904188-1-0-3': 'We find that the corresponding generating functionals are non-polynomial in the derivatives of the fields with respect to the space-like variable.', 'hep-th-9904188-1-1-0': 'After constructing models with three- and two-dimensional targets, we study renormalization group flows in this context.', 'hep-th-9904188-1-1-1': 'In the ultraviolet, in some cases, the target space of the theory reduces to a coset space or there is a fixed point where the theory becomes free.', 'hep-th-9904188-1-2-0': 'CERN-TH/99-112', 'hep-th-9904188-1-3-0': 'April 1999', 'hep-th-9904188-1-4-0': '16 pt', 'hep-th-9904188-1-5-0': '# Introduction', 'hep-th-9904188-1-6-0': 'Cosets [MATH] as target spaces in 2-dim field theories have been extensively studied in the literature, as they provide examples of spaces other than group manifolds, which give rise to integrable models.', 'hep-th-9904188-1-6-1': 'It is always of interest to find integrable deformations [CITATION]-[CITATION] of such models and if possible classify them.', 'hep-th-9904188-1-6-2': 'In the ordinary (undeformed) coset models one starts with the usual Wess-Zumino action for a group, with Lagrangian density proportional to [MATH]-[MATH], and then restricts the trace to the coset space only.', 'hep-th-9904188-1-6-3': 'Hence, this construction, but not the corresponding models, is quite trivial.', 'hep-th-9904188-1-6-4': 'Having in mind 2-dim field theories, with targets spaces representing continuous deformations of the latter coset spaces, we need models with non-trivial moduli as a starting point.', 'hep-th-9904188-1-6-5': 'Such an example was considered in [CITATION]; we present in this paper the generalization of this to a class of theories.', 'hep-th-9904188-1-7-0': 'We found natural to start, in section 2, with 2-dim models related by Poisson-Lie T-duality [CITATION], since these have indeed a non-trivial moduli space and, moreover, their classical equivalence has been established [CITATION].', 'hep-th-9904188-1-7-1': 'Also, in some examples, there are hints that point towards the classical equivalence promoted into a quantum one at 1-loop in perturbation theory [CITATION].', 'hep-th-9904188-1-7-2': 'We will show that in some points in this moduli space a local (gauge-like) invariance is developing.', 'hep-th-9904188-1-7-3': 'Hence, at these points the configuration space is lower-dimensional and we discover in a unifying manner spaces that are deformations of the usual coset spaces.', 'hep-th-9904188-1-7-4': 'In addition, as a byproduct, we will obtain duals of these models that are classically canonically equivalent to them as 2-dim field theories.', 'hep-th-9904188-1-7-5': 'This equivalence is encoded in infinite-dimensional current algebras of the parafermionic type that we construct.', 'hep-th-9904188-1-7-6': 'We derive these from the infinite-dimensional algebras with a central extension, which were found in the proof of canonical equivalence of the Poisson-Lie T-duality-related models in [CITATION].', 'hep-th-9904188-1-7-7': 'The corresponding generating functionals have the new feature that they are not linear in the derivatives of the fields with respect to the space-like variable.', 'hep-th-9904188-1-7-8': 'This is in contrast with the cases of Abelian duality [CITATION], non-Abelian duality in Principal Criral [CITATION] and more general [CITATION] models, as well as for Poisson-Lie T-duality (and its possible generalizations) [CITATION].', 'hep-th-9904188-1-7-9': 'They are, instead, non-polynomial functions of these derivatives.', 'hep-th-9904188-1-7-10': 'Many of these aspects are explicitly demonstrated in section 3, with a particular example.', 'hep-th-9904188-1-7-11': 'In section 4 we discuss the renormalization group (RG) flow in this context.', 'hep-th-9904188-1-7-12': 'As in [CITATION], we emphasize that taking the classical limit that leads to the lower-dimensional models and then studying the RG flow does not necessarily imply that this limit would correspond to a fixed point of the RG flow, i.e. the two procedures do not commute.', 'hep-th-9904188-1-7-13': 'There is, however, a particular domain in parameter space, where for a wide range of energies in the UV, the description is effectively perturbative with a UV-stable fixed point corresponding to the point where the gauge invariance develops.', 'hep-th-9904188-1-7-14': 'Then the model becomes effectively a two-dimensional one.', 'hep-th-9904188-1-8-0': 'We end the paper with section 5, containing concluding remarks and a discussion on future directions of this research.', 'hep-th-9904188-1-8-1': 'We have also written an appendix, where some mathematical aspects of our proofs are worked out explicitly.', 'hep-th-9904188-1-9-0': '# General formulation', 'hep-th-9904188-1-10-0': 'In this section we first show how new 2-dim field theories, with target spaces representing deformed coset spaces, arise in the context of Poisson-Lie T-duality-related [MATH]-models.', 'hep-th-9904188-1-10-1': 'We then present a duality-invariant formulation and show that canonical equivalences are encoded into algebras of the parafermionic-type in loop space.', 'hep-th-9904188-1-11-0': '## Formulation using Poisson-Lie T-duality-related [MATH]-models', 'hep-th-9904188-1-12-0': 'The form of 2-dim [MATH]-model actions related by Poisson-Lie T-duality (in the absence of spectator fields) is [CITATION] [EQUATION] and [EQUATION]', 'hep-th-9904188-1-12-1': 'The field variables in ([REF]) are [MATH], [MATH] and parametrize an element [MATH] of a group [MATH].', 'hep-th-9904188-1-12-2': 'We also introduce representation matrices [MATH], with [MATH] and the components of the left-invariant Maurer-Cartan forms [MATH].', 'hep-th-9904188-1-12-3': 'The light-cone coordinates on the 2-dim space-time are [MATH], whereas [MATH] denotes the overall coupling constant, which is assumed to be positive.', 'hep-th-9904188-1-12-4': 'Similarly, for ([REF]) the field variables are [MATH], where [MATH], [MATH], parametrize a different group [MATH], whose dimension is, however, equal to that of [MATH].', 'hep-th-9904188-1-12-5': 'Accordingly, we introduce a different set of representation matrices [MATH], with [MATH], and the corresponding components of the left-invariant Maurer-Cartan forms [MATH].', 'hep-th-9904188-1-12-6': 'In ([REF]) and ([REF]), [MATH] is a constant [MATH] matrix, whereas [MATH] and [MATH] are antisymmetric matrices with the same dimension as [MATH], but they depend on the variables [MATH] and [MATH] via the corresponding group elements [MATH] and [MATH].', 'hep-th-9904188-1-12-7': 'They are defined as [CITATION] [EQUATION] where the matrices [MATH], [MATH] are constructed using [EQUATION] and similarly for [MATH] and [MATH].', 'hep-th-9904188-1-12-8': 'Consistency restricts these to obey [EQUATION] and similarly for the tilded ones.', 'hep-th-9904188-1-12-9': 'There is also a bilinear invariant [MATH] with the various generators obeying [EQUATION]', 'hep-th-9904188-1-12-10': 'Finally, we note that the choice of possible groups [MATH] and [MATH] is restricted by the fact that [CITATION] their corresponding Lie algebras must form a pair of maximally isotropic subalgebras into which the Lie algebra of a larger group [MATH], known as the Drinfeld double, can be decomposed [CITATION].', 'hep-th-9904188-1-13-0': 'Let us consider two subgroups [MATH] and [MATH] with [MATH].', 'hep-th-9904188-1-13-1': 'Accordingly we split the Lie-algebra indices as [MATH], where Latin and Greek indices refer to subgroup and coset spaces, respectively.', 'hep-th-9904188-1-13-2': 'Then we may separate the various matrices appearing in ([REF]) and ([REF]) into blocks as [EQUATION] and [EQUATION]', 'hep-th-9904188-1-13-3': 'We would like to take a limit in the model ([REF]) and its dual ([REF]) such that the number of fields [MATH] (and [MATH]) is reduced by [MATH].', 'hep-th-9904188-1-13-4': 'We would call the remaining variables by [MATH] (and [MATH]) with [MATH].', 'hep-th-9904188-1-13-5': 'Consider the limit [EQUATION] in a uniform way for all matrix elements.', 'hep-th-9904188-1-13-6': 'This means that ratios of matrix elements remain constant in this limit.', 'hep-th-9904188-1-13-7': 'Using ([REF]) we find that in the limit ([REF]) [EQUATION]', 'hep-th-9904188-1-13-8': 'Then, the actions ([REF]) and ([REF]) take the form [EQUATION] and [EQUATION]', 'hep-th-9904188-1-13-9': 'Notice that in ([REF]) [MATH] are elements of a [MATH] matrix, whereas in ([REF]) [MATH] are elements of a [MATH] one.', 'hep-th-9904188-1-13-10': 'We have anticipated that the number of variables in ([REF]) and ([REF]) has been reduced to [MATH] upon taking the limit ([REF]).', 'hep-th-9904188-1-13-11': 'However, this does not happen automatically, but depends on whether or not certain conditions, as we will next prove, are fulfilled.', 'hep-th-9904188-1-13-12': 'In order to reduce the dimensionality of ([REF]) we should prove that, after taking the limit ([REF]), a local gauge invariance develops, which suffices to gauge-fix [MATH] degrees of freedom in the actions ([REF]) and similarly for ([REF]).', 'hep-th-9904188-1-13-13': 'For ([REF]) consider the transformation [EQUATION]', 'hep-th-9904188-1-13-14': 'In its infinitesimal form it reads [MATH].', 'hep-th-9904188-1-13-15': 'We may show that this induces the following transformations: [EQUATION]', 'hep-th-9904188-1-13-16': 'Using these and the relation (A.6) of [CITATION], specialized for coset space indices [EQUATION] we may prove that ([REF]) is invariant under the gauge transformation ([REF]), provided that the following condition holds: [EQUATION] or equivalently [EQUATION] where we have denoted by [MATH] and [MATH] the symmetric and antisymmetric parts of the matrix [MATH].', 'hep-th-9904188-1-13-17': 'When the conditions ([REF]) are satisfied then we may gauge-fix [MATH] parameters in the group element [MATH].', 'hep-th-9904188-1-13-18': 'The most efficient way is to parametrize the group element [MATH] as [MATH], where [MATH] and [MATH], and then set [MATH].', 'hep-th-9904188-1-13-19': 'It can be easily seen that this completely fixes the gauge freedom.', 'hep-th-9904188-1-14-0': 'There are [MATH] algebraic conditions in ([REF]) for the [MATH] elements of the matrix [MATH].', 'hep-th-9904188-1-14-1': 'Hence, it is not at all obvious that they can be fulfilled for a general Drinfeld double and then for any arbitrary choice of the subgroup [MATH].', 'hep-th-9904188-1-14-2': 'An obvious simplification occurs when [MATH] is an Abelian group.', 'hep-th-9904188-1-14-3': 'Then [MATH], [MATH] and eq. ([REF]) is solved by [MATH].', 'hep-th-9904188-1-14-4': 'Then ([REF]) with [MATH] takes the form of the usual [MATH]-model action on the coset [MATH] space.', 'hep-th-9904188-1-14-5': 'Accordingly ([REF]) represents its usual non-Abelian dual.', 'hep-th-9904188-1-14-6': 'Hence, when both groups [MATH] and [MATH] are non-Abelian, the models ([REF]) and ([REF]) are deformations of the usual models on coset spaces [MATH] and of their non-Abelian duals.', 'hep-th-9904188-1-15-0': '## Duality-invariant formulation', 'hep-th-9904188-1-16-0': 'We would like to find a duality-invariant action, from which the [MATH]-models ([REF]) and ([REF]) originate.', 'hep-th-9904188-1-16-1': 'It is natural to start with the manifestly Poisson-Lie T-duality-invariant action of [CITATION] from which the [MATH]-models ([REF]) and ([REF]) originate.', 'hep-th-9904188-1-16-2': 'This action is defined in the Drinfeld double as [CITATION], [EQUATION] where [MATH] is the WZW action for a group element [MATH].', 'hep-th-9904188-1-16-3': 'The operator [MATH] is defined as [CITATION] [EQUATION] with [EQUATION] where we have used the notation [MATH] and [MATH].', 'hep-th-9904188-1-16-4': 'In the limit ([REF]) we have [EQUATION]', 'hep-th-9904188-1-16-5': 'Using this and the conditions ([REF]), one can show that ([REF]), in the limit ([REF]), develops the gauge invariance [EQUATION] provided that the following constraint is obeyed [EQUATION]', 'hep-th-9904188-1-16-6': 'In order to avoid introducing this constraint we may use gauge fields instead.', 'hep-th-9904188-1-16-7': 'Indeed, consider the action [EQUATION] where [MATH] takes values in the Lie algebra of [MATH], i.e. [MATH].', 'hep-th-9904188-1-16-8': 'The operator [MATH] is defined as the restriction in [MATH] of the corresponding operator in ([REF]) [EQUATION] where [EQUATION] and [MATH] is the inverse matrix of [MATH].', 'hep-th-9904188-1-16-9': 'It can be shown that ([REF]) is gauge-invariant under ([REF]) and the corresponding transformation for the gauge field [EQUATION] provided that [MATH] is invariant under the similarity transformation [EQUATION]', 'hep-th-9904188-1-16-10': 'In order to prove ([REF]) we first show that [EQUATION] for some [MATH]-dependent matrix [MATH].', 'hep-th-9904188-1-16-11': 'After repeatedly using ([REF]) and a lengthy computation we find that such a matrix exists and is given by [EQUATION] provided that the following condition holds: [EQUATION] or equivalently, splitting into the symmetric and antisymmetric parts, [EQUATION]', 'hep-th-9904188-1-16-12': 'At first sight it seems that ([REF]) is more restrictive than the corresponding conditions in ([REF]), since, unlike ([REF]), they are valid for finite-gauge transformations.', 'hep-th-9904188-1-16-13': 'However, we show in the appendix that ([REF]) actually implies ([REF]).', 'hep-th-9904188-1-17-0': 'In the remainder of this subsection, we consider the classical equations of motion for the (manifestly) duality and gauge-invariant action ([REF]).', 'hep-th-9904188-1-17-1': 'Its variation with respect to all fields is [EQUATION]', 'hep-th-9904188-1-17-2': 'Specializing to subgroup and coset space indices, we find the equations of motion [EQUATION] where we have used also the fact that, because of ([REF]), [MATH].', 'hep-th-9904188-1-17-3': 'Hence, the constraint ([REF]) follows as the equation of motion for [MATH].', 'hep-th-9904188-1-17-4': 'Using ([REF]), the equations of motion in ([REF]) can be cast into the form [MATH]-[MATH].', 'hep-th-9904188-1-17-5': 'These have the same form as the equations of motion for the action ([REF]) [CITATION].', 'hep-th-9904188-1-18-0': 'We finally note that the action ([REF]) is manifestly invariant under the transformation [MATH] for some [MATH]-dependent group element [MATH] [CITATION].', 'hep-th-9904188-1-18-1': 'By introducing gauge fields this symmetry can be promoted into a gauge symmetry with [MATH] a function of [MATH] and [MATH].', 'hep-th-9904188-1-18-2': 'This type of gauge invariance, though interesting enough in its own right to be further investigated, has no apparent relation to the one we have just discussed.', 'hep-th-9904188-1-19-0': '## The canonical transformation', 'hep-th-9904188-1-20-0': 'Poisson-Lie T-duality-related models are canonically equivalent under the transformation [CITATION] [EQUATION]', 'hep-th-9904188-1-20-1': 'This transformation preserves the equal-time Poisson brackets of the conjugate pairs of variables [MATH] and [MATH] given by [CITATION] [EQUATION] and [EQUATION] where [MATH] is the antisymmetric step function that equals [MATH]) for [MATH]).', 'hep-th-9904188-1-20-2': 'Notice that the above Poisson brackets are independent of the details of the [MATH]-models related by Poisson-Lie T-duality.', 'hep-th-9904188-1-20-3': 'They are simply the central extensions, in loop space, of the usual Lie-(bi-)algebras defined in the Drinfeld double.', 'hep-th-9904188-1-20-4': 'One may also show that the Hamiltonians of the two dual actions ([REF]) and ([REF]) are equal [CITATION] as required for canonical transformation with no explicit [MATH]-dependence.', 'hep-th-9904188-1-20-5': 'After some algebraic manipulations, these Hamiltonians can be written as [EQUATION] and [EQUATION] where [MATH] and [MATH] are the symmetric and antisymmetric parts of [MATH] and similarly [MATH] and [MATH] are the symmetric and antisymmetric parts of [MATH]-[MATH].', 'hep-th-9904188-1-20-6': 'Notice that in the limit ([REF]) the conjugate momenta [MATH] vanish.', 'hep-th-9904188-1-20-7': 'This is consistent with the development of a local gauge invariance ([REF]).', 'hep-th-9904188-1-20-8': 'At the level of the Poisson brackets the vanishing of [MATH], together with its conjugate [MATH], has to be imposed as a constraint.', 'hep-th-9904188-1-20-9': 'In fact they form a set [MATH] of second-class constraints.', 'hep-th-9904188-1-20-10': 'We may see that in the limit ([REF]) and upon using ([REF]), the Hamiltonians ([REF]) and ([REF]) reduce to [EQUATION] and [EQUATION]', 'hep-th-9904188-1-20-11': 'We may show, with the help of ([REF]) and ([REF]), that [MATH] (weakly).', 'hep-th-9904188-1-20-12': 'Hence, no new constraints are generated by the time [MATH]-evolution.', 'hep-th-9904188-1-21-0': 'In general (see, for instance, [CITATION]), in the presence of a set of second-class constraints [MATH], one computes the antisymmetric matrix associated with their Poisson brackets [MATH].', 'hep-th-9904188-1-21-1': 'When [MATH] is invertible one simply postulates that the usual Poisson brackets are replaced by Dirac brackets, defined as [EQUATION] for any two phase-space variables [MATH] and [MATH].', 'hep-th-9904188-1-21-2': 'In our case we compute the (infinite-dimensional) matrix [EQUATION] with inverse [EQUATION]', 'hep-th-9904188-1-21-3': 'Then the Dirac brackets can be computed using ([REF]).', 'hep-th-9904188-1-21-4': 'We find (for notational convenience in the rest of the paper, we omit the subscript [MATH] from the Dirac brackets): [EQUATION]', 'hep-th-9904188-1-21-5': 'Notice the parafermionic character of this algebra, which is encoded in the terms containing [MATH].', 'hep-th-9904188-1-21-6': 'The Dirac brackets for the pair [MATH] are obtained from ([REF])-([REF]) by replacing untilded symbols by tilded ones and vice versa.', 'hep-th-9904188-1-21-7': 'It is instructive to write down the Dirac brackets for the case that the group [MATH] is Abelian, i.e. [MATH].', 'hep-th-9904188-1-21-8': 'We find [EQUATION]', 'hep-th-9904188-1-21-9': 'The above Dirac brackets can also be obtained from the ones in ([REF])-([REF]) via a contraction that Abelianizes the group [MATH], i.e. [MATH], [MATH], [MATH].', 'hep-th-9904188-1-22-0': '# An explicit example', 'hep-th-9904188-1-23-0': 'In this section we explicitly demonstrate many of the general aspects developed in section 2, using 3- and 2-dim models related by Poisson-Lie T-duality.', 'hep-th-9904188-1-23-1': 'That includes the explicit construction of the metric and antisymmetric tensor fields, of the Dirac-bracket algebra for canonical equivalence, and also of the corresponding generating functional.', 'hep-th-9904188-1-24-0': '## The Drinfeld double', 'hep-th-9904188-1-25-0': 'Our example will be based on the 6-dim Drinfeld double considered in [CITATION], which we first review by following [CITATION].', 'hep-th-9904188-1-25-1': 'It is just the non-compact group [MATH] with [MATH] and dual [MATH] given by the Iwasawa decomposition of [MATH] [CITATION].', 'hep-th-9904188-1-25-2': 'The associated 3-dim algebras [MATH] and [MATH] have generators denoted by [MATH] and [MATH], where [MATH].', 'hep-th-9904188-1-25-3': 'Leaving aside the details we only present the elements that are necessary in this paper.', 'hep-th-9904188-1-25-4': 'It is convenient to split the index [MATH], [MATH].', 'hep-th-9904188-1-25-5': 'The non-vanishing structure constants for the algebras [MATH] and [MATH] are [EQUATION] where our normalization is such that [MATH].', 'hep-th-9904188-1-25-6': 'We parametrize the [MATH] group element in terms of the three Euler angles [MATH], [MATH] and [MATH].', 'hep-th-9904188-1-25-7': 'It is represented by the [MATH] block-diagonal matrix [EQUATION] where [EQUATION]', 'hep-th-9904188-1-25-8': 'Also the group element of [MATH] is parametrized in terms of three variables [MATH], [MATH] and [MATH] and represented by the following [MATH] block-diagonal matrix [EQUATION] where [EQUATION]', 'hep-th-9904188-1-25-9': 'The Maurer-Cartan forms in the parametrization of the [MATH] group element ([REF]) are [EQUATION]', 'hep-th-9904188-1-25-10': 'Similarly, using the parametrization ([REF]) for the [MATH] group element we find [EQUATION]', 'hep-th-9904188-1-25-11': 'The antisymmetric matrices [MATH] and [MATH] are [EQUATION] and [EQUATION]', 'hep-th-9904188-1-26-0': '## Explicit three- and two-dimensional models', 'hep-th-9904188-1-27-0': 'Consider the [MATH]-model action ([REF]) for the case of our double based on [MATH].', 'hep-th-9904188-1-27-1': 'Let us single-out the 1-dim subgroup [MATH] that is generated by [MATH].', 'hep-th-9904188-1-27-2': 'For our purposes it will be sufficient to use the following form for the [MATH] matrix [MATH]-[MATH] [EQUATION] where we have kept the conventions of ([REF]) for the enumeration of the matrix elements.', 'hep-th-9904188-1-27-3': 'Using ([REF]), ([REF]) and ([REF]), it is the easy to compute the metric and antisymmetric tensor fields corresponding to ([REF]).', 'hep-th-9904188-1-27-4': 'We find a metric given by [EQUATION] and an antisymmetric tensor given by [EQUATION] where [EQUATION]', 'hep-th-9904188-1-27-5': 'Notice that the antisymmetric tensor can be (locally) gauged away since the corresponding 3-form field strength is zero.', 'hep-th-9904188-1-27-6': 'Also, for our purposes, we will not need the explicit expressions for the metric and antisymmetric tensor corresponding to the dual [MATH]-model ([REF]).', 'hep-th-9904188-1-27-7': 'For [MATH], but general [MATH] and [MATH], the above example (with its dual) was considered in [CITATION] (also in [CITATION] for [MATH] and [MATH]).', 'hep-th-9904188-1-28-0': 'We would like to take the analogue of the limit ([REF]).', 'hep-th-9904188-1-28-1': 'It is clear that in our case this corresponds to letting [MATH].', 'hep-th-9904188-1-28-2': 'Comparing ([REF]) to ([REF]) we see that the [MATH] matrix [MATH] is [EQUATION]', 'hep-th-9904188-1-28-3': 'It is easily seen that this is the most general [MATH] matrix that solves ([REF]), with structure constants given by ([REF]).', 'hep-th-9904188-1-28-4': 'In agreement with our general discussion, the [MATH]-model action with metric ([REF]) and antisymmetric tensor ([REF]) develops a local invariance under the transformation [EQUATION]', 'hep-th-9904188-1-28-5': 'This allows to gauge-fix the variable [MATH].', 'hep-th-9904188-1-28-6': 'Explicitly computing ([REF]) we find that the metric and antisymmetric tensors are given by [EQUATION]', 'hep-th-9904188-1-28-7': 'Equivalently, the same result follows if we set [MATH] directly into the expressions for the metric ([REF]) and antisymmetric ([REF]) tensors.', 'hep-th-9904188-1-28-8': 'Similarly, the dual model action ([REF]) is invariant under the local transformation [EQUATION]', 'hep-th-9904188-1-28-9': 'Hence, we may evaluate ([REF]) in the gauge [MATH].', 'hep-th-9904188-1-28-10': 'The corresponding metric (the antisymmetric tensor turns out to be zero) is found to be [EQUATION] where we have changed variables as [MATH]', 'hep-th-9904188-1-29-0': 'and [MATH].', 'hep-th-9904188-1-29-1': 'The metric in ([REF]) is free of singularities (since ([REF]) has no fixed point) and represents a deformed 2-sphere.', 'hep-th-9904188-1-29-2': 'In contrast, ([REF]) is singular for [MATH].', 'hep-th-9904188-1-29-3': 'This is related to the fact that [MATH] is a fixed point of the gauge transformation ([REF]).', 'hep-th-9904188-1-29-4': 'The singularity at [MATH] is only a coordinate singularity and can be removed by an appropriate change of variables.', 'hep-th-9904188-1-30-0': 'It is worth while to consider some analytic continuations of the models ([REF]) and its dual ([REF]).', 'hep-th-9904188-1-30-1': 'If we let [MATH], where [MATH], and also we change the sign of the overall coupling constant [MATH], then ([REF]) becomes [EQUATION]', 'hep-th-9904188-1-30-2': 'The corresponding analytic continuation in the dual metric ([REF]) should be [MATH], with a parallel change of sign in the overall coupling constant.', 'hep-th-9904188-1-30-3': 'The metric in ([REF]) is reduced to the Euclidean [MATH] metric if we rescale the coupling constant [MATH] and then take the limit [MATH] (keeping the new coupling finite).', 'hep-th-9904188-1-30-4': 'However, for generic values of the constant [MATH], it represents a space that is topologically a cigar.', 'hep-th-9904188-1-30-5': 'Indeed, for [MATH] we get the 2-dim Euclidean space [MATH] in polar coordinates, whereas for [MATH] we get, after an appropriate change of variables, [MATH].', 'hep-th-9904188-1-30-6': 'For [MATH], the cigar-shaped space develops a "pump" corresponding to the maximum of the metric components [MATH] at [EQUATION]', 'hep-th-9904188-1-30-7': 'We note that the cigar-shape topology is also a characteristic of the Euclidean black hole corresponding to the coset [MATH] exact conformal field theory [CITATION].', 'hep-th-9904188-1-30-8': 'However, in our case the model ([REF]) is not conformal.', 'hep-th-9904188-1-31-0': 'The Drinfeld double for ([REF]) and its dual model is [MATH], with [MATH], instead of [MATH].', 'hep-th-9904188-1-32-0': '## The Dirac brackets and the generating functional', 'hep-th-9904188-1-33-0': 'The Dirac brackets for the conjugate variables in our example are most easily written down in the basis [MATH] and [MATH], where the non-zero structure constants are [MATH], [MATH] and [MATH].', 'hep-th-9904188-1-33-1': 'Using ([REF])-([REF]) we obtain [EQUATION] where the underlined terms should be omitted in the Abelian limit of the dual group [MATH].', 'hep-th-9904188-1-33-2': 'In this case the above algebra provides a canonical equivalence between the [MATH]-model for [MATH] and its non-Abelian dual with respect to the left (or right) action of [MATH].', 'hep-th-9904188-1-33-3': 'Note also that the generators [MATH] form a subalgebra ([REF]).', 'hep-th-9904188-1-34-0': 'The generating functional that demonstrates the classical equivalence between [MATH]-models related by Poisson-Lie T-duality based on our Drinfeld double was explicitly constructed in [CITATION].', 'hep-th-9904188-1-34-1': 'In a slightly different form than that in [CITATION], it reads [EQUATION]', 'hep-th-9904188-1-34-2': 'Notice that the above generating functional depends only on the combination [MATH]; it is therefore invariant under the [MATH] gauge transformation [MATH] and [MATH].', 'hep-th-9904188-1-34-3': 'The generating functional for the deformed coset models ([REF]) is obtained by solving the equation [MATH] (equivalently [MATH]) for [MATH] and inserting the result back into ([REF]).', 'hep-th-9904188-1-34-4': 'The result is a generating functional, which is non-polynomial in derivatives with respect to [MATH].', 'hep-th-9904188-1-34-5': 'The obtained expressions are quite complicated and not very illustrative, so that we decided to present the corresponding result for the [MATH]-model for [MATH] and its non-Abelian dual.', 'hep-th-9904188-1-34-6': 'We start with the generating functional corresponding to the 2-dim [MATH]-models for [MATH] and its non-Abelian dual with respect to the left (or right) action of [MATH] that was obtained in [CITATION].', 'hep-th-9904188-1-34-7': 'In our notation it is given by [MATH].', 'hep-th-9904188-1-34-8': 'This is easily modified to depend on the angles [MATH] and [MATH] only through the combination [MATH], by adding the term [MATH].', 'hep-th-9904188-1-34-9': 'Such a term, being dependent on the variables of only one of the dual models, can be absorbed as total derivative into the corresponding action and hence it does not affect the classical dynamics.', 'hep-th-9904188-1-34-10': 'Explicitly, the resulting generating functional is [EQUATION]', 'hep-th-9904188-1-34-11': 'The variation of [MATH] with respect to [MATH] gives [EQUATION]', 'hep-th-9904188-1-34-12': 'Substituting back into ([REF]) we obtain [EQUATION] where [MATH] is given by ([REF]).', 'hep-th-9904188-1-34-13': 'The generating functional ([REF]) is non-polynomial in the derivatives of the fields with respect to [MATH].', 'hep-th-9904188-1-34-14': 'In that sense it belongs to a new class of generating functionals, which depend not only on the fields of the two dual [MATH]-models, but also on their first derivatives with respect to the space-like variable in a non-trivial way.', 'hep-th-9904188-1-34-15': 'For comparison, up to now, either in the case of non-Abelian duality [CITATION] or for Poisson-Lie T-duality (and its possible generalizations) [CITATION], there was no dependence of the generating functional on more than the first power of these derivatives (see, for example ([REF])).', 'hep-th-9904188-1-34-16': 'Finally, we note that, according to the work in [CITATION], generating functionals of the form ([REF]), being non-linear, are expected to receive quantum corrections.', 'hep-th-9904188-1-34-17': 'Consequently, the corresponding duality rules relating the 2-dim field theories, as well as the algebra ([REF])-([REF]), are expected to be quantum-corrected.', 'hep-th-9904188-1-35-0': '# Renormalization group flow', 'hep-th-9904188-1-36-0': 'In this section we study the 1-loop RG equations corresponding to the three-dimensional model ([REF]), ([REF]).', 'hep-th-9904188-1-36-1': 'We will show that there are no fixed points in the flow and also that the correct description of the models is a non-perturbative one.', 'hep-th-9904188-1-36-2': 'However, for large domains in parameter space and for a wide range of energies in the UV, the description is effectively perturbative and the model becomes a two-dimensional one.', 'hep-th-9904188-1-37-0': 'Finally, by performing some analytic continuations we will find three- and two-dimensional models with fixed points under the RG flow, where the theory becomes free.', 'hep-th-9904188-1-38-0': 'We begin this section with a short review of RG flow in 2-dim field theories with curved target spaces.', 'hep-th-9904188-1-38-1': 'and ([REF]).', 'hep-th-9904188-1-38-2': 'The 2-dim [MATH]-model corresponding to the metric ([REF]) and antisymmetric tensor ([REF]) is of the form [EQUATION]', 'hep-th-9904188-1-38-3': "It will be renormalizable if the corresponding counter-terms, at a given order in a loop expansion, can be absorbed into a renormalization of the coupling constant [MATH] and (or) of some parameters labelled collectively by [MATH], [MATH] In addition, we allow for general field redefinitions of the [MATH]'s, which are coordinate reparametrizations in the target space.", 'hep-th-9904188-1-38-4': 'This definition of renormalizability of [MATH]-models is quite strict and similar to that for ordinary field theories.', 'hep-th-9904188-1-38-5': 'A natural extension of this is to allow for the manifold to vary with the mass scale and the RG to act in the infinite-dimensional space of all metrics and torsions [CITATION].', 'hep-th-9904188-1-38-6': 'Further discussion of this generalized renormalizability will not be needed for our purposes.', 'hep-th-9904188-1-38-7': 'Perturbatively, in powers of [MATH], we express the bare quantities, denoted by a zero as a subscript, as [EQUATION]', 'hep-th-9904188-1-38-8': 'The ellipses stand for higher-order loop- and pole-terms in [MATH] and [MATH] respectively.', 'hep-th-9904188-1-38-9': 'Then, the beta-functions up to one loop are given by [MATH] and [MATH], where, as usual, [MATH], [MATH] and [MATH].', 'hep-th-9904188-1-39-0': 'The equations to be satisfied by appropriately choosing [MATH] and [MATH] are given by [EQUATION] where [MATH] are the components of the "generalized" Ricci tensor defined with a connection that includes the torsion, i.e. with [MATH].', 'hep-th-9904188-1-39-1': 'The corresponding counter-terms were computed in the dimensional regularization scheme (see, for instance, [CITATION]).', 'hep-th-9904188-1-40-0': '## Models with no fixed points', 'hep-th-9904188-1-41-0': '### Three-dimensional models', 'hep-th-9904188-1-42-0': 'In the metric ([REF]) there are three parameters [MATH], [MATH] and [MATH] and the three Euler angles [MATH] and [MATH] will be denoted by [MATH].', 'hep-th-9904188-1-42-1': 'Also for the antisymmetric tensor in ([REF]) we have [MATH].', 'hep-th-9904188-1-42-2': 'Examining ([REF]) we find that the coupling [MATH] and the coordinates [MATH] do not renormalize and therefore the corresponding beta-functions are zero.', 'hep-th-9904188-1-42-3': 'In contrast, for the parameters [MATH] and [MATH] we find [EQUATION]', 'hep-th-9904188-1-42-4': 'This system of coupled non-linear equations can be considerably simplified.', 'hep-th-9904188-1-42-5': 'First, using ([REF]), we may easily show that there is a RG-flow-invariant defined as [EQUATION] which implies that [EQUATION]', 'hep-th-9904188-1-42-6': 'Without loss of generality we may assume that [MATH] since ([REF]) remains invariant under [MATH] and [MATH].', 'hep-th-9904188-1-42-7': 'Then, using the last two equations in ([REF]) we may derive an equation for [MATH] as a function of [MATH] whose solution is [EQUATION] where [MATH] is a real constant, which is determined by the initial conditions for [MATH] and [MATH].', 'hep-th-9904188-1-42-8': 'The sign in front of the square root in ([REF]) is changed when [MATH], in order to ensure the continuity of [MATH] as a function of the energy scale [MATH].', 'hep-th-9904188-1-43-0': 'Hence, the only differential equation we still have to solve is the one for [MATH], which, after using ([REF]), takes the form [EQUATION] where [MATH] and [MATH] are determined by ([REF]) and ([REF]).', 'hep-th-9904188-1-43-1': 'Since the RG equations are real, [MATH] will stay strictly non-negative and therefore [MATH] will oscillate with [MATH] between its minimum and maximum values [MATH] and [MATH], where [MATH].', 'hep-th-9904188-1-43-2': 'When [MATH], for finite values of the overall coupling constant [MATH], the curvature for the metric ([REF]) approaches infinity and the perturbative expansion of the RG equations becomes meaningless.', 'hep-th-9904188-1-44-0': 'We have seen that the correct description of the theory is a genuine non-perturbative one.', 'hep-th-9904188-1-44-1': 'Neverthelss, for [MATH] we will show that there exists a wide range of energies in the UV, where the description is effectively perturbative.', 'hep-th-9904188-1-44-2': 'Moreover, there exists a fixed point at [MATH] where the theory has effectively a 2-dim target space.', 'hep-th-9904188-1-45-0': 'Indeed, using ([REF]), we have that [MATH] when [MATH].', 'hep-th-9904188-1-45-1': 'Hence, in that limit and after redefining [MATH] we may simplify the RG eqs. ([REF]) as [EQUATION]', 'hep-th-9904188-1-45-2': 'Then the metric ([REF]) becomes [EQUATION] which is the deformed [MATH] Principal Chiral model considered in [CITATION].', 'hep-th-9904188-1-45-3': 'Also the first two of the above equations are those derived in [CITATION] for the corresponding coupling [MATH] and deformation parameter [MATH].', 'hep-th-9904188-1-45-4': 'In the UV the solution of ([REF]) is [EQUATION]', 'hep-th-9904188-1-45-5': 'Hence, in the UV [MATH].', 'hep-th-9904188-1-45-6': 'Therefore if the condition [EQUATION] is fulfilled, then [MATH] and the model is indeed described perturbatively by ([REF]).', 'hep-th-9904188-1-45-7': 'The point [MATH] is a UV-fixed point, where the metric ([REF]) becomes [MATH].', 'hep-th-9904188-1-45-8': 'However, outside the validity of ([REF]) the correct description is non-perturbative.', 'hep-th-9904188-1-46-0': '### Two-dimensional models', 'hep-th-9904188-1-47-0': 'Let us now return to the 2-dim models ([REF]) and ([REF]).', 'hep-th-9904188-1-47-1': 'As before, there is no wave-function renormalization for [MATH] and [MATH], and the beta-function for the coupling [MATH] is zero.', 'hep-th-9904188-1-47-2': 'For the couplings [MATH] and [MATH] the corresponding beta-functions can be obtained by simply setting [MATH] into ([REF]).', 'hep-th-9904188-1-47-3': 'The reason why such a procedure is consistent seems to be intimately related to the local invariance that reduces the 3-dim models into 2-dim ones.', 'hep-th-9904188-1-47-4': 'Hence, we have [EQUATION] which are nothing but the beta-functions for the 2-dim model ([REF]) as well as for its dual ([REF]).', 'hep-th-9904188-1-47-5': 'This is a strong hint that their classical equivalence can be promoted into a quantum one as well.', 'hep-th-9904188-1-47-6': 'Having said that we note, once again, that [MATH] is not a fixed point of the ([REF]) in the UV.', 'hep-th-9904188-1-47-7': 'Since ([REF]) is still a RG invariant of ([REF]), it is clear that one variable between [MATH] and [MATH] is an independent one.', 'hep-th-9904188-1-47-8': 'Eliminating [MATH] from ([REF]) using ([REF]), we obtain [EQUATION]', 'hep-th-9904188-1-47-9': 'Hence, the solution for [MATH] as a function of the energy scale [MATH] oscillates between [MATH] and [MATH] as [EQUATION] where [MATH] is an arbitrary reference scale.', 'hep-th-9904188-1-47-10': 'This means that the corresponding [MATH]-model actions do not define local field theories and can be considered at most as effective actions for scales such that [MATH] stays away from [MATH].', 'hep-th-9904188-1-48-0': 'The usual [MATH] metric and its non-Abelian dual with respect to the right (or left) action of [MATH] are obtained from ([REF]) and ([REF]) if we rescale the coupling constant [MATH] and then take the limit [MATH] (keeping the new coupling finite).', 'hep-th-9904188-1-48-1': 'However, this limit is problematic at the quantum level since the corresponding [MATH]-functions do not tend to the beta-function obtained by studying the 2-dim field theories based on [MATH] (and its non-Abelian dual) by themselves [CITATION].', 'hep-th-9904188-1-48-2': 'The latter is, at one-loop, just [MATH] and is consistent with the fact that these models are asymptotically free.', 'hep-th-9904188-1-48-3': 'It is formally obtained by the first of ([REF]) in the limit [MATH] after we rescale [MATH] as described above.', 'hep-th-9904188-1-48-4': 'This limit does not correspond to any fixed point of ([REF]).', 'hep-th-9904188-1-48-5': 'It is easily seen that, from a RG theory view point, these models offer an effective description of the more general models ([REF]) and ([REF]) in the case of [MATH], which, according to ([REF]), occurs at scales [MATH].', 'hep-th-9904188-1-49-0': '## Models with fixed points', 'hep-th-9904188-1-50-0': '### Three-dimensional models', 'hep-th-9904188-1-51-0': 'We have seen that our model ([REF]), ([REF]) does not have a true fixed point under the 1-loop RG eqs. ([REF]).', 'hep-th-9904188-1-51-1': 'Consider, however, the analytic continuation [MATH] and [MATH].', 'hep-th-9904188-1-51-2': 'Then the metric and antisymmetric tensors become [EQUATION] and [EQUATION] where instead of ([REF]) the function [MATH] is given by [EQUATION]', 'hep-th-9904188-1-51-3': 'The fact that the antisymmetric tensor is imaginary is bothersome if we want to describe models in 2-dim Minkowskian space-times.', 'hep-th-9904188-1-51-4': 'However, for Euclidean ones, the (locally) exact 2-form measures the charge of non-trivial instanton-like configurations.', 'hep-th-9904188-1-51-5': 'The perturbative expansion is completely independent of the antisymmetric tensor, but this will definitely play a role in a, yet lacking, non-perturbative formulation of the model.', 'hep-th-9904188-1-51-6': 'The 1-loop RG equations for the metric ([REF]) are obtained from ([REF]) by the analytic continuation we have described above.', 'hep-th-9904188-1-51-7': 'Then the analogue of ([REF]) is given by [EQUATION] where now [EQUATION] and [MATH] is still given by ([REF]).', 'hep-th-9904188-1-51-8': 'As before, we will assume that [MATH] with no loss of generality.', 'hep-th-9904188-1-51-9': 'However, now [MATH] does not have to be larger than or equal to 1, as in ([REF]), in order to ensure reality for [MATH].', 'hep-th-9904188-1-51-10': 'If [MATH] then [MATH] are complex conjugate of each other and, unlike the case when they are real, [MATH] can take any real value without spoiling the reality of the parameter [MATH].', 'hep-th-9904188-1-51-11': 'However, now the condition [MATH] has to be fulfilled in order for [MATH] to remain real.', 'hep-th-9904188-1-51-12': 'If on the other hand [MATH], then [MATH] are both real and the reality condition for [MATH] requires that [MATH] or [MATH].', 'hep-th-9904188-1-51-13': 'Since [MATH], it turns out that there are fixed points for initial conditions where [MATH] is less than [MATH].', 'hep-th-9904188-1-51-14': 'Consider first the RG eq. ([REF]) near the point with [MATH], [MATH] and [MATH].', 'hep-th-9904188-1-51-15': 'It can be written as (we take the lower sign in ([REF])): [EQUATION]', 'hep-th-9904188-1-51-16': 'The same equation near the different point with [MATH], [MATH] and [MATH] takes the form [EQUATION]', 'hep-th-9904188-1-51-17': 'For [MATH] we have [MATH].', 'hep-th-9904188-1-51-18': 'Hence, for [MATH] we have an IR-stable point at [MATH] as well as a UV-stable point at [MATH].', 'hep-th-9904188-1-52-0': 'For [MATH], we have that [MATH].', 'hep-th-9904188-1-52-1': 'Therefore, for [MATH] there are two UV-stable points at [MATH] and at [MATH].', 'hep-th-9904188-1-53-0': 'In all cases the background ([REF]), ([REF]) flows, either in the IR or in the UV, towards the background with [EQUATION] where [MATH] represents any of the two fixed points [MATH] or [MATH].', 'hep-th-9904188-1-53-1': 'This represents a free theory, as can be seen by changing variables as [MATH].', 'hep-th-9904188-1-53-2': 'It is interesting to note that in the case [MATH] the signature of the metric in ([REF]) is [MATH] in the IR fixed point [MATH] and [MATH] in the UV fixed point [MATH].', 'hep-th-9904188-1-54-0': 'Also in the case of [MATH] the signature at the [MATH] UV-stable point is [MATH], but in the other UV-stable point at [MATH] it is [MATH].', 'hep-th-9904188-1-54-1': 'Hence, only at [MATH] the metric has Euclidean signature and we expect a well-defined field-theoretical description.', 'hep-th-9904188-1-55-0': 'Let us also note that for [MATH] the RG flow is described, as before, by ([REF]), ([REF]) and the corresponding [MATH]-model is again ([REF]), provided ([REF]) is satisfied.', 'hep-th-9904188-1-56-0': '### Two-dimensional models', 'hep-th-9904188-1-57-0': 'Now we turn to the 2-dim model ([REF]) after the same analytic continuation as before, [MATH] and [MATH]: [EQUATION]', 'hep-th-9904188-1-57-1': 'The 1-loop RG equation corresponding to ([REF]) is [EQUATION]', 'hep-th-9904188-1-57-2': 'The form of the solution for [MATH] as a function of the energy scale [MATH] depends on whether or not [MATH] is smaller or larger than 1.', 'hep-th-9904188-1-57-3': 'We find [EQUATION] where [MATH] denotes again an arbitrary reference scale.', 'hep-th-9904188-1-57-4': 'We see that in the UV there is a fixed point at [MATH] (and [MATH]).', 'hep-th-9904188-1-57-5': 'The lower bound for [MATH] above is needed for [MATH] to stay positive, since only then is ([REF]) a solution of ([REF]).', 'hep-th-9904188-1-57-6': 'For the case of [MATH], we have to distinguish the solutions between those with [MATH] and those with [MATH].', 'hep-th-9904188-1-57-7': 'In the former case we obtain [EQUATION] and [EQUATION] where, as before, [MATH] and [MATH] are arbitrary reference scales.', 'hep-th-9904188-1-57-8': 'For the trajectory given by ([REF]), [MATH] stays positive.', 'hep-th-9904188-1-57-9': 'It starts at [MATH] for [MATH], and ends at [MATH] for [MATH].', 'hep-th-9904188-1-57-10': 'For the trajectory given by ([REF]), [MATH] is always negative and starts at [MATH] for [MATH] and ends at [MATH] for [MATH].', 'hep-th-9904188-1-57-11': 'Hence, we see that [MATH] is a UV fixed point.', 'hep-th-9904188-1-57-12': 'Also as we lower the scale [MATH] towards the IR, the solution becomes singular in both cases.', 'hep-th-9904188-1-57-13': 'In any case, we then run into non-perturbative regimes.', 'hep-th-9904188-1-57-14': 'For trajectories in the region [MATH], the solution is still given by ([REF]), but with [MATH].', 'hep-th-9904188-1-57-15': 'In the lower limit [MATH] and in the upper limit [MATH].', 'hep-th-9904188-1-57-16': 'Hence in that case we have a singular behaviour of the 1-loop RG equations towards the IR as well as the UV.', 'hep-th-9904188-1-57-17': 'As we have mentioned, in those cases the corresponding 2-dim field theory is not well defined at the quantum level and can be considered only as an effective field theory at scales away from the singularities.', 'hep-th-9904188-1-58-0': 'The 2-dim model corresponding to the fixed point at [MATH] is obtained by setting [MATH] in ([REF]) [EQUATION]', 'hep-th-9904188-1-58-1': 'The fact that ([REF]) approaches a free-field conformal field theory at the fixed point is similar to the case of an integrable model (different from ([REF])), representing also a 1-parameter deformation of [MATH], that was considered in [CITATION].', 'hep-th-9904188-1-58-2': 'It is interesting to investigate whether or not ([REF]) represents also an integrable perturbation of [MATH].', 'hep-th-9904188-1-59-0': '# Concluding remarks', 'hep-th-9904188-1-60-0': 'We have constructed a new class of 2-dim field theories with target spaces corresponding to deformations of coset spaces [MATH].', 'hep-th-9904188-1-60-1': 'Our models correspond to special points of the classical moduli space of models related by Poisson-Lie T-duality, where a local invariance develops.', 'hep-th-9904188-1-60-2': 'A classification of all possible models that arise with such a procedure is an interesting open problem and can be done by analyzing the general conditions ([REF]), or equivalently ([REF]).', 'hep-th-9904188-1-60-3': 'By construction these models come in dual pairs.', 'hep-th-9904188-1-60-4': 'The corresponding generating functionals depend non-polynomially on the derivatives of the fields with respect to the space-like variable.', 'hep-th-9904188-1-60-5': 'The latter feature is also manifested in an underlying infinite-dimensional algebra with a central extension of the parafermionic type.', 'hep-th-9904188-1-60-6': 'It would also be interesting to uncover the relation of our models to those in [CITATION].', 'hep-th-9904188-1-61-0': 'We have also performed a quite general RG flow analysis using specific models with 3- and 2-dim target spaces.', 'hep-th-9904188-1-61-1': 'As in [CITATION], we conclude that quantum aspects of the lower dimensional models do not necessarily follow by taking the same classical limit as that used to relate the corresponding 2-dim field-theoretical classical actions.', 'hep-th-9904188-1-61-2': 'Concretely, the beta-function equations for the lower-dimensional models follow from those of the original models by just setting some parameters to their prescribed values (see ([REF]) and ([REF])).', 'hep-th-9904188-1-61-3': 'However, these values do not necessarily correspond to any fixed points of the solutions of these equations.', 'hep-th-9904188-1-61-4': 'Using our 3-dim example we saw that in a large domain in parameter space, and for a wide range of energies in the UV, the description is effectively perturbative with a UV-fixed point exactly where the local gauge invariance develops.', 'hep-th-9904188-1-62-0': 'We believe that this feature will persist for more general models related by Poisson-Lie T-duality.', 'hep-th-9904188-1-62-1': 'In that respect it would be very interesting to study the RG flow in general using ([REF]) and ([REF]) and possibly to formulate this flow in a duality-invariant way.'}

{'hep-th-9904188-2-0-0': '0,2cm We construct a new class of two-dimensional field theories with target spaces that are finite multiparameter deformations of the usual coset [MATH]-spaces.', 'hep-th-9904188-2-0-1': 'They arise naturally, when certain models, related by Poisson-Lie T-duality, develop a local gauge invariance at specific points of their classical moduli space.', 'hep-th-9904188-2-0-2': 'We show that canonical equivalences in this context can be formulated in loop space in terms of parafermionic-type algebras with a central extension.', 'hep-th-9904188-2-0-3': 'We find that the corresponding generating functionals are non-polynomial in the derivatives of the fields with respect to the space-like variable.', 'hep-th-9904188-2-1-0': 'After constructing models with three- and two-dimensional targets, we study renormalization group flows in this context.', 'hep-th-9904188-2-1-1': 'In the ultraviolet, in some cases, the target space of the theory reduces to a coset space or there is a fixed point where the theory becomes free.', 'hep-th-9904188-2-2-0': 'CERN-TH/99-112', 'hep-th-9904188-2-3-0': 'April 1999', 'hep-th-9904188-2-4-0': '16 pt', 'hep-th-9904188-2-5-0': '# Introduction', 'hep-th-9904188-2-6-0': 'Cosets [MATH] as target spaces in 2-dim field theories have been extensively studied in the literature, as they provide examples of spaces other than group manifolds, which give rise to integrable models.', 'hep-th-9904188-2-6-1': 'It is always of interest to find integrable deformations [CITATION]-[CITATION] of such models and if possible classify them.', 'hep-th-9904188-2-6-2': 'In the ordinary (undeformed) coset models one starts with the usual Wess-Zumino action for a group, with Lagrangian density proportional to [MATH]-[MATH], and then restricts the trace to the coset space only.', 'hep-th-9904188-2-6-3': 'Hence, this construction, but not the corresponding models, is quite trivial.', 'hep-th-9904188-2-6-4': 'Having in mind 2-dim field theories, with targets spaces representing continuous deformations of the latter coset spaces, we need models with non-trivial moduli as a starting point.', 'hep-th-9904188-2-6-5': 'Such an example was considered in [CITATION]; we present in this paper the generalization of this to a class of theories.', 'hep-th-9904188-2-7-0': 'We found natural to start, in section 2, with 2-dim models related by Poisson-Lie T-duality [CITATION], since these have indeed a non-trivial moduli space and, moreover, their classical equivalence has been established [CITATION].', 'hep-th-9904188-2-7-1': 'Also, in some examples, there are hints that point towards the classical equivalence promoted into a quantum one at 1-loop in perturbation theory [CITATION].', 'hep-th-9904188-2-7-2': 'We will show that in some points in this moduli space a local (gauge-like) invariance is developing.', 'hep-th-9904188-2-7-3': 'Hence, at these points the configuration space is lower-dimensional and we discover in a unifying manner spaces that are deformations of the usual coset spaces.', 'hep-th-9904188-2-7-4': 'In addition, as a byproduct, we will obtain duals of these models that are classically canonically equivalent to them as 2-dim field theories.', 'hep-th-9904188-2-7-5': 'This equivalence is encoded in infinite-dimensional current algebras of the parafermionic type that we construct.', 'hep-th-9904188-2-7-6': 'We derive these from the infinite-dimensional algebras with a central extension, which were found in the proof of canonical equivalence of the Poisson-Lie T-duality-related models in [CITATION].', 'hep-th-9904188-2-7-7': 'The corresponding generating functionals have the new feature that they are not linear in the derivatives of the fields with respect to the space-like variable.', 'hep-th-9904188-2-7-8': 'This is in contrast with the cases of Abelian duality [CITATION], non-Abelian duality in Principal Criral [CITATION] and more general [CITATION] models, as well as for Poisson-Lie T-duality (and its possible generalizations) [CITATION].', 'hep-th-9904188-2-7-9': 'They are, instead, non-polynomial functions of these derivatives.', 'hep-th-9904188-2-7-10': 'Many of these aspects are explicitly demonstrated in section 3, with a particular example.', 'hep-th-9904188-2-7-11': 'In section 4 we discuss the renormalization group (RG) flow in this context.', 'hep-th-9904188-2-7-12': 'As in [CITATION], we emphasize that taking the classical limit that leads to the lower-dimensional models and then studying the RG flow does not necessarily imply that this limit would correspond to a fixed point of the RG flow, i.e. the two procedures do not commute.', 'hep-th-9904188-2-7-13': 'There is, however, a particular domain in parameter space, where for a wide range of energies in the UV, the description is effectively perturbative with a UV-stable fixed point corresponding to the point where the gauge invariance develops.', 'hep-th-9904188-2-7-14': 'Then the model becomes effectively a two-dimensional one.', 'hep-th-9904188-2-8-0': 'We end the paper with section 5, containing concluding remarks and a discussion on future directions of this research.', 'hep-th-9904188-2-8-1': 'We have also written an appendix, where some mathematical aspects of our proofs are worked out explicitly.', 'hep-th-9904188-2-9-0': '# General formulation', 'hep-th-9904188-2-10-0': 'In this section we first show how new 2-dim field theories, with target spaces representing deformed coset spaces, arise in the context of Poisson-Lie T-duality-related [MATH]-models.', 'hep-th-9904188-2-10-1': 'We then present a duality-invariant formulation and show that canonical equivalences are encoded into algebras of the parafermionic-type in loop space.', 'hep-th-9904188-2-11-0': '## Formulation using Poisson-Lie T-duality-related [MATH]-models', 'hep-th-9904188-2-12-0': 'The form of 2-dim [MATH]-model actions related by Poisson-Lie T-duality (in the absence of spectator fields) is [CITATION] [EQUATION] and [EQUATION]', 'hep-th-9904188-2-12-1': 'The field variables in ([REF]) are [MATH], [MATH] and parametrize an element [MATH] of a group [MATH].', 'hep-th-9904188-2-12-2': 'We also introduce representation matrices [MATH], with [MATH] and the components of the left-invariant Maurer-Cartan forms [MATH].', 'hep-th-9904188-2-12-3': 'The light-cone coordinates on the 2-dim space-time are [MATH], whereas [MATH] denotes the overall coupling constant, which is assumed to be positive.', 'hep-th-9904188-2-12-4': 'Similarly, for ([REF]) the field variables are [MATH], where [MATH], [MATH], parametrize a different group [MATH], whose dimension is, however, equal to that of [MATH].', 'hep-th-9904188-2-12-5': 'Accordingly, we introduce a different set of representation matrices [MATH], with [MATH], and the corresponding components of the left-invariant Maurer-Cartan forms [MATH].', 'hep-th-9904188-2-12-6': 'In ([REF]) and ([REF]), [MATH] is a constant [MATH] matrix, whereas [MATH] and [MATH] are antisymmetric matrices with the same dimension as [MATH], but they depend on the variables [MATH] and [MATH] via the corresponding group elements [MATH] and [MATH].', 'hep-th-9904188-2-12-7': 'They are defined as [CITATION] [EQUATION] where the matrices [MATH], [MATH] are constructed using [EQUATION] and similarly for [MATH] and [MATH].', 'hep-th-9904188-2-12-8': 'Consistency restricts these to obey [EQUATION] and similarly for the tilded ones.', 'hep-th-9904188-2-12-9': 'There is also a bilinear invariant [MATH] with the various generators obeying [EQUATION]', 'hep-th-9904188-2-12-10': 'Finally, we note that the choice of possible groups [MATH] and [MATH] is restricted by the fact that [CITATION] their corresponding Lie algebras must form a pair of maximally isotropic subalgebras into which the Lie algebra of a larger group [MATH], known as the Drinfeld double, can be decomposed [CITATION].', 'hep-th-9904188-2-13-0': 'Let us consider two subgroups [MATH] and [MATH] with [MATH].', 'hep-th-9904188-2-13-1': 'Accordingly we split the Lie-algebra indices as [MATH], where Latin and Greek indices refer to subgroup and coset spaces, respectively.', 'hep-th-9904188-2-13-2': 'Then we may separate the various matrices appearing in ([REF]) and ([REF]) into blocks as [EQUATION] and [EQUATION]', 'hep-th-9904188-2-13-3': 'We would like to take a limit in the model ([REF]) and its dual ([REF]) such that the number of fields [MATH] (and [MATH]) is reduced by [MATH].', 'hep-th-9904188-2-13-4': 'We would call the remaining variables by [MATH] (and [MATH]) with [MATH].', 'hep-th-9904188-2-13-5': 'Consider the limit [EQUATION] in a uniform way for all matrix elements.', 'hep-th-9904188-2-13-6': 'This means that ratios of matrix elements remain constant in this limit.', 'hep-th-9904188-2-13-7': 'Using ([REF]) we find that in the limit ([REF]) [EQUATION]', 'hep-th-9904188-2-13-8': 'Then, the actions ([REF]) and ([REF]) take the form [EQUATION] and [EQUATION]', 'hep-th-9904188-2-13-9': 'Notice that in ([REF]) [MATH] are elements of a [MATH] matrix, whereas in ([REF]) [MATH] are elements of a [MATH] one.', 'hep-th-9904188-2-13-10': 'We have anticipated that the number of variables in ([REF]) and ([REF]) has been reduced to [MATH] upon taking the limit ([REF]).', 'hep-th-9904188-2-13-11': 'However, this does not happen automatically, but depends on whether or not certain conditions, as we will next prove, are fulfilled.', 'hep-th-9904188-2-13-12': 'In order to reduce the dimensionality of ([REF]) we should prove that, after taking the limit ([REF]), a local gauge invariance develops, which suffices to gauge-fix [MATH] degrees of freedom in the actions ([REF]) and similarly for ([REF]).', 'hep-th-9904188-2-13-13': 'For ([REF]) consider the transformation [EQUATION]', 'hep-th-9904188-2-13-14': 'In its infinitesimal form it reads [MATH].', 'hep-th-9904188-2-13-15': 'We may show that this induces the following transformations: [EQUATION]', 'hep-th-9904188-2-13-16': 'Using these and the relation (A.6) of [CITATION], specialized for coset space indices [EQUATION] we may prove that ([REF]) is invariant under the gauge transformation ([REF]), provided that the following condition holds: [EQUATION] or equivalently [EQUATION] where we have denoted by [MATH] and [MATH] the symmetric and antisymmetric parts of the matrix [MATH].', 'hep-th-9904188-2-13-17': 'When the conditions ([REF]) are satisfied then we may gauge-fix [MATH] parameters in the group element [MATH].', 'hep-th-9904188-2-13-18': 'The most efficient way is to parametrize the group element [MATH] as [MATH], where [MATH] and [MATH], and then set [MATH].', 'hep-th-9904188-2-13-19': 'It can be easily seen that this completely fixes the gauge freedom.', 'hep-th-9904188-2-14-0': 'There are [MATH] algebraic conditions in ([REF]) for the [MATH] elements of the matrix [MATH].', 'hep-th-9904188-2-14-1': 'Hence, it is not at all obvious that they can be fulfilled for a general Drinfeld double and then for any arbitrary choice of the subgroup [MATH].', 'hep-th-9904188-2-14-2': 'An obvious simplification occurs when [MATH] is an Abelian group.', 'hep-th-9904188-2-14-3': 'Then [MATH], [MATH] and eq. ([REF]) is solved by [MATH].', 'hep-th-9904188-2-14-4': 'Then ([REF]) with [MATH] takes the form of the usual [MATH]-model action on the coset [MATH] space.', 'hep-th-9904188-2-14-5': 'Accordingly ([REF]) represents its usual non-Abelian dual.', 'hep-th-9904188-2-14-6': 'Hence, when both groups [MATH] and [MATH] are non-Abelian, the models ([REF]) and ([REF]) are deformations of the usual models on coset spaces [MATH] and of their non-Abelian duals.', 'hep-th-9904188-2-15-0': '## Duality-invariant formulation', 'hep-th-9904188-2-16-0': 'We would like to find a duality-invariant action, from which the [MATH]-models ([REF]) and ([REF]) originate.', 'hep-th-9904188-2-16-1': 'It is natural to start with the manifestly Poisson-Lie T-duality-invariant action of [CITATION] from which the [MATH]-models ([REF]) and ([REF]) originate.', 'hep-th-9904188-2-16-2': 'This action is defined in the Drinfeld double as [CITATION], [EQUATION] where [MATH] is the WZW action for a group element [MATH].', 'hep-th-9904188-2-16-3': 'The operator [MATH] is defined as [CITATION] [EQUATION] with [EQUATION] where we have used the notation [MATH] and [MATH].', 'hep-th-9904188-2-16-4': 'In the limit ([REF]) we have [EQUATION]', 'hep-th-9904188-2-16-5': 'Using this and the conditions ([REF]), one can show that ([REF]), in the limit ([REF]), develops the gauge invariance [EQUATION] provided that the following constraint is obeyed [EQUATION]', 'hep-th-9904188-2-16-6': 'In order to avoid introducing this constraint we may use gauge fields instead.', 'hep-th-9904188-2-16-7': 'Indeed, consider the action [EQUATION] where [MATH] takes values in the Lie algebra of [MATH], i.e. [MATH].', 'hep-th-9904188-2-16-8': 'The operator [MATH] is defined as the restriction in [MATH] of the corresponding operator in ([REF]) [EQUATION] where [EQUATION] and [MATH] is the inverse matrix of [MATH].', 'hep-th-9904188-2-16-9': 'It can be shown that ([REF]) is gauge-invariant under ([REF]) and the corresponding transformation for the gauge field [EQUATION] provided that [MATH] is invariant under the similarity transformation [EQUATION]', 'hep-th-9904188-2-16-10': 'In order to prove ([REF]) we first show that [EQUATION] for some [MATH]-dependent matrix [MATH].', 'hep-th-9904188-2-16-11': 'After repeatedly using ([REF]) and a lengthy computation we find that such a matrix exists and is given by [EQUATION] provided that the following condition holds: [EQUATION] or equivalently, splitting into the symmetric and antisymmetric parts, [EQUATION]', 'hep-th-9904188-2-16-12': 'At first sight it seems that ([REF]) is more restrictive than the corresponding conditions in ([REF]), since, unlike ([REF]), they are valid for finite-gauge transformations.', 'hep-th-9904188-2-16-13': 'However, we show in the appendix that ([REF]) actually implies ([REF]).', 'hep-th-9904188-2-17-0': 'In the remainder of this subsection, we consider the classical equations of motion for the (manifestly) duality and gauge-invariant action ([REF]).', 'hep-th-9904188-2-17-1': 'Its variation with respect to all fields is [EQUATION]', 'hep-th-9904188-2-17-2': 'Specializing to subgroup and coset space indices, we find the equations of motion [EQUATION] where we have used also the fact that, because of ([REF]), [MATH].', 'hep-th-9904188-2-17-3': 'Hence, the constraint ([REF]) follows as the equation of motion for [MATH].', 'hep-th-9904188-2-17-4': 'Using ([REF]), the equations of motion in ([REF]) can be cast into the form [MATH]-[MATH].', 'hep-th-9904188-2-17-5': 'These have the same form as the equations of motion for the action ([REF]) [CITATION].', 'hep-th-9904188-2-18-0': 'We finally note that the action ([REF]) is manifestly invariant under the transformation [MATH] for some [MATH]-dependent group element [MATH] [CITATION].', 'hep-th-9904188-2-18-1': 'By introducing gauge fields this symmetry can be promoted into a gauge symmetry with [MATH] a function of [MATH] and [MATH].', 'hep-th-9904188-2-18-2': 'This type of gauge invariance, though interesting enough in its own right to be further investigated, has no apparent relation to the one we have just discussed.', 'hep-th-9904188-2-19-0': '## The canonical transformation', 'hep-th-9904188-2-20-0': 'Poisson-Lie T-duality-related models are canonically equivalent under the transformation [CITATION] [EQUATION]', 'hep-th-9904188-2-20-1': 'This transformation preserves the equal-time Poisson brackets of the conjugate pairs of variables [MATH] and [MATH] given by [CITATION] [EQUATION] and [EQUATION] where [MATH] is the antisymmetric step function that equals [MATH]) for [MATH]).', 'hep-th-9904188-2-20-2': 'Notice that the above Poisson brackets are independent of the details of the [MATH]-models related by Poisson-Lie T-duality.', 'hep-th-9904188-2-20-3': 'They are simply the central extensions, in loop space, of the usual Lie-(bi-)algebras defined in the Drinfeld double.', 'hep-th-9904188-2-20-4': 'One may also show that the Hamiltonians of the two dual actions ([REF]) and ([REF]) are equal [CITATION] as required for canonical transformation with no explicit [MATH]-dependence.', 'hep-th-9904188-2-20-5': 'After some algebraic manipulations, these Hamiltonians can be written as [EQUATION] and [EQUATION] where [MATH] and [MATH] are the symmetric and antisymmetric parts of [MATH] and similarly [MATH] and [MATH] are the symmetric and antisymmetric parts of [MATH]-[MATH].', 'hep-th-9904188-2-20-6': 'Notice that in the limit ([REF]) the conjugate momenta [MATH] vanish.', 'hep-th-9904188-2-20-7': 'This is consistent with the development of a local gauge invariance ([REF]).', 'hep-th-9904188-2-20-8': 'At the level of the Poisson brackets the vanishing of [MATH], together with its conjugate [MATH], has to be imposed as a constraint.', 'hep-th-9904188-2-20-9': 'In fact they form a set [MATH] of second-class constraints.', 'hep-th-9904188-2-20-10': 'We may see that in the limit ([REF]) and upon using ([REF]), the Hamiltonians ([REF]) and ([REF]) reduce to [EQUATION] and [EQUATION]', 'hep-th-9904188-2-20-11': 'We may show, with the help of ([REF]) and ([REF]), that [MATH] (weakly).', 'hep-th-9904188-2-20-12': 'Hence, no new constraints are generated by the time [MATH]-evolution.', 'hep-th-9904188-2-21-0': 'In general (see, for instance, [CITATION]), in the presence of a set of second-class constraints [MATH], one computes the antisymmetric matrix associated with their Poisson brackets [MATH].', 'hep-th-9904188-2-21-1': 'When [MATH] is invertible one simply postulates that the usual Poisson brackets are replaced by Dirac brackets, defined as [EQUATION] for any two phase-space variables [MATH] and [MATH].', 'hep-th-9904188-2-21-2': 'In our case we compute the (infinite-dimensional) matrix [EQUATION] with inverse [EQUATION]', 'hep-th-9904188-2-21-3': 'Then the Dirac brackets can be computed using ([REF]).', 'hep-th-9904188-2-21-4': 'We find (for notational convenience in the rest of the paper, we omit the subscript [MATH] from the Dirac brackets): [EQUATION]', 'hep-th-9904188-2-21-5': 'Notice the parafermionic character of this algebra, which is encoded in the terms containing [MATH].', 'hep-th-9904188-2-21-6': 'The Dirac brackets for the pair [MATH] are obtained from ([REF])-([REF]) by replacing untilded symbols by tilded ones and vice versa.', 'hep-th-9904188-2-21-7': 'It is instructive to write down the Dirac brackets for the case that the group [MATH] is Abelian, i.e. [MATH].', 'hep-th-9904188-2-21-8': 'We find [EQUATION]', 'hep-th-9904188-2-21-9': 'The above Dirac brackets can also be obtained from the ones in ([REF])-([REF]) via a contraction that Abelianizes the group [MATH], i.e. [MATH], [MATH], [MATH].', 'hep-th-9904188-2-22-0': '# An explicit example', 'hep-th-9904188-2-23-0': 'In this section we explicitly demonstrate many of the general aspects developed in section 2, using 3- and 2-dim models related by Poisson-Lie T-duality.', 'hep-th-9904188-2-23-1': 'That includes the explicit construction of the metric and antisymmetric tensor fields, of the Dirac-bracket algebra for canonical equivalence, and also of the corresponding generating functional.', 'hep-th-9904188-2-24-0': '## The Drinfeld double', 'hep-th-9904188-2-25-0': 'Our example will be based on the 6-dim Drinfeld double considered in [CITATION], which we first review by following [CITATION].', 'hep-th-9904188-2-25-1': 'It is just the non-compact group [MATH] with [MATH] and dual [MATH] given by the Iwasawa decomposition of [MATH] [CITATION].', 'hep-th-9904188-2-25-2': 'The associated 3-dim algebras [MATH] and [MATH] have generators denoted by [MATH] and [MATH], where [MATH].', 'hep-th-9904188-2-25-3': 'Leaving aside the details we only present the elements that are necessary in this paper.', 'hep-th-9904188-2-25-4': 'It is convenient to split the index [MATH], [MATH].', 'hep-th-9904188-2-25-5': 'The non-vanishing structure constants for the algebras [MATH] and [MATH] are [EQUATION] where our normalization is such that [MATH].', 'hep-th-9904188-2-25-6': 'We parametrize the [MATH] group element in terms of the three Euler angles [MATH], [MATH] and [MATH].', 'hep-th-9904188-2-25-7': 'It is represented by the [MATH] block-diagonal matrix [EQUATION] where [EQUATION]', 'hep-th-9904188-2-25-8': 'Also the group element of [MATH] is parametrized in terms of three variables [MATH], [MATH] and [MATH] and represented by the following [MATH] block-diagonal matrix [EQUATION] where [EQUATION]', 'hep-th-9904188-2-25-9': 'The Maurer-Cartan forms in the parametrization of the [MATH] group element ([REF]) are [EQUATION]', 'hep-th-9904188-2-25-10': 'Similarly, using the parametrization ([REF]) for the [MATH] group element we find [EQUATION]', 'hep-th-9904188-2-25-11': 'The antisymmetric matrices [MATH] and [MATH] are [EQUATION] and [EQUATION]', 'hep-th-9904188-2-26-0': '## Explicit three- and two-dimensional models', 'hep-th-9904188-2-27-0': 'Consider the [MATH]-model action ([REF]) for the case of our double based on [MATH].', 'hep-th-9904188-2-27-1': 'Let us single-out the 1-dim subgroup [MATH] that is generated by [MATH].', 'hep-th-9904188-2-27-2': 'For our purposes it will be sufficient to use the following form for the [MATH] matrix [MATH]-[MATH] [EQUATION] where we have kept the conventions of ([REF]) for the enumeration of the matrix elements.', 'hep-th-9904188-2-27-3': 'Using ([REF]), ([REF]) and ([REF]), it is the easy to compute the metric and antisymmetric tensor fields corresponding to ([REF]).', 'hep-th-9904188-2-27-4': 'We find a metric given by [EQUATION] and an antisymmetric tensor given by [EQUATION] where [EQUATION]', 'hep-th-9904188-2-27-5': 'Notice that the antisymmetric tensor can be (locally) gauged away since the corresponding 3-form field strength is zero.', 'hep-th-9904188-2-27-6': 'Also, for our purposes, we will not need the explicit expressions for the metric and antisymmetric tensor corresponding to the dual [MATH]-model ([REF]).', 'hep-th-9904188-2-27-7': 'For [MATH], but general [MATH] and [MATH], the above example (with its dual) was considered in [CITATION] (also in [CITATION] for [MATH] and [MATH]).', 'hep-th-9904188-2-28-0': 'We would like to take the analogue of the limit ([REF]).', 'hep-th-9904188-2-28-1': 'It is clear that in our case this corresponds to letting [MATH].', 'hep-th-9904188-2-28-2': 'Comparing ([REF]) to ([REF]) we see that the [MATH] matrix [MATH] is [EQUATION]', 'hep-th-9904188-2-28-3': 'It is easily seen that this is the most general [MATH] matrix that solves ([REF]), with structure constants given by ([REF]).', 'hep-th-9904188-2-28-4': 'In agreement with our general discussion, the [MATH]-model action with metric ([REF]) and antisymmetric tensor ([REF]) develops a local invariance under the transformation [EQUATION]', 'hep-th-9904188-2-28-5': 'This allows to gauge-fix the variable [MATH].', 'hep-th-9904188-2-28-6': 'Explicitly computing ([REF]) we find that the metric and antisymmetric tensors are given by [EQUATION]', 'hep-th-9904188-2-28-7': 'Equivalently, the same result follows if we set [MATH] directly into the expressions for the metric ([REF]) and antisymmetric ([REF]) tensors.', 'hep-th-9904188-2-28-8': 'Similarly, the dual model action ([REF]) is invariant under the local transformation [EQUATION]', 'hep-th-9904188-2-28-9': 'Hence, we may evaluate ([REF]) in the gauge [MATH].', 'hep-th-9904188-2-28-10': 'The corresponding metric (the antisymmetric tensor turns out to be zero) is found to be [EQUATION] where we have changed variables as [MATH]', 'hep-th-9904188-2-29-0': 'and [MATH].', 'hep-th-9904188-2-29-1': 'The metric in ([REF]) is free of singularities (since ([REF]) has no fixed point) and represents a deformed 2-sphere.', 'hep-th-9904188-2-29-2': 'In contrast, ([REF]) is singular for [MATH].', 'hep-th-9904188-2-29-3': 'This is related to the fact that [MATH] is a fixed point of the gauge transformation ([REF]).', 'hep-th-9904188-2-29-4': 'The singularity at [MATH] is only a coordinate singularity and can be removed by an appropriate change of variables.', 'hep-th-9904188-2-30-0': 'It is worth while to consider some analytic continuations of the models ([REF]) and its dual ([REF]).', 'hep-th-9904188-2-30-1': 'If we let [MATH], where [MATH], and also we change the sign of the overall coupling constant [MATH], then ([REF]) becomes [EQUATION]', 'hep-th-9904188-2-30-2': 'The corresponding analytic continuation in the dual metric ([REF]) should be [MATH], with a parallel change of sign in the overall coupling constant.', 'hep-th-9904188-2-30-3': 'The metric in ([REF]) is reduced to the Euclidean [MATH] metric if we rescale the coupling constant [MATH] and then take the limit [MATH] (keeping the new coupling finite).', 'hep-th-9904188-2-30-4': 'However, for generic values of the constant [MATH], it represents a space that is topologically a cigar.', 'hep-th-9904188-2-30-5': 'Indeed, for [MATH] we get the 2-dim Euclidean space [MATH] in polar coordinates, whereas for [MATH] we get, after an appropriate change of variables, [MATH].', 'hep-th-9904188-2-30-6': 'For [MATH], the cigar-shaped space develops a "pump" corresponding to the maximum of the metric components [MATH] at [EQUATION]', 'hep-th-9904188-2-30-7': 'We note that the cigar-shape topology is also a characteristic of the Euclidean black hole corresponding to the coset [MATH] exact conformal field theory [CITATION].', 'hep-th-9904188-2-30-8': 'However, in our case the model ([REF]) is not conformal.', 'hep-th-9904188-2-31-0': 'The Drinfeld double for ([REF]) and its dual model is [MATH], with [MATH], instead of [MATH].', 'hep-th-9904188-2-32-0': '## The Dirac brackets and the generating functional', 'hep-th-9904188-2-33-0': 'The Dirac brackets for the conjugate variables in our example are most easily written down in the basis [MATH] and [MATH], where the non-zero structure constants are [MATH], [MATH] and [MATH].', 'hep-th-9904188-2-33-1': 'Using ([REF])-([REF]) we obtain [EQUATION] where the underlined terms should be omitted in the Abelian limit of the dual group [MATH].', 'hep-th-9904188-2-33-2': 'In this case the above algebra provides a canonical equivalence between the [MATH]-model for [MATH] and its non-Abelian dual with respect to the left (or right) action of [MATH].', 'hep-th-9904188-2-33-3': 'Note also that the generators [MATH] form a subalgebra ([REF]).', 'hep-th-9904188-2-34-0': 'The generating functional that demonstrates the classical equivalence between [MATH]-models related by Poisson-Lie T-duality based on our Drinfeld double was explicitly constructed in [CITATION].', 'hep-th-9904188-2-34-1': 'In a slightly different form than that in [CITATION], it reads [EQUATION]', 'hep-th-9904188-2-34-2': 'Notice that the above generating functional depends only on the combination [MATH]; it is therefore invariant under the [MATH] gauge transformation [MATH] and [MATH].', 'hep-th-9904188-2-34-3': 'The generating functional for the deformed coset models ([REF]) is obtained by solving the equation [MATH] (equivalently [MATH]) for [MATH] and inserting the result back into ([REF]).', 'hep-th-9904188-2-34-4': 'The result is a generating functional, which is non-polynomial in derivatives with respect to [MATH].', 'hep-th-9904188-2-34-5': 'The obtained expressions are quite complicated and not very illustrative, so that we decided to present the corresponding result for the [MATH]-model for [MATH] and its non-Abelian dual.', 'hep-th-9904188-2-34-6': 'We start with the generating functional corresponding to the 2-dim [MATH]-models for [MATH] and its non-Abelian dual with respect to the left (or right) action of [MATH] that was obtained in [CITATION].', 'hep-th-9904188-2-34-7': 'In our notation it is given by [MATH].', 'hep-th-9904188-2-34-8': 'This is easily modified to depend on the angles [MATH] and [MATH] only through the combination [MATH], by adding the term [MATH].', 'hep-th-9904188-2-34-9': 'Such a term, being dependent on the variables of only one of the dual models, can be absorbed as total derivative into the corresponding action and hence it does not affect the classical dynamics.', 'hep-th-9904188-2-34-10': 'Explicitly, the resulting generating functional is [EQUATION]', 'hep-th-9904188-2-34-11': 'The variation of [MATH] with respect to [MATH] gives [EQUATION]', 'hep-th-9904188-2-34-12': 'Substituting back into ([REF]) we obtain [EQUATION] where [MATH] is given by ([REF]).', 'hep-th-9904188-2-34-13': 'The generating functional ([REF]) is non-polynomial in the derivatives of the fields with respect to [MATH].', 'hep-th-9904188-2-34-14': 'In that sense it belongs to a new class of generating functionals, which depend not only on the fields of the two dual [MATH]-models, but also on their first derivatives with respect to the space-like variable in a non-trivial way.', 'hep-th-9904188-2-34-15': 'For comparison, up to now, either in the case of non-Abelian duality [CITATION] or for Poisson-Lie T-duality (and its possible generalizations) [CITATION], there was no dependence of the generating functional on more than the first power of these derivatives (see, for example ([REF])).', 'hep-th-9904188-2-34-16': 'Finally, we note that, according to the work in [CITATION], generating functionals of the form ([REF]), being non-linear, are expected to receive quantum corrections.', 'hep-th-9904188-2-34-17': 'Consequently, the corresponding duality rules relating the 2-dim field theories, as well as the algebra ([REF])-([REF]), are expected to be quantum-corrected.', 'hep-th-9904188-2-35-0': '# Renormalization group flow', 'hep-th-9904188-2-36-0': 'In this section we study the 1-loop RG equations corresponding to the three-dimensional model ([REF]), ([REF]).', 'hep-th-9904188-2-36-1': 'We will show that there are no fixed points in the flow and also that the correct description of the models is a non-perturbative one.', 'hep-th-9904188-2-36-2': 'However, for large domains in parameter space and for a wide range of energies in the UV, the description is effectively perturbative and the model becomes a two-dimensional one.', 'hep-th-9904188-2-37-0': 'Finally, by performing some analytic continuations we will find three- and two-dimensional models with fixed points under the RG flow, where the theory becomes free.', 'hep-th-9904188-2-38-0': 'We begin this section with a short review of RG flow in 2-dim field theories with curved target spaces.', 'hep-th-9904188-2-38-1': 'and ([REF]).', 'hep-th-9904188-2-38-2': 'The 2-dim [MATH]-model corresponding to the metric ([REF]) and antisymmetric tensor ([REF]) is of the form [EQUATION]', 'hep-th-9904188-2-38-3': "It will be renormalizable if the corresponding counter-terms, at a given order in a loop expansion, can be absorbed into a renormalization of the coupling constant [MATH] and (or) of some parameters labelled collectively by [MATH], [MATH] In addition, we allow for general field redefinitions of the [MATH]'s, which are coordinate reparametrizations in the target space.", 'hep-th-9904188-2-38-4': 'This definition of renormalizability of [MATH]-models is quite strict and similar to that for ordinary field theories.', 'hep-th-9904188-2-38-5': 'A natural extension of this is to allow for the manifold to vary with the mass scale and the RG to act in the infinite-dimensional space of all metrics and torsions [CITATION].', 'hep-th-9904188-2-38-6': 'Further discussion of this generalized renormalizability will not be needed for our purposes.', 'hep-th-9904188-2-38-7': 'Perturbatively, in powers of [MATH], we express the bare quantities, denoted by a zero as a subscript, as [EQUATION]', 'hep-th-9904188-2-38-8': 'The ellipses stand for higher-order loop- and pole-terms in [MATH] and [MATH] respectively.', 'hep-th-9904188-2-38-9': 'Then, the beta-functions up to one loop are given by [MATH] and [MATH], where, as usual, [MATH], [MATH] and [MATH].', 'hep-th-9904188-2-39-0': 'The equations to be satisfied by appropriately choosing [MATH] and [MATH] are given by [EQUATION] where [MATH] are the components of the "generalized" Ricci tensor defined with a connection that includes the torsion, i.e. with [MATH].', 'hep-th-9904188-2-39-1': 'The corresponding counter-terms were computed in the dimensional regularization scheme (see, for instance, [CITATION]).', 'hep-th-9904188-2-40-0': '## Models with no fixed points', 'hep-th-9904188-2-41-0': '### Three-dimensional models', 'hep-th-9904188-2-42-0': 'In the metric ([REF]) there are three parameters [MATH], [MATH] and [MATH] and the three Euler angles [MATH] and [MATH] will be denoted by [MATH].', 'hep-th-9904188-2-42-1': 'Also for the antisymmetric tensor in ([REF]) we have [MATH].', 'hep-th-9904188-2-42-2': 'Examining ([REF]) we find that the coupling [MATH] and the coordinates [MATH] do not renormalize and therefore the corresponding beta-functions are zero.', 'hep-th-9904188-2-42-3': 'In contrast, for the parameters [MATH] and [MATH] we find [EQUATION]', 'hep-th-9904188-2-42-4': 'This system of coupled non-linear equations can be considerably simplified.', 'hep-th-9904188-2-42-5': 'First, using ([REF]), we may easily show that there is a RG-flow-invariant defined as [EQUATION] which implies that [EQUATION]', 'hep-th-9904188-2-42-6': 'Without loss of generality we may assume that [MATH] since ([REF]) remains invariant under [MATH] and [MATH].', 'hep-th-9904188-2-42-7': 'Then, using the last two equations in ([REF]) we may derive an equation for [MATH] as a function of [MATH] whose solution is [EQUATION] where [MATH] is a real constant, which is determined by the initial conditions for [MATH] and [MATH].', 'hep-th-9904188-2-42-8': 'The sign in front of the square root in ([REF]) is changed when [MATH], in order to ensure the continuity of [MATH] as a function of the energy scale [MATH].', 'hep-th-9904188-2-43-0': 'Hence, the only differential equation we still have to solve is the one for [MATH], which, after using ([REF]), takes the form [EQUATION] where [MATH] and [MATH] are determined by ([REF]) and ([REF]).', 'hep-th-9904188-2-43-1': 'Since the RG equations are real, [MATH] will stay strictly non-negative and therefore [MATH] will oscillate with [MATH] between its minimum and maximum values [MATH] and [MATH], where [MATH].', 'hep-th-9904188-2-43-2': 'When [MATH], for finite values of the overall coupling constant [MATH], the curvature for the metric ([REF]) approaches infinity and the perturbative expansion of the RG equations becomes meaningless.', 'hep-th-9904188-2-44-0': 'We have seen that the correct description of the theory is a genuine non-perturbative one.', 'hep-th-9904188-2-44-1': 'Neverthelss, for [MATH] we will show that there exists a wide range of energies in the UV, where the description is effectively perturbative.', 'hep-th-9904188-2-44-2': 'Moreover, there exists a fixed point at [MATH] where the theory has effectively a 2-dim target space.', 'hep-th-9904188-2-45-0': 'Indeed, using ([REF]), we have that [MATH] when [MATH].', 'hep-th-9904188-2-45-1': 'Hence, in that limit and after redefining [MATH] we may simplify the RG eqs. ([REF]) as [EQUATION]', 'hep-th-9904188-2-45-2': 'Then the metric ([REF]) becomes [EQUATION] which is the deformed [MATH] Principal Chiral model considered in [CITATION].', 'hep-th-9904188-2-45-3': 'Also the first two of the above equations are those derived in [CITATION] for the corresponding coupling [MATH] and deformation parameter [MATH].', 'hep-th-9904188-2-45-4': 'In the UV the solution of ([REF]) is [EQUATION]', 'hep-th-9904188-2-45-5': 'Hence, in the UV [MATH].', 'hep-th-9904188-2-45-6': 'Therefore if the condition [EQUATION] is fulfilled, then [MATH] and the model is indeed described perturbatively by ([REF]).', 'hep-th-9904188-2-45-7': 'The point [MATH] is a UV-fixed point, where the metric ([REF]) becomes [MATH].', 'hep-th-9904188-2-45-8': 'However, outside the validity of ([REF]) the correct description is non-perturbative.', 'hep-th-9904188-2-46-0': '### Two-dimensional models', 'hep-th-9904188-2-47-0': 'Let us now return to the 2-dim models ([REF]) and ([REF]).', 'hep-th-9904188-2-47-1': 'As before, there is no wave-function renormalization for [MATH] and [MATH], and the beta-function for the coupling [MATH] is zero.', 'hep-th-9904188-2-47-2': 'For the couplings [MATH] and [MATH] the corresponding beta-functions can be obtained by simply setting [MATH] into ([REF]).', 'hep-th-9904188-2-47-3': 'The reason why such a procedure is consistent seems to be intimately related to the local invariance that reduces the 3-dim models into 2-dim ones.', 'hep-th-9904188-2-47-4': 'Hence, we have [EQUATION] which are nothing but the beta-functions for the 2-dim model ([REF]) as well as for its dual ([REF]).', 'hep-th-9904188-2-47-5': 'This is a strong hint that their classical equivalence can be promoted into a quantum one as well.', 'hep-th-9904188-2-47-6': 'Having said that we note, once again, that [MATH] is not a fixed point of the ([REF]) in the UV.', 'hep-th-9904188-2-47-7': 'Since ([REF]) is still a RG invariant of ([REF]), it is clear that one variable between [MATH] and [MATH] is an independent one.', 'hep-th-9904188-2-47-8': 'Eliminating [MATH] from ([REF]) using ([REF]), we obtain [EQUATION]', 'hep-th-9904188-2-47-9': 'Hence, the solution for [MATH] as a function of the energy scale [MATH] oscillates between [MATH] and [MATH] as [EQUATION] where [MATH] is an arbitrary reference scale.', 'hep-th-9904188-2-47-10': 'This means that the corresponding [MATH]-model actions do not define local field theories and can be considered at most as effective actions for scales such that [MATH] stays away from [MATH].', 'hep-th-9904188-2-48-0': 'The usual [MATH] metric and its non-Abelian dual with respect to the right (or left) action of [MATH] are obtained from ([REF]) and ([REF]) if we rescale the coupling constant [MATH] and then take the limit [MATH] (keeping the new coupling finite).', 'hep-th-9904188-2-48-1': 'However, this limit is problematic at the quantum level since the corresponding [MATH]-functions do not tend to the beta-function obtained by studying the 2-dim field theories based on [MATH] (and its non-Abelian dual) by themselves [CITATION].', 'hep-th-9904188-2-48-2': 'The latter is, at one-loop, just [MATH] and is consistent with the fact that these models are asymptotically free.', 'hep-th-9904188-2-48-3': 'It is formally obtained by the first of ([REF]) in the limit [MATH] after we rescale [MATH] as described above.', 'hep-th-9904188-2-48-4': 'This limit does not correspond to any fixed point of ([REF]).', 'hep-th-9904188-2-48-5': 'It is easily seen that, from a RG theory view point, these models offer an effective description of the more general models ([REF]) and ([REF]) in the case of [MATH], which, according to ([REF]), occurs at scales [MATH].', 'hep-th-9904188-2-49-0': '## Models with fixed points', 'hep-th-9904188-2-50-0': '### Three-dimensional models', 'hep-th-9904188-2-51-0': 'We have seen that our model ([REF]), ([REF]) does not have a true fixed point under the 1-loop RG eqs. ([REF]).', 'hep-th-9904188-2-51-1': 'Consider, however, the analytic continuation [MATH] and [MATH].', 'hep-th-9904188-2-51-2': 'Then the metric and antisymmetric tensors become [EQUATION] and [EQUATION] where instead of ([REF]) the function [MATH] is given by [EQUATION]', 'hep-th-9904188-2-51-3': 'The fact that the antisymmetric tensor is imaginary is bothersome if we want to describe models in 2-dim Minkowskian space-times.', 'hep-th-9904188-2-51-4': 'However, for Euclidean ones, the (locally) exact 2-form measures the charge of non-trivial instanton-like configurations.', 'hep-th-9904188-2-51-5': 'The perturbative expansion is completely independent of the antisymmetric tensor, but this will definitely play a role in a, yet lacking, non-perturbative formulation of the model.', 'hep-th-9904188-2-51-6': 'The 1-loop RG equations for the metric ([REF]) are obtained from ([REF]) by the analytic continuation we have described above.', 'hep-th-9904188-2-51-7': 'Then the analogue of ([REF]) is given by [EQUATION] where now [EQUATION] and [MATH] is still given by ([REF]).', 'hep-th-9904188-2-51-8': 'As before, we will assume that [MATH] with no loss of generality.', 'hep-th-9904188-2-51-9': 'However, now [MATH] does not have to be larger than or equal to 1, as in ([REF]), in order to ensure reality for [MATH].', 'hep-th-9904188-2-51-10': 'If [MATH] then [MATH] are complex conjugate of each other and, unlike the case when they are real, [MATH] can take any real value without spoiling the reality of the parameter [MATH].', 'hep-th-9904188-2-51-11': 'However, now the condition [MATH] has to be fulfilled in order for [MATH] to remain real.', 'hep-th-9904188-2-51-12': 'If on the other hand [MATH], then [MATH] are both real and the reality condition for [MATH] requires that [MATH] or [MATH].', 'hep-th-9904188-2-51-13': 'Since [MATH], it turns out that there are fixed points for initial conditions where [MATH] is less than [MATH].', 'hep-th-9904188-2-51-14': 'Consider first the RG eq. ([REF]) near the point with [MATH], [MATH] and [MATH].', 'hep-th-9904188-2-51-15': 'It can be written as (we take the lower sign in ([REF])): [EQUATION]', 'hep-th-9904188-2-51-16': 'The same equation near the different point with [MATH], [MATH] and [MATH] takes the form [EQUATION]', 'hep-th-9904188-2-51-17': 'For [MATH] we have [MATH].', 'hep-th-9904188-2-51-18': 'Hence, for [MATH] we have an IR-stable point at [MATH] as well as a UV-stable point at [MATH].', 'hep-th-9904188-2-52-0': 'For [MATH], we have that [MATH].', 'hep-th-9904188-2-52-1': 'Therefore, for [MATH] there are two UV-stable points at [MATH] and at [MATH].', 'hep-th-9904188-2-53-0': 'In all cases the background ([REF]), ([REF]) flows, either in the IR or in the UV, towards the background with [EQUATION] where [MATH] represents any of the two fixed points [MATH] or [MATH].', 'hep-th-9904188-2-53-1': 'This represents a free theory, as can be seen by changing variables as [MATH].', 'hep-th-9904188-2-53-2': 'It is interesting to note that in the case [MATH] the signature of the metric in ([REF]) is [MATH] in the IR fixed point [MATH] and [MATH] in the UV fixed point [MATH].', 'hep-th-9904188-2-54-0': 'Also in the case of [MATH] the signature at the [MATH] UV-stable point is [MATH], but in the other UV-stable point at [MATH] it is [MATH].', 'hep-th-9904188-2-54-1': 'Hence, only at [MATH] the metric has Euclidean signature and we expect a well-defined field-theoretical description.', 'hep-th-9904188-2-55-0': 'Let us also note that for [MATH] the RG flow is described, as before, by ([REF]), ([REF]) and the corresponding [MATH]-model is again ([REF]), provided ([REF]) is satisfied.', 'hep-th-9904188-2-56-0': '### Two-dimensional models', 'hep-th-9904188-2-57-0': 'Now we turn to the 2-dim model ([REF]) after the same analytic continuation as before, [MATH] and [MATH]: [EQUATION]', 'hep-th-9904188-2-57-1': 'The 1-loop RG equation corresponding to ([REF]) is [EQUATION]', 'hep-th-9904188-2-57-2': 'The form of the solution for [MATH] as a function of the energy scale [MATH] depends on whether or not [MATH] is smaller or larger than 1.', 'hep-th-9904188-2-57-3': 'We find [EQUATION] where [MATH] denotes again an arbitrary reference scale.', 'hep-th-9904188-2-57-4': 'We see that in the UV there is a fixed point at [MATH] (and [MATH]).', 'hep-th-9904188-2-57-5': 'The lower bound for [MATH] above is needed for [MATH] to stay positive, since only then is ([REF]) a solution of ([REF]).', 'hep-th-9904188-2-57-6': 'For the case of [MATH], we have to distinguish the solutions between those with [MATH] and those with [MATH].', 'hep-th-9904188-2-57-7': 'In the former case we obtain [EQUATION] and [EQUATION] where, as before, [MATH] and [MATH] are arbitrary reference scales.', 'hep-th-9904188-2-57-8': 'For the trajectory given by ([REF]), [MATH] stays positive.', 'hep-th-9904188-2-57-9': 'It starts at [MATH] for [MATH], and ends at [MATH] for [MATH].', 'hep-th-9904188-2-57-10': 'For the trajectory given by ([REF]), [MATH] is always negative and starts at [MATH] for [MATH] and ends at [MATH] for [MATH].', 'hep-th-9904188-2-57-11': 'Hence, we see that [MATH] is a UV fixed point.', 'hep-th-9904188-2-57-12': 'Also as we lower the scale [MATH] towards the IR, the solution becomes singular in both cases.', 'hep-th-9904188-2-57-13': 'In any case, we then run into non-perturbative regimes.', 'hep-th-9904188-2-57-14': 'For trajectories in the region [MATH], the solution is still given by ([REF]), but with [MATH].', 'hep-th-9904188-2-57-15': 'In the lower limit [MATH] and in the upper limit [MATH].', 'hep-th-9904188-2-57-16': 'Hence in that case we have a singular behaviour of the 1-loop RG equations towards the IR as well as the UV.', 'hep-th-9904188-2-57-17': 'As we have mentioned, in those cases the corresponding 2-dim field theory is not well defined at the quantum level and can be considered only as an effective field theory at scales away from the singularities.', 'hep-th-9904188-2-58-0': 'The 2-dim model corresponding to the fixed point at [MATH] is obtained by setting [MATH] in ([REF]) [EQUATION]', 'hep-th-9904188-2-58-1': 'The fact that ([REF]) approaches a free-field conformal field theory at the fixed point is similar to the case of an integrable model (different from ([REF])), representing also a 1-parameter deformation of [MATH], that was considered in [CITATION].', 'hep-th-9904188-2-58-2': 'It is interesting to investigate whether or not ([REF]) represents also an integrable perturbation of [MATH].', 'hep-th-9904188-2-59-0': '# Concluding remarks', 'hep-th-9904188-2-60-0': 'We have constructed a new class of 2-dim field theories with target spaces corresponding to deformations of coset spaces [MATH].', 'hep-th-9904188-2-60-1': 'Our models correspond to special points of the classical moduli space of models related by Poisson-Lie T-duality, where a local invariance develops.', 'hep-th-9904188-2-60-2': 'A classification of all possible models that arise with such a procedure is an interesting open problem and can be done by analyzing the general conditions ([REF]), or equivalently ([REF]).', 'hep-th-9904188-2-60-3': 'By construction these models come in dual pairs.', 'hep-th-9904188-2-60-4': 'The corresponding generating functionals depend non-polynomially on the derivatives of the fields with respect to the space-like variable.', 'hep-th-9904188-2-60-5': 'The latter feature is also manifested in an underlying infinite-dimensional algebra with a central extension of the parafermionic type.', 'hep-th-9904188-2-60-6': 'It would also be interesting to uncover the relation of our models to those in [CITATION].', 'hep-th-9904188-2-61-0': 'We have also performed a quite general RG flow analysis using specific models with 3- and 2-dim target spaces.', 'hep-th-9904188-2-61-1': 'As in [CITATION], we conclude that quantum aspects of the lower dimensional models do not necessarily follow by taking the same classical limit as that used to relate the corresponding 2-dim field-theoretical classical actions.', 'hep-th-9904188-2-61-2': 'Concretely, the beta-function equations for the lower-dimensional models follow from those of the original models by just setting some parameters to their prescribed values (see ([REF]) and ([REF])).', 'hep-th-9904188-2-61-3': 'However, these values do not necessarily correspond to any fixed points of the solutions of these equations.', 'hep-th-9904188-2-61-4': 'Using our 3-dim example we saw that in a large domain in parameter space, and for a wide range of energies in the UV, the description is effectively perturbative with a UV-fixed point exactly where the local gauge invariance develops.', 'hep-th-9904188-2-62-0': 'We believe that this feature will persist for more general models related by Poisson-Lie T-duality.', 'hep-th-9904188-2-62-1': 'In that respect it would be very interesting to study the RG flow in general using ([REF]) and ([REF]) and possibly to formulate this flow in a duality-invariant way.'}

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'hep-th-9904188-2-51-10'], ['hep-th-9904188-1-51-11', 'hep-th-9904188-2-51-11'], ['hep-th-9904188-1-51-12', 'hep-th-9904188-2-51-12'], ['hep-th-9904188-1-51-13', 'hep-th-9904188-2-51-13'], ['hep-th-9904188-1-51-14', 'hep-th-9904188-2-51-14'], ['hep-th-9904188-1-51-15', 'hep-th-9904188-2-51-15'], ['hep-th-9904188-1-51-16', 'hep-th-9904188-2-51-16'], ['hep-th-9904188-1-51-17', 'hep-th-9904188-2-51-17'], ['hep-th-9904188-1-51-18', 'hep-th-9904188-2-51-18'], ['hep-th-9904188-1-52-0', 'hep-th-9904188-2-52-0'], ['hep-th-9904188-1-52-1', 'hep-th-9904188-2-52-1'], ['hep-th-9904188-1-57-0', 'hep-th-9904188-2-57-0'], ['hep-th-9904188-1-57-1', 'hep-th-9904188-2-57-1'], ['hep-th-9904188-1-57-2', 'hep-th-9904188-2-57-2'], ['hep-th-9904188-1-57-3', 'hep-th-9904188-2-57-3'], ['hep-th-9904188-1-57-4', 'hep-th-9904188-2-57-4'], ['hep-th-9904188-1-57-5', 'hep-th-9904188-2-57-5'], ['hep-th-9904188-1-57-6', 'hep-th-9904188-2-57-6'], ['hep-th-9904188-1-57-7', 'hep-th-9904188-2-57-7'], ['hep-th-9904188-1-57-8', 'hep-th-9904188-2-57-8'], ['hep-th-9904188-1-57-9', 'hep-th-9904188-2-57-9'], ['hep-th-9904188-1-57-10', 'hep-th-9904188-2-57-10'], ['hep-th-9904188-1-57-11', 'hep-th-9904188-2-57-11'], ['hep-th-9904188-1-57-12', 'hep-th-9904188-2-57-12'], ['hep-th-9904188-1-57-13', 'hep-th-9904188-2-57-13'], ['hep-th-9904188-1-57-14', 'hep-th-9904188-2-57-14'], ['hep-th-9904188-1-57-15', 'hep-th-9904188-2-57-15'], ['hep-th-9904188-1-57-16', 'hep-th-9904188-2-57-16'], ['hep-th-9904188-1-57-17', 'hep-th-9904188-2-57-17'], ['hep-th-9904188-1-39-0', 'hep-th-9904188-2-39-0'], ['hep-th-9904188-1-39-1', 'hep-th-9904188-2-39-1'], ['hep-th-9904188-1-42-0', 'hep-th-9904188-2-42-0'], ['hep-th-9904188-1-42-1', 'hep-th-9904188-2-42-1'], ['hep-th-9904188-1-42-2', 'hep-th-9904188-2-42-2'], ['hep-th-9904188-1-42-3', 'hep-th-9904188-2-42-3'], ['hep-th-9904188-1-42-4', 'hep-th-9904188-2-42-4'], ['hep-th-9904188-1-42-5', 'hep-th-9904188-2-42-5'], ['hep-th-9904188-1-42-6', 'hep-th-9904188-2-42-6'], ['hep-th-9904188-1-42-7', 'hep-th-9904188-2-42-7'], ['hep-th-9904188-1-42-8', 'hep-th-9904188-2-42-8'], ['hep-th-9904188-1-20-0', 'hep-th-9904188-2-20-0'], ['hep-th-9904188-1-20-1', 'hep-th-9904188-2-20-1'], ['hep-th-9904188-1-20-2', 'hep-th-9904188-2-20-2'], ['hep-th-9904188-1-20-3', 'hep-th-9904188-2-20-3'], ['hep-th-9904188-1-20-4', 'hep-th-9904188-2-20-4'], ['hep-th-9904188-1-20-5', 'hep-th-9904188-2-20-5'], ['hep-th-9904188-1-20-6', 'hep-th-9904188-2-20-6'], ['hep-th-9904188-1-20-7', 'hep-th-9904188-2-20-7'], ['hep-th-9904188-1-20-8', 'hep-th-9904188-2-20-8'], ['hep-th-9904188-1-20-9', 'hep-th-9904188-2-20-9'], ['hep-th-9904188-1-20-10', 'hep-th-9904188-2-20-10'], ['hep-th-9904188-1-20-11', 'hep-th-9904188-2-20-11'], ['hep-th-9904188-1-20-12', 'hep-th-9904188-2-20-12'], ['hep-th-9904188-1-58-0', 'hep-th-9904188-2-58-0'], ['hep-th-9904188-1-58-1', 'hep-th-9904188-2-58-1'], ['hep-th-9904188-1-58-2', 'hep-th-9904188-2-58-2'], ['hep-th-9904188-1-47-0', 'hep-th-9904188-2-47-0'], ['hep-th-9904188-1-47-1', 'hep-th-9904188-2-47-1'], ['hep-th-9904188-1-47-2', 'hep-th-9904188-2-47-2'], ['hep-th-9904188-1-47-3', 'hep-th-9904188-2-47-3'], ['hep-th-9904188-1-47-4', 'hep-th-9904188-2-47-4'], ['hep-th-9904188-1-47-5', 'hep-th-9904188-2-47-5'], ['hep-th-9904188-1-47-6', 'hep-th-9904188-2-47-6'], ['hep-th-9904188-1-47-7', 'hep-th-9904188-2-47-7'], ['hep-th-9904188-1-47-8', 'hep-th-9904188-2-47-8'], ['hep-th-9904188-1-47-9', 'hep-th-9904188-2-47-9'], ['hep-th-9904188-1-47-10', 'hep-th-9904188-2-47-10'], ['hep-th-9904188-1-62-0', 'hep-th-9904188-2-62-0'], ['hep-th-9904188-1-62-1', 'hep-th-9904188-2-62-1'], ['hep-th-9904188-1-61-0', 'hep-th-9904188-2-61-0'], ['hep-th-9904188-1-61-1', 'hep-th-9904188-2-61-1'], ['hep-th-9904188-1-61-2', 'hep-th-9904188-2-61-2'], ['hep-th-9904188-1-61-3', 'hep-th-9904188-2-61-3'], ['hep-th-9904188-1-61-4', 'hep-th-9904188-2-61-4'], ['hep-th-9904188-1-44-0', 'hep-th-9904188-2-44-0'], ['hep-th-9904188-1-44-1', 'hep-th-9904188-2-44-1'], ['hep-th-9904188-1-44-2', 'hep-th-9904188-2-44-2'], ['hep-th-9904188-1-45-0', 'hep-th-9904188-2-45-0'], ['hep-th-9904188-1-45-1', 'hep-th-9904188-2-45-1'], ['hep-th-9904188-1-45-2', 'hep-th-9904188-2-45-2'], ['hep-th-9904188-1-45-3', 'hep-th-9904188-2-45-3'], ['hep-th-9904188-1-45-4', 'hep-th-9904188-2-45-4'], ['hep-th-9904188-1-45-5', 'hep-th-9904188-2-45-5'], ['hep-th-9904188-1-45-6', 'hep-th-9904188-2-45-6'], ['hep-th-9904188-1-45-7', 'hep-th-9904188-2-45-7'], ['hep-th-9904188-1-45-8', 'hep-th-9904188-2-45-8'], ['hep-th-9904188-1-23-0', 'hep-th-9904188-2-23-0'], ['hep-th-9904188-1-23-1', 'hep-th-9904188-2-23-1'], ['hep-th-9904188-1-0-0', 'hep-th-9904188-2-0-0'], ['hep-th-9904188-1-0-1', 'hep-th-9904188-2-0-1'], ['hep-th-9904188-1-0-2', 'hep-th-9904188-2-0-2'], ['hep-th-9904188-1-0-3', 'hep-th-9904188-2-0-3']]

[]

[]

[]

[]

['hep-th-9904188-1-2-0', 'hep-th-9904188-1-3-0', 'hep-th-9904188-1-4-0', 'hep-th-9904188-1-14-3', 'hep-th-9904188-1-21-8', 'hep-th-9904188-1-29-0', 'hep-th-9904188-1-38-1', 'hep-th-9904188-2-2-0', 'hep-th-9904188-2-3-0', 'hep-th-9904188-2-4-0', 'hep-th-9904188-2-14-3', 'hep-th-9904188-2-21-8', 'hep-th-9904188-2-29-0', 'hep-th-9904188-2-38-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-th/9904188

1808.05001

{'1808.05001-1-0-0': 'We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature.', '1808.05001-1-0-1': 'When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor.', '1808.05001-1-0-2': 'In the general case, the Weyl- type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry.', '1808.05001-1-0-3': 'We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics.', '1808.05001-1-0-4': 'Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function.', '1808.05001-1-0-5': 'This way we provide a Finslerian version of Beltrami Theorem.', '1808.05001-1-1-0': '[2000]53C60, 53B40, 58E30, 49N45', '1808.05001-1-2-0': '# Introduction', '1808.05001-1-3-0': 'For a Riemannian manifold of dimension greater than or equal to [MATH], the projective Weyl curvature tensor vanishes if and only if the Riemannian metric has constant sectional curvature.', '1808.05001-1-3-1': 'Since the projective Weyl tensor is a projective invariant, it follows that projective deformations preserve the property of having constant sectional curvature and hence Beltrami Theorem is true in Riemannian geometry.', '1808.05001-1-4-0': 'In Finsler geometry this theorem is no longer valid due to the nonlinearity of the projective factor.', '1808.05001-1-4-1': 'There are examples of two projectively related Finsler metrics, which do not have, both of them, constant curvature, [CITATION], [CITATION].', '1808.05001-1-4-2': 'In this paper we prove the following Finslerian version of Beltrami Theorem (Theorem [REF]):', '1808.05001-1-5-0': 'Consider two projectively related Finsler metrics and assume that one of them has constant curvature.', '1808.05001-1-5-1': 'Then, the other Finsler metric has constant curvature if and only if the projective factor is a Hamel function.', '1808.05001-1-6-0': 'The key ingredient in our work is a Weyl-type curvature tensor that provides in Theorem [REF], in dimension greater than or equal to [MATH], a characterisation for Finsler metrics of constant flag curvature.', '1808.05001-1-6-1': 'We show that in Riemannian geometry the projective factor is always a Hamel function and hence we provide a Finslerian proof of classic Beltrami Theorem.', '1808.05001-1-7-0': 'For a Finsler manifold, of dimension greater than or equal to [MATH], we define in [REF] a Weyl-type curvature tensor that characterises Finsler metrics of constant flag curvature.', '1808.05001-1-7-1': 'The Weyl-type curvature tensor is inspired by condition [MATH] of [CITATION] for characterising sprays metrisable by Finsler functions of constant curvature.', '1808.05001-1-7-2': 'This tensor has been used also by Li and Shen [CITATION] to characterise sprays of isotropic curvature.', '1808.05001-1-7-3': 'The Weyl-type curvature tensor extends the Riemannian projective Weyl curvature tensor and it reduces to the Finslerian projective Weyl curvature tensor only in the case of constant flag curvature.', '1808.05001-1-7-4': 'In the Finslerian context, the class of projective deformations is larger than in the Riemannian case and we prove, in Lemma [REF], that the Weyl-type curvature tensor is preserved only by those projective deformations that satisfy the Hamel equation.', '1808.05001-1-7-5': 'There are some other proposals for Weyl-type curvature tensors that characterise Finsler metrics of constant curvature, [CITATION].', '1808.05001-1-7-6': 'For an extensive survey on the projective geometry in the Riemannian case we refer to [CITATION].', '1808.05001-1-8-0': '# A geometric setting for sprays', '1808.05001-1-9-0': 'In this work [MATH] represents a differentiable manifold of dimension greater than or equal to [MATH].', '1808.05001-1-9-1': 'We denote by [MATH] the tangent bundle of [MATH], while [MATH] denotes the tangent bundle with the zero section removed.', '1808.05001-1-9-2': 'Local coordinates on the base manifold [MATH] will be denoted by [MATH], while induced local coordinates on [MATH] or [MATH] will be denoted by [MATH].', '1808.05001-1-9-3': 'On [MATH] there is a canonical vector field, the Liouville vector field, and a canonical tensor field, the tangent structure, given by [EQUATION]', '1808.05001-1-9-4': 'A vector field [MATH] is called a spray if it is a second order vector field, which means that [MATH], and positively [MATH]-homogeneous in the fiber coordinates, which means [MATH].', '1808.05001-1-10-0': 'Locally, a spray [MATH] can be represented as [EQUATION] where the locally defined functions [MATH] are positively [MATH]-homogeneous with respect to the fiber coordinates.', '1808.05001-1-10-1': 'A regular curve [MATH] is called a geodesic of the spray [MATH], given by formula [REF], if [MATH], which means that it satisfies the system of second order ordinary differential equations: [EQUATION]', '1808.05001-1-10-2': 'A spray [MATH], given by formula [REF], is affine if the functions [MATH] are quadratic in the fiber coordinates.', '1808.05001-1-10-3': 'In other words, [MATH], where [MATH] are the coefficients of an affine connection on the base manifold.', '1808.05001-1-10-4': 'For an affine spray, its geodesics, solutions of the system [REF], are geodesics of the corresponding affine connection.', '1808.05001-1-11-0': 'In this work, we use the Frolicher-Nijenhuis theory of derivations associated to various vector-valued forms defined on [MATH] or [MATH], [CITATION], to define a geometric setting for studying sprays and Finsler spaces.', '1808.05001-1-12-0': 'For a given spray [MATH] we consider the induced geometric setting consisting of: the horizontal projector [MATH], the Jacobi endomorphism [MATH] and the curvature tensor [MATH] of the nonlinear connection, given by [EQUATION]', '1808.05001-1-12-1': 'Locally, these geometric structures are given by [EQUATION]', '1808.05001-1-12-2': 'The two curvature tensors, [MATH] and [MATH], are related by the following formulae [EQUATION]', '1808.05001-1-12-3': 'For an affine spray [MATH], the components of the Jacobi endomorphism are quadratic in the fiber coordinates, [MATH], where [MATH] are the curvature components of the affine connection.', '1808.05001-1-13-0': 'For a spray [MATH] and a function [MATH] on [MATH], consider the following semi-basic [MATH]-form, called the Euler-Lagrange [MATH]-form, [EQUATION]', '1808.05001-1-13-1': 'The function [MATH] is called a Lagrangian and a spray [MATH] that satisfies the Euler-Lagrange equation [MATH] is called a geodesic spray.', '1808.05001-1-14-0': '# Finsler functions and projective metrizability', '1808.05001-1-15-0': 'In Finsler geometry, the Lagrangian [MATH] from the Euler-Lagrange equation [MATH] is either a Finsler function [MATH] or the kinetic energy [MATH] of a Finsler function.', '1808.05001-1-16-0': 'A Finsler function is a continuous and positive function [MATH] that satisfies:', '1808.05001-1-17-0': 'The homogeneity of the Finsler function and the Euler Theorem on homogeneous functions implies that [MATH] and [MATH].', '1808.05001-1-17-1': 'When the metric tensor [REF] does not depend on the fiber coordinates, we say that the Finsler metric reduces to a Riemannian metric.', '1808.05001-1-18-0': 'For a Finsler metric [MATH], the regularity condition iii) in the Definition [REF] assures that there is a unique spray [MATH] satisfying the Euler-Lagrange equation [MATH], which in view of formulae [REF] is equivalent to [MATH].', '1808.05001-1-18-1': 'We call [MATH] the geodesic spray of the Finsler metric.', '1808.05001-1-18-2': 'The geodesics of the Finsler metric are the geodesics of the corresponding spray [MATH].', '1808.05001-1-19-0': 'A Finsler metric [MATH] has scalar flag curvature [MATH] if the Jacobi endomorphism, of its geodesic spray, or the curvature of the nonlinear connection, are given by [EQUATION]', '1808.05001-1-19-1': 'Locally, Finsler metrics of scalar flag curvature [MATH] are characterised by the following form of the Jacobi endomorphism [EQUATION]', '1808.05001-1-19-2': 'When the scalar flag curvature [MATH] is a constant, we say that [MATH] has constant flag curvature.', '1808.05001-1-19-3': 'Finsler metrics of constant curvature [MATH] are characterised by the following form of the curvature tensor [EQUATION]', '1808.05001-1-19-4': 'Locally, for such Finsler metrics, the curvature tensor takes the form [EQUATION]', '1808.05001-1-19-5': 'If [MATH] is the geodesic spray of a Riemannian metric then the condition [REF] reduces to the isotropy condition [MATH].', '1808.05001-1-20-0': 'Two sprays (or Finsler metrics) are projectively related if their geodesics coincide as oriented non-parametrised curves.', '1808.05001-1-20-1': 'Two sprays [MATH] and [MATH] are projectively related if and only if there exists a positively [MATH]-homogeneous function [MATH] such that [EQUATION]', '1808.05001-1-20-2': 'A Finsler metric [MATH] is projectively related to a spray [MATH] if and only if it satisfies the Hamel equation [MATH].', '1808.05001-1-20-3': 'In this case, the geodesic spray [MATH] of [MATH] and [MATH] are related by the projective factor [MATH].', '1808.05001-1-21-0': 'There are various characterisations for the projective equivalence of two sprays (Finsler metrics), the corresponding equations are known as Rapcsak equations, [CITATION].', '1808.05001-1-21-1': 'In the next theorem we provide new characterisations for the projective metrizability that include the projective factor [MATH].', '1808.05001-1-22-0': 'We assume that the equation [MATH] is true and hence the spray [MATH] is projectively related to the geodesic spray [MATH] of the Finsler metric [MATH].', '1808.05001-1-22-1': 'In view of formulae [REF] we have that [MATH] is equivalent to [MATH].', '1808.05001-1-22-2': 'Since [MATH] and [MATH] are projectively related, the projective factor is given by [MATH].', '1808.05001-1-22-3': 'We replace [MATH] in the previous equation and obtain [MATH], which is equation [MATH].', '1808.05001-1-23-0': 'If we evaluate both sides of equation [MATH] on the spray [MATH] we obtain [MATH] and hence [MATH].', '1808.05001-1-23-1': 'Multiplying [MATH] by [MATH] we obtain [MATH] and hence [MATH].', '1808.05001-1-23-2': 'Again using formulae [REF], last equation can be written as [MATH], which is equation [MATH].', '1808.05001-1-24-0': 'We start with equation [MATH] that is equivalent to [MATH].', '1808.05001-1-24-1': 'We apply [MATH] to both sides of this equation, use the commutation [MATH] and [MATH] and hence we obtain the first equation [MATH].', '1808.05001-1-24-2': 'If we evaluate the semi-basic [MATH]-forms from both sides of equation [MATH] on the spray [MATH] we obtain [MATH], which gives the second equation [MATH].', '1808.05001-1-25-0': 'Finally, we prove that the two equations [MATH] imply the projective metrizability condition [MATH].', '1808.05001-1-25-1': 'From first equation [MATH] we obtain [EQUATION]', '1808.05001-1-25-2': 'We use the commutation formula [MATH] [CITATION] and the fact that [MATH] to obtain [EQUATION]', '1808.05001-1-25-3': 'We reformulate this equation in terms of the Finsler function [MATH] and use the second equation [MATH] to replace the projective factor [MATH] and to obtain [EQUATION]', '1808.05001-1-25-4': 'We divide both sides of the above equation by [MATH] and therefore [EQUATION]', '1808.05001-1-25-5': 'Using formulae [REF], the left hand side of the above equation represents the semi-basic [MATH]-form [MATH].', '1808.05001-1-25-6': 'The right hand side can be written as [MATH].', '1808.05001-1-25-7': 'Therefore, last equation implies [MATH] and hence we proved that the Hamel equation [MATH] is satisfied.', '1808.05001-1-26-0': 'Using the definition of the curvature tensor, last formula [REF], we have [MATH].', '1808.05001-1-26-1': 'If we apply [MATH] of both sides of first formula [MATH] from Theorem [REF], we obtain that formula [REF] is true.', '1808.05001-1-27-0': 'If the Finsler metric [MATH] has constant curvature then the curvature tensor [MATH] is given by formula [REF].', '1808.05001-1-27-1': 'Using this, we have that [EQUATION] and hence formula [REF] implies formula [REF].', '1808.05001-1-28-0': 'If both Finsler metrics [MATH] and [MATH] are reducible to Riemannian metrics, then their geodesic sprays are quadratic and hence the projective factor [MATH] is linear in the fiber coordinates.', '1808.05001-1-28-1': 'Assume that [MATH].', '1808.05001-1-28-2': 'It follows that [MATH] is a basic [MATH]-form and hence [EQUATION]', '1808.05001-1-28-3': 'If [MATH] has constant curvature then formula [REF] is satisfied.', '1808.05001-1-28-4': 'If the Riemannian metric [MATH] has the expression [MATH] then formula [REF] can be written as follows [EQUATION]', '1808.05001-1-28-5': 'The linearity in [MATH] of the above formula implies [EQUATION]', '1808.05001-1-28-6': 'We multiply last formula by [MATH] and obtain [EQUATION]', '1808.05001-1-28-7': 'In this formula we make the contraction [MATH] and therefore [MATH], which gives the desired conclusion, [MATH].', '1808.05001-1-29-0': '# Weyl-type curvature tensor and Finsler metrics of constant curvature', '1808.05001-1-30-0': 'In Riemannian geometry, a metric has constant sectional curvature if and only if the projective Weyl curvature tensor vanishes.', '1808.05001-1-30-1': 'Since the projective Weyl tensor is a projective invariant it follows that projective deformations between Riemannian metrics preserve the property of having constant curvature, and this result is known as Beltrami Theorem.', '1808.05001-1-31-0': 'In Finsler geometry, the projective Weyl tensor, which is a projective invariant, characterises the metrics of scalar (not-necessarily constant) flag curvature, [CITATION].', '1808.05001-1-31-1': 'There have been several proposals for a Weyl-type curvature tensors, of [MATH]-type, that characterise Finsler metrics of constant flag curvature, [CITATION], none of them being invariant under projective deformations.', '1808.05001-1-32-0': 'In this section, we define yet another Weyl-type curvature tensor, of [MATH]-type, that characterises Finsler metrics of constant curvature in Theorem [REF].', '1808.05001-1-32-1': 'The Weyl-type curvature tensor coincides with the projective Weyl tensor if and only if the Finsler metric has constant curvature and it reduces to the classic projective Weyl tensor in the Riemannian context.', '1808.05001-1-33-0': 'Consider [MATH] a spray with Jacobi endomorphism [MATH].', '1808.05001-1-33-1': 'Inspired by [CITATION], we define the Weyl-type curvature tensor [EQUATION]', '1808.05001-1-33-2': 'This tensor has been used in [CITATION] to characterise sprays with scalar curvature that does not depend on fiber coordinates.', '1808.05001-1-33-3': 'Locally, the Weyl-type curvature tensor is given by [EQUATION]', '1808.05001-1-33-4': 'The Weyl-type curvature tensor is traceless, [MATH], and satisfies [MATH], for any spray [MATH].', '1808.05001-1-34-0': 'If the spray [MATH] is affine, the Weyl-type curvature tensor [REF] is quadratic in the fiber coordinates and can be expressed as follows: [EQUATION] where [MATH] is the classic projective Weyl tensor [EQUATION] and [MATH] is the Ricci tensor.', '1808.05001-1-34-1': 'Formula [REF] shows that in the Riemannian context one can recover the projective Weyl tensor from the Weyl-type tensor [REF].', '1808.05001-1-35-0': 'The classic Weyl tensor in Finsler geometry, which is a projective invariant, characterises Finsler metrics of scalar flag curvature, [CITATION], and it is given by: [EQUATION]', '1808.05001-1-35-1': 'Next theorem provides a characterisation for Finsler metrics of constant curvature using the Weyl-type curvature tensor [REF].', '1808.05001-1-36-0': 'For the implication [MATH], we assume that the Finsler metric [MATH] has constant flag curvature [MATH].', '1808.05001-1-36-1': 'Therefore, the Jacobi endomorphism is given by formula [REF], with [MATH] constant.', '1808.05001-1-36-2': 'It follows that [MATH], [MATH] and hence the Weyl-type curvature tensor [REF] vanishes.', '1808.05001-1-37-0': 'For the implication [MATH], we assume that the Weyl-type curvature tensor [REF] vanishes, which means that the Jacobi endomorphism is given by [EQUATION]', '1808.05001-1-37-1': 'Consider [MATH] the geodesic spray of the Finsler metric [MATH], which means that it satisfies [MATH].', '1808.05001-1-37-2': 'Using second formula [REF] we obtain [MATH].', '1808.05001-1-37-3': 'In the last formula, we use the expression [REF] of the Jacobi endomorphism to get [EQUATION]', '1808.05001-1-37-4': 'If [MATH], which is equivalent to [MATH], we have that [MATH].', '1808.05001-1-37-5': 'Assume now that [MATH].', '1808.05001-1-38-0': 'One can reformulate formula [REF] as follows: [EQUATION]', '1808.05001-1-38-1': 'Therefore, the Jacobi endomorphism [REF] can be written as [EQUATION]', '1808.05001-1-38-2': 'If we consider the function [EQUATION] the Jacobi endomorphism [MATH] is given by formula [REF].', '1808.05001-1-38-3': 'Using formula [REF], it follows that the function [MATH], defined by formula [REF], satisfies [MATH].', '1808.05001-1-38-4': 'Therefore [MATH] is constant along the fibers of the tangent bundle and, in view of the Finslerian version of the Schur Lemma [CITATION], we obtain that [MATH] is constant and hence the Finsler metric has constant flag curvature.', '1808.05001-1-39-0': 'We prove now the equivalence of the two conditions [MATH] and [MATH].', '1808.05001-1-39-1': 'The Weyl tensors [REF] and [REF] coincide if and only if [EQUATION] which is equivalent to [EQUATION]', '1808.05001-1-39-2': 'We assume now that the Finsler metric has scalar flag curvature, which means that the Jacobi endomorphism is given by formula [REF].', '1808.05001-1-39-3': 'Therefore, we have [MATH] and [MATH].', '1808.05001-1-39-4': 'For the left hand side of formula [REF], we have [EQUATION]', '1808.05001-1-39-5': 'Using the homogeneity of the terms, we have for the right side of formula [REF]: [EQUATION]', '1808.05001-1-39-6': 'Using formulae [REF] and [REF] we have that [REF] is true if and only if [MATH], which, in view of the Finslerian version of the Schur Lemma [CITATION], it means that the Finsler metric has constant flag curvature.', '1808.05001-1-40-0': 'If the Finslerian metric reduces to a Riemannian metric then Theorem [REF] reduces to the following characterisation of Riemannian metrics of constant curvature.', '1808.05001-1-41-0': 'A Riemannian metric has constant sectional curvature if and only if the Weyl-type curvature tensor [REF] vanishes.', '1808.05001-1-42-0': 'For an affine spray, the Weyl-type curvature tensor [REF] is given by formula [REF].', '1808.05001-1-42-1': 'Therefore, if for a Riemannian metric [MATH], the Weyl-type curvature tensor [REF] vanishes, then the tensor [REF] vanishes as well.', '1808.05001-1-42-2': 'Since [MATH], in formula [REF], is quadratic in the fiber coordinates, then [MATH] implies [MATH], the skew-symmetry in first and third covariant indices.', '1808.05001-1-42-3': 'Using this symmetry and the properties of the projective Weyl tensor (skew-symmetry in the last two covariant indices and algebraic Bianchi identity) we obtain [EQUATION]', '1808.05001-1-42-4': 'Consequently, the projective Weyl tensor [REF] vanishes, which means that the Riemannian curvature tensor is given by [EQUATION] which is equivalent to [EQUATION]', '1808.05001-1-42-5': 'The symmetry of the curvature tensor [MATH] implies the Ricci tensor [MATH] and the metric tensor [MATH] are proportional, say [MATH].', '1808.05001-1-42-6': 'Therefore, the curvature tensor is given by [EQUATION] which in view of Schur Lemma we have that [MATH] is constant and hence the Riemannian metric has constant curvature [MATH].', '1808.05001-1-43-0': 'Formula [REF] shows how to recover the scalar curvature of a Finsler metric from the fact that the Weyl-type tensor [REF] vanishes.', '1808.05001-1-43-1': 'In the Riemannian context, formula [REF] reduces to [EQUATION] which is true due to the fact that the vanishing of the projective Weyl tensor [REF] implies [MATH].', '1808.05001-1-44-0': '# A Finslerian version and a Finslerian proof of classic Beltrami Theorem', '1808.05001-1-45-0': 'Although the Weyl-type curvature tensor [REF] is not a projective invariant, we find the class of projective deformations that preserve Weyl-type curvature tensors and therefore we obtain the class of projective deformations in Lemma [REF], which makes Beltrami Theorem work in Finslerian setting, see Theorem [REF].', '1808.05001-1-46-0': 'Consider [MATH] and [MATH] two projectively related sprays.', '1808.05001-1-46-1': 'The corresponding Weyl-curvature tensors are related by [EQUATION]', '1808.05001-1-46-2': 'For two projectively related sprays [MATH], the corresponding Jacobi endomorphisms are related by the following formula [CITATION]: [EQUATION]', '1808.05001-1-46-3': 'This implies [MATH].', '1808.05001-1-46-4': 'Using these formulae, the Weyl-type curvature tensor [REF] of the spray [MATH] can be written as follows: [EQUATION] and hence formula [REF] is true.', '1808.05001-1-47-0': 'We can formulate the following version of Beltrami Theorem in Finsler geometry.', '1808.05001-1-47-1': '(Finslerian version of Beltrami Theorem) Consider [MATH] and [MATH] two projectively related Finsler metrics.', '1808.05001-1-48-0': 'For two projectively related Finsler metrics [MATH] and [MATH], we consider the corresponding Weyl-type curvature tensors [MATH] and [MATH], which are related by formula [REF].', '1808.05001-1-48-1': 'Using Theorem [REF] it follows that one of the two Finsler metrics has constant curvature if and only if the corresponding Weyl-type curvature tensor vanishes.', '1808.05001-1-48-2': 'Now, using formula [REF] the other Weyl-type curvature tensor vanishes, and hence the other Finsler metric has constant curvature, if and only if [MATH].', '1808.05001-1-49-0': 'It remains to prove the equivalence of the two conditions [MATH] and [MATH].', '1808.05001-1-49-1': 'From formula [REF] we have [MATH] and hence [MATH].', '1808.05001-1-49-2': 'Therefore [MATH] implies [MATH].', '1808.05001-1-50-0': 'For the converse we assume [MATH].', '1808.05001-1-50-1': 'Using the commutation formula for the two derivations [MATH] and [MATH], see [CITATION], we have [EQUATION]', '1808.05001-1-50-2': 'In the above formula we did use that [MATH] is a [MATH]-homogeneous function and hence [MATH], [MATH] and [MATH].', '1808.05001-1-51-0': 'A function [MATH], positively [MATH]-homogeneous in the fiber coordinates that satisfies one of the two equivalent conditions i) or ii) of Theorem [REF] is called a Hamel function.', '1808.05001-1-52-0': 'If one of the Finsler functions in Theorem [REF] is the Euclidean one then we obtain the following characterisation for projectively flat Finsler functions of constant flag curvature.', '1808.05001-1-53-0': 'Consider [MATH] a projectively flat Finsler metric.', '1808.05001-1-53-1': 'Then [MATH] has constant curvature if and only if if the projective factor [MATH] satisfies either one of the following equivalent Hamel equations:', '1808.05001-1-54-0': 'where [MATH] is the flat spray and [MATH] the corresponding horizontal projector.', '1808.05001-1-55-0': 'Corollary [REF] corresponds to [CITATION].', '1808.05001-1-56-0': 'According to Proposition [REF], in Riemannian geometry, the Hamel condition [REF] means that for the linear projective factor [MATH], the basic [MATH]-form [MATH] is closed.', '1808.05001-1-56-1': 'We can use these aspects and Theorem [REF] to provide a Finslerian proof of classic Beltrami Theorem.', '1808.05001-1-57-0': '(Beltrami Theorem) Consider [MATH] and [MATH] two projectively related Finsler metrics, each of them being reducible to a Riemannian metric.', '1808.05001-1-57-1': 'Then, [MATH] has constant curvature if and only if [MATH] has constant curvature.', '1808.05001-1-58-0': 'The only think we have to prove is that the projective factor [MATH] satisfies the condition [MATH] of Theorem [REF].', '1808.05001-1-58-1': 'It follows from the fact that [MATH] is linear in the fiber coordinates and satisfies formula [REF] of Proposition [REF].', '1808.05001-1-59-0': '# Examples', '1808.05001-1-60-0': 'In this section we use some examples to test the Hamel equations i) and ii) of Theorem [REF] for the projective factor of two projectively related Finsler functions.', '1808.05001-1-61-0': '## Projectively flat Finsler metric of non-constant curvature', '1808.05001-1-62-0': 'We provide now an example of a projectively flat Finsler metric, which does not have constant curvature.', '1808.05001-1-62-1': 'In view of Theorem [REF] this can be explained by the fact that the projective factor does not satisfy the Hamel equation [MATH].', '1808.05001-1-62-2': 'We consider the following Numata metric, [CITATION], on the unit Euclidean ball [MATH]: [EQUATION]', '1808.05001-1-62-3': 'The metric written above is a projectively flat Finsler metric, whose geodesic spray is given by: [EQUATION] where [MATH] is the flat spray, the geodesic spray of the Euclidean metric.', '1808.05001-1-63-0': 'The projective factor is given by [EQUATION]', '1808.05001-1-63-1': 'By a direct computation we have [EQUATION]', '1808.05001-1-63-2': 'Since [MATH] is the flat spray and the projective factor does not satisfy the Hamel equation [MATH], it follows by Theorem [REF] that the Numata metric does not have constant curvature.', '1808.05001-1-63-3': 'This can be checked independently, since its scalar flag curvature is given by [EQUATION]', '1808.05001-1-64-0': '## Projectively flat Finsler metric of constant curvature', '1808.05001-1-65-0': 'In this example, we consider a projectively flat Finsler metric [MATH] with the projective factor satisfying the Hamel equation [MATH].', '1808.05001-1-65-1': 'Using the Finslerian version of Beltrami Theorem [REF] it follows that the Finsler function [MATH] has constant curvature.', '1808.05001-1-66-0': 'The Funk function on the Euclidean ball [MATH], [CITATION], [EQUATION] is a Finsler metric, projectively related to the Euclidean metric.', '1808.05001-1-66-1': 'Its geodesic spray is given by [EQUATION] where [MATH] is the flat spray.', '1808.05001-1-66-2': 'The projective factor [MATH] is given by [EQUATION]', '1808.05001-1-66-3': 'Recall that [MATH] is projectively flat, therefore it satisfy the Hamel equation [MATH].', '1808.05001-1-66-4': 'It follows that the projective factor [MATH] satisfies also the equation [MATH] and due to the Finslerian version of Beltrami Theorem [REF] it follows that the Finsler metric [MATH] has constant curvature.', '1808.05001-1-66-5': 'One can independently check that the Finsler metric [MATH] has constant flag curvature [MATH].', '1808.05001-1-67-0': '## Projectively related Finsler metrics with linear projective factor', '1808.05001-1-68-0': 'In this example we consider two projectively related Finsler metric with a linear projective factor [MATH].', '1808.05001-1-68-1': 'If the first Finsler metric has constant curvature, then the obstruction for the second Finsler metric to be of constant curvature reduces to the fact that the basic [MATH]-form [MATH] is closed.', '1808.05001-1-69-0': 'It is known that the Funk metric [MATH], on the Euclidean unit ball [MATH], [EQUATION] is a projectively flat Finsler metric of constant flag curvature [MATH] with the geodesic spray: [EQUATION]', '1808.05001-1-69-1': 'From [CITATION], we know that for any constant vector [MATH], [MATH], the function [EQUATION] is a projective Finsler metric on [MATH].', '1808.05001-1-70-0': 'The two Finsler metrics [MATH] and [MATH] are projectively related with the projective factor linear in the fiber coordinates: [EQUATION]', '1808.05001-1-70-1': 'It follows that the basic [MATH]-form [MATH] is exact, [EQUATION]', '1808.05001-1-70-2': 'Therefore, the obstruction ii) for the projective factor [MATH] in Theorem [REF] is satisfied and hence the Finsler metric [MATH] has constant curvature as well.', '1808.05001-1-70-3': 'This can be checked directly, the flag curvature of [MATH] is [MATH].'}

{'1808.05001-2-0-0': 'We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature.', '1808.05001-2-0-1': 'When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classical projective Weyl tensor.', '1808.05001-2-0-2': 'In the general case, the Weyl- type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry.', '1808.05001-2-0-3': 'We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics.', '1808.05001-2-0-4': 'Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function.', '1808.05001-2-0-5': 'This way we provide a Finslerian version of Beltrami Theorem.', '1808.05001-2-1-0': '[2000]53C60, 53B40, 58E30, 49N45', '1808.05001-2-2-0': '# Introduction', '1808.05001-2-3-0': 'For a Riemannian manifold of dimension greater than or equal to [MATH], the projective Weyl curvature tensor vanishes if and only if the Riemannian metric has constant sectional curvature.', '1808.05001-2-3-1': 'Since the projective Weyl tensor is a projective invariant, it follows that projective deformations preserve the property of having constant sectional curvature and hence Beltrami Theorem is true in Riemannian geometry.', '1808.05001-2-4-0': 'In Finsler geometry this theorem is no longer valid due to the nonlinearity of the projective factor.', '1808.05001-2-4-1': 'There are examples of two projectively related Finsler metrics, which do not have, both of them, constant curvature, [CITATION], [CITATION].', '1808.05001-2-4-2': 'In this paper we prove the following Finslerian version of Beltrami Theorem (Theorem [REF]):', '1808.05001-2-5-0': 'Consider two projectively related Finsler metrics and assume that one of them has constant curvature.', '1808.05001-2-5-1': 'Then, the other Finsler metric has constant curvature if and only if the projective factor is a Hamel function.', '1808.05001-2-6-0': 'The key ingredient in our work is a Weyl-type curvature tensor that provides in Theorem [REF], in dimension greater than or equal to [MATH], a characterisation for Finsler metrics of constant flag curvature.', '1808.05001-2-6-1': 'We show that in Riemannian geometry the projective factor is always a Hamel function and hence we provide a Finslerian proof of classical Beltrami Theorem.', '1808.05001-2-7-0': 'For a Finsler manifold, of dimension greater than or equal to [MATH], we define in [REF] a Weyl-type curvature tensor that characterises Finsler metrics of constant flag curvature.', '1808.05001-2-7-1': 'The Weyl-type curvature tensor is inspired by condition [MATH] of [CITATION] for characterising sprays metrisable by Finsler metric of constant curvature.', '1808.05001-2-7-2': 'This tensor has been used also by Li and Shen [CITATION] to characterise sprays of isotropic curvature.', '1808.05001-2-7-3': 'The Weyl-type curvature tensor extends the Riemannian projective Weyl curvature tensor and it reduces to the Finslerian projective Weyl curvature tensor only in the case of constant flag curvature.', '1808.05001-2-7-4': 'In the Finslerian context, the class of projective deformations is larger than in the Riemannian case and we prove, in Lemma [REF], that the Weyl-type curvature tensor is preserved only by those projective deformations that satisfy the Hamel equation.', '1808.05001-2-7-5': 'There are some other proposals for Weyl-type curvature tensors that characterise Finsler metrics of constant curvature, [CITATION].', '1808.05001-2-7-6': 'For an extensive survey on the projective geometry in the Riemannian case we refer to [CITATION].', '1808.05001-2-8-0': '# A geometric setting for sprays', '1808.05001-2-9-0': 'In this work [MATH] represents a differentiable manifold of dimension greater than or equal to [MATH].', '1808.05001-2-9-1': 'We denote by [MATH] the tangent bundle of [MATH], while [MATH] denotes the tangent bundle with the zero section removed.', '1808.05001-2-9-2': 'Local coordinates on the base manifold [MATH] will be denoted by [MATH], while induced local coordinates on [MATH] or [MATH] will be denoted by [MATH].', '1808.05001-2-9-3': 'On [MATH] there is a canonical vector field, the Liouville vector field, and a canonical tensor field, the tangent structure, given by [EQUATION]', '1808.05001-2-9-4': 'A vector field [MATH] is called a spray if it is a second order vector field, which means that [MATH], and positively [MATH]-homogeneous in the fiber coordinates, which means [MATH].', '1808.05001-2-10-0': 'Locally, a spray [MATH] can be represented as [EQUATION] where the locally defined functions [MATH] are positively [MATH]-homogeneous with respect to the fiber coordinates.', '1808.05001-2-10-1': 'A regular curve [MATH] is called a geodesic of the spray [MATH], given by formula [REF], if [MATH], which means that it satisfies the system of second order ordinary differential equations: [EQUATION]', '1808.05001-2-10-2': 'A spray [MATH], given by formula [REF], is affine if the functions [MATH] are quadratic in the fiber coordinates.', '1808.05001-2-10-3': 'In other words, [MATH], where [MATH] are the coefficients of an affine connection on the base manifold.', '1808.05001-2-10-4': 'For an affine spray, its geodesics, solutions of the system [REF], are geodesics of the corresponding affine connection.', '1808.05001-2-11-0': 'In this work, we use the Frolicher-Nijenhuis theory of derivations associated to various vector-valued forms defined on [MATH] or [MATH], [CITATION], to define a geometric setting for studying sprays and Finsler spaces.', '1808.05001-2-12-0': 'For a given spray [MATH] we consider the induced geometric setting consisting of: the horizontal projector [MATH], the Jacobi endomorphism [MATH] and the curvature tensor [MATH] of the nonlinear connection, given by [EQUATION]', '1808.05001-2-12-1': 'Locally, these geometric structures are given by [EQUATION]', '1808.05001-2-12-2': 'The two curvature tensors, [MATH] and [MATH], are related by the following formulae [EQUATION]', '1808.05001-2-12-3': 'For an affine spray [MATH], the components of the Jacobi endomorphism are quadratic in the fiber coordinates, [MATH], where [MATH] are the curvature components of the affine connection.', '1808.05001-2-13-0': 'For a spray [MATH] and a function [MATH] on [MATH], consider the following semi-basic [MATH]-form, called the Euler-Lagrange [MATH]-form, [EQUATION]', '1808.05001-2-13-1': 'The function [MATH] is called a Lagrangian and a spray [MATH] that satisfies the Euler-Lagrange equation [MATH] is called a geodesic spray.', '1808.05001-2-14-0': '# Finsler metrics and projective metrizability', '1808.05001-2-15-0': 'In Finsler geometry, the Lagrangian [MATH] from the Euler-Lagrange equation [MATH] is either a Finsler metric [MATH] or the kinetic energy [MATH] of a Finsler metric.', '1808.05001-2-16-0': 'A Finsler metric is a continuous and positive function [MATH] that satisfies:', '1808.05001-2-17-0': 'The homogeneity of the Finsler metric and the Euler Theorem on homogeneous functions implies that [MATH] and [MATH].', '1808.05001-2-17-1': 'When the metric tensor [REF] does not depend on the fiber coordinates, we say that the Finsler metric reduces to a Riemannian metric.', '1808.05001-2-18-0': 'For a Finsler metric [MATH], the regularity condition iii) in the Definition [REF] assures that there is a unique spray [MATH] satisfying the Euler-Lagrange equation [MATH], which in view of formulae [REF] is equivalent to [MATH].', '1808.05001-2-18-1': 'We call [MATH] the geodesic spray of the Finsler metric.', '1808.05001-2-18-2': 'The geodesics of the Finsler metric are the geodesics of the corresponding spray [MATH].', '1808.05001-2-19-0': 'A Finsler metric [MATH] has scalar flag curvature [MATH] if the Jacobi endomorphism, of its geodesic spray, or the curvature of the nonlinear connection, are given by [EQUATION]', '1808.05001-2-19-1': 'Locally, Finsler metrics of scalar flag curvature [MATH] are characterised by the following form of the Jacobi endomorphism [EQUATION]', '1808.05001-2-19-2': 'When the scalar flag curvature [MATH] is a constant, we say that [MATH] has constant flag curvature.', '1808.05001-2-19-3': 'Finsler metrics of constant curvature [MATH] are characterised by the following form of the curvature tensor [EQUATION]', '1808.05001-2-19-4': 'Locally, for such Finsler metrics, the curvature tensor takes the form [EQUATION]', '1808.05001-2-19-5': 'If [MATH] is the geodesic spray of a Riemannian metric then the condition [REF] reduces to the isotropy condition [MATH].', '1808.05001-2-20-0': 'Two sprays (or Finsler metrics) are projectively related if their geodesics coincide as oriented non-parametrised curves.', '1808.05001-2-20-1': 'Two sprays [MATH] and [MATH] are projectively related if and only if there exists a positively [MATH]-homogeneous function [MATH] such that [EQUATION]', '1808.05001-2-20-2': 'A Finsler metric [MATH] is projectively related to a spray [MATH] if and only if it satisfies the Hamel equation [MATH].', '1808.05001-2-20-3': 'In this case, the geodesic spray [MATH] of [MATH] and [MATH] are related by the projective factor [MATH].', '1808.05001-2-21-0': 'There are various characterisations for the projective equivalence of two sprays (Finsler metrics), the corresponding equations are known as Rapcsak equations, [CITATION].', '1808.05001-2-21-1': 'In the next theorem we provide new characterisations for the projective metrizability that include the projective factor [MATH].', '1808.05001-2-22-0': 'We assume that the relation [MATH] is true and hence the spray [MATH] is projectively related to the geodesic spray [MATH] of the Finsler metric [MATH].', '1808.05001-2-22-1': 'In view of formulae [REF] we have that [MATH] is equivalent to [MATH].', '1808.05001-2-22-2': 'Since [MATH] and [MATH] are projectively related, the projective factor is given by [MATH].', '1808.05001-2-22-3': 'We replace [MATH] in the previous equation and obtain [MATH], which is relation [MATH].', '1808.05001-2-23-0': 'If we evaluate both sides of relation [MATH] on the spray [MATH] we obtain [MATH] and hence [MATH].', '1808.05001-2-23-1': 'Multiplying [MATH] by [MATH] we obtain [MATH] and hence [MATH].', '1808.05001-2-23-2': 'Again using formulae [REF], last formula can be written as [MATH], which is relation [MATH].', '1808.05001-2-24-0': 'We start with relation [MATH] that is equivalent to [MATH].', '1808.05001-2-24-1': 'We apply [MATH] to both sides of this relation, use the commutation [MATH] and [MATH] and hence we obtain the first relation [MATH].', '1808.05001-2-24-2': 'If we evaluate the semi-basic [MATH]-forms from both sides of relation [MATH] on the spray [MATH] we obtain [MATH], which gives the second relation [MATH].', '1808.05001-2-25-0': 'Finally, we prove that the two relations [MATH] imply the projective metrizability condition [MATH].', '1808.05001-2-25-1': 'From first relation [MATH] we obtain [EQUATION]', '1808.05001-2-25-2': 'We use the commutation formula [MATH] [CITATION] and the fact that [MATH] to obtain [EQUATION]', '1808.05001-2-25-3': 'We reformulate this formula in terms of the Finsler metric [MATH] and use the second relation [MATH] to replace the projective factor [MATH] and to obtain [EQUATION]', '1808.05001-2-25-4': 'We divide both sides of the above formula by [MATH] and therefore [EQUATION]', '1808.05001-2-25-5': 'Using formulae [REF], the left hand side of the above relation represents the semi-basic [MATH]-form [MATH].', '1808.05001-2-25-6': 'The right hand side can be written as [MATH].', '1808.05001-2-25-7': 'Therefore, last formula implies [MATH] and hence we proved that the Hamel condition [MATH] is satisfied.', '1808.05001-2-26-0': 'Using the definition of the curvature tensor, last formula [REF], we have [MATH].', '1808.05001-2-26-1': 'If we apply [MATH] of both sides of first formula [MATH] from Theorem [REF], we obtain that formula [REF] is true.', '1808.05001-2-27-0': 'If the Finsler metric [MATH] has constant curvature then the curvature tensor [MATH] is given by formula [REF].', '1808.05001-2-27-1': 'Using this, we have that [EQUATION] and hence formula [REF] implies formula [REF].', '1808.05001-2-28-0': 'If both Finsler metrics [MATH] and [MATH] are reducible to Riemannian metrics, then their geodesic sprays are quadratic and hence the projective factor [MATH] is linear in the fiber coordinates.', '1808.05001-2-28-1': 'Assume that [MATH].', '1808.05001-2-28-2': 'It follows that [MATH] is a basic [MATH]-form and hence [EQUATION]', '1808.05001-2-28-3': 'If [MATH] has constant curvature then formula [REF] is satisfied.', '1808.05001-2-28-4': 'If the Riemannian metric [MATH] has the expression [MATH] then formula [REF] can be written as follows [EQUATION]', '1808.05001-2-28-5': 'The linearity in [MATH] of the above formula implies [EQUATION]', '1808.05001-2-28-6': 'We multiply last formula by [MATH] and obtain [EQUATION]', '1808.05001-2-28-7': 'In this formula we make the contraction [MATH] and therefore [MATH], which gives the desired conclusion, [MATH].', '1808.05001-2-29-0': '# Weyl-type curvature tensor and Finsler metrics of constant curvature', '1808.05001-2-30-0': 'In Riemannian geometry, a metric has constant sectional curvature if and only if the projective Weyl curvature tensor vanishes.', '1808.05001-2-30-1': 'Since the projective Weyl tensor is a projective invariant it follows that projective deformations between Riemannian metrics preserve the property of having constant curvature, and this result is known as Beltrami Theorem.', '1808.05001-2-31-0': 'In Finsler geometry, the projective Weyl tensor, which is a projective invariant, characterises the metrics of scalar (not-necessarily constant) flag curvature, [CITATION].', '1808.05001-2-31-1': 'There have been several proposals for a Weyl-type curvature tensors, of [MATH]-type, that characterise Finsler metrics of constant flag curvature, [CITATION], none of them being invariant under projective deformations.', '1808.05001-2-32-0': 'In this section, we define yet another Weyl-type curvature tensor, of [MATH]-type, that characterises Finsler metrics of constant curvature in Theorem [REF].', '1808.05001-2-32-1': 'The Weyl-type curvature tensor coincides with the projective Weyl tensor if and only if the Finsler metric has constant curvature and it reduces to the classical projective Weyl tensor in the Riemannian context.', '1808.05001-2-33-0': 'Consider [MATH] a spray with Jacobi endomorphism [MATH].', '1808.05001-2-33-1': 'Inspired by [CITATION], we define the Weyl-type curvature tensor [EQUATION]', '1808.05001-2-33-2': 'This tensor has been used in [CITATION] to characterise sprays with scalar curvature that does not depend on fiber coordinates.', '1808.05001-2-33-3': 'Locally, the Weyl-type curvature tensor is given by [EQUATION]', '1808.05001-2-33-4': 'The Weyl-type curvature tensor is traceless, [MATH], and satisfies [MATH], for any spray [MATH].', '1808.05001-2-34-0': 'If the spray [MATH] is affine, the Weyl-type curvature tensor [REF] is quadratic in the fiber coordinates and can be expressed as follows: [EQUATION] where [MATH] is the classical projective Weyl tensor [EQUATION] and [MATH] is the Ricci tensor.', '1808.05001-2-34-1': 'Formula [REF] shows that in the Riemannian context one can recover the projective Weyl tensor from the Weyl-type tensor [REF].', '1808.05001-2-35-0': 'The classical Weyl tensor in Finsler geometry, which is a projective invariant, characterises Finsler metrics of scalar flag curvature, [CITATION], and it is given by: [EQUATION]', '1808.05001-2-35-1': 'Next theorem provides a characterisation for Finsler metrics of constant curvature using the Weyl-type curvature tensor [REF].', '1808.05001-2-36-0': 'For the implication [MATH], we assume that the Finsler metric [MATH] has constant flag curvature [MATH].', '1808.05001-2-36-1': 'Therefore, the Jacobi endomorphism is given by formula [REF], with [MATH] constant.', '1808.05001-2-36-2': 'It follows that [MATH], [MATH] and hence the Weyl-type curvature tensor [REF] vanishes.', '1808.05001-2-37-0': 'For the implication [MATH], we assume that the Weyl-type curvature tensor [REF] vanishes, which means that the Jacobi endomorphism is given by [EQUATION]', '1808.05001-2-37-1': 'Consider [MATH] the geodesic spray of the Finsler metric [MATH], which means that it satisfies [MATH].', '1808.05001-2-37-2': 'Using second formula [REF] we obtain [MATH].', '1808.05001-2-37-3': 'In the last formula, we use the expression [REF] of the Jacobi endomorphism to get [EQUATION]', '1808.05001-2-37-4': 'If [MATH], which is equivalent to [MATH], we have that [MATH].', '1808.05001-2-37-5': 'Assume now that [MATH].', '1808.05001-2-38-0': 'One can reformulate formula [REF] as follows: [EQUATION]', '1808.05001-2-38-1': 'Therefore, the Jacobi endomorphism [REF] can be written as [EQUATION]', '1808.05001-2-38-2': 'If we consider the function [EQUATION] the Jacobi endomorphism [MATH] is given by formula [REF].', '1808.05001-2-38-3': 'Using formula [REF], it follows that the function [MATH], defined by formula [REF], satisfies [MATH].', '1808.05001-2-38-4': 'Therefore [MATH] is constant along the fibers of the tangent bundle and, in view of the Finslerian version of the Schur Lemma [CITATION], we obtain that [MATH] is constant and hence the Finsler metric has constant flag curvature.', '1808.05001-2-39-0': 'We prove now the equivalence of the two conditions [MATH] and [MATH].', '1808.05001-2-39-1': 'The Weyl tensors [REF] and [REF] coincide if and only if [EQUATION] which is equivalent to [EQUATION]', '1808.05001-2-39-2': 'We assume now that the Finsler metric has scalar flag curvature, which means that the Jacobi endomorphism is given by formula [REF].', '1808.05001-2-39-3': 'Therefore, we have [MATH] and [MATH].', '1808.05001-2-39-4': 'For the left hand side of formula [REF], we have [EQUATION]', '1808.05001-2-39-5': 'Using the homogeneity of the terms, we have for the right side of formula [REF]: [EQUATION]', '1808.05001-2-39-6': 'Using formulae [REF] and [REF] we have that [REF] is true if and only if [MATH], which, in view of the Finslerian version of the Schur Lemma [CITATION], it means that the Finsler metric has constant flag curvature.', '1808.05001-2-40-0': 'If the Finslerian metric reduces to a Riemannian metric then Theorem [REF] reduces to the following characterisation of Riemannian metrics of constant curvature.', '1808.05001-2-41-0': 'A Riemannian metric has constant sectional curvature if and only if the Weyl-type curvature tensor [REF] vanishes.', '1808.05001-2-42-0': 'For an affine spray, the Weyl-type curvature tensor [REF] is given by formula [REF].', '1808.05001-2-42-1': 'Therefore, if for a Riemannian metric [MATH], the Weyl-type curvature tensor [REF] vanishes, then the tensor [REF] vanishes as well.', '1808.05001-2-42-2': 'Since [MATH], in formula [REF], is quadratic in the fiber coordinates, then [MATH] implies [MATH], the skew-symmetry in first and third covariant indices.', '1808.05001-2-42-3': 'Using this symmetry and the properties of the projective Weyl tensor (skew-symmetry in the last two covariant indices and algebraic Bianchi identity) we obtain [EQUATION]', '1808.05001-2-42-4': 'Consequently, the projective Weyl tensor [REF] vanishes, which means that the Riemannian curvature tensor is given by [EQUATION] which is equivalent to [EQUATION]', '1808.05001-2-42-5': 'The symmetry of the curvature tensor [MATH] implies the Ricci tensor [MATH] and the metric tensor [MATH] are proportional, say [MATH].', '1808.05001-2-42-6': 'Therefore, the curvature tensor is given by [EQUATION] which in view of Schur Lemma we have that [MATH] is constant and hence the Riemannian metric has constant curvature [MATH].', '1808.05001-2-43-0': 'Formula [REF] shows how to recover the scalar curvature of a Finsler metric from the fact that the Weyl-type tensor [REF] vanishes.', '1808.05001-2-43-1': 'In the Riemannian context, formula [REF] reduces to [EQUATION] which is true due to the fact that the vanishing of the projective Weyl tensor [REF] implies [MATH].', '1808.05001-2-44-0': '# A Finslerian version and a Finslerian proof of classical Beltrami Theorem', '1808.05001-2-45-0': 'Although the Weyl-type curvature tensor [REF] is not a projective invariant, we find the class of projective deformations that preserve Weyl-type curvature tensors and therefore we obtain the class of projective deformations in Lemma [REF], which makes Beltrami Theorem work in Finslerian setting, see Theorem [REF].', '1808.05001-2-46-0': 'The corresponding Weyl-type curvature tensors of two projectively related sprays [MATH] and [MATH] are related by [EQUATION]', '1808.05001-2-46-1': 'For two projectively related sprays [MATH] and [MATH], the corresponding Jacobi endomorphisms are related by the following formula [CITATION]: [EQUATION]', '1808.05001-2-46-2': 'This implies [MATH].', '1808.05001-2-46-3': 'Using these formulae, the Weyl-type curvature tensor [REF] of the spray [MATH] can be written as follows: [EQUATION] and hence formula [REF] is true.', '1808.05001-2-47-0': 'We can formulate the following version of Beltrami Theorem in Finsler geometry.', '1808.05001-2-47-1': '(Finslerian version of Beltrami Theorem) Consider two projectively related Finsler metrics [MATH] and [MATH].', '1808.05001-2-48-0': 'For two projectively related Finsler metrics [MATH] and [MATH], we consider the corresponding Weyl-type curvature tensors [MATH] and [MATH], which are related by formula [REF].', '1808.05001-2-48-1': 'Using Theorem [REF] it follows that one of the two Finsler metrics has constant curvature if and only if the corresponding Weyl-type curvature tensor vanishes.', '1808.05001-2-48-2': 'Now, using formula [REF] the other Weyl-type curvature tensor vanishes, and hence the other Finsler metric has constant curvature, if and only if [MATH].', '1808.05001-2-49-0': 'It remains to prove the equivalence of the two conditions [MATH] and [MATH].', '1808.05001-2-49-1': 'From formula [REF] we have [MATH] and hence [MATH].', '1808.05001-2-49-2': 'Therefore [MATH] implies [MATH].', '1808.05001-2-50-0': 'For the converse we assume [MATH].', '1808.05001-2-50-1': 'Using the commutation formula for the two derivations [MATH] and [MATH], see [CITATION], we have [EQUATION]', '1808.05001-2-50-2': 'In the above formula we did use that [MATH] is a [MATH]-homogeneous function and hence [MATH], [MATH] and [MATH].', '1808.05001-2-51-0': 'A function [MATH], positively [MATH]-homogeneous in the fiber coordinates that satisfies one of the two equivalent conditions i) or ii) of Theorem [REF] is called a Hamel function.', '1808.05001-2-52-0': 'If one of the Finsler metrics in Theorem [REF] is the Euclidean one then we obtain the following characterisation for projectively flat Finsler metrics of constant flag curvature.', '1808.05001-2-53-0': 'A projectively flat Finsler metric [MATH] has constant curvature if and only if the projective factor [MATH] satisfies either one of the following equivalent Hamel conditions:', '1808.05001-2-54-0': 'where [MATH] is the flat spray and [MATH] the corresponding horizontal projector.', '1808.05001-2-55-0': 'Corollary [REF] corresponds to [CITATION].', '1808.05001-2-56-0': 'According to Proposition [REF], in Riemannian geometry, the Hamel condition [REF] means that for the linear projective factor [MATH], the basic [MATH]-form [MATH] is closed.', '1808.05001-2-56-1': 'We can use these aspects and Theorem [REF] to provide a Finslerian proof of classical Beltrami Theorem.', '1808.05001-2-57-0': '(Beltrami Theorem) Consider two projectively related Finsler metrics [MATH] and [MATH], each of them being reducible to a Riemannian metric.', '1808.05001-2-57-1': 'Then, [MATH] has constant curvature if and only if [MATH] has constant curvature.', '1808.05001-2-58-0': 'The only think we have to prove is that the projective factor [MATH] satisfies the condition [MATH] of Theorem [REF].', '1808.05001-2-58-1': 'It follows from the fact that [MATH] is linear in the fiber coordinates and satisfies formula [REF] of Proposition [REF].', '1808.05001-2-59-0': '# Examples', '1808.05001-2-60-0': 'In this section we use some examples to test the Hamel conditions i) and ii) of Theorem [REF] for the projective factor of two projectively related Finsler metrics.', '1808.05001-2-61-0': '## Projectively flat Finsler metric of non-constant curvature', '1808.05001-2-62-0': 'We provide now an example of a projectively flat Finsler metric, which does not have constant curvature.', '1808.05001-2-62-1': 'In view of Theorem [REF] this can be explained by the fact that the projective factor does not satisfy the Hamel equation [MATH].', '1808.05001-2-62-2': 'We consider the following Numata metric, [CITATION], on the unit Euclidean ball [MATH]: [EQUATION]', '1808.05001-2-62-3': 'The metric written above is a projectively flat Finsler metric, whose geodesic spray is given by: [EQUATION] where [MATH] is the flat spray, the geodesic spray of the Euclidean metric.', '1808.05001-2-63-0': 'The projective factor is given by [EQUATION]', '1808.05001-2-63-1': 'By a direct computation we have [EQUATION]', '1808.05001-2-63-2': 'Since [MATH] is the flat spray and the projective factor does not satisfy the Hamel equation [MATH], it follows by Theorem [REF] that the Numata metric does not have constant curvature.', '1808.05001-2-63-3': 'This can be checked independently, since its scalar flag curvature is given by [EQUATION]', '1808.05001-2-64-0': '## Projectively flat Finsler metric of constant curvature', '1808.05001-2-65-0': 'In this example, we consider a projectively flat Finsler metric [MATH] with the projective factor satisfying the Hamel equation [MATH].', '1808.05001-2-65-1': 'Using the Finslerian version of Beltrami Theorem [REF] it follows that the Finsler metric [MATH] has constant curvature.', '1808.05001-2-66-0': 'The Funk function on the Euclidean ball [MATH], [CITATION], [EQUATION] is a Finsler metric, projectively related to the Euclidean metric.', '1808.05001-2-66-1': 'Its geodesic spray is given by [EQUATION] where [MATH] is the flat spray.', '1808.05001-2-66-2': 'The projective factor [MATH] is given by [EQUATION]', '1808.05001-2-66-3': 'Recall that [MATH] is projectively flat, therefore it satisfy the Hamel equation [MATH].', '1808.05001-2-66-4': 'It follows that the projective factor [MATH] satisfies also the equation [MATH] and due to the Finslerian version of Beltrami Theorem [REF] it follows that the Finsler metric [MATH] has constant curvature.', '1808.05001-2-66-5': 'One can independently check that the Finsler metric [MATH] has constant flag curvature [MATH].', '1808.05001-2-67-0': '## Projectively related Finsler metrics with linear projective factor', '1808.05001-2-68-0': 'In this example we consider two projectively related Finsler metric with a linear projective factor [MATH].', '1808.05001-2-68-1': 'If the first Finsler metric has constant curvature, then the obstruction for the second Finsler metric to be of constant curvature reduces to the fact that the basic [MATH]-form [MATH] is closed.', '1808.05001-2-69-0': 'It is known that the Funk metric [MATH], on the Euclidean unit ball [MATH], [EQUATION] is a projectively flat Finsler metric of constant flag curvature [MATH] with the geodesic spray: [EQUATION]', '1808.05001-2-69-1': 'From [CITATION], we know that for any constant vector [MATH], [MATH], the function [EQUATION] is a projective Finsler metric on [MATH].', '1808.05001-2-70-0': 'The two Finsler metrics [MATH] and [MATH] are projectively related with the projective factor linear in the fiber coordinates: [EQUATION]', '1808.05001-2-70-1': 'It follows that the basic [MATH]-form [MATH] is exact, [EQUATION]', '1808.05001-2-70-2': 'Therefore, the obstruction ii) for the projective factor [MATH] in Theorem [REF] is satisfied and hence the Finsler metric [MATH] has constant curvature as well.', '1808.05001-2-70-3': 'This can be checked directly, the flag curvature of [MATH] is [MATH].'}

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['1808.05001-1-27-1', '1808.05001-2-27-1'], ['1808.05001-1-11-0', '1808.05001-2-11-0'], ['1808.05001-1-6-0', '1808.05001-2-6-0'], ['1808.05001-1-70-0', '1808.05001-2-70-0'], ['1808.05001-1-70-1', '1808.05001-2-70-1'], ['1808.05001-1-70-2', '1808.05001-2-70-2'], ['1808.05001-1-70-3', '1808.05001-2-70-3'], ['1808.05001-1-25-2', '1808.05001-2-25-2'], ['1808.05001-1-25-6', '1808.05001-2-25-6'], ['1808.05001-1-33-0', '1808.05001-2-33-0'], ['1808.05001-1-33-1', '1808.05001-2-33-1'], ['1808.05001-1-33-2', '1808.05001-2-33-2'], ['1808.05001-1-33-3', '1808.05001-2-33-3'], ['1808.05001-1-33-4', '1808.05001-2-33-4'], ['1808.05001-1-37-0', '1808.05001-2-37-0'], ['1808.05001-1-37-1', '1808.05001-2-37-1'], ['1808.05001-1-37-2', '1808.05001-2-37-2'], ['1808.05001-1-37-3', '1808.05001-2-37-3'], ['1808.05001-1-37-4', '1808.05001-2-37-4'], ['1808.05001-1-37-5', '1808.05001-2-37-5'], ['1808.05001-1-39-0', '1808.05001-2-39-0'], ['1808.05001-1-39-1', '1808.05001-2-39-1'], ['1808.05001-1-39-2', '1808.05001-2-39-2'], ['1808.05001-1-39-3', '1808.05001-2-39-3'], ['1808.05001-1-39-4', '1808.05001-2-39-4'], ['1808.05001-1-39-5', '1808.05001-2-39-5'], ['1808.05001-1-39-6', '1808.05001-2-39-6'], ['1808.05001-1-54-0', '1808.05001-2-54-0'], ['1808.05001-1-21-0', '1808.05001-2-21-0'], ['1808.05001-1-21-1', '1808.05001-2-21-1'], ['1808.05001-1-35-1', '1808.05001-2-35-1'], ['1808.05001-1-51-0', '1808.05001-2-51-0'], ['1808.05001-1-0-0', '1808.05001-2-0-0'], ['1808.05001-1-0-2', '1808.05001-2-0-2'], ['1808.05001-1-0-3', '1808.05001-2-0-3'], ['1808.05001-1-0-4', '1808.05001-2-0-4'], ['1808.05001-1-0-5', '1808.05001-2-0-5'], ['1808.05001-1-41-0', '1808.05001-2-41-0'], ['1808.05001-1-56-0', '1808.05001-2-56-0']]

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[]

[['1808.05001-1-46-0', '1808.05001-2-46-0'], ['1808.05001-1-46-1', '1808.05001-2-46-0']]

[]

['1808.05001-1-1-0', '1808.05001-1-4-2', '1808.05001-1-16-0', '1808.05001-1-28-1', '1808.05001-1-46-3', '1808.05001-1-53-0', '1808.05001-1-53-1', '1808.05001-1-55-0', '1808.05001-2-1-0', '1808.05001-2-4-2', '1808.05001-2-16-0', '1808.05001-2-28-1', '1808.05001-2-46-2', '1808.05001-2-53-0', '1808.05001-2-55-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1808.05001

1901.02860

{'1901.02860-1-0-0': 'Transformer networks have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling.', '1901.02860-1-0-1': 'As a solution, we propose a novel neural architecture, Transformer-XL, that enables Transformer to learn dependency beyond a fixed length without disrupting temporal coherence.', '1901.02860-1-0-2': 'Concretely, it consists of a segment-level recurrence mechanism and a novel positional encoding scheme.', '1901.02860-1-0-3': 'Our method not only enables capturing longer-term dependency, but also resolves the problem of context fragmentation.', '1901.02860-1-0-4': 'As a result, Transformer-XL learns dependency that is about 80% longer than RNNs and 450% longer than vanilla Transformers, achieves better performance on both short and long sequences, and is up to 1,800+ times faster than vanilla Transformer during evaluation.', '1901.02860-1-0-5': 'Additionally, we improve the state-of-the-art (SoTA) results of bpc/perplexity from 1.06 to 0.99 on enwiki8, from 1.13 to 1.08 on text8, from 20.5 to 18.3 on WikiText-103, from 23.7 to 21.8 on One Billion Word, and from 55.3 to 54.5 on Penn Treebank (without finetuning).', '1901.02860-1-0-6': 'Our code, pretrained models, and hyperparameters are available in both Tensorflow and PyTorch.', '1901.02860-1-1-0': '# Introduction', '1901.02860-1-2-0': 'Language modeling is among the important problems that require modeling long-term dependency, with successful applications such as unsupervised pretraining .', '1901.02860-1-2-1': 'However, it has been a challenge to equip neural networks with the capability to model long-term dependency in sequential data.', '1901.02860-1-2-2': 'Recurrent neural networks (RNNs), in particular Long Short-Term Memory (LSTM) networks , have been a standard solution to language modeling and obtained strong results on multiple benchmarks.', '1901.02860-1-2-3': 'Despite the wide adaption, RNNs are difficult to optimize due to gradient vanishing and explosion , and the introduction of gating in LSTMs and the gradient clipping technique might not be sufficient to fully address this issue.', '1901.02860-1-2-4': 'Empirically, previous work has found that LSTM language models use 200 context words on average , indicating room for further improvement.', '1901.02860-1-3-0': 'On the other hand, the direct connections between long-distance word pairs baked in attention mechanisms might ease optimization and enable the learning of long-term dependency .', '1901.02860-1-3-1': 'Recently, [CITATION] designed a set of auxiliary losses to train deep Transformer networks for character-level language modeling, which outperform LSTMs by a large margin.', '1901.02860-1-3-2': 'Despite the success, the LM training in [CITATION] is performed on separated fixed-length segments of a few hundred characters, without any information flow across segments.', '1901.02860-1-3-3': 'As a consequence of the fixed context length, the model cannot capture any longer-term dependency beyond the predefined context length.', '1901.02860-1-3-4': 'In addition, the fixed-length segments are created by selecting a consecutive chunk of symbols without respecting the sentence or any other semantic boundary.', '1901.02860-1-3-5': 'Hence, the model lacks necessary contextual information needed to well predict the first few symbols, leading to inefficient optimization and inferior performance.', '1901.02860-1-3-6': 'We refer to this problem as context fragmentation.', '1901.02860-1-4-0': 'To address the aforementioned limitations of fixed-length contexts, we propose a new architecture called Transformer-XL (meaning extra long).', '1901.02860-1-4-1': 'We introduce the notion of recurrence into our deep self-attention network.', '1901.02860-1-4-2': 'In particular, instead of computing the hidden states from scratch for each new segment, we reuse the hidden states obtained in previous segments.', '1901.02860-1-4-3': 'The reused hidden states serve as memory for the current segment, which builds up a recurrent connection between the segments.', '1901.02860-1-4-4': 'As a result, modeling very long-term dependency becomes possible because information can be propagated through the recurrent connections.', '1901.02860-1-4-5': 'Meanwhile, passing information from the previous segment can also resolve the problem of context fragmentation.', '1901.02860-1-4-6': 'More importantly, we show the necessity of using relative positional encodings rather than absolute ones, in order to enable state reuse without causing temporal confusion.', '1901.02860-1-4-7': 'Hence, as an additional technical contribution, we introduce a simple but more effective relative positional encoding formulation that generalizes to attention lengths longer than the one observed during training.', '1901.02860-1-5-0': 'Transformer-XL obtained strong results on five datasets, varying from word-level to character-level language modeling.', '1901.02860-1-5-1': 'Transformer-XL improves the previous state-of-the-art (SoTA) results from 1.06 to 0.99 in bpc on enwiki8, from 1.13 to 1.08 in bpc on text8, from 20.5 to 18.3 in perplexity on WikiText-103, and from 23.7 to 21.8 in perplexity on One Billion Word.', '1901.02860-1-5-2': 'On small data, Transformer-XL also achieves a perplexity of 54.5 on Penn Treebank without finetuning, which is SoTA when comparable settings are considered.', '1901.02860-1-6-0': 'We use two methods to quantitatively study the effective lengths of Transformer-XL and the baselines.', '1901.02860-1-6-1': 'Similar to [CITATION], we gradually increase the attention length at test time until no further noticeable improvement ([MATH]0.1% relative gains) can be observed.', '1901.02860-1-6-2': 'Our best model in this settings use attention lengths of 1,600 and 3,800 on WikiText-103 and enwiki8 respectively.', '1901.02860-1-6-3': 'In addition, we devise a metric called Relative Effective Context Length (RECL) that aims to perform a fair comparison of the gains brought by increasing the context lengths for different models.', '1901.02860-1-6-4': 'In this setting, Transformer-XL learns a RECL of 900 words on WikiText-103, while the numbers for recurrent networks and Transformer are only 500 and 128.', '1901.02860-1-7-0': '# Related Work', '1901.02860-1-8-0': 'In the last few years, the field of language modeling has witnessed many significant advances, including but not limited to devising novel architectures to better encode the context , improving regularization and optimization algorithms [CITATION], speeding up the Softmax computation , and enriching the output distribution family .', '1901.02860-1-9-0': 'To capture the long-range context in language modeling, a line of work directly feeds a representation of the wider context into the network as an additional input.', '1901.02860-1-9-1': 'Existing works range from ones where context representations are manually defined to others that rely on document-level topics learned from data .', '1901.02860-1-10-0': 'More broadly, in generic sequence modeling, how to capture long-term dependency has been a long-standing research problem.', '1901.02860-1-10-1': 'From this perspective, since the ubiquitous adaption of LSTM, many efforts have been spent on relieving the vanishing gradient problem, including better initialization , additional loss signal , augmented memory structure and others that modify the internal architecture of RNNs to ease the optimization [CITATION].', '1901.02860-1-10-2': 'Different from them, our work is based on the Transformer architecture and shows that language modeling as a real-world task benefits from the ability to learn longer-term dependency.', '1901.02860-1-11-0': '# Model', '1901.02860-1-12-0': 'Given a corpus of tokens [MATH], the task of language modeling is to estimate the joint probability [MATH], which is often auto-regressively factorized as [MATH].', '1901.02860-1-12-1': 'With the factorization, the problem reduces to estimating each conditional factor.', '1901.02860-1-12-2': 'In this work, we stick to the standard neural approach to modeling the conditional probability.', '1901.02860-1-12-3': 'Specifically, a trainable neural network is used to encode the context [MATH] into a fixed size hidden state, which is multiplied with the word embeddings to obtain the logits.', '1901.02860-1-12-4': 'The logits are then fed into the Softmax function, yielding a categorical probability distribution over the next token.', '1901.02860-1-13-0': '## Vanilla Transformer Language Models', '1901.02860-1-14-0': 'In order to apply Transformer or self-attention to language modeling, the central problem is how to train a Transformer to effectively encode an arbitrarily long context into a fixed size representation.', '1901.02860-1-14-1': 'Given infinite memory and computation, a simple solution would be to process the entire context sequence using an unconditional Transformer decoder, similar to a feed-forward neural network.', '1901.02860-1-14-2': 'However, this is usually infeasible with the limited resource in practice.', '1901.02860-1-15-0': 'One feasible but crude approximation is to split the entire corpus into shorter segments of manageable sizes, and only train the model within each segment, ignoring all contextual information from previous segments.', '1901.02860-1-15-1': 'This is the idea adopted by [CITATION].', '1901.02860-1-15-2': 'We call it the vanilla model and visualize it in Fig. [REF].', '1901.02860-1-15-3': 'Under this training paradigm, information never flows across segments in either the forward or backward pass.', '1901.02860-1-15-4': 'There are two critical limitations of using a fixed-length context.', '1901.02860-1-15-5': 'First, the largest possible dependency length is upper bounded by the segment length, which is a few hundred on character-level language modeling .', '1901.02860-1-15-6': 'Therefore, although the self-attention mechanism is less affected by the vanishing gradient problem compared to RNNs, the vanilla model is not able to fully exploit this optimization advantage.', '1901.02860-1-15-7': 'Second, though it is possible to use padding to respect the sentence or other semantic boundaries, in practice it has been standard practice to simply chunk long text into fixed-length segments due to improved efficiency .', '1901.02860-1-15-8': 'However, simply chunking a sequence into fixed-length segments will lead to the context fragmentation problem as discussed in Section [REF].', '1901.02860-1-16-0': 'During evaluation, at each step, the vanilla model also consumes a segment of the same length as in training, but only makes one prediction at the last position.', '1901.02860-1-16-1': 'Then, at the next step, the segment is shifted to the right by only one position, and the new segment has to be processed all from scratch.', '1901.02860-1-16-2': 'As shown in Fig. [REF], this procedure ensures that each prediction utilizes the longest possible context exposed during training, and also relieves context fragmentation issue encountered in training.', '1901.02860-1-16-3': 'However, this evaluation procedure is extremely expensive.', '1901.02860-1-16-4': 'We will show that our proposed architecture is able to substantially improve the evaluation speed.', '1901.02860-1-17-0': '## Segment-Level Recurrence with State Reuse', '1901.02860-1-18-0': 'To address the limitations of using a fixed-length context, we propose to introduce a recurrence mechanism to the Transformer architecture.', '1901.02860-1-18-1': 'During training, the hidden state sequence computed for the previous segment is fixed and cached to be reused as an extended context when the model processes the next new segment, as shown in Fig. [REF].', '1901.02860-1-18-2': 'Although the gradient still remains within a segment, this additional input allows the network to exploit information in the history, leading to an ability of modeling longer-term dependency and avoiding context fragmentation.', '1901.02860-1-18-3': 'Formally, let the two consecutive segments of length [MATH] be [MATH] and [MATH] respectively.', '1901.02860-1-18-4': 'Denoting the [MATH]-th layer hidden state sequence produced for the [MATH]-th segment [MATH] by [MATH], where [MATH] is the hidden dimension.', '1901.02860-1-18-5': 'Then, the [MATH]-th layer hidden state for segment [MATH] is produced (schematically) as follows, [EQUATION] where the function [MATH] stands for stop-gradient, the notation [MATH] indicates the concatenation of two hidden sequences along the length dimension, and [MATH] denotes model parameters.', '1901.02860-1-18-6': 'Compared to the standard Transformer, the critical difference lies in that the key [MATH] and value [MATH] are conditioned on the extended context [MATH] and hence [MATH] cached from the previous segment.', '1901.02860-1-18-7': 'We emphasize this particular design by the green paths in Fig. [REF].', '1901.02860-1-19-0': 'With this recurrence mechanism applied to every two consecutive segments of a corpus, it essentially creates a segment-level recurrence in the hidden states.', '1901.02860-1-19-1': 'As a result, the effective context being utilized can go way beyond just two segments.', '1901.02860-1-19-2': 'However, notice that the recurrent dependency between [MATH] and [MATH] shifts one layer downwards per-segment, which differs from the same-layer recurrence in conventional RNN-LMs.', '1901.02860-1-19-3': 'Consequently, the largest possible dependency length grows linearly w.r.t. the number of layers as well as the segment length, i.e., [MATH], as visualized by the shaded area in Fig. [REF].', '1901.02860-1-19-4': 'This is analogous to truncated BPTT , a technique developed for training RNN-LMs.', '1901.02860-1-19-5': 'However, different from truncated BPTT, our method caches a sequence of hidden states instead of the last one, and should be applied together with the relative positional encoding technique described in Section [REF].', '1901.02860-1-20-0': 'Besides achieving extra long context and resolving fragmentation, another benefit that comes with the recurrence scheme is significantly faster evaluation.', '1901.02860-1-20-1': 'Specifically, during evaluation, the representations from the previous segments can be reused instead of being computed from scratch as in the case of the vanilla model.', '1901.02860-1-20-2': 'In our experiments on enwiki8, Transformer-XL is up to 1,800+ times faster than the vanilla model during evaluation (see Section [REF]).', '1901.02860-1-21-0': 'Finally, notice that the recurrence scheme does not need to be restricted to only the previous segment.', '1901.02860-1-21-1': 'In theory, we can cache as many previous segments as the GPU memory allows, and reuse all of them as the extra context when processing the current segment.', '1901.02860-1-21-2': 'Thus, we can cache a predefined length-[MATH] old hidden states spanning (possibly) multiple segments, and refer to them as the memory [MATH], due to a clear connection to the memory augmented neural networks .', '1901.02860-1-21-3': 'In our experiments, we set [MATH] equal to the segment length during training, and increase it by multiple times during evaluation.', '1901.02860-1-22-0': '## Relative Positional Encodings', '1901.02860-1-23-0': "While we found the idea presented in the previous subsection very appealing, there is a crucial technical challenge we haven't solved in order to reuse the hidden states.", '1901.02860-1-23-1': 'That is, how can we keep the positional information coherent when we reuse the states?', '1901.02860-1-23-2': 'Recall that, in the standard Transformer, the information of sequence order is provided by a set of positional encodings, denoted as [MATH], where the [MATH]-th row [MATH] corresponds to the [MATH]-th absolute position within a segment and [MATH] prescribes the maximum possible length to be modeled.', '1901.02860-1-23-3': 'Then, the actual input to the Transformer is the element-wise addition of the word embeddings and the positional encodings.', '1901.02860-1-23-4': 'If we simply adapt this positional encoding to our recurrence mechanism introduced above, the hidden state sequence would be computed schematically by [EQUATION] where [MATH] is the word embedding sequence of [MATH], and [MATH] represents a transformation function.', '1901.02860-1-23-5': 'Notice that, both [MATH] and [MATH] are associated with the same positional encoding [MATH].', '1901.02860-1-23-6': 'As a result, the model has no information to distinguish the positional difference between [MATH] and [MATH] for any [MATH], resulting in a sheer performance loss.', '1901.02860-1-24-0': 'In order to avoid this failure mode, the fundamental idea is to only encode the relative positional information in the hidden states.', '1901.02860-1-24-1': 'Conceptually, the positional encoding gives the model a temporal clue or "bias" about how information should be gathered, i.e., where to attend.', '1901.02860-1-24-2': 'For the same purpose, instead of incorporating bias statically into the initial embedding, one can inject the same information into the attention score of each layer.', '1901.02860-1-24-3': 'More importantly, it is more intuitive and generalizable to define the temporal bias in a relative manner.', '1901.02860-1-24-4': 'For instance, when a query vector [MATH] attends on the key vectors [MATH], it does not need to know the absolute position of each key vector to identify the temporal order of the segment.', '1901.02860-1-24-5': 'Instead, it suffices to know the relative distance between each key vector [MATH] and itself [MATH], i.e. [MATH].', '1901.02860-1-24-6': 'Practically, one can create a set of relative positional encodings [MATH], where the [MATH]-th row [MATH] indicates a relative distance of [MATH] between two positions.', '1901.02860-1-24-7': 'By injecting the relative distance dynamically into the attention score, the query vector can easily distinguish the representations of [MATH] and [MATH] from their different distances, making the state reuse mechanism feasible.', '1901.02860-1-24-8': "Meanwhile, we won't lose any temporal information, as the absolute position can be recovered recursively from relative distances.", '1901.02860-1-25-0': 'Previously, the idea of relative positional encodings has been explored in the context of machine translation and music generation .', '1901.02860-1-25-1': 'Here, we offer a different derivation, arriving at a new form of relative positional encodings, which not only has a one-to-one correspondence to its absolute counterpart but also enjoys much better generalization empirically (see Section [REF]).', '1901.02860-1-25-2': 'Firstly, in the standard Transformer , the attention score between query [MATH] and key vector [MATH] within the same segment can be decomposed as [EQUATION]', '1901.02860-1-25-3': 'Following the idea of only relying on relative positional information, we propose to re-parameterize the four terms as follows [EQUATION]', '1901.02860-1-25-4': 'Under the new parameterization, each term has an intuitive meaning: term [MATH] represents content-based addressing, term [MATH] captures a content-dependent positional bias, term [MATH] governs a global content bias, and [MATH] encodes a global positional bias.', '1901.02860-1-26-0': 'In comparison, the formulation in [CITATION] only has terms [MATH] and [MATH], dropping the two bias terms [MATH] and [MATH].', '1901.02860-1-26-1': 'Moreover, [CITATION] merge the multiplication [MATH] into a single trainable matrix [MATH], which abandons the inductive bias built into the original sinusoid positional encoding .', '1901.02860-1-26-2': 'In contrast, our relative positional embedding [MATH] adapts the sinusoid formulation.', '1901.02860-1-26-3': 'As a benefit of the inductive bias, a model trained on a memory of some certain length can automatically generalize to a memory several times longer during evaluation.', '1901.02860-1-27-0': 'Equipping the recurrence mechanism with our proposed relative positional embedding, we finally arrive at the Transformer-XL architecture.', '1901.02860-1-27-1': 'For completeness, we summarize the computational procedure for a [MATH]-layer Transformer-XL with a single attention head below:', '1901.02860-1-28-0': 'with [MATH] defined as the word embedding sequence.', '1901.02860-1-28-1': 'In addition, it is worth mentioning that a naive way to compute [MATH] requires computing [MATH] for all pairs [MATH], whose cost is quadratic w.r.t. the sequence length.', '1901.02860-1-28-2': 'However, noticing that the value of [MATH] only ranges from zero to the sequence length, we show a simple computation procedure in Appendix [REF], which reduces the cost to be linear w.r.t. the sequence length.', '1901.02860-1-29-0': '# Experiments', '1901.02860-1-30-0': '## Main Results', '1901.02860-1-31-0': 'We apply Transformer-XL to a variety of datasets on both word-level and character-level language modeling to have a comparison with state-of-the-art systems, including WikiText-103 , enwiki8 , text8 , One Billion Word , and Penn Treebank .', '1901.02860-1-32-0': 'WikiText-103 is the largest available word-level language modeling benchmark with long-term dependency.', '1901.02860-1-32-1': 'It contains 103M training tokens from 28K articles, with an average length of 3.6K tokens per article, which allows testing the ability of long-term dependency modeling.', '1901.02860-1-32-2': 'We set the attention length to 384 during training and 1600 during evaluation.', '1901.02860-1-32-3': 'We adopted adaptive softmax and input representations .', '1901.02860-1-32-4': 'As shown in Table [REF], Transformer-XL reduces the previous SoTA perplexity from 20.5 to 18.3, which demonstrates the superiority of the Transformer-XL architecture.', '1901.02860-1-33-0': 'The dataset enwiki8 contains 100M bytes of unprocessed Wikipedia text.', '1901.02860-1-33-1': 'We compare our architecture with the previous results in Table [REF].', '1901.02860-1-33-2': 'Under the model size constraint, the 12-layer Transformer-XL achieves a new SoTA result, outperforming the 12-layer vanilla Transformer from [CITATION] by 0.05, while both Transformer variants have a large margin over conventional RNN-based models.', '1901.02860-1-33-3': 'Notably, our 12-layer architecture achieves the same result as the 64-layer network from [CITATION], using only 17% of the parameter budget.', '1901.02860-1-33-4': 'In order to see whether better performances can be obtained by increasing the model size, we train 18-layer and 24-layer Transformer-XLs with increased model sizes.', '1901.02860-1-33-5': 'With the attention length 784 during training and 3,800 during evaluation, we obtained a new SoTA result and our method is the first to break through 1.0 on widely-studied character-level benchmarks.', '1901.02860-1-33-6': 'Different from [CITATION], Transformer-XL does not need any auxiliary losses, and thus all benefits are credited to a better architecture.', '1901.02860-1-34-0': 'Similar to but different from enwiki8, text8 contains 100M processed Wikipedia characters created by lowering case the text and removing any character other than the 26 letters a through z, and space.', '1901.02860-1-34-1': 'Due to the similarity, we simply adapt the best model and the same hyper-parameters on enwiki8 to text8 without further tuning.', '1901.02860-1-34-2': 'The comparison with previous methods is summarized in Table [REF].', '1901.02860-1-34-3': 'Again, Transformer-XL achieves the new SoTA result with a clear margin.', '1901.02860-1-35-0': 'One Billion Word does not preserve any long-term dependency because sentences have been shuffled.', '1901.02860-1-35-1': 'Consequently, this dataset mainly tests the ability of modeling only short-term dependency.', '1901.02860-1-35-2': 'The comparison between Transformer-XL and the other methods is shown in Table [REF].', '1901.02860-1-35-3': 'Although Transformer-XL is mainly designed to better capture longer-term dependency, it dramatically improves the single-model SoTA from 23.7 to 21.8.', '1901.02860-1-35-4': 'Specifically, Transformer-XL significantly outperforms a contemporary method using vanilla Transformers [CITATION], suggesting the advantage of Transformer-XL is generalizable to modeling short sequences.', '1901.02860-1-36-0': 'We also report the results on word-level Penn Treebank in Table [REF].', '1901.02860-1-36-1': 'Similar to AWD-LSTM , we apply variational dropout and weight average to Transformer-XL.', '1901.02860-1-36-2': 'With proper regularization, Transformer-XL achieves a new SoTA result among models without two-step finetuning.', '1901.02860-1-36-3': 'Penn Treebank has only 1M training tokens, which implies that Transformer-XL also generalizes well even on small datasets.', '1901.02860-1-37-0': '## Ablation Study', '1901.02860-1-38-0': 'We conduct two sets of ablation studies to examine the effects of two proposed techniques used in Transformer-XL: the recurrence mechanism and the new positional encoding scheme.', '1901.02860-1-39-0': 'The first study is performed on WikiText-103, which requires modeling long-term dependency.', '1901.02860-1-39-1': 'The results are reported in Table [REF].', '1901.02860-1-39-2': 'Among the compared encoding schemes, [CITATION] is relative, while [CITATION] and [CITATION] are absolute.', '1901.02860-1-39-3': '"Full" and "half" losses refer to applying a cross entropy loss to all or the recent half positions in the segment.', '1901.02860-1-39-4': 'We found that absolute encodings only work well with half losses because half losses exclude positions with very short attention lengths during training for better generalization.', '1901.02860-1-39-5': 'Table [REF] shows that both the recurrence mechanism and our encoding scheme are necessary to achieve the best performance, as well as generalizing to longer attention sequences during evaluation time.', '1901.02860-1-39-6': 'Although the backpropagation length during training is only 128, with the two techniques the attention length can be increased to 640 at test time.', '1901.02860-1-39-7': 'In the standard setting with 151M parameters, the perplexity decreases as the attention length increases.', '1901.02860-1-40-0': 'Since the recurrence mechanism costs additional memory, we also compare Transformer-XL with baselines under the same GPU memory constraints.', '1901.02860-1-40-1': 'As shown in Table [REF] in Appendix [REF], despite using a shorter backpropagation length, Transformer-XL remains superior to the baselines.', '1901.02860-1-41-0': 'The second study targets at isolating the effects of resolving the context fragmentation problem from the benefit of capturing longer context length.', '1901.02860-1-41-1': 'In order to achieve this goal, we deliberately choose a dataset that does not require long-term dependency, so that any improvement from establishing the recurrence can be attributed to solving the context fragmentation.', '1901.02860-1-41-2': 'Specifically, we perform this controlled experiment on the One Billion Word dataset, which can only benefit from removing the context fragmentation.', '1901.02860-1-41-3': 'We train a 20-layer Transformer-XL with [MATH]0.3B parameters for 400K steps.', '1901.02860-1-41-4': 'As shown in Table [REF], using segment-level recurrence substantially improves performance even when long-term dependency is not needed, which is consistent with our previous discussion that the recurrence mechanism resolves the context fragmentation problem.', '1901.02860-1-41-5': 'Moreover, our relative positional encodings is also superior to [CITATION] on short sequences.', '1901.02860-1-42-0': '## Relative Effective Context Length', '1901.02860-1-43-0': '[CITATION] proposed a method to evaluate the Effective Context Length (ECL) of a sequence model.', '1901.02860-1-43-1': 'ECL is the longest length to which increasing the context span would lead to a gain more than a threshold.', '1901.02860-1-43-2': 'However, ECL ignores the fact that it is harder to get improvement when a model already achieves a lower perplexity using only a shorter context, and thus it is not suitable for fair comparison among multiple models.', '1901.02860-1-43-3': 'We instead propose a new metric called Relative Effective Context Length (RECL).', '1901.02860-1-43-4': 'RECL is defined on a model group instead of a single model, and the gain of a long context is measure by the relative improvement over the best short context model.', '1901.02860-1-43-5': 'As such, the model group shares the same baseline to enable fair comparison.', '1901.02860-1-43-6': 'RECL also has a parameter [MATH], which means constraining the comparison on top-[MATH] hard examples.', '1901.02860-1-43-7': 'See Appedix [REF] for more details about RECL.', '1901.02860-1-43-8': 'As shown in Table [REF], Transformer-XL manages to model dependency of 900 words long on average with [MATH].', '1901.02860-1-43-9': 'The RECL of Transformer-XL is 80% and 450% longer than recurrent networks and Transformer respectively.', '1901.02860-1-43-10': 'Both the recurrence mechanism and our positional encodings contribute to a longer RECL.', '1901.02860-1-43-11': 'This further substantiates our argument that Transformer-XL is able to model longer-term dependency.', '1901.02860-1-44-0': '## Evaluation Speed', '1901.02860-1-45-0': 'Finally, we compare the evaluation speed of the proposed model with the vanilla Transformer model [CITATION].', '1901.02860-1-45-1': 'As shown in Table [REF], due to the state reuse scheme, Transformer-XL achieves an up to 1,874 times speedup during evaluation compared to the architecture in [CITATION].', '1901.02860-1-46-0': '# Conclusions', '1901.02860-1-47-0': 'We propose a novel architecture, Transformer-XL, for language modeling with self-attention architectures beyond a fixed-length context.', '1901.02860-1-47-1': 'Our main technical contributions include introducing the notion of recurrence in a purely self-attentive model and deriving a novel positional encoding scheme.', '1901.02860-1-47-2': 'These two techniques form a complete set of solutions, as any one of them alone does not address the issue of fixed-length contexts.', '1901.02860-1-47-3': 'Transformer-XL is the first self-attention model that achieves substantially better results than RNNs on both character-level and word-level language modeling.', '1901.02860-1-47-4': 'Transformer-XL is also able to model longer-term dependency than RNNs and Transformer, and achieves substantial speedup during evaluation compared to vanilla Transformers.'}

{'1901.02860-2-0-0': 'Transformer networks have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling.', '1901.02860-2-0-1': 'As a solution, we propose a novel neural architecture, Transformer-XL, that enables Transformer to learn dependency beyond a fixed length without disrupting temporal coherence.', '1901.02860-2-0-2': 'Concretely, it consists of a segment-level recurrence mechanism and a novel positional encoding scheme.', '1901.02860-2-0-3': 'Our method not only enables capturing longer-term dependency, but also resolves the problem of context fragmentation.', '1901.02860-2-0-4': 'As a result, Transformer-XL learns dependency that is about 80% longer than RNNs and 450% longer than vanilla Transformers, achieves better performance on both short and long sequences, and is up to 1,800+ times faster than vanilla Transformer during evaluation.', '1901.02860-2-0-5': 'Additionally, we improve the state-of-the-art (SoTA) results of bpc/perplexity from 1.06 to 0.99 on enwiki8, from 1.13 to 1.08 on text8, from 20.5 to 18.3 on WikiText-103, from 23.7 to 21.8 on One Billion Word, and from 55.3 to 54.5 on Penn Treebank (without finetuning).', '1901.02860-2-0-6': 'Our code, pretrained models, and hyperparameters are available in both Tensorflow and PyTorch.', '1901.02860-2-1-0': '# Introduction', '1901.02860-2-2-0': 'Language modeling is among the important problems that require modeling long-term dependency, with successful applications such as unsupervised pretraining .', '1901.02860-2-2-1': 'However, it has been a challenge to equip neural networks with the capability to model long-term dependency in sequential data.', '1901.02860-2-2-2': 'Recurrent neural networks (RNNs), in particular Long Short-Term Memory (LSTM) networks , have been a standard solution to language modeling and obtained strong results on multiple benchmarks.', '1901.02860-2-2-3': 'Despite the wide adaption, RNNs are difficult to optimize due to gradient vanishing and explosion , and the introduction of gating in LSTMs and the gradient clipping technique might not be sufficient to fully address this issue.', '1901.02860-2-2-4': 'Empirically, previous work has found that LSTM language models use 200 context words on average , indicating room for further improvement.', '1901.02860-2-3-0': 'On the other hand, the direct connections between long-distance word pairs baked in attention mechanisms might ease optimization and enable the learning of long-term dependency .', '1901.02860-2-3-1': 'Recently, [CITATION] designed a set of auxiliary losses to train deep Transformer networks for character-level language modeling, which outperform LSTMs by a large margin.', '1901.02860-2-3-2': 'Despite the success, the LM training in [CITATION] is performed on separated fixed-length segments of a few hundred characters, without any information flow across segments.', '1901.02860-2-3-3': 'As a consequence of the fixed context length, the model cannot capture any longer-term dependency beyond the predefined context length.', '1901.02860-2-3-4': 'In addition, the fixed-length segments are created by selecting a consecutive chunk of symbols without respecting the sentence or any other semantic boundary.', '1901.02860-2-3-5': 'Hence, the model lacks necessary contextual information needed to well predict the first few symbols, leading to inefficient optimization and inferior performance.', '1901.02860-2-3-6': 'We refer to this problem as context fragmentation.', '1901.02860-2-4-0': 'To address the aforementioned limitations of fixed-length contexts, we propose a new architecture called Transformer-XL (meaning extra long).', '1901.02860-2-4-1': 'We introduce the notion of recurrence into our deep self-attention network.', '1901.02860-2-4-2': 'In particular, instead of computing the hidden states from scratch for each new segment, we reuse the hidden states obtained in previous segments.', '1901.02860-2-4-3': 'The reused hidden states serve as memory for the current segment, which builds up a recurrent connection between the segments.', '1901.02860-2-4-4': 'As a result, modeling very long-term dependency becomes possible because information can be propagated through the recurrent connections.', '1901.02860-2-4-5': 'Meanwhile, passing information from the previous segment can also resolve the problem of context fragmentation.', '1901.02860-2-4-6': 'More importantly, we show the necessity of using relative positional encodings rather than absolute ones, in order to enable state reuse without causing temporal confusion.', '1901.02860-2-4-7': 'Hence, as an additional technical contribution, we introduce a simple but more effective relative positional encoding formulation that generalizes to attention lengths longer than the one observed during training.', '1901.02860-2-5-0': 'Transformer-XL obtained strong results on five datasets, varying from word-level to character-level language modeling.', '1901.02860-2-5-1': 'Transformer-XL improves the previous state-of-the-art (SoTA) results from 1.06 to 0.99 in bpc on enwiki8, from 1.13 to 1.08 in bpc on text8, from 20.5 to 18.3 in perplexity on WikiText-103, and from 23.7 to 21.8 in perplexity on One Billion Word.', '1901.02860-2-5-2': 'On small data, Transformer-XL also achieves a perplexity of 54.5 on Penn Treebank without finetuning, which is SoTA when comparable settings are considered.', '1901.02860-2-6-0': 'We use two methods to quantitatively study the effective lengths of Transformer-XL and the baselines.', '1901.02860-2-6-1': 'Similar to [CITATION], we gradually increase the attention length at test time until no further noticeable improvement ([MATH]0.1% relative gains) can be observed.', '1901.02860-2-6-2': 'Our best model in this settings use attention lengths of 1,600 and 3,800 on WikiText-103 and enwiki8 respectively.', '1901.02860-2-6-3': 'In addition, we devise a metric called Relative Effective Context Length (RECL) that aims to perform a fair comparison of the gains brought by increasing the context lengths for different models.', '1901.02860-2-6-4': 'In this setting, Transformer-XL learns a RECL of 900 words on WikiText-103, while the numbers for recurrent networks and Transformer are only 500 and 128.', '1901.02860-2-7-0': '# Related Work', '1901.02860-2-8-0': 'In the last few years, the field of language modeling has witnessed many significant advances, including but not limited to devising novel architectures to better encode the context , improving regularization and optimization algorithms [CITATION], speeding up the Softmax computation , and enriching the output distribution family .', '1901.02860-2-9-0': 'To capture the long-range context in language modeling, a line of work directly feeds a representation of the wider context into the network as an additional input.', '1901.02860-2-9-1': 'Existing works range from ones where context representations are manually defined to others that rely on document-level topics learned from data .', '1901.02860-2-10-0': 'More broadly, in generic sequence modeling, how to capture long-term dependency has been a long-standing research problem.', '1901.02860-2-10-1': 'From this perspective, since the ubiquitous adaption of LSTM, many efforts have been spent on relieving the vanishing gradient problem, including better initialization , additional loss signal , augmented memory structure and others that modify the internal architecture of RNNs to ease the optimization [CITATION].', '1901.02860-2-10-2': 'Different from them, our work is based on the Transformer architecture and shows that language modeling as a real-world task benefits from the ability to learn longer-term dependency.', '1901.02860-2-11-0': '# Model', '1901.02860-2-12-0': 'Given a corpus of tokens [MATH], the task of language modeling is to estimate the joint probability [MATH], which is often auto-regressively factorized as [MATH].', '1901.02860-2-12-1': 'With the factorization, the problem reduces to estimating each conditional factor.', '1901.02860-2-12-2': 'In this work, we stick to the standard neural approach to modeling the conditional probability.', '1901.02860-2-12-3': 'Specifically, a trainable neural network is used to encode the context [MATH] into a fixed size hidden state, which is multiplied with the word embeddings to obtain the logits.', '1901.02860-2-12-4': 'The logits are then fed into the Softmax function, yielding a categorical probability distribution over the next token.', '1901.02860-2-13-0': '## Vanilla Transformer Language Models', '1901.02860-2-14-0': 'In order to apply Transformer or self-attention to language modeling, the central problem is how to train a Transformer to effectively encode an arbitrarily long context into a fixed size representation.', '1901.02860-2-14-1': 'Given infinite memory and computation, a simple solution would be to process the entire context sequence using an unconditional Transformer decoder, similar to a feed-forward neural network.', '1901.02860-2-14-2': 'However, this is usually infeasible with the limited resource in practice.', '1901.02860-2-15-0': 'One feasible but crude approximation is to split the entire corpus into shorter segments of manageable sizes, and only train the model within each segment, ignoring all contextual information from previous segments.', '1901.02860-2-15-1': 'This is the idea adopted by [CITATION].', '1901.02860-2-15-2': 'We call it the vanilla model and visualize it in Fig. [REF].', '1901.02860-2-15-3': 'Under this training paradigm, information never flows across segments in either the forward or backward pass.', '1901.02860-2-15-4': 'There are two critical limitations of using a fixed-length context.', '1901.02860-2-15-5': 'First, the largest possible dependency length is upper bounded by the segment length, which is a few hundred on character-level language modeling .', '1901.02860-2-15-6': 'Therefore, although the self-attention mechanism is less affected by the vanishing gradient problem compared to RNNs, the vanilla model is not able to fully exploit this optimization advantage.', '1901.02860-2-15-7': 'Second, though it is possible to use padding to respect the sentence or other semantic boundaries, in practice it has been standard practice to simply chunk long text into fixed-length segments due to improved efficiency .', '1901.02860-2-15-8': 'However, simply chunking a sequence into fixed-length segments will lead to the context fragmentation problem as discussed in Section [REF].', '1901.02860-2-16-0': 'During evaluation, at each step, the vanilla model also consumes a segment of the same length as in training, but only makes one prediction at the last position.', '1901.02860-2-16-1': 'Then, at the next step, the segment is shifted to the right by only one position, and the new segment has to be processed all from scratch.', '1901.02860-2-16-2': 'As shown in Fig. [REF], this procedure ensures that each prediction utilizes the longest possible context exposed during training, and also relieves context fragmentation issue encountered in training.', '1901.02860-2-16-3': 'However, this evaluation procedure is extremely expensive.', '1901.02860-2-16-4': 'We will show that our proposed architecture is able to substantially improve the evaluation speed.', '1901.02860-2-17-0': '## Segment-Level Recurrence with State Reuse', '1901.02860-2-18-0': 'To address the limitations of using a fixed-length context, we propose to introduce a recurrence mechanism to the Transformer architecture.', '1901.02860-2-18-1': 'During training, the hidden state sequence computed for the previous segment is fixed and cached to be reused as an extended context when the model processes the next new segment, as shown in Fig. [REF].', '1901.02860-2-18-2': 'Although the gradient still remains within a segment, this additional input allows the network to exploit information in the history, leading to an ability of modeling longer-term dependency and avoiding context fragmentation.', '1901.02860-2-18-3': 'Formally, let the two consecutive segments of length [MATH] be [MATH] and [MATH] respectively.', '1901.02860-2-18-4': 'Denoting the [MATH]-th layer hidden state sequence produced for the [MATH]-th segment [MATH] by [MATH], where [MATH] is the hidden dimension.', '1901.02860-2-18-5': 'Then, the [MATH]-th layer hidden state for segment [MATH] is produced (schematically) as follows, [EQUATION] where the function [MATH] stands for stop-gradient, the notation [MATH] indicates the concatenation of two hidden sequences along the length dimension, and [MATH] denotes model parameters.', '1901.02860-2-18-6': 'Compared to the standard Transformer, the critical difference lies in that the key [MATH] and value [MATH] are conditioned on the extended context [MATH] and hence [MATH] cached from the previous segment.', '1901.02860-2-18-7': 'We emphasize this particular design by the green paths in Fig. [REF].', '1901.02860-2-19-0': 'With this recurrence mechanism applied to every two consecutive segments of a corpus, it essentially creates a segment-level recurrence in the hidden states.', '1901.02860-2-19-1': 'As a result, the effective context being utilized can go way beyond just two segments.', '1901.02860-2-19-2': 'However, notice that the recurrent dependency between [MATH] and [MATH] shifts one layer downwards per-segment, which differs from the same-layer recurrence in conventional RNN-LMs.', '1901.02860-2-19-3': 'Consequently, the largest possible dependency length grows linearly w.r.t. the number of layers as well as the segment length, i.e., [MATH], as visualized by the shaded area in Fig. [REF].', '1901.02860-2-19-4': 'This is analogous to truncated BPTT , a technique developed for training RNN-LMs.', '1901.02860-2-19-5': 'However, different from truncated BPTT, our method caches a sequence of hidden states instead of the last one, and should be applied together with the relative positional encoding technique described in Section [REF].', '1901.02860-2-20-0': 'Besides achieving extra long context and resolving fragmentation, another benefit that comes with the recurrence scheme is significantly faster evaluation.', '1901.02860-2-20-1': 'Specifically, during evaluation, the representations from the previous segments can be reused instead of being computed from scratch as in the case of the vanilla model.', '1901.02860-2-20-2': 'In our experiments on enwiki8, Transformer-XL is up to 1,800+ times faster than the vanilla model during evaluation (see Section [REF]).', '1901.02860-2-21-0': 'Finally, notice that the recurrence scheme does not need to be restricted to only the previous segment.', '1901.02860-2-21-1': 'In theory, we can cache as many previous segments as the GPU memory allows, and reuse all of them as the extra context when processing the current segment.', '1901.02860-2-21-2': 'Thus, we can cache a predefined length-[MATH] old hidden states spanning (possibly) multiple segments, and refer to them as the memory [MATH], due to a clear connection to the memory augmented neural networks .', '1901.02860-2-21-3': 'In our experiments, we set [MATH] equal to the segment length during training, and increase it by multiple times during evaluation.', '1901.02860-2-22-0': '## Relative Positional Encodings', '1901.02860-2-23-0': "While we found the idea presented in the previous subsection very appealing, there is a crucial technical challenge we haven't solved in order to reuse the hidden states.", '1901.02860-2-23-1': 'That is, how can we keep the positional information coherent when we reuse the states?', '1901.02860-2-23-2': 'Recall that, in the standard Transformer, the information of sequence order is provided by a set of positional encodings, denoted as [MATH], where the [MATH]-th row [MATH] corresponds to the [MATH]-th absolute position within a segment and [MATH] prescribes the maximum possible length to be modeled.', '1901.02860-2-23-3': 'Then, the actual input to the Transformer is the element-wise addition of the word embeddings and the positional encodings.', '1901.02860-2-23-4': 'If we simply adapt this positional encoding to our recurrence mechanism introduced above, the hidden state sequence would be computed schematically by [EQUATION] where [MATH] is the word embedding sequence of [MATH], and [MATH] represents a transformation function.', '1901.02860-2-23-5': 'Notice that, both [MATH] and [MATH] are associated with the same positional encoding [MATH].', '1901.02860-2-23-6': 'As a result, the model has no information to distinguish the positional difference between [MATH] and [MATH] for any [MATH], resulting in a sheer performance loss.', '1901.02860-2-24-0': 'In order to avoid this failure mode, the fundamental idea is to only encode the relative positional information in the hidden states.', '1901.02860-2-24-1': 'Conceptually, the positional encoding gives the model a temporal clue or "bias" about how information should be gathered, i.e., where to attend.', '1901.02860-2-24-2': 'For the same purpose, instead of incorporating bias statically into the initial embedding, one can inject the same information into the attention score of each layer.', '1901.02860-2-24-3': 'More importantly, it is more intuitive and generalizable to define the temporal bias in a relative manner.', '1901.02860-2-24-4': 'For instance, when a query vector [MATH] attends on the key vectors [MATH], it does not need to know the absolute position of each key vector to identify the temporal order of the segment.', '1901.02860-2-24-5': 'Instead, it suffices to know the relative distance between each key vector [MATH] and itself [MATH], i.e. [MATH].', '1901.02860-2-24-6': 'Practically, one can create a set of relative positional encodings [MATH], where the [MATH]-th row [MATH] indicates a relative distance of [MATH] between two positions.', '1901.02860-2-24-7': 'By injecting the relative distance dynamically into the attention score, the query vector can easily distinguish the representations of [MATH] and [MATH] from their different distances, making the state reuse mechanism feasible.', '1901.02860-2-24-8': "Meanwhile, we won't lose any temporal information, as the absolute position can be recovered recursively from relative distances.", '1901.02860-2-25-0': 'Previously, the idea of relative positional encodings has been explored in the context of machine translation and music generation .', '1901.02860-2-25-1': 'Here, we offer a different derivation, arriving at a new form of relative positional encodings, which not only has a one-to-one correspondence to its absolute counterpart but also enjoys much better generalization empirically (see Section [REF]).', '1901.02860-2-25-2': 'Firstly, in the standard Transformer , the attention score between query [MATH] and key vector [MATH] within the same segment can be decomposed as [EQUATION]', '1901.02860-2-25-3': 'Following the idea of only relying on relative positional information, we propose to re-parameterize the four terms as follows [EQUATION]', '1901.02860-2-25-4': 'Under the new parameterization, each term has an intuitive meaning: term [MATH] represents content-based addressing, term [MATH] captures a content-dependent positional bias, term [MATH] governs a global content bias, and [MATH] encodes a global positional bias.', '1901.02860-2-26-0': 'In comparison, the formulation in [CITATION] only has terms [MATH] and [MATH], dropping the two bias terms [MATH] and [MATH].', '1901.02860-2-26-1': 'Moreover, [CITATION] merge the multiplication [MATH] into a single trainable matrix [MATH], which abandons the inductive bias built into the original sinusoid positional encoding .', '1901.02860-2-26-2': 'In contrast, our relative positional embedding [MATH] adapts the sinusoid formulation.', '1901.02860-2-26-3': 'As a benefit of the inductive bias, a model trained on a memory of some certain length can automatically generalize to a memory several times longer during evaluation.', '1901.02860-2-27-0': 'Equipping the recurrence mechanism with our proposed relative positional embedding, we finally arrive at the Transformer-XL architecture.', '1901.02860-2-27-1': 'For completeness, we summarize the computational procedure for a [MATH]-layer Transformer-XL with a single attention head below:', '1901.02860-2-28-0': 'with [MATH] defined as the word embedding sequence.', '1901.02860-2-28-1': 'In addition, it is worth mentioning that a naive way to compute [MATH] requires computing [MATH] for all pairs [MATH], whose cost is quadratic w.r.t. the sequence length.', '1901.02860-2-28-2': 'However, noticing that the value of [MATH] only ranges from zero to the sequence length, we show a simple computation procedure in Appendix [REF], which reduces the cost to be linear w.r.t. the sequence length.', '1901.02860-2-29-0': '# Experiments', '1901.02860-2-30-0': '## Main Results', '1901.02860-2-31-0': 'We apply Transformer-XL to a variety of datasets on both word-level and character-level language modeling to have a comparison with state-of-the-art systems, including WikiText-103 , enwiki8 , text8 , One Billion Word , and Penn Treebank .', '1901.02860-2-32-0': 'WikiText-103 is the largest available word-level language modeling benchmark with long-term dependency.', '1901.02860-2-32-1': 'It contains 103M training tokens from 28K articles, with an average length of 3.6K tokens per article, which allows testing the ability of long-term dependency modeling.', '1901.02860-2-32-2': 'We set the attention length to 384 during training and 1600 during evaluation.', '1901.02860-2-32-3': 'We adopted adaptive softmax and input representations .', '1901.02860-2-32-4': 'As shown in Table [REF], Transformer-XL reduces the previous SoTA perplexity from 20.5 to 18.3, which demonstrates the superiority of the Transformer-XL architecture.', '1901.02860-2-33-0': 'The dataset enwiki8 contains 100M bytes of unprocessed Wikipedia text.', '1901.02860-2-33-1': 'We compare our architecture with the previous results in Table [REF].', '1901.02860-2-33-2': 'Under the model size constraint, the 12-layer Transformer-XL achieves a new SoTA result, outperforming the 12-layer vanilla Transformer from [CITATION] by 0.05, while both Transformer variants have a large margin over conventional RNN-based models.', '1901.02860-2-33-3': 'Notably, our 12-layer architecture achieves the same result as the 64-layer network from [CITATION], using only 17% of the parameter budget.', '1901.02860-2-33-4': 'In order to see whether better performances can be obtained by increasing the model size, we train 18-layer and 24-layer Transformer-XLs with increased model sizes.', '1901.02860-2-33-5': 'With the attention length 784 during training and 3,800 during evaluation, we obtained a new SoTA result and our method is the first to break through 1.0 on widely-studied character-level benchmarks.', '1901.02860-2-33-6': 'Different from [CITATION], Transformer-XL does not need any auxiliary losses, and thus all benefits are credited to a better architecture.', '1901.02860-2-34-0': 'Similar to but different from enwiki8, text8 contains 100M processed Wikipedia characters created by lowering case the text and removing any character other than the 26 letters a through z, and space.', '1901.02860-2-34-1': 'Due to the similarity, we simply adapt the best model and the same hyper-parameters on enwiki8 to text8 without further tuning.', '1901.02860-2-34-2': 'The comparison with previous methods is summarized in Table [REF].', '1901.02860-2-34-3': 'Again, Transformer-XL achieves the new SoTA result with a clear margin.', '1901.02860-2-35-0': 'One Billion Word does not preserve any long-term dependency because sentences have been shuffled.', '1901.02860-2-35-1': 'Consequently, this dataset mainly tests the ability of modeling only short-term dependency.', '1901.02860-2-35-2': 'The comparison between Transformer-XL and the other methods is shown in Table [REF].', '1901.02860-2-35-3': 'Although Transformer-XL is mainly designed to better capture longer-term dependency, it dramatically improves the single-model SoTA from 23.7 to 21.8.', '1901.02860-2-35-4': 'Specifically, Transformer-XL significantly outperforms a contemporary method using vanilla Transformers [CITATION], suggesting the advantage of Transformer-XL is generalizable to modeling short sequences.', '1901.02860-2-36-0': 'We also report the results on word-level Penn Treebank in Table [REF].', '1901.02860-2-36-1': 'Similar to AWD-LSTM , we apply variational dropout and weight average to Transformer-XL.', '1901.02860-2-36-2': 'With proper regularization, Transformer-XL achieves a new SoTA result among models without two-step finetuning.', '1901.02860-2-36-3': 'Penn Treebank has only 1M training tokens, which implies that Transformer-XL also generalizes well even on small datasets.', '1901.02860-2-37-0': '## Ablation Study', '1901.02860-2-38-0': 'We conduct two sets of ablation studies to examine the effects of two proposed techniques used in Transformer-XL: the recurrence mechanism and the new positional encoding scheme.', '1901.02860-2-39-0': 'The first study is performed on WikiText-103, which requires modeling long-term dependency.', '1901.02860-2-39-1': 'The results are reported in Table [REF].', '1901.02860-2-39-2': 'Among the compared encoding schemes, [CITATION] is relative, while [CITATION] and [CITATION] are absolute.', '1901.02860-2-39-3': '"Full" and "half" losses refer to applying a cross entropy loss to all or the recent half positions in the segment.', '1901.02860-2-39-4': 'We found that absolute encodings only work well with half losses because half losses exclude positions with very short attention lengths during training for better generalization.', '1901.02860-2-39-5': 'Table [REF] shows that both the recurrence mechanism and our encoding scheme are necessary to achieve the best performance, as well as generalizing to longer attention sequences during evaluation time.', '1901.02860-2-39-6': 'Although the backpropagation length during training is only 128, with the two techniques the attention length can be increased to 640 at test time.', '1901.02860-2-39-7': 'In the standard setting with 151M parameters, the perplexity decreases as the attention length increases.', '1901.02860-2-40-0': 'Since the recurrence mechanism costs additional memory, we also compare Transformer-XL with baselines under the same GPU memory constraints.', '1901.02860-2-40-1': 'As shown in Table [REF] in Appendix [REF], despite using a shorter backpropagation length, Transformer-XL remains superior to the baselines.', '1901.02860-2-41-0': 'The second study targets at isolating the effects of resolving the context fragmentation problem from the benefit of capturing longer context length.', '1901.02860-2-41-1': 'In order to achieve this goal, we deliberately choose a dataset that does not require long-term dependency, so that any improvement from establishing the recurrence can be attributed to solving the context fragmentation.', '1901.02860-2-41-2': 'Specifically, we perform this controlled experiment on the One Billion Word dataset, which can only benefit from removing the context fragmentation.', '1901.02860-2-41-3': 'We train a 20-layer Transformer-XL with [MATH]0.3B parameters for 400K steps.', '1901.02860-2-41-4': 'As shown in Table [REF], using segment-level recurrence substantially improves performance even when long-term dependency is not needed, which is consistent with our previous discussion that the recurrence mechanism resolves the context fragmentation problem.', '1901.02860-2-41-5': 'Moreover, our relative positional encodings is also superior to [CITATION] on short sequences.', '1901.02860-2-42-0': '## Relative Effective Context Length', '1901.02860-2-43-0': '[CITATION] proposed a method to evaluate the Effective Context Length (ECL) of a sequence model.', '1901.02860-2-43-1': 'ECL is the longest length to which increasing the context span would lead to a gain more than a threshold.', '1901.02860-2-43-2': 'However, ECL ignores the fact that it is harder to get improvement when a model already achieves a lower perplexity using only a shorter context, and thus it is not suitable for fair comparison among multiple models.', '1901.02860-2-43-3': 'We instead propose a new metric called Relative Effective Context Length (RECL).', '1901.02860-2-43-4': 'RECL is defined on a model group instead of a single model, and the gain of a long context is measure by the relative improvement over the best short context model.', '1901.02860-2-43-5': 'As such, the model group shares the same baseline to enable fair comparison.', '1901.02860-2-43-6': 'RECL also has a parameter [MATH], which means constraining the comparison on top-[MATH] hard examples.', '1901.02860-2-43-7': 'See Appedix [REF] for more details about RECL.', '1901.02860-2-43-8': 'As shown in Table [REF], Transformer-XL manages to model dependency of 900 words long on average with [MATH].', '1901.02860-2-43-9': 'The RECL of Transformer-XL is 80% and 450% longer than recurrent networks and Transformer respectively.', '1901.02860-2-43-10': 'Both the recurrence mechanism and our positional encodings contribute to a longer RECL.', '1901.02860-2-43-11': 'This further substantiates our argument that Transformer-XL is able to model longer-term dependency.', '1901.02860-2-44-0': '## Evaluation Speed', '1901.02860-2-45-0': 'Finally, we compare the evaluation speed of the proposed model with the vanilla Transformer model [CITATION].', '1901.02860-2-45-1': 'As shown in Table [REF], due to the state reuse scheme, Transformer-XL achieves an up to 1,874 times speedup during evaluation compared to the architecture in [CITATION].', '1901.02860-2-46-0': '# Conclusions', '1901.02860-2-47-0': 'We propose a novel architecture, Transformer-XL, for language modeling with self-attention architectures beyond a fixed-length context.', '1901.02860-2-47-1': 'Our main technical contributions include introducing the notion of recurrence in a purely self-attentive model and deriving a novel positional encoding scheme.', '1901.02860-2-47-2': 'These two techniques form a complete set of solutions, as any one of them alone does not address the issue of fixed-length contexts.', '1901.02860-2-47-3': 'Transformer-XL is the first self-attention model that achieves substantially better results than RNNs on both character-level and word-level language modeling.', '1901.02860-2-47-4': 'Transformer-XL is also able to model longer-term dependency than RNNs and Transformer, and achieves substantial speedup during evaluation compared to vanilla Transformers.'}

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'1901.02860-3-43-4'], ['1901.02860-2-39-5', '1901.02860-3-43-5'], ['1901.02860-2-39-6', '1901.02860-3-43-6'], ['1901.02860-2-39-7', '1901.02860-3-43-7'], ['1901.02860-2-12-0', '1901.02860-3-12-0'], ['1901.02860-2-12-1', '1901.02860-3-12-1'], ['1901.02860-2-12-2', '1901.02860-3-12-2'], ['1901.02860-2-12-3', '1901.02860-3-12-3'], ['1901.02860-2-12-4', '1901.02860-3-12-4'], ['1901.02860-2-23-0', '1901.02860-3-24-0'], ['1901.02860-2-23-1', '1901.02860-3-24-1'], ['1901.02860-2-23-2', '1901.02860-3-24-2'], ['1901.02860-2-23-3', '1901.02860-3-24-3'], ['1901.02860-2-23-5', '1901.02860-3-25-1'], ['1901.02860-2-23-6', '1901.02860-3-25-2'], ['1901.02860-2-47-1', '1901.02860-3-6-0'], ['1901.02860-2-47-2', '1901.02860-3-6-1'], ['1901.02860-2-47-3', '1901.02860-3-6-2'], ['1901.02860-2-25-0', '1901.02860-3-27-0'], ['1901.02860-2-25-1', '1901.02860-3-27-1'], ['1901.02860-2-25-4', '1901.02860-3-29-0'], ['1901.02860-2-18-0', '1901.02860-3-18-0'], ['1901.02860-2-18-1', '1901.02860-3-18-1'], ['1901.02860-2-18-2', '1901.02860-3-18-2'], ['1901.02860-2-18-3', '1901.02860-3-18-3'], ['1901.02860-2-18-4', '1901.02860-3-18-4'], ['1901.02860-2-18-6', '1901.02860-3-19-1'], ['1901.02860-2-18-7', '1901.02860-3-19-2']]

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[]

[['1901.02860-2-23-4', '1901.02860-3-24-4'], ['1901.02860-2-23-4', '1901.02860-3-25-0'], ['1901.02860-2-47-4', '1901.02860-3-53-0'], ['1901.02860-2-18-5', '1901.02860-3-18-5'], ['1901.02860-2-18-5', '1901.02860-3-19-0']]

[]

['1901.02860-1-27-1', '1901.02860-2-27-1', '1901.02860-3-31-2']

{'1': 'http://creativecommons.org/licenses/by-nc-sa/4.0/', '2': 'http://creativecommons.org/licenses/by-nc-sa/4.0/', '3': 'http://creativecommons.org/licenses/by-nc-sa/4.0/'}

https://arxiv.org/abs/1901.02860

{'1901.02860-3-0-0': 'Transformers have a potential of learning longer-term dependency, but are limited by a fixed-length context in the setting of language modeling.', '1901.02860-3-0-1': 'We propose a novel neural architecture Transformer-XL that enables learning dependency beyond a fixed length without disrupting temporal coherence.', '1901.02860-3-0-2': 'It consists of a segment-level recurrence mechanism and a novel positional encoding scheme.', '1901.02860-3-0-3': 'Our method not only enables capturing longer-term dependency, but also resolves the context fragmentation problem.', '1901.02860-3-0-4': 'As a result, Transformer-XL learns dependency that is 80% longer than RNNs and 450% longer than vanilla Transformers, achieves better performance on both short and long sequences, and is up to 1,800+ times faster than vanilla Transformers during evaluation.', '1901.02860-3-0-5': 'Notably, we improve the state-of-the-art results of bpc/perplexity to 0.99 on enwiki8, 1.08 on text8, 18.3 on WikiText-103, 21.8 on One Billion Word, and 54.5 on Penn Treebank (without finetuning).', '1901.02860-3-0-6': 'When trained only on WikiText-103, Transformer-XL manages to generate reasonably coherent, novel text articles with thousands of tokens.', '1901.02860-3-0-7': 'Our code, pretrained models, and hyperparameters are available in both Tensorflow and PyTorch.', '1901.02860-3-1-0': '# Introduction', '1901.02860-3-2-0': 'Language modeling is among the important problems that require modeling long-term dependency, with successful applications such as unsupervised pretraining .', '1901.02860-3-2-1': 'However, it has been a challenge to equip neural networks with the capability to model long-term dependency in sequential data.', '1901.02860-3-2-2': 'Recurrent neural networks (RNNs), in particular Long Short-Term Memory (LSTM) networks , have been a standard solution to language modeling and obtained strong results on multiple benchmarks.', '1901.02860-3-2-3': 'Despite the wide adaption, RNNs are difficult to optimize due to gradient vanishing and explosion , and the introduction of gating in LSTMs and the gradient clipping technique might not be sufficient to fully address this issue.', '1901.02860-3-2-4': 'Empirically, previous work has found that LSTM language models use 200 context words on average , indicating room for further improvement.', '1901.02860-3-3-0': 'On the other hand, the direct connections between long-distance word pairs baked in attention mechanisms might ease optimization and enable the learning of long-term dependency .', '1901.02860-3-3-1': 'Recently, [CITATION] designed a set of auxiliary losses to train deep Transformer networks for character-level language modeling, which outperform LSTMs by a large margin.', '1901.02860-3-3-2': 'Despite the success, the LM training in [CITATION] is performed on separated fixed-length segments of a few hundred characters, without any information flow across segments.', '1901.02860-3-3-3': 'As a consequence of the fixed context length, the model cannot capture any longer-term dependency beyond the predefined context length.', '1901.02860-3-3-4': 'In addition, the fixed-length segments are created by selecting a consecutive chunk of symbols without respecting the sentence or any other semantic boundary.', '1901.02860-3-3-5': 'Hence, the model lacks necessary contextual information needed to well predict the first few symbols, leading to inefficient optimization and inferior performance.', '1901.02860-3-3-6': 'We refer to this problem as context fragmentation.', '1901.02860-3-4-0': 'To address the aforementioned limitations of fixed-length contexts, we propose a new architecture called Transformer-XL (meaning extra long).', '1901.02860-3-4-1': 'We introduce the notion of recurrence into our deep self-attention network.', '1901.02860-3-4-2': 'In particular, instead of computing the hidden states from scratch for each new segment, we reuse the hidden states obtained in previous segments.', '1901.02860-3-4-3': 'The reused hidden states serve as memory for the current segment, which builds up a recurrent connection between the segments.', '1901.02860-3-4-4': 'As a result, modeling very long-term dependency becomes possible because information can be propagated through the recurrent connections.', '1901.02860-3-4-5': 'Meanwhile, passing information from the previous segment can also resolve the problem of context fragmentation.', '1901.02860-3-4-6': 'More importantly, we show the necessity of using relative positional encodings rather than absolute ones, in order to enable state reuse without causing temporal confusion.', '1901.02860-3-4-7': 'Hence, as an additional technical contribution, we introduce a simple but more effective relative positional encoding formulation that generalizes to attention lengths longer than the one observed during training.', '1901.02860-3-5-0': 'Transformer-XL obtained strong results on five datasets, varying from word-level to character-level language modeling.', '1901.02860-3-5-1': 'Transformer-XL is also able to generate relatively coherent long text articles with thousands of tokens (see Appendix [REF]), trained on only 100M tokens.', '1901.02860-3-6-0': 'Our main technical contributions include introducing the notion of recurrence in a purely self-attentive model and deriving a novel positional encoding scheme.', '1901.02860-3-6-1': 'These two techniques form a complete set of solutions, as any one of them alone does not address the issue of fixed-length contexts.', '1901.02860-3-6-2': 'Transformer-XL is the first self-attention model that achieves substantially better results than RNNs on both character-level and word-level language modeling.', '1901.02860-3-7-0': '# Related Work', '1901.02860-3-8-0': 'In the last few years, the field of language modeling has witnessed many significant advances, including but not limited to devising novel architectures to better encode the context , improving regularization and optimization algorithms [CITATION] , speeding up the Softmax computation , and enriching the output distribution family .', '1901.02860-3-9-0': 'To capture the long-range context in language modeling, a line of work directly feeds a representation of the wider context into the network as an additional input.', '1901.02860-3-9-1': 'Existing works range from ones where context representations are manually defined to others that rely on document-level topics learned from data .', '1901.02860-3-10-0': 'More broadly, in generic sequence modeling, how to capture long-term dependency has been a long-standing research problem.', '1901.02860-3-10-1': 'From this perspective, since the ubiquitous adaption of LSTM, many efforts have been spent on relieving the vanishing gradient problem, including better initialization , additional loss signal , augmented memory structure and others that modify the internal architecture of RNNs to ease the optimization [CITATION].', '1901.02860-3-10-2': 'Different from them, our work is based on the Transformer architecture and shows that language modeling as a real-world task benefits from the ability to learn longer-term dependency.', '1901.02860-3-11-0': '# Model', '1901.02860-3-12-0': 'Given a corpus of tokens [MATH], the task of language modeling is to estimate the joint probability [MATH], which is often auto-regressively factorized as [MATH].', '1901.02860-3-12-1': 'With the factorization, the problem reduces to estimating each conditional factor.', '1901.02860-3-12-2': 'In this work, we stick to the standard neural approach to modeling the conditional probability.', '1901.02860-3-12-3': 'Specifically, a trainable neural network is used to encode the context [MATH] into a fixed size hidden state, which is multiplied with the word embeddings to obtain the logits.', '1901.02860-3-12-4': 'The logits are then fed into the Softmax function, yielding a categorical probability distribution over the next token.', '1901.02860-3-13-0': '## Vanilla Transformer Language Models', '1901.02860-3-14-0': 'In order to apply Transformer or self-attention to language modeling, the central problem is how to train a Transformer to effectively encode an arbitrarily long context into a fixed size representation.', '1901.02860-3-14-1': 'Given infinite memory and computation, a simple solution would be to process the entire context sequence using an unconditional Transformer decoder, similar to a feed-forward neural network.', '1901.02860-3-14-2': 'However, this is usually infeasible with the limited resource in practice.', '1901.02860-3-15-0': 'One feasible but crude approximation is to split the entire corpus into shorter segments of manageable sizes, and only train the model within each segment, ignoring all contextual information from previous segments.', '1901.02860-3-15-1': 'This is the idea adopted by [CITATION].', '1901.02860-3-15-2': 'We call it the vanilla model and visualize it in Fig. [REF].', '1901.02860-3-15-3': 'Under this training paradigm, information never flows across segments in either the forward or backward pass.', '1901.02860-3-15-4': 'There are two critical limitations of using a fixed-length context.', '1901.02860-3-15-5': 'First, the largest possible dependency length is upper bounded by the segment length, which is a few hundred on character-level language modeling .', '1901.02860-3-15-6': 'Therefore, although the self-attention mechanism is less affected by the vanishing gradient problem compared to RNNs, the vanilla model is not able to fully exploit this optimization advantage.', '1901.02860-3-15-7': 'Second, though it is possible to use padding to respect the sentence or other semantic boundaries, in practice it has been standard practice to simply chunk long text into fixed-length segments due to improved efficiency .', '1901.02860-3-15-8': 'However, simply chunking a sequence into fixed-length segments will lead to the context fragmentation problem as discussed in Section [REF].', '1901.02860-3-16-0': 'During evaluation, at each step, the vanilla model also consumes a segment of the same length as in training, but only makes one prediction at the last position.', '1901.02860-3-16-1': 'Then, at the next step, the segment is shifted to the right by only one position, and the new segment has to be processed all from scratch.', '1901.02860-3-16-2': 'As shown in Fig. [REF], this procedure ensures that each prediction utilizes the longest possible context exposed during training, and also relieves context fragmentation issue encountered in training.', '1901.02860-3-16-3': 'However, this evaluation procedure is extremely expensive.', '1901.02860-3-16-4': 'We will show that our proposed architecture is able to substantially improve the evaluation speed.', '1901.02860-3-17-0': '## Segment-Level Recurrence with State Reuse', '1901.02860-3-18-0': 'To address the limitations of using a fixed-length context, we propose to introduce a recurrence mechanism to the Transformer architecture.', '1901.02860-3-18-1': 'During training, the hidden state sequence computed for the previous segment is fixed and cached to be reused as an extended context when the model processes the next new segment, as shown in Fig. [REF].', '1901.02860-3-18-2': 'Although the gradient still remains within a segment, this additional input allows the network to exploit information in the history, leading to an ability of modeling longer-term dependency and avoiding context fragmentation.', '1901.02860-3-18-3': 'Formally, let the two consecutive segments of length [MATH] be [MATH] and [MATH] respectively.', '1901.02860-3-18-4': 'Denoting the [MATH]-th layer hidden state sequence produced for the [MATH]-th segment [MATH] by [MATH], where [MATH] is the hidden dimension.', '1901.02860-3-18-5': 'Then, the [MATH]-th layer hidden state for segment [MATH] is produced (schematically) as follows, -0.5em [EQUATION] -1.5em', '1901.02860-3-19-0': 'where the function [MATH] stands for stop-gradient, the notation [MATH] indicates the concatenation of two hidden sequences along the length dimension, and [MATH] denotes model parameters.', '1901.02860-3-19-1': 'Compared to the standard Transformer, the critical difference lies in that the key [MATH] and value [MATH] are conditioned on the extended context [MATH] and hence [MATH] cached from the previous segment.', '1901.02860-3-19-2': 'We emphasize this particular design by the green paths in Fig. [REF].', '1901.02860-3-20-0': 'With this recurrence mechanism applied to every two consecutive segments of a corpus, it essentially creates a segment-level recurrence in the hidden states.', '1901.02860-3-20-1': 'As a result, the effective context being utilized can go way beyond just two segments.', '1901.02860-3-20-2': 'However, notice that the recurrent dependency between [MATH] and [MATH] shifts one layer downwards per-segment, which differs from the same-layer recurrence in conventional RNN-LMs.', '1901.02860-3-20-3': 'Consequently, the largest possible dependency length grows linearly w.r.t. the number of layers as well as the segment length, i.e., [MATH], as visualized by the shaded area in Fig. [REF].', '1901.02860-3-20-4': 'This is analogous to truncated BPTT , a technique developed for training RNN-LMs.', '1901.02860-3-20-5': 'However, different from truncated BPTT, our method caches a sequence of hidden states instead of the last one, and should be applied together with the relative positional encoding technique described in Section [REF].', '1901.02860-3-21-0': 'Besides achieving extra long context and resolving fragmentation, another benefit that comes with the recurrence scheme is significantly faster evaluation.', '1901.02860-3-21-1': 'Specifically, during evaluation, the representations from the previous segments can be reused instead of being computed from scratch as in the case of the vanilla model.', '1901.02860-3-21-2': 'In our experiments on enwiki8, Transformer-XL is up to 1,800+ times faster than the vanilla model during evaluation (see Section [REF]).', '1901.02860-3-22-0': 'Finally, notice that the recurrence scheme does not need to be restricted to only the previous segment.', '1901.02860-3-22-1': 'In theory, we can cache as many previous segments as the GPU memory allows, and reuse all of them as the extra context when processing the current segment.', '1901.02860-3-22-2': 'Thus, we can cache a predefined length-[MATH] old hidden states spanning (possibly) multiple segments, and refer to them as the memory [MATH], due to a clear connection to the memory augmented neural networks .', '1901.02860-3-22-3': 'In our experiments, we set [MATH] equal to the segment length during training, and increase it by multiple times during evaluation.', '1901.02860-3-23-0': '## Relative Positional Encodings', '1901.02860-3-24-0': "While we found the idea presented in the previous subsection very appealing, there is a crucial technical challenge we haven't solved in order to reuse the hidden states.", '1901.02860-3-24-1': 'That is, how can we keep the positional information coherent when we reuse the states?', '1901.02860-3-24-2': 'Recall that, in the standard Transformer, the information of sequence order is provided by a set of positional encodings, denoted as [MATH], where the [MATH]-th row [MATH] corresponds to the [MATH]-th absolute position within a segment and [MATH] prescribes the maximum possible length to be modeled.', '1901.02860-3-24-3': 'Then, the actual input to the Transformer is the element-wise addition of the word embeddings and the positional encodings.', '1901.02860-3-24-4': 'If we simply adapt this positional encoding to our recurrence mechanism, the hidden state sequence would be computed schematically by -0.5em [EQUATION] -1.5em', '1901.02860-3-25-0': 'where [MATH] is the word embedding sequence of [MATH], and [MATH] represents a transformation function.', '1901.02860-3-25-1': 'Notice that, both [MATH] and [MATH] are associated with the same positional encoding [MATH].', '1901.02860-3-25-2': 'As a result, the model has no information to distinguish the positional difference between [MATH] and [MATH] for any [MATH], resulting in a sheer performance loss.', '1901.02860-3-26-0': 'In order to avoid this failure mode, the fundamental idea is to only encode the relative positional information in the hidden states.', '1901.02860-3-26-1': 'Conceptually, the positional encoding gives the model a temporal clue or "bias" about how information should be gathered, i.e., where to attend.', '1901.02860-3-26-2': 'For the same purpose, instead of incorporating bias statically into the initial embedding, one can inject the same information into the attention score of each layer.', '1901.02860-3-26-3': 'More importantly, it is more intuitive and generalizable to define the temporal bias in a relative manner.', '1901.02860-3-26-4': 'For instance, when a query vector [MATH] attends on the key vectors [MATH], it does not need to know the absolute position of each key vector to identify the temporal order of the segment.', '1901.02860-3-26-5': 'Instead, it suffices to know the relative distance between each key vector [MATH] and itself [MATH], i.e. [MATH].', '1901.02860-3-26-6': 'Practically, one can create a set of relative positional encodings [MATH], where the [MATH]-th row [MATH] indicates a relative distance of [MATH] between two positions.', '1901.02860-3-26-7': 'By injecting the relative distance dynamically into the attention score, the query vector can easily distinguish the representations of [MATH] and [MATH] from their different distances, making the state reuse mechanism feasible.', '1901.02860-3-26-8': "Meanwhile, we won't lose any temporal information, as the absolute position can be recovered recursively from relative distances.", '1901.02860-3-27-0': 'Previously, the idea of relative positional encodings has been explored in the context of machine translation and music generation .', '1901.02860-3-27-1': 'Here, we offer a different derivation, arriving at a new form of relative positional encodings, which not only has a one-to-one correspondence to its absolute counterpart but also enjoys much better generalization empirically (see Section [REF]).', '1901.02860-3-27-2': 'Firstly, in the standard Transformer , the attention score between query [MATH] and key vector [MATH] within the same segment can be decomposed as -0.5em [EQUATION] -1em', '1901.02860-3-28-0': 'Following the idea of only relying on relative positional information, we propose to re-parameterize the four terms as follows -0.5em [EQUATION] -1em', '1901.02860-3-29-0': 'Under the new parameterization, each term has an intuitive meaning: term [MATH] represents content-based addressing, term [MATH] captures a content-dependent positional bias, term [MATH] governs a global content bias, and [MATH] encodes a global positional bias.', '1901.02860-3-30-0': 'In comparison, the formulation in [CITATION] only has terms [MATH] and [MATH], dropping the two bias terms [MATH] and [MATH].', '1901.02860-3-30-1': 'Moreover, [CITATION] merge the multiplication [MATH] into a single trainable matrix [MATH], which abandons the inductive bias built into the original sinusoid positional encoding .', '1901.02860-3-30-2': 'In contrast, our relative positional embedding [MATH] adapts the sinusoid formulation.', '1901.02860-3-30-3': 'As a benefit of the inductive bias, a model trained on a memory of some certain length can automatically generalize to a memory several times longer during evaluation.', '1901.02860-3-31-0': 'Equipping the recurrence mechanism with our proposed relative positional embedding, we finally arrive at the Transformer-XL architecture.', '1901.02860-3-31-1': 'For completeness, we summarize the computational procedure for a [MATH]-layer Transformer-XL with a single attention head here.', '1901.02860-3-31-2': 'For [MATH]: -0.5em [EQUATION] -1.5em', '1901.02860-3-32-0': 'with [MATH] defined as the word embedding sequence.', '1901.02860-3-32-1': 'In addition, it is worth mentioning that a naive way to compute [MATH] requires computing [MATH] for all pairs [MATH], whose cost is quadratic w.r.t. the sequence length.', '1901.02860-3-32-2': 'However, noticing that the value of [MATH] only ranges from zero to the sequence length, we show a simple computation procedure in Appendix [REF], which reduces the cost to be linear w.r.t. the sequence length.', '1901.02860-3-33-0': '# Experiments', '1901.02860-3-34-0': '## Main Results', '1901.02860-3-35-0': 'We apply Transformer-XL to a variety of datasets on both word-level and character-level language modeling to have a comparison with state-of-the-art systems, including WikiText-103 , enwik8 , text8 , One Billion Word , and Penn Treebank .', '1901.02860-3-36-0': 'WikiText-103 is the largest available word-level language modeling benchmark with long-term dependency.', '1901.02860-3-36-1': 'It contains 103M training tokens from 28K articles, with an average length of 3.6K tokens per article, which allows testing the ability of long-term dependency modeling.', '1901.02860-3-36-2': 'We set the attention length to 384 during training and 1600 during evaluation.', '1901.02860-3-36-3': 'We adopted adaptive softmax and input representations .', '1901.02860-3-36-4': 'As shown in Table [REF], Transformer-XL reduces the previous state-of-the-art (SoTA) perplexity from 20.5 to 18.3, which demonstrates the superiority of the Transformer-XL architecture.', '1901.02860-3-37-0': 'The dataset enwik8 contains 100M bytes of unprocessed Wikipedia text.', '1901.02860-3-37-1': 'We compare our architecture with the previous results in Table [REF].', '1901.02860-3-37-2': 'Under the model size constraint, the 12-layer Transformer-XL achieves a new SoTA result, outperforming the 12-layer vanilla Transformer from [CITATION] by 0.05, while both Transformer variants have a large margin over conventional RNN-based models.', '1901.02860-3-37-3': 'Notably, our 12-layer architecture achieves the same result as the 64-layer network from [CITATION], using only 17% of the parameter budget.', '1901.02860-3-37-4': 'In order to see whether better performances can be obtained by increasing the model size, we train 18-layer and 24-layer Transformer-XLs with increased model sizes.', '1901.02860-3-37-5': 'With the attention length 784 during training and 3,800 during evaluation, we obtained a new SoTA result and our method is the first to break through 1.0 on widely-studied character-level benchmarks.', '1901.02860-3-37-6': 'Different from [CITATION], Transformer-XL does not need any auxiliary losses, and thus all benefits are credited to a better architecture.', '1901.02860-3-38-0': 'Similar to but different from enwik8, text8 contains 100M processed Wikipedia characters created by lowering case the text and removing any character other than the 26 letters a through z, and space.', '1901.02860-3-38-1': 'Due to the similarity, we simply adapt the best model and the same hyper-parameters on enwik8 to text8 without further tuning.', '1901.02860-3-38-2': 'The comparison with previous methods is summarized in Table [REF].', '1901.02860-3-38-3': 'Again, Transformer-XL achieves the new SoTA result with a clear margin.', '1901.02860-3-39-0': 'One Billion Word does not preserve any long-term dependency because sentences have been shuffled.', '1901.02860-3-39-1': 'Consequently, this dataset mainly tests the ability of modeling only short-term dependency.', '1901.02860-3-39-2': 'The comparison between Transformer-XL and the other methods is shown in Table [REF].', '1901.02860-3-39-3': 'Although Transformer-XL is mainly designed to better capture longer-term dependency, it dramatically improves the single-model SoTA from 23.7 to 21.8.', '1901.02860-3-39-4': 'Specifically, Transformer-XL significantly outperforms a contemporary method using vanilla Transformers [CITATION], suggesting the advantage of Transformer-XL is generalizable to modeling short sequences.', '1901.02860-3-40-0': 'We also report the results on word-level Penn Treebank in Table [REF].', '1901.02860-3-40-1': 'Similar to AWD-LSTM , we apply variational dropout and weight average to Transformer-XL.', '1901.02860-3-40-2': 'With proper regularization, Transformer-XL achieves a new SoTA result among models without two-step finetuning.', '1901.02860-3-40-3': 'Penn Treebank has only 1M training tokens, which implies that Transformer-XL also generalizes well even on small datasets.', '1901.02860-3-41-0': '## Ablation Study', '1901.02860-3-42-0': 'We conduct two sets of ablation studies to examine the effects of two proposed techniques used in Transformer-XL: the recurrence mechanism and the new positional encoding scheme.', '1901.02860-3-43-0': 'The first study is performed on WikiText-103, which requires modeling long-term dependency.', '1901.02860-3-43-1': 'The results are reported in Table [REF].', '1901.02860-3-43-2': 'Among the compared encoding schemes, [CITATION] is relative, while [CITATION] and [CITATION] are absolute.', '1901.02860-3-43-3': '"Full" and "half" losses refer to applying a cross entropy loss to all or the recent half positions in the segment.', '1901.02860-3-43-4': 'We found that absolute encodings only work well with half losses because half losses exclude positions with very short attention lengths during training for better generalization.', '1901.02860-3-43-5': 'Table [REF] shows that both the recurrence mechanism and our encoding scheme are necessary to achieve the best performance, as well as generalizing to longer attention sequences during evaluation time.', '1901.02860-3-43-6': 'Although the backpropagation length during training is only 128, with the two techniques the attention length can be increased to 640 at test time.', '1901.02860-3-43-7': 'In the standard setting with 151M parameters, the perplexity decreases as the attention length increases.', '1901.02860-3-44-0': 'Since the recurrence mechanism costs additional memory, we also compare Transformer-XL with baselines under the same GPU memory constraints.', '1901.02860-3-44-1': 'As shown in Table [REF] in Appendix [REF], despite using a shorter backpropagation length, Transformer-XL remains superior to the baselines.', '1901.02860-3-45-0': 'The second study targets at isolating the effects of resolving the context fragmentation problem from the benefit of capturing longer context length.', '1901.02860-3-45-1': 'In order to achieve this goal, we deliberately choose a dataset that does not require long-term dependency, so that any improvement from establishing the recurrence can be attributed to solving the context fragmentation.', '1901.02860-3-45-2': 'Specifically, we perform this controlled experiment on the One Billion Word dataset, which can only benefit from removing the context fragmentation.', '1901.02860-3-45-3': 'We train a 20-layer Transformer-XL with [MATH]0.3B parameters for 400K steps.', '1901.02860-3-45-4': 'As shown in Table [REF], using segment-level recurrence substantially improves performance even when long-term dependency is not needed, which is consistent with our previous discussion that the recurrence mechanism resolves the context fragmentation problem.', '1901.02860-3-45-5': 'Moreover, our relative positional encodings is also superior to [CITATION] on short sequences.', '1901.02860-3-46-0': '## Relative Effective Context Length', '1901.02860-3-47-0': '[CITATION] proposed a method to evaluate the Effective Context Length (ECL) of a sequence model.', '1901.02860-3-47-1': 'ECL is the longest length to which increasing the context span would lead to a gain more than a threshold.', '1901.02860-3-47-2': 'However, ECL ignores the fact that it is harder to get improvement when a model already achieves a lower perplexity using only a shorter context, and thus it is not suitable for fair comparison among multiple models.', '1901.02860-3-47-3': 'We instead propose a new metric called Relative Effective Context Length (RECL).', '1901.02860-3-47-4': 'RECL is defined on a model group instead of a single model, and the gain of a long context is measure by the relative improvement over the best short context model.', '1901.02860-3-47-5': 'As such, the model group shares the same baseline to enable fair comparison.', '1901.02860-3-47-6': 'RECL also has a parameter [MATH], which means constraining the comparison on top-[MATH] hard examples.', '1901.02860-3-47-7': 'See Appedix [REF] for more details about RECL.', '1901.02860-3-47-8': 'As shown in Table [REF], Transformer-XL manages to model dependency of 900 words long on average with [MATH].', '1901.02860-3-47-9': 'The RECL of Transformer-XL is 80% and 450% longer than recurrent networks and Transformer respectively.', '1901.02860-3-47-10': 'Both the recurrence mechanism and our positional encodings contribute to a longer RECL.', '1901.02860-3-47-11': 'This further substantiates our argument that Transformer-XL is able to model longer-term dependency.', '1901.02860-3-48-0': '## Generated Text', '1901.02860-3-49-0': 'Trained only on WikiText-103 which is medium-sized, Transformer-XL is already able to generate relatively coherent articles with thousands of tokens without manual cherry picking, despite minor flaws.', '1901.02860-3-49-1': 'Please refer to Appendix [REF] for samples.', '1901.02860-3-50-0': '## Evaluation Speed', '1901.02860-3-51-0': 'Finally, we compare the evaluation speed of our model with the vanilla Transformer model [CITATION].', '1901.02860-3-51-1': 'As shown in Table [REF], due to the state reuse scheme, Transformer-XL achieves an up to 1,874 times speedup during evaluation.', '1901.02860-3-52-0': '# Conclusions', '1901.02860-3-53-0': 'Transformer-XL obtains strong perplexity results, models longer-term dependency than RNNs and Transformer, achieves substantial speedup during evaluation, and is able to generate coherent text articles.', '1901.02860-3-53-1': 'We envision interesting applications of Transformer-XL in the fields of text generation, unsupervised feature learning, image and speech modeling.'}

1511.00980

{'1511.00980-1-0-0': 'Interactions between many-body atomic systems and light in cavities induce new atomic dynamics, which we show can be tailored by projective light measurement backaction, leading to collective effects such as density-density interactions, perfectly-correlated atomic tunneling, superexchange, and effective pair creation and annihilation.', '1511.00980-1-0-1': 'These can be long- and short-range, with tunable strengths, based on the optical setup.', '1511.00980-1-0-2': 'We show this provides a framework to enhance quantum simulations of novel physical phenomena, including reservoir models and dynamical gauge fields, beyond current methods.', '1511.00980-1-1-0': 'The study of quantum gases trapped and controlled by optical potentials has expanded rapidly in recent years, [CITATION] as they provide a clean and versatile way to realize and observe many-body quantum dynamics, enabling quantum simulation of models from condensed matter and particle physics, and beyond.', '1511.00980-1-1-1': 'In parallel, studies of quantum light, such as cavity quantum-electrodynamics [CITATION] have yielded fascinating results, including controlled state preparation and quantum non-demolition measurement.', '1511.00980-1-1-2': 'Uniting these fields [CITATION], broadens both, and goes beyond when either light or matter is treated classically.', '1511.00980-1-1-3': 'Experimental [CITATION] and theoretical works in this regime have revealed many interesting phenomena, such as preparation of atomic states and dynamics [CITATION], non-destructive measurement [CITATION], many-body light-matter entanglement [CITATION], self-organization and other new phases [CITATION].', '1511.00980-1-1-4': 'Going beyond this, we study atomic dynamics due to interactions with quantum light, from the interplay of cavity and measurement backaction.', '1511.00980-1-2-0': 'We consider ultracold (bosonic) atoms trapped in an optical lattice, probed by light.', '1511.00980-1-2-1': 'Light scattering by the atoms can be enhanced by optical cavities, which, once populated, can drive atomic dynamics.', '1511.00980-1-2-2': 'By engineering the light modes dynamics beyond the standard Bose-Hubbard model can be realized, which we show includes perfectly correlated tunneling, effective pair creation and annihilation, long-range tunneling, and long- and short-range density-density interactions that can be tuned independently.', '1511.00980-1-2-3': 'We demonstrate that the backaction from light measurement enables control of these synthesized processes to highlight desired effects through quantum Zeno dynamics [CITATION], and moreover, discuss how this provides a framework to enhance quantum simulations.', '1511.00980-1-3-0': 'We study a generalized Bose-Hubbard model [CITATION] where light is scattered by atoms in a (classical) optical lattice into cavities [Fig. [REF]], a key feature being that both light and atoms are dynamical quantum fields.', '1511.00980-1-3-1': 'After adiabatic elimination of the atomic excited state, the Hamiltonian can be written [MATH], where (in natural units) [MATH], [MATH], where [MATH] creates photons in light mode [MATH] with frequency [MATH] and [MATH] creates bosons at lattice site [MATH].', '1511.00980-1-3-2': '[MATH] parameterizes on-site interactions between atoms, and [MATH] the tunneling rates between sites due to the classical potential.', '1511.00980-1-3-3': 'The fully quantum light-matter interaction Hamiltonian is [MATH] where [MATH] are the light-matter coupling constants, and [MATH] is the atomic transition frequency [MATH] detuning from a reference frequency [MATH] (e.g. that of the pump).', '1511.00980-1-3-4': 'The interactions are parameterized by [MATH] where [MATH] is the light mode function of mode [MATH] and [MATH] is the Wannier function of an atom at site [MATH].', '1511.00980-1-3-5': 'Adiabatically eliminating the light fields by finding their steady state, when pumping a single mode [MATH] in a coherent state of amplitude [MATH] with occupation much larger than other modes, the effective atomic Hamiltonian is [CITATION] [EQUATION] where we define [MATH], [MATH], [MATH] is the cavity decay rate, [MATH] is the cavity mode detuning, [MATH], and a small dispersive frequency shift of cavity modes is assumed ([MATH] for [MATH]).', '1511.00980-1-4-0': 'The additional terms introduce long-range correlations and interactions between lattice sites.', '1511.00980-1-4-1': 'Additional pumps will manifest further dynamics of the same form.', '1511.00980-1-4-2': 'The choice of light mode functions leads to different dynamics, and are tunable by a variety of methods, including changing the wavelength and angle of the lasers, and the angle and size of the cavities.', '1511.00980-1-4-3': 'We highlight that multimode, or even mulitple cavities allow flexibility beyond a single cavity mode.', '1511.00980-1-5-0': 'On-site terms typically dominate [MATH] [CITATION], whence these terms can be replaced by [MATH].', '1511.00980-1-5-1': 'Complex phases can be imparted onto the [MATH]), generating a matter mode structure of sites scattering light with the same phase [CITATION].', '1511.00980-1-5-2': 'Illumination in the diffraction maximum ([MATH]) gives [MATH] (number of atoms illuminated by both pump and mode [MATH]); the light-induced dynamics is then [MATH].', '1511.00980-1-5-3': 'These form effective chemical potentials and (long-range) density-density interactions between all illuminated sites.', '1511.00980-1-6-0': 'The density-density interactions can be tuned to have different long- and short-range strengths.', '1511.00980-1-6-1': 'With a single pump and mode, these strengths are not independent; each region has identical short-range coupling strengths, and the long-range term cannot exceed twice this (denoting the interaction between sites in regions [MATH] and [MATH] as [MATH], we have short-range [MATH] and long-range [MATH], where [MATH] is the phase difference between the [MATH] for sites in the two regions [see Fig. [REF]]).', '1511.00980-1-6-2': 'Using additional pumps each illuminating just one region allows short-range interactions to have different magnitudes, but identical sign, and the restriction on the long-range coupling size remains.', '1511.00980-1-6-3': 'However, with multiple cavity modes, the detuning allows contributions from each mode to have differing signs, enabling the [MATH] to be tuned fully independently, both in magnitude and sign.', '1511.00980-1-6-4': 'For a single pump, and three cavity modes [MATH] illuminating regions 1, 2 and both respectively, we have [MATH], [MATH] and [MATH].', '1511.00980-1-6-5': 'Additional pumps allow further flexibility in tuning the relative magnitudes.', '1511.00980-1-7-0': 'In contrast to the above, if light modes are concentrated between the lattice sites, nearest-neighbor terms in [MATH] can be more significant than on-site terms [CITATION].', '1511.00980-1-7-1': 'Thence, we can replace the light-matter interaction terms by [MATH], leading to light-induced tunneling.', '1511.00980-1-7-2': 'The coefficients [MATH] are in general different for each pair of sites, leading to spatially-dependent tunneling rates, though when pump intensity [MATH] and lattice sites are commensurate, [MATH] has equal coupling for each pair, allowing one-body tunneling terms in [MATH] to be suppressed or enhanced.', '1511.00980-1-7-3': 'This also admits elimination of single-body tunneling events, leaving only the two-body terms, with rates which may be engineered semi-independently from the one-body term.', '1511.00980-1-7-4': 'These two-body processes have the form [MATH].', '1511.00980-1-7-5': 'In each case, while [MATH] and [MATH] must be pairs of neighboring sites, the two pairs may be distributed anywhere within the illuminated region of the lattice.', '1511.00980-1-7-6': 'As illustrated in Fig. [REF], these give to rise effects that include correlated pair tunneling to/from the same sites, effective next-nearest neighbor tunneling across a site, correlated pair tunneling into (out of) one site from (to) two neighboring sites, and correlated pair tunneling between neighboring sites in two entirely distinct locations.', '1511.00980-1-8-0': 'Two-body tunneling processes have been suggested as a way to simulate superexchange interactions in spin models [CITATION].', '1511.00980-1-8-1': 'Here we have the advantage that to achieve such processes we do not rely on second-order effects, and further, implementing the correlated tunneling with a cavity provides additional tunability.', '1511.00980-1-8-2': 'By following the original proposal and dividing into pairs of sites with a superlattice potential, each containing two atoms of different (pseudo-)spin species, and introducing the correlated tunneling as above (with one cavity mode [MATH]) leads to each double well having Hamiltonian [MATH] where nn denotes nearest-neighbor sites.', '1511.00980-1-8-3': 'Equivalently, expressing this in terms of spin operators ([MATH]), we have [EQUATION] where [MATH].', '1511.00980-1-8-4': 'Single tunneling events can be eliminated by on-site interactions as in the original proposal.', '1511.00980-1-8-5': 'Critically however, since the magnitude of the exchange term here does not have the [MATH] dependence of the second-order process in the original proposal, the on-site repulsion can here be increased without suppressing the exchange interaction.', '1511.00980-1-9-0': 'Previously [CITATION], we discussed how light measurement backaction can selectively suppress atomic dynamics.', '1511.00980-1-9-1': 'We go beyond this, to incorporate the light-induced dynamics, explicating the potential to enhance quantum simulations.', '1511.00980-1-9-2': 'Crucially, measurement backaction is achieved through an additional cavity (and possibly pump) where the scattered light it leaks is measured: for frequent, persistent measurement, the light field state is reduced [CITATION] and subsequently pinned (due to the quantum Zeno effect [CITATION]) to a particular coherent state [MATH], confining the matter to evolve within only a particular subspace of its full Hilbert space, determined by the measurement cavity and pump mode functions [CITATION], i.e. quantum Zeno dynamics [CITATION].', '1511.00980-1-9-3': 'The two-body terms present here arise as first order terms in the system evolution and are hence perfectly correlated, in contrast to the earlier work, where it is only a second-order process [CITATION].', '1511.00980-1-9-4': 'The two cases are fundamentally different; here increasing pump strength increases two-body tunneling rates, rather than suppressing them.', '1511.00980-1-10-0': 'Measurement at the diffraction minimum ([MATH]; [MATH]) can freeze the occupation number difference between odd and even sites (modes), forbidding nearest neighbor tunneling.', '1511.00980-1-10-1': 'In the absence of a further cavity driving dynamics this leaves next-nearest neighbor tunneling as the leading process.', '1511.00980-1-10-2': 'Though this typically occurs at a much slower rate and is thus often ignored, this is now the dominant term, is no longer negligible, and can have the rate at which it occurs increased via light-induced dynamics.', '1511.00980-1-10-3': 'With additional dynamics spawned from the cavity field, this process can also be augmented with the two-body terms introduced above, namely, those that preserve matter mode occupation number difference.', '1511.00980-1-10-4': 'Events in which two atoms tunnel into the same site (and the reverse) are forbidden by the measurement, as well as certain long-range pair tunneling events.', '1511.00980-1-10-5': 'The latter processes form an effective long-range tunneling event in each mode, where in the simplest case (uniform illumination) the rate is independent of the site separation.', '1511.00980-1-10-6': 'Akin to the density-density interactions, spatially-dependent long-range tunneling rates can be engineered with multiple cavity modes.', '1511.00980-1-10-7': 'Additionally, the flexibility in the choice of light-modes allows a range of configurations of the two-body terms to be engineered through measurement backaction.', '1511.00980-1-10-8': 'For example, when light measurement fixes occupation numbers for multiple matter modes [CITATION], the only allowed tunneling events are of the form of process 2, and a subset of process 6 type (those which preserve the total mode occupations).', '1511.00980-1-11-0': 'This architecture naturally lends itself to quantum simulations.', '1511.00980-1-11-1': 'One such example would be systems that involve (particle) reservoirs, where we assign a subset of modes/sites as the reservoirs.', '1511.00980-1-11-2': 'For example, consider the conceptual three-site model shown in Fig. [REF](a).', '1511.00980-1-11-3': 'A cavity drives dynamics to generate correlated tunneling between the sites, and the outer two sites are designated as the reservoirs, prepared in coherent states [MATH].', '1511.00980-1-11-4': 'With the occupation number difference between reservoirs fixed through persistent light measurement (e.g. by illuminating reservoirs with coherent pumps in antiphase, destructively interfering at the central site), the central site is described by the Hamiltonian [EQUATION] where [MATH].', '1511.00980-1-11-5': 'On-site interactions and chemical potential are omitted; these can be tuned by driving on-site dynamics.', '1511.00980-1-11-6': 'The two-body terms then effect pair creation/annihilation dynamics for the central site.', '1511.00980-1-12-0': 'Extending this to include an additional site between the reservoirs, as depicted in Fig. [REF](b), with the reservoirs having larger occupation than the central sites, the dynamics of the two non-reservoir modes obey a (generalized) Dicke Hamiltonian [CITATION] [EQUATION] where chemical potentials [MATH] can be tuned as above, and [EQUATION]', '1511.00980-1-12-1': 'In contrast to the original model these parameters can be tuned independently and could even be set to vanish individually, allowing each of the terms to be studied in isolation.', '1511.00980-1-12-2': 'Interfering pump beams at different angles enables the amplitude of the measurement probe to be shaped such that the occupation of the central sites does not contribute to the measurement, and complex phases can be induced with additional pump beams.', '1511.00980-1-12-3': 'To our knowledge, this generalized model has not yet been solved in the fully quantum regime, though classical treatments show bifurcations when varying [MATH] [CITATION], suggesting possible novel phase behaviour.', '1511.00980-1-12-4': 'Light measurement can also be used to fix the difference in occupation number between a larger number of modes [CITATION]; applying this to the above, multiple reservoir modes can be generated, allowing for extensions with additional simulated modes, realizing (again generalized) multimode Dicke models, in which the simulation can now have multiple species of atoms, as well as multiple cavities; in Fig. [REF](c) an example schematic for extending to simulate a second atomic species is given, where measurement fixing [MATH] and [MATH] and [MATH] are synthetic atomic modes, [MATH] synthetic light and [MATH] the reservoirs) results in a Hamiltonian of the form [EQUATION].', '1511.00980-1-12-5': 'This setup could be achieved by crossing two one-dimensional lattices each implementing the single-species model above, with both lattices sharing a common site for the synthetic light mode.', '1511.00980-1-12-6': 'Other possible extensions using additional reservoirs and coupling more sites (for example, arranged in an array) could be to realize exotic Hubbard models with pair creation/annihiliation effects.', '1511.00980-1-13-0': 'Another application of this framework to quantum simulation is in the synthesis of artificial dynamical gauge fields.', '1511.00980-1-13-1': 'Current proposals for this purpose are typically based on quantum link models [CITATION], where changes in the gauge field due matter motion is simulated by motion of another particle species [CITATION].', '1511.00980-1-13-2': 'The long-range interactions and correlated tunneling processes made possible by the coupling through the cavity modes can be augmented with these proposals allow for a rich range of further new phenomena to be simulated.', '1511.00980-1-13-3': 'Beyond this, the long-range nature of the interactions also present the opportunity to realize such dynamical gauge fields where the links are global, as the cavity-mediated correlated motion need not be local, and hence motion across all sites can be controlled by a common link.', '1511.00980-1-13-4': 'As a (conceptually) simple example of this, consider a 1D lattice with light measurement of a site-dependent strength (e.g. by a gradiated intensity of the measurement pump) such that the measured light state has amplitude [MATH] for some constant [MATH].', '1511.00980-1-13-5': 'With two auxiliary neighboring sites [MATH] and [MATH] that atoms can tunnel between to form the link (and tunneling out of this pair of sites is suppressed), these sites also contributing an effective [MATH] (plus constant) to the measurement, freezing due the measurement then only permits dynamics involving correlated tunneling events (induced by a cavity) where an atom tunnels in the main lattice simultaneously with a tunneling event in the link.', '1511.00980-1-13-6': 'The Hamiltonian is then [MATH], or equivalently, by mapping the link sites to a spin with [MATH], [EQUATION] where [MATH] and [MATH] is the ratio of the intensity of the pump driving correlated dynamics at the link sites compared to the rest of the lattice.', '1511.00980-1-13-7': 'This forms a global-link dynamical matter-gauge field interaction.', '1511.00980-1-13-8': 'Strictly, there are pair correlated tunneling events allowed within the main lattice independent of the link, provided they occur in opposite directions, though the current of particles across the lattice is wholly dependent on the link.', '1511.00980-1-13-9': 'Terms bypassing the link can be made less significant by adjusting [MATH].', '1511.00980-1-13-10': 'The gauge field energy terms [MATH] can be engineered through implementing the density-density interactions discussed above between the link sites.', '1511.00980-1-13-11': 'Unlike the local link models of high-energy physics, the global link here leads to a peculiar effect where motion of a particle at any site will significantly affect the field experienced by all particles, at all sites.', '1511.00980-1-14-0': 'In summary, we have shown that interactions of quantum gases with quantum light in cavities give rise to a panoply of new dynamics beyond those possible with classical light, which may then be controlled through measurement.', '1511.00980-1-14-1': 'These include tunable effects such as correlated tunneling, pair creation/annihilation, and density-density interactions.', '1511.00980-1-14-2': 'We discussed how this provides a variety of possibilities for extending quantum simulations.', '1511.00980-1-14-3': 'Current experiments have trapped Bose-Einstein condensates inside cavities, without a lattice [CITATION], while other experiments have scattered light from quantum gases trapped in lattice potentials, but with no cavity [CITATION].', '1511.00980-1-14-4': 'Amalgamating these would allow for a realization of our proposal, and very recent efforts towards this have proved successful [CITATION].', '1511.00980-1-14-5': 'Additionally, there have been recent experiments that demonstrated examples of the quantum Zeno effect and dynamics in similar but less versatile settings [CITATION].', '1511.00980-1-14-6': 'Furthermore, it may be possible to implement the dynamical effects discussed above by using other systems that involve off-resonant scattering, such as molecules [CITATION], fermions [CITATION], spins [CITATION], ions [CITATION], and semiconductor [CITATION] and superconducting qubits [CITATION].', '1511.00980-1-15-0': 'The authors thank EPSRC for financial support (DTA and EP/I004394/1).'}

{'1511.00980-2-0-0': 'Interactions between many-body atomic systems in optical lattices and light in cavities induce long-range and correlated atomic dynamics beyond the standard Bose-Hubbard model, due to the global nature of the light modes.', '1511.00980-2-0-1': 'We characterise these processes, and show that uniting such phenomena with dynamical constraints enforced by the backaction resultant from strong light measurement leads to a synergy that enables the atomic dynamics to be tailored, based on the particular optical geometry, exploiting the additional structure imparted by the quantum light field.', '1511.00980-2-0-2': 'This leads to a range of novel, tunable effects such as long-range density-density interactions, perfectly-correlated atomic tunnelling, superexchange, and effective pair processes.', '1511.00980-2-0-3': 'We further show that this provides a framework for enhancing quantum simulations to include such long-range and correlated processes, including reservoir models and dynamical global gauge fields.', '1511.00980-2-1-0': '# Introduction', '1511.00980-2-2-0': 'The study of quantum gases trapped and controlled by optical potentials has expanded rapidly in recent years [CITATION], as they provide a clean and versatile way to realise and observe many-body quantum dynamics, enabling quantum simulation of models from condensed matter and particle physics, and beyond.', '1511.00980-2-2-1': 'In parallel, studies of quantum light, such as cavity quantum-electrodynamics [CITATION], have yielded fascinating results, including controlled state preparation and quantum non-demolition measurement.', '1511.00980-2-2-2': 'Uniting these fields [CITATION] broadens both, and goes beyond the cases when either the light or matter are treated classically.', '1511.00980-2-2-3': 'Experimental [CITATION] and theoretical works in this regime have revealed many interesting phenomena, such as the preparation of atomic states and dynamics [CITATION], non-destructive measurement [CITATION], many-body light-matter entanglement [CITATION], self-organisation, and other new quantum phases [CITATION].', '1511.00980-2-3-0': 'In the aforementioned works, these effects occur due to either the collective behavior arising from the cavity-mediated interactions, or the suppression of atomic dynamics by light measurement backaction.', '1511.00980-2-3-1': 'We go beyond this, and for the first time study the union of these mechanisms.', '1511.00980-2-3-2': 'In doing so, we show that their interplay enables a selective engineering of the cavity-mediated processes, which may then be used to orchestrate dynamics for quantum simulation purposes.', '1511.00980-2-4-0': 'Specifically, we consider a system of ultracold (bosonic) atoms trapped in an optical lattice, probed by light.', '1511.00980-2-4-1': 'The introduction of optical cavities [Fig. [REF]] enhances the light scattering from the atoms, and these cavities, once populated, can drive atomic dynamics.', '1511.00980-2-4-2': 'The light fields inside the cavities are dynamical quantum fields, and hence form a quantum potential for the atoms.', '1511.00980-2-4-3': 'By engineering the light modefunctions, atomic dynamics beyond the standard Bose-Hubbard model can then be realised.', '1511.00980-2-5-0': 'In this article, we begin by investigating the form of these dynamics, and provide a characterisation of the constituent terms.', '1511.00980-2-5-1': 'We show that this classification reveals processes that include perfectly correlated tunnelling, effective pair creation and annihilation, long-range tunnelling, superexchange, and independently tunable long- and short-range density-density interactions.', '1511.00980-2-5-2': 'We illustrate how these different effects may be controlled and tuned by the optical geometry.', '1511.00980-2-6-0': 'Following this, we introduce the backaction effect that arises from measurement of the light leaking from the cavity to these extended dynamics.', '1511.00980-2-6-1': 'We demonstrate how this enables further control of the processes, by imparting additional structure to the lattice, allowing for the highlighting of desired effects (by suppression of others) through quantum Zeno dynamics [CITATION].', '1511.00980-2-6-2': 'We describe how this united formalism provides a framework to enhance quantum simulations, through the incorporation of these correlated and long-range processes that are not accessible in other systems with finite-range interactions, through the introduction of building block components, which include reservoir models and dynamical global gauge fields.', '1511.00980-2-7-0': '# Cavity-Mediated Dynamics', '1511.00980-2-8-0': '## The Model', '1511.00980-2-9-0': 'In this article we study a extended form of the Bose-Hubbard model [CITATION] in which interactions with an additional set of modes possessing long-range spatial extent over the lattice are included.', '1511.00980-2-9-1': 'Physically, this can correspond to the scenario where the lattice is a (classical) optical trap containing bosonic atoms, embedded within an optical cavity, with light from an external laser scattered by the atoms into the cavity modes.', '1511.00980-2-9-2': 'A key feature of this model is that both the light and the atoms are dynamical quantum fields.', '1511.00980-2-9-3': 'It forms the cornerstone of many of the aforementioned works in the field of fully-quantum many-body light-matter interactions, and in its general form describes a wide range of possible optical geometries.', '1511.00980-2-9-4': 'We outline the main steps in the derivation of the effective atomic Hamiltonian; more detailed treatments may be found in, e.g. [CITATION].', '1511.00980-2-10-0': 'The full Hamiltonian can be written [MATH], where (in natural units) [EQUATION] are the bare Hamiltonians describing the light and matter respectively (the matter Hamiltonian being the standard Bose-Hubbard Hamiltonian for atoms in a lattice when [MATH] is restricted to nearest-neighbour terms only).', '1511.00980-2-10-1': 'Here, [MATH] creates photons in light mode [MATH] with frequency [MATH] and modefunction [MATH], while [MATH] creates bosons at lattice site [MATH] with Wannier function [MATH].', '1511.00980-2-10-2': 'On-site interactions between atoms are parameterised by [MATH], and [MATH] are the (classical) tunnelling rates between sites due to the classical potential.', '1511.00980-2-11-0': 'The final term [MATH] describes the fully quantum light-matter interactions between the light and atomic modes.', '1511.00980-2-11-1': 'It follows from a many-atom generalisation of the Jaynes-Cummings Hamiltonian, with the excited atomic state adiabatically eliminated [CITATION].', '1511.00980-2-11-2': 'To obtain this, consider first the single-particle Hamiltonian for an atom interacting with many light modes: the interaction part of the Hamiltonian can be written [EQUATION] where [MATH] raises the atom to its excited state, and [MATH] is the light-matter coupling constant for mode [MATH].', '1511.00980-2-11-3': 'To perform the adiabatic elimination, we assume that the detuning is sufficiently large that the excited state population is negligible, and from the Heisenberg equation [MATH] set the time dependence of this operator to vanish in a frame rotating at a reference frequency [MATH] (e.g. that of an external pump laser) leading to [MATH], where [MATH] is the atomic transition frequency [MATH] detuning from [MATH].', '1511.00980-2-11-4': 'Inserting this into the single-particle Hamiltonian, we have [EQUATION]', '1511.00980-2-11-5': 'Finally, to obtain the many-body form of the Hamiltonian, we express the single-particle state in terms of the localised atomic basis states [MATH], resulting in [CITATION] [EQUATION]', '1511.00980-2-11-6': 'The interactions are parameterised by [EQUATION] describing the overlap between the atomic Wannier functions and light modefunctions.', '1511.00980-2-11-7': 'These coefficients thus encompass the dependence of the dynamics on the particular optical setup used in the system.', '1511.00980-2-11-8': 'We note also that due to the complex nature of light modefunctions, these coefficients may too be complex.', '1511.00980-2-11-9': 'Taking the Wannier functions to be real-valued, these coefficients satisfy the properties [MATH].', '1511.00980-2-12-0': 'Let us now consider one of the light modes, which we label [MATH], to be a pump mode sourced from an external laser.', '1511.00980-2-12-1': 'We describe this mode by a coherent state of amplitude [MATH], with an occupation much larger than the cavity modes, such that we can replace [MATH].', '1511.00980-2-12-2': 'Assuming that the light scattering occurs on timescales much faster than the atomic dynamics, we can obtain the time dependence of the cavity modes (in the frame rotating at the pump frequency) from the Heisenberg equation: [EQUATION]', '1511.00980-2-12-3': 'In this expression, the cavity detuning [MATH], and we have phenomenologically introduced a cavity decay for photon loss with rate [MATH], and defined for shorthand [MATH] and [EQUATION]', '1511.00980-2-12-4': 'These light-matter coupling operators [MATH] inherit certain properties from the coupling coefficients [MATH]: [MATH].', '1511.00980-2-13-0': 'We now further assume also that there is only a small dispersive frequency shift of the cavity modes [MATH] for [MATH]), and use that [MATH] due to the large pump amplitude compared to the cavity occupations.', '1511.00980-2-13-1': 'Thus, the steady states of the cavity modes are hence [EQUATION]', '1511.00980-2-13-2': 'When these steady states are reached on timescales much faster than the atomic dynamics (i.e. [MATH]), we can perform a further adiabatic elimination, to remove the cavity modes from the Hamiltonian.', '1511.00980-2-13-3': 'Replacing the cavity mode operators with their steady state values in [MATH] results in the effective atomic Hamiltonian [CITATION] [EQUATION]', '1511.00980-2-13-4': 'Note that the terms in Eq. [REF] containing products of two cavity modes are neglected, as they are much smaller than the pump-pump and pump-cavity product terms.', '1511.00980-2-13-5': 'Note also that the symmetric splitting of the products of [MATH] and its conjugate into two parts of opposite order are necessary to preserve the form of the Heisenberg equations for the atomic modes [MATH] before and after the elimination, as the ordering freedom of [MATH] and [MATH] is lost after the steady state replacement (see Appendix) [CITATION].', '1511.00980-2-13-6': 'This regime in which the cavity steady state is reached much faster than the timescales of atomic dynamics is readily accessible in experiments, and indeed has already been demonstrated [CITATION]; typical tunnelling rates [MATH] between nearest-neighbour sites are [MATH]Hz) [CITATION], while cavities with decay rates [MATH]Hz) are found in experimental use [CITATION].', '1511.00980-2-14-0': 'Beyond the standard Bose-Hubbard Hamiltonian, the first additional term in this effective Hamiltonian is the pump-pump term, due to the interaction between the atoms and the (classical) light from the pump laser.', '1511.00980-2-14-1': 'Such terms can be derived straightforwardly from semi-classical treatments of light-matter interactions, and is essentially of the form of a Raman transition, giving rise to light-induced tunnelling and effective chemical potentials.', '1511.00980-2-14-2': 'The inclusion of additional pumps will manifest similar such terms, including terms mixing different pump modes, which have previously been used to introduce complex phases to atomic tunnelling (e.g. [CITATION]).', '1511.00980-2-14-3': 'The second set of additional terms, the cavity-pump terms, arise due to interaction of the atoms with the quantised cavity light fields, and do not appear in semi-classical treatments, as they are inherently defined by the backaction of the atomic state on the cavity population.', '1511.00980-2-14-4': 'These terms, due to the long-range spatial extent of the cavity modes, induce effective long-range correlations and interactions between lattice sites.', '1511.00980-2-14-5': 'We shall primarily focus on the use of these two-body dynamical processes.', '1511.00980-2-15-0': 'The choice of light modefunctions leads to different dynamics, and are tunable by a variety of methods, including changing the wavelength and angle of the lasers, and the angle and size of the cavities.', '1511.00980-2-15-1': 'We highlight that multimode, or even multiple cavities allow flexibility beyond a single cavity mode.', '1511.00980-2-15-2': 'We now proceed to characterise the induced processes, focussing on the two dominant contributions, from the on-site, and neighbouring inter-site terms.', '1511.00980-2-15-3': 'In contrast to previous works investigating these additional terms, which treat them as a whole and consider them generically as four-point correlations [CITATION], we introduce a characterisation that becomes meaningful and important when we later introduce a method to distinguish between and selectively tailor such processes, through the measurement backaction.', '1511.00980-2-16-0': '## On-site Terms', '1511.00980-2-17-0': 'The on-site terms will typically dominate in the light-matter interaction operators [MATH] [CITATION], and thus these operators can often be approximated with replacement by their on-site counterparts [EQUATION]', '1511.00980-2-17-1': 'Due to the perfect overlap of the Wannier functions (as they are identical), the corresponding light-matter interaction coefficients [MATH] have close to unit magnitude when the light modefunctions are at peak intensity at the centre of lattice sites [CITATION].', '1511.00980-2-17-2': 'These coefficients may be imprinted with complex phases through the phase difference of the incoming and outgoing light modefunctions.', '1511.00980-2-17-3': 'This allows for the generation of a matter mode structure, in which the lattice is partitioned into sets of sites scattering light with the same phase, and the modes are defined by atoms occupying these sets of sites in which they scatter light with the same phase [CITATION].', '1511.00980-2-17-4': 'For example, when considering the main diffraction maximum, we have that [MATH], and hence all odd sites scatter light with the same phase (thus forming one matter mode), while all even sites scatter light with the same, opposing phase (hence forming the second matter mode).', '1511.00980-2-18-0': 'Focussing first on the special case of illumination in the diffraction maximum, in which the light-matter interaction coefficients are all identically [MATH] (and similarly, [MATH]) the interaction terms become [MATH], where [MATH] is the number of atoms in total that occupy any of the sites illuminated by both the pump, and cavity mode [MATH].', '1511.00980-2-18-1': 'Analogously, we have that [MATH].', '1511.00980-2-18-2': 'Thus, the light-induced dynamics in the effective atomic Hamiltonian Eq. [REF] is given by [EQUATION] where the first term forms an effective chemical potential, and the second set of terms mediate density-density interactions between sites illuminated by the pump and their respective cavity mode.', '1511.00980-2-18-3': 'These latter interactions occur irrespective of the spatial separation of the sites, and thus exemplify the long-range nature of the cavity-induced processes.', '1511.00980-2-19-0': 'When considering the inclusion of multiple cavity modes, the tunability of these interactions will exceed what is possible with a single cavity.', '1511.00980-2-19-1': 'One such possibility this provides is that the long- and short-range interaction strengths may be tuned independently, as can be seen by noting that the sign of the cavity detuning determines whether the long-range density-density interactions are repulsive or attractive, and thus different cavities can have different contributions to the overall dynamics.', '1511.00980-2-20-0': 'As can be seen from Eq. [REF], a single pump and cavity mode will produce density-density interactions with a strength [MATH] between atoms on all sites illuminated by pump and mode [MATH].', '1511.00980-2-20-1': 'An illustration of how short- and long-range interactions between two regions may be varied can be seen by considering three cavity modes, which we label [MATH], [MATH], and [MATH], which illuminate regions 1, 2, and both respectively (see Fig. [REF]) at the diffraction maximum, with both regions illuminated by a common pump.', '1511.00980-2-20-2': 'Denoting the light-mediated density-density interaction strength between an atom in region [MATH] and an atom in region [MATH] as [MATH], we hence have [EQUATION]', '1511.00980-2-20-3': 'Thus, the three interaction strengths can all be tuned independently of each other, through the respective [MATH] and [MATH] of each cavity mode.', '1511.00980-2-20-4': 'We note also that in general, as the regions are defined by the matter mode structure, which is in turn defined by the light-matter interaction coefficients [MATH], the regions considered here need not be spatially contiguous, and indeed, the modes can have a very non-trivial spatial overlap with each other [CITATION].', '1511.00980-2-20-5': 'In this sense, one can more generally consider the interaction strengths as being classed as inter- and intra-mode, rather than short- and long-range.', '1511.00980-2-21-0': 'For more general, arbitrary illumination patterns, all of the sites in the illuminated region will still be part of the resultant density-density interactions.', '1511.00980-2-21-1': 'The interaction strength between two particular sites (or modes) is determined by their [MATH], and for a cavity mode [MATH], the associated interaction strength between modes [MATH] and [MATH] is [EQUATION] where [MATH] is the difference between the phases of the associated [MATH] of the two modes.', '1511.00980-2-21-2': 'For multiple cavity modes, one sums up the contributions from each cavity individually, while with multiple pumps one must sum over the [MATH] generated by each pump, and also consider the additional pump-pump terms in the chemical potential, which may now differ from unit strength.', '1511.00980-2-22-0': 'The effective atomic interactions resulting from such cavity-induced processes have already been demonstrated experimentally for the special case of illumination at the main diffraction minimum [CITATION].', '1511.00980-2-22-1': 'In these experiments, the resulting interactions were of the form [MATH], with the detuning chosen such that this term favours a population imbalance between even and odd states, leading to cavity-induced atomic self-organisation for large enough interaction strengths.', '1511.00980-2-22-2': 'The experiment demonstrated that this interaction strength can be made comparable to, and even stronger than the tunnelling from the standard Bose-Hubbard model by one or two orders of magnitude (i.e. up to [MATH]Hz))[CITATION].', '1511.00980-2-22-3': 'As the other more general schemes based on using the on-site terms in the light-matter interaction operator typically involve only changing the phase of the [MATH] through adjustment of the optical geometry, the size of such interaction strengths can be expected to remain of a similar size.', '1511.00980-2-23-0': '## Inter-site Terms', '1511.00980-2-24-0': 'In contrast to the on-site case discussed above, when the overlap of the light modes are arranged to be concentrated between lattice sites the nearest-neighbour terms in [MATH] can be made more significant than the on-site terms [CITATION].', '1511.00980-2-24-1': 'Specifically, when the modefunction of cavity [MATH] and the pump modefunction are given by standing waves with wavenumbers [MATH] being the lattice spacing), at opposing angles to the lattice [MATH], then the on-site pump-cavity light-matter coupling coefficients vanish ([MATH]), while the nearest-neighbour coefficients [MATH] (the subscript nn denoting nearest-neighbour sites) take a constant value at all site pairs with bonds in a given direction (see [CITATION] for further details).', '1511.00980-2-24-2': 'Intuitively, this can be seen to occur because the Wannier functions of a site are symmetric, while the product of light modefunctions [MATH] are antisymmetric, and periodic across two lattice sites, hence leading to a cancellation of their overlap with each on-site product of Wannier functions, but not the inter-site products.', '1511.00980-2-24-3': 'Note that because [MATH] is a positive-definite function, such a cancellation does not occur for the [MATH]; however, for such an illumination pattern, this coefficient is equal for all illuminated lattice sites, forming a constant effective potential within this region, and hence when the pump illuminates the entire lattice this may generally be neglected in the effective Hamiltonian Eq. [REF].', '1511.00980-2-24-4': 'The pump-pump inter-site coefficients [MATH] will then also take a constant value.', '1511.00980-2-25-0': 'Using this, in this regime we replace the light-matter coupling operators by these inter-site terms alone: [EQUATION] where [MATH] indicate neighbouring site pairs.', '1511.00980-2-25-1': 'These terms describe light-induced atomic tunnelling events.', '1511.00980-2-25-2': 'While in general the [MATH] are different for each pair of sites, leading to spatially-dependent tunnelling rates, which may be tuned as with the density-density interactions, we shall here focus on the aforementioned case where they are homogeneous.', '1511.00980-2-25-3': 'This allows for the one-body tunnelling terms (as would be expected from semiclassical treatments) to be suppressed or enhanced by the pump light, or potentially even eliminated.', '1511.00980-2-25-4': 'This latter case would leave only the two-body terms, the rates of which may be tuned semi-independently of the one-body terms.', '1511.00980-2-26-0': 'These two-body processes are of the form [EQUATION] where [MATH] and [MATH] must be pairs of neighbouring sites, though the two pairs may be distributed anywhere within the illuminated regions.', '1511.00980-2-26-1': 'Each pair corresponds to a tunnelling process, and hence the complete two-body processes correspond to correlated tunnelling events.', '1511.00980-2-26-2': 'These can be classified according to the relationship between the two site pairs [Fig. [REF]], giving rise to (1) pair tunnelling, (2) pair exchange, (3) effective next-nearest-neighbour tunnelling, (4 and 5) effective pair processes, and (6) general long-range correlated tunnelling.', '1511.00980-2-26-3': 'Due to the smaller inter-site overlap of Wannier functions compared to on-site overlaps, the inter-site light-matter coupling coefficients will necessarily be smaller than the on-site coefficients at their respective maxima.', '1511.00980-2-26-4': 'Their relative magnitude has previously been studied in [CITATION], where they are shown to typically be separated by approximately an order of magnitude.', '1511.00980-2-26-5': 'Thus, the complete two-body correlated tunnelling terms can have a strength approximately two orders of magnitude less than that which may be achieved for the light-induced density-density interactions, and hence can occur at rates comparable to the classical tunnelling [MATH] (i.e. [MATH]Hz)).', '1511.00980-2-27-0': '## Simulation of Superexchange Interactions', '1511.00980-2-28-0': 'Before we introduce the measurement backaction as a method of control for these processes, we shall first suggest an alternative approach for exerting additional tunability beyond the optical geometry.', '1511.00980-2-28-1': 'One such way is to impart an additional structure to the atomic system by shaping the underlying lattice, so that it is no longer homogeneous at every site.', '1511.00980-2-28-2': 'We provide as an example of this a proposal for how superexchange interactions in spin models may be simulated using the setup.', '1511.00980-2-29-0': 'Two-body atomic tunnelling processes have previously been used to implement such simulations [CITATION].', '1511.00980-2-29-1': 'However, in these earlier proposals, the superexchange occurs as a second-order perturbative process.', '1511.00980-2-29-2': 'In contrast, here we suggest using the cavity-induced pair exchange processes for the same purpose.', '1511.00980-2-29-3': 'Our proposal follows the original by dividing the lattice into pairs of double wells with a superlattice potential, with each site pair containing two atoms of different (pseudo)spin species, and strong interparticle interactions enforcing a one-particle-per-site constraint (we also assume the use of the same method for the initial state preparation).', '1511.00980-2-29-4': 'Introducing the correlated tunnelling as above for the inter-site case (through use of a single cavity mode [MATH]), the superlattice potential and limit on site occupation effects constraints on the allowed dynamical processes, permitting only pair exchange processes between atoms in each site pair.', '1511.00980-2-30-0': 'To see this, consider each of the possible tunnelling events.', '1511.00980-2-30-1': 'The superlattice potential only permits tunnelling of the atoms into the corresponding other site in the double well pair.', '1511.00980-2-30-2': 'The one-particle-per-site constraint suppresses any process in which an atom tunnels to an already occupied site (as is the case in each of the double wells).', '1511.00980-2-30-3': 'However, when we consider the correlated processes, such tunnelling events can take place when coupled with another tunnelling that preserves the unit occupation of each site.', '1511.00980-2-30-4': 'There are two such processes: those in which the atom of the other species in the site pair tunnels in the opposite direction, and those in which the same atom tunnels in the reverse direction (see Fig. [REF]).', '1511.00980-2-30-5': 'The latter processes take the form [MATH], where [MATH] denotes the corresponding site in the well, and [MATH] the spin species.', '1511.00980-2-30-6': 'Due to the unit occupation of each site pair by each spin species, these terms have a constant value, and so may be neglected.', '1511.00980-2-31-0': 'Thus, each double well (up to constant terms) behaves according to the Hamiltonian [EQUATION]', '1511.00980-2-31-1': 'This can equivalently be expressed in terms of spin operators ([MATH]), to give [EQUATION] where [EQUATION]', '1511.00980-2-31-2': 'Critically, the exchange term here does not suffer the [MATH] dependence of the second-order process in the original proposal, and hence the interparticle interactions necessary for enforcing the single-particle-per-site constraint can here be increased without suppressing the exchange interaction.', '1511.00980-2-31-3': 'Indeed, as noted above, these correlated tunnelling processes can occur at rates comparable to the standard tunnelling [MATH] from the classical optical lattice potential.', '1511.00980-2-32-0': '# Inclusion of the Light Measurement Backaction', '1511.00980-2-33-0': 'A drawback of the above method of imparting additional structure by altering the lattice geometry is that it bears an adverse consequence whereby the suppression of a particular tunnelling event will also necessitate such a suppression when it would otherwise have occurred as part of a correlated tunnelling event.', '1511.00980-2-33-1': 'For example, if the process [MATH] is suppressed by the lattice geometry causing it to occur with low amplitude, then any process of the form [MATH] also has a correspondingly low amplitude.', '1511.00980-2-33-2': 'In contrast to this, the matter mode structure discussed above offers an alternative avenue for imprinting further structure onto the atoms, without having to suffer such a penalty.', '1511.00980-2-34-0': 'In previous works [CITATION], we have discussed how the light measurement backaction from the photons leaked by a cavity can be used to selectively suppress atomic dynamics in the standard Bose-Hubbard model, through constraints on the dynamics imposed by the rate at which photons are detected.', '1511.00980-2-34-1': 'We now go beyond this, and incorporate the cavity light-induced dynamics into such methods, thus uniting the measurement backaction effect with the cavity backaction for the first time.', '1511.00980-2-34-2': 'We will then evince the potential of this union for enhancing quantum simulations with atomic systems.', '1511.00980-2-34-3': 'Crucially, the measurement backaction allows for a process to be forbidden as a single event, but permitted when correlated with another particular event (or events) commensurate with the measurement outcome; in this case the suppression is achieved through constraints on the matter mode occupations.', '1511.00980-2-35-0': 'This measurement backaction is realised through the introduction of an additional cavity (and possibly pump) mode, where measurement is made of the scattered light that it leaks.', '1511.00980-2-35-1': 'The cavity leaks photons at a rate that is proportional to its occupation, which is determined by the particular atomic state through Eq. [REF].', '1511.00980-2-35-2': 'We will here assume that this measurement cavity and associated pump is arranged such that the on-site terms dominate the light-matter interactions, and hence for a given atomic Fock state configuration [MATH], the cavity mode [MATH] has a well-defined amplitude [EQUATION]', '1511.00980-2-35-3': 'For such a configuration, the (average) rate of photon leakage from the cavity is constant.', '1511.00980-2-35-4': 'More generally, the atomic configuration is a superposition of such Fock states.', '1511.00980-2-35-5': 'Consider a state where the Fock states [MATH] occur with initial amplitudes [MATH].', '1511.00980-2-35-6': 'Applying the quantum jump measurement formalism, these amplitudes then evolve according to [CITATION] [EQUATION] where [MATH] is the cavity field amplitude for the given Fock state [MATH], and [MATH] is a normalisation factor.', '1511.00980-2-35-7': 'In this expression, the first factor represents the effect of the quantum jumps occurring at each of the [MATH] photodetection events of the leaked photons, while the second factor gives the non-Hermitian evolution occurring between such jumps during the elapsed time [MATH].', '1511.00980-2-35-8': 'This evolution then enacts a natural selection of sorts, reducing the relative probabilities of states not consistent with the observed leakage rate.', '1511.00980-2-35-9': 'For intense, persistent measurement of this form, the distribution of amplitudes is compressed, and consequently the light field state converges towards a particular coherent state [MATH].', '1511.00980-2-35-10': 'When this convergence to a single state happens on timescales much shorter than the atomic dynamics, the light field is ultimately pinned to this state (through the quantum Zeno effect [CITATION]).', '1511.00980-2-35-11': 'The requisite condition on the timescales can be expressed as [MATH] [CITATION].', '1511.00980-2-35-12': 'Note that while this regime is not reached in the aforementioned example experiment [CITATION] (because their intent was to study the cavity backaction alone), it would be feasible by, e.g. a modest reduction of the cavity detuning, or an increase in the pump power, both of which have been performed in previous incarnations of the same setup [CITATION].', '1511.00980-2-36-0': 'The matter is then confined to evolve within only a subspace of its full Hilbert space, which is determined by the particular [MATH]; specifically, it must remain within the subspace of states [MATH] for which [MATH].', '1511.00980-2-36-1': 'This set of states is defined by the matter mode structure, and hence depends on the modefunctions of the measurement cavity and pump [CITATION].', '1511.00980-2-36-2': 'The ensuing atomic dynamics must take place within this measurement subspace, thus undergoing quantum Zeno dynamics [CITATION].', '1511.00980-2-36-3': 'In this regime, the dynamics is described by the appropriate Zeno Hamiltonian, defined as [EQUATION] where [MATH] is the projector that describes the subspace of states [MATH] consistent with the measured light state [CITATION].', '1511.00980-2-36-4': 'This result follows generally from considerations of a system subject to very frequent measurement: in the limit that the number of measurements [MATH] in a fixed time [MATH], the evolution [MATH] can be expanded approximately as [CITATION] [EQUATION] with the second line following from the definition of the exponential function [MATH], and the idempotence of the projector ([MATH]).', '1511.00980-2-36-5': 'We shall now drop the [MATH] subscript from the Zeno Hamiltonians for the remainder of this article.', '1511.00980-2-37-0': 'Thus, utilising this formalism, we can use the measurement backaction to selectively eliminate particular atomic dynamics, as determined by the measurement cavity geometry and the associated projection operators.', '1511.00980-2-37-1': 'In contrast to earlier work incorporating measurement backaction, in which the two-body terms that appear due to measurement are only second-order processes [CITATION], in this scheme such terms now arise here at first order in the system evolution, as they are directly induced by the cavity backaction (rather than simply being higher-order processes which are not suppressed by the measurement), and are hence perfectly correlated.', '1511.00980-2-37-2': 'The two cases are thus fundamentally different, and here increasing pump strength increases the two-body tunnelling rates, rather than suppressing them.', '1511.00980-2-37-3': 'Note that because the dynamics of the system is now constrained to the subspace of states consistent with a single value of the measurement cavity light-matter interaction operator, the dynamics induced by the measurement cavity may be disregarded, as they are identical for all states in the subspace, and hence form a constant energy shift for all states.', '1511.00980-2-38-0': 'As a straightforward example of such a scheme, consider the case of measurement made at the diffraction minimum ([MATH]; [MATH]) across the lattice.', '1511.00980-2-38-1': 'This freezes the occupation number difference between odd and even sites, and thus when this difference is given by [MATH], the allowed states are superpositions of states of the form [MATH] with [MATH] and [MATH], for integer [MATH].', '1511.00980-2-38-2': 'The dynamics is restricted to this subspace of states with associated projector [EQUATION] by the Zeno dynamics, and since single nearest neighbour tunnelling events change the [MATH] of a state, such processes are forbidden from taking place by the measurement.', '1511.00980-2-38-3': 'In the absence of a further cavity driving dynamics this would leave next-nearest neighbour tunnelling as the leading process, which typically occurs at a much slower rate than nearest-neighbour tunnelling in the standard Bose-Hubbard model (and thus is often ignored), but may now be no longer negligible.', '1511.00980-2-39-0': 'Considering also the cavity backaction when a cavity is present to drive dynamics, this scenario may be augmented with the two-body terms introduced above (namely, those that preserve matter mode occupation number difference).', '1511.00980-2-39-1': 'The correlated processes in which two atoms tunnel into the same site (and the reverse process) are forbidden by the measurement as they violate the constraint on [MATH], as are certain of the long-range correlated tunnelling events.', '1511.00980-2-39-2': 'The allowed processes of the latter form lead to effective long-range tunnelling events within each mode.', '1511.00980-2-39-3': 'All of the light-induced effective nearest-neighbour tunnelling events are permitted, as they simply move atoms between neighbouring sites in each of the two modes, as are the pair exchange events, as they do not change the site occupation numbers.', '1511.00980-2-39-4': 'Fig. [REF] illustrates examples of each of the allowed processes in this two mode example.', '1511.00980-2-39-5': 'In the simplest case of uniform illumination, the rate of the long-range correlated tunnelling processes are independent of the site separation, though as noted above, akin to the density-density interactions, spatially-dependent long-range tunnelling rates can be tuned with multiple cavity modes.', '1511.00980-2-40-0': 'The flexibility of the light-modes allows a range of configurations of the two-body terms to be engineered through measurement backaction; for example, when light measurement fixes occupation numbers for multiple matter modes [CITATION], the only allowed correlated tunnelling events are pair exchange, and the long-range which preserve the total mode occupations.', '1511.00980-2-41-0': '# Framework for Quantum Simulations', '1511.00980-2-42-0': 'The architecture based on the union of measurement backaction and cavity backaction that we have detailed above naturally lends itself to quantum simulations.', '1511.00980-2-42-1': 'In particular, it offers opportunities to mimic correlated processes and long-range interactions of the atoms that would be difficult (or even not possible) to realise in systems with finite-range interactions.', '1511.00980-2-42-2': "We now proceed by outlining a framework to this end, by describing 'building block' components for implementing reservoir and dynamical global gauge field models, which can be used to enhance methods of optical lattice quantum simulation beyond current methods, by incorporating such phenomena.", '1511.00980-2-43-0': '## Reservoir Models', '1511.00980-2-44-0': 'The matter mode structure defined by the light geometry allows the lattice to be partitioned into sets of sites corresponding to each mode.', '1511.00980-2-44-1': 'We can assign a subset of these modes as reservoirs, and investigate the dynamics of the remaining sites, subject to the presence of the reservoirs.', '1511.00980-2-44-2': 'Consider the conceptual three-site model shown in Fig. [REF](a).', '1511.00980-2-44-3': 'In this scenario, a cavity is used to drive dynamics that generate (homogenous amplitude) two-body correlated tunnelling between the sites, as per Eq. [REF] with a single cavity [MATH] and uniform light-matter interaction coefficients [MATH], such that the system is described by [EQUATION]', '1511.00980-2-44-4': 'Neglecting the effective (uniform) chemical potential due to the pump light (which can be tuned away by e.g. using another classical light source), and the on-site interactions (which can be tuned away by Feshbach resonances [CITATION]), this becomes [EQUATION]', '1511.00980-2-44-5': 'We label the sites [MATH], and designate the outer two sites as the reservoirs.', '1511.00980-2-44-6': 'The measurement cavity is arranged at the maxima of two coherent antiphase pump lasers antisymmetric about the central site, such that they fully destructively interfere at this site, and have opposing contributions at the corresponding site pairs about the centre.', '1511.00980-2-44-7': 'The resulting total pump mode function is [MATH], and thus the measured operator is [MATH].', '1511.00980-2-44-8': 'An example set of appropriate pump modefunctions are [MATH], where [MATH] is the lattice spacing and [MATH] is the lattice axis, such modefunctions being achievable by using travelling waves angled at [MATH] to the lattice (the incommensurate nature of the pump with the lattice ensures that any other sites adjacent to the reservoirs do not have vanishing contributions to the measurement, preventing events where one atom tunnels into the central site simultaneously with an atom tunnelling to an external site).', '1511.00980-2-44-9': 'The measurement then constrains the system to remain in a subspace of states in which this is constant.', '1511.00980-2-44-10': 'When the measured value for this operator is [MATH], the appropriate projectors applied to the system are (using the designation [MATH]) [EQUATION]', '1511.00980-2-44-11': 'Applying these projectors to the Hamiltonian Eq. [REF], we obtain the Zeno Hamiltonian [MATH].', '1511.00980-2-44-12': 'The projectors eliminate all of the single-atom tunnelling terms, and preserve only the two-body tunnellings in which [MATH] is conserved.', '1511.00980-2-44-13': 'These are the following terms: [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; and [MATH].', '1511.00980-2-44-14': 'The first four of these terms correspond to processes where an atom leaves one of the reservoirs, concurrent with another atom entering the same reservoir, while the latter four describe events where atoms simultaneously leave (or enter) both reservoirs.', '1511.00980-2-45-0': 'Consider now the reservoirs to both be prepared in coherent states (as would be approximately expected for a system initially in a superfluid state from the Gutzwiller ansatz [CITATION]).', '1511.00980-2-45-1': 'Replacing the operators for the two reservoir sites by their coherent amplitudes [MATH] and [MATH], we now only have one dynamical variable, describing the occupation of the central site.', '1511.00980-2-45-2': 'Relabelling this as [MATH], and defining [MATH], the effective Hamiltonian becomes [EQUATION]', '1511.00980-2-45-3': 'Discarding the constant term, and further tuning on-site terms to eliminate the effective chemical potential, we arrive at a Hamiltonian that describes effective pair creation and annihilation dynamics at the central site: [EQUATION] where [EQUATION]', '1511.00980-2-45-4': 'With the experimental parameters considered above, and for typical occupations [MATH], these processes can be of a comparable size to the classical tunnelling rate [MATH] (or even slightly larger depending on [MATH]).', '1511.00980-2-46-0': 'Now consider the extension of this to include an additional site between the reservoirs, as depicted in Fig. [REF](b).', '1511.00980-2-46-1': 'Again, the outer sites are designated as the reservoirs, and their occupation number difference is the subject of measurement.', '1511.00980-2-46-2': 'One way to achieve such a mode structure is to again use the interference of two coherent pump beams, now arranged to have vanishing contributions on the central two sites.', '1511.00980-2-46-3': 'Thus, the measured operator is (labelling the central two sites 1 and 2, and the reservoirs Res1 and Res2) given by [MATH], again fixed at some value [MATH], now with the associated projectors [EQUATION]', '1511.00980-2-46-4': 'As before, this measurement imposes constraints on the allowed tunnelling events.', '1511.00980-2-46-5': 'Unlike the previous case however, some single atom tunnelling events survive: those between the two central sites [MATH] and [MATH].', '1511.00980-2-46-6': 'The permitted correlated tunnelling events involving the reservoirs are analogous to the previous case, with the processes involving atoms simultaneously crossing the same boundary between a reservoir and central site in opposite directions, or the simultaneous tunnelling of a particle from (to) each of the reservoirs in to (from) each of the central sites.', '1511.00980-2-47-0': 'However, there are also now present correlated events where both tunnelling events take place between the central two sites, these being of the form [MATH], [MATH], [MATH], and [MATH].', '1511.00980-2-47-1': 'Once again replacing the operators for the reservoirs by their coherent state amplitudes, we can write the Hamiltonian describing the dynamics of the central two sites (again for now neglecting the effective chemical potentials) as [EQUATION]', '1511.00980-2-47-2': 'Consider now the reservoirs to have a large occupation compared to the central sites (e.g. [MATH] atom per central site, and [MATH] per reservoir; such filling factors are available in typical experiments [CITATION]).', '1511.00980-2-47-3': 'In this case, the terms in the third line of Eq. [REF], corresponding to the correlated processes between atoms in the central sites alone, become negligible compared to the reservoir-based correlated tunnellings.', '1511.00980-2-47-4': 'Reintroducing the chemical potential (which, as noted before, can be tuned with additional semiclassical light sources), the effective Hamiltonian describing the central two sites is now a generalised Dicke Hamiltonian [CITATION]: [EQUATION] where [EQUATION]', '1511.00980-2-47-5': 'In contrast to the original model, and its corresponding realisation in optical cavities [CITATION], the parameters [MATH], which represent the co- and counter-rotating terms respectively, can be tuned independently, by adjusting the optical geometry; their magnitude is controlled by e.g. increasing the pump strength or adjusting the reservoir populations [MATH], and complex phases can be induced with the use of additional pump beams.', '1511.00980-2-48-0': 'In addition to the well-known phase diagrams for [MATH] (traditional Dicke) and [MATH] (Tavis-Cummings) [CITATION] in the quantum case, classical treatments have shown bifurcations when varying [MATH] [CITATION], suggesting possible further novel phase behavior.', '1511.00980-2-48-1': 'We explore part of this extended parameter space by using exact diagonalisation methods to find the ground state average occupation of each of the modes.', '1511.00980-2-48-2': 'Specifically, we do this by limiting the occupation of the two modes to 20 atoms per site, and with this limitation we can construct the Hamiltonian Eq. [REF], and obtain the ground state.', '1511.00980-2-48-3': 'We do this for the regime in which [MATH], and find that a general phase boundary [Fig. [REF]] for the onset of superradiance (that is, the transition from [MATH] to diverging (in the full non-occupation-limited case)) occurs at [MATH], with the standard transition at [MATH] being a special case of this.', '1511.00980-2-49-0': 'This can be further extended by using the light measurement to fix occupation number differences between larger numbers of modes [CITATION], allowing for additional reservoir modes to be generated.', '1511.00980-2-49-1': 'These can be used to increase the number of sites coupled to reservoirs, and increase the number of simulated modes in our model (for example, to have multiple atomic species).', '1511.00980-2-49-2': 'As a concrete example of how these building blocks we propose can be put together to form more complex systems, we now describe how to realise a generalised Dicke model with two (synthetic) atomic species and one synthetic light mode.', '1511.00980-2-50-0': 'Consider an amalgamation of two copies of the above setup for the one-species generalised Dicke model.', '1511.00980-2-50-1': 'When these setups are crossed perpendicularly, with both setups sharing a common site for the synthetic light mode (see Fig. [REF]), the two-species generalised Dicke model can be realised.', '1511.00980-2-50-2': 'As before, cavities are used to induce the correlated tunnelling dynamics, one in each of the setups (that is, the correlated events both involve tunnellings within the same individual setup).', '1511.00980-2-50-3': 'By using also two measurement cavities, to fix each the difference in occupation of the reservoir sites in one of the setups (to [MATH] and [MATH] respectively), the resulting projectors are [EQUATION] where [MATH] and [MATH] are the simulated atomic species, [MATH] is the synthetic light mode, [MATH] and [MATH] are the reservoir modes associated with the simulated light and atomic modes respectively in the first setup (and [MATH] and [MATH] for the second), and the states are labelled as [MATH].', '1511.00980-2-51-0': 'Following the analysis for the single-species Dicke model Eq. [REF], replacing the reservoirs by coherent states, and noting that both setups have a central site in common (corresponding to the synthetic light mode), the resulting Hamiltonian in the Zeno subspace for the central sites may be written [EQUATION] where the [MATH] are the same as for the one-species setup, with the appropriate coefficients [MATH], [MATH], [MATH] and [MATH] as for each of the individual setups.', '1511.00980-2-52-0': 'Yet more exotic setups can be envisaged within this framework, through the use of additional reservoirs, or with multiple sites coupling to each reservoir.', '1511.00980-2-52-1': 'In the extreme case of each site in the simulated system being coupled to its own two reservoirs with fixed occupation differences (i.e. a lattice of copies of the setup Fig. [REF](a)), one would obtain an extended Bose-Hubbard model with additional pair creation/annihilation effects present at each site.', '1511.00980-2-52-2': 'Perhaps more straightforward to realise experimentally, consider a similar setup with two reservoir modes (again with fixed occupation difference) common to all sites.', '1511.00980-2-52-3': 'This could be achieved by again taking the setup of Fig. [REF](a), where instead of three individual sites we instead have three connected lattices (which may be one- or two-dimensional), with each of the lattices forming one of the three modes.', '1511.00980-2-52-4': 'Arranging the measurement cavity to scatter light with the same phase from all sites in a given reservoir mode renders them indistinguishable to the measurement, as per the matter mode structure, and so they behave like a collective reservoir.', '1511.00980-2-52-5': 'In the superfluid regime, each site in reservoir [MATH] may be approximated again by the Gutzwiller ansatz, and described by a coherent state of amplitude [MATH], where [MATH] is the filling factor of the lattice.', '1511.00980-2-52-6': 'As the sites in each reservoir mode are indistinguishable, the correlated tunnellings from each of the reservoirs to/from the central lattice may occur due to any of the sites in each reservoir.', '1511.00980-2-52-7': 'Thus, the effective pair creation/annihilation in the central lattice may have the pair of atoms far apart in the lattice from each other.', '1511.00980-2-52-8': 'We have analogous projectors to the three-site case Eq. [REF], but with each number now representing the occupation across the whole of the respective lattice, rather than individual sites.', '1511.00980-2-52-9': 'Note that the standard Bose-Hubbard dynamics for atoms tunnelling between sites in the central lattice is unaffected by the measurement, and thus the resulting system in the central lattice obeys a Bose-Hubbard model with long-range pair creation/annihilation; [EQUATION] where [EQUATION]', '1511.00980-2-53-0': '## Dynamical Global Gauge Fields', '1511.00980-2-54-0': 'Another application of this framework for quantum simulation is in the synthesis of artificial dynamical gauge fields.', '1511.00980-2-54-1': 'Current proposals to this end also typically employ ultracold atoms in optical lattices, focussing on local gauge fields based on quantum link models [CITATION], where changes in the gauge field due to the motion of matter are simulated by the motion of another particle species that plays the role of the gauge field [CITATION].', '1511.00980-2-54-2': 'In addition to the possibility of extending the variety of such models by the inclusion of long-range interactions, the introduction of the cavity-mediated dynamics presents a further opportunity: the long-range nature of the interactions allows for the correlated atomic motion itself to be long-range, and hence the links need not be local.', '1511.00980-2-54-3': 'As such, this paves the way for realising global dynamical gauge fields, where motion across all sites is controlled by a common link.', '1511.00980-2-55-0': 'For example, consider a one-dimensional lattice, with light measurement of a site-dependent strength.', '1511.00980-2-55-1': 'Specifically (by, e.g. a gradiated intensity of the measurement pump), the measured light state has an amplitude that is proportional to [MATH], for some constant [MATH].', '1511.00980-2-55-2': 'We consider two auxiliary sites [MATH] and [MATH], between which atoms can tunnel (and tunnelling out of this pair is suppressed), which are also measured in the same fashion, contributing an effective [MATH] (plus constant) to the measurement value (because their conserved total occupation [MATH]).', '1511.00980-2-55-3': 'In this case, sites [MATH] and [MATH] form a link that mediates the dynamics in the rest of the lattice, as the Zeno Hamiltonian will then only contain the dynamics from the cavity-mediated correlated tunnelling events, where an atom tunnels in the main lattice simultaneously with a tunnelling event either in the link, or in the opposite direction in the main lattice.', '1511.00980-2-55-4': 'This follows from considering that a tunnelling [MATH] (or [MATH]) increases the value of [MATH] by [MATH], while a tunnelling [MATH] (or [MATH]) decreases it by [MATH], and so only when one of each of these processes occur simultaneously is the measurement outcome value preserved.', '1511.00980-2-56-0': 'The resulting Hamiltonian after applying this constraint is [EQUATION] where [EQUATION] and [MATH] is the ratio of the intensity of the pump used to drive dynamics at the link sites compared to the rest of the lattice.', '1511.00980-2-56-1': 'Equivalently, by mapping the link to a spin with [MATH], this can be written [EQUATION]', '1511.00980-2-56-2': 'This [MATH] is comparable in size to the equivalent parameter considered in the reservoir models (e.g. Eq. [REF]), and so too may also be of a similar magnitude to the standard tunnelling [MATH], with its precise value depending on the pump strength.', '1511.00980-2-57-0': 'This forms a global-link dynamical matter-gauge field interaction, in contrast to current proposals for dynamical gauge fields, which are limited to local links by their finite-range interactions.', '1511.00980-2-57-1': 'Strictly, we note that pair-correlated tunnelling events are allowed within the main lattice independent of the link, provided that they occur in opposite directions, though the particle current across the lattice is wholly dependent on the link.', '1511.00980-2-57-2': 'These terms bypassing the link can be made less significant by adjustment of [MATH], which also controls the gauge field energy terms [MATH].', '1511.00980-2-57-3': 'These gauge field energy terms can be further engineered through the density-density interactions discussed above.', '1511.00980-2-57-4': 'Unlike the local link models ubiquitous in high-energy physics, the common global link here leads to a peculiar effect where the motion of a particle at any site can significantly affect the field experienced by all other particles, at all other sites.', '1511.00980-2-58-0': '# Conclusions', '1511.00980-2-59-0': 'In summary, we have characterised the new dynamics manifest by the interactions of atomic quantum gases with quantum light in optical cavities, exhibiting effects beyond those possible with classical light, and subsequently shown that these may then be controlled through measurement of the light leaked from the cavity.', '1511.00980-2-59-1': 'These effects include long-range correlated tunnelling, effective pair processes, and density-density interactions.', '1511.00980-2-59-2': 'Further, we have discussed how this provides opportunities for the enhancement of quantum simulations, by using these correlated processes.', '1511.00980-2-59-3': 'Specifically, we have demonstrated how the formalism can mimic superexchange interactions, reservoir models, and dynamical global gauge fields.', '1511.00980-2-59-4': 'This invites a wealth of opportunities for further study, such as combining the various simulation building blocks presented here to generate yet more exotic and interesting systems for study, finding additional building blocks to expand the simulation framework, and further characterisation of cavity-induced processes.', '1511.00980-2-60-0': 'As discussed above, our proposal should be feasible with current state-of-the-art experimental setups.', '1511.00980-2-60-1': 'So far, several groups have trapped Bose-Einstein condensates inside cavities, without a lattice potential [CITATION], while others have scattered light from quantum gases trapped in lattice potentials, but with no cavity present [CITATION].', '1511.00980-2-60-2': 'The amalgamation of these proposals has recently been achieved, where the light scattered into an optical cavity by the atoms generates a further, quantum potential for the atoms, which is dynamically evolving conditional on the atomic state [CITATION].', '1511.00980-2-60-3': 'These experiments already exhibit a particular case of the cavity-induced dynamics, where the the cavity field causes the atoms to self-organise into charge density wave and supersolid phases.', '1511.00980-2-61-0': 'Additionally, many recent experiments have demonstrated examples of quantum Zeno physics in similar, but less versatile settings, such as cavity QED and optical lattices [CITATION].', '1511.00980-2-61-1': 'Thus, the possibility to use measurement for such a selective suppression of dynamics is well verified.', '1511.00980-2-61-2': 'Furthermore, it may be possible to implement the dynamical effects discussed here using other types of systems that are also based on off-resonant scattering, such as molecules [CITATION], fermions [CITATION], spins [CITATION], ions [CITATION], and semiconductor [CITATION] and superconducting qubits [CITATION], as their dynamics are based on similar mathematical structures.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1511.00980

0812.3358

{'0812.3358-1-0-0': 'We report on the results from Suzaku broadband X-ray observations of the galactic binary source LS 5039.', '0812.3358-1-0-1': 'The Suzaku data, which covered continuously more than one orbital period, show strong modulation of X-ray emission at the orbital period of this TeV gamma-ray emitting system.', '0812.3358-1-0-2': 'The X-ray emission shows a minimum at the orbital phase [MATH], close to the so-called superior conjunction of the compact object, and a maximum at phase [MATH], very close to the inferior conjunction of the compact object.', '0812.3358-1-0-3': 'The X-ray data up to 70 keV are described by a hard power-law spectrum with a phase-dependent photon index which varies within [MATH]-1.6.', '0812.3358-1-0-4': 'The amplitude of flux variation amounts to a factor of 2.5, but is significantly less than that of the TeV flux which is as large as a factor of 8.', '0812.3358-1-0-5': 'Otherwise the two light curves are similar, but not identical.', '0812.3358-1-0-6': 'Although periodic X-ray emission has been found from many galactic binary systems, the Suzaku result implies a phenomenon different from the "standard" origin of X-rays related to the emission of the hot accretion plasma formed around the compact companion object.', '0812.3358-1-0-7': 'The X-ray radiation of LS 5039 is likely to be linked to very high energy electrons which are also responsible for the TeV gama-ray emission.', '0812.3358-1-0-8': 'While the gamma-rays are the result of inverse Compton scattering by electrons on the photons of the optical star, X-rays are produced via synchrotron radiation.', '0812.3358-1-0-9': 'Yet, while the modulation of the TeV gamma-ray signal can be naturally explained by the photon-photon pair production and anisotropic inverse Compton scattering, the observed modulation of synchrotron X-rays requires an additional process, being the most natural one, adiabatic expansion in the radiation production region.', '0812.3358-1-1-0': '# Introduction', '0812.3358-1-2-0': 'LS 5039 is a high-mass X-ray binary with extended radio jets .', '0812.3358-1-2-1': 'This system is formed by a main sequence O type star and a compact object of disputed nature that has been claimed to be a black hole or a neutron star/pulsar .', '0812.3358-1-2-2': 'The compact object is moving around the companion star in a moderately elliptic orbit (eccentricity [MATH]) with an orbital period of [MATH] day .', '0812.3358-1-3-0': 'As summarized in [CITATION], LS 5039 has been observed several times in the X-ray energy band for limited period in the phase.', '0812.3358-1-3-1': 'Flux variations on time scale of days and sometimes on much shorter timescales have been reported.', '0812.3358-1-3-2': 'The spectrum was always well represented by a power-law model with a photon index ranging [MATH]-1.6 up to [MATH] keV, with fluxes changing moderately around [MATH].', '0812.3358-1-3-3': 'Softer spectra and larger fluxes had been also inferred from RXTE observations, although background contamination was probably behind these differences (see ).', '0812.3358-1-3-4': 'Also, Chandra data taken in 2002 and 2005 showed spectra significantly harder than 1.5, but such a hard spectrum was probably an artifact produced by photon pile-up.', '0812.3358-1-3-5': 'Recently, [CITATION] reported the results of INTEGRAL observations at hard X-rays.', '0812.3358-1-3-6': 'The source was detected in the range between 25 and 60 keV.', '0812.3358-1-3-7': 'The flux was estimated to be [MATH] around the inferior conjunction (INFC) of the compact object, and a flux upper limit was placed in [MATH] at the superior conjunction of the compact object (SUPC).', '0812.3358-1-4-0': 'LS 5039 has also been detected at very high-energy gamma-rays , exhibiting a periodic signal modulated with the orbital period .', '0812.3358-1-4-1': 'There are two other binary systems with robust detections in the VHE range: PSR B1259[MATH]63 and LS I [MATH]303 .', '0812.3358-1-4-2': 'An evidence of TeV detection has been found in Cygnus X-1 .', '0812.3358-1-4-3': 'PSR B1259[MATH]63 is a clear case of a high-mass binary system containing a non-accreting pulsar , whereas Cygnus X-1 is a well known accreting black hole system .', '0812.3358-1-4-4': 'Otherwise, the nature of the compact object in LS 5039 is not yet established, and the origin of the VHE emitting electrons is unclear.', '0812.3358-1-4-5': 'They can be related to a pulsar wind or to a black hole with a (sub)relativistic jet.', '0812.3358-1-4-6': 'The severe photon-photon absorption expected for VHE gamma-rays makes the explanation of the detection of the VHE radiation problematic, at least at the position corresponding to the SUPC in the standard pulsar-wind scenario, in which the emitter is expected to be located between the compact object and the star.', '0812.3358-1-4-7': 'Yet, particles can be accelerated in a relativistic outflow formed at the interaction of the pulsar and the stellar winds and radiate far from the compact object, still making the pulsar wind scenario a viable option.', '0812.3358-1-4-8': 'Regarding the microquasar scenario, the lack of accretion features in the X-ray spectrum could be a problem unless the major fraction of the accretion power is released in the form of kinetic energy of the outflow instead of thermal emission of the accretion, as in the case of SS 433 .', '0812.3358-1-4-9': 'At this stage, we cannot give a preference to any of these scenarios, but new data, in particular those obtained with the Suzaku satellite, allow us to make an important step towards the understanding of the nature of the non-thermal processes of acceleration and radiation in this misterious object.', '0812.3358-1-5-0': '# Observation', '0812.3358-1-6-0': 'The temporal and spectral characteristics of the X-ray emission from LS 5039 along the orbit should give us important clues for understanding the acceleration/radiation processes in this source.', '0812.3358-1-6-1': 'The fact that all previous X-ray observations of this object have incomplete coverage of the orbital period, or suffered from background contamination, is therefore rather unsatisfactory.', '0812.3358-1-6-2': 'This motivated our long, [MATH] ks observation with the Suzaku X-ray observatory, which gives us an unprecendent coverage of more than one orbital period, continuously from 2007 September 9 to 15 (see Table [REF]).', '0812.3358-1-6-3': 'Suzaku has four sets of X-ray telescopes each with a focal-plane X-ray CCD camera that are sensitive in the energy range of 0.3-12 keV.', '0812.3358-1-6-4': 'Three of the XIS detectors (XIS0, 2 and 3) have front-illuminated (FI) CCDs, whereas XIS1 utilizes a back-illuminated (BI) CCD.', '0812.3358-1-6-5': 'The merit of the BI CCD is its improved sensitivity in the soft X-ray energy band below 1 keV.', '0812.3358-1-6-6': 'Suzaku contains also a non-imaging collimated Hard X-ray Detector , which covers the 10-600 keV energy band with Si PIN photodiodes (10-70 keV) and GSO scintillation detectors (40-600 keV).', '0812.3358-1-6-7': 'Suzaku has two default pointing positions, the XIS nominal position and the HXD nominal position.', '0812.3358-1-6-8': 'In this observation, we used the HXD nominal position, in which the effective area of HXD is maximized, whereas that of the XIS is reduced to on average [MATH] 88%.', '0812.3358-1-6-9': 'In what follows, results from XIS2 are not reported since it has not been in operation since an anomaly the sensor underwent in November 2006.', '0812.3358-1-6-10': 'In addition, we do not describe the detail of analysis on HXD-GSO data, since the HXD-GSO detected no significant signal from the source.', '0812.3358-1-7-0': '# Data Reduction', '0812.3358-1-8-0': 'We used data sets processed using the software of the Suzaku data processing pipeline (version 2.1.6.16).', '0812.3358-1-8-1': 'Reduction and analysis of the data were performed following the standard procedure using the HEADAS v6.4 software package, and spectral fitting was performed with XSPEC v.11.3.2.', '0812.3358-1-9-0': 'For the XIS data analysis, we accumulated cleaned events over good time intervals that were selected by removing spacecraft passages through the South Atlantic Anomaly (SAA).', '0812.3358-1-9-1': "Further, we screened the data with the following criteria - (1) cut-off rigidity is larger than 6 GV and (2) elevation angle from the Earth's rim is larger than 5[MATH].", '0812.3358-1-9-2': 'The source photons were accumulated from a circular region with a radius of [MATH], whereas the background region was chosen in the same field of view with the same radius and an offset of [MATH] from the source region.', '0812.3358-1-9-3': 'We have co-added the data from the two FI-CCDs (XIS0 and XIS3) to increase statistics.', '0812.3358-1-9-4': 'The response (RMF) files and the auxiliary response (ARF) files used in this paper were produced using xisrmfgen and xissimarfgen, respectively.', '0812.3358-1-10-0': 'For the HXD data,"uncleaned event files" were screened with the standard event screening criteria: the cut-off rigidity is larger than 6 GV, the elapsed time after the passage of the SAA is more than 500 s and the time to the next SAA passage is more than 180 s, high voltages from all eight HV units are within the normal value and elevation angle from the Earth\'s rim is more than [MATH]; we also discarded the telemetry-saturated time intervals.', '0812.3358-1-10-1': 'In the spectral analysis in 3, we used the response file for a point-like source at the HXD-nominal position (aehxdpinhxnome420070914.rsp), which is released as a part of CALDB.', '0812.3358-1-11-0': 'The HXD-PIN spectrum is dominated by the time-variable instrumental background induced by cosmic rays and trapped charged particles in the satellite orbit.', '0812.3358-1-11-1': 'To estimate the instrumental background component, we used the the time-dependent non-X-ray background (NXB) event files released by the HXD instrument team, whose reproducibility is reported by [CITATION].', '0812.3358-1-11-2': 'In order to estimate the systematic uncertainty of the NXB model during our observation, we compare the NXB model spectrum during Earth occultation with the observed spectrum of the same time interval.', '0812.3358-1-11-3': 'The event selection criteria for this study are the same as those of the cleaned event except for the criterium about the Earth elevation angle, which was chosen to to be less than [MATH].', '0812.3358-1-11-4': 'The estimated uncertainty obtained is [MATH]%, which is consistent with the values reported by [CITATION].', '0812.3358-1-12-0': 'Another component of the HXD-PIN background is the Cosmic X-ray background (CXB).', '0812.3358-1-12-1': 'In our analysis, we assumed the CXB spectrum reported by : [EQUATION] where [MATH] and [MATH].', '0812.3358-1-12-2': 'The CXB spectrum observed with HXD-PIN was simulated by using a PIN response file for isotropic diffuse emission (aehxdpinflate420070914.rsp) and added to the NXB spectrum.', '0812.3358-1-12-3': 'Based on this, the contribution from the CXB flux amounts to [MATH]% of the NXB.', '0812.3358-1-13-0': 'Since LS 5039 is located close to the Galactic plane, the contribution from the Galactic ridge X-ray emission (GRXE) must be examined, especially for HXD-PIN spectra.', '0812.3358-1-13-1': 'In order to model the shape of the GRXE, data from Suzaku observations of the Galactic ridge region (ObsID: 500009010 and 500009020) were analyzed.', '0812.3358-1-13-2': 'The Suzaku spectrum from 3 keV to 50 keV can be well fitted with the Raymond-Smith plasma with a temperature of [MATH] keV and a power law function with [MATH].', '0812.3358-1-13-3': 'Although we also tried a power law with exponential cutoff following the results by the INTEGRAL IBIS , it turned out that the assumption on the spectral shapes of the GRXE has negligible effects on the spectral parameters of LS 5039.', '0812.3358-1-13-4': 'The normalization of the GRXE spectrum contaminated in the HXD-PIN spectrum of LS 5039 is determined from the XIS spectrum of the LS 5039 observation by excluding an encircled region with a radius of [MATH] centered on the LS 5039 location.', '0812.3358-1-13-5': 'The flux of the GRXE is estimated to be [MATH] of the contribution from the CXB.', '0812.3358-1-13-6': 'In Figure [REF], we show the time averaged HXD-PIN spectrum plotted with the NXB model, the CXB model and the GRXE model.', '0812.3358-1-14-0': '# Analysis and Results', '0812.3358-1-15-0': '## Temporal analysis', '0812.3358-1-16-0': 'The light curve obtained from the XIS detector is shown in the top panel of Figure [REF].', '0812.3358-1-16-1': 'The continuous coverage in X-rays, longer than the orbital period of the LS 5039 system, reveals a smooth variation of a factor 2 in the 1-10 keV count rate.', '0812.3358-1-16-2': 'The light curve is drawn over two orbital periods.', '0812.3358-1-16-3': 'The orbital phase is calculated with the period of 3.90603 days, and [MATH] with reference epoch [MATH]) taken from [CITATION].', '0812.3358-1-16-4': 'The light curve from phase [MATH] to 1.5, which was obtained in the last part of the observation, smoothly overlaps with the one obtained at the beginning of the observation ([MATH]-0.5).', '0812.3358-1-17-0': 'In the middle panel of Figure [REF], we present the light curve obtained with the HXD-PIN for the energy range 15-40 keV.', '0812.3358-1-17-1': 'Although the statistical errors are larger, the modulation behavior is similar to that of the XIS.', '0812.3358-1-17-2': 'The amplitude of the modulation is roughly the same between the XIS and HXD-PIN, indicating small changes of spectral shapes depending on orbital phase.', '0812.3358-1-17-3': 'The spectral parameters obtained for each orbital phase are reported in the following section.', '0812.3358-1-18-0': 'The light curves obtained with Suzaku show that the X-ray flux minimum appears around phase 0.0-0.3 and it reaches maximum around phase 0.5-0.8.', '0812.3358-1-18-1': 'This is similar to the structure discovered in the phase diagram of integral fluxes at energies [MATH] TeV obtained on a run-by-run basis from HESS data (2004 to 2005) .', '0812.3358-1-18-2': 'The temporal X-ray behavior was already suggested by RXTE data, as well as by a compilation of all the previous X-ray data obtained with imaging instruments .', '0812.3358-1-19-0': 'In addition to the continuously changing component with respect to the orbital phase, wiggle-like structures are found around [MATH] and [MATH].', '0812.3358-1-19-1': 'The unabsorbed flux changes about a 30% in [MATH].', '0812.3358-1-19-2': 'A dip structure can be seen around [MATH] in the top panel of Figure [REF].', '0812.3358-1-19-3': 'In comparison with the data obtained in the first half of the observation, whereas the overall flux is at the same level, the flux decreased [MATH]50% only in this phase.', '0812.3358-1-19-4': 'The time duration of this dip corresponds to [MATH].', '0812.3358-1-19-5': 'These structures may reflect features of the (possibly changing) environment of the X-ray emitting region, it is therefore of importance to test if they are persistent features with further observations.', '0812.3358-1-20-0': '## Spectral Analysis', '0812.3358-1-21-0': 'Firstly we study time-resolved (phase-resolved) X-ray spectra.', '0812.3358-1-21-1': 'The data are divided into data segments with respect to the assigned phase, and model fitting is performed for XIS spectra for each segment with [MATH].', '0812.3358-1-22-0': 'A single power-law function with photoelectric absorption, provides a good fit for all the segments.', '0812.3358-1-22-1': 'In order to study the possible changes of the amount of photoelectric absorption, we here fit the data by allowing [MATH] to be free.', '0812.3358-1-22-2': 'The best-fit parameters are presented in Table [REF].', '0812.3358-1-22-3': 'The derived values of the photon index and absorption column density are consistent with previous observations .', '0812.3358-1-23-0': 'The photon index ([MATH]) values are plotted as a function of orbital phase in the middle panel of Figure [REF].', '0812.3358-1-23-1': 'The spectral shape varied such that the spectrum is steep around SUPC ([MATH]) and becomes hard ([MATH]) around apastron.', '0812.3358-1-23-2': 'The modulation behavior of [MATH] is somewhat different from that observed by HESS in the VHE range.', '0812.3358-1-23-3': 'The amplitude of the variation is [MATH], which is much smaller than the change of [MATH]0.6 in the VHE region .', '0812.3358-1-23-4': 'The 1-10 keV flux changes from [MATH]) to [MATH].', '0812.3358-1-23-5': '([MATH]).', '0812.3358-1-24-0': 'In all the data segments, the source is significantly detected with the HXD-PIN, indicating that hard X-ray emission extends at least up to 70 keV.', '0812.3358-1-24-1': 'Note also that although the XIS and HXD-PIN spectra do not overlap, they seem to be smoothly connected within the gap between 10 and 15 keV.', '0812.3358-1-25-0': 'To study the shape of the spectrum above 10 keV, the XIS spectra and the PIN spectrum in the range 15-70 keV after subtraction of background (NXB [MATH] CXB [MATH] GRXE) are jointly fitted (Figure [REF] Top).', '0812.3358-1-25-1': 'The spectra are well represented by an absorbed power-law model with [MATH]= 1.51 [MATH]0.02 with reduced [MATH] (235 degrees of freedom).', '0812.3358-1-25-2': 'We find no cutoff structure in the energy range of the HXD-PIN.', '0812.3358-1-25-3': 'The spectra within the phase intervals [[MATH]] and [[MATH]], which correspond to the INFC and SUPC, respectively, are also shown in Figure [REF].', '0812.3358-1-26-0': 'The presence of a possible line feature around 6.7 keV was earlier claimed based on RXTE observations (see however the discussion on this issue in , where it is suggedted that this line would be a background artifact).', '0812.3358-1-26-1': 'The [MATH] data, however, do not support this result.', '0812.3358-1-26-2': 'In an attempt to find the possible signature of iron emission lines from LS 5039, we analyzed the phase-averaged spectrum.', '0812.3358-1-26-3': 'The upper limits on iron line structures are determined by fitting a Gaussian at various energies and line widths at which Fe emission is expected.', '0812.3358-1-26-4': 'The power-law continuum model parameters are fixed with the best fit values and a Gaussian component is added to the power-law function.', '0812.3358-1-26-5': 'The central energy of the Gaussian line is swept over 6.0 keV to 7.1 keV with a step of 0.1 keV.', '0812.3358-1-26-6': 'Line widths are changed from 0.01 keV to 0.09 keV with a step of 0.01 keV, together with lines with larger widths of 0.15 and 0.20 keV.', '0812.3358-1-26-7': 'An equivalent width is determined at each grid point.', '0812.3358-1-26-8': 'The resulting upper-limit on the equivalent width is 40 eV.', '0812.3358-1-27-0': '# Discussion', '0812.3358-1-28-0': 'The X-ray emission observed with Suzaku is characterized by (1) a hard power law with [MATH] extending from soft X-rays to [MATH] keV, (2) clear orbital modulation in flux and photon index, (3) a moderate X-ray luminosity of [MATH], (4) a small and constant absorbing column density, and (5) lack of detectable emission lines.', '0812.3358-1-29-0': 'Although variable X-ray emission has been found from more than two hundred galactic binary systems, the Suzaku data can be hardly explained within the "standard" scenario when X-rays are produced by a hot thermal (comptonized) accretion plasma around the compact object.', '0812.3358-1-29-1': 'Indeed, the hard [MATH] type energy spectrum of the X-ray continuum extending from soft X-rays up to 70 keV, as well as the lack of X-ray line features in the spectrum, suggests a non-thermal origin of X-rays.', '0812.3358-1-29-2': 'This conclusion is supported by the general similarities between the properties of the observed X-rays and TeV gamma-rays.', '0812.3358-1-29-3': 'Namely, both radiation components require a rather hard energy distribution of parent electrons with a power-law index of [MATH].', '0812.3358-1-29-4': 'This directly follows from the photon index of the synchrotron radiation [MATH], and agrees quite well with the currently most favored interpretation of TeV gamma-rays, in which they would be produced by IC scattering of the anisotropic photon field of the massive companion star.', '0812.3358-1-29-5': 'In this framework, and accounting for photon-photon absorption, a power-law distribution of electrons with the power-law index [MATH] is responsible for the hard TeV spectra observed with HESS.', '0812.3358-1-30-0': 'The apparent similarities between the X-ray and gamma-ray light curves can be also considered as an argument in favor of a common origin of the X-rays and the TeV gamma-rays.', '0812.3358-1-30-1': 'However we should note that the interpretation of the lightcurves in these two energy bands is not trivial, as it may seem at first glance.', '0812.3358-1-30-2': 'Indeed, both the production (through anisotropic IC) and the attenuation (due to photon-photon absorption) of gamma-rays have a strong orbital phase dependence, whereas X-rays can be produced isotropically and do not suffer strong absorption in the system.', '0812.3358-1-31-0': 'Analysis of the observed fluxes allows some important conclusions on the magnetic field strength.', '0812.3358-1-31-1': 'The energy fluxes of X-rays are close, on average, to the TeV energy fluxes.', '0812.3358-1-31-2': 'Assuming that the TeV gamma-ray production region is located at a distance from the companion star of [MATH] cm (i.e. the binary system size), and taking into account that gamma-rays are produced in deep Klein-Nishina (KN) regime with significantly suppressed cross-section, for the well known luminosity of the optical star [MATH], one can estimate quite robustly the strength of the emitter magnetic field.', '0812.3358-1-31-3': 'The numerical calculations show that the field should be around a few Gauss (see e.g. Fig. 5).', '0812.3358-1-31-4': 'For such a magnetic field strength, the energy intervals of electrons responsible for the two emission components largely overlap, as shown in Figure [REF].', '0812.3358-1-31-5': 'Therefore, most likely, we deal with the same population of parent electrons, which should be located at large distances from the compact object, in the system periphery, to prevent the severe absorption of the TeV radiation and the subsequent intense emission from the pair-created secondaries .', '0812.3358-1-32-0': 'Figure [REF] shows the synchrotron and IC cooling times of electrons, as a function of the electron energy calculated for the stellar photon density at [MATH] and for a magnetic field [MATH] G.', '0812.3358-1-32-1': 'It is seen that synchrotron losses dominate over IC ones at [MATH] TeV.', '0812.3358-1-32-2': 'Note that the TeV gamma-ray production takes place in the deep KN regime.', '0812.3358-1-32-3': 'This implies that the cooling time, [MATH], of electrons generating GeV gamma-rays via IC scattering (Thomson regime) is shorter than the cooling time of TeV electrons responsible for producing very high-energy gamma-rays (KN).', '0812.3358-1-32-4': 'The same applies for synchrotron cooling time of multi-TeV electrons that produce by synchrotron radiation low-energy (MeV) gamma-rays.', '0812.3358-1-32-5': 'Therefore one should expect significantly higher MeV (synchrotron) and GeV (IC) fluxes than at X- and TeV gamma-rays, provided that the acceleration spectrum of electrons extends from low energies to very high energies.', '0812.3358-1-32-6': 'However, in the case of existence of a low-energy and very-high-energy cutoffs in the acceleration spectrum, the gamma-ray fluxes [MATH] MeV and at GeV energies would be significantly suppressed.', '0812.3358-1-33-0': 'To understand better the energy intervals of electrons responsible for X-ray and gamma-ray production in different energy bands, in Figure [REF] we show the energy zones of electrons relevant to the Suzaku, Fermi, and HESS radiation domains.', '0812.3358-1-33-1': 'Note that the reconstruction of the average energy of electrons responsible for the IC gamma-rays depends only on the well known temperature of the companion star [MATH] K.', '0812.3358-1-33-2': 'The light green zone in Figure [REF] marked as "Suzaku synchrotron", corresponds to electrons responsible for the synchrotron photons produced in the energy interval [MATH].', '0812.3358-1-33-3': 'For a reasonable range of magnetic field values, the energy interval of electrons relevant for Suzaku data overlap on one hand with the HESS energy interval, and can overlap with the Fermi one.', '0812.3358-1-33-4': 'This should allow us, in the case of detection of MeV/GeV gamma-rays by Fermi, to reduce robustly the parameter space, in particular, to better localize the X- and gamma-ray production regions from electromagnetic cascade constraints, and derive the broadband energy spectrum of electrons and the strength of the magnetic field as functions of the orbital phase.', '0812.3358-1-34-0': 'Formally, when X-rays and TeV gamma-rays are produced by the same population of very high energy electrons, one should expect a general correlation between the light curves obtained by Suzaku and HESS.', '0812.3358-1-34-1': 'In this regard, the similarity between the Suzaku and HESS light curves seems to be natural.', '0812.3358-1-34-2': 'However, such an interpretation is not straightforward in the sense that two major mechanisms that might cause modulation of the TeV gamma-ray signal are related to interactions of electrons and gamma-rays with the photons of the companion star, i.e. anisotropic IC scattering and photon-photon pair production , and thus they cannot contribute to the X-ray modulation.', '0812.3358-1-34-3': 'This modulation requires periodic changes of the strength of the ambient magnetic field or number of relativistic electrons.', '0812.3358-1-34-4': 'Note, however, that the change of magnetic field would not have a strong impact as long as the radiation proceeds in the saturation regime and synchrotron losses dominate in the relevant energy interval.', '0812.3358-1-34-5': 'One can expect otherwise modulation of the synchrotron X-ray flux if the energy losses of electrons are dominated by IC scattering, although in such a case we should observe significantly lower X-ray fluxes.', '0812.3358-1-35-0': 'Finally we note that the observed X-ray fluxes can be hardly explained by synchrotron emission produced by the secondary (pair-produced) electron and positrons.', '0812.3358-1-35-1': 'Since the pair production cross-section has strong energy dependence with a distinct maximum, for the target photons of typical energy of [MATH] eV, the major fraction of the absorbed energy will be released in the form of [MATH] GeV electrons.', '0812.3358-1-35-2': 'Thus secondary pair synchrotron emission must show a spectral break in the Suzaku energy band unless one assumes unreasonably high magnetic fields, [MATH] kG, in the surroundings of the gamma-ray emitter .', '0812.3358-1-36-0': 'A more natural reason for the modulation of the synchrotron fluxes would come from dominant adiabatic losses.', '0812.3358-1-36-1': 'The adiabatic cooling of electrons in binary systems can be realized through complex (magneto)hydrodynamical processes, e.g. due to interactions between a black hole jet or a pulsar wind with the dense stellar wind of a massive companion star (see e.g. , ).', '0812.3358-1-36-2': 'The orbital motion could naturally render the modulation of adiabatic cooling of electrons along the orbit (see e.g. ).', '0812.3358-1-36-3': 'Note that because of the relatively small variation of the X-ray flux over the orbit, a factor of two, the requirements to the model are quite modest.', '0812.3358-1-36-4': 'In any case, the adiabatic losses should dominate over the radiative, synchrotron plus IC, losses.', '0812.3358-1-36-5': 'We note that dominant adiabatic losses have been invoked by [CITATION] to explain the variations of the X-ray and TeV gamma-ray fluxes from the binary pulsar PSR B1259[MATH]63.', '0812.3358-1-37-0': 'The detected power-law spectrum of X-rays with photon index around [MATH] implies that the established energy spectrum of electrons is also a power-law with index [MATH].', '0812.3358-1-37-1': 'This agrees perfectly with the hypothesis of dominance of adiabatic losses, because the adiabatic losses do not change the initial spectrum of electrons.', '0812.3358-1-37-2': 'Thus the required power-law index [MATH] implies a reasonable acceleration spectrum [MATH].', '0812.3358-1-37-3': 'Otherwise, in an environment dominated by synchrotron losses, the acceleration spectrum should be very hard, with a power-law index [MATH], or should have an unreasonably large low-energy cutoff at [MATH] TeV to explain the observed X-ray spectra.', '0812.3358-1-38-0': 'Obviously, adiabatic losses modulate the IC gamma-ray flux in a similar manner.', '0812.3358-1-38-1': 'However, unlike X-rays, which escape the system without significant absorption, the TeV gamma-rays suffer significant distortion due to photon-photon absorption (see e.g. ) and anisotropic IC scattering (with its strong hardening of the gamma-ray spectrum ).', '0812.3358-1-38-2': 'All this leads to additional orbital modulation of the gamma-ray signal, and it is likely that these two additional processes are responsible for the strong change of gamma-ray flux, by a factor of several more pronounced than in X-rays (see Fig. [REF]).', '0812.3358-1-38-3': 'The Suzaku data presented in this paper implies the key additional assumption, namely one should allow the accelerated electrons to loose their energy adiabatically, before they cool radiatively.', '0812.3358-1-39-0': 'In order to demonstrate that the suggested scenario can explain satisfactorily the combined Suzaku X-ray and the HESS gamma-ray data, we performed calculations of the broad-band spectral energy distributions (SEDs) of the synchrotron and IC emission, assuming a simple model in which the same population of electrons is responsible for both X-rays and TeV gamma-rays.', '0812.3358-1-39-1': 'We also assumed that the emission region has homogeneous physical conditions.', '0812.3358-1-39-2': 'This is a reasonable assumption given that we deal with very short cooling timescales ([MATH] s), thus electrons cannot travel significant distances while emitting.', '0812.3358-1-39-3': 'We used the fluxes detected with Suzaku to reconstruct the rates and orbital phase dependence of adiabatic losses.', '0812.3358-1-40-0': 'In the regime dominated by adiabatic energy losses, the synchrotron X-ray flux is proportional to [MATH].', '0812.3358-1-40-1': 'The X-ray modulation seen by Suzaku is then described by the modulation of the adiabatic loss rate.', '0812.3358-1-40-2': 'In Fig. [REF], we show [MATH], i.e. the orbital modulation of the adiabatic cooling time as inferred from the X-ray data.', '0812.3358-1-40-3': 'As seen from the plot, the required adiabatic cooling times are [MATH] s. Any consistent calculation of the adiabatic cooling requires the solution of the corresponding hydrodynamical problem, and one needs to know detailed information concerning the source nature.', '0812.3358-1-40-4': 'At the present stage, we suggest to study the obtained adiabatic cooling on a simple example of a relativistically expanding source.', '0812.3358-1-40-5': 'In such a case, the adiabatic loss rate can be written as: [MATH] with [MATH] sec, where [MATH] is the characteristic size of the source.', '0812.3358-1-40-6': 'Thus the required variation of the adiabatic cooling is reduced to the modulation of the size of the radiation region ([MATH]).', '0812.3358-1-40-7': 'The size in its turn depends on the external pressure exerted by, e.g., the stellar wind from the massive star.', '0812.3358-1-40-8': 'The expected weaker external pressure around apastron implicitly assumed in our model would be broadly consistent with the radial dependence of the wind pressure.', '0812.3358-1-41-0': 'In Fig. [REF] we show the broad-band spectral distribution of non-thermal radiation of electrons consisting of synchrotron and IC components averaged over the INFC ([MATH]-[MATH]) and SUPC ([MATH], [MATH]) phase intervals.', '0812.3358-1-41-1': 'The corresponding gamma-ray data have been reported by HESS [CITATION].', '0812.3358-1-41-2': 'It should be noted that in LS 5039 both the absolute flux and the energy spectrum of TeV gamma-rays vary rapidly with phase, therefore in order to compare the theoretical predictions with observations we should use the fluxes obtained simultaneously in X-ray and gamma-ray bands, ideally speaking during the time intervals [MATH] s corresponding to the characteristic cooling timescales of electrons.', '0812.3358-1-41-3': 'Because of lack of such observations one may use the X-ray and gamma-ray data integrated in the same phase intervals.', '0812.3358-1-41-4': 'While this compromise does not allow us to perform quantitative studies, it can be invoked nevertheless for qualitative comparison of the model calculations with observational data.', '0812.3358-1-41-5': 'For this reason the Suzaku X-ray data shown in Fig. [REF] are averaged over the same gamma-ray phase intervals.', '0812.3358-1-41-6': 'Although, the theoretical calculations do not perfectly match the gamma-ray fluxes, the agreement can be treated reasonable, given the reason mentioned above as well as taking into account that the gamma-ray and X-ray data have been taken during completely different epochs.', '0812.3358-1-41-7': 'One can improve the fits by introducing slight phase-dependent changes in the spectra of accelerated electrons, but the detailed modeling is beyond the scopes of this paper and, more importantly, it cannot be presently supported by the quality of available data.', '0812.3358-1-41-8': 'We should also note that the calculations of low energy gamma-rays (in the Fermi domain), shown in the bottom panel of Fig. [REF], are performed assuming that the injection spectrum of electrons continues down to [MATH] GeV.', '0812.3358-1-41-9': 'If this is not the case, for example the electron acceleration spectrum has a lower energy cutoff at energies higher than [MATH] GeV, the gamma-ray fluxes in the Fermi energy domain may be significantly suppressed.', '0812.3358-1-41-10': 'On the other hand the detection of gamma-rays by Fermi would allow us to recover the spectrum of electrons in a very broad energy interval, and thus distinguish between different acceleration models.', '0812.3358-1-41-11': 'Another important feature in this scenario is the hard synchrotron spectrum extending up to a few MeV in energy, is required by the robust detection of [MATH] TeV gamma-rays from the system.', '0812.3358-1-41-12': 'Future detection of the emission in this energy band may bring important information on the presence of highest energy particles in the system.', '0812.3358-1-42-0': 'The reproduction, at least qualitatively, of the observed spectral and temporal features of the nonthermal radiation with the simple toy model described above supports the production of X-ray and TeV gamma-rays by the same population of parent particles and allows us to derive several principal conclusions and predictions for future observations.', '0812.3358-1-42-1': 'In particular, the electron energy distribution should be a power-law with an almost constant index of [MATH] to explain the X-ray spectra.', '0812.3358-1-42-2': 'Note that in a hot isotropic photon gas when the Comton scattering takes place in the deep KN regime such an electron distribution results in a quite steep TeV spectrum with photon index [MATH].', '0812.3358-1-42-3': 'This does not agree with HESS observations.', '0812.3358-1-42-4': 'However, the anisotropic IC provides a remarkable hardening of the gamma-ray spectrum , in particular, a [MATH] would be expected for the INFC, and it has been indeed observed by HESS .', '0812.3358-1-42-5': 'We also notice that, to give a proper account of the VHE spectrum at SUPC, we have to assume that the emitter is located the at the distance [MATH] cm from the compact object in the direction perpendicular to the orbital plane.', '0812.3358-1-42-6': 'In the standard pulsar scenario, the production region cannot be located far away from the compact object, and even invoking electromagnetic cascading (see e.g. ), one cannot reproduce the reported fluxes around SUPC .', '0812.3358-1-42-7': 'Regarding the magnetic field, we have found that the field strength cannot deviate much from a few G.', '0812.3358-1-42-8': 'We can also derive an emitter size constraint, imposing a maximum expansion speed of c (light speed), and the Hillas criterion , in which the minimum size of the source capable of accelerating particles to the given energy [MATH] is [MATH] (where [MATH] cm), the Larmor radius.', '0812.3358-1-42-9': 'This estimate gives [MATH]-[MATH] cm, which agrees quite well with the estimate based on the required timescale of adiabatic cooling.', '0812.3358-1-42-10': 'Note that the requirement of fast adiabatic losses imposes a strong constraint on the acceleration rate of electrons.', '0812.3358-1-42-11': 'Indeed, the acceleration timescale can be expressed as: [MATH], where [MATH] parametrizes the acceleration efficiency.', '0812.3358-1-42-12': 'In the extreme accelerators with the maximum possible rate allowed by classical electrodynamics [MATH].', '0812.3358-1-42-13': 'The HESS spectrum provides evidence of electron acceleration well above 10 TeV.', '0812.3358-1-42-14': 'Therefore, [MATH] s is required at [MATH] TeV, which translates into [MATH] for [MATH] G. Thus we arrive at the conclusion that an extremely efficient acceleration with [MATH] should operate in a compact region of [MATH].', '0812.3358-1-43-0': 'Finally, we would like to emphasize that in the scenario described here different energy intervals of radiation are characterized by essentially different light curves.', '0812.3358-1-43-1': 'While the synchrotron X-ray modulation is caused by adiabatic losses, the light curve of gamma-rays significantly depends, in addition, on the effects related to the interactions with the optical photons of the companion star.', '0812.3358-1-43-2': 'Two of these effects, photon-photon pair production and anisotropic Compton scattering are equally important for the formation of the light curve of TeV gamma-rays.', '0812.3358-1-43-3': 'On the other hand, only the effect of anisotropic Compton scattering has an impact on the formation of the light curve of GeV gamma-rays.', '0812.3358-1-43-4': 'The difference of light curves in the X-ray and GeV and TeV gamma-ray intervals is shown in Figure [REF].', '0812.3358-1-44-0': '# Summary', '0812.3358-1-45-0': 'The Suzaku X-ray satellite has observed for the first time LS 5039 with imaging capabilities along one orbit and a half, and the results show a similar (although not fully identical) X-ray and TeV behavior.', '0812.3358-1-45-1': 'The close correlation of the X-ray and TeV gamma-ray light curves can be interpreted as evidence of production of these two radiation components by the same electron population via synchrotron radiation and IC scattering, respectively.', '0812.3358-1-45-2': 'In this regard, the origin of the X-ray modulation in this source is completely different from that of the thermal X-rays observed from tens of compact galactic X-ray binaries and interpreted in terms of radiation of the accretion plasma around a black hole or a neutron star.', '0812.3358-1-45-3': 'Whereas there are at least two reasons for the formation of a periodic TeV gamma-ray light curve, both related to the interaction with photons from the normal companion star (photon-photon absorption of VHE gamma-rays and IC scattering in an anisotropic photon field), the modulated X-ray signal requires an additional effect.', '0812.3358-1-45-4': 'A simple and natural reason for the modulaton in X-rays seems to be adiabatic losses which should dominate over the radiative, synchrotron and IC losses of electrons.', '0812.3358-1-45-5': 'We demonstrate that this assumption allows us to explain, at least qualitatively, the combined Suzaku and HESS data concerning both the spectral characteristics and temporal variations.', '0812.3358-1-45-6': 'In particular, the introduction of adiabatic losses not only provides a quite natural explanation for the rather stable photon index of the X-ray spectrum [MATH], but also allows one to properly reproduce the TeV gamma-ray spectra.', '0812.3358-1-46-0': 'The gamma-ray data require a location of the production region at the periphery of the binary system at [MATH] cm.', '0812.3358-1-46-1': 'This constraint allows a quite robust estimate of the magnetic field of about few Gauss derived directly from the X/TeV flux ratio, and an adiabatic loss time of a few seconds to provide the dominance of adiabatic losses.', '0812.3358-1-46-2': 'In the case of a relativistically expanding source, the typical size of the production region should not exceed [MATH] cm.', '0812.3358-1-46-3': 'The adiabatic cooling cannot be shorter than several seconds, and correspondingly, the size of the production region cannot be much smaller than [MATH] cm, since otherwise the electrons could not be accelerated up to energies beyond 10 TeV, even assuming an extreme acceleration rate close to the fundamental limit determined by quantum electrodynamics.', '0812.3358-1-46-4': 'In any case, it is clear that we are dealing with an extreme accelerator.', '0812.3358-1-46-5': 'There is little doubt that future simultaneous observations of LS 5039 with the Suzaku, Fermi, and HESS telescopes will provide key information for understanding the nature of this mysterious non-thermal source.', '0812.3358-1-47-0': 'T. Kishishita and T. Tanaka are supported by research fellowships of the Japan Society for the Promotion of Science for Young Scientists.'}

{'0812.3358-2-0-0': 'We report on the results from Suzaku broadband X-ray observations of the galactic binary source LS 5039.', '0812.3358-2-0-1': 'The Suzaku data, which have continuous coverage of more than one orbital period, show strong modulation of the X-ray emission at the orbital period of this TeV gamma-ray emitting system.', '0812.3358-2-0-2': 'The X-ray emission shows a minimum at orbital phase [MATH], close to the so-called superior conjunction of the compact object, and a maximum at phase [MATH], very close to the inferior conjunction of the compact object.', '0812.3358-2-0-3': 'The X-ray spectral data up to 70 keV are described by a hard power-law with a phase-dependent photon index which varies within [MATH]- 1.61.', '0812.3358-2-0-4': 'The amplitude of the flux variation is a factor of 2.5, but is significantly less than that of the factor [MATH]8 variation in the TeV flux.', '0812.3358-2-0-5': 'Otherwise the two light curves are similar, but not identical.', '0812.3358-2-0-6': 'Although periodic X-ray emission has been found from many galactic binary systems, the Suzaku result implies a phenomenon different from the "standard" origin of X-rays related to the emission of the hot accretion plasma formed around the compact companion object.', '0812.3358-2-0-7': 'The X-ray radiation of LS 5039 is likely to be linked to very-high-energy electrons which are also responsible for the TeV gamma-ray emission.', '0812.3358-2-0-8': 'While the gamma-rays are the result of inverse Compton scattering by electrons on optical stellar photons, X-rays are produced via synchrotron radiation.', '0812.3358-2-0-9': 'Yet, while the modulation of the TeV gamma-ray signal can be naturally explained by the photon-photon pair production and anisotropic inverse Compton scattering, the observed modulation of synchrotron X-rays requires an additional process, the most natural one being adiabatic expansion in the radiation production region.', '0812.3358-2-1-0': '# Introduction', '0812.3358-2-2-0': 'LS 5039 is a high-mass X-ray binary with extended radio emission .', '0812.3358-2-2-1': 'This system is formed by a main sequence O type star and a compact object of disputed nature that has been claimed to be both a black hole and a neutron star/pulsar .', '0812.3358-2-2-2': 'The compact object is moving around the companion star in a moderately elliptic orbit (eccentricity [MATH]) with an orbital period of [MATH] days .', '0812.3358-2-3-0': 'As summarized in [CITATION], LS 5039 has been observed several times in the X-ray energy band for limited phases in the orbital period.', '0812.3358-2-3-1': 'Flux variations on time scale of days and sometimes on much shorter timescales have been reported.', '0812.3358-2-3-2': 'The spectrum was always well represented by a power-law model with a photon index ranging [MATH]-1.6 up to [MATH] keV, with fluxes changing moderately around [MATH].', '0812.3358-2-3-3': 'Softer spectra and larger fluxes had been also inferred from RXTE observations, although background contamination was probably behind these differences (see ).', '0812.3358-2-3-4': 'Also, Chandra data taken in 2002 and 2005 showed spectra significantly harder than 1.5 , but such a hard spectrum was probably an artifact produced by photon pile-up.', '0812.3358-2-3-5': 'Recently, [CITATION] reported the results of INTEGRAL observations in hard X-rays.', '0812.3358-2-3-6': 'The source was detected at energies between 25 and 60 keV.', '0812.3358-2-3-7': 'The flux was estimated to be [MATH] (90 % confidence level) around the inferior conjunction (INFC) of the compact object, and a flux upper limit of [MATH] (90 % confidence level) was derived at the superior conjunction of the compact object (SUPC).', '0812.3358-2-4-0': 'LS 5039 has also been detected in very-high-energy (VHE: [MATH] TeV) gamma-rays , exhibiting a periodic signal modulated with the orbital period .', '0812.3358-2-4-1': 'There are two other binary systems with robust detections in the VHE range: PSR B1259[MATH]63 and LS I [MATH]303 .', '0812.3358-2-4-2': 'Evidence for TeV emission has been found also in Cygnus X-1 .', '0812.3358-2-4-3': 'PSR B1259[MATH]63 is a clear case of a high-mass binary system containing a non-accreting pulsar , whereas Cygnus X-1 is a well known accreting black hole system .', '0812.3358-2-4-4': 'The nature of the compact object in LS 5039 is not yet established, and the origin of the VHE emitting electrons is unclear.', '0812.3358-2-4-5': 'They may be related to a pulsar wind or to a black hole with a (sub)relativistic jet.', '0812.3358-2-4-6': 'In the standard pulsar-wind scenario, the severe photon-photon absorption makes the explanation of the detection of the VHE radiation problematic, at least at the position corresponding to the orbital phase [MATH] (see Figs. 16 and 4 from Sierpowska-Bartosik Torres 2008 and Dubus et al. 2008 and compare with Fig. 5 in Aharonian et al. 2006), in which the emitter is expected to be located between the compact object and the star.', '0812.3358-2-4-7': 'However, particles may be accelerated in a relativistic outflow formed at the interaction of the pulsar and the stellar winds and radiate far from the compact object, making the pulsar wind scenario a viable option.', '0812.3358-2-4-8': 'In the microquasar scenario, the lack of accretion features in the X-ray spectrum may be a problem unless the bulk of the accretion power is released in the form of kinetic energy of the outflow, rather than thermal emission during accretion, as in the case of SS 433 .', '0812.3358-2-4-9': 'At this stage, we cannot give a preference to any of these scenarios, but new data, in particular those obtained with the Suzaku satellite, allow us to make an important step towards the understanding of the nature of the non-thermal processes of acceleration and radiation in this mysterious object.', '0812.3358-2-5-0': '# Observation', '0812.3358-2-6-0': 'The temporal and spectral characteristics of the X-ray emission from LS 5039 along the orbit should provide important clues for understanding the acceleration/radiation processes in this source.', '0812.3358-2-6-1': 'The fact that all previous X-ray observations of this object have incomplete coverage of the orbital period, or suffered from background contamination, is therefore rather unsatisfactory.', '0812.3358-2-6-2': 'This motivated our long, [MATH] ks observation with the Suzaku X-ray observatory, which gives us unprecedented coverage of more than one orbital period, continuously from 2007 September 9 to 15 (see Table [REF]).', '0812.3358-2-6-3': 'Suzaku has four sets of X-ray telescopes each with a focal-plane X-ray CCD camera that are sensitive in the energy range of 0.3-12 keV.', '0812.3358-2-6-4': 'Three of the XIS detectors (XIS0, 2 and 3) have front-illuminated (FI) CCDs, whereas XIS1 utilizes a back-illuminated (BI) CCD.', '0812.3358-2-6-5': 'The merit of the BI CCD is its improved sensitivity in the soft X-ray energy band below 1 keV.', '0812.3358-2-6-6': 'Suzaku contains also a non-imaging collimated Hard X-ray Detector , which covers the 10-600 keV energy band with Si PIN photodiodes (10-70 keV) and GSO scintillation detectors (40-600 keV).', '0812.3358-2-6-7': 'Suzaku has two default pointing positions, the XIS nominal position and the HXD nominal position.', '0812.3358-2-6-8': 'In this observation, we used the HXD nominal position, in which the effective area of HXD is maximized, whereas that of the XIS is reduced to on average [MATH] 88%.', '0812.3358-2-6-9': 'Results from XIS2 are not reported here since it has not been in operation since an anomaly in November 2006.', '0812.3358-2-6-10': 'In addition, we do not describe in detail the analysis of HXD-GSO data, since the HXD-GSO detected no significant signal from the source.', '0812.3358-2-7-0': '# Data Reduction', '0812.3358-2-8-0': 'We used data sets processed using the software of the Suzaku data processing pipeline (version 2.1.6.16).', '0812.3358-2-8-1': 'Reduction and analysis of the data were performed following the standard procedure using the HEADAS v6.4 software package, and spectral fitting was performed with XSPEC v.11.3.2.', '0812.3358-2-9-0': 'For the XIS data analysis, we accumulated cleaned events over good time intervals that were selected by removing spacecraft passages through the South Atlantic Anomaly (SAA).', '0812.3358-2-9-1': "Further, we screened the data with the following criteria - (1) cut-off rigidity is larger than 6 GV and (2) elevation angle from the Earth's rim is larger than 5[MATH].", '0812.3358-2-9-2': 'The source photons were accumulated from a circular region with a radius of [MATH].', '0812.3358-2-9-3': 'The background region was chosen in the same field of view with the same radius and an offset of [MATH] from the source region.', '0812.3358-2-9-4': 'We have co-added the data from the two FI-CCDs (XIS0 and XIS3) to increase statistics.', '0812.3358-2-9-5': 'The response (RMF) files and the auxiliary response (ARF) files used in this paper were produced using xisrmfgen and xissimarfgen, respectively.', '0812.3358-2-10-0': 'For the HXD data,"uncleaned event files" were screened with the standard event screening criteria: the cut-off rigidity is larger than 6 GV, the elapsed time after the passage of the SAA is more than 500 s and the time to the next SAA passage is more than 180 s, high voltages from all eight HV units are within the normal range and the elevation angle from the Earth\'s rim is more than [MATH].', '0812.3358-2-10-1': 'We also discarded telemetry-saturated time intervals.', '0812.3358-2-10-2': 'In the spectral analysis in 3, we used the response file for a point-like source at the HXD-nominal position (aehxdpinhxnome420070914.rsp), which is released as a part of CALDB.', '0812.3358-2-11-0': 'The HXD-PIN spectrum is dominated by the time-variable instrumental background ( non-X-ray background (NXB) ) induced by cosmic rays and trapped charged particles in the satellite orbit.', '0812.3358-2-11-1': 'To estimate the instrumental background component, we used the the time-dependent NXB event files released by the HXD instrument team, whose reproducibility is reported by [CITATION].', '0812.3358-2-11-2': 'In order to estimate the systematic uncertainty of the NXB model during our observation, we compare the NXB model spectrum during Earth occultation with the observed spectrum of the same time interval.', '0812.3358-2-11-3': 'The event selection criteria for this study are the same as those of the cleaned event except for the criterion on Earth elevation angle, which was chosen to to be less than [MATH].', '0812.3358-2-11-4': 'The estimated uncertainty obtained is [MATH]%, which is consistent with the values reported by [CITATION].', '0812.3358-2-12-0': 'Another component of the HXD-PIN background is the cosmic X-ray background (CXB).', '0812.3358-2-12-1': 'In our analysis, we assumed the CXB spectrum reported by : [EQUATION] where [MATH] and [MATH].', '0812.3358-2-12-2': 'The CXB spectrum observed with HXD-PIN was simulated by using a PIN response file for isotropic diffuse emission (aehxdpinflate420070914.rsp) and added to the NXB spectrum.', '0812.3358-2-12-3': 'Based on this approach, the contribution from the CXB flux is [MATH]% of the NXB.', '0812.3358-2-13-0': 'Since LS 5039 is located close to the Galactic plane, the contribution from the Galactic ridge X-ray emission (GRXE) must be examined, especially for HXD-PIN spectra.', '0812.3358-2-13-1': 'In order to model the shape of the GRXE, data from Suzaku observations of the Galactic ridge region (ObsID: 500009010 and 500009020) were analyzed.', '0812.3358-2-13-2': 'The Suzaku spectrum from 3 keV to 50 keV can be well fitted with the Raymond-Smith plasma with a temperature of [MATH] keV and a power-law function with [MATH].', '0812.3358-2-13-3': 'Although we also tried a power law with exponential cutoff, following the results from the INTEGRAL IBIS , it turned out that the assumption on the spectral shape of the GRXE has negligible effect on the spectral parameters of LS 5039.', '0812.3358-2-13-4': 'The normalization of the GRXE spectrum component in the HXD-PIN spectrum of LS 5039 is determined from the XIS spectrum of the LS 5039 observation by excluding an encircled region with a radius of [MATH] centered on the LS 5039 location.', '0812.3358-2-13-5': 'The flux of the GRXE is estimated to be [MATH] of the contribution from the CXB.', '0812.3358-2-13-6': 'In Figure [REF], we show the time averaged HXD-PIN spectrum plotted together with models for the NXB, CXB and GRXE.', '0812.3358-2-14-0': '# Analysis and Results', '0812.3358-2-15-0': '## Temporal analysis', '0812.3358-2-16-0': 'The light curve obtained from the XIS detector is shown in the top panel of Figure [REF].', '0812.3358-2-16-1': 'The continuous coverage in X-rays, longer than the orbital period of the LS 5039 system, reveals a smooth variation of a factor 2 in the 1-10 keV count rate.', '0812.3358-2-16-2': 'The light curve is drawn over two orbital periods.', '0812.3358-2-16-3': 'The orbital phase is calculated with the period of 3.90603 days, and [MATH] with reference epoch [MATH]) taken from [CITATION].', '0812.3358-2-16-4': 'The light curve from phase [MATH] to 1.5, which was obtained in the last part of the observation, smoothly overlaps with the one obtained at the beginning of the observation ([MATH]-0.5).', '0812.3358-2-17-0': 'In the middle panel of Figure [REF], we present the light curve obtained with the HXD-PIN for the energy range 15-40 keV.', '0812.3358-2-17-1': 'Although the statistical errors are larger, the modulation behavior is similar to that of the XIS.', '0812.3358-2-17-2': 'The amplitude of the modulation is roughly the same between the XIS and HXD-PIN, indicating small changes of spectral shape depending on orbital phase.', '0812.3358-2-17-3': 'The spectral parameters obtained for each orbital phase are reported in the following section.', '0812.3358-2-18-0': 'The light curves obtained with Suzaku show that the X-ray flux minimum appears around phase 0.0-0.3 and it reaches maximum around phase 0.5-0.8.', '0812.3358-2-18-1': 'In order to quantify the amplitude of the flux variations, we fitted the XIS light curve with a simple sinusoidal function.', '0812.3358-2-18-2': 'Due to structures in the light curve, the fit converges with large chi-square ([MATH] = 4.92 (121)).', '0812.3358-2-18-3': 'However, the general trend is well represented by a sinusoidal function: [EQUATION] where [MATH] is the count rate as a function of phase [MATH].', '0812.3358-2-18-4': 'The ratios between the minimum and maximum count rates are 2.21 [MATH] counts s[MATH] for XIS and 2.02 [MATH] counts s[MATH] for HXD-PIN.', '0812.3358-2-18-5': 'Structures of the X-ray and hard X-ray light curves are similar to that discovered in the phase diagram of integral fluxes at energies [MATH] TeV obtained on a run-by-run basis from HESS data (2004 to 2005) .', '0812.3358-2-18-6': 'The temporal X-ray behavior was already suggested by RXTE data, as well as by a compilation of all the previous X-ray data obtained with imaging instruments .', '0812.3358-2-19-0': 'In addition to the continuously changing component with respect to the orbital phase, short timescale structures are found around [MATH] and [MATH].', '0812.3358-2-19-1': 'The unabsorbed flux changes about a 30% in [MATH].', '0812.3358-2-19-2': 'A significant dip can be seen around [MATH] in the top panel of Figure [REF].', '0812.3358-2-19-3': 'In comparison with the data at around [MATH] obtained in the first half of the observation, the flux decreased [MATH]50% only in this phase.', '0812.3358-2-19-4': 'The time duration of this dip corresponds to [MATH].', '0812.3358-2-19-5': 'These structures may reflect features of the (possibly changing) environment of the X-ray emitting region, it is therefore of importance to test with further observations if these are persistent features.', '0812.3358-2-20-0': '## Spectral Analysis', '0812.3358-2-21-0': 'Firstly we study time-resolved (phase-resolved) X-ray spectra.', '0812.3358-2-21-1': 'The data are divided into data segments with respect to the assigned phase, and model fitting is performed for XIS spectra for each segment with [MATH].', '0812.3358-2-22-0': 'A single power-law function with photoelectric absorption, provides a good fit for all the segments.', '0812.3358-2-22-1': 'In order to study the possible changes of the amount of photoelectric absorption, we here fit the data with [MATH] free.', '0812.3358-2-22-2': 'The best-fit parameters are presented in Table [REF].', '0812.3358-2-22-3': 'The derived values of the photon index and absorption column density are consistent with previous observations .', '0812.3358-2-22-4': 'When we fix the [MATH] to the value obtained from the time averaged spectrum, resultant photon indices stay same within statistical error.', '0812.3358-2-23-0': 'The photon index ([MATH]) values are plotted as a function of orbital phase in the top panel of Figure [REF].', '0812.3358-2-23-1': 'The spectral shape varied such that the spectrum is steep around SUPC ([MATH]) and becomes hard ([MATH]) around apastron.', '0812.3358-2-23-2': 'The modulation behavior of [MATH] is somewhat different from that observed using HESS in the VHE range.', '0812.3358-2-23-3': 'The amplitude of the variation is [MATH], which is much smaller than the change of [MATH]0.6 in the VHE region .', '0812.3358-2-23-4': 'The 1-10 keV flux changes from [MATH]) to [MATH].', '0812.3358-2-23-5': '([MATH]).', '0812.3358-2-24-0': 'In all the data segments, the source is significantly detected with the HXD-PIN, indicating that hard X-ray emission extends at least up to 70 keV.', '0812.3358-2-24-1': 'Note also that although the XIS and HXD-PIN spectra do not overlap, they seem to be smoothly connected in the gap between 10 and 15 keV.', '0812.3358-2-25-0': 'To study the shape of the spectrum above 10 keV, the XIS spectra and the PIN spectrum in the range 15-70 keV after subtraction of background (NXB [MATH] CXB [MATH] GRXE) are jointly fitted (Figure [REF] Top).', '0812.3358-2-25-1': 'The time-averaged spectra are well represented by an absorbed power-law model with [MATH]= 1.51 [MATH]0.02 with reduced [MATH] (235 degrees of freedom).', '0812.3358-2-25-2': 'We find no cutoff structure in the energy range of the HXD-PIN.', '0812.3358-2-25-3': 'The spectra within the phase intervals [[MATH]] and [[MATH]], which correspond to the INFC and SUPC, respectively, are also shown in Figure [REF].', '0812.3358-2-25-4': 'The best-fit parameters are presented in Table [REF].', '0812.3358-2-26-0': 'Althought earlier observations by RXTE suggested the presence of an iron emission line at 6.7 keV (Ribo et al. 1999), later observations by Chandra and XMM could not find evidence of it (e.g. ).', '0812.3358-2-26-1': 'A careful study of new and longer RXTE observations, using slew data to account for background emission, revealed that the earlierly reported 6.7 keV emission line is likely a background feature .', '0812.3358-2-26-2': 'The Suzaku data confirm this result.', '0812.3358-2-26-3': 'In an attempt to find the possible signature of iron emission lines from LS 5039, we analyzed the phase-averaged spectrum.', '0812.3358-2-26-4': 'The upper limits on iron line structures are determined by fitting a Gaussian at various energies and line widths at which Fe emission might be expected.', '0812.3358-2-26-5': 'The power-law continuum model parameters are fixed with the best fit values and a Gaussian component is added to the power-law function.', '0812.3358-2-26-6': 'The central energy of the Gaussian line is swept from 6.0 keV to 7.1 keV in steps of 0.1 keV.', '0812.3358-2-26-7': 'Line widths are changed from 0.01 keV to 0.09 keV in steps of 0.01 keV, together with lines with larger widths of 0.15 and 0.20 keV.', '0812.3358-2-26-8': 'An equivalent width is determined at each grid point.', '0812.3358-2-26-9': 'The resulting upper-limit on the equivalent width is 40 eV with 90% confidence level.', '0812.3358-2-27-0': '# Discussion', '0812.3358-2-28-0': 'The X-ray emission observed with Suzaku is characterized by (1) a hard power law with [MATH] extending from soft X-rays to [MATH] keV, (2) clear orbital modulation in flux and photon index, (3) a moderate X-ray luminosity of [MATH], (4) a small and constant absorbing column density, and (5) a lack of detectable emission lines.', '0812.3358-2-29-0': 'Although variable X-ray emission has been found from more than two hundred galactic binary systems, the Suzaku data hardly can be explained within the "standard accretion" scenario where X-rays are produced by a hot thermal (comptonized) accretion plasma around the compact object.', '0812.3358-2-29-1': 'The lack of X-ray emission lines (at the level of sensitivity of Suzaku) as well as the hard [MATH] type energy spectrum of the X-ray continuum, extending from soft X-rays up to 70 keV, favors a non-thermal origin of the X-rays.', '0812.3358-2-29-2': 'This conclusion is supported by the general similarities between the properties of the observed X-rays and TeV gamma-rays.', '0812.3358-2-29-3': 'Namely, both radiation components require a rather hard energy distribution of parent electrons with a power-law index of [MATH].', '0812.3358-2-29-4': 'This directly follows from the photon index of the synchrotron radiation [MATH], and agrees quite well with the currently most favored interpretation of the TeV gamma-rays, in which they would be produced by IC scattering off the anisotropic photon field of the massive companion star.', '0812.3358-2-30-0': 'Assuming that the TeV gamma-ray production region is located at a distance from the companion star of [MATH] cm (i.e. the binary system size), and taking into account that gamma-rays are produced in the deep Klein-Nishina (KN) regime with significantly suppressed cross-section, for the well known luminosity of the optical star [MATH], one can estimate quite robustly the strength of the magnetic field in the emission region.', '0812.3358-2-30-1': 'The numerical calculations show that the field should be around a few Gauss (see e.g. Fig. 5).', '0812.3358-2-30-2': 'For such a magnetic field strength, the energy intervals of electrons responsible for the two emission components overlap substantially, as shown in Figure [REF].', '0812.3358-2-30-3': 'Therefore, we are most likely dealing with the same population of parent electrons, which should be located at large distances from the compact object, in the system periphery, to prevent the severe absorption of the TeV radiation and the subsequent intense emission from the pair-created secondaries .', '0812.3358-2-31-0': 'It should be noted that the observed X-ray emission is very difficult to explain as synchrotron emission produced by secondary (pair-produced) electron and positrons.', '0812.3358-2-31-1': 'Since the pair production cross-section has strong energy dependence with a distinct maximum, for the target photons of typical energy of [MATH] eV, the major fraction of the absorbed energy will be released in the form of [MATH] GeV electrons.', '0812.3358-2-31-2': 'Thus secondary pair synchrotron emission must show a spectral break in the Suzaku energy band unless one assumes unreasonably high magnetic fields, [MATH] kG, in the surroundings of the gamma-ray emission region .', '0812.3358-2-32-0': 'Figure [REF] shows the synchrotron and IC cooling times of electrons, as a function of electron energy, calculated for the stellar photon density at [MATH] and for a magnetic field [MATH] G.', '0812.3358-2-32-1': 'It can be seen that synchrotron losses dominate over IC losses at [MATH] TeV.', '0812.3358-2-32-2': 'Note that the TeV gamma-ray production takes place in the deep KN regime.', '0812.3358-2-32-3': 'This implies that the cooling time, [MATH], of electrons generating GeV gamma-rays via IC scattering (Thomson regime) is shorter than the cooling time of TeV electrons responsible for producing very high-energy gamma-rays (KN).', '0812.3358-2-32-4': 'The same applies for synchrotron cooling time of multi-TeV electrons that produce low-energy (MeV) gamma-rays by synchrotron radiation.', '0812.3358-2-32-5': 'One should therefore expect significantly higher MeV (synchrotron) and GeV (IC) fluxes than at keV and TeV energies, provided that the acceleration spectrum of electrons extends from low energies to very high energies.', '0812.3358-2-32-6': 'However, in the case of existence of low-energy and very-high-energy cutoffs in the acceleration spectrum, the gamma-ray fluxes [MATH] MeV and at GeV energies would be significantly suppressed.', '0812.3358-2-33-0': 'To better understand the energy ranges of the electrons responsible for X-ray and gamma-ray production, we show in Figure [REF] the energy zones of electrons relevant to the Suzaku, Fermi, and HESS radiation domains.', '0812.3358-2-33-1': 'Note that the reconstruction of the average energy of electrons responsible for the IC gamma-rays depends only on the well known temperature of the companion star [MATH] K.', '0812.3358-2-33-2': 'The light green zone in Figure [REF] marked as "Suzaku synchrotron", corresponds to electrons responsible for the synchrotron photons produced in the energy interval [MATH].', '0812.3358-2-33-3': 'For a reasonable range of magnetic field values, the energy interval of electrons relevant for Suzaku data overlap on one hand with the HESS energy interval, and can overlap with the Fermi one.', '0812.3358-2-33-4': 'This should allow us, in the case of detection of MeV/GeV gamma-rays by Fermi, to considerably reduce the parameter space, in particular, to better localize the X- and gamma-ray production regions from electromagnetic cascade constraints, and derive the broadband energy spectrum of electrons and the strength of the magnetic field, both as a function of the orbital phase.', '0812.3358-2-34-0': 'Formally, when X-rays and TeV gamma-rays are produced by the same population of very-high-energy electrons, one should expect a general correlation between the light curves obtained by Suzaku and HESS.', '0812.3358-2-34-1': 'In this regard, the similarity between the Suzaku and HESS light curves seems to be natural.', '0812.3358-2-34-2': 'However, such an interpretation is not straightforward in the sense that two major mechanisms that might cause modulation of the TeV gamma-ray signal are related to interactions of electrons and gamma-rays with the photons of the companion star, i.e. anisotropic IC scattering and photon-photon pair production , and thus cannot contribute to the X-ray modulation.', '0812.3358-2-34-3': 'The X-ray modulation requires periodic changes of the strength of the ambient magnetic field or the number of relativistic electrons.', '0812.3358-2-34-4': 'Note, however, that the change of magnetic field would not have a strong impact as long as the radiation proceeds in the saturation regime and synchrotron losses dominate in the relevant energy interval.', '0812.3358-2-34-5': 'One would also expect modulation of the synchrotron X-ray flux if the energy losses of electrons are dominated by IC scattering, although in such a case we should observe significantly lower X-ray fluxes.', '0812.3358-2-35-0': 'A more natural reason for the modulation of the synchrotron fluxes would come from dominantly adiabatic losses.', '0812.3358-2-35-1': 'The adiabatic cooling of electrons in binary systems can be realized through complex (magneto)hydrodynamical processes, e.g. due to interactions between a black hole jet or a pulsar wind with the dense stellar wind of a massive companion star (see e.g. , ).', '0812.3358-2-35-2': 'The orbital motion could naturally produce the modulation of adiabatic cooling of electrons around the orbit (see e.g. ).', '0812.3358-2-35-3': 'Note that because of the relatively small variation of the X-ray flux over the orbit, a factor of only two, the requirements for this scenario are quite modest.', '0812.3358-2-35-4': 'We note that dominant adiabatic losses have been invoked by [CITATION] to explain the variations of the X-ray and TeV gamma-ray fluxes from the binary pulsar PSR B1259[MATH]63.', '0812.3358-2-36-0': 'The detected power-law spectrum of X-rays with photon index around [MATH] implies that the established energy spectrum of electrons is also a power-law with index [MATH].', '0812.3358-2-36-1': 'This agrees well with the hypothesis of dominance of adiabatic losses, because the adiabatic losses do not change the initial spectrum of electrons.', '0812.3358-2-36-2': 'Thus the required power-law index [MATH] implies a reasonable acceleration spectrum [MATH].', '0812.3358-2-36-3': 'Otherwise, in an environment dominated by synchrotron losses, the acceleration spectrum should be very hard, with a power-law index [MATH], or should have an unreasonably large low-energy cutoff at [MATH] TeV to explain the observed X-ray spectra.', '0812.3358-2-37-0': 'Obviously, adiabatic losses modulate the IC gamma-ray flux in a similar manner.', '0812.3358-2-37-1': 'However, unlike X-rays, the TeV gamma-rays suffer significant distortion due to photon-photon absorption (see e.g. ) and anisotropic IC scattering with its strong hardening of the gamma-ray spectrum .', '0812.3358-2-37-2': 'All this leads to additional orbital modulation of the gamma-ray signal, and it is likely that these two additional processes are responsible for the strong change of gamma-ray flux, much more pronounced than that seen in X-rays (see Fig. [REF]).', '0812.3358-2-37-3': 'The Suzaku data presented in this paper implies a key additional assumption, namely that the accelerated electrons must loose their energy adiabatically before they cool radiatively.', '0812.3358-2-38-0': 'In order to demonstrate that the suggested scenario can satisfactorily explain the combined Suzaku X-ray and HESS gamma-ray data, we performed calculations of the broad-band spectral energy distributions (SEDs) of the synchrotron and IC emission, assuming a simple model in which the same population of electrons is responsible for both X-rays and TeV gamma-rays.', '0812.3358-2-38-1': 'We also assumed that the emission region has homogeneous physical conditions.', '0812.3358-2-38-2': 'This is a reasonable assumption given that we deal with very short cooling timescales ([MATH] s), thus electrons cannot travel significant distances while emitting.', '0812.3358-2-39-0': 'In the regime dominated by adiabatic energy losses, the synchrotron X-ray flux is proportional to [MATH].', '0812.3358-2-39-1': 'The X-ray modulation seen by Suzaku is then described by the modulation of the adiabatic loss rate.', '0812.3358-2-39-2': 'In Fig. [REF], we show [MATH] that is inferred from the X-ray data.', '0812.3358-2-39-3': 'The required adiabatic cooling timescales are [MATH] s. Any consistent calculation of the adiabatic cooling requires the solution of the corresponding hydrodynamical problem, and one needs to know in detail the nature of the source.', '0812.3358-2-39-4': 'At the present stage, we consider the simple example of adiabatic cooling in a relativistically expanding source.', '0812.3358-2-39-5': 'In such a case, the adiabatic loss rate can be written as: [MATH] with [MATH] sec, where [MATH] is the characteristic size of the source.', '0812.3358-2-39-6': 'The required variation of the adiabatic cooling is thus reduced to the modulation of the size of the radiation region ([MATH]).', '0812.3358-2-39-7': 'The size in turn depends on the external pressure exerted by, e.g., the stellar wind from the massive star.', '0812.3358-2-39-8': 'The expected weaker external pressure around apastron implicitly assumed in our model would be broadly consistent with the radial dependence of the wind pressure.', '0812.3358-2-40-0': 'In Fig. [REF] we show the SEDs averaged over the INFC ([MATH]-[MATH]) and SUPC ([MATH], [MATH]) phase intervals.', '0812.3358-2-40-1': 'The corresponding gamma-ray data have been previously reported by HESS .', '0812.3358-2-40-2': 'Since both the absolute flux and the energy spectrum of TeV gamma-rays vary rapidly with phase, in order to compare the theoretical predictions with observations, we should use smaller phase bins, ideally speaking with the time intervals [MATH] s corresponding to the characteristic cooling timescales of electrons.', '0812.3358-2-40-3': 'Because of the lack of the relevant gamma-ray data available to us, we here use the X-ray and gamma-ray data integrated over [MATH].', '0812.3358-2-40-4': 'While this compromise does not allow us to perform quantitative studies, it can be used to make a qualitative comparison of the model calculations with observational data.', '0812.3358-2-41-0': 'The theoretical calculations of the SEDs are in a reasonable agreement with the observed spectra, though they do not perfectly match the gamma-ray fluxes.', '0812.3358-2-41-1': 'One can improve the fits by introducing slight phase-dependent changes in the spectra of accelerated electrons, but it is beyond the scope of this paper given the caveat mentioned above.', '0812.3358-2-41-2': 'We should also note that the calculations of low energy gamma-rays (in the Fermi domain) are performed assuming that the injection spectrum of electrons continues down to [MATH] GeV.', '0812.3358-2-41-3': 'If this is not the case, the gamma-ray fluxes in the Fermi energy domain may be significantly suppressed.', '0812.3358-2-41-4': 'On the other hand, the detection of gamma-rays by Fermi would allow us to recover the spectrum of electrons in a very broad energy interval, and thus distinguish between different acceleration models.', '0812.3358-2-41-5': 'Another important feature in this scenario is that a hard synchrotron spectrum extending up to a few MeV, is required by the robust detection of [MATH] TeV gamma-rays from the system.', '0812.3358-2-41-6': 'A future detection of the emission at MeV energies may bring important information on the presence of highest energy particles in the system.', '0812.3358-2-42-0': 'The reproduction, at least qualitatively, of the observed spectral and temporal features of the nonthermal radiation with the simple toy model supports the production of X-ray and TeV gamma-rays by the same population of parent particles and allows us to derive several principal conclusions.', '0812.3358-2-42-1': 'The electron energy distribution should be a power-law with an almost constant index of [MATH] to explain the X-ray spectra.', '0812.3358-2-42-2': 'Note that in an isotropic photon gas when the Compton scattering takes place in the deep KN regime such an electron distribution results in a quite steep TeV spectrum with photon index [MATH].', '0812.3358-2-42-3': 'This does not agree with HESS observations.', '0812.3358-2-42-4': 'However, the anisotropic IC provides a remarkable hardening of the gamma-ray spectrum , in particular, [MATH] would be expected for the INFC, and it has indeed been observed using HESS .', '0812.3358-2-42-5': 'We also note that, to explain the VHE spectrum at SUPC, we have to assume that the emission region is located at a distance [MATH] cm from the compact object, in the direction perpendicular to the orbital plane.', '0812.3358-2-42-6': 'In the standard pulsar scenario, the production region cannot be located far away from the compact object, and even invoking electromagnetic cascading (see e.g. Fig. 16 in ), one cannot reproduce the reported fluxes around orbital phase 0.0 .', '0812.3358-2-42-7': 'We have found that the magnetic field strength cannot deviate much from a few G.', '0812.3358-2-42-8': 'We can also derive a constraint on the size of the emission region, imposing a maximum expansion speed of [MATH] (the speed of light), and the Hillas criterion , in which the minimum size of a source capable of accelerating particles to a given energy [MATH] is [MATH] (where [MATH] cm), the Larmor radius.', '0812.3358-2-42-9': 'This estimate yields a size of [MATH]-[MATH] cm, which agrees quite well with the estimate based on the required timescale of adiabatic cooling.', '0812.3358-2-42-10': 'Note that the requirement of fast adiabatic losses imposes a strong constraint on the acceleration rate of electrons.', '0812.3358-2-42-11': 'Indeed, the acceleration timescale can be expressed as: [MATH], where [MATH] parametrizes the acceleration efficiency.', '0812.3358-2-42-12': 'In extreme accelerators with the maximum possible rate allowed by classical electrodynamics [MATH].', '0812.3358-2-42-13': 'The HESS spectrum provides evidence of electron acceleration well above 10 TeV.', '0812.3358-2-42-14': 'Therefore, [MATH] s is required at [MATH] TeV, which translates into [MATH] for [MATH] G. Thus we arrive at the conclusion that an extremely efficient acceleration with [MATH] should operate in a compact region of [MATH].', '0812.3358-2-43-0': 'Finally, we would like to emphasize that in the scenario described here different radiation energy intervals are characterized by fundamentally different light curves.', '0812.3358-2-43-1': 'While the synchrotron X-ray modulation is caused by adiabatic losses, the light curve in gamma-rays depends critically, in addition, on effects related to interactions with the optical photons of the companion star.', '0812.3358-2-43-2': 'Two of these effects, photon-photon pair production and anisotropic Compton scattering are equally important for the formation of the light curve of TeV gamma-rays.', '0812.3358-2-43-3': 'On the other hand, only the effect of anisotropic Compton scattering has an impact on the formation of the light curve of GeV gamma-rays.', '0812.3358-2-43-4': 'The difference of light curves in the X-ray and GeV and TeV gamma-ray intervals in this scenario is shown in Figure [REF].', '0812.3358-2-44-0': '# Summary', '0812.3358-2-45-0': 'The Suzaku X-ray satellite has observed LS 5039 for the first time with imaging capabilities over one and a half orbits.', '0812.3358-2-45-1': 'The Suzaku data show strong modulation of the X-ray emission at the orbital period of the system and the X-ray spectral data are described by a hard power-law up to 70 keV.', '0812.3358-2-45-2': 'We found the close correlation of the X-ray and TeV gamma-ray light curves, which can be interpreted as evidence of production of these two radiation components by the same electron population via synchrotron radiation and IC scattering, respectively.', '0812.3358-2-45-3': 'Whereas there are at least two reasons for the formation of a periodic TeV gamma-ray light curve, both related to the interaction with photons from the companion star (photon-photon absorption of VHE gamma-rays and IC scattering in an anisotropic photon field), the modulated X-ray signal requires an additional effect.', '0812.3358-2-45-4': 'A simple and natural reason for the modulation in X-rays seems to be adiabatic losses which should dominate over the radiative (synchrotron and IC) losses of electrons.', '0812.3358-2-45-5': 'We demonstrate that this assumption allows us to explain, at least qualitatively, the spectral and temporal characteristics of the combined Suzaku and HESS data.', '0812.3358-2-45-6': 'In particular, the introduction of adiabatic losses not only provides a natural explanation for the rather stable photon index of the X-ray spectrum [MATH], but also allows one to approximately reproduce the TeV gamma-ray spectra.', '0812.3358-2-46-0': 'The gamma-ray data require a location of the production region at the periphery of the binary system at [MATH] cm.', '0812.3358-2-46-1': 'This constraint allows a quite robust estimate of the magnetic field of a few Gauss to be derived directly from the X/TeV flux ratio, and an adiabatic loss time of a few seconds to provide the dominance of adiabatic losses.', '0812.3358-2-46-2': 'In the case of a relativistically expanding source, the size of the production region should not exceed [MATH] cm.', '0812.3358-2-46-3': 'The adiabatic cooling cannot be shorter than several seconds, and correspondingly, the size of the production region cannot be much smaller than [MATH] cm, since otherwise the electrons could not be accelerated up to energies beyond 10 TeV, even assuming an extreme acceleration rate close to the fundamental limit determined by quantum electrodynamics.', '0812.3358-2-46-4': 'There is little doubt that future simultaneous observations of LS 5039 with the Suzaku, Fermi, and HESS telescopes will provide key information for understanding the nature of this mysterious non-thermal source.', '0812.3358-2-47-0': 'T. Kishishita and T. Tanaka are supported by research fellowships of the Japan Society for the Promotion of Science for Young Scientists.', '0812.3358-2-47-1': 'The authors acknowledge support by the Spanish DGI of MEC under grant AYA2007-6803407171-C03-01, as well as partial support by the European Regional Development Fund (ERDF/FEDER).', '0812.3358-2-47-2': 'V.B-R. acknowledges support by the Spanish DGI of MEC under grant AYA2007-6803407171-C03-01, as well as partial support by the European Regional Development Fund (ERDF/FEDER).'}

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[]

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[]

['0812.3358-1-23-5', '0812.3358-2-23-5', '0812.3358-3-23-5']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0812.3358

{'0812.3358-3-0-0': 'We report on the results from Suzaku broadband X-ray observations of the galactic binary source LS 5039.', '0812.3358-3-0-1': 'The Suzaku data, which have continuous coverage of more than one orbital period, show strong modulation of the X-ray emission at the orbital period of this TeV gamma-ray emitting system.', '0812.3358-3-0-2': 'The X-ray emission shows a minimum at orbital phase [MATH], close to the so-called superior conjunction of the compact object, and a maximum at phase [MATH], very close to the inferior conjunction of the compact object.', '0812.3358-3-0-3': 'The X-ray spectral data up to 70 keV are described by a hard power-law with a phase-dependent photon index which varies within [MATH]- 1.61.', '0812.3358-3-0-4': 'The amplitude of the flux variation is a factor of 2.5, but is significantly less than that of the factor [MATH]8 variation in the TeV flux.', '0812.3358-3-0-5': 'Otherwise the two light curves are similar, but not identical.', '0812.3358-3-0-6': 'Although periodic X-ray emission has been found from many galactic binary systems, the Suzaku result implies a phenomenon different from the "standard" origin of X-rays related to the emission of the hot accretion plasma formed around the compact companion object.', '0812.3358-3-0-7': 'The X-ray radiation of LS 5039 is likely to be linked to very-high-energy electrons which are also responsible for the TeV gamma-ray emission.', '0812.3358-3-0-8': 'While the gamma-rays are the result of inverse Compton scattering by electrons on optical stellar photons, X-rays are produced via synchrotron radiation.', '0812.3358-3-0-9': 'Yet, while the modulation of the TeV gamma-ray signal can be naturally explained by the photon-photon pair production and anisotropic inverse Compton scattering, the observed modulation of synchrotron X-rays requires an additional process, the most natural one being adiabatic expansion in the radiation production region.', '0812.3358-3-1-0': '# Introduction', '0812.3358-3-2-0': 'LS 5039 is a high-mass X-ray binary with extended radio emission .', '0812.3358-3-2-1': 'This system is formed by a main sequence O type star and a compact object of disputed nature that has been claimed to be both a black hole and a neutron star/pulsar .', '0812.3358-3-2-2': 'The compact object is moving around the companion star in a moderately elliptic orbit (eccentricity [MATH]) with an orbital period of [MATH] days .', '0812.3358-3-3-0': 'As summarized in [CITATION], LS 5039 has been observed several times in the X-ray energy band for limited phases in the orbital period.', '0812.3358-3-3-1': 'Flux variations on time scale of days and sometimes on much shorter timescales have been reported.', '0812.3358-3-3-2': 'The spectrum was always well represented by a power-law model with a photon index ranging [MATH]-1.6 up to [MATH] keV, with fluxes changing moderately around [MATH].', '0812.3358-3-3-3': 'Softer spectra and larger fluxes had been also inferred from RXTE observations, although background contamination was probably behind these differences (see ).', '0812.3358-3-3-4': 'Also, Chandra data taken in 2002 and 2005 showed spectra significantly harder than 1.5 , but such a hard spectrum was probably an artifact produced by photon pile-up.', '0812.3358-3-3-5': 'Recently, [CITATION] reported the results of INTEGRAL observations in hard X-rays.', '0812.3358-3-3-6': 'The source was detected at energies between 25 and 60 keV.', '0812.3358-3-3-7': 'The source was detected at energies between 25 and 60 keV.', '0812.3358-3-3-8': 'The flux was estimated to be [MATH] (90 % confidence level) around the inferior conjunction (INFC) of the compact object, and a flux upper limit of [MATH] (90 % confidence level) was derived at the superior conjunction of the compact object (SUPC).', '0812.3358-3-4-0': 'LS 5039 has also been detected in very-high-energy (VHE: [MATH] TeV) gamma-rays , exhibiting a periodic signal modulated with the orbital period .', '0812.3358-3-4-1': 'There are two other binary systems with robust detections in the VHE range: PSR B1259[MATH]63 and LS I [MATH]303 .', '0812.3358-3-4-2': 'Evidence for TeV emission has been found also in Cygnus X-1 .', '0812.3358-3-4-3': 'PSR B1259[MATH]63 is a clear case of a high-mass binary system containing a non-accreting pulsar , whereas Cygnus X-1 is a well known accreting black hole system .', '0812.3358-3-4-4': 'The nature of the compact object in LS 5039 is not yet established, and the origin of the VHE emitting electrons is unclear.', '0812.3358-3-4-5': 'They may be related to a pulsar wind or to a black hole with a (sub)relativistic jet.', '0812.3358-3-4-6': 'In the standard pulsar-wind scenario, the severe photon-photon absorption makes the explanation of the detection of the VHE radiation problematic, at least at the position corresponding to the orbital phase [MATH] (see Figs. 16 and 4 from Sierpowska-Bartosik Torres 2008 and Dubus et al. 2008 and compare with Fig. 5 in Aharonian et al. 2006), in which the emitter is expected to be located between the compact object and the star.', '0812.3358-3-4-7': 'However, particles may be accelerated in a relativistic outflow formed at the interaction of the pulsar and the stellar winds and radiate far from the compact object, making the pulsar wind scenario a viable option.', '0812.3358-3-4-8': 'In the microquasar scenario, the lack of accretion features in the X-ray spectrum may be a problem unless the bulk of the accretion power is released in the form of kinetic energy of the outflow, rather than thermal emission during accretion, as in the case of SS 433 .', '0812.3358-3-4-9': 'At this stage, we cannot give a preference to any of these scenarios, but new data, in particular those obtained with the Suzaku satellite, allow us to make an important step towards the understanding of the nature of the non-thermal processes of acceleration and radiation in this mysterious object.', '0812.3358-3-5-0': '# Observation', '0812.3358-3-6-0': 'The temporal and spectral characteristics of the X-ray emission from LS 5039 along the orbit should provide important clues for understanding the acceleration/radiation processes in this source.', '0812.3358-3-6-1': 'The fact that all previous X-ray observations of this object have incomplete coverage of the orbital period, or suffered from background contamination, is therefore rather unsatisfactory.', '0812.3358-3-6-2': 'This motivated our long, [MATH] ks observation with the Suzaku X-ray observatory , which gives us unprecedented coverage of more than one orbital period, continuously from 2007 September 9 to 15 (see Table [REF]).', '0812.3358-3-6-3': 'Suzaku has four sets of X-ray telescopes each with a focal-plane X-ray CCD camera that are sensitive in the energy range of 0.3-12 keV.', '0812.3358-3-6-4': 'Three of the XIS detectors (XIS0, 2 and 3) have front-illuminated (FI) CCDs, whereas XIS1 utilizes a back-illuminated (BI) CCD.', '0812.3358-3-6-5': 'The merit of the BI CCD is its improved sensitivity in the soft X-ray energy band below 1 keV.', '0812.3358-3-6-6': 'Suzaku contains also a non-imaging collimated Hard X-ray Detector , which covers the 10-600 keV energy band with Si PIN photodiodes (10-70 keV) and GSO scintillation detectors (40-600 keV).', '0812.3358-3-6-7': 'Suzaku has two default pointing positions, the XIS nominal position and the HXD nominal position.', '0812.3358-3-6-8': 'In this observation, we used the HXD nominal position, in which the effective area of HXD is maximized, whereas that of the XIS is reduced to on average [MATH] 88%.', '0812.3358-3-6-9': 'Results from XIS2 are not reported here since it has not been in operation since an anomaly in November 2006.', '0812.3358-3-6-10': 'In addition, we do not describe in detail the analysis of HXD-GSO data, since the HXD-GSO detected no significant signal from the source.', '0812.3358-3-7-0': '# Data Reduction', '0812.3358-3-8-0': 'We used data sets processed using the software of the Suzaku data processing pipeline (version 2.1.6.16).', '0812.3358-3-8-1': 'Reduction and analysis of the data were performed following the standard procedure using the HEADAS v6.4 software package, and spectral fitting was performed with XSPEC v.11.3.2.', '0812.3358-3-9-0': 'For the XIS data analysis, we accumulated cleaned events over good time intervals that were selected by removing spacecraft passages through the South Atlantic Anomaly (SAA).', '0812.3358-3-9-1': "Further, we screened the data with the following criteria - (1) cut-off rigidity is larger than 6 GV and (2) elevation angle from the Earth's rim is larger than 5[MATH].", '0812.3358-3-9-2': 'The source photons were accumulated from a circular region with a radius of [MATH].', '0812.3358-3-9-3': 'The background region was chosen in the same field of view with the same radius and an offset of [MATH] from the source region.', '0812.3358-3-9-4': 'We have co-added the data from the two FI-CCDs (XIS0 and XIS3) to increase statistics.', '0812.3358-3-9-5': 'The response (RMF) files and the auxiliary response (ARF) files used in this paper were produced using xisrmfgen and xissimarfgen, respectively.', '0812.3358-3-10-0': 'For the HXD data,"uncleaned event files" were screened with the standard event screening criteria: the cut-off rigidity is larger than 6 GV, the elapsed time after the passage of the SAA is more than 500 s and the time to the next SAA passage is more than 180 s, high voltages from all eight HV units are within the normal range and the elevation angle from the Earth\'s rim is more than [MATH].', '0812.3358-3-10-1': 'We also discarded telemetry-saturated time intervals.', '0812.3358-3-10-2': 'In the spectral analysis in 3, we used the response file for a point-like source at the HXD-nominal position (aehxdpinhxnome420070914.rsp), which is released as a part of CALDB (Suzaku calibration data base).', '0812.3358-3-11-0': 'The HXD-PIN spectrum is dominated by the time-variable instrumental background ( non-X-ray background (NXB) ) induced by cosmic rays and trapped charged particles in the satellite orbit.', '0812.3358-3-11-1': 'To estimate the instrumental background component, we used the the time-dependent NXB event files released by the HXD instrument team, whose reproducibility is reported by [CITATION].', '0812.3358-3-11-2': 'In order to estimate the systematic uncertainty of the NXB model during our observation, we compare the NXB model spectrum during Earth occultation with the observed spectrum of the same time interval.', '0812.3358-3-11-3': 'The event selection criteria for this study are the same as those of the cleaned event except for the criterion on Earth elevation angle, which was chosen to to be less than [MATH].', '0812.3358-3-11-4': 'The estimated uncertainty obtained is [MATH]%, which is consistent with the values reported by [CITATION].', '0812.3358-3-12-0': 'Another component of the HXD-PIN background is the cosmic X-ray background (CXB).', '0812.3358-3-12-1': 'In our analysis, we assumed the CXB spectrum reported by : [EQUATION] where [MATH] and [MATH].', '0812.3358-3-12-2': 'The CXB spectrum observed with HXD-PIN was simulated by using a PIN response file for isotropic diffuse emission (aehxdpinflate420070914.rsp) and added to the NXB spectrum.', '0812.3358-3-12-3': 'Based on this approach, the contribution from the CXB flux is [MATH]% of the NXB.', '0812.3358-3-13-0': 'Since LS 5039 is located close to the Galactic plane, the contribution from the Galactic ridge X-ray emission (GRXE) must be examined, especially for HXD-PIN spectra.', '0812.3358-3-13-1': 'In order to model the shape of the GRXE, data from Suzaku observations of the Galactic ridge region (ObsID: 500009010 and 500009020) were analyzed.', '0812.3358-3-13-2': 'The Suzaku spectrum from 3 keV to 50 keV can be well fitted with the Raymond-Smith plasma with a temperature of [MATH] keV and a power-law function with [MATH].', '0812.3358-3-13-3': 'Although we also tried a power law with exponential cutoff, following the results from the INTEGRAL IBIS , it turned out that the assumption on the spectral shape of the GRXE has negligible effect on the spectral parameters of LS 5039.', '0812.3358-3-13-4': 'The normalization of the GRXE spectrum component in the HXD-PIN spectrum of LS 5039 is determined from the XIS spectrum of the LS 5039 observation by excluding an encircled region with a radius of [MATH] centered on the LS 5039 location.', '0812.3358-3-13-5': 'The flux of the GRXE is estimated to be [MATH] of the contribution from the CXB.', '0812.3358-3-13-6': 'In Figure [REF], we show the time averaged HXD-PIN spectrum plotted together with models for the NXB, CXB and GRXE.', '0812.3358-3-14-0': '# Analysis and Results', '0812.3358-3-15-0': '## Temporal analysis', '0812.3358-3-16-0': 'The light curve obtained from the XIS detector is shown in the top panel of Figure [REF].', '0812.3358-3-16-1': 'The continuous coverage in X-rays, longer than the orbital period of the LS 5039 system, reveals a smooth variation of a factor 2 in the 1-10 keV count rate.', '0812.3358-3-16-2': 'The light curve is drawn over two orbital periods.', '0812.3358-3-16-3': 'The orbital phase is calculated with the period of 3.90603 days, and [MATH] with reference epoch [MATH]) taken from [CITATION].', '0812.3358-3-16-4': 'The light curve from phase [MATH] to 1.5, which was obtained in the last part of the observation, smoothly overlaps with the one obtained at the beginning of the observation ([MATH]-0.5).', '0812.3358-3-17-0': 'In the middle panel of Figure [REF], we present the light curve obtained with the HXD-PIN for the energy range 15-40 keV.', '0812.3358-3-17-1': 'Although the statistical errors are larger, the modulation behavior is similar to that of the XIS.', '0812.3358-3-17-2': 'The amplitude of the modulation is roughly the same between the XIS and HXD-PIN, indicating small changes of spectral shape depending on orbital phase.', '0812.3358-3-17-3': 'The spectral parameters obtained for each orbital phase are reported in the following section.', '0812.3358-3-18-0': 'The light curves obtained with Suzaku show that the X-ray flux minimum appears around phase 0.0-0.3 and it reaches maximum around phase 0.5-0.8.', '0812.3358-3-18-1': 'In order to quantify the amplitude of the flux variations, we fitted the XIS light curve with a simple sinusoidal function.', '0812.3358-3-18-2': 'Due to structures in the light curve, the fit converges with large chi-square ([MATH] = 4.92 (121)).', '0812.3358-3-18-3': 'However, the general trend is well represented by a sinusoidal function: [EQUATION] where [MATH] is the count rate as a function of phase [MATH].', '0812.3358-3-18-4': 'The ratios between the minimum and maximum count rates are 2.21 [MATH] counts s[MATH] for XIS and 2.02 [MATH] counts s[MATH] for HXD-PIN.', '0812.3358-3-18-5': 'Structures of the X-ray and hard X-ray light curves are similar to that discovered in the phase diagram of integral fluxes at energies [MATH] TeV obtained on a run-by-run basis from HESS data (2004 to 2005) .', '0812.3358-3-18-6': 'The temporal X-ray behavior was already suggested by RXTE data, as well as by a compilation of all the previous X-ray data obtained with imaging instruments .', '0812.3358-3-19-0': 'In addition to the continuously changing component with respect to the orbital phase, short timescale structures are found around [MATH] and [MATH].', '0812.3358-3-19-1': 'The unabsorbed flux changes about a 30% in [MATH].', '0812.3358-3-19-2': 'A significant dip can be seen around [MATH] in the top panel of Figure [REF].', '0812.3358-3-19-3': 'In comparison with the data at around [MATH] obtained in the first half of the observation, the flux decreased [MATH]50% only in this phase.', '0812.3358-3-19-4': 'The time duration of this dip corresponds to [MATH].', '0812.3358-3-19-5': 'These structures may reflect features of the (possibly changing) environment of the X-ray emitting region, it is therefore of importance to test with further observations if these are persistent features.', '0812.3358-3-20-0': '## Spectral Analysis', '0812.3358-3-21-0': 'Firstly we study time-resolved (phase-resolved) X-ray spectra.', '0812.3358-3-21-1': 'The data are divided into data segments with respect to the assigned phase, and model fitting is performed for XIS spectra for each segment with [MATH].', '0812.3358-3-22-0': 'A single power-law function with photoelectric absorption, provides a good fit for all the segments.', '0812.3358-3-22-1': 'In order to study the possible changes of the amount of photoelectric absorption, we here fit the data with [MATH] free.', '0812.3358-3-22-2': 'The best-fit parameters are presented in Table [REF].', '0812.3358-3-22-3': 'The derived values of the photon index and absorption column density are consistent with previous observations .', '0812.3358-3-22-4': 'When we fix the [MATH] to the value obtained from the time averaged spectrum, resultant photon indices stay same within statistical error.', '0812.3358-3-23-0': 'The photon index ([MATH]) values are plotted as a function of orbital phase in the top panel of Figure [REF].', '0812.3358-3-23-1': 'The spectral shape varied such that the spectrum is steep around SUPC ([MATH]) and becomes hard ([MATH]) around apastron.', '0812.3358-3-23-2': 'The modulation behavior of [MATH] is somewhat different from that observed using HESS in the VHE range.', '0812.3358-3-23-3': 'The amplitude of the variation is [MATH], which is much smaller than the change of [MATH]0.6 in the VHE region .', '0812.3358-3-23-4': 'The 1-10 keV flux changes from [MATH]) to [MATH].', '0812.3358-3-23-5': '([MATH]).', '0812.3358-3-24-0': 'In all the data segments, the source is significantly detected with the HXD-PIN, indicating that hard X-ray emission extends at least up to 70 keV.', '0812.3358-3-24-1': 'Note also that although the XIS and HXD-PIN spectra do not overlap, they seem to be smoothly connected in the gap between 10 and 15 keV.', '0812.3358-3-25-0': 'To study the shape of the spectrum above 10 keV, the XIS spectra and the PIN spectrum in the range 15-70 keV after subtraction of background (NXB [MATH] CXB [MATH] GRXE) are jointly fitted (Figure [REF] Top).', '0812.3358-3-25-1': 'The time-averaged spectra are well represented by an absorbed power-law model with [MATH]= 1.51 [MATH]0.02 with reduced [MATH] (235 degrees of freedom).', '0812.3358-3-25-2': 'We find no cutoff structure in the energy range of the HXD-PIN.', '0812.3358-3-25-3': 'The spectra within the phase intervals [[MATH]] and [[MATH]], which correspond to the INFC and SUPC, respectively, are also shown in Figure [REF].', '0812.3358-3-25-4': 'The best-fit parameters are presented in Table [REF].', '0812.3358-3-26-0': 'Althought earlier observations by RXTE suggested the presence of an iron emission line at 6.7 keV (Ribo et al. 1999), later observations by Chandra and XMM could not find evidence of it (e.g. ).', '0812.3358-3-26-1': 'A careful study of new and longer RXTE observations, using slew data to account for background emission, revealed that the earlierly reported 6.7 keV emission line is likely a background feature .', '0812.3358-3-26-2': 'The Suzaku data confirm this result.', '0812.3358-3-26-3': 'In an attempt to find the possible signature of iron emission lines from LS 5039, we analyzed the phase-averaged spectrum.', '0812.3358-3-26-4': 'The upper limits on iron line structures are determined by fitting a Gaussian at various energies and line widths at which Fe emission might be expected.', '0812.3358-3-26-5': 'The power-law continuum model parameters are fixed with the best fit values and a Gaussian component is added to the power-law function.', '0812.3358-3-26-6': 'The central energy of the Gaussian line is swept from 6.0 keV to 7.1 keV in steps of 0.1 keV.', '0812.3358-3-26-7': 'Line widths are changed from 0.01 keV to 0.09 keV in steps of 0.01 keV, together with lines with larger widths of 0.15 and 0.20 keV.', '0812.3358-3-26-8': 'An equivalent width is determined at each grid point.', '0812.3358-3-26-9': 'The resulting upper-limit on the equivalent width is 40 eV with 90% confidence level.', '0812.3358-3-27-0': '# Discussion', '0812.3358-3-28-0': 'The X-ray emission observed with Suzaku is characterized by (1) a hard power law with [MATH] extending from soft X-rays to [MATH] keV, (2) clear orbital modulation in flux and photon index, (3) a moderate X-ray luminosity of [MATH], (4) a small and constant absorbing column density, and (5) a lack of detectable emission lines.', '0812.3358-3-29-0': 'Although variable X-ray emission has been found from more than two hundred galactic binary systems, the Suzaku data hardly can be explained within the "standard accretion" scenario where X-rays are produced by a hot thermal (comptonized) accretion plasma around the compact object.', '0812.3358-3-29-1': 'The lack of X-ray emission lines (at the level of sensitivity of Suzaku) as well as the hard [MATH] type energy spectrum of the X-ray continuum, extending from soft X-rays up to 70 keV, favors a non-thermal origin of the X-rays.', '0812.3358-3-29-2': 'This conclusion is supported by the general similarities between the properties of the observed X-rays and TeV gamma-rays.', '0812.3358-3-29-3': 'Namely, both radiation components require a rather hard energy distribution of parent electrons with a power-law index of [MATH].', '0812.3358-3-29-4': 'This directly follows from the photon index of the synchrotron radiation [MATH], and agrees quite well with the currently most favored interpretation of the TeV gamma-rays, in which they would be produced by IC scattering off the anisotropic photon field of the massive companion star.', '0812.3358-3-30-0': 'Assuming that the TeV gamma-ray production region is located at a distance from the companion star of [MATH] cm (i.e. the binary system size), and taking into account that gamma-rays are produced in the deep Klein-Nishina (KN) regime with significantly suppressed cross-section, for the well known luminosity of the optical star [MATH], one can estimate quite robustly the strength of the magnetic field in the emission region.', '0812.3358-3-30-1': 'The numerical calculations show that the field should be around a few Gauss (see e.g. Fig. 5).', '0812.3358-3-30-2': 'For such a magnetic field strength, the energy intervals of electrons responsible for the two emission components overlap substantially, as shown in Figure [REF].', '0812.3358-3-30-3': 'Therefore, we are most likely dealing with the same population of parent electrons, which should be located at large distances from the compact object, in the system periphery, to prevent the severe absorption of the TeV radiation and the subsequent intense emission from the pair-created secondaries .', '0812.3358-3-31-0': 'It should be noted that the observed X-ray emission is very difficult to explain as synchrotron emission produced by secondary (pair-produced) electron and positrons.', '0812.3358-3-31-1': 'Since the pair production cross-section has strong energy dependence with a distinct maximum, for the target photons of typical energy of [MATH] eV, the major fraction of the absorbed energy will be released in the form of [MATH] GeV electrons.', '0812.3358-3-31-2': 'Thus secondary pair synchrotron emission must show a spectral break in the Suzaku energy band unless one assumes unreasonably high magnetic fields, [MATH] kG, in the surroundings of the gamma-ray emission region .', '0812.3358-3-32-0': 'Figure [REF] shows the synchrotron and IC cooling times of electrons, as a function of electron energy, calculated for the stellar photon density at [MATH] and for a magnetic field [MATH] G.', '0812.3358-3-32-1': 'It can be seen that synchrotron losses dominate over IC losses at [MATH] TeV.', '0812.3358-3-32-2': 'Note that the TeV gamma-ray production takes place in the deep KN regime.', '0812.3358-3-32-3': 'This implies that the cooling time, [MATH], of electrons generating GeV gamma-rays via IC scattering (Thomson regime) is shorter than the cooling time of TeV electrons responsible for producing very high-energy gamma-rays (KN).', '0812.3358-3-32-4': 'The same applies for synchrotron cooling time of multi-TeV electrons that produce low-energy (MeV) gamma-rays by synchrotron radiation.', '0812.3358-3-32-5': 'One should therefore expect significantly higher MeV (synchrotron) and GeV (IC) fluxes than at keV and TeV energies, provided that the acceleration spectrum of electrons extends from low energies to very high energies.', '0812.3358-3-32-6': 'However, in the case of existence of low-energy and very-high-energy cutoffs in the acceleration spectrum, the gamma-ray fluxes [MATH] MeV and at GeV energies would be significantly suppressed.', '0812.3358-3-33-0': 'To better understand the energy ranges of the electrons responsible for X-ray and gamma-ray production, we show in Figure [REF] the energy zones of electrons relevant to the Suzaku, Fermi, and HESS radiation domains.', '0812.3358-3-33-1': 'Note that the reconstruction of the average energy of electrons responsible for the IC gamma-rays depends only on the well known temperature of the companion star [MATH] K.', '0812.3358-3-33-2': 'The light green zone in Figure [REF] marked as "Suzaku synchrotron", corresponds to electrons responsible for the synchrotron photons produced in the energy interval [MATH].', '0812.3358-3-33-3': 'For a reasonable range of magnetic field values, the energy interval of electrons relevant for Suzaku data overlap on one hand with the HESS energy interval, and can overlap with the Fermi one.', '0812.3358-3-33-4': 'This should allow us, in the case of detection of MeV/GeV gamma-rays by Fermi, to considerably reduce the parameter space, in particular, to better localize the X- and gamma-ray production regions from electromagnetic cascade constraints, and derive the broadband energy spectrum of electrons and the strength of the magnetic field, both as a function of the orbital phase.', '0812.3358-3-34-0': 'Formally, when X-rays and TeV gamma-rays are produced by the same population of very-high-energy electrons, one should expect a general correlation between the light curves obtained by Suzaku and HESS.', '0812.3358-3-34-1': 'In this regard, the similarity between the Suzaku and HESS light curves seems to be natural.', '0812.3358-3-34-2': 'However, such an interpretation is not straightforward in the sense that two major mechanisms that might cause modulation of the TeV gamma-ray signal are related to interactions of electrons and gamma-rays with the photons of the companion star, i.e. anisotropic IC scattering and photon-photon pair production , and thus cannot contribute to the X-ray modulation.', '0812.3358-3-34-3': 'The X-ray modulation requires periodic changes of the strength of the ambient magnetic field or the number of relativistic electrons.', '0812.3358-3-34-4': 'Note, however, that the change of magnetic field would not have a strong impact as long as the radiation proceeds in the saturation regime and synchrotron losses dominate in the relevant energy interval.', '0812.3358-3-34-5': 'One would also expect modulation of the synchrotron X-ray flux if the energy losses of electrons are dominated by IC scattering, although in such a case we should observe significantly lower X-ray fluxes.', '0812.3358-3-35-0': 'A more natural reason for the modulation of the synchrotron fluxes would come from dominantly adiabatic losses.', '0812.3358-3-35-1': 'The adiabatic cooling of electrons in binary systems can be realized through complex (magneto)hydrodynamical processes, e.g. due to interactions between a black hole jet or a pulsar wind with the dense stellar wind of a massive companion star (see e.g. , ).', '0812.3358-3-35-2': 'The orbital motion could naturally produce the modulation of adiabatic cooling of electrons around the orbit (see e.g. ).', '0812.3358-3-35-3': 'Note that because of the relatively small variation of the X-ray flux over the orbit, a factor of only two, the requirements for this scenario are quite modest.', '0812.3358-3-35-4': 'We note that dominant adiabatic losses have been invoked by [CITATION] to explain the variations of the X-ray and TeV gamma-ray fluxes from the binary pulsar PSR B1259[MATH]63.', '0812.3358-3-36-0': 'The detected power-law spectrum of X-rays with photon index around [MATH] implies that the established energy spectrum of electrons is also a power-law with index [MATH].', '0812.3358-3-36-1': 'This agrees well with the hypothesis of dominance of adiabatic losses, because the adiabatic losses do not change the initial spectrum of electrons.', '0812.3358-3-36-2': 'Thus the required power-law index [MATH] implies a reasonable acceleration spectrum [MATH].', '0812.3358-3-36-3': 'Otherwise, in an environment dominated by synchrotron losses, the acceleration spectrum should be very hard, with a power-law index [MATH], or should have an unreasonably large low-energy cutoff at [MATH] TeV to explain the observed X-ray spectra.', '0812.3358-3-37-0': 'Obviously, adiabatic losses modulate the IC gamma-ray flux in a similar manner.', '0812.3358-3-37-1': 'However, unlike X-rays, the TeV gamma-rays suffer significant distortion due to photon-photon absorption (see e.g. ) and anisotropic IC scattering with its strong hardening of the gamma-ray spectrum .', '0812.3358-3-37-2': 'All this leads to additional orbital modulation of the gamma-ray signal, and it is likely that these two additional processes are responsible for the strong change of gamma-ray flux, much more pronounced than that seen in X-rays (see Fig. [REF]).', '0812.3358-3-37-3': 'The Suzaku data presented in this paper implies a key additional assumption, namely that the accelerated electrons must loose their energy adiabatically before they cool radiatively.', '0812.3358-3-38-0': 'In order to demonstrate that the suggested scenario can satisfactorily explain the combined Suzaku X-ray and HESS gamma-ray data, we performed calculations of the broad-band spectral energy distributions (SEDs) of the synchrotron and IC emission, assuming a simple model in which the same population of electrons is responsible for both X-rays and TeV gamma-rays.', '0812.3358-3-38-1': 'We also assumed that the emission region has homogeneous physical conditions.', '0812.3358-3-38-2': 'This is a reasonable assumption given that we deal with very short cooling timescales ([MATH] s), thus electrons cannot travel significant distances while emitting.', '0812.3358-3-39-0': 'In the regime dominated by adiabatic energy losses, the synchrotron X-ray flux is proportional to [MATH].', '0812.3358-3-39-1': 'The X-ray modulation seen by Suzaku is then described by the modulation of the adiabatic loss rate.', '0812.3358-3-39-2': 'In Fig. [REF], we show [MATH] that is inferred from the X-ray data.', '0812.3358-3-39-3': 'The required adiabatic cooling timescales are [MATH] s. Any consistent calculation of the adiabatic cooling requires the solution of the corresponding hydrodynamical problem, and one needs to know in detail the nature of the source.', '0812.3358-3-39-4': 'At the present stage, we consider the simple example of adiabatic cooling in a relativistically expanding source.', '0812.3358-3-39-5': 'In such a case, the adiabatic loss rate can be written as: [MATH] with [MATH] sec, where [MATH] is the characteristic size of the source.', '0812.3358-3-39-6': 'The required variation of the adiabatic cooling is thus reduced to the modulation of the size of the radiation region ([MATH]).', '0812.3358-3-39-7': 'The size in turn depends on the external pressure exerted by, e.g., the stellar wind from the massive star.', '0812.3358-3-39-8': 'The expected weaker external pressure around apastron implicitly assumed in our model would be broadly consistent with the radial dependence of the wind pressure.', '0812.3358-3-40-0': 'In Fig. [REF] we show the SEDs averaged over the INFC ([MATH]-[MATH]) and SUPC ([MATH], [MATH]) phase intervals.', '0812.3358-3-40-1': 'The corresponding gamma-ray data have been previously reported by HESS .', '0812.3358-3-40-2': 'Since both the absolute flux and the energy spectrum of TeV gamma-rays vary rapidly with phase, in order to compare the theoretical predictions with observations, we should use smaller phase bins, ideally speaking with the time intervals [MATH] s corresponding to the characteristic cooling timescales of electrons.', '0812.3358-3-40-3': 'Because of the lack of the relevant gamma-ray data available to us, we here use the X-ray and gamma-ray data integrated over [MATH].', '0812.3358-3-40-4': 'While this compromise does not allow us to perform quantitative studies, it can be used to make a qualitative comparison of the model calculations with observational data.', '0812.3358-3-41-0': 'The theoretical calculations of the SEDs are in a reasonable agreement with the observed spectra, though they do not perfectly match the gamma-ray fluxes.', '0812.3358-3-41-1': 'One can improve the fits by introducing slight phase-dependent changes in the spectra of accelerated electrons, but it is beyond the scope of this paper given the caveat mentioned above.', '0812.3358-3-41-2': 'We should also note that the calculations of low energy gamma-rays (in the Fermi domain) are performed assuming that the injection spectrum of electrons continues down to [MATH] GeV.', '0812.3358-3-41-3': 'If this is not the case, the gamma-ray fluxes in the Fermi energy domain may be significantly suppressed.', '0812.3358-3-41-4': 'On the other hand, the detection of gamma-rays by Fermi would allow us to recover the spectrum of electrons in a very broad energy interval, and thus distinguish between different acceleration models.', '0812.3358-3-41-5': 'Another important feature in this scenario is that a hard synchrotron spectrum extending up to a few MeV, is required by the robust detection of [MATH] TeV gamma-rays from the system.', '0812.3358-3-41-6': 'A future detection of the emission at MeV energies may bring important information on the presence of highest energy particles in the system.', '0812.3358-3-42-0': 'The reproduction, at least qualitatively, of the observed spectral and temporal features of the nonthermal radiation with the simple toy model supports the production of X-ray and TeV gamma-rays by the same population of parent particles and allows us to derive several principal conclusions.', '0812.3358-3-42-1': 'The electron energy distribution should be a power-law with an almost constant index of [MATH] to explain the X-ray spectra.', '0812.3358-3-42-2': 'Note that in an isotropic photon gas when the Compton scattering takes place in the deep KN regime such an electron distribution results in a quite steep TeV spectrum with photon index [MATH].', '0812.3358-3-42-3': 'This does not agree with HESS observations.', '0812.3358-3-42-4': 'However, the anisotropic IC provides a remarkable hardening of the gamma-ray spectrum , in particular, [MATH] would be expected for the INFC, and it has indeed been observed using HESS .', '0812.3358-3-42-5': 'We also note that, to explain the VHE spectrum at SUPC, we have to assume that the emission region is located at a distance [MATH] cm from the compact object, in the direction perpendicular to the orbital plane.', '0812.3358-3-42-6': 'In the standard pulsar scenario, the production region cannot be located far away from the compact object, and even invoking electromagnetic cascading (see e.g. Fig. 16 in ), one cannot reproduce the reported fluxes around orbital phase 0.0 .', '0812.3358-3-42-7': 'We have found that the magnetic field strength cannot deviate much from a few G.', '0812.3358-3-42-8': 'We can also derive a constraint on the size of the emission region, imposing a maximum expansion speed of [MATH] (the speed of light), and the Hillas criterion , in which the minimum size of a source capable of accelerating particles to a given energy [MATH] is [MATH] (where [MATH] cm), the Larmor radius.', '0812.3358-3-42-9': 'This estimate yields a size of [MATH]-[MATH] cm, which agrees quite well with the estimate based on the required timescale of adiabatic cooling.', '0812.3358-3-42-10': 'Note that the requirement of fast adiabatic losses imposes a strong constraint on the acceleration rate of electrons.', '0812.3358-3-42-11': 'Indeed, the acceleration timescale can be expressed as: [MATH], where [MATH] parametrizes the acceleration efficiency.', '0812.3358-3-42-12': 'In extreme accelerators with the maximum possible rate allowed by classical electrodynamics [MATH].', '0812.3358-3-42-13': 'The HESS spectrum provides evidence of electron acceleration well above 10 TeV.', '0812.3358-3-42-14': 'Therefore, [MATH] s is required at [MATH] TeV, which translates into [MATH] for [MATH] G. Thus we arrive at the conclusion that an extremely efficient acceleration with [MATH] should operate in a compact region of [MATH].', '0812.3358-3-43-0': 'Finally, we would like to emphasize that in the scenario described here different radiation energy intervals are characterized by fundamentally different light curves.', '0812.3358-3-43-1': 'While the synchrotron X-ray modulation is caused by adiabatic losses, the light curve in gamma-rays depends critically, in addition, on effects related to interactions with the optical photons of the companion star.', '0812.3358-3-43-2': 'Two of these effects, photon-photon pair production and anisotropic Compton scattering are equally important for the formation of the light curve of TeV gamma-rays.', '0812.3358-3-43-3': 'On the other hand, only the effect of anisotropic Compton scattering has an impact on the formation of the light curve of GeV gamma-rays.', '0812.3358-3-43-4': 'The difference of light curves in the X-ray and GeV and TeV gamma-ray intervals in this scenario is shown in Figure [REF].', '0812.3358-3-44-0': '# Summary', '0812.3358-3-45-0': 'The Suzaku X-ray satellite has observed LS 5039 for the first time with imaging capabilities over one and a half orbits.', '0812.3358-3-45-1': 'The Suzaku data show strong modulation of the X-ray emission at the orbital period of the system and the X-ray spectral data are described by a hard power-law up to 70 keV.', '0812.3358-3-45-2': 'We found the close correlation of the X-ray and TeV gamma-ray light curves, which can be interpreted as evidence of production of these two radiation components by the same electron population via synchrotron radiation and IC scattering, respectively.', '0812.3358-3-45-3': 'Whereas there are at least two reasons for the formation of a periodic TeV gamma-ray light curve, both related to the interaction with photons from the companion star (photon-photon absorption of VHE gamma-rays and IC scattering in an anisotropic photon field), the modulated X-ray signal requires an additional effect.', '0812.3358-3-45-4': 'A simple and natural reason for the modulation in X-rays seems to be adiabatic losses which should dominate over the radiative (synchrotron and IC) losses of electrons.', '0812.3358-3-45-5': 'We demonstrate that this assumption allows us to explain, at least qualitatively, the spectral and temporal characteristics of the combined Suzaku and HESS data.', '0812.3358-3-45-6': 'In particular, the introduction of adiabatic losses not only provides a natural explanation for the rather stable photon index of the X-ray spectrum [MATH], but also allows one to approximately reproduce the TeV gamma-ray spectra.', '0812.3358-3-46-0': 'The gamma-ray data require a location of the production region at the periphery of the binary system at [MATH] cm.', '0812.3358-3-46-1': 'This constraint allows a quite robust estimate of the magnetic field of a few Gauss to be derived directly from the X/TeV flux ratio, and an adiabatic loss time of a few seconds to provide the dominance of adiabatic losses.', '0812.3358-3-46-2': 'In the case of a relativistically expanding source, the size of the production region should not exceed [MATH] cm.', '0812.3358-3-46-3': 'The adiabatic cooling cannot be shorter than several seconds, and correspondingly, the size of the production region cannot be much smaller than [MATH] cm, since otherwise the electrons could not be accelerated up to energies beyond 10 TeV, even assuming an extreme acceleration rate close to the fundamental limit determined by quantum electrodynamics.', '0812.3358-3-46-4': 'There is little doubt that future simultaneous observations of LS 5039 with the Suzaku, Fermi, and HESS telescopes will provide key information for understanding the nature of this mysterious non-thermal source.', '0812.3358-3-47-0': 'T. Kishishita and T. Tanaka are supported by research fellowships of the Japan Society for the Promotion of Science for Young Scientists.', '0812.3358-3-47-1': 'The authors acknowledge support by the Spanish DGI of MEC under grant AYA2007-6803407171-C03-01, as well as partial support by the European Regional Development Fund (ERDF/FEDER).', '0812.3358-3-47-2': 'V.B-R. acknowledges support by the Spanish DGI of MEC under grant AYA2007-6803407171-C03-01, as well as partial support by the European Regional Development Fund (ERDF/FEDER).'}

1607.08685

{'1607.08685-1-0-0': 'This study concerns posterior inference on the state of reaction networks, conditioned on noisy and partial measurements.', '1607.08685-1-0-1': 'The difficulty in deriving the equation that the posterior probability distribution of the state satisfies stems from the fact that the master equation, which governs the evolution of the reaction networks, is analytically intractable.', '1607.08685-1-0-2': 'The linear noise approximation (LNA) technique, which is widely used in the analysis of reaction networks, has recently been applied to develop approximate posterior inference.', '1607.08685-1-0-3': 'Here, we propose another approximation scheme, based on the Gaussian projection (GP), and compare it to LNA in terms of their derivations and performance in posterior inference.', '1607.08685-1-0-4': 'We illustrate our method on two reaction networks that include higher-order reactions and show that the GP outperforms the LNA.', '1607.08685-1-1-0': '# Introduction', '1607.08685-1-2-0': 'Stochastic reaction networks provide probabilistic descriptions of the evolution of interacting species.', '1607.08685-1-2-1': 'They are used for modeling phenomena in a wide range of disciplines; those species can represent molecules in chemical reactions [CITATION], RNA, DNA and proteins in gene regulatory networks [CITATION], animal species in ecology [CITATION], susceptibles and infectives in epidemic models [CITATION], and information packets in telecommunication networks [CITATION].', '1607.08685-1-3-0': 'The evolution of a network is modeled by a continuous-time Markov jump process, for which the probability distribution of the number of individuals of each species obeys the master equation [CITATION].', '1607.08685-1-3-1': 'Here, we consider a situation wherein only noisy and partial measurements of underlying reaction networks are available.', '1607.08685-1-3-2': 'Our objective is to infer the number of individuals of species from the observations obtained up to the current time.', '1607.08685-1-3-3': 'In the literature on signal processing, this problem is called filtering [CITATION].', '1607.08685-1-4-0': 'The filtering equation, which governs the posterior distribution conditioned on the observations, is not analytically obtainable due to the intractability of the master equation.', '1607.08685-1-4-1': 'It is possible to perform exact numerical simulation and obtain samples from the Markov jump processes using a stochastic simulation algorithm (SSA) [CITATION].', '1607.08685-1-4-2': 'Simulating many "particles" with the SSA and sampling the weighted particles in the favor of the observations, we could obtain samples from the posterior distribution.', '1607.08685-1-4-3': 'This technique is known as the sequential Monte Carlo method or particle filtering [CITATION].', '1607.08685-1-4-4': 'However, the SSA is often too slow.', '1607.08685-1-4-5': 'Moreover, particle filtering sufficiently requires many particles to obtain precise posterior expectations.', '1607.08685-1-4-6': 'Thus, particle filtering might not be efficient for performing online posterior inference.', '1607.08685-1-5-0': 'An alternative approach is to consider the suitable approximations of the Markov jump processes.', '1607.08685-1-5-1': 'In the linear noise approximation (LNA), which is most widely used in such analysis, a Gaussian process whose mean obeys the deterministic rate equation approximates a Markov jump process [CITATION].', '1607.08685-1-5-2': 'LNA is valid under the assumption that the number of individuals of a species is large [CITATION].', '1607.08685-1-5-3': 'As the Gaussian process is tractable, LNA allows us to derive an analytical expression of the approximate filtering equation [CITATION].', '1607.08685-1-6-0': 'In this study, we propose applying the projection method [CITATION] to derive an approximate filter.', '1607.08685-1-6-1': 'In this method, the evolution of the probability distributions is constrained on a finite-dimensional family of densities through orthogonal projection onto the tangent space with respect to the Fisher metric.', '1607.08685-1-6-2': 'By choosing the Gaussian distributions for a finite-dimensional manifold, we obtain another Gaussian process that approximates the original Markov jump process.', '1607.08685-1-6-3': 'We label this approximation as "Gaussian projection (GP)".', '1607.08685-1-7-0': 'We contrast the two approximate filters based on the LNA and the GP in terms of their derivations and filtering performance.', '1607.08685-1-7-1': 'We demonstrate on two reaction networks that the approximate filter based on the GP outperforms that based on the LNA; the superiority of the GP over the LNA stands out when the observation noise is increased.', '1607.08685-1-8-0': '# Method', '1607.08685-1-9-0': '## Reaction networks', '1607.08685-1-10-0': 'Throughout the study, the transpose of a matrix [MATH] is written [MATH].', '1607.08685-1-10-1': 'Let [MATH] be [MATH] species, and consider [MATH] reactions among these species described by [EQUATION] where [MATH] and [MATH] are stoichiometric coefficients of reactants and products, respectively, and [MATH] is the reaction rate constant.', '1607.08685-1-10-2': 'We denote by [MATH] the discrete composition vector whose [MATH]th component, [MATH], is the number of individuals of species [MATH].', '1607.08685-1-10-3': 'Let [MATH] be an [MATH] matrix, called the net effect matrix, whose [MATH] element, [MATH], is the change in the number of individuals of the [MATH]th species after one step of the [MATH]th reaction.', '1607.08685-1-10-4': 'Let [MATH] be the vector whose [MATH]th component, [MATH], is the rate of [MATH]th reaction, given as [EQUATION]', '1607.08685-1-10-5': 'From the Markov property, it follows that the probability distribution over [MATH] at time [MATH], [MATH], is governed by the master equation [CITATION]: [EQUATION]', '1607.08685-1-10-6': 'Stochastic processes described by Eq. ([REF]) are related to an ordinary differential equation (ODE), called the rate equation, via the thermodynamic limit.', '1607.08685-1-10-7': 'To see this, we introduce a scale factor [MATH] (typically taken to be "volume"), and rescale the composition vector and the reaction rate as [EQUATION]', '1607.08685-1-10-8': 'Accordingly, the reaction rate constants are rescaled as [EQUATION]', '1607.08685-1-10-9': 'With these rescaled parameters, the ODE governing the evolution of [MATH] is [EQUATION]', '1607.08685-1-10-10': 'It has been proved that solutions of the stochastic model ([REF]) converges in probability to solutions of ([REF]) as [MATH] [CITATION].', '1607.08685-1-11-0': '## State space model and filtering', '1607.08685-1-12-0': 'We consider a situation wherein the system of interest is given by a stochastic reaction network, whose state is not directly observable, but instead, we have noisy and partial measurements at discrete time points [CITATION]; this situation is formulated within the framework of state space models.', '1607.08685-1-12-1': 'In state space modeling, the state process, [MATH], is given by the master equation ([REF]), and the measurement model is assumed to be [EQUATION] where [MATH], [MATH], and [MATH] is a [MATH]-dimensional Gaussian random variable with zero mean and covariance matrix [MATH].', '1607.08685-1-12-2': 'The goal of a filtering problem is to compute the posterior probability of the state [MATH] at time [MATH], when the observations [MATH] are given.', '1607.08685-1-13-0': '## Approximation methods', '1607.08685-1-14-0': 'We apply two approximation methods, the LNA and the GP, through which Gaussian processes approximate the original state process.', '1607.08685-1-14-1': "By combining these Gaussian processes with the measurement model ([REF]) via Bayes' rule, we can derive approximate filtering algorithms.", '1607.08685-1-15-0': '### Diffusion approximation', '1607.08685-1-16-0': 'We first describe the diffusion approximation of the Markov jump process [CITATION]; based on this, the LNA and the GP will be introduced in the sequel.', '1607.08685-1-16-1': 'Let [MATH] denote the number of [MATH]th reactions that occur in the interval [MATH].', '1607.08685-1-16-2': 'The number of [MATH]th species at time [MATH] is described as [EQUATION]', '1607.08685-1-16-3': 'We assume that [MATH] is sufficiently small that the reaction rate [MATH] remains approximately constant over the interval [MATH].', '1607.08685-1-16-4': 'This implies that [MATH] follows a Poisson distribution with mean [MATH].', '1607.08685-1-16-5': 'We also assume that [MATH] is sufficiently large such that the [MATH]th reaction occurs many more times than once over the interval [MATH], which further implies that the Poisson distribution can be approximated by a Gaussian distribution with the same mean and variance.', '1607.08685-1-16-6': 'Under these conditions, Eq. ([REF]) can be approximated by [EQUATION] where [MATH] is an independent Gaussian random variable with zero mean and variance [MATH].', '1607.08685-1-16-7': 'If we assume that there exists a domain of macroscopically infinitesimal time interval [MATH] satisfying the above conditions, then Eq. ([REF]) can be rewritten as [EQUATION] where [MATH] and [MATH] is an [MATH]-dimensional standard Wiener process.', '1607.08685-1-17-0': '### Linear noise approximation', '1607.08685-1-18-0': 'We introduce the LNA based on the diffusion approximation ([REF]) [CITATION].', '1607.08685-1-18-1': 'Using Eqs. ([REF]) and ([REF]), we rewrite Eq. ([REF]) as [EQUATION] from which we see that the fluctuations are of the order of [MATH].', '1607.08685-1-18-2': 'To obtain the LNA, the solution of Eq. ([REF]) is assumed to be [EQUATION]', '1607.08685-1-18-3': 'Substituting Eq. ([REF]) into [MATH] and expanding around [MATH] leads to [EQUATION] where [EQUATION] is the Jacobian matrix of [MATH].', '1607.08685-1-18-4': 'Substituting Eq. ([REF]) back into Eq. ([REF]) and collecting terms of [MATH], we obtain the rate equation for [MATH]: [EQUATION]', '1607.08685-1-18-5': 'Collecting terms of [MATH] leads to the equation for [MATH] as [EQUATION]', '1607.08685-1-18-6': 'The solution of Eq. ([REF]) is a Gaussian process with mean [MATH] and covariance matrix [MATH] that satisfy the following ODEs: [EQUATION]', '1607.08685-1-18-7': 'A particular solution for ([REF]) is found to be [MATH] by choosing a zero initial condition, with which the LNA solution of Eq. ([REF]) is obtained as a Gaussian process whose mean [MATH] and covariance [MATH] are given by [MATH] and [MATH], respectively.', '1607.08685-1-18-8': 'By substituting [MATH] into [MATH], the ODEs for [MATH] and [MATH] are expressed as [EQUATION]', '1607.08685-1-19-0': '### Gaussian projection', '1607.08685-1-20-0': 'Another approximation we use is the projection method proposed in [CITATION].', '1607.08685-1-20-1': 'The key idea is to introduce a finite-dimensional family of probability densities [MATH], [MATH], onto which the probability density of [MATH] defined by Eq. ([REF]) is projected.', '1607.08685-1-21-0': 'Let [MATH] be a space of square-integrable functions, and consider the square roots of the probability densities, [MATH].', '1607.08685-1-21-1': 'The tangent space of [MATH] at [MATH] is given by [EQUATION]', '1607.08685-1-21-2': 'The [MATH] inner product of any two bases of [MATH] is defined as [EQUATION] where [MATH] is the Fisher information matrix.', '1607.08685-1-21-3': 'Then, the orthogonal projection of [MATH] onto [MATH] is given by [EQUATION] where [MATH] is the inverse of the Fisher information matrix.', '1607.08685-1-22-0': 'Using Eq. ([REF]), we project the Fokker-Plank equation associated with Eq. ([REF]) onto [MATH] as follows: Using the chain rule, we obtain the equation for [MATH] as [EQUATION] where [MATH] is the Fokker-Planck operator: [EQUATION]', '1607.08685-1-22-1': 'Applying the orthogonal projection ([REF]) to Eq. ([REF]), we obtain ODEs for [MATH] as [EQUATION] where [MATH] is the expectation of [MATH] with respect to [MATH].', '1607.08685-1-22-2': 'If a multivariate Gaussian distribution with mean vector [MATH] and covariance matrix [MATH] is chosen for [MATH], then Eq. ([REF]) becomes [EQUATION]', '1607.08685-1-22-3': 'See [REF] for the derivation.', '1607.08685-1-23-0': 'Notice the difference between GP ([REF])-([REF]) and LNA ([REF])-([REF]).', '1607.08685-1-23-1': 'In the GP, the expectation of [MATH] is taken outside of [MATH] and [MATH], while it is taken inside of these functions in the LNA.', '1607.08685-1-23-2': 'Hence, these two approximations are equivalent for first-order reactions; they differ for second- and higher-order reactions.', '1607.08685-1-24-0': '### Bayesian update', '1607.08685-1-25-0': 'Let [MATH] be the Gaussian distribution whose mean vector [MATH] and covariance matrix [MATH] are obtained by solving the ODEs (Eqs. ([REF])-([REF]) for the LNA or Eqs. ([REF])-([REF]) for the GP) from time [MATH] to [MATH].', '1607.08685-1-25-1': "At time [MATH], the observation [MATH] is combined with [MATH] through Bayes' rule, leading to the posterior distribution of [MATH]: [EQUATION] where [MATH] is the likelihood function of [MATH], given as [EQUATION]", '1607.08685-1-25-2': 'Since both [MATH] and [MATH] are Gaussian distributions, [MATH] is also Gaussian, and its mean vector [MATH] and covariance matrix [MATH] are computed using the standard Kalman filter recursion as [EQUATION] where [EQUATION] is the Kalman gain [CITATION].', '1607.08685-1-26-0': 'The filtering algorithm is summarized in the following two steps:', '1607.08685-1-27-0': 'Solve the ODEs (Eqs. ([REF])-([REF]) for the LNA or Eqs. ([REF])-([REF]) for the GP) from time [MATH] to [MATH] with initial conditions [MATH] and [MATH], to obtain [MATH] and [MATH].', '1607.08685-1-28-0': 'Compute the posterior mean [MATH] and covariance matrix [MATH] at time [MATH] by Eqs. ([REF])-([REF]).', '1607.08685-1-29-0': 'Filtering is performed by executing these two steps recursively from time [MATH] to [MATH].', '1607.08685-1-29-1': 'The estimate of a state path, [MATH], is given by [EQUATION]', '1607.08685-1-30-0': '# Results', '1607.08685-1-31-0': 'We illustrate the two approximate filters based on the GP and the LNA on two reaction networks, and compare their filtering performances.', '1607.08685-1-32-0': '## Bistable system', '1607.08685-1-33-0': 'We first consider the following reaction network consisting of a single species [CITATION]: [EQUATION]', '1607.08685-1-33-1': 'The net effect matrix and the reaction rate vector, respectively, are given by [EQUATION] and [EQUATION]', '1607.08685-1-33-2': 'The rate equation ([REF]) for [MATH] is given by [EQUATION] where [MATH] is the potential: [EQUATION] with the recalled rate constants: [EQUATION]', '1607.08685-1-33-3': 'The parameter values were considered to be [MATH], [MATH], [MATH] and [MATH], with which the potential ([REF]) has two local minima (Figure [REF]a).', '1607.08685-1-33-4': 'The stochastic version of the reaction network with [MATH] was simulated using the SSA.', '1607.08685-1-33-5': 'A sample path is shown in Figure [REF]b (gray line) wherein we see that the reaction network exhibits stochastic switching between the two states that correspond to the two local minima of the potential.', '1607.08685-1-34-0': 'The LNA and the GP for this reaction network are obtained by substituting Eqs. ([REF])-([REF]) into Eqs. ([REF])-([REF]) for the LNA, and into ([REF])-([REF]) for the GP.', '1607.08685-1-34-1': 'To compare the performance of these methods, a numerical study was conducted using the following steps: First, the reaction network was simulated with the SSA in a time interval [MATH] to generate a sample path, [MATH] (Figure [REF]b, gray line).', '1607.08685-1-34-2': 'The observations, [MATH], were simulated using Eq. ([REF]), where the parameter values were taken to be [MATH] and [MATH].', '1607.08685-1-34-3': 'The variance of the observation noise, [MATH], ranged from [MATH] to [MATH] (Figure [REF]b; crosses represent the observations with [MATH]).', '1607.08685-1-34-4': 'The two approximate filters based on the LNA and the GP were then performed to estimate the simulated path from the observations.', '1607.08685-1-35-0': 'To quantify the extent to which the approximate filters estimate the true path, we computed the mean squared error (MSE) between the true and estimated paths: [EQUATION]', '1607.08685-1-35-1': 'Figure [REF] plots the MSE for the two approximate filters as a function of the noise variance of the observations.', '1607.08685-1-35-2': 'The difference in the MSE between the LNA and the GP is small, when the observation noise is small ([MATH]).', '1607.08685-1-35-3': 'Sample estimated paths for [MATH] are shown in Figure [REF]a wherein it is observed that the two estimated paths are very similar.', '1607.08685-1-35-4': 'The MSE for the LNA filter, however, increases abruptly as [MATH] is increased ([MATH]).', '1607.08685-1-35-5': 'Figure [REF]b depicts the sample paths estimated by the LNA and GP filters for [MATH]; as seen in this figure, while the GP filter can capture the jumps from one local equilibrium state to the other, the LNA filter fails, resulting in the large estimation error.', '1607.08685-1-36-0': '## Reaction network near a saddle-node bifurcation', '1607.08685-1-37-0': 'Next, we consider a reaction network consisting of two species, [MATH], which follow a set of seven reactions [CITATION]: [EQUATION]', '1607.08685-1-37-1': 'The net effect matrix and the reaction rate vector, respectively, are given by [EQUATION]', '1607.08685-1-37-2': 'The rate equation ([REF]) for [MATH] is derived as [EQUATION] where the reaction rate constants are rescaled as [EQUATION]', '1607.08685-1-37-3': 'The values of the rate constants are chosen as [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1607.08685-1-37-4': 'Figure [REF]a depicts the phase space wherein the nullclines of Eq. ([REF]), defined by [MATH] and [MATH], are plotted (dashed and dotted lines).', '1607.08685-1-37-5': 'These nullclines intersect at three points, denoted by SN (stable node), saddle, and UN (unstable node), in accordance with [CITATION].', '1607.08685-1-37-6': 'We also plot an illustrative path that starts close to the saddle, travels around the phase space, and then converges to the SN (Figure [REF]a; thick black line).', '1607.08685-1-37-7': 'Figure [REF]b depicts the time course of [MATH] corresponding to the sample path plotted in Figure [REF]a, showing that the reaction network exhibits "spiking" behavior.', '1607.08685-1-37-8': 'The stochastic version of the reaction network with [MATH] was simulated with the SSA.', '1607.08685-1-37-9': 'A sample path of the rescaled variable [MATH] plotted in Figure [REF] exhibits a stochastic oscillation.', '1607.08685-1-38-0': 'A numerical study for this reaction network was performed using the same procedure as for the bistable system.', '1607.08685-1-38-1': 'The duration of the simulation interval was chosen as [MATH].', '1607.08685-1-38-2': 'The parameters of the observation model ([REF]) were considered to be [MATH] and [MATH].', '1607.08685-1-38-3': 'The variance of the observation noise, [MATH], ranged from [MATH] to [MATH].', '1607.08685-1-39-0': 'Figure [REF] depicts the MSE between the true and estimated paths as a function of [MATH] for the GP (solid line) and for the LNA (dashed line).', '1607.08685-1-39-1': 'We see that the MSE for the GP is smaller than that for the LNA over the range of [MATH].', '1607.08685-1-39-2': 'To look in more detail, Figure [REF]a shows an illustrative sample path (gray line) together with its estimates (blue and red lines) wherein we see that the estimated paths could not capture particular spikes.', '1607.08685-1-39-3': 'To quantify the extent to which the approximate filters can capture spikes, a spike was defined if [MATH], and we counted the number of spikes in estimated paths across 20 repeated simulations.', '1607.08685-1-39-4': 'Figure [REF]b shows the ratio of the number of detected spikes in the estimated paths to that in the true paths.', '1607.08685-1-39-5': 'The rate of spike detection for the GP is greater than that for the LNA, and the latter deteriorates more than that of the former as the noise variance is increased.', '1607.08685-1-39-6': 'This result demonstrates that the approximate filter based on the GP can detect spikes more reliably than that based on the LNA for noisy observations.', '1607.08685-1-40-0': '# Discussion and Conclusion', '1607.08685-1-41-0': 'This study concerned the filtering problem for stochastic reaction networks.', '1607.08685-1-41-1': 'The difficulty in deriving filtering algorithms stems from the analytical intractability of the master equation.', '1607.08685-1-41-2': 'To derive the approximate filters, we applied two approximation methods: the LNA and the GP.', '1607.08685-1-42-0': 'The LNA, which involves the leading-order terms of the system-size expansion, has successfully been applied to the analysis of reaction networks [CITATION].', '1607.08685-1-42-1': 'The GP, on the other hand, is derived based on the orthogonal projection of the Fokker-Planck equation onto a Gaussian manifold with respect to the Fisher metric.', '1607.08685-1-42-2': 'To the best of our knowledge, this is the first time the GP has been applied to reaction networks.', '1607.08685-1-42-3': 'Since the dynamics of reaction networks are approximated by Gaussian processes in both methods, they are easily combined with the observations to derive the filtering algorithms (i.e., the Kalman filter).', '1607.08685-1-43-0': 'We compared two filters based on the GP and the LNA in terms of their filtering performance and showed using numerical simulations that the former outperforms the latter.', '1607.08685-1-43-1': 'In particular, the filtering performance of the LNA filter deteriorates more than that of the GP filter as the observation noise is increased; in other words, the GP filter is more robust than the LNA filter against observation noise.', '1607.08685-1-44-0': 'In general, the difference in approximations of the state process is insignificant when the observation noise is small [CITATION].', '1607.08685-1-44-1': "This can be understood intuitively using Bayes' rule ([REF]): when the observation noise is small, the likelihood is dominant over the state process (i.e., the prior) with the effect that the error caused by approximations in the state process is negligible.", '1607.08685-1-44-2': 'Therefore, approximate methods for the state process must be developed carefully when the observation noise is large.', '1607.08685-1-45-0': 'Note that in the projection method, the parametric family of probability distributions onto which the evolution of the reaction networks is projected is not restricted to Gaussian distributions.', '1607.08685-1-45-1': 'In principle, any probability distribution in exponential families fits this method [CITATION]; the choice of a suitable distribution depends on the underlying reaction networks.', '1607.08685-1-46-0': 'We considered the filtering problem wherein the objective is to estimate the state paths from the observations; another important problem is to infer the model parameters [CITATION].', '1607.08685-1-46-1': 'The LNA has been applied to develop algorithms for inferring the reaction rate constants as well [CITATION].', '1607.08685-1-46-2': 'The projection method must be applicable to develop approximate algorithms in the same manner.', '1607.08685-1-46-3': 'It would be interesting to examine the extent to which the projection method enhances the parameter estimation for reaction networks.'}

{'1607.08685-2-0-0': 'This study concerns online inference (i.e., filtering) on the state of reaction networks, conditioned on noisy and partial measurements.', '1607.08685-2-0-1': 'The difficulty in deriving the equation that the conditional probability distribution of the state satisfies stems from the fact that the master equation, which governs the evolution of the reaction networks, is analytically intractable.', '1607.08685-2-0-2': 'The linear noise approximation (LNA) technique, which is widely used in the analysis of reaction networks, has recently been applied to develop approximate inference.', '1607.08685-2-0-3': 'Here, we apply the projection method to derive approximate filters, and compare them to a filter based on the LNA numerically in their filtering performance.', '1607.08685-2-0-4': 'We also contrast the projection method with moment-closure techniques in terms of approximating the evolution of stochastic reaction networks.', '1607.08685-2-1-0': '# Introduction', '1607.08685-2-2-0': 'Stochastic reaction networks provide probabilistic descriptions of the evolution of interacting species.', '1607.08685-2-2-1': 'They are used for modeling phenomena in a wide range of disciplines; those species can represent molecules in chemical reactions [CITATION], animal species in ecology [CITATION], susceptibles and infectives in epidemic models [CITATION], and information packets in telecommunication networks [CITATION].', '1607.08685-2-3-0': 'The evolution of a network is modeled by a continuous-time Markov jump process, for which the probability distribution of the number of individuals of each species obeys the master equation [CITATION].', '1607.08685-2-3-1': 'Here, we consider a situation wherein only noisy and partial measurements of underlying reaction networks are available.', '1607.08685-2-3-2': 'Our objective is to infer the number of individuals of species from the observations obtained up to the current time.', '1607.08685-2-3-3': 'In the literature on signal processing, this problem is called filtering [CITATION].', '1607.08685-2-4-0': 'The filtering equation, which governs the posterior distribution conditioned on the observations, is not analytically obtainable due to the intractability of the master equation.', '1607.08685-2-4-1': 'It is possible to perform exact numerical simulation and obtain samples from the Markov jump processes using a stochastic simulation algorithm (SSA) [CITATION].', '1607.08685-2-4-2': 'Simulating many "particles" with the SSA and sampling the weighted particles in the favor of the observations, we could obtain samples from the posterior distribution.', '1607.08685-2-4-3': 'This technique is known as the sequential Monte Carlo method or particle filtering [CITATION].', '1607.08685-2-4-4': 'However, the SSA is often too slow.', '1607.08685-2-4-5': 'Moreover, particle filtering sufficiently requires many particles to obtain precise posterior expectations.', '1607.08685-2-4-6': 'Thus, particle filtering might not be efficient for performing online inference.', '1607.08685-2-5-0': 'An alternative approach is to consider the suitable approximations of the Markov jump processes.', '1607.08685-2-5-1': 'In the linear noise approximation (LNA), which is most widely used in such analysis, a Gaussian process whose mean obeys the deterministic rate equation approximates a Markov jump process [CITATION].', '1607.08685-2-5-2': 'The LNA is valid under the assumption that the number of individuals of a species is large [CITATION].', '1607.08685-2-5-3': 'It is also exact for all systems with affine propensities as well as for some systems with nonlinear propensities [CITATION].', '1607.08685-2-5-4': 'As the Gaussian process is tractable, The LNA allows us to derive an analytical expression of the approximate filtering equation [CITATION].', '1607.08685-2-5-5': 'In addition to the LNA, a number of approximation techniques have been proposed such as system-size expansions [CITATION], moment-closure approximations [CITATION] and conditional moment equations [CITATION], and have been applied to inference of model parameters [CITATION].', '1607.08685-2-6-0': 'In this study, we propose applying the projection method [CITATION] to derive approximate filters.', '1607.08685-2-6-1': 'In this method, the evolution of the probability distributions is constrained on a finite-dimensional family of densities through orthogonal projection onto the tangent space with respect to the Fisher metric.', '1607.08685-2-6-2': 'We derive the projection-based filter for stochastic reaction networks, and compare it to an approximate filter based on the LNA numerically in their filtering performance.', '1607.08685-2-6-3': 'We also contrast between the projection method and moment-closure techniques in terms of approximating the master equation.', '1607.08685-2-7-0': '# Method', '1607.08685-2-8-0': '## Reaction networks', '1607.08685-2-9-0': 'Throughout the study, the transpose of a matrix [MATH] is written [MATH].', '1607.08685-2-9-1': 'Let [MATH] be [MATH] species, and consider [MATH] reactions among these species described by [EQUATION] where [MATH] and [MATH] are stoichiometric coefficients of reactants and products, respectively, and [MATH] is the reaction rate constant.', '1607.08685-2-9-2': 'We denote by [MATH] the discrete composition vector whose [MATH]th component, [MATH], is the number of individuals of species [MATH].', '1607.08685-2-9-3': 'Let [MATH] be an [MATH] matrix, called the net effect matrix, whose [MATH] element, [MATH], is the change in the number of individuals of the [MATH]th species after one step of the [MATH]th reaction.', '1607.08685-2-9-4': 'Let [MATH] be the vector whose [MATH]th component, [MATH], is the rate of [MATH]th reaction, given as [EQUATION]', '1607.08685-2-9-5': 'From the Markov property, it follows that the probability distribution over [MATH] at time [MATH], [MATH], is governed by the master equation [CITATION]: [EQUATION]', '1607.08685-2-9-6': 'Stochastic processes described by Eq. ([REF]) are related to an ordinary differential equation (ODE), called the rate equation, via the thermodynamic limit.', '1607.08685-2-9-7': 'To see this, we introduce a scale factor [MATH] (typically taken to be "volume"), and rescale the composition vector and the reaction rate as [EQUATION]', '1607.08685-2-9-8': 'Accordingly, the reaction rate constants are rescaled as [EQUATION]', '1607.08685-2-9-9': 'With these rescaled parameters, it has been proved in [CITATION] that [MATH] as [MATH] in probability, where [MATH] satisfies the rate equation: [EQUATION]', '1607.08685-2-10-0': '## State space model and filtering', '1607.08685-2-11-0': 'We consider a situation wherein the system of interest is given by a stochastic reaction network, whose state is not directly observable, but instead, we have noisy and partial measurements at discrete time points [CITATION]; this situation is formulated within the framework of state space models.', '1607.08685-2-11-1': 'In state space modeling, the state process, [MATH], is given by the master equation ([REF]), and the measurement model is assumed to be [EQUATION] where [MATH], [MATH], and [MATH] is a [MATH]-dimensional Gaussian random variable with zero mean and covariance matrix [MATH].', '1607.08685-2-11-2': 'The goal of a filtering problem is to compute the posterior probability of the state [MATH] at time [MATH], when the observations [MATH] are given.', '1607.08685-2-12-0': '## Projection-based filter', '1607.08685-2-13-0': '### Projection method', '1607.08685-2-14-0': 'We apply the projection method proposed in [CITATION] to derive approximate filters.', '1607.08685-2-14-1': 'To apply the projection method, we need a Fokker-Planck equation derived from the master equation ([REF]).', '1607.08685-2-14-2': 'By taking up to the second-order terms in the Kramers-Moyal expansion of the master equation, a Fokker-Planck equation is obtained as [EQUATION] where [MATH] is the probability density of [MATH] at time [MATH] [CITATION].', '1607.08685-2-14-3': 'We apply the projection method to Eq. ([REF]).', '1607.08685-2-14-4': 'The key idea is to introduce a finite-dimensional family of probability densities [MATH], where [MATH] is the parameter characterizing the probability distributions, and to project the evolution of the probability density [MATH] onto the space of [MATH]; the resulting ODE for [MATH] approximates the master equation.', '1607.08685-2-15-0': 'Let [MATH] be a space of square-integrable functions, and consider the square roots of the probability densities, [MATH].', '1607.08685-2-15-1': 'The tangent space of [MATH] at [MATH] is given by [EQUATION]', '1607.08685-2-15-2': 'The [MATH] inner product of any two bases of [MATH] is defined as [EQUATION] where [MATH] is the Fisher information matrix.', '1607.08685-2-15-3': 'Then, the orthogonal projection of [MATH] onto [MATH] is given by [EQUATION] where [MATH] is the inverse of the Fisher information matrix.', '1607.08685-2-16-0': 'Using Eq. ([REF]), we project the Fokker-Plank equation ([REF]) onto [MATH] as follows: Using the chain rule, we obtain the equation for [MATH] as [EQUATION]', '1607.08685-2-16-1': 'Applying the orthogonal projection ([REF]) to Eq. ([REF]), we obtain an ODE for [MATH] as [EQUATION] where [MATH] is the expectation of [MATH] with respect to [MATH].', '1607.08685-2-16-2': 'We further assume that [MATH] is an exponential family of probability densities [CITATION]: [EQUATION] where [MATH] is the natural parameter, [MATH] is the sufficient statistic for [MATH] and [MATH] is the normalization factor.', '1607.08685-2-16-3': 'Substituting Eq. ([REF]) into Eq. ([REF]) leads to the projection approximation onto the exponential family: [EQUATION] where [MATH] is the backward diffusion operator: [EQUATION]', '1607.08685-2-17-0': '### Bayesian update', '1607.08685-2-18-0': 'Let [MATH] be the solution of Eq. ([REF]) at time [MATH].', '1607.08685-2-18-1': "At time [MATH], the observation [MATH] is combined with [MATH] through Bayes' rule, leading to the posterior probability density of [MATH]: [EQUATION] where [MATH] is the likelihood function of the observation model ([REF]).", '1607.08685-2-18-2': "If [MATH] is a conjugate family for [MATH], then the posterior probability density is in the same exponential family ([REF]): [EQUATION] where [MATH] is the parameter updated by Bayes' rule.", '1607.08685-2-19-0': 'The filtering algorithm is summarized in the following two steps:', '1607.08685-2-20-0': '(Prediction step) Solve the ODE ([REF]) from time [MATH] to [MATH] with initial conditions [MATH] to obtain [MATH].', '1607.08685-2-20-1': "(Correction step) Update the parameter [MATH] to [MATH] by Bayes' rule ([REF]).", '1607.08685-2-21-0': 'Filtering is performed by executing these two steps recursively from time [MATH] to [MATH].', '1607.08685-2-22-0': '## Choice of probability distributions', '1607.08685-2-23-0': 'We use two specific probability distributions for [MATH] to illustrate our method.', '1607.08685-2-24-0': '### Gaussian distribution', '1607.08685-2-25-0': 'Consider a multi-dimensional Gaussian distribution with mean vector [MATH] and covariance matrix [MATH]: [EQUATION]', '1607.08685-2-25-1': 'It is easily confirmed that the Gaussian distribution belongs to the exponential families ([REF]).', '1607.08685-2-25-2': 'The projection approximation ([REF]) is obtained as (see [REF]) [EQUATION] where [EQUATION] is the Jacobian matrix of [MATH].', '1607.08685-2-25-3': 'Note that Eqs. ([REF])-([REF]) are expressed with [MATH] instead of the natural parameter [MATH] of the exponential family.', '1607.08685-2-25-4': 'For systems with reactions of order three or higher, [MATH] contains polynomials in the variables of order three or higher, so that Eqs. ([REF])-([REF]) depend on moments of order three or larger; these moments can be computed with [MATH] and [MATH] due to the Gaussian assumption, and therefore Eqs. ([REF]) and ([REF]) are closed for such systems.', '1607.08685-2-25-5': 'We also point out that the Gaussian projection is equivalent to the normal moment-closure approximation (see [REF] for proof).', '1607.08685-2-26-0': 'Since both [MATH] and [MATH] in Eq. ([REF]) are Gaussian distributions, [MATH] is also Gaussian, and its mean vector [MATH] and covariance matrix [MATH] are computed using the standard Kalman filter recursion as [EQUATION] where [EQUATION] is the Kalman gain [CITATION].', '1607.08685-2-27-0': '### Quartic polynomial', '1607.08685-2-28-0': 'Another example is an exponential family of probability distributions with quartic polynomials in the exponent: [MATH] and [MATH].', '1607.08685-2-28-1': 'A characteristic of this exponential family is that it allows bimodality.', '1607.08685-2-28-2': 'We briefly summarize how to compute the Fisher information matrix [MATH] and the moments [MATH] that are required to solve the ODE ([REF]) (see [CITATION] for details).', '1607.08685-2-29-0': 'For [MATH], compute the following integral numerically: [EQUATION] and [MATH].', '1607.08685-2-30-0': 'Compute recursively the higher-order moments [MATH], [MATH] by [EQUATION]', '1607.08685-2-31-0': 'Compute the Fisher information matrix [MATH] where [EQUATION]', '1607.08685-2-31-1': "For this exponential family distribution, the parameter update through Bayes' rule ([REF]) becomes [EQUATION]", '1607.08685-2-32-0': '# Results', '1607.08685-2-33-0': 'We illustrate our method on two reaction networks, and compare it to an approximate filter based on the LNA in their filtering performances.', '1607.08685-2-33-1': 'The LNA-based filter is briefly summarized in [REF].', '1607.08685-2-33-2': 'Hereafter, we label the projection-based filter onto Gaussian distributions "GPF" and that onto quartic polynomial exponential distributions "QPF".', '1607.08685-2-34-0': '## Bistable system', '1607.08685-2-35-0': 'We first consider the following reaction network consisting of a single species [CITATION]: [EQUATION]', '1607.08685-2-35-1': 'The net effect matrix and the reaction rate vector, respectively, are given by [EQUATION] and [EQUATION]', '1607.08685-2-35-2': 'The rate equation ([REF]) for [MATH] is given by [EQUATION] where [MATH] is the potential: [EQUATION] with the rescaled rate constants: [EQUATION]', '1607.08685-2-35-3': 'The parameter values were considered to be [MATH], [MATH], [MATH] and [MATH], with which the potential ([REF]) has two local minima (Figure [REF]a).', '1607.08685-2-35-4': 'The stochastic version of the reaction network with [MATH] was simulated using the SSA.', '1607.08685-2-35-5': 'A sample path is shown in Figure [REF]b (gray line) wherein we see that the reaction network exhibits stochastic switching between the two states that correspond to the two local minima of the potential.', '1607.08685-2-36-0': 'For this reaction network, we applied the GPF, QPF and LNA.', '1607.08685-2-36-1': 'A numerical study was conducted using the following steps: First, the reaction network was simulated with the SSA in a time interval [MATH] to generate a sample path, [MATH] (Figure [REF]b, gray line).', '1607.08685-2-36-2': 'The observations, [MATH], were simulated using Eq. ([REF]), where we set [MATH].', '1607.08685-2-36-3': 'The inter-observation interval, [MATH], ranged from [MATH] to [MATH], and the variance of the observation noise, [MATH], ranged from [MATH] to [MATH] (Figure [REF]b; crosses represent the observations with [MATH] and [MATH]).', '1607.08685-2-36-4': 'The three approximate filters were then performed to estimate the simulated path from the observations.', '1607.08685-2-37-0': 'To quantify the extent to which the approximate filters estimate the true path, we employed a maximum a posteriori (MAP) estimate, [MATH], for each filter, and computed the mean squared error (MSE) between the true and estimated paths: [EQUATION]', '1607.08685-2-37-1': 'We plotted the MSE for the three approximate filters as a function of [MATH] (Figure [REF]a) and as a function of [MATH] (Figure [REF]b).', '1607.08685-2-37-2': 'The difference in the MSE among the three filters is small when [MATH] or [MATH] is small.', '1607.08685-2-37-3': 'The MSE for the LNA increases more than that for the GPF and QPF as [MATH] or [MATH] is increased.', '1607.08685-2-37-4': 'In particular, the MSE for the QPF remains relatively small over the range of [MATH] and [MATH].', '1607.08685-2-37-5': 'Figure [REF] depicts sample paths estimated by the three filters for [MATH] and [MATH]; as seen in this figure, while the QPF can capture the sharp transitions from one local equilibrium state to the other, the GPF and LNA fail, resulting in the large estimation error.', '1607.08685-2-37-6': 'These results suggest that for the reaction network with bistability, the QPF performs better that the GPF and LNA; the superiority of the QPF over the others stands out for noisy and sparse observations.', '1607.08685-2-38-0': '## Reaction network with limit cycle', '1607.08685-2-39-0': 'Next, we consider a reaction network consisting of three species, [MATH], which follow a set of five reactions [CITATION]: [EQUATION]', '1607.08685-2-39-1': 'The net effect matrix and the reaction rate vector, respectively, are given by [EQUATION]', '1607.08685-2-39-2': 'The rate equation ([REF]) is derived as [EQUATION] where the reaction rate constants are rescaled as [EQUATION]', '1607.08685-2-39-3': 'The values of the rate constants were chosen as [MATH], [MATH], [MATH], [MATH] and [MATH].', '1607.08685-2-39-4': 'Figure [REF] depicts the phase space [MATH] wherein an illustrative path of the rate equation is plotted (black line), showing that it converges to the limit cycle.', '1607.08685-2-39-5': 'The stochastic version of the reaction network with [MATH] was simulated with the SSA.', '1607.08685-2-39-6': 'A sample path of the rescaled variable [MATH] was also plotted in Figure [REF] (gray line).', '1607.08685-2-40-0': 'We applied the GPF and the LNA for this reaction network.', '1607.08685-2-40-1': 'A numerical study for this reaction network was performed using the same procedure as for the bistable system.', '1607.08685-2-40-2': 'The duration of the simulation interval was chosen as [MATH].', '1607.08685-2-40-3': 'The parameter of the observation model ([REF]) was considered to be [MATH].', '1607.08685-2-40-4': 'The inter-observation interval, [MATH], ranged from [MATH] to [MATH], and the variance of the observation noise, [MATH], ranged from [MATH] to [MATH].', '1607.08685-2-40-5': 'We plotted the MSE between the true and estimated paths as a function of [MATH] (Figure [REF]a) and as a function of [MATH] (Figure [REF]b) for the GPF (solid line) and for the LNA (dashed line).', '1607.08685-2-40-6': 'We see that the MSE for the GPF is smaller than that for the LNA.', '1607.08685-2-40-7': 'However, a very little difference in the MSE between these two methods is observed.', '1607.08685-2-41-0': '# Discussion', '1607.08685-2-42-0': 'In this section, we compared between the projection and moment-closure approximations.', '1607.08685-2-42-1': 'As seen in the section [REF] and [REF], the projection approximation onto Gaussian distributions is equivalent to the moment-closure approximation based on the same Gaussian distributions.', '1607.08685-2-42-2': 'However, the projection approximation does not always coincide with moment-closure approximations even if these share a common probability distribution.', '1607.08685-2-42-3': 'A difference between the two approximation techniques is that while moment-closures yield ODEs for the moments [MATH], the projection method produces ODEs for the natural parameter [MATH] of exponential family distributions, which is related to the expectation of the sufficient statistic [MATH] [CITATION].', '1607.08685-2-43-0': 'We illustrate this difference using a reaction network consisting of single species and at most bimolecular reactions: [EQUATION] and using gamma distributions for the base probability distributions.', '1607.08685-2-43-1': 'The probability density of a gamma distribution is given by [EQUATION] whose mean and variance are [MATH] and [MATH], respectively.', '1607.08685-2-43-2': 'Eq. ([REF]) can be rewritten in the form ([REF]) with the natural parameter [MATH] and the sufficient statistic [MATH].', '1607.08685-2-43-3': 'The expectations of [MATH] is expressed with [MATH] as [EQUATION] where [MATH] is the digamma function.', '1607.08685-2-43-4': 'The Fisher information matrix of the gamma distribution with respect to [MATH] is given by [EQUATION]', '1607.08685-2-43-5': 'Using these quantities, the projection approximation of the reaction network onto the gamma distributions is derived as [EQUATION]', '1607.08685-2-43-6': 'On the other hand, the moment-closure approximation based on the gamma distributions yields a set of ODEs for [MATH] and [MATH]: [EQUATION] where we used [MATH] to derive Eq. ([REF]).', '1607.08685-2-44-0': '# Conclusion', '1607.08685-2-45-0': 'This study concerned the filtering problem for stochastic reaction networks.', '1607.08685-2-45-1': 'The difficulty in deriving filtering algorithms stems from the analytical intractability of the master equation.', '1607.08685-2-45-2': 'We applied the projection method to derive approximate filters.', '1607.08685-2-46-0': 'The projection method provides a flexible framework for approximating reaction networks, as any probability distribution in exponential families fits this method.', '1607.08685-2-46-1': 'We demonstrated it on the two reaction networks.', '1607.08685-2-46-2': 'In particular, the projection-based filter with quartic polynomials exhibited much better performance than the other methods for the reaction system with bistability (Figure [REF]), due to its capability to accommodate bimodal distributions.', '1607.08685-2-47-0': 'We note that numerical methods based on particle filtering have been proposed for the inference of reaction networks [CITATION], which would be applicable for the considered molecule numbers.', '1607.08685-2-47-1': 'It would be interesting to compare the projection-based filter with these methods in terms of the balance between accuracy and computational time of estimation.', '1607.08685-2-48-0': 'We considered the filtering problem wherein the objective is to estimate the state paths from the observations obtained up to the current time; another related problem is smoothing, which aims to estimate the state paths from the whole observations [CITATION].', '1607.08685-2-48-1': 'The smoothing equation is not analytically tractable except in the case of linear Gaussian systems, hence approximate methods must be developed along the same line.', '1607.08685-2-49-0': 'It is also an important issue to infer the model parameters [CITATION].', '1607.08685-2-49-1': 'Methods for estimating the reaction rate constants have been developed using the LNA, the system-size expansion and moment-closure approximations [CITATION].', '1607.08685-2-49-2': 'In addition, it is difficult to distinguish between process and measurement noise; the simultaneous estimation of the noise parameters would render the problem substantially more challenging.', '1607.08685-2-49-3': 'We leave it for future research.'}

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[]

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['1607.08685-1-23-0', '1607.08685-1-26-0', '1607.08685-1-27-0', '1607.08685-1-28-0', '1607.08685-2-19-0', '1607.08685-2-29-0', '1607.08685-2-30-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1607.08685

quant-ph-9808015

{'quant-ph-9808015-1-0-0': "We try to obtain Born's principle as a result of a subquantum heat death, using classical [MATH]-theorem and the definition of a proper quantum [MATH]-theorem, within the framwork of Bohm's theory.", 'quant-ph-9808015-1-0-1': "We shall show the possibility of solving the problem of action-reaction asymmetry present in Bohm's theory and the arrow of time problem in our procedure.", 'quant-ph-9808015-1-1-0': '# Introduction', 'quant-ph-9808015-1-2-0': "Bohm's theory is a casual interpretation of quantum mechanics that was initially introduced by de Broglie[1] and then developed by David Bohm[2].", 'quant-ph-9808015-1-2-1': 'This theory is claimed to be equivalent to the standard quantum theory without having the conceptual problems of the latter[3].', 'quant-ph-9808015-1-2-2': 'Yet, there are some difficulties with this theory.', 'quant-ph-9808015-1-2-3': 'In this theory the velocity of a particle is given by [EQUATION] where [MATH] is the phase of the wave function ([MATH]).', 'quant-ph-9808015-1-2-4': 'The wave function [MATH] itself is a solution of Schrodinger equation: [EQUATION]', 'quant-ph-9808015-1-2-5': 'By substituting [MATH] for [MATH] in (2) , we shall have the following equations [EQUATION] where (3) is the classical Hamilton-Jacobi equation with an additional term, [MATH], which is called quantum potential.', 'quant-ph-9808015-1-2-6': "By differentiating (1) with respect to [MATH] and making use of (3), we obtain [EQUATION] which shows that the [MATH]-wave affects particle's motion through the [MATH] term.", 'quant-ph-9808015-1-2-7': "To secure the action-reaction (AR) symmetry, we expect the presence of a term corresponding to particle's reaction on the wave, in the wave equation of motion (2).", 'quant-ph-9808015-1-2-8': 'This term is not present there[4,5].', 'quant-ph-9808015-1-2-9': 'Some people argue that AR symmetry is a classical principle, and it is not necessarily applicable to quantum mechanics.', 'quant-ph-9808015-1-2-10': 'This is not, in our view, a strong argument.', 'quant-ph-9808015-1-2-11': 'Because, if one follows this line of thought, one might question the applicablicity of the concept of path (i. e. relation (1)) in Bohmian quantum mechanics.', 'quant-ph-9808015-1-2-12': 'Then, we have to stick to the standard formulation of quantum mechanics.', 'quant-ph-9808015-1-2-13': "Another difficulty with Bohm's theory is Born's statistical principle.", 'quant-ph-9808015-1-2-14': 'In this theory the field [MATH] enters as a guiding field for the motion of particles but at same time it is required by the experimental facts to represent a probability density through [MATH].', 'quant-ph-9808015-1-2-15': 'The question is why the probability for the ensemble of particles has to be equal to [MATH].', 'quant-ph-9808015-1-3-0': 'One of foundational problems in physics is that of temporal asymmetry or the existence of the arrow of time.', 'quant-ph-9808015-1-3-1': 'The laws of physics do not distinguish between the two directions of time.', 'quant-ph-9808015-1-3-2': 'In spite of this, there is a very obvious difference between the future and past directions of time in our universe.', 'quant-ph-9808015-1-3-3': 'This means that there is an arrow of time, i. e. time flows in one direction.', 'quant-ph-9808015-1-3-4': 'Recently Wootters claimed that: time asymmetry is logically prior to quantum mechanics and that:" the attempt to envision a sub-quantum theory suggests a picture of world in which time-asymmetric events are taken as fundamental.', 'quant-ph-9808015-1-3-5': 'Unitary evolution would appear as a derived concept and would of course not be universal"[6].', 'quant-ph-9808015-1-3-6': "Our approach exemplifies Wootters's suggestion.", 'quant-ph-9808015-1-4-0': "Our paper is organized as follows: After reviewing Valentini's quantum [MATH]-theorem in section 2, we introduce our new quantum [MATH]-theorem in section 3 and, finally, we show how the AR problem and the arrow of time problem are solved by our proposal.", 'quant-ph-9808015-1-5-0': '# quantum [MATH]-theorem', 'quant-ph-9808015-1-6-0': 'Recently, A.Valentini has tried to derive the relation [MATH] as the result of a statistical subquantum [MATH]-theorem[7].', 'quant-ph-9808015-1-6-1': "He looked for a proper quantum function [MATH] that, like the classical N-particle distribution function, satisfies Liouville's equation.", 'quant-ph-9808015-1-6-2': 'He considered an ensemble of N-body systems which could be described by the wave function [MATH] and the distribution function [MATH].', 'quant-ph-9808015-1-6-3': 'Because [MATH] and [MATH] must equalize during the assumed heat death, in general [MATH] and one can write [EQUATION] where [MATH] measures the ratio of [MATH] to [MATH] at the point [MATH]), at time [MATH].', 'quant-ph-9808015-1-6-4': 'Since both [MATH] and [MATH] satisfy the continuity equation ([MATH] due to its being a solution of the Schrodinger equation and [MATH] by its very definition) one can easily show that [MATH] satisfies the following equation [EQUATION] where [MATH] denotes [MATH] as before.', 'quant-ph-9808015-1-6-5': 'Thus, Valentini defined his quantum [MATH]-function in the following way [EQUATION]', 'quant-ph-9808015-1-6-6': 'The only difference with the classical one is that [MATH] is defined in the configuration space while the classical [MATH] is defined in the phase space and [MATH].', 'quant-ph-9808015-1-6-7': "Valentini used Ehrenfest's coarse-graining method[8].", 'quant-ph-9808015-1-6-8': 'He claims that [MATH][MATH], where [MATH] is the coarse-graind [MATH]-function and the equality holds in the equilibrium state, where [MATH] or [MATH].', 'quant-ph-9808015-1-6-9': 'Here [MATH] and [MATH] are coarse grained forms of [MATH] and [MATH] respectivily.', 'quant-ph-9808015-1-6-10': 'Valentini termed this process a subquantumic heath death.', 'quant-ph-9808015-1-6-11': 'Then, he showed that if a single particle is extracted from a large system and prepared in a state with a wavefunction [MATH], its probability density [MATH] will be equal to [MATH], provided that [MATH] holds for the large system.', 'quant-ph-9808015-1-6-12': 'Notice that a one-body system not in quantum equilibrium can never relax to quantum equilibrium (when it is left to itself).', 'quant-ph-9808015-1-6-13': 'But any one-body system extracted from a large system can do.', 'quant-ph-9808015-1-6-14': "Here, we want to modify Valentini's procedure so that one can directly obtain Born's principle for a one-body system.", 'quant-ph-9808015-1-6-15': "In this case we will be forced to use Boltzmann's procedure, and that naturally leads to a solution of the AR problem.", 'quant-ph-9808015-1-7-0': '# An alternative quantum [MATH]-theorem', 'quant-ph-9808015-1-8-0': 'Consider an ensemble of one-body systems.', 'quant-ph-9808015-1-8-1': 'Suppose that all these systems are in the [MATH] state and that their distribution function is [MATH].', 'quant-ph-9808015-1-8-2': 'Furthermore, suppose that at [MATH] we have [MATH], and define [EQUATION]', 'quant-ph-9808015-1-8-3': "In Valentini's procedure the complexity of systems leads to heat death , but our systems are simple (one-body) ones.", 'quant-ph-9808015-1-8-4': 'Thus, if we want to have heat death, we must assume that [MATH] satisfies a quantum Boltzmann equation [EQUATION] where [MATH] is related to the particle reaction on its associated wave.', 'quant-ph-9808015-1-8-5': 'Since [MATH] satisfies a continuity equation, as the result of its definition, thus (5) and (6) imply that [MATH] does not satisfy the continuity equation any more.', 'quant-ph-9808015-1-8-6': 'In fact, the continuity equation for [MATH] is changed to [EQUATION]', 'quant-ph-9808015-1-8-7': 'This means that [MATH] is a soluation of a nonlinear Schrodinger equation.', 'quant-ph-9808015-1-8-8': 'If we want to have the quantum potential in its regular form, i.e. [MATH], we must choose the nonlinear term in a particular form.', 'quant-ph-9808015-1-8-9': 'The proper selection is [EQUATION] where [MATH] is a real function of [MATH].', 'quant-ph-9808015-1-8-10': 'With the substitution [MATH] we shall have [EQUATION]', 'quant-ph-9808015-1-8-11': 'By comparing (10) with (7) we have [MATH].', 'quant-ph-9808015-1-8-12': 'Now, we have to select [MATH] in such a way that it leads to the equality of [MATH] and [MATH] (i.e. [MATH]).', 'quant-ph-9808015-1-8-13': 'Some requirements for such a function is:', 'quant-ph-9808015-1-9-0': '1- It must be invariant under [MATH].', 'quant-ph-9808015-1-10-0': '2- It must change its sign for [MATH].', 'quant-ph-9808015-1-11-0': 'If we fined systems for which subquantum heat death has not occured[9], we shall obtain the actual form of the [MATH] function.', 'quant-ph-9808015-1-11-1': 'A proper selection is [MATH] where [MATH] is a constant.', 'quant-ph-9808015-1-11-2': 'Then, (10) gives (with [MATH]) [EQUATION]', 'quant-ph-9808015-1-11-3': 'Now, consider the right hand side of (11) as the source of [MATH] field.', 'quant-ph-9808015-1-11-4': 'At any point of space where [MATH] (i.e. [MATH] 1), the source of [MATH] is positive and, therefore, [MATH] increases at that point.', 'quant-ph-9808015-1-11-5': 'On the other hand, at any point of space where [MATH] (i.e. [MATH] 1), the source of [MATH] is negative and therefore [MATH] decreases at that point.', 'quant-ph-9808015-1-11-6': 'The variation of [MATH] continues until [MATH] becomes equal to [MATH].', 'quant-ph-9808015-1-11-7': 'After that, since both [MATH] and [MATH] evolve under the same velocity field ([MATH]), they remain equal.', 'quant-ph-9808015-1-11-8': 'To prove [MATH] exactly, we introduce the following [MATH] function: [EQUATION]', 'quant-ph-9808015-1-11-9': 'Since [MATH][MATH][MATH] for all [MATH], we have [MATH] - the equality being relevant to the case [MATH].', 'quant-ph-9808015-1-11-10': 'If we show that for the forgoing [MATH] one has [MATH][MATH], where again the equality is to relevant to [MATH] (i.e. [MATH]) state, we have shown that [MATH] becomes equal to [MATH] finally.', 'quant-ph-9808015-1-11-11': 'We write (12) in the form [EQUATION] where [MATH].', 'quant-ph-9808015-1-11-12': 'Now, we have for [MATH] [EQUATION] where we have done an integration by parts.', 'quant-ph-9808015-1-11-13': 'Passing to the limit of large volumes and dropping the surface term in (13) leads to [EQUATION]', 'quant-ph-9808015-1-11-14': 'The quantity in [MATH], is negative for [MATH] and positive for [MATH].', 'quant-ph-9808015-1-11-15': 'Now, since the [MATH] (i.e. [MATH]) is positive for [MATH] and negative for [MATH], the integrand is negative or zero for all values of [MATH] and we have [EQUATION] where the equality holds for [MATH].', 'quant-ph-9808015-1-11-16': "The existance of a quantity that decreases continuously to its minimum value at [MATH] (i.e. [MATH]) guarantees Born's principle.", 'quant-ph-9808015-1-12-0': '# The action-reaction problem', 'quant-ph-9808015-1-13-0': "In the classical gravity, matter fixes space-time geometry and correspondingly matter's motion is determined by the space-time geometry.", 'quant-ph-9808015-1-13-1': 'This means that matter and space-time affect each other.', 'quant-ph-9808015-1-13-2': 'Thus, the action-reaction symmetry is preserved.', 'quant-ph-9808015-1-13-3': 'In fact, the nonlinearity of Einstein equations is the result of this mutual action-reaction.', 'quant-ph-9808015-1-13-4': "In the same way, mutual action-reaction between wave and particle in Bohm's theory, leads to a nonlinear Schrodinger equation.", 'quant-ph-9808015-1-13-5': 'But nonlinearity does not necessarily mean particle reaction on the wave.', 'quant-ph-9808015-1-13-6': "Indeed nonlinear terms must contain some information about particle's position too.", 'quant-ph-9808015-1-13-7': "We claim that the non-linear term present in (8) arises from particle's reaction on its associated wave, and it represents information about particle's position.", 'quant-ph-9808015-1-13-8': "Considering the statistical nature of [MATH], the question arises as to why particle's associated wave is affected by an ensemble of particles.", 'quant-ph-9808015-1-13-9': 'This can be answered in the following ways:', 'quant-ph-9808015-1-14-0': "1- It is natural to expect particle's associated wave to be a function of particle's position, satisfing a non-linear equation of the following form: [EQUATION] where [MATH] represents particle's position, and the subscript [MATH] on the operator [MATH] is an indication of the non-linearity of the equation with respect to [MATH].", 'quant-ph-9808015-1-14-1': 'Consider an ensemble of particles with the same [MATH], but with different [MATH], distributed according to [MATH].', 'quant-ph-9808015-1-14-2': 'One can look a [MATH], giving the average motion of the ensemble of particles instead of individual motions and satisfying the following equation: [EQUATION]', 'quant-ph-9808015-1-14-3': 'This equation is of type (8).', 'quant-ph-9808015-1-14-4': 'It is not a linear conbination of functions [MATH].', 'quant-ph-9808015-1-14-5': 'Rather, it is an average field that gives the evolution of the ensemble correctly.', 'quant-ph-9808015-1-14-6': 'Of course, it may not give the correct path of individual particles.', 'quant-ph-9808015-1-14-7': "Note, [MATH] in (8) is the reaction of the ensemble of particles on the wave that represents ensemble's evolution.", 'quant-ph-9808015-1-14-8': 'After quantum heat death and the equality of [MATH] with [MATH], this equation takes the form of the Schrodinger equation: [EQUATION]', 'quant-ph-9808015-1-14-9': "This means that the AR problem is a result of the establishment of Born's principle.", 'quant-ph-9808015-1-14-10': 'In fact this is similar to the argument that Valentini presents[10] to show that the signal-locality (i.e. the absence of practical instantaneous signalling) and the uncertainty principle are valid if and only if [MATH].', 'quant-ph-9808015-1-15-0': '(2) The information of a wave about particles position could be incomplete and this defect could be a result of the form of the interaction between the two.', 'quant-ph-9808015-1-15-1': 'Thus, [MATH] does not represent a statistical distribution for an ensemble of particles.', 'quant-ph-9808015-1-15-2': "Rather, it represents the maximum amount of informatiom of particle's associated wave about particle.", 'quant-ph-9808015-1-15-3': 'Then, if we have an ensemble of particles having a same [MATH] and [MATH] we can take [MATH] as the distribution function for the ensemble.', 'quant-ph-9808015-1-16-0': '# Arrow of time problem', 'quant-ph-9808015-1-17-0': 'The Arrow of time problem can essentially be seen as arising from the different behavior of the time asymmetric evolution of macroscopic irreversible processes and the time symmetric evolution of the reversible processes governing the underlying dynamics of the atomic constituents.', 'quant-ph-9808015-1-17-1': 'The apparent paradox is a direct consequence of the symmetric nature of the basic laws of physics with respect to time inversion.', 'quant-ph-9808015-1-18-0': 'The nonlinear Schrodinger equation that we introduced is not time reversal, although after the subquantum heat death with the vanishing of the [MATH] term, it seems time reversal.', 'quant-ph-9808015-1-18-1': 'In other words, the equation (8) for one of the two directions of time leads to the common Schrodinger equation that describes real microscopic world, but for the other direction of time, the nonlinear term [MATH] grows up and (8) could not describe the real world.', 'quant-ph-9808015-1-18-2': 'Now, if [MATH] is a solution of the common Schrodinger equation, so is [MATH], and both [MATH] and [MATH] are solutions of our nonlinear equation in the subquantum equilibrium state.', 'quant-ph-9808015-1-18-3': 'But in the framework of our theory [MATH] and [MATH] are not time reversal of each other.', 'quant-ph-9808015-1-18-4': 'Nevertheless, we can formally relate them to each other with a time reversal transformation (ignoring the nonlinear term in subquantum equilibrium state).', 'quant-ph-9808015-1-19-0': '# conclusion', 'quant-ph-9808015-1-20-0': "We have shown that the Born's principle can be the result of the presence of a nonlinear term in Schrodinger equation, with special characteristics.", 'quant-ph-9808015-1-20-1': "Then, we have shown that this nonlinear term can be considered as indicator of particle's reaction on the wave.", 'quant-ph-9808015-1-20-2': 'Besides, our nonlinear equation introduces an arrow of time.'}

{'quant-ph-9808015-2-0-0': "We try to obtain Born's principle as a result of a subquantum heat death, using classical [MATH]-theorem and the definition of a proper quantum [MATH]-theorem, within the framwork of Bohm's theory.", 'quant-ph-9808015-2-0-1': "We shall show the possibility of solving the problem of action-reaction asymmetry present in Bohm's theory and the arrow of time problem in our procedure.", 'quant-ph-9808015-2-1-0': "Key words: Born's principle, Bohmian mechanics, action-reaction asymmetry, arrow of time, quantum [MATH]-theorem", 'quant-ph-9808015-2-2-0': '1.', 'quant-ph-9808015-2-2-1': 'INTRODUCTION', 'quant-ph-9808015-2-3-0': "Bohm's theory is a casual interpretation of quantum mechanics that was initially introduced by de Broglie[1] and then developed by David Bohm[2].", 'quant-ph-9808015-2-3-1': 'This theory is claimed to be equivalent to the standard quantum theory without having the conceptual problems of the latter[3].', 'quant-ph-9808015-2-3-2': 'Yet, there are some difficulties with this theory.', 'quant-ph-9808015-2-3-3': 'In this theory the velocity of a particle is given by [EQUATION] where [MATH] is the phase of the wave function ([MATH]).', 'quant-ph-9808015-2-3-4': 'The wave function [MATH] itself is a solution of Schrodinger equation: [EQUATION]', 'quant-ph-9808015-2-3-5': 'By substituting [MATH] for [MATH] in (2) , we shall have the following equations [EQUATION] where (3) is the classical Hamilton-Jacobi equation with an additional term, [MATH], which is called quantum potential.', 'quant-ph-9808015-2-3-6': "By differentiating (1) with respect to [MATH] and making use of (3), we obtain [EQUATION] which shows that the [MATH]-wave affects particle's motion through the [MATH] term.", 'quant-ph-9808015-2-3-7': "To secure the action-reaction (AR) symmetry, we expect the presence of a term corresponding to particle's reaction on the wave, in the wave equation of motion (2).", 'quant-ph-9808015-2-3-8': 'This term is not present there[4,5].', 'quant-ph-9808015-2-3-9': 'Some people argue that AR symmetry is a classical principle, and it is not necessarily applicable to quantum mechanics.', 'quant-ph-9808015-2-3-10': 'This is not, in our view, a strong argument.', 'quant-ph-9808015-2-3-11': 'Because, if one follows this line of thought, one might question the applicablicity of the concept of path (i. e. relation (1)) in Bohmian quantum mechanics.', 'quant-ph-9808015-2-3-12': 'Then, we have to stick to the standard formulation of quantum mechanics.', 'quant-ph-9808015-2-3-13': "Another difficulty with Bohm's theory is Born's statistical principle.", 'quant-ph-9808015-2-3-14': 'In this theory the field [MATH] enters as a guiding field for the motion of particles but at same time it is required by the experimental facts to represent a probability density through [MATH].', 'quant-ph-9808015-2-3-15': 'The question is why the probability for the ensemble of particles has to be equal to [MATH].', 'quant-ph-9808015-2-3-16': 'One of foundational problems in physics is that of temporal asymmetry or the existence of the arrow of time.', 'quant-ph-9808015-2-3-17': 'The laws of physics do not distinguish between the two directions of time.', 'quant-ph-9808015-2-3-18': 'In spite of this, there is a very obvious difference between the future and past directions of time in our universe.', 'quant-ph-9808015-2-3-19': 'This means that there is an arrow of time, i. e. time flows in one direction.', 'quant-ph-9808015-2-3-20': 'Recently Wootters claimed that: time asymmetry is logically prior to quantum mechanics and that:" the attempt to envision a sub-quantum theory suggests a picture of world in which time-asymmetric events are taken as fundamental.', 'quant-ph-9808015-2-3-21': 'Unitary evolution would appear as a derived concept and would of course not be universal "[6].', 'quant-ph-9808015-2-3-22': "Our approach exemplifies Wootters's suggestion.", 'quant-ph-9808015-2-4-0': "Our paper is organized as follows: After reviewing Valentini's quantum [MATH]-theorem in section 2, we introduce our new quantum [MATH]-theorem in section 3 and, finally, we show how the AR problem and the arrow of time problem are solved by our proposal.", 'quant-ph-9808015-2-5-0': '2.', 'quant-ph-9808015-2-5-1': 'QUANTUM [MATH]-THEOREM', 'quant-ph-9808015-2-6-0': 'Recently, A.Valentini has tried to derive the relation [MATH] as the result of a statistical subquantum [MATH]-theorem[7].', 'quant-ph-9808015-2-6-1': "He looked for a proper quantum function [MATH] that, like the classical N-particle distribution function, satisfies Liouville's equation.", 'quant-ph-9808015-2-6-2': 'He considered an ensemble of N-body systems which could be described by the wave function [MATH] and the distribution function [MATH].', 'quant-ph-9808015-2-6-3': 'Because [MATH] and [MATH] must equalize during the assumed heat death, in general [MATH] and one can write [EQUATION] where [MATH] measures the ratio of [MATH] to [MATH] at the point [MATH]), at time [MATH].', 'quant-ph-9808015-2-6-4': 'Since both [MATH] and [MATH] satisfy the continuity equation ([MATH] due to its being a solution of the Schrodinger equation and [MATH] by its very definition) one can easily show that [MATH] satisfies the following equation [EQUATION] where [MATH] denotes [MATH] as before.', 'quant-ph-9808015-2-6-5': 'Thus, Valentini defined his quantum [MATH]-function in the following way [EQUATION]', 'quant-ph-9808015-2-6-6': 'The only difference with the classical one is that [MATH] is defined in the configuration space while the classical [MATH] is defined in the phase space and [MATH].', 'quant-ph-9808015-2-6-7': "Valentini used Ehrenfest's coarse-graining method[8].", 'quant-ph-9808015-2-6-8': 'He claims that [MATH][MATH], where [MATH] is the coarse-graind [MATH]-function and the equality holds in the equilibrium state, where [MATH] or [MATH].', 'quant-ph-9808015-2-6-9': 'Here [MATH] and [MATH] are coarse grained forms of [MATH] and [MATH] respectivily.', 'quant-ph-9808015-2-6-10': 'Valentini termed this process a subquantumic heath death.', 'quant-ph-9808015-2-6-11': 'Then, he showed that if a single particle is extracted from a large system and prepared in a state with a wavefunction [MATH], its probability density [MATH] will be equal to [MATH], provided that [MATH] holds for the large system.', 'quant-ph-9808015-2-6-12': 'Notice that a one-body system not in quantum equilibrium can never relax to quantum equilibrium (when it is left to itself).', 'quant-ph-9808015-2-6-13': 'But any one-body system extracted from a large system can do.', 'quant-ph-9808015-2-6-14': "Here, we want to modify Valentini's procedure so that one can directly obtain Born's principle for a one-body system.", 'quant-ph-9808015-2-6-15': "In this case we will be forced to use Boltzmann's procedure, and that naturally leads to a solution of the AR problem.", 'quant-ph-9808015-2-7-0': '3.', 'quant-ph-9808015-2-7-1': 'AN ALTERNATIVE QUANTUM [MATH]-THEOREM', 'quant-ph-9808015-2-8-0': 'Consider an ensemble of one-body systems.', 'quant-ph-9808015-2-8-1': 'Suppose that all these systems are in the [MATH] state and that their distribution function is [MATH].', 'quant-ph-9808015-2-8-2': 'Furthermore, suppose that at [MATH] we have [MATH], and define [EQUATION]', 'quant-ph-9808015-2-8-3': "In Valentini's procedure the complexity of systems leads to heat death, but our systems are simple (one-body) ones.", 'quant-ph-9808015-2-8-4': 'Thus, if we want to have heat death, we must assume that [MATH] satisfies a quantum Boltzmann equation [EQUATION] where [MATH] is related to the particle reaction on its associated wave.', 'quant-ph-9808015-2-8-5': 'Since [MATH] satisfies a continuity equation, as the result of its definition, thus (5) and (6) imply that [MATH] does not satisfy the continuity equation any more.', 'quant-ph-9808015-2-8-6': 'In fact, the continuity equation for [MATH] is changed to [EQUATION]', 'quant-ph-9808015-2-8-7': 'This means that [MATH] is a soluation of a nonlinear Schrodinger equation.', 'quant-ph-9808015-2-8-8': 'If we want to have the quantum potential in its regular form, i.e. [MATH], we must choose the nonlinear term in a particular form.', 'quant-ph-9808015-2-8-9': 'The proper selection is [EQUATION] where [MATH] is a real function of [MATH].', 'quant-ph-9808015-2-8-10': 'With the substitution [MATH] we shall have [EQUATION]', 'quant-ph-9808015-2-8-11': 'By comparing (10) with (7) we have [MATH].', 'quant-ph-9808015-2-8-12': 'Now, we have to select [MATH] in such a way that it leads to the equality of [MATH] and [MATH] (i.e. [MATH]).', 'quant-ph-9808015-2-8-13': 'Some requirements for such a function is:', 'quant-ph-9808015-2-9-0': '1- It must be invariant under [MATH].', 'quant-ph-9808015-2-10-0': '2- It must change its sign for [MATH].', 'quant-ph-9808015-2-11-0': 'If we fined systems for which subquantum heat death has not occured[9], we shall obtain the actual form of the [MATH] function.', 'quant-ph-9808015-2-11-1': 'A proper selection is [MATH] where [MATH] is a constant.', 'quant-ph-9808015-2-11-2': 'Then, (10) gives (with[MATH]) [EQUATION]', 'quant-ph-9808015-2-11-3': 'Now, consider the right hand side of (11) as the source of [MATH] field.', 'quant-ph-9808015-2-11-4': 'At any point of space where [MATH] (i.e. [MATH] 1), the source of [MATH] is positive and, therefore, [MATH] increases at that point.', 'quant-ph-9808015-2-11-5': 'On the other hand, at any point of space where [MATH] (i.e.[MATH] 1), the source of [MATH] is negative and therefore [MATH] decreases at that point.', 'quant-ph-9808015-2-11-6': 'The variation of [MATH] continues until [MATH] becomes equal to [MATH].', 'quant-ph-9808015-2-11-7': 'After that, since both [MATH] and [MATH] evolve under the same velocity field ([MATH]), they remain equal.', 'quant-ph-9808015-2-11-8': 'To prove [MATH] exactly, we introduce the following [MATH] function: [EQUATION]', 'quant-ph-9808015-2-11-9': 'Since [MATH][MATH][MATH] for all [MATH], we have [MATH] - the equality being relevant to the case [MATH].', 'quant-ph-9808015-2-11-10': 'If we show that for the forgoing [MATH] one has [MATH][MATH], where again the equality is to relevant to [MATH] (i.e. [MATH]) state, we have shown that [MATH] becomes equal to [MATH] finally.', 'quant-ph-9808015-2-11-11': 'We write (12) in the form [EQUATION] where [MATH].', 'quant-ph-9808015-2-11-12': 'Now, we have for [MATH] [EQUATION] where we have done an integration by parts.', 'quant-ph-9808015-2-11-13': 'Passing to the limit of large volumes and dropping the surface term in (13) leads to [EQUATION]', 'quant-ph-9808015-2-11-14': 'The quantity in [MATH], is negative for [MATH] and positive for [MATH].', 'quant-ph-9808015-2-11-15': 'Now, since the [MATH] (i.e. [MATH]) is positive for [MATH] and negative for [MATH], the integrand is negative or zero for all values of [MATH] and we have [EQUATION] where the equality holds for [MATH].', 'quant-ph-9808015-2-11-16': "The existance of a quantity that decreases continuously to its minimum value at [MATH] (i.e. [MATH]) guarantees Born's principle.", 'quant-ph-9808015-2-12-0': '4.', 'quant-ph-9808015-2-12-1': 'THE ACTION-REACTION PROBLEM', 'quant-ph-9808015-2-13-0': "In the classical gravity, matter fixes space-time geometry and correspondingly matter's motion is determined by the space-time Thus, the action-reaction symmetry is preserved.", 'quant-ph-9808015-2-13-1': 'In fact, the nonlinearity of Einstein equations is the result of this mutual action-reaction.', 'quant-ph-9808015-2-13-2': "In the same way, mutual action-reaction between wave and particle in Bohm's theory, leads to a nonlinear Schrodinger equation.", 'quant-ph-9808015-2-13-3': 'But nonlinearity does not necessarily mean particle reaction on the wave.', 'quant-ph-9808015-2-13-4': "Indeed nonlinear terms must contain some information about particle's position too.", 'quant-ph-9808015-2-13-5': "We claim that the non-linear term present in (8) arises from particle's reaction on its associated wave, and it represents information about particle's position.", 'quant-ph-9808015-2-13-6': "Considering the statistical nature of [MATH], the question arises as to why particle's associated wave is affected by an ensemble of particles.", 'quant-ph-9808015-2-13-7': 'This can be answered in the following ways:', 'quant-ph-9808015-2-14-0': "1- It is natural to expect particle's associated wave to be a function of particle's position, satisfing a non-linear equation of the following form: [EQUATION] where [MATH] represents particle's position, and the subscript [MATH] on the operator [MATH] is an indication of the non-linearity of the equation with respect to [MATH].", 'quant-ph-9808015-2-14-1': 'Consider an ensemble of particles with the same [MATH], but with different [MATH], distributed according to [MATH].', 'quant-ph-9808015-2-14-2': 'One can look a [MATH], giving the average motion of the ensemble of particles instead of individual motions and satisfying the following equation: [EQUATION]', 'quant-ph-9808015-2-14-3': 'This equation is of type (8).', 'quant-ph-9808015-2-14-4': 'The [MATH] is not a linear conbination of functions [MATH].', 'quant-ph-9808015-2-14-5': 'Rather, it is an average field that gives the evolution of the ensemble correctly.', 'quant-ph-9808015-2-14-6': 'Of course, it may not give the correct path of individual particles.', 'quant-ph-9808015-2-14-7': "Note, [MATH] in (8) is the reaction of the ensemble of particles on the wave that represents ensemble's evolution.", 'quant-ph-9808015-2-14-8': 'After quantum heat death and the equality of [MATH] with [MATH], this equation takes the form of the Schrodinger equation: [EQUATION]', 'quant-ph-9808015-2-14-9': "This means that the AR problem is a result of the establishment of Born's principle.", 'quant-ph-9808015-2-14-10': 'In fact this is similar to the argument that Valentini presents[10] to show that the signal-locality (i.e. the absence of practical instantaneous signalling) and the uncertainty principle are valid if and only if [MATH].', 'quant-ph-9808015-2-15-0': '2- The information of a wave about particles position could be incomplete and this defect could be a result of the form of the interaction between the two.', 'quant-ph-9808015-2-15-1': 'Thus, [MATH] does not represent a statistical distribution for an ensemble of particles.', 'quant-ph-9808015-2-15-2': "Rather, it represents the maximum amount of informatiom of particle's associated wave about particle.", 'quant-ph-9808015-2-15-3': 'Then, if we have an ensemble of particles having a same [MATH] and [MATH] we can take [MATH] as the distribution function for the ensemble.', 'quant-ph-9808015-2-16-0': '5.', 'quant-ph-9808015-2-16-1': 'ARROW OF TIME PROBLEM', 'quant-ph-9808015-2-17-0': 'The Arrow of time problem can essentially be seen as arising from the different behavior of the time asymmetric evolution of macroscopic irreversible processes and the time symmetric evolution of the reversible processes governing the underlying dynamics of the atomic constituents.', 'quant-ph-9808015-2-17-1': 'The apparent paradox is a direct consequence of the symmetric nature of the basic laws of physics with respect to time inversion.', 'quant-ph-9808015-2-17-2': 'The nonlinear Schrodinger equation that we introduced is not time reversal, although after the subquantum heat death with the vanishing of the [MATH] term, it seems time reversal.', 'quant-ph-9808015-2-17-3': 'In other words, the equation (8) for one of the two directions of time leads to the common Schrodinger equation that describes real microscopic world, but for the other direction of time, the nonlinear term [MATH] grows up and (8) could not describe the real world.', 'quant-ph-9808015-2-17-4': 'Now, if [MATH] is a solution of the common Schrodinger equation, so is [MATH], and both [MATH] and [MATH] are solutions of our nonlinear equation in the subquantum equilibrium state.', 'quant-ph-9808015-2-17-5': 'But in the framework of our theory [MATH] and [MATH] are not time reversal of each other.', 'quant-ph-9808015-2-17-6': 'Nevertheless, we can formally relate them to each other with a time reversal transformation (ignoring the nonlinear term in subquantum equilibrium state).', 'quant-ph-9808015-2-18-0': '6.', 'quant-ph-9808015-2-18-1': 'CONCLUSIONS', 'quant-ph-9808015-2-19-0': "We have shown that the Born's principle can be the result of the presence of a nonlinear term in Schrodinger equation, with special characteristics.", 'quant-ph-9808015-2-19-1': "Then, we have shown that this nonlinear term can be considered as indicator of particle's reaction on the wave.", 'quant-ph-9808015-2-19-2': 'Besides, our nonlinear equation introduces an arrow of time.', 'quant-ph-9808015-2-20-0': 'ACKNOWLEDGMENTS', 'quant-ph-9808015-2-21-0': 'The authors would like to thank Prof. J. T. Cushing for useful comments on an earlier draft of this paper.', 'quant-ph-9808015-2-22-0': 'REFERENCE', 'quant-ph-9808015-2-23-0': '1.', 'quant-ph-9808015-2-23-1': 'L. de Broglie, J.Phys.', 'quant-ph-9808015-2-23-2': '(Paris) 6, 225 (1927).', 'quant-ph-9808015-2-24-0': '2.', 'quant-ph-9808015-2-24-1': 'D. Bohm, Phys.', 'quant-ph-9808015-2-24-2': 'Rev. 85, 166-179, 180-193 (1952).', 'quant-ph-9808015-2-25-0': '3.', 'quant-ph-9808015-2-25-1': 'P. Holland.', 'quant-ph-9808015-2-25-2': 'The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993).', 'quant-ph-9808015-2-26-0': '4.', 'quant-ph-9808015-2-26-1': 'J. Anandan and H. R. Brown, Found.', 'quant-ph-9808015-2-26-2': 'Phys.', 'quant-ph-9808015-2-26-3': '25, 349 (1995).', 'quant-ph-9808015-2-27-0': '5.', 'quant-ph-9808015-2-27-1': 'E. Squires, The Mystery of Quantum World (Institute of Physics Publishing, London, 1994).', 'quant-ph-9808015-2-28-0': '6.', 'quant-ph-9808015-2-28-1': 'W. K. Wootters, "Is Time Asymmetry Logically Prior to Quantum Mechanics?"', 'quant-ph-9808015-2-28-2': ", in Physical Origins of Time, edited by J. J. Halliwell, J. P'erez-Mercader and W. H. Zurek, (Cambridge University Press, Cambridge, 1994).", 'quant-ph-9808015-2-29-0': '7.', 'quant-ph-9808015-2-29-1': 'A. Valentini, Phys.', 'quant-ph-9808015-2-29-2': 'Lett.', 'quant-ph-9808015-2-29-3': 'A156, 5 (1991).', 'quant-ph-9808015-2-30-0': '8.', 'quant-ph-9808015-2-30-1': 'R. L. Liboff.', 'quant-ph-9808015-2-30-2': 'Kinetic Theory.', 'quant-ph-9808015-2-30-3': '(Prentice-Hall, New Jersey, 1990); H. J. Kreuzer.', 'quant-ph-9808015-2-30-4': 'Nonequilibrium Thermodinamics and its Statistical Foundations (Oxford University Press, New York, 1984).', 'quant-ph-9808015-2-31-0': '9.', 'quant-ph-9808015-2-31-1': 'A. Valentini, On the pilot-wave theory of classical, quantum and subquantum physics (Springer-Verlag, Berlin, 1996).', 'quant-ph-9808015-2-32-0': '10.', 'quant-ph-9808015-2-32-1': 'A. 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[]

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['quant-ph-9808015-1-8-11', 'quant-ph-9808015-1-8-13', 'quant-ph-9808015-1-9-0', 'quant-ph-9808015-1-10-0', 'quant-ph-9808015-1-11-2', 'quant-ph-9808015-1-13-9', 'quant-ph-9808015-2-2-0', 'quant-ph-9808015-2-2-1', 'quant-ph-9808015-2-5-0', 'quant-ph-9808015-2-5-1', 'quant-ph-9808015-2-7-0', 'quant-ph-9808015-2-7-1', 'quant-ph-9808015-2-8-11', 'quant-ph-9808015-2-8-13', 'quant-ph-9808015-2-9-0', 'quant-ph-9808015-2-10-0', 'quant-ph-9808015-2-11-2', 'quant-ph-9808015-2-12-0', 'quant-ph-9808015-2-12-1', 'quant-ph-9808015-2-13-7', 'quant-ph-9808015-2-16-0', 'quant-ph-9808015-2-16-1', 'quant-ph-9808015-2-18-0', 'quant-ph-9808015-2-18-1', 'quant-ph-9808015-2-20-0', 'quant-ph-9808015-2-22-0', 'quant-ph-9808015-2-23-0', 'quant-ph-9808015-2-23-1', 'quant-ph-9808015-2-23-2', 'quant-ph-9808015-2-24-0', 'quant-ph-9808015-2-24-1', 'quant-ph-9808015-2-24-2', 'quant-ph-9808015-2-25-0', 'quant-ph-9808015-2-25-1', 'quant-ph-9808015-2-25-2', 'quant-ph-9808015-2-26-0', 'quant-ph-9808015-2-26-1', 'quant-ph-9808015-2-26-2', 'quant-ph-9808015-2-26-3', 'quant-ph-9808015-2-27-0', 'quant-ph-9808015-2-28-0', 'quant-ph-9808015-2-28-1', 'quant-ph-9808015-2-28-2', 'quant-ph-9808015-2-29-0', 'quant-ph-9808015-2-29-1', 'quant-ph-9808015-2-29-2', 'quant-ph-9808015-2-29-3', 'quant-ph-9808015-2-30-0', 'quant-ph-9808015-2-30-1', 'quant-ph-9808015-2-30-2', 'quant-ph-9808015-2-30-3', 'quant-ph-9808015-2-31-0', 'quant-ph-9808015-2-32-0', 'quant-ph-9808015-2-32-1', 'quant-ph-9808015-2-32-2', 'quant-ph-9808015-2-32-3']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/quant-ph/9808015

1701.08393

{'1701.08393-1-0-0': 'We propose a deep convolutional neural network (CNN) for face detection leveraging on facial attributes based supervision.', '1701.08393-1-0-1': 'We observe a phenomenon that part detectors emerge within CNN trained to classify attributes from uncropped face images, without any explicit part supervision.', '1701.08393-1-0-2': 'The observation motivates a new method for finding faces through scoring facial parts responses by their spatial structure and arrangement.', '1701.08393-1-0-3': 'The scoring mechanism is data-driven, and carefully formulated considering challenging cases where faces are only partially visible.', '1701.08393-1-0-4': 'This consideration allows our network to detect faces under severe occlusion and unconstrained pose variations.', '1701.08393-1-0-5': 'Our method achieves promising performance on popular benchmarks including FDDB, PASCAL Faces, AFW, and WIDER FACE.', '1701.08393-1-1-0': 'Face Detection, Deep Learning, Convolutional Neural Network.', '1701.08393-1-2-0': '# Introduction', '1701.08393-1-3-0': 'Face detection is an important and long-standing problem in computer vision.', '1701.08393-1-3-1': 'A number of methods have been proposed in the past, including neural network based methods [CITATION], cascade structures [CITATION] and deformable part models (DPM) [CITATION] detectors.', '1701.08393-1-3-2': 'There has been a resurgence of interest in applying convolutional neural networks (CNN) on this classic problem [CITATION].', '1701.08393-1-3-3': 'Many of these methods follow a cascade object detection framework [CITATION], some of which directly adopt the effective generic object detection framework Faster-R-CNN [CITATION] as the backbone network, with very deep networks (e.g., 101-layer ResNet) to leverage the remarkable representation learning capacity of deep CNN [CITATION].', '1701.08393-1-4-0': 'While face bounding boxes have been used as a standard supervisory source for learning a face detector, the usefulness of facial attributes remains little explored.', '1701.08393-1-4-1': 'In this study, we show that facial attributes based supervision can effectively enhance the capability of a face detection network in handling severe occlusions.', '1701.08393-1-4-2': 'As depicted in Fig. [REF], a CNN supervised with facial attributes can detect faces even when more than half of the face region is occluded.', '1701.08393-1-4-3': 'In addition, the CNN is capable of detecting faces with large pose variation, e.g., profile view without training separate models under different viewpoints.', '1701.08393-1-4-4': 'Such compelling results are hard to achieve by using supervision based on face bounding boxes alone, especially when the training dataset has limited scene diversity and pose variations.', '1701.08393-1-5-0': 'In this study, we show the benefits of facial attributes supervision through the following considerations:', '1701.08393-1-6-0': '(1) Discovering facial parts responses supervised by facial attributes: Human face has a unique structure.', '1701.08393-1-6-1': 'We believe the reasoning of the unique structure of local facial parts (e.g., eyes, nose, mouth) help detecting faces under unconstrained environments.', '1701.08393-1-6-2': 'We observe an interesting phenomenon that one can actually obtain part detectors within a CNN by training it to classify part-level binary attributes (e.g., mouth attributes including big lips, opened mouth, smiling, wearing lipstick) from uncropped face images, without any explicit part supervision.', '1701.08393-1-6-3': 'The trained CNN is then capable of generating high-quality facial part responses in its deep layers that strongly indicate the locations of the face parts.', '1701.08393-1-6-4': "The examples depicted in Fig. [REF](b) show the response maps (known as 'partness map' in our paper) of five different face parts.", '1701.08393-1-7-0': "(2) Computing faceness score from responses configurations: Given the parts' responses, we formulate an effective method to reason the degree of face likeliness (which we call faceness score) through analyzing their spatial arrangement.", '1701.08393-1-7-1': 'For instance, the hair should appear above the eyes, and the mouth should only appear below the nose.', '1701.08393-1-7-2': 'Any inconsistency would be penalized.', '1701.08393-1-7-3': 'Faceness scores will be derived and used to re-rank candidate windows to obtain a set of face proposals.', '1701.08393-1-7-4': 'Our face proposal approach enjoys a high recall with just a modest number of proposals (over 90% of face recall with around [MATH] proposals, [MATH]0.5% of full sliding windows, and [MATH]10% of generic object proposals [CITATION], measured on the FDDB dataset [CITATION]).', '1701.08393-1-8-0': '(3) Refining the face hypotheses - Both the aforementioned components offer a chance to find a face even under severe occlusion and pose variations.', '1701.08393-1-8-1': 'The output of these components is a small set of high-quality face bounding box proposals that cover most faces in an image.', '1701.08393-1-8-2': 'Given the face proposals, we design a multi-task CNN in the second stage to refine the hypotheses further, by simultaneously recognizing the true faces and estimating more precise face locations.', '1701.08393-1-9-0': "Our main contribution in this study is the novel use of CNN and attributes supervision for discovering facial parts' responses.", '1701.08393-1-9-1': 'We show that part detectors emerge within a CNN trained to classify attributes from uncropped face images, without any explicit part supervision.', '1701.08393-1-9-2': "The parts' responses are subsequently employed to generate high-quality proposals for training a face detector that is robust to severe occlusion.", '1701.08393-1-9-3': 'The findings aforementioned are new in the literature.', '1701.08393-1-9-4': 'It is worth pointing out that our network is trained on datasets that are not targeted for face detection (CelebA [CITATION] for face recognition, and AFLW [CITATION] for face alignment) and with simple background.', '1701.08393-1-9-5': 'Nevertheless, it still achieves promising performance on various face detection benchmarks including FDDB, PASCAL Faces, AFW, and the challenging WIDER FACE dataset.', '1701.08393-1-10-0': 'In comparison to our earlier version of this work [CITATION], we present a more effective design of CNN to achieve improved performance and speed.', '1701.08393-1-10-1': 'Firstly, in contrast to our previous work that requires independent convolutional networks for learning responses of different facial parts, we now share feature representations between these attribute-aware networks.', '1701.08393-1-10-2': 'The sharing of low and mid-levels representations largely reduce the number of parameters in our framework ([MATH]83% fewer parameters), while improving the robustness of the feature representation.', '1701.08393-1-10-3': 'Secondly, our previous framework relies on external generic object proposal generators such as selective search [CITATION] and EdgeBox [CITATION] for proposing candidate windows.', '1701.08393-1-10-4': 'Inspired by region proposal network presented in [CITATION], in this study we directly generate proposals from our attribute-aware networks, thus proposal generation becomes an inherent part of the framework.', '1701.08393-1-10-5': 'This design not only leads to improved computation efficiency but also higher recall rate compared with generic object proposal algorithms.', '1701.08393-1-10-6': 'Thirdly, we compare our face detector pre-trained on the task of facial attributes classification with that pre-trained on ImageNet large-scale object classification.', '1701.08393-1-10-7': 'Apart from the above major changes, we also provide more technical details and discussions.', '1701.08393-1-10-8': 'Additional experiments are conducted on the challenging WIDER FACE dataset [CITATION].', '1701.08393-1-11-0': '# Related Work', '1701.08393-1-12-0': 'There is a long history of using neural network for the task of face detection [CITATION].', '1701.08393-1-12-1': 'An early face detection survey [CITATION] provides an extensive coverage on relevant methods.', '1701.08393-1-12-2': 'Here we highlight a few notable studies.', '1701.08393-1-12-3': 'Rowley et al. [CITATION] exploit a set of neural network-based filters to detect presence of faces in multiple scales, and merge the detections from individual filters.', '1701.08393-1-12-4': 'Osadchy et al. [CITATION] demonstrate that a joint learning of face detection and pose estimation significantly improves the performance of face detection.', '1701.08393-1-12-5': 'The seminal work of Vaillant et al. [CITATION] adopt a two-stage coarse-to-fine detection.', '1701.08393-1-12-6': 'Specifically, the first stage approximately locates the face region, whilst the second stage provides a more precise localization.', '1701.08393-1-12-7': 'Our approach is inspired by these studies, but we introduce innovations on many aspects.', '1701.08393-1-12-8': 'For instance, our first stage network is conceptually different from that of [CITATION], and many recent deep learning detection frameworks - we train attribute-aware networks to achieve precise localization of facial parts, and exploit their spatial structure for inferring face likeliness.', '1701.08393-1-12-9': 'This concept is new and it allows our model to detect faces under severe occlusion and pose variations.', '1701.08393-1-12-10': 'While great efforts have been devoted for addressing face detection under occlusion [CITATION], these methods are all confined to frontal faces.', '1701.08393-1-12-11': 'In contrast, our model can discover faces under variations of both pose and occlusion.', '1701.08393-1-13-0': 'In the last decades, cascade based [CITATION] and deformable part models (DPM) detectors dominate face detection approaches.', '1701.08393-1-13-1': 'Viola and Jones [CITATION] introduced fast Haar-like features computation via integral image and boosted cascade classifier.', '1701.08393-1-13-2': 'Various studies thereafter follow a similar pipeline.', '1701.08393-1-13-3': 'Among the variants, SURF cascade [CITATION] was one of the top performers.', '1701.08393-1-13-4': 'Later Chen et al. [CITATION] demonstrate state-of-the-art face detection performance by learning face detection and face alignment jointly in the same cascade framework.', '1701.08393-1-13-5': 'Deformable part models define face as a collection of parts.', '1701.08393-1-13-6': 'Latent Support Vector Machine is typically used to find the parts and their relationships.', '1701.08393-1-13-7': 'DPM is shown more robust to occlusion than the cascade based methods.', '1701.08393-1-13-8': 'A recent study [CITATION] demonstrates state-of-the-art performance with just a vanilla DPM, achieving better results than more sophisticated DPM variants [CITATION].', '1701.08393-1-14-0': 'Recent studies [CITATION] show that face detection can be further improved by using deep learning.', '1701.08393-1-14-1': 'The network proposed by [CITATION] does not have an explicit mechanism to handle occlusion, the face detector therefore fails to detect faces with heavy occlusions, as acknowledged by the authors.', '1701.08393-1-14-2': 'Cascade based convolutional neural networks [CITATION] replace boosting classifiers with a set of small CNNs to quickly reject negative samples in the early stage.', '1701.08393-1-14-3': 'Recent studies [CITATION] exploit facial landmarks as supervision signals to improve face detection performance.', '1701.08393-1-14-4': 'In this study, we show that facial attributes can serve as an important source too for learning a robust face detector.', '1701.08393-1-15-0': 'The first stage of our model is partially inspired by generic object proposal approaches [CITATION].', '1701.08393-1-15-1': 'Generic object proposal generators are commonly used in standard object detection algorithms for providing high-quality and category-independent bounding boxes.', '1701.08393-1-15-2': 'These methods typically involve redundant computations over regions that are covered by multiple proposals.', '1701.08393-1-15-3': 'To reduce computation, Ren et al. [CITATION] propose Regional Proposal Network (RPN) to generate proposals from high-level response maps in a CNN through a set of predefined anchor boxes.', '1701.08393-1-15-4': 'Both generic object proposal and RPN methods do not consider the unique structure and parts on the face.', '1701.08393-1-15-5': 'Hence, no principled mechanism is available to recall faces when the face is only partially visible.', '1701.08393-1-15-6': 'These shortcomings motivate us to formulate the new faceness measure to achieve high recall on faces, while reducing the number of candidate windows to half the original (compared to the original RPN [CITATION]).', '1701.08393-1-16-0': '# Faceness-Net', '1701.08393-1-17-0': 'This section introduces the baseline Faceness-Net.', '1701.08393-1-17-1': 'We first briefly overview the entire pipeline and then discuss the details.', '1701.08393-1-17-2': 'As shown in Fig. [REF], Faceness-Net consists of two stages, i.e., (i) generating face proposals from partness maps by ranking candidate windows using faceness scores, and (ii) refining face proposals for face detection.', '1701.08393-1-18-0': 'First stage.', '1701.08393-1-18-1': 'A full image [MATH] is used as an input to a CNN to generate the partness map for each face part.', '1701.08393-1-18-2': 'A set of CNNs, known as attribute-aware networks, are used to generate the partness map of different parts independently.', '1701.08393-1-18-3': 'The partness map is obtained by weighted averaging over all the response maps at its top convolutional layer.', '1701.08393-1-18-4': 'The map indicates the location of a specific facial component presented in the image, e.g., hair, eyes, nose, mouth, and beard denoted by [MATH], [MATH], [MATH], [MATH], and [MATH], respectively.', '1701.08393-1-18-5': "For illustration, we add all these maps into a face label map [MATH], which clearly suggests faces' locations", '1701.08393-1-19-0': 'Given a set of candidate windows [MATH] that are generated by existing object proposal methods such as [CITATION], or a region proposal network (RPN) [CITATION], we rank these windows according to their faceness scores, [MATH], which are derived from the partness maps with respect to different facial parts configurations, as illustrated at the bottom of Fig. [REF](a).', '1701.08393-1-19-1': "For example, as visualized in Fig. [REF](a), a candidate window 'A' covers a local region of [MATH] (i.e., hair) and its faceness score is calculated by dividing the values at its upper part with respect to the values at its lower part, because hair is more likely to present at the top of a face region.", '1701.08393-1-19-2': 'The bottom part of Fig. [REF](a) illustrates the spatial configurations of five facial parts.', '1701.08393-1-19-3': 'The facial configurations can be learned from the training data.', '1701.08393-1-19-4': "A final faceness score of 'A' is obtained by averaging over the scores of these parts.", '1701.08393-1-19-5': 'In this case, large number of false positive windows can be pruned.', '1701.08393-1-19-6': 'To further reduce the number of the proposed windows, we apply non-maximum suppression (NMS) to smooth the scores by leveraging the spatial relations among these windows.', '1701.08393-1-19-7': "The proposed approach is capable of coping with severe face occlusions, as shown in Fig. [REF](b), where face windows 'A' and 'E' can be retrieved by objectness [CITATION] only if large amount of windows are proposed, while they rank top [MATH] by using our method.", '1701.08393-1-20-0': 'Second stage.', '1701.08393-1-20-1': 'The face proposals are refined by training a multi-task CNN, where face classification and bounding box regression are jointly optimized (Fig. [REF]).', '1701.08393-1-21-0': '## Attribute-Aware Networks', '1701.08393-1-22-0': 'The first stage of the baseline Faceness-Net consists of multiple attribute-aware networks for generating response maps of different parts (Fig. [REF]).', '1701.08393-1-22-1': 'Five networks are needed to cover all five pre-defined facial components, i.e., hair, eyes, nose, mouth, and beard.', '1701.08393-1-22-2': 'These attribute-aware networks share the same structure.', '1701.08393-1-22-3': 'Next, we first discuss the network structure and subsequently show that these networks can share representation to save parameters.', '1701.08393-1-23-0': 'Network structure.', '1701.08393-1-23-1': 'The choice of network structure for extracting partness maps is flexible.', '1701.08393-1-23-2': 'Figure [REF](a) depicts the structure and hyper-parameters of the CNN used in the baseline Faceness-Net.', '1701.08393-1-23-3': 'This convolutional structure is inspired by the AlexNet [CITATION], which is proposed for object categorization.', '1701.08393-1-23-4': 'Specifically, the network stacks seven convolutional layers (Conv1 to Conv7) and two max-pooling layers (Max1 and Max2).', '1701.08393-1-23-5': 'The hyper-parameters of each layer is specified in Fig. [REF](a).', '1701.08393-1-24-0': 'Once the attribute networks are trained (training details are provided in Sec. [REF]), we examine the response maps obtained from the attribute-aware networks.', '1701.08393-1-24-1': 'As observed from Fig. [REF](b), the feature maps of the first few convolutional layers do not clearly indicate the locations of facial parts.', '1701.08393-1-24-2': 'However, clear indication of the facial component can be seen from the average-pooled responses of Conv7.', '1701.08393-1-24-3': 'Consequently, we obtain the initial partness map from the output layer, which contains a single filter with a size of [MATH].', '1701.08393-1-24-4': 'The role of this filter is to perform weighted average on the feature maps of Conv7.', '1701.08393-1-24-5': "The final partness map that matches the input image's size is obtained through unpooling operation [CITATION].", '1701.08393-1-25-0': 'Shared representation.', '1701.08393-1-25-1': "It is observed that the feature maps of earlier layers across the different attribute-aware networks are almost identical and they are not indicative of parts' locations.", '1701.08393-1-25-2': 'Motivated by these observations, instead of designating separate attribute-aware networks for different facial components, we share early convolutional layers of these networks to reduce parameters.', '1701.08393-1-25-3': "Specifically, the first four convolutional layers that do not clearly suggests parts' locations are shared, followed by five branches, each of which consists of two convolutional layers responsible for a facial component, as shown in Fig. [REF].", '1701.08393-1-25-4': 'Note that in comparison to the structure presented in Fig. [REF](a), we additionally remove a convolutional layer and trim the number of filters in other layers to reduce parameters.', '1701.08393-1-25-5': 'The sharing of representation and filter reduction lead to a single attribute-aware network with [MATH] fewer parameters than the original five attribute-aware networks.', '1701.08393-1-25-6': 'We denote a Faceness-Net with shared representation as Faceness-Net-SR.', '1701.08393-1-25-7': 'We will show that this network structure not only reduces computations, but also improves the robustness of feature representation for face detection.', '1701.08393-1-26-0': '## Learning to Generate Partness Maps', '1701.08393-1-27-0': 'Pre-training the attribute-aware networks.', '1701.08393-1-27-1': 'Pre-training generally helps improving the performance of a deep network.', '1701.08393-1-27-2': 'There are two plausible pre-training options depending upon whether we share the representations across attribute-aware networks or not.', '1701.08393-1-28-0': 'The first option is to pre-train our attribute-aware networks with massive general object categories in ImageNet [CITATION].', '1701.08393-1-28-1': 'From our observations, this option works well when the representations across networks are not shared.', '1701.08393-1-28-2': 'Since each attribute-aware network originally has access only to a particular group of data specific to a certain attribute, the larger-scale ImageNet data helps to mitigate the overfitting issue that is caused by insufficient data.', '1701.08393-1-29-0': 'The second option omits the ImageNet pre-training stage and directly trains a network on the task of facial attributes classification.', '1701.08393-1-29-1': 'This option works best when we adopt the shared representation scheme discussed in Sec. [REF].', '1701.08393-1-29-2': 'Thanks to the sharing of representation, the attribute-aware network requires relatively smaller quantity of training data.', '1701.08393-1-29-3': 'Thus, no overfitting is observed despite we use the facial attributes dataset, which is much smaller in scale, i.e., 180,000 images compared to 1 million images in ImageNet.', '1701.08393-1-30-0': 'Fine-tuning the attribute-aware networks.', '1701.08393-1-30-1': 'Once an attribute-network is pre-trained, we can fine-tune it to generate the desired partness maps.', '1701.08393-1-30-2': 'There are different fine-tuning strategies, but not all of them can generate meaningful partness maps for deriving a robust faceness score.', '1701.08393-1-31-0': "As shown in Fig. [REF](b), a deep network trained on generic objects, e.g., AlexNet [CITATION], is not capable of providing us with precise faces' locations, let alone partness map.", '1701.08393-1-31-1': 'To generate accurate partness maps, we explore multiple ways for learning an attribute-aware network.', '1701.08393-1-31-2': 'The most straight-forward manner is to use the image and its pixel-wise segmentation label map as input and target, respectively.', '1701.08393-1-31-3': 'This setting is widely employed in image labeling [CITATION].', '1701.08393-1-31-4': 'However, it requires label maps with pixelwise annotations, which are expensive to collect.', '1701.08393-1-31-5': 'Another setting is image-level classification (i.e., faces and non-faces), as shown in Fig. [REF](c).', '1701.08393-1-31-6': 'It works well where the training images are well-aligned, such as face recognition [CITATION].', '1701.08393-1-31-7': 'Nevertheless, it suffers from complex background clutter because the supervisory information is not sufficient to account for rich and diverse face variations.', '1701.08393-1-31-8': "Its learned feature maps contain too much noises, which overwhelm the actual faces' locations.", '1701.08393-1-31-9': 'Attribute learning in Fig. [REF](d) extends the binary classification in (c) to the extreme by using a combination of attributes to capture face variations.', '1701.08393-1-31-10': "For instance, an 'Asian' face can be distinguished from a 'European' face.", '1701.08393-1-31-11': 'However, our experiments demonstrate that the setting is not robust to occlusion.', '1701.08393-1-32-0': 'Figure [REF](e) shows the partness maps obtained by the baseline FacenessNet, of which the attribute networks do not share representations.', '1701.08393-1-32-1': 'The strategy we propose extends (d) by partitioning attributes into groups based on facial components.', '1701.08393-1-32-2': "For instance, 'black hair', 'blond hair', 'bald', and 'bangs' are grouped together, as all of them are related to hair.", '1701.08393-1-32-3': 'The grouped attributes are summarized in Table [REF].', '1701.08393-1-32-4': 'In this case, face parts are modeled separately.', '1701.08393-1-32-5': 'If one part is occluded, the face region can still be localized by the other parts.', '1701.08393-1-32-6': 'We take the Hair-Branch shown in the stage one of Fig. [REF] as an example to illustrate the learning procedure.', '1701.08393-1-32-7': 'Let [MATH] be a set of full face images and the attribute labels of hair, where [MATH] and [MATH], implying that each full image is rescaled to [MATH] and there is nine attributes related to hair as listed in Table [REF].', '1701.08393-1-32-8': 'Learning is formulated as a multi-variate classification problem by minimizing the cross-entropy loss, [EQUATION] where [MATH] is modeled as a sigmoid function, i.e. [MATH], indicating the probability of the presence of the attributes.', '1701.08393-1-32-9': 'The features of [MATH] is denoted as [MATH] and [MATH] represents a vector with all elements equal one.', '1701.08393-1-32-10': 'To facilitate the learning, we stack two fully-connected layers on top of the last convolutional layer of the structure shown in Fig. [REF].', '1701.08393-1-32-11': 'After which we optimize the loss function by using the stochastic gradient descent with back-propagation.', '1701.08393-1-32-12': 'After training the attribute-aware network, the fully-connected layers are removed to make the network fully convolutional again.', '1701.08393-1-33-0': 'Figure [REF](f) shows the partness maps that are generated from the networks with shared representation, i.e., Faceness-Net-SR (see Fig. [REF]).', '1701.08393-1-33-1': 'Visually, the partness maps generated by this model are more noisy compared to Fig. [REF](e).', '1701.08393-1-33-2': 'The key reason is that the Faceness-Net-SR is not pre-trained using ImageNet data but directly trained on the attribute classification task.', '1701.08393-1-33-3': "Despite the noisy partness maps, they actually capture more subtle parts' responses and therefore lead to higher recall rate in the subsequent face proposal stage, provided that the number of proposals is sufficiently large.", '1701.08393-1-34-0': '## Generating Candidate Windows', '1701.08393-1-35-0': 'Face detection can be improved if the inputs are formed by moderate number of proposals with a high-recall rate.', '1701.08393-1-35-1': 'To produce the required proposals, we will explore two plausible choices to generate the initial set of candidate windows.', '1701.08393-1-36-0': 'Generic object proposal.', '1701.08393-1-36-1': 'Generic object scoring is primarily employed to reduce the computational cost of a detector.', '1701.08393-1-36-2': 'It has also been shown improving detection accuracy due to reduction of spurious false positives [CITATION].', '1701.08393-1-36-3': 'A variety of cues have been proposed to quantify the objectness of an image window, e.g., norm of the gradient [CITATION], edges [CITATION], or integration of a number of low-level features [CITATION].', '1701.08393-1-36-4': 'Other popular methods include super-pixel based approaches, e.g., selective search [CITATION], randomized Prim [CITATION], and multi-scale combinatorial grouping [CITATION].', '1701.08393-1-36-5': 'Our framework can readily employ these generic candidate windows for ranking using the proposed faceness score (Sec. [REF]).', '1701.08393-1-37-0': 'Region proposal network.', '1701.08393-1-37-1': 'Previous study [CITATION] employs deep neural network to generate region proposals through conducting object/non-object classification and bounding box regression.', '1701.08393-1-37-2': 'We adapt this idea to our framework with a few changes.', '1701.08393-1-38-0': '(a) Proposing regions on partness maps - instead of using the generic object feature maps, we exploit the partness maps produced by face attribute-aware networks for proposing the initial candidate windows.', '1701.08393-1-39-0': '(b) Omitting face/non-face classification and bounding box regression - Different from RPN [CITATION], we skip the learning from face/non-face classification and bounding box regression at this stage.', '1701.08393-1-39-1': 'The partness maps that are learned from face attributes classification already provide high-quality response maps to propose face candidate windows.', '1701.08393-1-40-0': 'We provide an example below on using a partness map of hair, [MATH], for region proposal.', '1701.08393-1-40-1': 'As shown in Fig. [REF], each value of location [MATH] on the partness map [MATH] indicates the probability of the appearance of the hair component.', '1701.08393-1-40-2': 'We select a set of [MATH] locations [MATH] with a probability [MATH] higher than [MATH].', '1701.08393-1-40-3': 'For each selected location, multiple region proposals are generated, where the number of maximum possible proposals for each location is fixed as [MATH].', '1701.08393-1-40-4': 'The proposals are obtained from predefined reference boxes, which we call anchors.', '1701.08393-1-40-5': 'For each face part, anchors are centered at different locations considering the structure of human face.', '1701.08393-1-40-6': 'In addition, they are associated with a specific scale and aspect ratio, as shown at the top of Fig. [REF].', '1701.08393-1-40-7': 'For instance, the anchors of the hair region are centered at [MATH] and the anchors of eyes are centered at [MATH], where [MATH] and [MATH] represent the width and height of an anchor.', '1701.08393-1-40-8': "Similar to previous work [CITATION], these anchors are translation invariant up to the network's total stride, and the method does not incur extra cost for addressing scales thanks to the multi-scale anchors.", '1701.08393-1-41-0': 'In our study, we define [MATH] scales and [MATH] aspect ratio, yielding [MATH] anchors at each selected position.', '1701.08393-1-41-1': 'Specifically, we use [MATH] scales with box areas of [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] pixels, and [MATH] aspect ratio of [MATH] (with width to height).', '1701.08393-1-41-2': 'The parameters of anchors, i.e., center location, scale and aspect ratio are selected by maximizing the recall rate given an average number of [MATH] proposal per image.', '1701.08393-1-41-3': 'In our study, we perform grid search on the training set to select the parameters.', '1701.08393-1-42-0': 'Discussion.', '1701.08393-1-42-1': 'Both the generic objectness measures and original RPN (trained on ImageNet) are devoted to generic objects therefore not suitable to propose windows specific to face.', '1701.08393-1-42-2': 'In particular, applying a generic proposal generator directly would produce enormous number of candidate windows but only minority of them contain faces.', '1701.08393-1-42-3': 'While RPN is computationally more efficient than generic object proposal generators, it cannot be directly applied to our problem too.', '1701.08393-1-42-4': 'Specifically, in order for the RPN to cope with faces with tiny size and various poses, a large number of anchor boxes are required, leading to enormous number of proposals.', '1701.08393-1-42-5': 'In the next section, we discuss a new faceness measure that can complement existing object proposal generators to achieve high recall on faces, while significantly reduce the number of candidate windows.', '1701.08393-1-43-0': '## Ranking Windows by Faceness Score', '1701.08393-1-44-0': 'After generating candidate windows based on the methods described in Sec. [REF], our approach computes faceness score on these windows to return a ranked set of top-scoring face proposals.', '1701.08393-1-44-1': 'Figure [REF] illustrates the procedure of deriving the faceness measure from the partness maps of hair and eyes.', '1701.08393-1-44-2': 'Let [MATH] be the faceness score of a window [MATH].', '1701.08393-1-44-3': 'For example, as shown in Fig. [REF](a), given a partness map of hair, [MATH], [MATH] is attained by dividing the sum of values in ABEF (green) by the sum of values in FECD.', '1701.08393-1-44-4': 'Similarly, Fig. [REF](b) shows that [MATH] is obtained by dividing the sum of values in EFGH (green) with respect to ABEF+HGCD of [MATH].', '1701.08393-1-44-5': 'For both of the above examples, a larger value of [MATH] indicates a higher overlapping ratio of [MATH] with a face.', '1701.08393-1-45-0': 'The spatial configurations, such as ABEF in Fig. [REF](a) and EFGH in Fig. [REF](b), can be learned from data.', '1701.08393-1-45-1': 'We take hair as an example.', '1701.08393-1-45-2': 'We need to learn the positions of points E and F, which can be represented by the [MATH]-coordinates of ABCD, i.e., the proposed window.', '1701.08393-1-45-3': 'For instance, the position of E in Fig. [REF](a) can be represented by [MATH] and [MATH], implying that the value of its [MATH]-axis is a linear combination of [MATH] and [MATH].', '1701.08393-1-45-4': 'With this representation, [MATH] can be efficiently computed by using the integral image (denoted as [MATH]) of the partness map.', '1701.08393-1-45-5': 'For instance, [MATH] in (a) is attained by [EQUATION] where [MATH] signifies the value at the location [MATH].', '1701.08393-1-46-0': 'Given a training set [MATH], where [MATH] and [MATH] denote the [MATH]-th window and its label (i.e. face/non-face), respectively.', '1701.08393-1-46-1': 'Let [MATH] be the cropped partness map with respect to the [MATH]-th window, e.g., region ABCD in [MATH].', '1701.08393-1-46-2': 'This problem can be formulated as maximum a posteriori (MAP) estimation [EQUATION] where [MATH] represents a set of parameters when learning the spatial configuration of hair (Fig. [REF](a)).', '1701.08393-1-46-3': 'The terms [MATH] and [MATH] denote the likelihood and prior, respectively.', '1701.08393-1-46-4': 'The likelihood of faceness can be modeled by a sigmoid function, i.e., [MATH], where [MATH] is a coefficient.', '1701.08393-1-46-5': 'This likelihood measures the confidence of partitioning the face and non-face, given a certain spatial configuration.', '1701.08393-1-46-6': 'The prior term can be factorized, [MATH], where [MATH] is a uniform distribution between zero and one, as it indicates the coefficients of linear combination, [MATH] models the prior of the candidate window, which can be generated by object proposal methods, and [MATH] is the partness map as obtained in Sec. [REF].', '1701.08393-1-46-7': 'Since [MATH] typically has a low dimension (e.g., one dimension of hair), it can be simply obtained by line search.', '1701.08393-1-46-8': 'Note that Eq. [REF] can be easily extended to model more complex spatial configurations.', '1701.08393-1-47-0': '## Face Detection', '1701.08393-1-48-0': 'The top candidate windows that are ranked by faceness measure attain a high recall rate.', '1701.08393-1-48-1': 'These face proposals can be subsequently fed to the multi-task CNN at stage 2 of the proposed pipeline (Fig. [REF]) for face detection.', '1701.08393-1-49-0': 'Pre-training.', '1701.08393-1-49-1': 'We directly use the earlier layers of attribute-aware networks (the stage-1 network with shared represention as shown in Fig. [REF]) up to Conv4 as the pre-trained model for the multi-task CNN of stage 2.', '1701.08393-1-49-2': 'After Conv4, as shown in Fig. [REF], the multi-task CNN forks into two branches, each of which consists of two convolutional layers and two fully connected layers.', '1701.08393-1-49-3': 'The two branches are optimized to handle different tasks, namely face classification and bounding box regression, respectively.', '1701.08393-1-50-0': 'It is worth pointing out that the multi-task CNN can be pre-trained on the ImageNet data, instead of reusing the parameters of the attribute-aware networks.', '1701.08393-1-50-1': 'Nevertheless, we found that the multi-task CNN converges much faster given the face attributes based pretrained model.', '1701.08393-1-50-2': 'Specifically, the attribute pretrained network only requires [MATH] iterations to converge during the face detection fine-tuning stage, in comparison to more than [MATH] iterations for the ImageNet pertrained network using the same mini-batch size.', '1701.08393-1-50-3': 'We conjecture that much less efforts are needed to transform the feature representations learned from the facial attribute classification task to the face detection task.', '1701.08393-1-51-0': 'Multi-task fine-tuning.', '1701.08393-1-51-1': 'We fine-tune the first branch of the multi-task CNN for face classification and the second branch for bounding box regresssion.', '1701.08393-1-51-2': 'Fine-tuning is performed using the face proposals obtained from the previous step (Sec [REF]).', '1701.08393-1-51-3': 'For face classification, we assign a face proposal to its closest ground truth bounding box based on the Euclidean distance between their respective center coordinates.', '1701.08393-1-51-4': 'A face proposal is considered positive if the Intersection over Union (IoU) between the proposal box and the assigned ground truth box is larger than [MATH]; otherwise it is negative.', '1701.08393-1-51-5': 'For bounding box regression, we train the second branch of the multi-task CNN to regress each proposal to the coordinates of its assigned ground truth box.', '1701.08393-1-51-6': 'If the proposed window is a positive sample, the regression target is generated by Eq. [REF].', '1701.08393-1-51-7': 'We use the following parameterizations of the 4 coordinates: [EQUATION] where [MATH] is a normalizing factor.', '1701.08393-1-51-8': 'The [MATH] denotes the top-left and bottom-right coordinates of a bounding box.', '1701.08393-1-51-9': 'Variables [MATH], [MATH], and [MATH] represent the ground truth box, proposed box, and regression target.', '1701.08393-1-51-10': 'If a proposed window is non-face, the CNN outputs a vector of [MATH].', '1701.08393-1-52-0': 'More implementation details are given below.', '1701.08393-1-52-1': 'During the training process, if the number of positive samples in a mini-batch is smaller than [MATH] of the total samples, we randomly crop the ground truth faces and add these samples as additional positive samples.', '1701.08393-1-52-2': 'Therefore, the ratio of positive samples and negative samples is kept not lower than [MATH].', '1701.08393-1-52-3': 'Meanwhile, we conduct bounding box NMS on the negative samples.', '1701.08393-1-52-4': 'The IoU for the NMS is set to [MATH].', '1701.08393-1-52-5': 'The proposed bounding boxes are cropped and then resized to [MATH].', '1701.08393-1-52-6': 'To handle blurry faces, we augment our training samples by applying Gaussian blur.', '1701.08393-1-52-7': 'The fine-tuning consumes [MATH] iterations with a batch size of [MATH] images.', '1701.08393-1-52-8': 'We adopt Euclidean loss and cross-entropy loss for bounding box regression and face classification, respectively.', '1701.08393-1-53-0': '# Experimental Settings', '1701.08393-1-54-0': 'Training datasets.', '1701.08393-1-54-1': '(i) We employ CelebA dataset [CITATION] to train our attribute-aware networks.', '1701.08393-1-54-2': 'The dataset contains [MATH] web-based images exclusive from the LFW [CITATION], FDDB [CITATION], AFW [CITATION] and PASCAL [CITATION] datasets.', '1701.08393-1-54-3': 'Every image in the dataset are labeled with [MATH] facial attributes.', '1701.08393-1-54-4': 'We select [MATH] facial attributes from CelebA dataset for each image and divide the attributes into five categories based on their respective facial parts as shown in Table [REF].', '1701.08393-1-54-5': 'We randomly select [MATH] images from the CelebA dataset for training and the remaining is reserved as the validation set.', '1701.08393-1-54-6': '(ii) For face detection training, we choose [MATH] face images from the AFLW dataset [CITATION] to ensure a balanced out-of-plane pose distribution.', '1701.08393-1-54-7': 'We observe a large number of missed annotated faces in the AFLW dataset, which could hamper the training of our face detector.', '1701.08393-1-54-8': 'Hence, we re-annotate face bounding boxes for those missing faces.', '1701.08393-1-54-9': 'The total number of faces in the re-annotated AFLW is [MATH] compared with [MATH] in the original data.', '1701.08393-1-54-10': 'As negative samples, we randomly select [MATH] person-free images from the PASCAL VOC 2007 dataset [CITATION].', '1701.08393-1-55-0': 'Part response test dataset.', '1701.08393-1-55-1': 'In Sec. [REF], we use LFW dataset [CITATION] for evaluating the quality of part response maps for part localization.', '1701.08393-1-55-2': 'Since the original dataset does not come with part-level bounding boxes, we label the boxes with the following scheme.', '1701.08393-1-55-3': 'We follow the annotations provided by [CITATION] on hairs and beard for a set of [MATH] LFW images.', '1701.08393-1-55-4': 'Hair bounding boxes are generated with minimal and maximal coordinates of hair superpixel as shown in Fig [REF].', '1701.08393-1-55-5': 'Using similar strategy, eye, nose and mouth bounding boxes are obtained from the manually labeled [MATH] dense facial landmarks [CITATION] on the original LFW [CITATION] images, as shown in Fig [REF].', '1701.08393-1-56-0': 'Face proposal and detection test datasets.', '1701.08393-1-56-1': 'In Sec. [REF] and Sec. [REF], we use the following datasets.', '1701.08393-1-56-2': '(i) FDDB [CITATION] dataset contains [MATH] faces in a set of [MATH] images.', '1701.08393-1-56-3': 'For the face proposal evaluation, we follow the standard evaluation protocol widely used in object proposal studies [CITATION] and transform the original FDDB ellipses ground truth into bounding boxes by minimal bounding rectangle.', '1701.08393-1-56-4': 'For the face detection evaluation, the original FDDB ellipse ground truth is used.', '1701.08393-1-56-5': '(ii) AFW [CITATION] dataset contains [MATH] Flickr images with [MATH] annotated faces of large variations in both face viewpoint and appearance.', '1701.08393-1-56-6': '(iii) PASCAL faces [CITATION] is a widely used face detection benchmark dataset.', '1701.08393-1-56-7': 'It consists of [MATH] images and [MATH] annotated faces.', '1701.08393-1-56-8': '(iv) WIDER FACE [CITATION] is the largest and extremely challenging face detection benchmark dataset.', '1701.08393-1-56-9': 'It consists of [MATH] images and [MATH] annotated faces.', '1701.08393-1-57-0': 'Evaluation settings.', '1701.08393-1-57-1': 'Following [CITATION], we employ the Intersection over Union (IoU) as our evaluation metric.', '1701.08393-1-57-2': 'We fix the IoU threshold to [MATH] following the strict PASCAL criterion.', '1701.08393-1-57-3': 'In particular, an object is considered being covered/detected by a proposal if the IoU is no less than [MATH].', '1701.08393-1-57-4': 'To evaluate the effectiveness of different object proposal algorithms, we use the detection rate (DR) given the number of proposals per image [CITATION].', '1701.08393-1-57-5': 'For face detection, we use standard precision and recall (PR) to evaluate the effectiveness of face detection algorithms.', '1701.08393-1-58-0': 'Baseline Faceness-Nets.', '1701.08393-1-58-1': 'We evaluate four variants of Faceness-Net:', '1701.08393-1-59-0': 'The discussion on generic object proposal and region proposal techniques can be found in Sec. [REF].', '1701.08393-1-60-0': '# Results', '1701.08393-1-61-0': '## Evaluating the Attribute-Aware Networks', '1701.08393-1-62-0': 'Robustness to unconstrained training input.', '1701.08393-1-62-1': 'The proposed attribute-aware networks do not assume well-cropped faces as input in both the training and test stages.', '1701.08393-1-62-2': 'To support this statement, we conduct an experiment by fine-tuning two attribute-aware networks as shown in the Fig. [REF](a), each of which taking different inputs: (1) uncropped images, which may include large portion of background clutter apart the face; and (2) cropped images, which encompass roughly the face and shoulder regions.', '1701.08393-1-62-3': 'Some examples of uncropped images are shown in Fig. [REF](a).', '1701.08393-1-62-4': 'The performance is measured based on the part detection rate.', '1701.08393-1-62-5': "Note that we combine the evaluation on 'Hair+Beard' to suit the ground truth provided by [CITATION] (see Sec. [REF]).", '1701.08393-1-63-0': 'The detection results are summarized in Table [REF].', '1701.08393-1-63-1': 'As can be observed, the proposed approach performs similarly given both the uncropped and cropped images as training inputs.', '1701.08393-1-63-2': 'The results suggest the robustness of the method in handling unconstrained images for training.', '1701.08393-1-63-3': 'In particular, thanks to the facial attribute-driven training, despite the use of uncropped images, the deep model is encouraged to discover and capture the facial part representation in the deep layers, it is therefore capable of generating response maps that precisely pinpoint the locations of parts.', '1701.08393-1-63-4': 'In the following experiments, all the proposed models are trained on uncropped images.', '1701.08393-1-63-5': 'Fig. [REF](a) shows the qualitative results.', '1701.08393-1-63-6': 'Note that facial parts can be discovered despite challenging poses.', '1701.08393-1-64-0': 'With and without sharing representation.', '1701.08393-1-64-1': 'As mentioned in Sec. [REF], we can train an attribute-aware network for each face part or we can train a single network for all the parts by sharing representation.', '1701.08393-1-64-2': 'We compare the proposal detection rate of these two options.', '1701.08393-1-64-3': 'Figure [REF] shows the proposal detection rate of attribute-aware network(s) trained with and without sharing representation, indicated by blue and red curves, respectively.', '1701.08393-1-64-4': 'Attribute-aware networks trained without sharing representation require fewer number of proposals but with a detection rate typically lower than [MATH] (given 150-200 proposals).', '1701.08393-1-64-5': 'On the contrary, the attribute-aware network that shares low-level and mid-level representations can achieve a higher detection rate but with an expense of larger number of proposals.', '1701.08393-1-65-0': 'The observations can be explained as follows.', '1701.08393-1-65-1': 'The networks without sharing representation tend to model the discrepancies between individual face part and background, while the network that shares representation is more likely to learn the differences between facial parts.', '1701.08393-1-65-2': 'Thus, the latter has poorer background modelling capacity thus leading to inferior performance when the number of proposal is small, in comparison to the former.', '1701.08393-1-65-3': 'Nevertheless, we found that the network that shares representation yields high responses for subtle facial parts.', '1701.08393-1-65-4': 'This high recall rate is essential to improve the performance of face detection in the later stage.', '1701.08393-1-66-0': 'Different fine-tuning strategies.', '1701.08393-1-66-1': 'As discussed in Sec. [REF], there are different fine-tuning strategies that can be considered for learning to generate a partness map, but not all of them are well-suited for deriving a robust faceness measure.', '1701.08393-1-66-2': 'Qualitative results have been provided in Fig. [REF].', '1701.08393-1-66-3': 'Here, we provide quantitative comparisons between the following fine-tuning approaches: (i) a network fine-tuned with a large number of face images from CelebA and non-face images, (ii) fine-tuning the network with 25 face attributes, and (iii) the proposed approach that fine-tunes attribute-aware networks with part-level attributes in accordance to Table [REF].', '1701.08393-1-66-4': 'It is evident from Fig. [REF] that our approach performs significantly better than approaches (i) and (ii).', '1701.08393-1-67-0': '## From Part Responses to Face Proposal', '1701.08393-1-68-0': 'Generic object proposal methods.', '1701.08393-1-68-1': 'In this experiment, we show the effectiveness of adapting different generic object proposal generators [CITATION] to produce face-specific proposals.', '1701.08393-1-68-2': 'Since the notion of face proposal is new, no suitable methods are comparable therefore we use the original generic methods as baselines.', '1701.08393-1-68-3': 'We first apply any object proposal generator to generate the candidate windows and we use our faceness scoring method described in Sec. [REF] to obtain the face proposals.', '1701.08393-1-68-4': 'We experiment with different parameters for the generic methods, and choose parameters that produce moderate number of proposals with very high recall.', '1701.08393-1-68-5': 'Evaluation is conducted following the standard protocol [CITATION].', '1701.08393-1-69-0': 'The results are shown in the first three rows of Fig. [REF].', '1701.08393-1-69-1': 'The green curves show the performance of baseline generic object proposal generators.', '1701.08393-1-69-2': 'It can be observed that our method consistently improves the baselines for proposing face candidate windows, under different IoU thresholds.', '1701.08393-1-69-3': 'Table [REF] shows that our method achieves high detection rate with small number of proposals.', '1701.08393-1-70-0': 'Region proposal method.', '1701.08393-1-70-1': 'The bottom row of Fig. [REF] depicts the results of using region proposal (Sec. [REF]) to produce face-specific proposals.', '1701.08393-1-70-2': 'The proposed region proposal technique achieves a high detection rate of over [MATH] when the average number of proposal is [MATH] with an IoU of 0.5.', '1701.08393-1-70-3': 'It is worth pointing out that high quality region proposals are typically generated by a region proposal network that is trained by conducting object/non-object classification and anchor box regression [CITATION].', '1701.08393-1-70-4': 'Our proposals, nonetheless, do not require such an explicit learning process (see Sec. [REF]), thanks to the high-quality responses of partness maps and faceness score.', '1701.08393-1-71-0': 'Evaluate the contributions of each face part.', '1701.08393-1-71-1': 'We factor the contributions of different face parts to proposing face.', '1701.08393-1-71-2': 'Specifically, we generate face proposals with partness maps from each face part individually using the same evaluation protocol in previous experiment.', '1701.08393-1-71-3': 'As can be observed from Fig. [REF], the hair, eye, and nose parts perform much better than mouth and beard.', '1701.08393-1-71-4': 'The lower part of the face is often occluded, making the mouth and beard less effective in proposing face windows.', '1701.08393-1-71-5': 'In contrast, hair, eye, and nose are visible in most cases.', '1701.08393-1-71-6': 'They therefore become important clues for face proposal.', '1701.08393-1-71-7': 'Nonetheless, mouth and beard could provide complementary cues.', '1701.08393-1-71-8': 'Thus combining all parts leads to better result than considering each part in isolation.', '1701.08393-1-72-0': '## From Face Proposal to Face Detection', '1701.08393-1-73-0': 'Next we compare the proposed Faceness-Net and its variants against state-of-the-art face detection approaches on four benchmark datasets including FDDB, AFW, PASCAL Faces, and WIDER FACE.', '1701.08393-1-74-0': 'We conduct extensive face detection experiments on four benchmark datasets FDDB [CITATION], AFW [CITATION], PASCAL faces [CITATION] and WIDER FACE [CITATION].', '1701.08393-1-74-1': 'Our baseline face detector, Faceness-Net, which involves five CNNs with the structure shown in the Fig. [REF], is trained with top [MATH] proposals by re-ranking MCG proposals following the process described in Sec. [REF].', '1701.08393-1-74-2': 'To factor the contributions of share representation and region proposal, we build another three variants of Faceness-Net as discussed in Sec. [REF].', '1701.08393-1-74-3': 'The variant Faceness-Net-RP is trained with top [MATH] region proposals that are re-ranked following Sec. [REF].', '1701.08393-1-75-0': 'We compare Faceness-Net and its variants against representative published methods [CITATION] on FDDB.', '1701.08393-1-75-1': 'For the PASCAL faces and AFW we compare with (1) deformable part based methods, e.g. structure model [CITATION] and Tree Parts Model (TSM) [CITATION]; (2) cascade-based methods, e.g., Headhunter [CITATION].', '1701.08393-1-75-2': 'For the WIDER FACE [CITATION] we compare with (1) aggregated channel feature method (ACF) [CITATION]; (2) deformable part based model [CITATION]; (3) cascaded-based method [CITATION].', '1701.08393-1-76-0': 'AFW dataset.', '1701.08393-1-76-1': 'Figures [REF] shows the precision and recall curves of the compared face detection algorithms on the AFW dataset.', '1701.08393-1-76-2': 'We observe that the Faceness-Net and its variants outperform all the previous approaches by a considerable margin.', '1701.08393-1-76-3': 'The Faceness-Net-SR and Faceness-Net-RP outperform baseline Faceness-Net showing the effectiveness of sharing representation and region proposal technique.', '1701.08393-1-76-4': 'Among all the Faceness-Net variants, Faceness-Net-SP-RP achieves the best performance with a high average precision of [MATH].', '1701.08393-1-77-0': 'PASCAL faces dataset.', '1701.08393-1-77-1': 'Fig. [REF] shows the precision and recall curves.', '1701.08393-1-77-2': 'The baseline Faceness-Net outperforms its variants and other previous face detection algorithms.', '1701.08393-1-77-3': 'Compared with other benchmark datasets, PASCAL faces dataset has fewer number of faces in each image, therefore only requires small number of proposals to achieve a high recall rate.', '1701.08393-1-77-4': 'As shown in Fig. [REF], the quality of proposals generated by the baseline Faceness-Net is higher than its variants when the number of proposal is less than [MATH], which leads to its better face detection performance on PASCAL face dataset.', '1701.08393-1-78-0': 'FDDB dataset.', '1701.08393-1-78-1': 'The results are shown in Fig. [REF] and Fig. [REF].', '1701.08393-1-78-2': 'Similar with AFW dataset, Faceness-Net and its variants achieve better performance compared with previous algorithms evaluated using the discrete score as shown in the Fig [REF].', '1701.08393-1-78-3': 'Faceness-Net baseline achieves [MATH] recall rate, while Faceness-Net-SR and Faceness-Net-RP outperforming the baseline Faceness-Net by [MATH] and [MATH], respectively.', '1701.08393-1-78-4': 'Faceness-Net-SR-RP performs best with a large improvement of [MATH] compared with the baseline Faceness-Net.', '1701.08393-1-78-5': 'As shown in Fig. [REF], the Joint Cascade [CITATION] achieves the best performance evaluated using continues score.', '1701.08393-1-79-0': 'WIDER FACE dataset.', '1701.08393-1-79-1': 'WIDER FACE dataset is currently the largest face detection benchmark dataset.', '1701.08393-1-79-2': 'We evaluate our algorithm using the external setting because our face detectors are trained without using images in the WIDER FACE dataset.', '1701.08393-1-79-3': 'Faceness-Net and its variants yield better performance in all three evaluation settings, namely "Easy", "Medium", and "Hard" as shown in Fig. [REF].', '1701.08393-1-79-4': 'The variants of Faceness-Net outperform baseline Faceness-Net by a considerable margin, suggesting the effectiveness of representation sharing and region proposal techniques.', '1701.08393-1-80-0': 'Discussion: Recent studies [CITATION] achieve better face detection performance on FDDB, AFW, and PASCAL faces datasets compared to our Faceness-Net.', '1701.08393-1-80-1': 'The performance gap between Faceness-Net and other methods arises from two aspects, namely, the better modeling of background clutter and stronger supervision signals.', '1701.08393-1-80-2': 'Table [REF] summarizes the training data and supervision signals used by different algorithms.', '1701.08393-1-80-3': 'Faceness-Net is trained on CelebA and AFLW datasets.', '1701.08393-1-80-4': 'These datasets are originally proposed for face recognition and facial landmark detection, respectively.', '1701.08393-1-80-5': 'The background in CelebA and AFLW is less cluttered and diverse compared with various backgrounds available in WIDER FACE and MS-COCO datasets.', '1701.08393-1-80-6': 'In addition, faces in CelebA and AFLW datasets have smaller variations, both in scale and poses, compared to those captured in the WIDER FACE dataset.', '1701.08393-1-80-7': 'We use [MATH]k face bounding boxes compared to more than [MATH]k face bounding boxes employed by other methods.', '1701.08393-1-81-0': 'To gain a fairer comparison, we train the Faster-RCNN model presented in [CITATION] using the same training sets (AFLW and CelebA) employed by Faceness-Net.', '1701.08393-1-81-1': 'Evaluation is performed on the FDDB dataset.', '1701.08393-1-81-2': 'The results are shown in Fig. [REF].', '1701.08393-1-81-3': 'The Faster-RCNN face detector achieves [MATH] detection rate on the FDDB dataset which is marginally lower than that of Faceness-Net.', '1701.08393-1-81-4': 'Note that, Faceness-SR-RP is not finetuned by using ImageNet data, but achieves better performance than Faster-RCNN.', '1701.08393-1-81-5': 'This is probably because attribute supervisions are more capable in modeling facial parts.', '1701.08393-1-82-0': 'Apart from using more challenging training images, both STN [CITATION] and MTCNN [CITATION] use facial landmarks to localize face.', '1701.08393-1-82-1': 'Facial landmarks indicate explicit location of face parts and thus provide stronger supervisory information than face attributes.', '1701.08393-1-82-2': 'Our method can benefit from these additional factors.', '1701.08393-1-82-3': 'Specifically, it is possible to obtain a stronger Faceness-Net detector using facial landmarks based supervision and datasets with more cluttered background.', '1701.08393-1-83-0': '# Runtime analysis', '1701.08393-1-84-0': 'The runtime of the proposed Faceness-Net-SR-RP is [MATH]ms on a single GPU.', '1701.08393-1-84-1': 'The time includes [MATH]ms to generate faceness proposals with the height of testing image no more than [MATH] pixels.', '1701.08393-1-84-2': 'The efficiency of Faceness-Net-SR-RP is clearly faster than the baseline Faceness-Net since the former shares the layers from Conv1 to Conv4 in its attribute-aware networks.', '1701.08393-1-84-3': 'Previous CNN based face detector [CITATION] achieves good runtime efficiency too.', '1701.08393-1-84-4': 'Our method differs significantly to this method in that we explicitly handle partial occlusion by inferring face likeliness through part responses.', '1701.08393-1-84-5': 'This difference leads to a significant margin of [MATH] in recall rate (Cascade-CNN [MATH], our method [MATH]) when the number of false positives is fixed at [MATH] on the FDDB dataset.', '1701.08393-1-84-6': 'The complete recall rate of the proposed Faceness-Net-SR-RP is [MATH] compared to [MATH] of Cascade-CNN.', '1701.08393-1-84-7': 'At the expense of recall rate, the fast version of Cascade-CNN achieves [MATH]fps on CPU and [MATH]fps on GPU for [MATH] VGA images.', '1701.08393-1-84-8': 'Our Faceness-Net-SR-RP can achieve practical runtime efficiency under the aggressive setting mentioned above, but still with a [MATH] higher recall rate than the Cascade-CNN.', '1701.08393-1-85-0': '# Conclusion', '1701.08393-1-86-0': 'Different from existing face detection studies, we explored the usefulness of face attributes based supervision for learning a robust face detector.', '1701.08393-1-86-1': 'We observed an interesting phenomenon that face part detectors can be obtained from a CNN that is trained on recognizing attributes from uncropped face images, without explicit part supervision.', '1701.08393-1-86-2': "Consequently, we introduced the notion of 'faceness' score, which was carefully formulated through considering facial parts responses and the associated spatial arrangements.", '1701.08393-1-86-3': 'The faceness score can be employed to re-rank candidate windows of any region proposal techniques to generate a modest set of high-quality face proposals with high recall.', '1701.08393-1-86-4': 'With the generated face proposals, we trained a strong face detector that demonstrated promising performance on various face detection benchmark datasets.', '1701.08393-1-87-0': 'Despite Faceness-Net achieves encouraging face detection performance on benchmark datasets, it may fail to detect tiny faces (with a resolution as low as 20 pixels height).', '1701.08393-1-87-1': 'The visual appearance between tiny and normal-size faces exhibits a huge difference.', '1701.08393-1-87-2': 'In particular, the facial parts such as eyes, nose or mouth can be barely distinguished from tiny faces, which makes responses produced by attribute-aware networks meaningless.', '1701.08393-1-87-3': 'In order to retrieve tiny faces, data augmentation and multi-scale inference may be adopted.', '1701.08393-1-87-4': 'Nonetheless, learning scale-invariant representation is still an open problem.', '1701.08393-1-87-5': 'In this study, we do not deal with tiny faces explicitly.', '1701.08393-1-87-6': 'It is part of our on-going works.'}

{'1701.08393-2-0-0': 'We propose a deep convolutional neural network (CNN) for face detection leveraging on facial attributes based supervision.', '1701.08393-2-0-1': 'We observe a phenomenon that part detectors emerge within CNN trained to classify attributes from uncropped face images, without any explicit part supervision.', '1701.08393-2-0-2': 'The observation motivates a new method for finding faces through scoring facial parts responses by their spatial structure and arrangement.', '1701.08393-2-0-3': 'The scoring mechanism is data-driven, and carefully formulated considering challenging cases where faces are only partially visible.', '1701.08393-2-0-4': 'This consideration allows our network to detect faces under severe occlusion and unconstrained pose variations.', '1701.08393-2-0-5': 'Our method achieves promising performance on popular benchmarks including FDDB, PASCAL Faces, AFW, and WIDER FACE.', '1701.08393-2-1-0': 'Face Detection, Deep Learning, Convolutional Neural Network.', '1701.08393-2-2-0': '# Introduction', '1701.08393-2-3-0': 'Face detection is an important and long-standing problem in computer vision.', '1701.08393-2-3-1': 'A number of methods have been proposed in the past, including neural network based methods [CITATION], cascade structures [CITATION] and deformable part models (DPM) [CITATION] detectors.', '1701.08393-2-3-2': 'There has been a resurgence of interest in applying convolutional neural networks (CNN) on this classic problem [CITATION].', '1701.08393-2-3-3': 'Many of these methods follow a cascade object detection framework [CITATION], some of which directly adopt the effective generic object detection framework RCNN [CITATION] and Faster-RCNN [CITATION] as the backbone network, with very deep networks (e.g., 101-layer ResNet) to leverage the remarkable representation learning capacity of deep CNN [CITATION].', '1701.08393-2-4-0': 'While face bounding boxes have been used as a standard supervisory source for learning a face detector, the usefulness of facial attributes remains little explored.', '1701.08393-2-4-1': 'In this study, we show that facial attributes based supervision can effectively enhance the capability of a face detection network in handling severe occlusions.', '1701.08393-2-4-2': 'As depicted in Fig. [REF], a CNN supervised with facial attributes can detect faces even when more than half of the face region is occluded.', '1701.08393-2-4-3': 'In addition, the CNN is capable of detecting faces with large pose variation, e.g., profile view without training separate models under different viewpoints.', '1701.08393-2-4-4': 'Such compelling results are hard to achieve by using supervision based on face bounding boxes alone, especially when the training dataset has limited scene diversity and pose variations.', '1701.08393-2-5-0': 'In this study, we show the benefits of facial attributes supervision through the following considerations:', '1701.08393-2-6-0': '(1) Discovering facial parts responses supervised by facial attributes: The human face has a unique structure.', '1701.08393-2-6-1': 'We believe the reasoning of the unique structure of local facial parts (e.g., eyes, nose, mouth) help detecting faces under unconstrained environments.', '1701.08393-2-6-2': 'We observe an interesting phenomenon that one can actually obtain part detectors within a CNN by training it to classify part-level binary attributes (e.g., mouth attributes including big lips, opened mouth, smiling, wearing lipstick) from uncropped face images, without any explicit part supervision.', '1701.08393-2-6-3': 'The trained CNN is then capable of generating high-quality facial part responses in its deep layers that strongly indicate the locations of the face parts.', '1701.08393-2-6-4': "The examples depicted in Fig. [REF](b) show the response maps (known as 'partness map' in our paper) of five different face parts.", '1701.08393-2-7-0': "(2) Computing faceness score from responses configurations: Given the parts' responses, we formulate an effective method to reason the degree of face likeliness (which we call faceness score) through analyzing their spatial arrangement.", '1701.08393-2-7-1': 'For instance, the hair should appear above the eyes, and the mouth should only appear below the nose.', '1701.08393-2-7-2': 'Any inconsistency would be penalized.', '1701.08393-2-7-3': 'Faceness scores will be derived and used to re-rank candidate windows to obtain a set of face proposals.', '1701.08393-2-7-4': 'Our face proposal approach enjoys a high recall with just a modest number of proposals (over 90% of face recall with around [MATH] proposals, [MATH]0.5% of full sliding windows, and [MATH]10% of generic object proposals [CITATION], measured on the FDDB dataset [CITATION]).', '1701.08393-2-8-0': '(3) Refining the face hypotheses - Both the aforementioned components offer a chance to find a face even under severe occlusion and pose variations.', '1701.08393-2-8-1': 'The output of these components is a small set of high-quality face bounding box proposals that cover most faces in an image.', '1701.08393-2-8-2': 'Given the face proposals, we design a multi-task CNN [CITATION] in the second stage to refine the hypotheses further, by simultaneously recognizing the true faces and estimating more precise face locations.', '1701.08393-2-9-0': "Our main contribution in this study is the novel use of CNN and attributes supervision for discovering facial parts' responses.", '1701.08393-2-9-1': 'We show that part detectors emerge within a CNN trained to classify attributes from uncropped face images, without any explicit part supervision.', '1701.08393-2-9-2': "The parts' responses are subsequently employed to generate high-quality proposals for training a face detector that is robust to severe occlusion.", '1701.08393-2-9-3': 'The findings aforementioned are new in the literature.', '1701.08393-2-9-4': 'It is worth pointing out that our network is trained on datasets that are not targeted for face detection (CelebA [CITATION] for face recognition, and AFLW [CITATION] for face alignment) and with simple backgrounds.', '1701.08393-2-9-5': 'Nevertheless, it still achieves promising performance on various face detection benchmarks including FDDB, PASCAL Faces, AFW, and the challenging WIDER FACE dataset.', '1701.08393-2-10-0': 'In comparison to our earlier version of this work [CITATION], we present a more effective design of CNN to achieve improved performance and speed.', '1701.08393-2-10-1': 'Firstly, in contrast to our previous work that requires independent convolutional networks for learning responses of different facial parts, we now share feature representations between these attribute-aware networks.', '1701.08393-2-10-2': 'The sharing of low and mid-levels representations largely reduce the number of parameters in our framework ([MATH]83% fewer parameters), while improving the robustness of the feature representation.', '1701.08393-2-10-3': 'Secondly, our previous framework relies on external generic object proposal generators such as selective search [CITATION] and EdgeBox [CITATION] for proposing candidate windows.', '1701.08393-2-10-4': 'Inspired by region proposal network presented in [CITATION], in this study we directly generate proposals from our attribute-aware networks, thus proposal generation becomes an inherent part of the framework.', '1701.08393-2-10-5': 'This design not only leads to improved computation efficiency but also higher recall rate compared with generic object proposal algorithms.', '1701.08393-2-10-6': 'Thirdly, we compare our face detector pre-trained on the task of facial attributes classification with that pre-trained on ImageNet large-scale object classification.', '1701.08393-2-10-7': 'Apart from the above major changes, we also provide more technical details and discussions.', '1701.08393-2-10-8': 'Additional experiments are conducted on the challenging WIDER FACE dataset [CITATION].', '1701.08393-2-11-0': '# Related Work', '1701.08393-2-12-0': 'There is a long history of using neural network for the task of face detection [CITATION].', '1701.08393-2-12-1': 'An early face detection survey [CITATION] provides an extensive coverage on relevant methods.', '1701.08393-2-12-2': 'Here we highlight a few notable studies.', '1701.08393-2-12-3': 'Rowley et al. [CITATION] exploit a set of neural network-based filters to detect the presence of faces in multiple scales and merge the detections from individual filters.', '1701.08393-2-12-4': 'Osadchy et al. [CITATION] demonstrate that a joint learning of face detection and pose estimation significantly improves the performance of face detection.', '1701.08393-2-12-5': 'The seminal work of Vaillant et al. [CITATION] adopt a two-stage coarse-to-fine detection.', '1701.08393-2-12-6': 'Specifically, the first stage approximately locates the face region, whilst the second stage provides a more precise localization.', '1701.08393-2-12-7': 'Our approach is inspired by these studies, but we introduce innovations on many aspects.', '1701.08393-2-12-8': 'For instance, our first stage network is conceptually different from that of [CITATION], and many recent deep learning detection frameworks - we train attribute-aware networks to achieve precise localization of facial parts and exploit their spatial structure for inferring face likeliness.', '1701.08393-2-12-9': 'This concept is new and it allows our model to detect faces under severe occlusion and pose variations.', '1701.08393-2-12-10': 'While great efforts have been devoted to addressing face detection under occlusion [CITATION], these methods are all confined to frontal faces.', '1701.08393-2-12-11': 'In contrast, our model can discover faces under variations of both pose and occlusion.', '1701.08393-2-13-0': 'In the last decades, cascade based [CITATION] and deformable part models (DPM) detectors dominate face detection approaches.', '1701.08393-2-13-1': 'Viola and Jones [CITATION] introduced fast Haar-like features computation via integral image and boosted cascade classifier.', '1701.08393-2-13-2': 'Various studies thereafter follow a similar pipeline.', '1701.08393-2-13-3': 'Among the variants, SURF cascade [CITATION] was one of the top performers.', '1701.08393-2-13-4': 'Later Chen et al. [CITATION] demonstrate state-of-the-art face detection performance by learning face detection and face alignment jointly in the same cascade framework.', '1701.08393-2-13-5': 'Deformable part models define face as a collection of parts.', '1701.08393-2-13-6': 'Latent Support Vector Machine is typically used to find the parts and their relationships.', '1701.08393-2-13-7': 'DPM is shown more robust to occlusion than the cascade based methods.', '1701.08393-2-13-8': 'A recent study [CITATION] demonstrates good performance with just a vanilla DPM, achieving better results than more sophisticated DPM variants [CITATION].', '1701.08393-2-14-0': 'Recent studies [CITATION] show that face detection can be further improved by using deep learning.', '1701.08393-2-14-1': 'The network proposed by [CITATION] does not have an explicit mechanism to handle occlusion, the face detector therefore fails to detect faces with heavy occlusions, as acknowledged by the authors.', '1701.08393-2-14-2': 'Cascade based convolutional neural networks [CITATION] replace boosting classifiers with a set of small CNNs to quickly reject negative samples in the early stage.', '1701.08393-2-14-3': 'Recent studies [CITATION] exploit facial landmarks as supervision signals to improve face detection performance.', '1701.08393-2-14-4': 'In this study, we show that facial attributes can serve as an important source too for learning a robust face detector.', '1701.08393-2-15-0': 'The first stage of our model is partially inspired by generic object proposal approaches [CITATION].', '1701.08393-2-15-1': 'Generic object proposal generators are commonly used in standard object detection algorithms for providing high-quality and category-independent bounding boxes.', '1701.08393-2-15-2': 'These methods typically involve redundant computations over regions that are covered by multiple proposals.', '1701.08393-2-15-3': 'To reduce computation, Ren et al. [CITATION] propose Region Proposal Network (RPN) to generate proposals from high-level response maps in a CNN through a set of predefined anchor boxes.', '1701.08393-2-15-4': 'Both generic object proposal and RPN methods do not consider the unique structure and parts on the face.', '1701.08393-2-15-5': 'Hence, no mechanism is available to recall faces when the face is only partially visible.', '1701.08393-2-15-6': 'These shortcomings motivate us to formulate the new faceness measure to achieve high recall on faces while reducing the number of candidate windows to half the original (compared to the original RPN [CITATION]).', '1701.08393-2-16-0': '# Faceness-Net', '1701.08393-2-17-0': 'This section introduces the baseline Faceness-Net.', '1701.08393-2-17-1': 'We first briefly overview the entire pipeline and then discuss the details.', '1701.08393-2-17-2': 'As shown in Fig. [REF], Faceness-Net consists of two stages, i.e., (i) generating face proposals from partness maps by ranking candidate windows using faceness scores, and (ii) refining face proposals for face detection.', '1701.08393-2-18-0': 'First stage.', '1701.08393-2-18-1': 'A full image [MATH] is used as an input to a CNN to generate the partness map for each face part.', '1701.08393-2-18-2': 'A set of CNNs, known as attribute-aware networks, are used to generate the partness map of different parts.', '1701.08393-2-18-3': 'The partness map is obtained by weighted averaging over all the response maps at its top convolutional layer.', '1701.08393-2-18-4': 'The map indicates the location of a specific facial component presented in the image, e.g., hair, eyes, nose, mouth, and beard denoted by [MATH], [MATH], [MATH], [MATH], and [MATH], respectively.', '1701.08393-2-18-5': "For illustration, we sum all these maps into a face label map [MATH], which clearly suggests faces' locations.", '1701.08393-2-19-0': 'Given a set of candidate windows [MATH] that are generated by existing object proposal methods such as [CITATION], or a region proposal network (RPN) [CITATION], we rank these windows according to their faceness scores, [MATH], which are derived from the partness maps with respect to different facial parts configurations, as illustrated at the bottom of Fig. [REF](a).', '1701.08393-2-19-1': "For example, as visualized in Fig. [REF](a), a candidate window 'A' covers a local region of [MATH] (i.e., hair) and its faceness score is calculated by dividing the values at its upper part with respect to the values at its lower part, because hair is more likely to present at the top of a face region.", '1701.08393-2-19-2': 'The bottom part of Fig. [REF](a) illustrates the spatial configurations of five facial parts.', '1701.08393-2-19-3': 'The facial configurations can be learned from the training data.', '1701.08393-2-19-4': 'To reduce the number of the proposed windows, we apply non-maximum suppression (NMS) to smooth the scores by leveraging the spatial relations among these windows.', '1701.08393-2-19-5': "A final faceness score of 'A' is obtained by averaging over the scores of these parts.", '1701.08393-2-19-6': 'We perform another round of NMS to further reduce the number of proposed windows using faceness score.', '1701.08393-2-19-7': 'In this case, a large number of false positive windows can be pruned.', '1701.08393-2-19-8': 'The proposed approach is capable of coping with severe face occlusions.', '1701.08393-2-19-9': "As shown in Fig. [REF](b), face windows 'A' and 'E' can be retrieved by objectness [CITATION] only if a lot of windows are proposed, while windows 'A' and 'E' rank top [MATH] by using our method.", '1701.08393-2-20-0': 'Second stage.', '1701.08393-2-20-1': 'The face proposals are refined by training a multi-task CNN, where face classification and bounding box regression are jointly optimized (Fig. [REF]).', '1701.08393-2-21-0': '## Attribute-Aware Networks', '1701.08393-2-22-0': 'The first stage of the baseline Faceness-Net consists of multiple attribute-aware networks for generating response maps of different parts (Fig. [REF]).', '1701.08393-2-22-1': 'Five networks are needed to cover all five pre-defined facial components, i.e., hair, eyes, nose, mouth, and beard.', '1701.08393-2-22-2': 'These attribute-aware networks share the same structure.', '1701.08393-2-22-3': 'Next, we first discuss the network structure and subsequently show that these networks can share representation to reduce parameters.', '1701.08393-2-23-0': 'Network structure.', '1701.08393-2-23-1': 'The choice of network structure for extracting partness maps is flexible.', '1701.08393-2-23-2': 'Figure [REF](a) depicts the structure and hyper-parameters of the CNN used in the baseline Faceness-Net.', '1701.08393-2-23-3': 'This convolutional structure is inspired by the AlexNet [CITATION], which was originally proposed for object categorization.', '1701.08393-2-23-4': 'Specifically, the network stacks seven convolutional layers (conv1 to conv7) and two max-pooling layers (max1 and max2).', '1701.08393-2-23-5': 'The hyper-parameters of each layer is specified in Fig. [REF](a).', '1701.08393-2-24-0': 'Once the attribute networks are trained (training details are provided in Sec. [REF]), we obtain the response map of a part through first applying [MATH] normalization on the feature map for each channel and then element-wise averaging along the channels.', '1701.08393-2-24-1': 'We examine the response maps obtained from the attribute-aware networks.', '1701.08393-2-24-2': 'As observed from Fig. [REF](b), the feature maps of the first few convolutional layers do not clearly indicate the locations of facial parts.', '1701.08393-2-24-3': 'However, a clear indication of the facial component can be seen from responses of conv7.', '1701.08393-2-24-4': 'Consequently, we obtain an initial response map from the conv7 layer.', '1701.08393-2-24-5': "The final partness map that matches the input image's size is obtained through performing unpooling [CITATION] on the conv7's response map.", '1701.08393-2-25-0': 'Shared representation.', '1701.08393-2-25-1': "It is observed that the feature maps of earlier layers across the different attribute-aware networks are almost identical and they are not indicative of parts' locations.", '1701.08393-2-25-2': 'Motivated by these observations, instead of designating separate attribute-aware networks for different facial components, we share early convolutional layers of these networks to reduce parameters.', '1701.08393-2-25-3': "Specifically, the first four convolutional layers that do not clearly suggests parts' locations are shared, followed by five branches, each of which consists of two convolutional layers responsible for a facial component, as shown in Fig. [REF].", '1701.08393-2-25-4': 'Note that in comparison to the structure presented in Fig. [REF](a), we additionally remove a convolutional layer and trim the number of filters in other layers to reduce parameters.', '1701.08393-2-25-5': 'The sharing of representation and filter reduction lead to a single attribute-aware network with [MATH] fewer parameters than the original five attribute-aware networks.', '1701.08393-2-25-6': 'We denote a Faceness-Net with shared representation as Faceness-Net-SR.', '1701.08393-2-25-7': 'We will show that this network structure not only reduces computations but also improves the robustness of feature representation for face detection.', '1701.08393-2-26-0': '## Learning to Generate Partness Maps', '1701.08393-2-27-0': 'Pre-training the attribute-aware networks.', '1701.08393-2-27-1': 'Pre-training generally helps to improve the performance of a deep network.', '1701.08393-2-27-2': 'There are two plausible pre-training options depending upon whether we share the representations across attribute-aware networks or not.', '1701.08393-2-28-0': 'The first option is to pre-train our attribute-aware networks with massive general object categories in ImageNet [CITATION].', '1701.08393-2-28-1': 'From our observations, this option works well when the representations across networks are not shared.', '1701.08393-2-28-2': 'Since each attribute-aware network originally has access only to a particular group of data specific to a certain attribute, the larger-scale ImageNet data helps to mitigate the overfitting issue that is caused by insufficient data.', '1701.08393-2-29-0': 'The second option omits the ImageNet pre-training stage and trains a network directly on the task of facial attributes classification.', '1701.08393-2-29-1': 'This option works best when we adopt the shared representation scheme discussed in Sec. [REF].', '1701.08393-2-29-2': 'Thanks to the sharing of representation, the attribute-aware network requires a relatively smaller quantity of training data.', '1701.08393-2-29-3': 'Thus, no overfitting is observed despite we use the facial attributes dataset, which is much smaller in scale, i.e., 180,000 images compared to 1 million images in ImageNet.', '1701.08393-2-30-0': 'Fine-tuning the attribute-aware networks.', '1701.08393-2-30-1': 'Once an attribute-network is pre-trained, we can fine-tune it to generate the desired partness maps.', '1701.08393-2-30-2': 'There are different fine-tuning strategies, but not all of them can generate meaningful partness maps for deriving a robust faceness score.', '1701.08393-2-31-0': "As shown in Fig. [REF](b), a deep network trained on generic objects, e.g., AlexNet [CITATION], is not capable of providing us with precise faces' locations, let alone partness map.", '1701.08393-2-31-1': 'To generate accurate partness maps, we explore multiple ways for learning an attribute-aware network.', '1701.08393-2-31-2': 'The most straightforward manner is to use the image and its pixel-wise segmentation label map as input and target, respectively.', '1701.08393-2-31-3': 'This setting is widely employed in image labeling [CITATION].', '1701.08393-2-31-4': 'However, it requires label maps with pixel-wise annotations, which are expensive to collect.', '1701.08393-2-31-5': 'Another setting is image-level classification (i.e., faces and non-faces), as shown in Fig. [REF](c).', '1701.08393-2-31-6': 'It works well where the training images are well-aligned, such as face recognition [CITATION].', '1701.08393-2-31-7': 'Nevertheless, it suffers from complex background clutter because the supervisory information is not sufficient to account for rich and diverse face variations.', '1701.08393-2-31-8': "Its learned feature maps contain too many noises, which overwhelm the actual faces' locations.", '1701.08393-2-31-9': 'Attribute learning in Fig. [REF](d) extends the binary classification in (c) to the extreme by using a combination of attributes to capture face variations.', '1701.08393-2-31-10': "For instance, an 'Asian' face can be distinguished from a 'European' face.", '1701.08393-2-31-11': 'However, our experiments demonstrate that this setting is not robust to occlusion.', '1701.08393-2-32-0': 'Figure [REF](e) shows the partness maps obtained by the baseline Faceness-Net, for which the attribute networks do not share representations.', '1701.08393-2-32-1': 'The strategy we propose extends (d) by partitioning attributes into groups based on facial components.', '1701.08393-2-32-2': "For instance, 'black hair', 'blond hair', 'bald', and 'bangs' are grouped together, as all of them are related to hair.", '1701.08393-2-32-3': 'The grouped attributes are summarized in Table [REF].', '1701.08393-2-32-4': 'In this case, face parts are modeled separately.', '1701.08393-2-32-5': 'If one part is occluded, the face region can still be localized by the other parts.', '1701.08393-2-32-6': 'We take the Hair-Branch shown in the stage one of Fig. [REF] as an example to illustrate the learning procedure.', '1701.08393-2-32-7': 'Let [MATH] be a set of full face images and the attribute labels of hair.', '1701.08393-2-32-8': 'Images are first resized to [MATH] where [MATH] and [MATH] indicate there are nine attributes ([MATH]) related to hair as listed in Table [REF].', '1701.08393-2-32-9': 'Learning is formulated as a multi-variate classification problem by minimizing the cross-entropy loss, [EQUATION] where [MATH] is modeled as a sigmoid function, i.e. [MATH], indicating the probability of the presence of [MATH] attributes.', '1701.08393-2-32-10': 'The features of [MATH] are denoted as [MATH].', '1701.08393-2-32-11': 'To facilitate the learning, we stack two fully-connected layers on top of the last convolutional layer of the structure shown in Fig. [REF].', '1701.08393-2-32-12': 'We optimize the loss function by using stochastic gradient descent with back-propagation.', '1701.08393-2-32-13': 'After training the attribute-aware network, the fully-connected layers are removed to make the network fully convolutional again.', '1701.08393-2-33-0': 'Figure [REF](f) shows the partness maps that are generated from the networks with shared representation, i.e., Faceness-Net-SR (see Fig. [REF]).', '1701.08393-2-33-1': 'Visually, the partness maps generated by this model are noisier compared to Fig. [REF](e).', '1701.08393-2-33-2': 'The key reason is that the Faceness-Net-SR is not pre-trained using ImageNet data but directly trained on the attribute classification task.', '1701.08393-2-33-3': "Despite the noisy partness maps, they actually capture more subtle parts' responses and therefore lead to higher recall rate in the subsequent face proposal stage, provided that the number of proposals is sufficiently large.", '1701.08393-2-34-0': '## Generating Candidate Windows', '1701.08393-2-35-0': 'Face detection can be improved if the inputs are formed by a moderate number of proposals with a high recall rate.', '1701.08393-2-35-1': 'To produce the required proposals, we will explore two plausible choices to generate the initial set of candidate windows.', '1701.08393-2-36-0': 'Generic object proposal.', '1701.08393-2-36-1': 'Generic object scoring is primarily employed to reduce the computational cost of a detector.', '1701.08393-2-36-2': 'It has also been shown improving detection accuracy due to the reduction of spurious false positives [CITATION].', '1701.08393-2-36-3': 'A variety of cues has been proposed to quantify the objectness of an image window, e.g., norm of the gradient [CITATION], edges [CITATION], or integration of a number of low-level features [CITATION].', '1701.08393-2-36-4': 'Other popular methods include super-pixel based approaches, e.g., selective search [CITATION], randomized Prim [CITATION], and multi-scale combinatorial grouping [CITATION].', '1701.08393-2-36-5': 'Our framework can readily employ these generic candidate windows for ranking using the proposed faceness score (Sec. [REF]).', '1701.08393-2-37-0': 'Template proposal.', '1701.08393-2-37-1': 'In order to decouple the dependence of object proposal algorithms to generate candidate windows, we propose a template proposal method in which candidate windows are generated from multiple predefined templates on feature maps.', '1701.08393-2-38-0': 'We provide an example below on using a partness map of hair, [MATH], for template proposal.', '1701.08393-2-38-1': 'As shown in Fig. [REF], each value of location [MATH] on the partness map [MATH] indicates the probability of the appearance of the hair component.', '1701.08393-2-38-2': 'We select a set of [MATH] locations [MATH] with a probability [MATH] higher than [MATH].', '1701.08393-2-38-3': 'For each selected location, multiple template proposals are generated, where the number of maximum possible proposals for each location is fixed as [MATH].', '1701.08393-2-38-4': 'The proposals are obtained from predefined reference boxes, which we call templates.', '1701.08393-2-38-5': 'For each face part, templates are centered at different locations considering the structure of the human face.', '1701.08393-2-38-6': 'In addition, they are associated with a specific scale and aspect ratio, as shown at the top of Fig. [REF].', '1701.08393-2-38-7': 'For instance, the templates of the hair region are centered at [MATH] and the templates of eyes are centered at [MATH], where [MATH] and [MATH] represent the width and height of an anchor.', '1701.08393-2-38-8': "Similar to previous work [CITATION], these templates are translation invariant up to the network's total stride, and the method does not incur extra cost for addressing scales thanks to the multi-scale templates.", '1701.08393-2-39-0': 'In our study, we define [MATH] scales and [MATH] aspect ratio, yielding [MATH] templates at each selected position.', '1701.08393-2-39-1': 'Specifically, we use [MATH] scales with box areas of [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] pixels, and [MATH] aspect ratio of [MATH] (with width to height).', '1701.08393-2-39-2': 'The parameters of templates, i.e., center location, scale and aspect ratio are selected by maximizing the recall rate given an average number of [MATH] proposals per image.', '1701.08393-2-39-3': 'In our study, we perform a grid search on the training set to select the parameters.', '1701.08393-2-40-0': 'Discussion.', '1701.08393-2-40-1': 'Both the generic objectness measures and RPN (trained on ImageNet) are devoted to generic objects therefore not suitable to propose windows specific to faces.', '1701.08393-2-40-2': 'In particular, applying a generic proposal generator directly would produce an enormous number of candidate windows but an only minority of them contain faces.', '1701.08393-2-40-3': 'While RPN is computationally more efficient than generic object proposal generators, it cannot be directly applied to our problem too.', '1701.08393-2-40-4': 'Specifically, in order for the RPN to cope with faces with tiny size and various poses, a large number of anchor boxes are required, leading to an enormous number of proposals.', '1701.08393-2-40-5': 'In the next section, we discuss a new faceness measure that can complement existing object proposal generators or the template proposal method to achieve high recall on faces, while significantly reduce the number of candidate windows.', '1701.08393-2-40-6': 'The proposed faceness measure scheme is in practice related to traditional face detector schemes based on Haar features, with the difference that here the Haar features pool CNN feature responses instead of pixel luminance values.', '1701.08393-2-41-0': '## Ranking Windows by Faceness Score', '1701.08393-2-42-0': 'After generating candidate windows based on the methods described in Sec. [REF], our approach computes a faceness score on these windows to return a ranked set of top-scoring face proposals.', '1701.08393-2-42-1': 'Figure [REF] illustrates the procedure of deriving the faceness measure from the partness maps of hair and eyes.', '1701.08393-2-42-2': 'Let [MATH] be the faceness score of a window [MATH].', '1701.08393-2-42-3': 'For example, as shown in Fig. [REF](a), given a partness map of hair, [MATH], [MATH] is attained by dividing the sum of values in ABEF (green) by the sum of values in FECD.', '1701.08393-2-42-4': 'Similarly, Fig. [REF](b) shows that [MATH] is obtained by dividing the sum of values in EFGH (green) with respect to ABEF+HGCD of [MATH].', '1701.08393-2-42-5': 'For both of the above examples, a larger value of [MATH] indicates a higher overlapping ratio of [MATH] with a face.', '1701.08393-2-42-6': 'The choice of method for computing the faceness score is flexible.', '1701.08393-2-42-7': 'It is possible to compute the faceness score using other forms of handcrafted features that can effectively capture the face structure through response maps.', '1701.08393-2-43-0': 'The spatial configurations, such as ABEF in Fig. [REF](a) and EFGH in Fig. [REF](b), can be learned from data.', '1701.08393-2-43-1': 'We take hair as an example.', '1701.08393-2-43-2': 'We need to learn the positions of points E and F, which can be represented by the [MATH]-coordinates of ABCD, i.e., the proposed window.', '1701.08393-2-43-3': 'For instance, the position of E in Fig. [REF](a) can be represented by [MATH] and [MATH], implying that the value of its [MATH]-axis is a linear combination of [MATH] and [MATH].', '1701.08393-2-43-4': 'With this representation, [MATH] can be efficiently computed by using the integral image (denoted as [MATH]) of the partness map.', '1701.08393-2-43-5': 'For instance, [MATH] in (a) is attained by [MATH].', '1701.08393-2-43-6': '[EQUATION] where [MATH] signifies the value at the location [MATH].', '1701.08393-2-44-0': 'Given a training set [MATH], where [MATH] and [MATH] denote the [MATH]-th window and its label (i.e. face/non-face), respectively.', '1701.08393-2-44-1': 'Let [MATH] be the cropped partness map with respect to the [MATH]-th window, e.g., region ABCD in [MATH].', '1701.08393-2-44-2': 'This problem can be formulated as maximum a posteriori (MAP) estimation [EQUATION] where [MATH] represents a set of parameters when learning the spatial configuration of hair (Fig. [REF](a)).', '1701.08393-2-44-3': 'The terms [MATH] and [MATH] denote the likelihood and prior, respectively.', '1701.08393-2-44-4': 'The likelihood of faceness can be modeled by a sigmoid function, i.e., [MATH], where [MATH] is a coefficient.', '1701.08393-2-44-5': 'This likelihood measures the confidence of partitioning the face and non-face, given a certain spatial configuration.', '1701.08393-2-44-6': 'The prior term can be factorized, [MATH], where [MATH] is a uniform distribution between zero and one, as it indicates the coefficients of linear combination, [MATH] models the prior of the candidate window, which can be generated by object proposal methods, and [MATH] is the partness map as obtained in Sec. [REF].', '1701.08393-2-44-7': 'Since [MATH] typically has a low dimension (e.g., one dimension of hair), it can be simply obtained by line search.', '1701.08393-2-44-8': 'Note that Eq. [REF] can be easily extended to model more complex spatial configurations.', '1701.08393-2-44-9': 'This process is similar with learning Haar templates using boosting classifier, but requires less computation while achieving good performance compared with more elaborated process.', '1701.08393-2-45-0': '## Face Detection', '1701.08393-2-46-0': 'The top candidate windows that are ranked by faceness score attain a high recall rate.', '1701.08393-2-46-1': 'These face proposals can be subsequently fed to the multi-task CNN at stage 2 of the proposed pipeline (Fig. [REF]) for face detection.', '1701.08393-2-47-0': 'Pre-training.', '1701.08393-2-47-1': 'We directly use the earlier layers of attribute-aware networks (the stage-1 network with shared representation as shown in Fig. [REF]) up to conv4 as the pre-trained model for the multi-task CNN of stage 2.', '1701.08393-2-47-2': 'After conv4, as shown in Fig. [REF], the multi-task CNN forks into two branches, each of which consists of two convolutional layers and two fully connected layers.', '1701.08393-2-47-3': 'The two branches are optimized to handle different tasks, namely face classification and bounding box regression, respectively.', '1701.08393-2-48-0': 'It is worth pointing out that the multi-task CNN can be pre-trained on the ImageNet data, instead of reusing the parameters of the attribute-aware networks.', '1701.08393-2-48-1': 'Nevertheless, we found that the multi-task CNN converges much faster given the face attributes based pretrained model.', '1701.08393-2-48-2': 'Specifically, the attribute pretrained network only requires [MATH] iterations to converge during the face detection fine-tuning stage, in comparison to more than [MATH] iterations for the ImageNet pertrained network using the same mini-batch size.', '1701.08393-2-48-3': 'We conjecture that much less effort is needed to transform the feature representations learned from the facial attribute classification task to the face detection task.', '1701.08393-2-49-0': 'Multi-task fine-tuning.', '1701.08393-2-49-1': 'We fine-tune the first branch of the multi-task CNN for face classification and the second branch for bounding box regresssion.', '1701.08393-2-49-2': 'Fine-tuning is performed using the face proposals obtained from the previous step (Sec [REF]).', '1701.08393-2-49-3': 'For face classification, we assign a face proposal to its closest ground truth bounding box based on the Euclidean distance between their respective center coordinates.', '1701.08393-2-49-4': 'A face proposal is considered positive if the Intersection over Union (IoU) between the proposal box and the assigned ground truth box is larger than [MATH]; otherwise it is negative.', '1701.08393-2-49-5': 'For bounding box regression, we train the second branch of the multi-task CNN to regress each proposal to the coordinates of its assigned ground truth box.', '1701.08393-2-49-6': 'If the proposed window is a positive sample, the regression target is generated by Eq. [REF].', '1701.08393-2-49-7': 'We use the following parameterizations of the 4 coordinates: [EQUATION] where [MATH] is a normalizing factor.', '1701.08393-2-49-8': 'The vector [MATH] denotes the top-left and bottom-right coordinates of a bounding box.', '1701.08393-2-49-9': 'Variables [MATH], [MATH], and [MATH] represent the ground truth box, proposed box, and regression target.', '1701.08393-2-49-10': 'This process normalizes regression target into a range of [MATH] which can be easily optimized by using least square loss.', '1701.08393-2-49-11': 'The standard bounding box regression targets [CITATION] and [MATH] loss are also applicable.', '1701.08393-2-49-12': 'If a proposed window is non-face, the CNN outputs a vector of [MATH] whose gradients will be ignored during back propagation.', '1701.08393-2-50-0': 'More implementation details are given below.', '1701.08393-2-50-1': 'During the training process, if the number of positive samples in a mini-batch is smaller than [MATH] of the total samples, we randomly crop the ground truth faces and add these samples as additional positive samples.', '1701.08393-2-50-2': 'Therefore, the ratio of positive samples and negative samples is kept not lower than [MATH].', '1701.08393-2-50-3': 'Meanwhile, we conduct bounding box NMS on the negative samples.', '1701.08393-2-50-4': 'The IoU for the NMS is set to [MATH].', '1701.08393-2-50-5': 'The proposed bounding boxes are cropped and then resized to [MATH].', '1701.08393-2-50-6': 'To handle blurry faces, we augment our training samples by applying Gaussian blurring.', '1701.08393-2-50-7': 'The fine-tuning consumes [MATH] iterations with a batch size of [MATH] images.', '1701.08393-2-50-8': 'We adopt Euclidean loss and cross-entropy loss for bounding box regression and face classification, respectively.', '1701.08393-2-51-0': '# Experimental Settings', '1701.08393-2-52-0': 'Training datasets.', '1701.08393-2-52-1': '(i) We employ CelebA dataset [CITATION] to train our attribute-aware networks.', '1701.08393-2-52-2': 'The dataset contains [MATH] web-based images exclusive from the LFW [CITATION], FDDB [CITATION], AFW [CITATION] and PASCAL [CITATION] datasets.', '1701.08393-2-52-3': 'Every image in the dataset are labeled with [MATH] facial attributes.', '1701.08393-2-52-4': 'We select [MATH] facial attributes from CelebA dataset for each image and divide the attributes into five categories based on their respective facial parts as shown in Table [REF].', '1701.08393-2-52-5': 'We randomly select [MATH] images from the CelebA dataset for training and the remaining is reserved as the validation set.', '1701.08393-2-52-6': '(ii) For face detection training, we choose [MATH] face images from the AFLW dataset [CITATION] to ensure a balanced out-of-plane pose distribution.', '1701.08393-2-52-7': 'We observe a large number of missed annotated faces in the AFLW dataset, which could hamper the training of our face detector.', '1701.08393-2-52-8': 'Hence, we re-annotate face bounding boxes for those missing faces.', '1701.08393-2-52-9': 'The total number of faces in the re-annotated AFLW is [MATH] compared with [MATH] in the original data.', '1701.08393-2-52-10': 'As negative samples, we randomly select [MATH] person-free images from the PASCAL VOC 2007 dataset [CITATION].', '1701.08393-2-53-0': 'Part response test dataset.', '1701.08393-2-53-1': 'We use LFW dataset [CITATION] for evaluating the quality of part response maps for part localization.', '1701.08393-2-53-2': 'Since the original dataset does not come with part-level bounding boxes, we label the boxes with the following scheme.', '1701.08393-2-53-3': 'We follow the annotations provided by [CITATION] on hairs and beard for a set of [MATH] LFW images.', '1701.08393-2-53-4': 'Hair bounding boxes are generated with minimal and maximal coordinates of hair superpixel as shown in Fig. [REF].', '1701.08393-2-53-5': 'Using a similar strategy, eye, nose and mouth bounding boxes are obtained from the manually labeled [MATH] dense facial landmarks [CITATION] on the original LFW [CITATION] images, as shown in Fig. [REF].', '1701.08393-2-54-0': 'Face proposal and detection test datasets.', '1701.08393-2-54-1': 'We use the following datasets.', '1701.08393-2-54-2': '(i) FDDB [CITATION] dataset contains [MATH] faces in a set of [MATH] images.', '1701.08393-2-54-3': 'For the face proposal evaluation, we follow the standard evaluation protocol widely used in object proposal studies [CITATION] and transform the original FDDB ellipses ground truth into bounding boxes by minimal bounding rectangle.', '1701.08393-2-54-4': 'For the face detection evaluation, the original FDDB ellipse ground truth is used.', '1701.08393-2-54-5': '(ii) AFW [CITATION] dataset contains [MATH] Flickr images with [MATH] annotated faces of large variations in both face viewpoint and appearance.', '1701.08393-2-54-6': '(iii) PASCAL faces [CITATION] is a widely used face detection benchmark dataset.', '1701.08393-2-54-7': 'It consists of [MATH] images and [MATH] annotated faces.', '1701.08393-2-54-8': '(iv) WIDER FACE [CITATION] is the largest and extremely challenging face detection benchmark dataset.', '1701.08393-2-54-9': 'It consists of [MATH] images and [MATH] annotated faces.', '1701.08393-2-55-0': 'Evaluation settings.', '1701.08393-2-55-1': 'Following [CITATION], we employ the Intersection over Union (IoU) as our evaluation metric.', '1701.08393-2-55-2': 'We fix the IoU threshold to [MATH] following the strict PASCAL criterion.', '1701.08393-2-55-3': 'In particular, an object is considered being covered/detected by a proposal if the IoU is no less than [MATH].', '1701.08393-2-55-4': 'To evaluate the effectiveness of different object proposal algorithms, we use the detection rate (DR) given the number of proposals per image [CITATION].', '1701.08393-2-55-5': 'For face detection, we use standard precision and recall (PR) to evaluate the effectiveness of face detection algorithms.', '1701.08393-2-56-0': 'Faceness-Net Variants.', '1701.08393-2-56-1': 'We evaluate four variants of Faceness-Net:', '1701.08393-2-57-0': 'The discussion on generic object proposal and template proposal techniques can be found in Sec. [REF].', '1701.08393-2-58-0': '# Results', '1701.08393-2-59-0': '## Evaluating the Attribute-Aware Networks', '1701.08393-2-60-0': 'Robustness to unconstrained training input.', '1701.08393-2-60-1': 'The proposed attribute-aware networks do not assume well-cropped faces as input in both the training and test stages.', '1701.08393-2-60-2': 'To support this statement, we conduct an experiment by fine-tuning two attribute-aware networks as shown in Fig. [REF](a), each of which taking different inputs: (1) cropped images, which encompass roughly the face and shoulder regions, and (2) uncropped images, which may include large portions of background apart the face.', '1701.08393-2-60-3': 'Some examples of cropped and uncropped images are shown in Fig. [REF].', '1701.08393-2-61-0': 'The performances of these two networks are measured based on the part detection rate.', '1701.08393-2-61-1': "Note that we combine the evaluation on 'Hair+Beard' to suit the ground truth provided by [CITATION] (see Sec. [REF]).", '1701.08393-2-61-2': 'We provide more details of part detection as follows.', '1701.08393-2-61-3': 'For each facial part, a total of five region templates are first defined using statistics obtained from the LFW training set.', '1701.08393-2-61-4': 'Non Maximum Suppression (NMS) is used to find the pixel locations with local maximum responses.', '1701.08393-2-61-5': 'We select the top [MATH] NMS points and propose region templates centered at the points.', '1701.08393-2-62-0': 'The detection results are summarized in Table [REF].', '1701.08393-2-62-1': 'As can be observed, the proposed approach performs similarly given both the cropped and uncropped images as training inputs.', '1701.08393-2-62-2': 'The results suggest the robustness of the method in handling unconstrained images for training.', '1701.08393-2-62-3': 'In particular, thanks to the facial attribute-driven training, despite the use of uncropped images, the deep model is encouraged to discover and capture the facial part representation in the deep layers, it is therefore capable of generating response maps that precisely pinpoint the locations of parts.', '1701.08393-2-62-4': 'The top row Fig. [REF](a) shows the partness maps generated from LFW images.', '1701.08393-2-62-5': 'The bottom row of Fig. [REF](a) shows the proposals which have the maximum overlap with the ground truth bounding boxes.', '1701.08393-2-62-6': 'Note that facial parts can be discovered despite challenging poses.', '1701.08393-2-62-7': 'In the following experiments, all the proposed models are trained on uncropped images.', '1701.08393-2-63-0': 'With and without sharing representation.', '1701.08393-2-63-1': 'As mentioned in Sec. [REF], we can train an attribute-aware network for each face part or we can train a single network for all the parts by sharing representation.', '1701.08393-2-63-2': 'We compare the proposal detection rate of these two options.', '1701.08393-2-63-3': 'Figure [REF] shows the proposal detection rate of the attribute-aware network(s) trained with and without sharing representation, indicated by blue and red curves, respectively.', '1701.08393-2-63-4': 'Attribute-aware networks trained without sharing representation require a fewer number of proposals but with a detection rate typically lower than [MATH] (given 150-200 proposals).', '1701.08393-2-63-5': 'On the contrary, the attribute-aware network that shares low-level and mid-level representations can achieve a higher detection rate but with an expense of a larger number of proposals.', '1701.08393-2-64-0': 'The observations can be explained as follows.', '1701.08393-2-64-1': 'The networks without sharing representation tend to model the discrepancies between individual parts and background, while the network that shares representation is more likely to learn the differences between facial parts.', '1701.08393-2-64-2': 'Thus, the latter has poorer background modelling capacity thus leading to inferior performance when the number of proposals is small, in comparison to the former.', '1701.08393-2-64-3': 'Nevertheless, we found that the network that shares representation yields high responses for subtle facial parts.', '1701.08393-2-64-4': 'This high recall rate is essential to improve the performance of face detection in the later stage.', '1701.08393-2-65-0': 'Different fine-tuning strategies.', '1701.08393-2-65-1': 'As discussed in Sec. [REF], there are different fine-tuning strategies that can be considered for learning to generate a partness map, but not all of them are well-suited for deriving a robust faceness measure.', '1701.08393-2-65-2': 'Qualitative results have been provided in Fig. [REF].', '1701.08393-2-65-3': 'Here, we provide quantitative comparisons of face proposal performance between the following fine-tuning approaches: (i) a network fine-tuned with a large number of face images from CelebA and non-face images, (ii) fine-tuning the network with 25 face attributes, and (iii) the proposed approach that fine-tunes attribute-aware networks with part-level attributes in accordance to Table [REF].', '1701.08393-2-65-4': 'It is evident from Fig. [REF] that our approach performs significantly better than approaches (i) and (ii).', '1701.08393-2-66-0': '## From Part Responses to Face Proposal', '1701.08393-2-67-0': 'Generic object proposal methods.', '1701.08393-2-67-1': 'In this experiment, we show the effectiveness of adapting different generic object proposal generators [CITATION] to produce face-specific proposals.', '1701.08393-2-67-2': 'Since the notion of face proposal is new, no suitable methods are comparable therefore we use the original generic methods as baselines.', '1701.08393-2-67-3': 'We first apply any object proposal generator to generate the candidate windows and we use our faceness scoring method described in Sec. [REF] to obtain the face proposals.', '1701.08393-2-67-4': 'We experiment with different parameters for the generic methods, and choose parameters that produce a moderate number of proposals with a very high recall.', '1701.08393-2-67-5': 'Evaluation is conducted following the standard protocol [CITATION].', '1701.08393-2-68-0': 'The results are shown in Fig. [REF].', '1701.08393-2-68-1': 'The green curves show the performance of baseline generic object proposal generators.', '1701.08393-2-68-2': 'It can be observed that our methods, both Faceness-Net and its variant Faceness-Net-SR, consistently improve the baselines for proposing face candidate windows, under different IoU thresholds.', '1701.08393-2-68-3': 'Table [REF] shows that our method achieves high detection rate with moderate number of proposals.', '1701.08393-2-69-0': 'Template proposal method.', '1701.08393-2-69-1': 'In this experiment, we compare face proposal performance by using three different methods for generating and scoring candidate windows:', '1701.08393-2-70-0': 'The original Faceness-Net in which an external generic object proposal generator is adopted for generating candidate windows.', '1701.08393-2-70-1': 'The candidate windows are re-ranked using the faceness score.', '1701.08393-2-70-2': 'The result is shown in Fig. [REF] indicated with a red curve.', '1701.08393-2-70-3': 'A variant of the Faceness-Net, named as Faceness-Net-TP, that adopts the template proposal technique (Section [REF]) to generate candidate windows.', '1701.08393-2-70-4': 'The candidate windows are re-ranked using the faceness score.', '1701.08393-2-70-5': 'The result is shown in Fig. [REF] indicated with a blue curve.', '1701.08393-2-70-6': 'The baseline is a Faceness-Net-TP, of which candidate windows are not re-ranked using the faceness score.', '1701.08393-2-70-7': 'Specifically, given a normalized partness map, we find pixel locations where response values are equal or higher than a threshold.', '1701.08393-2-70-8': 'Then, templates are applied centered at selected locations to generate template proposals without using faceness score to re-scoring and re-ranking proposals.', '1701.08393-2-70-9': 'The result is shown in Fig. [REF] indicated with a green curve.', '1701.08393-2-71-0': 'One can observe from Fig. [REF] that Faceness-Net outperforms Faceness-Net-TP when the number of proposals is fewer than [MATH] (the low-recall region).', '1701.08393-2-71-1': 'The performance gap is likely caused by the quality of initial candidate windows generated by the generic object proposal (used by Faceness-Net) and template proposal (used by Faceness-Net-TP).', '1701.08393-2-71-2': 'The former, such as EdgeBox, employs Structured Edges as informative representation to generate the initial set of candidate windows.', '1701.08393-2-71-3': 'The latter, on the other hand, starts with a set of pre-determined template boxes.', '1701.08393-2-71-4': 'Despite the lower performance at low-recall region, Faceness-Net-TP achieves a high detection rate of over [MATH] at high-recall region, which is not achievable using Faceness-Net that employs generic object proposal.', '1701.08393-2-71-5': 'Moreover, the computation cost of template proposal is much lower than generic object proposal.', '1701.08393-2-72-0': 'Evaluate the contributions of each face part.', '1701.08393-2-72-1': 'We factor the contributions of different face parts to proposing face.', '1701.08393-2-72-2': 'Specifically, we generate face proposals with partness maps from each face part individually using the same evaluation protocol in previous experiment.', '1701.08393-2-72-3': 'As can be observed from Fig. [REF], the hair, eye, and nose parts perform much better than mouth and beard.', '1701.08393-2-72-4': 'The lower part of the face is often occluded, making the mouth and beard less effective in proposing face windows.', '1701.08393-2-72-5': 'In contrast, hair, eye, and nose are visible in most cases.', '1701.08393-2-72-6': 'They therefore become important clues for face proposal.', '1701.08393-2-72-7': 'Nonetheless, mouth and beard could provide complementary cues.', '1701.08393-2-72-8': 'Thus combining all parts leads to better result than considering each part in isolation.', '1701.08393-2-73-0': '## From Face Proposal to Face Detection', '1701.08393-2-74-0': 'Next, we compare the proposed Faceness-Net and its variants against state-of-the-art face detection approaches on four benchmark datasets FDDB [CITATION], AFW [CITATION], PASCAL faces [CITATION] and WIDER FACE [CITATION].', '1701.08393-2-74-1': 'Our baseline face detector, Faceness-Net, which involves five CNNs with the structure shown in the Fig. [REF], is trained with the top [MATH] proposals by re-ranking MCG proposals following the process described in Sec. [REF].', '1701.08393-2-74-2': 'To factor the contributions of share representation and template proposal, we build another three variants of Faceness-Net as discussed in Sec. [REF].', '1701.08393-2-74-3': 'The variant Faceness-Net-TP is trained with the top [MATH] template proposals that are re-ranked following Sec. [REF].', '1701.08393-2-75-0': 'We compare Faceness-Net and its variants against representative published methods [CITATION] on FDDB.', '1701.08393-2-75-1': 'For the PASCAL faces and AFW we compare with (1) deformable part based methods, e.g. structure model [CITATION] and Tree Parts Model (TSM) [CITATION]; (2) cascade-based methods, e.g., Headhunter [CITATION].', '1701.08393-2-75-2': 'For the WIDER FACE [CITATION] we compare with (1) aggregated channel feature method (ACF) [CITATION]; (2) deformable part based model [CITATION]; (3) cascaded-based method [CITATION].', '1701.08393-2-76-0': 'AFW dataset.', '1701.08393-2-76-1': 'Figures [REF] shows the precision and recall curves of the compared face detection algorithms on the AFW dataset.', '1701.08393-2-76-2': 'We observe that Faceness-Net and its variants outperform all the compared approaches by a considerable margin.', '1701.08393-2-76-3': 'The Faceness-Net-SR and Faceness-Net-TP outperform baseline Faceness-Net, suggesting the effectiveness of sharing representation and template proposal technique.', '1701.08393-2-76-4': 'Among all the Faceness-Net variants, Faceness-Net-SP-TP achieves the best performance with a high average precision of [MATH].', '1701.08393-2-77-0': 'PASCAL faces dataset.', '1701.08393-2-77-1': 'Figure [REF] shows the precision and recall curves.', '1701.08393-2-77-2': 'The baseline Faceness-Net outperforms its variants and other compared face detection algorithms.', '1701.08393-2-77-3': 'Compared with other benchmark datasets, PASCAL faces dataset has a fewer number of faces in each image, therefore only a small number of proposals is required to achieve a high recall rate.', '1701.08393-2-77-4': 'As shown in Fig. [REF], the quality of proposals generated by the baseline Faceness-Net is higher than its variants when the number of proposals is lower than [MATH], which leads to its better face detection performance on PASCAL face dataset.', '1701.08393-2-78-0': 'FDDB dataset.', '1701.08393-2-78-1': 'The results are shown in Fig. [REF] and Fig. [REF].', '1701.08393-2-78-2': 'Faceness-Net and its variants achieve competitive performance compared with existing algorithms evaluated using the discrete score as shown in the Fig. [REF].', '1701.08393-2-78-3': 'Faceness-Net baseline achieves [MATH] recall rate, while Faceness-Net-SR and Faceness-Net-TP outperform the baseline Faceness-Net by [MATH] and [MATH], respectively.', '1701.08393-2-78-4': 'Faceness-Net-SR-TP performs best with a large improvement of [MATH] compared with the baseline Faceness-Net.', '1701.08393-2-79-0': 'WIDER FACE dataset.', '1701.08393-2-79-1': 'WIDER FACE dataset is currently the largest face detection benchmark dataset.', '1701.08393-2-79-2': 'The dataset has two evaluation protocols.', '1701.08393-2-79-3': 'The internal protocol evaluates face detection algorithms that use WIDER FACE data during training.', '1701.08393-2-79-4': 'In contrast, the external protocol evaluates face detection algorithms that are trained on external data.', '1701.08393-2-79-5': 'Since Faceness-Net and its variants are trained on CelebA and AFLW datasets without using images in the WIDER FACE dataset, we evaluate our algorithm using the external setting.', '1701.08393-2-79-6': 'Faceness-Net and its variants yield better performance in all three evaluation settings compared with baseline method, namely "Easy", "Medium", and "Hard" as shown in Fig. [REF].', '1701.08393-2-79-7': 'The variants of Faceness-Net outperform baseline Faceness-Net by a considerable margin, suggesting the effectiveness of representation sharing and template proposal techniques.', '1701.08393-2-79-8': 'Although Faceness-Net and its variants outperform baseline methods under external setting, there exist large gap between Faceness-Net and recent state-of-the-art methods [CITATION].', '1701.08393-2-79-9': 'These methods are trained using the WIDER FACE dataset and thus they can deal with more challenging cases.', '1701.08393-2-79-10': 'On the contrary, our method is trained on datasets that are not targeted for face detection (CelebA for face recognition, and AFLW for face alignment) and with simple backgrounds.', '1701.08393-2-79-11': 'Nevertheless, it still achieves promising performance.', '1701.08393-2-79-12': 'We provide more discussion below.', '1701.08393-2-80-0': 'Discussion: Recent studies [CITATION] achieve better face detection performance on FDDB, AFW, and PASCAL faces datasets compared to our Faceness-Net.', '1701.08393-2-80-1': 'The performance gap between Faceness-Net and other methods arises from two aspects, namely, the better modeling of background clutter and stronger supervision signals.', '1701.08393-2-80-2': 'Table [REF] summarizes the training data and supervision signals used by different algorithms.', '1701.08393-2-80-3': 'Faceness-Net is trained on CelebA and AFLW datasets.', '1701.08393-2-80-4': 'These datasets are originally proposed for face recognition and facial landmark detection, respectively.', '1701.08393-2-80-5': 'The background in CelebA and AFLW is less cluttered and diverse compared with various backgrounds available in WIDER FACE and MS-COCO datasets.', '1701.08393-2-80-6': 'In addition, faces in CelebA and AFLW datasets have smaller variations, both in scale and poses, compared to those captured in the WIDER FACE dataset.', '1701.08393-2-80-7': 'We use [MATH]k face bounding boxes compared to more than [MATH]k face bounding boxes employed by other methods.', '1701.08393-2-81-0': 'To gain a fairer comparison, we train the Faster-RCNN model presented in [CITATION] using the same training sets (AFLW and CelebA) employed by Faceness-Net.', '1701.08393-2-81-1': 'Evaluation is performed on the FDDB dataset.', '1701.08393-2-81-2': 'The results are shown in Fig. [REF].', '1701.08393-2-81-3': 'The Faster-RCNN face detector achieves [MATH] detection rate on the FDDB dataset which is marginally lower than that of Faceness-Net.', '1701.08393-2-81-4': 'Note that, Faceness-SR-TP is not finetuned by using ImageNet data, but still achieves better performance than Faster-RCNN.', '1701.08393-2-81-5': 'This is probably because attribute supervisions are more capable of modeling facial parts.', '1701.08393-2-82-0': 'Apart from using more challenging training images, both STN [CITATION] and MTCNN [CITATION] use facial landmarks to localize face.', '1701.08393-2-82-1': 'Facial landmarks indicate the explicit location of face parts and thus provide stronger supervisory information than face attributes.', '1701.08393-2-82-2': 'Our method can benefit from these additional factors.', '1701.08393-2-82-3': 'Specifically, it is possible to obtain a stronger Faceness-Net detector using facial landmarks based supervision and datasets with a more cluttered background.', '1701.08393-2-83-0': 'Finally, we show some qualitative examples in Fig. [REF].', '1701.08393-2-83-1': 'Some failure cases are provided in Fig. [REF].', '1701.08393-2-83-2': 'The failures are mainly caused by blurring, illumination, tiny face scale, and missed annotations.', '1701.08393-2-83-3': 'Among the various causes, tiny faces (with a resolution as low as 20 pixels height) remain one of the hardest issues that we wish to further resolve.', '1701.08393-2-83-4': 'The visual appearances between tiny and normal-size faces exhibit a huge difference.', '1701.08393-2-83-5': 'In particular, the facial parts such as eyes, nose or mouth can be barely distinguished from tiny faces, which makes responses produced by attribute-aware networks meaningless.', '1701.08393-2-83-6': 'In order to recall tiny faces, data augmentation and multi-scale inference may be adopted.', '1701.08393-2-83-7': 'Nonetheless, learning scale-invariant representation is still an open problem.', '1701.08393-2-83-8': 'In this study, we do not deal with tiny faces explicitly.', '1701.08393-2-83-9': 'It is part of our on-going work [CITATION].', '1701.08393-2-84-0': '# Runtime analysis', '1701.08393-2-85-0': 'The runtime of the proposed Faceness-Net-SR-TP is [MATH]ms on a single GPU.', '1701.08393-2-85-1': 'The time includes [MATH]ms to generate faceness proposals with the height of testing image no more than [MATH] pixels.', '1701.08393-2-85-2': 'The efficiency of Faceness-Net-SR-TP is clearly faster than the baseline Faceness-Net since the former shares the layers from conv1 to conv4 in its attribute-aware networks.', '1701.08393-2-85-3': 'Previous CNN based face detector [CITATION] achieves good runtime efficiency too.', '1701.08393-2-85-4': 'Our method differs significantly to this method in that we explicitly handle partial occlusion by inferring face likeliness through part responses.', '1701.08393-2-85-5': 'This difference leads to a significant margin of [MATH] in recall rate (Cascade-CNN [MATH], our method [MATH]) when the number of false positives is fixed at [MATH] on the FDDB dataset.', '1701.08393-2-85-6': 'The complete recall rate of the proposed Faceness-Net-SR-TP is [MATH] compared to [MATH] of Cascade-CNN.', '1701.08393-2-85-7': 'At the expense of recall rate, the fast version of Cascade-CNN achieves [MATH]fps on CPU and [MATH]fps on GPU for [MATH] VGA images.', '1701.08393-2-85-8': 'Our Faceness-Net-SR-TP can achieve practical runtime efficiency under the aggressive setting mentioned above, but still with a [MATH] higher recall rate than the Cascade-CNN.', '1701.08393-2-86-0': '# Conclusion', '1701.08393-2-87-0': 'Different from existing face detection studies, we explored the usefulness of face attributes based supervision for learning a robust face detector.', '1701.08393-2-87-1': 'We observed an interesting phenomenon that face part detectors can be obtained from a CNN that is trained on recognizing attributes from uncropped face images, without explicit part supervision.', '1701.08393-2-87-2': "Consequently, we introduced the notion of 'faceness' score, which was carefully formulated through considering facial parts responses and the associated spatial arrangements.", '1701.08393-2-87-3': 'The faceness score can be employed to re-rank candidate windows of any region proposal techniques to generate a modest set of high-quality face proposals with high recall.', '1701.08393-2-87-4': 'With the generated face proposals, we trained a strong face detector that demonstrated promising performance on various face detection benchmark datasets.', '1701.08393-2-88-0': '# Acknowledgments', '1701.08393-2-89-0': 'regular IEEE prefers the singular form'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1701.08393

{'1701.08393-3-0-0': 'We propose a deep convolutional neural network (CNN) for face detection leveraging on facial attributes based supervision.', '1701.08393-3-0-1': 'We observe a phenomenon that part detectors emerge within CNN trained to classify attributes from uncropped face images, without any explicit part supervision.', '1701.08393-3-0-2': 'The observation motivates a new method for finding faces through scoring facial parts responses by their spatial structure and arrangement.', '1701.08393-3-0-3': 'The scoring mechanism is data-driven, and carefully formulated considering challenging cases where faces are only partially visible.', '1701.08393-3-0-4': 'This consideration allows our network to detect faces under severe occlusion and unconstrained pose variations.', '1701.08393-3-0-5': 'Our method achieves promising performance on popular benchmarks including FDDB, PASCAL Faces, AFW, and WIDER FACE.', '1701.08393-3-1-0': 'Face Detection, Deep Learning, Convolutional Neural Network.', '1701.08393-3-2-0': '# Introduction', '1701.08393-3-3-0': 'Face detection is an important and long-standing problem in computer vision.', '1701.08393-3-3-1': 'A number of methods have been proposed in the past, including neural network based methods [CITATION], cascade structures [CITATION] and deformable part models (DPM) [CITATION] detectors.', '1701.08393-3-3-2': 'There has been a resurgence of interest in applying convolutional neural networks (CNN) on this classic problem [CITATION].', '1701.08393-3-3-3': 'Many of these methods follow a cascade object detection framework [CITATION], some of which directly adopt the effective generic object detection framework RCNN [CITATION] and Faster-RCNN [CITATION] as the backbone network, with very deep networks (e.g., 101-layer ResNet) to leverage the remarkable representation learning capacity of deep CNN [CITATION].', '1701.08393-3-4-0': 'While face bounding boxes have been used as a standard supervisory source for learning a face detector, the usefulness of facial attributes remains little explored.', '1701.08393-3-4-1': 'In this study, we show that facial attributes based supervision can effectively enhance the capability of a face detection network in handling severe occlusions.', '1701.08393-3-4-2': 'As depicted in Fig. [REF], a CNN supervised with facial attributes can detect faces even when more than half of the face region is occluded.', '1701.08393-3-4-3': 'In addition, the CNN is capable of detecting faces with large pose variation, e.g., profile view without training separate models under different viewpoints.', '1701.08393-3-4-4': 'Such compelling results are hard to achieve by using supervision based on face bounding boxes alone, especially when the training dataset has limited scene diversity and pose variations.', '1701.08393-3-5-0': 'In this study, we show the benefits of facial attributes supervision through the following considerations:', '1701.08393-3-6-0': '(1) Discovering facial parts responses supervised by facial attributes: The human face has a unique structure.', '1701.08393-3-6-1': 'We believe the reasoning of the unique structure of local facial parts (e.g., eyes, nose, mouth) help detecting faces under unconstrained environments.', '1701.08393-3-6-2': 'We observe an interesting phenomenon that one can actually obtain part detectors within a CNN by training it to classify part-level binary attributes (e.g., mouth attributes including big lips, opened mouth, smiling, wearing lipstick) from uncropped face images, without any explicit part supervision.', '1701.08393-3-6-3': 'The trained CNN is then capable of generating high-quality facial part responses in its deep layers that strongly indicate the locations of the face parts.', '1701.08393-3-6-4': "The examples depicted in Fig. [REF](b) show the response maps (known as 'partness map' in our paper) of five different face parts.", '1701.08393-3-7-0': "(2) Computing faceness score from responses configurations: Given the parts' responses, we formulate an effective method to reason the degree of face likeliness (which we call faceness score) through analyzing their spatial arrangement.", '1701.08393-3-7-1': 'For instance, the hair should appear above the eyes, and the mouth should only appear below the nose.', '1701.08393-3-7-2': 'Any inconsistency would be penalized.', '1701.08393-3-7-3': 'Faceness scores will be derived and used to re-rank candidate windows to obtain a set of face proposals.', '1701.08393-3-7-4': 'Our face proposal approach enjoys a high recall with just a modest number of proposals (over 90% of face recall with around [MATH] proposals, [MATH]0.5% of full sliding windows, and [MATH]10% of generic object proposals [CITATION], measured on the FDDB dataset [CITATION]).', '1701.08393-3-8-0': '(3) Refining the face hypotheses - Both the aforementioned components offer a chance to find a face even under severe occlusion and pose variations.', '1701.08393-3-8-1': 'The output of these components is a small set of high-quality face bounding box proposals that cover most faces in an image.', '1701.08393-3-8-2': 'Given the face proposals, we design a multi-task CNN [CITATION] in the second stage to refine the hypotheses further, by simultaneously recognizing the true faces and estimating more precise face locations.', '1701.08393-3-9-0': "Our main contribution in this study is the novel use of CNN and attributes supervision for discovering facial parts' responses.", '1701.08393-3-9-1': 'We show that part detectors emerge within a CNN trained to classify attributes from uncropped face images, without any explicit part supervision.', '1701.08393-3-9-2': "The parts' responses are subsequently employed to generate high-quality proposals for training a face detector that is robust to severe occlusion.", '1701.08393-3-9-3': 'The findings aforementioned are new in the literature.', '1701.08393-3-9-4': 'It is worth pointing out that our network is trained on datasets that are not targeted for face detection (CelebA [CITATION] for face recognition, and AFLW [CITATION] for face alignment) and with simple backgrounds.', '1701.08393-3-9-5': 'Nevertheless, it still achieves promising performance on various face detection benchmarks including FDDB, PASCAL Faces, AFW, and the challenging WIDER FACE dataset.', '1701.08393-3-10-0': 'In comparison to our earlier version of this work [CITATION], we present a more effective design of CNN to achieve improved performance and speed.', '1701.08393-3-10-1': 'Firstly, in contrast to our previous work that requires independent convolutional networks for learning responses of different facial parts, we now share feature representations between these attribute-aware networks.', '1701.08393-3-10-2': 'The sharing of low and mid-levels representations largely reduce the number of parameters in our framework ([MATH]83% fewer parameters), while improving the robustness of the feature representation.', '1701.08393-3-10-3': 'Secondly, our previous framework relies on external generic object proposal generators such as selective search [CITATION] and EdgeBox [CITATION] for proposing candidate windows.', '1701.08393-3-10-4': 'Inspired by region proposal network presented in [CITATION], in this study we directly generate proposals from our attribute-aware networks, thus proposal generation becomes an inherent part of the framework.', '1701.08393-3-10-5': 'This design not only leads to improved computation efficiency but also higher recall rate compared with generic object proposal algorithms.', '1701.08393-3-10-6': 'Thirdly, we compare our face detector pre-trained on the task of facial attributes classification with that pre-trained on ImageNet large-scale object classification.', '1701.08393-3-10-7': 'Apart from the above major changes, we also provide more technical details and discussions.', '1701.08393-3-10-8': 'Additional experiments are conducted on the challenging WIDER FACE dataset [CITATION].', '1701.08393-3-11-0': '# Related Work', '1701.08393-3-12-0': 'There is a long history of using neural network for the task of face detection [CITATION].', '1701.08393-3-12-1': 'An early face detection survey [CITATION] provides an extensive coverage on relevant methods.', '1701.08393-3-12-2': 'Here we highlight a few notable studies.', '1701.08393-3-12-3': 'Rowley et al. [CITATION] exploit a set of neural network-based filters to detect the presence of faces in multiple scales and merge the detections from individual filters.', '1701.08393-3-12-4': 'Osadchy et al. [CITATION] demonstrate that a joint learning of face detection and pose estimation significantly improves the performance of face detection.', '1701.08393-3-12-5': 'The seminal work of Vaillant et al. [CITATION] adopt a two-stage coarse-to-fine detection.', '1701.08393-3-12-6': 'Specifically, the first stage approximately locates the face region, whilst the second stage provides a more precise localization.', '1701.08393-3-12-7': 'Our approach is inspired by these studies, but we introduce innovations on many aspects.', '1701.08393-3-12-8': 'For instance, our first stage network is conceptually different from that of [CITATION], and many recent deep learning detection frameworks - we train attribute-aware networks to achieve precise localization of facial parts and exploit their spatial structure for inferring face likeliness.', '1701.08393-3-12-9': 'This concept is new and it allows our model to detect faces under severe occlusion and pose variations.', '1701.08393-3-12-10': 'While great efforts have been devoted to addressing face detection under occlusion [CITATION], these methods are all confined to frontal faces.', '1701.08393-3-12-11': 'In contrast, our model can discover faces under variations of both pose and occlusion.', '1701.08393-3-13-0': 'In the last decades, cascade based [CITATION] and deformable part models (DPM) detectors dominate face detection approaches.', '1701.08393-3-13-1': 'Viola and Jones [CITATION] introduced fast Haar-like features computation via integral image and boosted cascade classifier.', '1701.08393-3-13-2': 'Various studies thereafter follow a similar pipeline.', '1701.08393-3-13-3': 'Among the variants, SURF cascade [CITATION] was one of the top performers.', '1701.08393-3-13-4': 'Later Chen et al. [CITATION] demonstrate state-of-the-art face detection performance by learning face detection and face alignment jointly in the same cascade framework.', '1701.08393-3-13-5': 'Deformable part models define face as a collection of parts.', '1701.08393-3-13-6': 'Latent Support Vector Machine is typically used to find the parts and their relationships.', '1701.08393-3-13-7': 'DPM is shown more robust to occlusion than the cascade based methods.', '1701.08393-3-13-8': 'A recent study [CITATION] demonstrates good performance with just a vanilla DPM, achieving better results than more sophisticated DPM variants [CITATION].', '1701.08393-3-14-0': 'Recent studies [CITATION] show that face detection can be further improved by using deep learning.', '1701.08393-3-14-1': 'The network proposed by [CITATION] does not have an explicit mechanism to handle occlusion, the face detector therefore fails to detect faces with heavy occlusions, as acknowledged by the authors.', '1701.08393-3-14-2': 'Cascade based convolutional neural networks [CITATION] replace boosting classifiers with a set of small CNNs to quickly reject negative samples in the early stage.', '1701.08393-3-14-3': 'Recent studies [CITATION] exploit facial landmarks as supervision signals to improve face detection performance.', '1701.08393-3-14-4': 'In this study, we show that facial attributes can serve as an important source too for learning a robust face detector.', '1701.08393-3-15-0': 'The first stage of our model is partially inspired by generic object proposal approaches [CITATION].', '1701.08393-3-15-1': 'Generic object proposal generators are commonly used in standard object detection algorithms for providing high-quality and category-independent bounding boxes.', '1701.08393-3-15-2': 'These methods typically involve redundant computations over regions that are covered by multiple proposals.', '1701.08393-3-15-3': 'To reduce computation, Ren et al. [CITATION] propose Region Proposal Network (RPN) to generate proposals from high-level response maps in a CNN through a set of predefined anchor boxes.', '1701.08393-3-15-4': 'Both generic object proposal and RPN methods do not consider the unique structure and parts on the face.', '1701.08393-3-15-5': 'Hence, no mechanism is available to recall faces when the face is only partially visible.', '1701.08393-3-15-6': 'These shortcomings motivate us to formulate the new faceness measure to achieve high recall on faces while reducing the number of candidate windows to half the original (compared to the original RPN [CITATION]).', '1701.08393-3-16-0': '# Faceness-Net', '1701.08393-3-17-0': 'This section introduces the baseline Faceness-Net.', '1701.08393-3-17-1': 'We first briefly overview the entire pipeline and then discuss the details.', '1701.08393-3-17-2': 'As shown in Fig. [REF], Faceness-Net consists of two stages, i.e., (i) generating face proposals from partness maps by ranking candidate windows using faceness scores, and (ii) refining face proposals for face detection.', '1701.08393-3-18-0': 'First stage.', '1701.08393-3-18-1': 'A full image [MATH] is used as an input to a CNN to generate the partness map for each face part.', '1701.08393-3-18-2': 'A set of CNNs, known as attribute-aware networks, are used to generate the partness map of different parts.', '1701.08393-3-18-3': 'The partness map is obtained by weighted averaging over all the response maps at its top convolutional layer.', '1701.08393-3-18-4': 'The map indicates the location of a specific facial component presented in the image, e.g., hair, eyes, nose, mouth, and beard denoted by [MATH], [MATH], [MATH], [MATH], and [MATH], respectively.', '1701.08393-3-18-5': "For illustration, we sum all these maps into a face label map [MATH], which clearly suggests faces' locations.", '1701.08393-3-19-0': 'Given a set of candidate windows [MATH] that are generated by existing object proposal methods such as [CITATION], or a region proposal network (RPN) [CITATION], we rank these windows according to their faceness scores, [MATH], which are derived from the partness maps with respect to different facial parts configurations, as illustrated at the bottom of Fig. [REF](a).', '1701.08393-3-19-1': "For example, as visualized in Fig. [REF](a), a candidate window 'A' covers a local region of [MATH] (i.e., hair) and its faceness score is calculated by dividing the values at its upper part with respect to the values at its lower part, because hair is more likely to present at the top of a face region.", '1701.08393-3-19-2': 'The bottom part of Fig. [REF](a) illustrates the spatial configurations of five facial parts.', '1701.08393-3-19-3': 'The facial configurations can be learned from the training data.', '1701.08393-3-19-4': 'To reduce the number of the proposed windows, we apply non-maximum suppression (NMS) to smooth the scores by leveraging the spatial relations among these windows.', '1701.08393-3-19-5': "A final faceness score of 'A' is obtained by averaging over the scores of these parts.", '1701.08393-3-19-6': 'We perform another round of NMS to further reduce the number of proposed windows using faceness score.', '1701.08393-3-19-7': 'In this case, a large number of false positive windows can be pruned.', '1701.08393-3-19-8': 'The proposed approach is capable of coping with severe face occlusions.', '1701.08393-3-19-9': "As shown in Fig. [REF](b), face windows 'A' and 'E' can be retrieved by objectness [CITATION] only if a lot of windows are proposed, while windows 'A' and 'E' rank top [MATH] by using our method.", '1701.08393-3-20-0': 'Second stage.', '1701.08393-3-20-1': 'The face proposals are refined by training a multi-task CNN, where face classification and bounding box regression are jointly optimized (Fig. [REF]).', '1701.08393-3-21-0': '## Attribute-Aware Networks', '1701.08393-3-22-0': 'The first stage of the baseline Faceness-Net consists of multiple attribute-aware networks for generating response maps of different parts (Fig. [REF]).', '1701.08393-3-22-1': 'Five networks are needed to cover all five pre-defined facial components, i.e., hair, eyes, nose, mouth, and beard.', '1701.08393-3-22-2': 'These attribute-aware networks share the same structure.', '1701.08393-3-22-3': 'Next, we first discuss the network structure and subsequently show that these networks can share representation to reduce parameters.', '1701.08393-3-23-0': 'Network structure.', '1701.08393-3-23-1': 'The choice of network structure for extracting partness maps is flexible.', '1701.08393-3-23-2': 'Figure [REF](a) depicts the structure and hyper-parameters of the CNN used in the baseline Faceness-Net.', '1701.08393-3-23-3': 'This convolutional structure is inspired by the AlexNet [CITATION], which was originally proposed for object categorization.', '1701.08393-3-23-4': 'Specifically, the network stacks seven convolutional layers (conv1 to conv7) and two max-pooling layers (max1 and max2).', '1701.08393-3-23-5': 'The hyper-parameters of each layer is specified in Fig. [REF](a).', '1701.08393-3-24-0': 'Once the attribute networks are trained (training details are provided in Sec. [REF]), we obtain the response map of a part through first applying [MATH] normalization on the feature map for each channel and then element-wise averaging along the channels.', '1701.08393-3-24-1': 'We examine the response maps obtained from the attribute-aware networks.', '1701.08393-3-24-2': 'As observed from Fig. [REF](b), the feature maps of the first few convolutional layers do not clearly indicate the locations of facial parts.', '1701.08393-3-24-3': 'However, a clear indication of the facial component can be seen from responses of conv7.', '1701.08393-3-24-4': 'Consequently, we obtain an initial response map from the conv7 layer.', '1701.08393-3-24-5': "The final partness map that matches the input image's size is obtained through performing unpooling [CITATION] on the conv7's response map.", '1701.08393-3-25-0': 'Shared representation.', '1701.08393-3-25-1': "It is observed that the feature maps of earlier layers across the different attribute-aware networks are almost identical and they are not indicative of parts' locations.", '1701.08393-3-25-2': 'Motivated by these observations, instead of designating separate attribute-aware networks for different facial components, we share early convolutional layers of these networks to reduce parameters.', '1701.08393-3-25-3': "Specifically, the first four convolutional layers that do not clearly suggests parts' locations are shared, followed by five branches, each of which consists of two convolutional layers responsible for a facial component, as shown in Fig. [REF].", '1701.08393-3-25-4': 'Note that in comparison to the structure presented in Fig. [REF](a), we additionally remove a convolutional layer and trim the number of filters in other layers to reduce parameters.', '1701.08393-3-25-5': 'The sharing of representation and filter reduction lead to a single attribute-aware network with [MATH] fewer parameters than the original five attribute-aware networks.', '1701.08393-3-25-6': 'We denote a Faceness-Net with shared representation as Faceness-Net-SR.', '1701.08393-3-25-7': 'We will show that this network structure not only reduces computations but also improves the robustness of feature representation for face detection.', '1701.08393-3-26-0': '## Learning to Generate Partness Maps', '1701.08393-3-27-0': 'Pre-training the attribute-aware networks.', '1701.08393-3-27-1': 'Pre-training generally helps to improve the performance of a deep network.', '1701.08393-3-27-2': 'There are two plausible pre-training options depending upon whether we share the representations across attribute-aware networks or not.', '1701.08393-3-28-0': 'The first option is to pre-train our attribute-aware networks with massive general object categories in ImageNet [CITATION].', '1701.08393-3-28-1': 'From our observations, this option works well when the representations across networks are not shared.', '1701.08393-3-28-2': 'Since each attribute-aware network originally has access only to a particular group of data specific to a certain attribute, the larger-scale ImageNet data helps to mitigate the overfitting issue that is caused by insufficient data.', '1701.08393-3-29-0': 'The second option omits the ImageNet pre-training stage and trains a network directly on the task of facial attributes classification.', '1701.08393-3-29-1': 'This option works best when we adopt the shared representation scheme discussed in Sec. [REF].', '1701.08393-3-29-2': 'Thanks to the sharing of representation, the attribute-aware network requires a relatively smaller quantity of training data.', '1701.08393-3-29-3': 'Thus, no overfitting is observed despite we use the facial attributes dataset, which is much smaller in scale, i.e., 180,000 images compared to 1 million images in ImageNet.', '1701.08393-3-30-0': 'Fine-tuning the attribute-aware networks.', '1701.08393-3-30-1': 'Once an attribute-network is pre-trained, we can fine-tune it to generate the desired partness maps.', '1701.08393-3-30-2': 'There are different fine-tuning strategies, but not all of them can generate meaningful partness maps for deriving a robust faceness score.', '1701.08393-3-31-0': "As shown in Fig. [REF](b), a deep network trained on generic objects, e.g., AlexNet [CITATION], is not capable of providing us with precise faces' locations, let alone partness map.", '1701.08393-3-31-1': 'To generate accurate partness maps, we explore multiple ways for learning an attribute-aware network.', '1701.08393-3-31-2': 'The most straightforward manner is to use the image and its pixel-wise segmentation label map as input and target, respectively.', '1701.08393-3-31-3': 'This setting is widely employed in image labeling [CITATION].', '1701.08393-3-31-4': 'However, it requires label maps with pixel-wise annotations, which are expensive to collect.', '1701.08393-3-31-5': 'Another setting is image-level classification (i.e., faces and non-faces), as shown in Fig. [REF](c).', '1701.08393-3-31-6': 'It works well where the training images are well-aligned, such as face recognition [CITATION].', '1701.08393-3-31-7': 'Nevertheless, it suffers from complex background clutter because the supervisory information is not sufficient to account for rich and diverse face variations.', '1701.08393-3-31-8': "Its learned feature maps contain too many noises, which overwhelm the actual faces' locations.", '1701.08393-3-31-9': 'Attribute learning in Fig. [REF](d) extends the binary classification in (c) to the extreme by using a combination of attributes to capture face variations.', '1701.08393-3-31-10': "For instance, an 'Asian' face can be distinguished from a 'European' face.", '1701.08393-3-31-11': 'However, our experiments demonstrate that this setting is not robust to occlusion.', '1701.08393-3-32-0': 'Figure [REF](e) shows the partness maps obtained by the baseline Faceness-Net, for which the attribute networks do not share representations.', '1701.08393-3-32-1': 'The strategy we propose extends (d) by partitioning attributes into groups based on facial components.', '1701.08393-3-32-2': "For instance, 'black hair', 'blond hair', 'bald', and 'bangs' are grouped together, as all of them are related to hair.", '1701.08393-3-32-3': 'The grouped attributes are summarized in Table [REF].', '1701.08393-3-32-4': 'In this case, face parts are modeled separately.', '1701.08393-3-32-5': 'If one part is occluded, the face region can still be localized by the other parts.', '1701.08393-3-32-6': 'We take the Hair-Branch shown in the stage one of Fig. [REF] as an example to illustrate the learning procedure.', '1701.08393-3-32-7': 'Let [MATH] be a set of full face images and the attribute labels of hair.', '1701.08393-3-32-8': 'Images are first resized to [MATH] where [MATH] and [MATH] indicate there are nine attributes ([MATH]) related to hair as listed in Table [REF].', '1701.08393-3-32-9': 'Learning is formulated as a multi-variate classification problem by minimizing the cross-entropy loss, [EQUATION] where [MATH] is modeled as a sigmoid function, i.e. [MATH], indicating the probability of the presence of [MATH] attributes.', '1701.08393-3-32-10': 'The features of [MATH] are denoted as [MATH].', '1701.08393-3-32-11': 'To facilitate the learning, we stack two fully-connected layers on top of the last convolutional layer of the structure shown in Fig. [REF].', '1701.08393-3-32-12': 'We optimize the loss function by using stochastic gradient descent with back-propagation.', '1701.08393-3-32-13': 'After training the attribute-aware network, the fully-connected layers are removed to make the network fully convolutional again.', '1701.08393-3-33-0': 'Figure [REF](f) shows the partness maps that are generated from the networks with shared representation, i.e., Faceness-Net-SR (see Fig. [REF]).', '1701.08393-3-33-1': 'Visually, the partness maps generated by this model are noisier compared to Fig. [REF](e).', '1701.08393-3-33-2': 'The key reason is that the Faceness-Net-SR is not pre-trained using ImageNet data but directly trained on the attribute classification task.', '1701.08393-3-33-3': "Despite the noisy partness maps, they actually capture more subtle parts' responses and therefore lead to higher recall rate in the subsequent face proposal stage, provided that the number of proposals is sufficiently large.", '1701.08393-3-34-0': '## Generating Candidate Windows', '1701.08393-3-35-0': 'Face detection can be improved if the inputs are formed by a moderate number of proposals with a high recall rate.', '1701.08393-3-35-1': 'To produce the required proposals, we will explore two plausible choices to generate the initial set of candidate windows.', '1701.08393-3-36-0': 'Generic object proposal.', '1701.08393-3-36-1': 'Generic object scoring is primarily employed to reduce the computational cost of a detector.', '1701.08393-3-36-2': 'It has also been shown improving detection accuracy due to the reduction of spurious false positives [CITATION].', '1701.08393-3-36-3': 'A variety of cues has been proposed to quantify the objectness of an image window, e.g., norm of the gradient [CITATION], edges [CITATION], or integration of a number of low-level features [CITATION].', '1701.08393-3-36-4': 'Other popular methods include super-pixel based approaches, e.g., selective search [CITATION], randomized Prim [CITATION], and multi-scale combinatorial grouping [CITATION].', '1701.08393-3-36-5': 'Our framework can readily employ these generic candidate windows for ranking using the proposed faceness score (Sec. [REF]).', '1701.08393-3-37-0': 'Template proposal.', '1701.08393-3-37-1': 'In order to decouple the dependence of object proposal algorithms to generate candidate windows, we propose a template proposal method in which candidate windows are generated from multiple predefined templates on feature maps.', '1701.08393-3-38-0': 'We provide an example below on using a partness map of hair, [MATH], for template proposal.', '1701.08393-3-38-1': 'As shown in Fig. [REF], each value of location [MATH] on the partness map [MATH] indicates the probability of the appearance of the hair component.', '1701.08393-3-38-2': 'We select a set of [MATH] locations [MATH] with a probability [MATH] higher than [MATH].', '1701.08393-3-38-3': 'For each selected location, multiple template proposals are generated, where the number of maximum possible proposals for each location is fixed as [MATH].', '1701.08393-3-38-4': 'The proposals are obtained from predefined reference boxes, which we call templates.', '1701.08393-3-38-5': 'For each face part, templates are centered at different locations considering the structure of the human face.', '1701.08393-3-38-6': 'In addition, they are associated with a specific scale and aspect ratio, as shown at the top of Fig. [REF].', '1701.08393-3-38-7': 'For instance, the templates of the hair region are centered at [MATH] and the templates of eyes are centered at [MATH], where [MATH] and [MATH] represent the width and height of an anchor.', '1701.08393-3-38-8': "Similar to previous work [CITATION], these templates are translation invariant up to the network's total stride, and the method does not incur extra cost for addressing scales thanks to the multi-scale templates.", '1701.08393-3-39-0': 'In our study, we define [MATH] scales and [MATH] aspect ratio, yielding [MATH] templates at each selected position.', '1701.08393-3-39-1': 'Specifically, we use [MATH] scales with box areas of [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] pixels, and [MATH] aspect ratio of [MATH] (with width to height).', '1701.08393-3-39-2': 'The parameters of templates, i.e., center location, scale and aspect ratio are selected by maximizing the recall rate given an average number of [MATH] proposals per image.', '1701.08393-3-39-3': 'In our study, we perform a grid search on the training set to select the parameters.', '1701.08393-3-40-0': 'Discussion.', '1701.08393-3-40-1': 'Both the generic objectness measures and RPN (trained on ImageNet) are devoted to generic objects therefore not suitable to propose windows specific to faces.', '1701.08393-3-40-2': 'In particular, applying a generic proposal generator directly would produce an enormous number of candidate windows but an only minority of them contain faces.', '1701.08393-3-40-3': 'While RPN is computationally more efficient than generic object proposal generators, it cannot be directly applied to our problem too.', '1701.08393-3-40-4': 'Specifically, in order for the RPN to cope with faces with tiny size and various poses, a large number of anchor boxes are required, leading to an enormous number of proposals.', '1701.08393-3-40-5': 'In the next section, we discuss a new faceness measure that can complement existing object proposal generators or the template proposal method to achieve high recall on faces, while significantly reduce the number of candidate windows.', '1701.08393-3-40-6': 'The proposed faceness measure scheme is in practice related to traditional face detector schemes based on Haar features, with the difference that here the Haar features pool CNN feature responses instead of pixel luminance values.', '1701.08393-3-41-0': '## Ranking Windows by Faceness Score', '1701.08393-3-42-0': 'After generating candidate windows based on the methods described in Sec. [REF], our approach computes a faceness score on these windows to return a ranked set of top-scoring face proposals.', '1701.08393-3-42-1': 'Figure [REF] illustrates the procedure of deriving the faceness measure from the partness maps of hair and eyes.', '1701.08393-3-42-2': 'Let [MATH] be the faceness score of a window [MATH].', '1701.08393-3-42-3': 'For example, as shown in Fig. [REF](a), given a partness map of hair, [MATH], [MATH] is attained by dividing the sum of values in ABEF (green) by the sum of values in FECD.', '1701.08393-3-42-4': 'Similarly, Fig. [REF](b) shows that [MATH] is obtained by dividing the sum of values in EFGH (green) with respect to ABEF+HGCD of [MATH].', '1701.08393-3-42-5': 'For both of the above examples, a larger value of [MATH] indicates a higher overlapping ratio of [MATH] with a face.', '1701.08393-3-42-6': 'The choice of method for computing the faceness score is flexible.', '1701.08393-3-42-7': 'It is possible to compute the faceness score using other forms of handcrafted features that can effectively capture the face structure through response maps.', '1701.08393-3-43-0': 'The spatial configurations, such as ABEF in Fig. [REF](a) and EFGH in Fig. [REF](b), can be learned from data.', '1701.08393-3-43-1': 'We take hair as an example.', '1701.08393-3-43-2': 'We need to learn the positions of points E and F, which can be represented by the [MATH]-coordinates of ABCD, i.e., the proposed window.', '1701.08393-3-43-3': 'For instance, the position of E in Fig. [REF](a) can be represented by [MATH] and [MATH], implying that the value of its [MATH]-axis is a linear combination of [MATH] and [MATH].', '1701.08393-3-43-4': 'With this representation, [MATH] can be efficiently computed by using the integral image (denoted as [MATH]) of the partness map.', '1701.08393-3-43-5': 'For instance, [MATH] in (a) is attained by [MATH].', '1701.08393-3-43-6': '[EQUATION] where [MATH] signifies the value at the location [MATH].', '1701.08393-3-44-0': 'Given a training set [MATH], where [MATH] and [MATH] denote the [MATH]-th window and its label (i.e. face/non-face), respectively.', '1701.08393-3-44-1': 'Let [MATH] be the cropped partness map with respect to the [MATH]-th window, e.g., region ABCD in [MATH].', '1701.08393-3-44-2': 'This problem can be formulated as maximum a posteriori (MAP) estimation [EQUATION] where [MATH] represents a set of parameters when learning the spatial configuration of hair (Fig. [REF](a)).', '1701.08393-3-44-3': 'The terms [MATH] and [MATH] denote the likelihood and prior, respectively.', '1701.08393-3-44-4': 'The likelihood of faceness can be modeled by a sigmoid function, i.e., [MATH], where [MATH] is a coefficient.', '1701.08393-3-44-5': 'This likelihood measures the confidence of partitioning the face and non-face, given a certain spatial configuration.', '1701.08393-3-44-6': 'The prior term can be factorized, [MATH], where [MATH] is a uniform distribution between zero and one, as it indicates the coefficients of linear combination, [MATH] models the prior of the candidate window, which can be generated by object proposal methods, and [MATH] is the partness map as obtained in Sec. [REF].', '1701.08393-3-44-7': 'Since [MATH] typically has a low dimension (e.g., one dimension of hair), it can be simply obtained by line search.', '1701.08393-3-44-8': 'Note that Eq. [REF] can be easily extended to model more complex spatial configurations.', '1701.08393-3-44-9': 'This process is similar with learning Haar templates using boosting classifier, but requires less computation while achieving good performance compared with more elaborated process.', '1701.08393-3-45-0': '## Face Detection', '1701.08393-3-46-0': 'The top candidate windows that are ranked by faceness score attain a high recall rate.', '1701.08393-3-46-1': 'These face proposals can be subsequently fed to the multi-task CNN at stage 2 of the proposed pipeline (Fig. [REF]) for face detection.', '1701.08393-3-47-0': 'Pre-training.', '1701.08393-3-47-1': 'We directly use the earlier layers of attribute-aware networks (the stage-1 network with shared representation as shown in Fig. [REF]) up to conv4 as the pre-trained model for the multi-task CNN of stage 2.', '1701.08393-3-47-2': 'After conv4, as shown in Fig. [REF], the multi-task CNN forks into two branches, each of which consists of two convolutional layers and two fully connected layers.', '1701.08393-3-47-3': 'The two branches are optimized to handle different tasks, namely face classification and bounding box regression, respectively.', '1701.08393-3-48-0': 'It is worth pointing out that the multi-task CNN can be pre-trained on the ImageNet data, instead of reusing the parameters of the attribute-aware networks.', '1701.08393-3-48-1': 'Nevertheless, we found that the multi-task CNN converges much faster given the face attributes based pretrained model.', '1701.08393-3-48-2': 'Specifically, the attribute pretrained network only requires [MATH] iterations to converge during the face detection fine-tuning stage, in comparison to more than [MATH] iterations for the ImageNet pertrained network using the same mini-batch size.', '1701.08393-3-48-3': 'We conjecture that much less effort is needed to transform the feature representations learned from the facial attribute classification task to the face detection task.', '1701.08393-3-49-0': 'Multi-task fine-tuning.', '1701.08393-3-49-1': 'We fine-tune the first branch of the multi-task CNN for face classification and the second branch for bounding box regresssion.', '1701.08393-3-49-2': 'Fine-tuning is performed using the face proposals obtained from the previous step (Sec [REF]).', '1701.08393-3-49-3': 'For face classification, we assign a face proposal to its closest ground truth bounding box based on the Euclidean distance between their respective center coordinates.', '1701.08393-3-49-4': 'A face proposal is considered positive if the Intersection over Union (IoU) between the proposal box and the assigned ground truth box is larger than [MATH]; otherwise it is negative.', '1701.08393-3-49-5': 'For bounding box regression, we train the second branch of the multi-task CNN to regress each proposal to the coordinates of its assigned ground truth box.', '1701.08393-3-49-6': 'If the proposed window is a positive sample, the regression target is generated by Eq. [REF].', '1701.08393-3-49-7': 'We use the following parameterizations of the 4 coordinates: [EQUATION] where [MATH] is a normalizing factor.', '1701.08393-3-49-8': 'The vector [MATH] denotes the top-left and bottom-right coordinates of a bounding box.', '1701.08393-3-49-9': 'Variables [MATH], [MATH], and [MATH] represent the ground truth box, proposed box, and regression target.', '1701.08393-3-49-10': 'This process normalizes regression target into a range of [MATH] which can be easily optimized by using least square loss.', '1701.08393-3-49-11': 'The standard bounding box regression targets [CITATION] and [MATH] loss are also applicable.', '1701.08393-3-49-12': 'If a proposed window is non-face, the CNN outputs a vector of [MATH] whose gradients will be ignored during back propagation.', '1701.08393-3-50-0': 'More implementation details are given below.', '1701.08393-3-50-1': 'During the training process, if the number of positive samples in a mini-batch is smaller than [MATH] of the total samples, we randomly crop the ground truth faces and add these samples as additional positive samples.', '1701.08393-3-50-2': 'Therefore, the ratio of positive samples and negative samples is kept not lower than [MATH].', '1701.08393-3-50-3': 'Meanwhile, we conduct bounding box NMS on the negative samples.', '1701.08393-3-50-4': 'The IoU for the NMS is set to [MATH].', '1701.08393-3-50-5': 'The proposed bounding boxes are cropped and then resized to [MATH].', '1701.08393-3-50-6': 'To handle blurry faces, we augment our training samples by applying Gaussian blurring.', '1701.08393-3-50-7': 'The fine-tuning consumes [MATH] iterations with a batch size of [MATH] images.', '1701.08393-3-50-8': 'We adopt Euclidean loss and cross-entropy loss for bounding box regression and face classification, respectively.', '1701.08393-3-51-0': '# Experimental Settings', '1701.08393-3-52-0': 'Training datasets.', '1701.08393-3-52-1': '(i) We employ CelebA dataset [CITATION] to train our attribute-aware networks.', '1701.08393-3-52-2': 'The dataset contains [MATH] web-based images exclusive from the LFW [CITATION], FDDB [CITATION], AFW [CITATION] and PASCAL [CITATION] datasets.', '1701.08393-3-52-3': 'Every image in the dataset are labeled with [MATH] facial attributes.', '1701.08393-3-52-4': 'We select [MATH] facial attributes from CelebA dataset for each image and divide the attributes into five categories based on their respective facial parts as shown in Table [REF].', '1701.08393-3-52-5': 'We randomly select [MATH] images from the CelebA dataset for training and the remaining is reserved as the validation set.', '1701.08393-3-52-6': '(ii) For face detection training, we choose [MATH] face images from the AFLW dataset [CITATION] to ensure a balanced out-of-plane pose distribution.', '1701.08393-3-52-7': 'We observe a large number of missed annotated faces in the AFLW dataset, which could hamper the training of our face detector.', '1701.08393-3-52-8': 'Hence, we re-annotate face bounding boxes for those missing faces.', '1701.08393-3-52-9': 'The total number of faces in the re-annotated AFLW is [MATH] compared with [MATH] in the original data.', '1701.08393-3-52-10': 'As negative samples, we randomly select [MATH] person-free images from the PASCAL VOC 2007 dataset [CITATION].', '1701.08393-3-53-0': 'Part response test dataset.', '1701.08393-3-53-1': 'We use LFW dataset [CITATION] for evaluating the quality of part response maps for part localization.', '1701.08393-3-53-2': 'Since the original dataset does not come with part-level bounding boxes, we label the boxes with the following scheme.', '1701.08393-3-53-3': 'We follow the annotations provided by [CITATION] on hairs and beard for a set of [MATH] LFW images.', '1701.08393-3-53-4': 'Hair bounding boxes are generated with minimal and maximal coordinates of hair superpixel as shown in Fig. [REF].', '1701.08393-3-53-5': 'Using a similar strategy, eye, nose and mouth bounding boxes are obtained from the manually labeled [MATH] dense facial landmarks [CITATION] on the original LFW [CITATION] images, as shown in Fig. [REF].', '1701.08393-3-54-0': 'Face proposal and detection test datasets.', '1701.08393-3-54-1': 'We use the following datasets.', '1701.08393-3-54-2': '(i) FDDB [CITATION] dataset contains [MATH] faces in a set of [MATH] images.', '1701.08393-3-54-3': 'For the face proposal evaluation, we follow the standard evaluation protocol widely used in object proposal studies [CITATION] and transform the original FDDB ellipses ground truth into bounding boxes by minimal bounding rectangle.', '1701.08393-3-54-4': 'For the face detection evaluation, the original FDDB ellipse ground truth is used.', '1701.08393-3-54-5': '(ii) AFW [CITATION] dataset contains [MATH] Flickr images with [MATH] annotated faces of large variations in both face viewpoint and appearance.', '1701.08393-3-54-6': '(iii) PASCAL faces [CITATION] is a widely used face detection benchmark dataset.', '1701.08393-3-54-7': 'It consists of [MATH] images and [MATH] annotated faces.', '1701.08393-3-54-8': '(iv) WIDER FACE [CITATION] is the largest and extremely challenging face detection benchmark dataset.', '1701.08393-3-54-9': 'It consists of [MATH] images and [MATH] annotated faces.', '1701.08393-3-55-0': 'Evaluation settings.', '1701.08393-3-55-1': 'Following [CITATION], we employ the Intersection over Union (IoU) as our evaluation metric.', '1701.08393-3-55-2': 'We fix the IoU threshold to [MATH] following the strict PASCAL criterion.', '1701.08393-3-55-3': 'In particular, an object is considered being covered/detected by a proposal if the IoU is no less than [MATH].', '1701.08393-3-55-4': 'To evaluate the effectiveness of different object proposal algorithms, we use the detection rate (DR) given the number of proposals per image [CITATION].', '1701.08393-3-55-5': 'For face detection, we use standard precision and recall (PR) to evaluate the effectiveness of face detection algorithms.', '1701.08393-3-56-0': 'Faceness-Net Variants.', '1701.08393-3-56-1': 'We evaluate four variants of Faceness-Net:', '1701.08393-3-57-0': 'The discussion on generic object proposal and template proposal techniques can be found in Sec. [REF].', '1701.08393-3-58-0': '# Results', '1701.08393-3-59-0': '## Evaluating the Attribute-Aware Networks', '1701.08393-3-60-0': 'Robustness to unconstrained training input.', '1701.08393-3-60-1': 'The proposed attribute-aware networks do not assume well-cropped faces as input in both the training and test stages.', '1701.08393-3-60-2': 'To support this statement, we conduct an experiment by fine-tuning two attribute-aware networks as shown in Fig. [REF](a), each of which taking different inputs: (1) cropped images, which encompass roughly the face and shoulder regions, and (2) uncropped images, which may include large portions of background apart the face.', '1701.08393-3-60-3': 'Some examples of cropped and uncropped images are shown in Fig. [REF].', '1701.08393-3-61-0': 'The performances of these two networks are measured based on the part detection rate.', '1701.08393-3-61-1': "Note that we combine the evaluation on 'Hair+Beard' to suit the ground truth provided by [CITATION] (see Sec. [REF]).", '1701.08393-3-61-2': 'We provide more details of part detection as follows.', '1701.08393-3-61-3': 'For each facial part, a total of five region templates are first defined using statistics obtained from the LFW training set.', '1701.08393-3-61-4': 'Non Maximum Suppression (NMS) is used to find the pixel locations with local maximum responses.', '1701.08393-3-61-5': 'We select the top [MATH] NMS points and propose region templates centered at the points.', '1701.08393-3-62-0': 'The detection results are summarized in Table [REF].', '1701.08393-3-62-1': 'As can be observed, the proposed approach performs similarly given both the cropped and uncropped images as training inputs.', '1701.08393-3-62-2': 'The results suggest the robustness of the method in handling unconstrained images for training.', '1701.08393-3-62-3': 'In particular, thanks to the facial attribute-driven training, despite the use of uncropped images, the deep model is encouraged to discover and capture the facial part representation in the deep layers, it is therefore capable of generating response maps that precisely pinpoint the locations of parts.', '1701.08393-3-62-4': 'The top row Fig. [REF](a) shows the partness maps generated from LFW images.', '1701.08393-3-62-5': 'The bottom row of Fig. [REF](a) shows the proposals which have the maximum overlap with the ground truth bounding boxes.', '1701.08393-3-62-6': 'Note that facial parts can be discovered despite challenging poses.', '1701.08393-3-62-7': 'In the following experiments, all the proposed models are trained on uncropped images.', '1701.08393-3-63-0': 'With and without sharing representation.', '1701.08393-3-63-1': 'As mentioned in Sec. [REF], we can train an attribute-aware network for each face part or we can train a single network for all the parts by sharing representation.', '1701.08393-3-63-2': 'We compare the proposal detection rate of these two options.', '1701.08393-3-63-3': 'Figure [REF] shows the proposal detection rate of the attribute-aware network(s) trained with and without sharing representation, indicated by blue and red curves, respectively.', '1701.08393-3-63-4': 'Attribute-aware networks trained without sharing representation require a fewer number of proposals but with a detection rate typically lower than [MATH] (given 150-200 proposals).', '1701.08393-3-63-5': 'On the contrary, the attribute-aware network that shares low-level and mid-level representations can achieve a higher detection rate but with an expense of a larger number of proposals.', '1701.08393-3-64-0': 'The observations can be explained as follows.', '1701.08393-3-64-1': 'The networks without sharing representation tend to model the discrepancies between individual parts and background, while the network that shares representation is more likely to learn the differences between facial parts.', '1701.08393-3-64-2': 'Thus, the latter has poorer background modelling capacity thus leading to inferior performance when the number of proposals is small, in comparison to the former.', '1701.08393-3-64-3': 'Nevertheless, we found that the network that shares representation yields high responses for subtle facial parts.', '1701.08393-3-64-4': 'This high recall rate is essential to improve the performance of face detection in the later stage.', '1701.08393-3-65-0': 'Different fine-tuning strategies.', '1701.08393-3-65-1': 'As discussed in Sec. [REF], there are different fine-tuning strategies that can be considered for learning to generate a partness map, but not all of them are well-suited for deriving a robust faceness measure.', '1701.08393-3-65-2': 'Qualitative results have been provided in Fig. [REF].', '1701.08393-3-65-3': 'Here, we provide quantitative comparisons of face proposal performance between the following fine-tuning approaches: (i) a network fine-tuned with a large number of face images from CelebA and non-face images, (ii) fine-tuning the network with 25 face attributes, and (iii) the proposed approach that fine-tunes attribute-aware networks with part-level attributes in accordance to Table [REF].', '1701.08393-3-65-4': 'It is evident from Fig. [REF] that our approach performs significantly better than approaches (i) and (ii).', '1701.08393-3-66-0': '## From Part Responses to Face Proposal', '1701.08393-3-67-0': 'Generic object proposal methods.', '1701.08393-3-67-1': 'In this experiment, we show the effectiveness of adapting different generic object proposal generators [CITATION] to produce face-specific proposals.', '1701.08393-3-67-2': 'Since the notion of face proposal is new, no suitable methods are comparable therefore we use the original generic methods as baselines.', '1701.08393-3-67-3': 'We first apply any object proposal generator to generate the candidate windows and we use our faceness scoring method described in Sec. [REF] to obtain the face proposals.', '1701.08393-3-67-4': 'We experiment with different parameters for the generic methods, and choose parameters that produce a moderate number of proposals with a very high recall.', '1701.08393-3-67-5': 'Evaluation is conducted following the standard protocol [CITATION].', '1701.08393-3-68-0': 'The results are shown in Fig. [REF].', '1701.08393-3-68-1': 'The green curves show the performance of baseline generic object proposal generators.', '1701.08393-3-68-2': 'It can be observed that our methods, both Faceness-Net and its variant Faceness-Net-SR, consistently improve the baselines for proposing face candidate windows, under different IoU thresholds.', '1701.08393-3-68-3': 'Table [REF] shows that our method achieves high detection rate with moderate number of proposals.', '1701.08393-3-69-0': 'Template proposal method.', '1701.08393-3-69-1': 'In this experiment, we compare face proposal performance by using three different methods for generating and scoring candidate windows:', '1701.08393-3-70-0': 'The original Faceness-Net in which an external generic object proposal generator is adopted for generating candidate windows.', '1701.08393-3-70-1': 'The candidate windows are re-ranked using the faceness score.', '1701.08393-3-70-2': 'The result is shown in Fig. [REF] indicated with a red curve.', '1701.08393-3-70-3': 'A variant of the Faceness-Net, named as Faceness-Net-TP, that adopts the template proposal technique (Section [REF]) to generate candidate windows.', '1701.08393-3-70-4': 'The candidate windows are re-ranked using the faceness score.', '1701.08393-3-70-5': 'The result is shown in Fig. [REF] indicated with a blue curve.', '1701.08393-3-70-6': 'The baseline is a Faceness-Net-TP, of which candidate windows are not re-ranked using the faceness score.', '1701.08393-3-70-7': 'Specifically, given a normalized partness map, we find pixel locations where response values are equal or higher than a threshold.', '1701.08393-3-70-8': 'Then, templates are applied centered at selected locations to generate template proposals without using faceness score to re-scoring and re-ranking proposals.', '1701.08393-3-70-9': 'The result is shown in Fig. [REF] indicated with a green curve.', '1701.08393-3-71-0': 'One can observe from Fig. [REF] that Faceness-Net outperforms Faceness-Net-TP when the number of proposals is fewer than [MATH] (the low-recall region).', '1701.08393-3-71-1': 'The performance gap is likely caused by the quality of initial candidate windows generated by the generic object proposal (used by Faceness-Net) and template proposal (used by Faceness-Net-TP).', '1701.08393-3-71-2': 'The former, such as EdgeBox, employs Structured Edges as informative representation to generate the initial set of candidate windows.', '1701.08393-3-71-3': 'The latter, on the other hand, starts with a set of pre-determined template boxes.', '1701.08393-3-71-4': 'Despite the lower performance at low-recall region, Faceness-Net-TP achieves a high detection rate of over [MATH] at high-recall region, which is not achievable using Faceness-Net that employs generic object proposal.', '1701.08393-3-71-5': 'Moreover, the computation cost of template proposal is much lower than generic object proposal.', '1701.08393-3-72-0': 'Evaluate the contributions of each face part.', '1701.08393-3-72-1': 'We factor the contributions of different face parts to proposing face.', '1701.08393-3-72-2': 'Specifically, we generate face proposals with partness maps from each face part individually using the same evaluation protocol in previous experiment.', '1701.08393-3-72-3': 'As can be observed from Fig. [REF], the hair, eye, and nose parts perform much better than mouth and beard.', '1701.08393-3-72-4': 'The lower part of the face is often occluded, making the mouth and beard less effective in proposing face windows.', '1701.08393-3-72-5': 'In contrast, hair, eye, and nose are visible in most cases.', '1701.08393-3-72-6': 'They therefore become important clues for face proposal.', '1701.08393-3-72-7': 'Nonetheless, mouth and beard could provide complementary cues.', '1701.08393-3-72-8': 'Thus combining all parts leads to better result than considering each part in isolation.', '1701.08393-3-73-0': '## From Face Proposal to Face Detection', '1701.08393-3-74-0': 'Next, we compare the proposed Faceness-Net and its variants against state-of-the-art face detection approaches on four benchmark datasets FDDB [CITATION], AFW [CITATION], PASCAL faces [CITATION] and WIDER FACE [CITATION].', '1701.08393-3-74-1': 'Our baseline face detector, Faceness-Net, which involves five CNNs with the structure shown in the Fig. [REF], is trained with the top [MATH] proposals by re-ranking MCG proposals following the process described in Sec. [REF].', '1701.08393-3-74-2': 'To factor the contributions of share representation and template proposal, we build another three variants of Faceness-Net as discussed in Sec. [REF].', '1701.08393-3-74-3': 'The variant Faceness-Net-TP is trained with the top [MATH] template proposals that are re-ranked following Sec. [REF].', '1701.08393-3-75-0': 'We compare Faceness-Net and its variants against representative published methods [CITATION] on FDDB.', '1701.08393-3-75-1': 'For the PASCAL faces and AFW we compare with (1) deformable part based methods, e.g. structure model [CITATION] and Tree Parts Model (TSM) [CITATION]; (2) cascade-based methods, e.g., Headhunter [CITATION].', '1701.08393-3-75-2': 'For the WIDER FACE [CITATION] we compare with (1) aggregated channel feature method (ACF) [CITATION]; (2) deformable part based model [CITATION]; (3) cascaded-based method [CITATION].', '1701.08393-3-76-0': 'AFW dataset.', '1701.08393-3-76-1': 'Figures [REF] shows the precision and recall curves of the compared face detection algorithms on the AFW dataset.', '1701.08393-3-76-2': 'We observe that Faceness-Net and its variants outperform all the compared approaches by a considerable margin.', '1701.08393-3-76-3': 'The Faceness-Net-SR and Faceness-Net-TP outperform baseline Faceness-Net, suggesting the effectiveness of sharing representation and template proposal technique.', '1701.08393-3-76-4': 'Among all the Faceness-Net variants, Faceness-Net-SP-TP achieves the best performance with a high average precision of [MATH].', '1701.08393-3-77-0': 'PASCAL faces dataset.', '1701.08393-3-77-1': 'Figure [REF] shows the precision and recall curves.', '1701.08393-3-77-2': 'The baseline Faceness-Net outperforms its variants and other compared face detection algorithms.', '1701.08393-3-77-3': 'Compared with other benchmark datasets, PASCAL faces dataset has a fewer number of faces in each image, therefore only a small number of proposals is required to achieve a high recall rate.', '1701.08393-3-77-4': 'As shown in Fig. [REF], the quality of proposals generated by the baseline Faceness-Net is higher than its variants when the number of proposals is lower than [MATH], which leads to its better face detection performance on PASCAL face dataset.', '1701.08393-3-78-0': 'FDDB dataset.', '1701.08393-3-78-1': 'The results are shown in Fig. [REF] and Fig. [REF].', '1701.08393-3-78-2': 'Faceness-Net and its variants achieve competitive performance compared with existing algorithms evaluated using the discrete score as shown in the Fig. [REF].', '1701.08393-3-78-3': 'Faceness-Net baseline achieves [MATH] recall rate, while Faceness-Net-SR and Faceness-Net-TP outperform the baseline Faceness-Net by [MATH] and [MATH], respectively.', '1701.08393-3-78-4': 'Faceness-Net-SR-TP performs best with a large improvement of [MATH] compared with the baseline Faceness-Net.', '1701.08393-3-79-0': 'WIDER FACE dataset.', '1701.08393-3-79-1': 'WIDER FACE dataset is currently the largest face detection benchmark dataset.', '1701.08393-3-79-2': 'The dataset has two evaluation protocols.', '1701.08393-3-79-3': 'The internal protocol evaluates face detection algorithms that use WIDER FACE data during training.', '1701.08393-3-79-4': 'In contrast, the external protocol evaluates face detection algorithms that are trained on external data.', '1701.08393-3-79-5': 'Since Faceness-Net and its variants are trained on CelebA and AFLW datasets without using images in the WIDER FACE dataset, we evaluate our algorithm using the external setting.', '1701.08393-3-79-6': 'Faceness-Net and its variants yield better performance in all three evaluation settings compared with baseline method, namely "Easy", "Medium", and "Hard" as shown in Fig. [REF].', '1701.08393-3-79-7': 'The variants of Faceness-Net outperform baseline Faceness-Net by a considerable margin, suggesting the effectiveness of representation sharing and template proposal techniques.', '1701.08393-3-79-8': 'Although Faceness-Net and its variants outperform baseline methods under external setting, there exist large gap between Faceness-Net and recent state-of-the-art methods [CITATION].', '1701.08393-3-79-9': 'These methods are trained using the WIDER FACE dataset and thus they can deal with more challenging cases.', '1701.08393-3-79-10': 'On the contrary, our method is trained on datasets that are not targeted for face detection (CelebA for face recognition, and AFLW for face alignment) and with simple backgrounds.', '1701.08393-3-79-11': 'Nevertheless, it still achieves promising performance.', '1701.08393-3-79-12': 'We provide more discussion below.', '1701.08393-3-80-0': 'Discussion: Recent studies [CITATION] achieve better face detection performance on FDDB, AFW, and PASCAL faces datasets compared to our Faceness-Net.', '1701.08393-3-80-1': 'The performance gap between Faceness-Net and other methods arises from two aspects, namely, the better modeling of background clutter and stronger supervision signals.', '1701.08393-3-80-2': 'Table [REF] summarizes the training data and supervision signals used by different algorithms.', '1701.08393-3-80-3': 'Faceness-Net is trained on CelebA and AFLW datasets.', '1701.08393-3-80-4': 'These datasets are originally proposed for face recognition and facial landmark detection, respectively.', '1701.08393-3-80-5': 'The background in CelebA and AFLW is less cluttered and diverse compared with various backgrounds available in WIDER FACE and MS-COCO datasets.', '1701.08393-3-80-6': 'In addition, faces in CelebA and AFLW datasets have smaller variations, both in scale and poses, compared to those captured in the WIDER FACE dataset.', '1701.08393-3-80-7': 'We use [MATH]k face bounding boxes compared to more than [MATH]k face bounding boxes employed by other methods.', '1701.08393-3-81-0': 'To gain a fairer comparison, we train the Faster-RCNN model presented in [CITATION] using the same training sets (AFLW and CelebA) employed by Faceness-Net.', '1701.08393-3-81-1': 'Evaluation is performed on the FDDB dataset.', '1701.08393-3-81-2': 'The results are shown in Fig. [REF].', '1701.08393-3-81-3': 'The Faster-RCNN face detector achieves [MATH] detection rate on the FDDB dataset which is marginally lower than that of Faceness-Net.', '1701.08393-3-81-4': 'Note that, Faceness-SR-TP is not finetuned by using ImageNet data, but still achieves better performance than Faster-RCNN.', '1701.08393-3-81-5': 'This is probably because attribute supervisions are more capable of modeling facial parts.', '1701.08393-3-82-0': 'Apart from using more challenging training images, both STN [CITATION] and MTCNN [CITATION] use facial landmarks to localize face.', '1701.08393-3-82-1': 'Facial landmarks indicate the explicit location of face parts and thus provide stronger supervisory information than face attributes.', '1701.08393-3-82-2': 'Our method can benefit from these additional factors.', '1701.08393-3-82-3': 'Specifically, it is possible to obtain a stronger Faceness-Net detector using facial landmarks based supervision and datasets with a more cluttered background.', '1701.08393-3-83-0': 'Finally, we show some qualitative examples in Fig. [REF].', '1701.08393-3-83-1': 'Some failure cases are provided in Fig. [REF].', '1701.08393-3-83-2': 'The failures are mainly caused by blurring, illumination, tiny face scale, and missed annotations.', '1701.08393-3-83-3': 'Among the various causes, tiny faces (with a resolution as low as 20 pixels height) remain one of the hardest issues that we wish to further resolve.', '1701.08393-3-83-4': 'The visual appearances between tiny and normal-size faces exhibit a huge difference.', '1701.08393-3-83-5': 'In particular, the facial parts such as eyes, nose or mouth can be barely distinguished from tiny faces, which makes responses produced by attribute-aware networks meaningless.', '1701.08393-3-83-6': 'In order to recall tiny faces, data augmentation and multi-scale inference may be adopted.', '1701.08393-3-83-7': 'Nonetheless, learning scale-invariant representation is still an open problem.', '1701.08393-3-83-8': 'In this study, we do not deal with tiny faces explicitly.', '1701.08393-3-83-9': 'It is part of our on-going work [CITATION].', '1701.08393-3-84-0': '# Runtime analysis', '1701.08393-3-85-0': 'The runtime of the proposed Faceness-Net-SR-TP is [MATH]ms on a single GPU.', '1701.08393-3-85-1': 'The time includes [MATH]ms to generate faceness proposals with the height of testing image no more than [MATH] pixels.', '1701.08393-3-85-2': 'The efficiency of Faceness-Net-SR-TP is clearly faster than the baseline Faceness-Net since the former shares the layers from conv1 to conv4 in its attribute-aware networks.', '1701.08393-3-85-3': 'Previous CNN based face detector [CITATION] achieves good runtime efficiency too.', '1701.08393-3-85-4': 'Our method differs significantly to this method in that we explicitly handle partial occlusion by inferring face likeliness through part responses.', '1701.08393-3-85-5': 'This difference leads to a significant margin of [MATH] in recall rate (Cascade-CNN [MATH], our method [MATH]) when the number of false positives is fixed at [MATH] on the FDDB dataset.', '1701.08393-3-85-6': 'The complete recall rate of the proposed Faceness-Net-SR-TP is [MATH] compared to [MATH] of Cascade-CNN.', '1701.08393-3-85-7': 'At the expense of recall rate, the fast version of Cascade-CNN achieves [MATH]fps on CPU and [MATH]fps on GPU for [MATH] VGA images.', '1701.08393-3-85-8': 'Our Faceness-Net-SR-TP can achieve practical runtime efficiency under the aggressive setting mentioned above, but still with a [MATH] higher recall rate than the Cascade-CNN.', '1701.08393-3-86-0': '# Conclusion', '1701.08393-3-87-0': 'Different from existing face detection studies, we explored the usefulness of face attributes based supervision for learning a robust face detector.', '1701.08393-3-87-1': 'We observed an interesting phenomenon that face part detectors can be obtained from a CNN that is trained on recognizing attributes from uncropped face images, without explicit part supervision.', '1701.08393-3-87-2': "Consequently, we introduced the notion of 'faceness' score, which was carefully formulated through considering facial parts responses and the associated spatial arrangements.", '1701.08393-3-87-3': 'The faceness score can be employed to re-rank candidate windows of any region proposal techniques to generate a modest set of high-quality face proposals with high recall.', '1701.08393-3-87-4': 'With the generated face proposals, we trained a strong face detector that demonstrated promising performance on various face detection benchmark datasets.', '1701.08393-3-88-0': '# Acknowledgments', '1701.08393-3-89-0': 'regular IEEE prefers the singular form'}

1710.09608

{'1710.09608-1-0-0': 'The emergence and nature of amplitude mediated chimera states, spatio-temporal patterns of co-existing coherent and incoherent regions, are investigated for a globally coupled system of active and inactive Ginzburg-Landau oscillators.', '1710.09608-1-0-1': 'The existence domain of such states is found to shrink and shift in parametric space as the fraction of inactive oscillators is increased.', '1710.09608-1-0-2': 'The role of inactive oscillators is found to be two fold - they get activated to form a separate region of coherent oscillations and in addition decrease the common collective frequency of the coherent regions by their presence.', '1710.09608-1-0-3': 'The dynamical origin of these effects is delineated through a detailed bifurcation analysis of a reduced model equation that is based on a mean field approximation.', '1710.09608-1-0-4': 'Our results may have practical implications for the robustness of such states in biological or physical systems where age related deterioration in the functionality of components can occur.', '1710.09608-1-1-0': '# Introduction', '1710.09608-1-2-0': 'The chimera state, a novel collective phenomenon observed in coupled oscillator systems, that spontaneously emerges as a spatio-temporal pattern of co-existing synchronous and asynchronous groups of oscillators, has attracted a great deal of attention in recent years [CITATION].', '1710.09608-1-2-1': 'First observed and identified by Battogtokh and Kuramoto [CITATION] for a model system of identical phase oscillators that are non-locally coupled, such states have now been shown to occur in a wide variety of systems [CITATION] and under less restrictive conditions than previously thought of [CITATION].', '1710.09608-1-2-2': 'Theoretical and numerical studies have established the existence of chimera states in neuronal models [CITATION], in systems with non-identical oscillators [CITATION], time delay coupled systems [CITATION] and globally coupled systems that retain the amplitude dynamics of the oscillators [CITATION].', '1710.09608-1-2-3': 'Chimera states have also been observed experimentally in chemical [CITATION], optical [CITATION], mechanical [CITATION], electronic [CITATION] and electro-chemical [CITATION] oscillator systems.', '1710.09608-1-2-4': 'Furthermore, chimera states have been associated with some natural phenomena such as unihemispheric sleep [CITATION] in certain birds and mammals during which one half of the brain is synchronized while the other half is in a de-synchronized state [CITATION].', '1710.09608-1-2-5': 'In fact, the electrical activity of the brain resulting in the collective dynamics of the cortical neurons provides a rich canvas for the application of chimera states and is the subject of many present studies [CITATION].', '1710.09608-1-2-6': 'It is believed that a prominent feature of the dynamics of the brain is the generation of a multitude of meta-stable chimera states that it keeps switching between.', '1710.09608-1-2-7': 'Such a process, presumably, is at the heart of our ability to respond to different stimuli and to much of our learning behavior.', '1710.09608-1-2-8': 'Chimera states are thus vital to the functioning of the brain and it is important therefore to investigate their existence conditions and robustness to changes in the system environment.', '1710.09608-1-2-9': 'An interesting question to ask is how the formation of chimera states can be influenced by the loss of functionality of some of the constituent components of the system.', '1710.09608-1-2-10': 'In the case of the brain it could be the damage suffered by some of the neuronal components due to aging or disease [CITATION].', '1710.09608-1-2-11': 'For a model system of coupled oscillators this can be the loss of oscillatory behavior of some of the oscillators.', '1710.09608-1-2-12': 'In this paper we address this question by studying the existence and characteristics of the amplitude mediated chimera (AMC) states in an ensemble of globally coupled Ginzburg-Landau oscillators some of which are in a non-oscillatory (inactive) state.', '1710.09608-1-2-13': 'The coupled set of Ginzburg-Landau oscillators display a much richer dynamics than the coupled phase oscillator systems on which many past studies on chimeras have been carried out.', '1710.09608-1-2-14': 'The conditions governing the emergence of the AMCs are also free of the topological and coupling constraints of the classical phase oscillator chimeras and may therefore have wider practical applications.', '1710.09608-1-2-15': 'Past studies on the robustness of the collective states of a population of coupled oscillators that are a mix of oscillatory and non-oscillatory (inactive) elements have been restricted to the synchronous state [CITATION].', '1710.09608-1-2-16': 'Our work extends such an analysis to the emergent dynamics of the amplitude mediated chimera state.', '1710.09608-1-2-17': 'We find that the presence of inactive elements in the system can significantly impact the existence domain and the nature of the AMCs.', '1710.09608-1-2-18': 'The existence domain of the AMC states is found to shrink and shift in parametric space as the fraction of inactive oscillators is increased in the system.', '1710.09608-1-2-19': 'Under the influence of the coupling the inactive oscillators experience a revival and turn oscillatory to form another separate coherent region.', '1710.09608-1-2-20': "They also inject a sort of 'inertia' in the system that decreases the collective frequency of the coherent regions.", '1710.09608-1-2-21': 'These results are established through extensive numerical simulations of the coupled system and by a systematic comparison with past results obtained in the absence of the inactive oscillators.', '1710.09608-1-2-22': 'To provide a deeper understanding of the numerical results and to trace the dynamical origins of these changes we also present a detailed bifurcation analysis of a reduced model system that has the form of a single driven oscillator equation with a forcing term obtained from a mean field approximation.', '1710.09608-1-3-0': 'The paper is organized as follows.', '1710.09608-1-3-1': 'In the next section (section II) we describe the model equations and summarize the past results on the AMC states obtained from these equations in the absence of any inactive oscillators.', '1710.09608-1-3-2': 'Section III details our numerical simulation results obtained for various fractions of inactive oscillators and discusses the consequent modifications in the existence regions and characteristics of the AMC states.', '1710.09608-1-3-3': 'In section IV we provide some analytic results based on a mean field theory to explain the findings of section III.', '1710.09608-1-3-4': 'Section V gives a brief summary and some concluding remarks on our results.', '1710.09608-1-4-0': '# Model Equations', '1710.09608-1-5-0': 'We consider a large population of globally coupled identical complex Ginzburg-Landau type oscillators whose time evolution is governed by the following set of equations [EQUATION] where, [MATH], [MATH] being the number of oscillators and [MATH] is the mean field.', '1710.09608-1-5-1': 'In Equation ([REF]) [MATH] is the complex amplitude of the [MATH] oscillator, overdot represents a differentiation w.r.t. time ([MATH]), [MATH] are real constants and [MATH] is a parameter specifying the distance from the Hopf bifurcation point.', '1710.09608-1-5-2': 'In the absence of coupling, the [MATH] oscillator exhibits a periodic oscillation if [MATH] while if [MATH] then, the [MATH] oscillator settles down to the fixed point [MATH] and does not oscillate.', '1710.09608-1-5-3': 'Oscillators with [MATH] are termed as active and those with [MATH] are termed as inactive or dead.', '1710.09608-1-5-4': 'Since the oscillators are identical and globally coupled, for the sake of convenience, and without any loss of generality, we set the group of inactive elements to be [MATH] and the active elements to be [MATH], where, [MATH].', '1710.09608-1-5-5': 'For simplicity we also assume, [MATH], [MATH] and [MATH], where both [MATH] and [MATH] are parameters.', '1710.09608-1-5-6': 'Eq. ([REF]), in the absence of the inactive oscillators ([MATH]) has been extensively studied in the past [CITATION] and shown to possess a variety of collective states including synchronous states, splay states, single and multi-cluster states, chaotic states and more recently, also, amplitude mediated chimera states [CITATION].', '1710.09608-1-5-7': 'Fig. ([REF]), reproduced from [CITATION] shows a phase diagram in the [MATH] space for [MATH] that summarizes these past results.', '1710.09608-1-5-8': 'Our objective in this paper is to study the effect of inactive elements on the nature of the AMCs and of modifications if any of their existence region (the shaded region of Fig.1).', '1710.09608-1-5-9': 'Accordingly we carry out an extensive numerical exploration of Eq. ([REF]) in the relevant parametric domain of [MATH] space and compare them to previous results obtained in the absence of inactive oscillators.', '1710.09608-1-6-0': '# Amplitude Mediated Chimera States', '1710.09608-1-7-0': 'In Fig. ([REF]) we display a snapshot of the time evolution of a typical AMC state obtained by a numerical solution of Eq. ([REF]) for [MATH], [MATH] and [MATH].', '1710.09608-1-7-1': 'The figure shows the distribution of the [MATH] oscillators in the complex plane of ([MATH]).', '1710.09608-1-7-2': 'We observe the typical signature of an AMC in the form of a string like object representing the incoherent oscillators and a cluster (represented by a black filled circle) marking the coherent oscillators which move together at a fixed frequency.', '1710.09608-1-7-3': 'The new feature compared to the standard AMC state [CITATION] is the presence of another cluster (marked by a red triangle) that represents the dynamics of the dead oscillators which are no longer inactive now but have acquired a frequency and form another coherent region in the AMC.', '1710.09608-1-7-4': 'These oscillators have a lower amplitude than the originally active oscillators but oscillate at the same common collective frequency as them.', '1710.09608-1-7-5': 'Thus the inactive components of the system experience a revival due to the global coupling and modulate the coherent portion of the AMC profile.', '1710.09608-1-8-0': 'This can be seen more clearly in the snapshots of the profiles of [MATH] and phase [MATH] shown in Figs. ([REF]) and ([REF]) respectively for [MATH], [MATH] and [MATH].', '1710.09608-1-8-1': 'The red solid line in Fig. ([REF]) marks the coherent region arising from the initially inactive oscillators.', '1710.09608-1-8-2': 'They have a smaller amplitude of oscillations than the initially active ones that form the coherent region shown by the blue solid line.', '1710.09608-1-8-3': 'The scattered dots show the incoherent region whose oscillators drift at different frequencies.', '1710.09608-1-8-4': 'The oscillator phases of the three regions are shown in Fig. ([REF]) where one observes that the phases of the oscillators in each coherent region remain the same and there is a finite phase difference between the two regions.', '1710.09608-1-8-5': 'The incoherent regions have a random distribution of phases among the oscillators.', '1710.09608-1-9-0': 'We have next carried out extensive numerical explorations to determine the extent and location of the parametric domain where such AMCs can occur in order to determine the changes if any of the shaded existence region shown in Fig. ([REF]).', '1710.09608-1-9-1': 'Our results are shown in Fig. ([REF]) for several different values of [MATH].', '1710.09608-1-9-2': 'We find that, for a fixed value of [MATH], as the value of [MATH] is increased the parametric domain of the existence region of AMC shrinks and also shifts upward and away from the [MATH] region towards a higher value of [MATH].', '1710.09608-1-9-3': 'This can be understood qualitatively by the argument that as the number of inactive oscillators increases one needs a higher amount of coupling to pull them in and create an AMC state.', '1710.09608-1-9-4': 'The area of this region asymptotically goes to zero as [MATH].', '1710.09608-1-9-5': 'Note that there is a finite parametric domain of existence even for a [MATH] value that is as large as [MATH] indicating that AMCs can occur even in the presence of a very large number of inactive oscillators and thus are very robust to aging related changes in the system environment.', '1710.09608-1-10-0': '# Mean Field Theory', '1710.09608-1-11-0': 'In order to gain a deeper understanding of the dynamical origin of our numerical results we now analyse a reduced model that has been employed in the past to study the chaotic and AMC states of Eqn. ([REF]) in the absence of inactive oscillators [CITATION].', '1710.09608-1-11-1': 'We divide the whole population ([MATH]) into two sub-populations,namely, [MATH] for the active oscillators with [MATH] and [MATH] for the inactive oscillators (to start with) having [MATH].', '1710.09608-1-11-2': 'Along with the general mean field parameter, [MATH], we next construct two other mean field parameters, one for the active oscillators and the other for the inactive ones, as follows, [EQUATION] where [MATH] is the amplitude of the mean field for all the oscillators, [MATH] is the amplitude of the mean field for the active oscillators and [MATH] is the same for the inactive oscillators.', '1710.09608-1-11-3': 'The nature of [MATH], [MATH] and [MATH] can be seen from our numerical simulation results shown in Fig. ([REF]) where we have plotted the time evolution of [MATH] and [MATH].', '1710.09608-1-11-4': 'We observe that the time variation of [MATH] is nearly periodic and identical for the active and inactive oscillators.', '1710.09608-1-11-5': 'A power spectrum plot of the numerical solutions given in Fig. ([REF]) further shows that the periodicity is primarily around a single frequency, [MATH], as denoted by the sharp peak in the power spectrum.', '1710.09608-1-11-6': 'The amplitudes [MATH] and [MATH] evolve on a slower time scale and are nearly constant with small fluctuations around a mean value.', '1710.09608-1-11-7': 'We can thus treat them as constants with [MATH].', '1710.09608-1-12-0': 'Defining [EQUATION] and re-scaling time as [EQUATION] equation ([REF]) can be reduced to [EQUATION] with [EQUATION]', '1710.09608-1-12-1': 'In the above relations, [MATH] and [MATH] are functions of [MATH] since [MATH] and [MATH] now vary with [MATH] as seen from Figs. ([REF]) and ([REF]).', '1710.09608-1-12-2': 'From our numerical simulation results the variation of [MATH] and [MATH] can be represented by the following approximate relations, [EQUATION] where for [MATH] = 0.7, [MATH] = -1, [MATH] = 2 and [MATH] = 1 we get, [MATH] = 1.291, [MATH] = 1.57, [MATH] = 0.63, [MATH] = 0.48 and [MATH] = 0.141.', '1710.09608-1-12-3': 'We can now try to understand the dynamical origin of the behaviour of the existence region of the AMC as a function of [MATH] by making use of the bifurcation diagram of eqn. ([REF]) and the relations ([REF]) and ([REF]).', '1710.09608-1-12-4': 'The bifurcation diagram of ([REF]) in the [MATH] space for [MATH], is shown in Fig. ([REF]) where the red (solid) line is the Hopf bifurcation line and the blue (dotted) line represents the saddle-node bifurcation line.', '1710.09608-1-12-5': 'As discussed in [CITATION] the AMC states arise from the coexistence of a stable node and a limit cycle or a spiral attractor close to the saddle-node curve.', '1710.09608-1-12-6': 'The fluctuations in the amplitude of the mean field then drive the oscillators towards these equilibrium points with those that go to the node constituting the coherent part of the AMC while those that populate the limit cycle or the stable spiral forming the incoherent part of the AMC.', '1710.09608-1-12-7': 'The distribution of the oscillators among these two sub-populations depends on the initial conditions and the kicks in phase space that they receive from the amplitude fluctuations.', '1710.09608-1-12-8': 'The location of these [MATH] AMCs are marked by the open and filled circles in the figure.', '1710.09608-1-12-9': 'With the increase of [MATH], [MATH] decreases, causing the determinant of the Jacobian ([MATH]) of Equation ([REF]) to monotonically decrease while keeping the trace of the Jacobian ([MATH]) to remain unchanged.', '1710.09608-1-12-10': 'This results in the loss of a spiral into a node leading the AMC to collapse into two or three coherent cluster states.', '1710.09608-1-12-11': 'Also, as [MATH] increases, [MATH] decreases and [MATH] decreases.', '1710.09608-1-12-12': 'This leads to the loss of a node and the generation of a spiral causing the AMC states near the Saddle-Node bifurcation line, for [MATH], to evolve into chaotic states.', '1710.09608-1-12-13': 'The above two reasons account for the shrinkage of the existence region of the AMC with the increase of [MATH].', '1710.09608-1-12-14': 'Furthermore due to the re-scaling of [MATH] and [MATH] for a given value of [MATH], the bifurcation plot also shifts in the [MATH] space leading to a relocation of the position of the new AMC states.', '1710.09608-1-12-15': 'The direction and amount of these shifts are indicated by the direction and length of arrows marked in Fig ([REF]) for different values of [MATH].', '1710.09608-1-13-0': 'We can further use expressions ([REF]) and ([REF]) to understand the shift in the existence domain of the AMC by a simple approximate analysis for small values of [MATH].', '1710.09608-1-13-1': 'We seek the shifts in the values of [MATH] and [MATH] for which a given AMC for [MATH] will still remain an AMC for a finite value of [MATH].', '1710.09608-1-13-2': 'This can be obtained by doing a linear perturbation analysis around the values of [MATH],[MATH], [MATH] and [MATH] for [MATH] in expressions ([REF]) and ([REF]).', '1710.09608-1-13-3': 'Writing, [EQUATION] (where the subscript [MATH] labels values at [MATH] and the terms with [MATH] represent small perturbations) substituting in ([REF]) and ([REF]) and retaining only the linear terms of the perturbed quantities, we get, [EQUATION]', '1710.09608-1-13-4': 'The above equations can be solved for [MATH] and [MATH] using the numerically obtained values for [MATH], [MATH] for a chosen small value of [MATH].', '1710.09608-1-13-5': 'For [MATH] we have [MATH] and [MATH].', '1710.09608-1-13-6': 'Using the above values at [MATH] and [MATH] we get [MATH] and [MATH].', '1710.09608-1-13-7': 'These shifts in the values of [MATH] and [MATH] are depicted by an arrow in Fig. ([REF]) and indicate the shift in the location of an AMC in the [MATH] space.', '1710.09608-1-13-8': 'The direction of the shift agrees quite well with the observed shift in the domain of the [MATH] AMC compared to the domain with [MATH].', '1710.09608-1-14-0': '# Summary and Discussion', '1710.09608-1-15-0': 'To summarize, we have studied the influence of a population of inactive oscillators on the formation and dynamical features of amplitude mediated chimera states in an ensemble of globally coupled Ginzburg-Landau oscillators.', '1710.09608-1-15-1': 'From our numerical investigations we find that the inactive oscillators influence the AMCs in several distinct ways.', '1710.09608-1-15-2': 'The coupling with the rest of the active oscillators revives their oscillatory properties and they become a part of the AMC as a separate coherent cluster thereby modulating the structure of the AMC.', '1710.09608-1-15-3': 'Their presence also introduces an additional inertia in the system that reduces the overall frequency of the coherent group of oscillators.', '1710.09608-1-15-4': 'Finally they shrink the existence region of the AMCs in the parametric space of the coupling strength [MATH] and the constant [MATH] (where [MATH] is the imaginary component of the coupling constant).', '1710.09608-1-15-5': 'This region continously shrinks and shifts in this parametric domain as a function of [MATH].', '1710.09608-1-15-6': 'Remarkably, the AMCs continue to exist (albeit in a very small parametric domain) even for [MATH] as large as [MATH] which is indicative of their robustness to aging effects in the system.', '1710.09608-1-15-7': 'Our numerical results can be well understood from an analytic study of a reduced model oscillator that is derived from a mean field theory and consists of a single driven nonlinear oscillator.', '1710.09608-1-15-8': 'The driving term is in fact the mean field arising from the global coupling and that effectively captures the dynamics of both the active and the inactive oscillators.', '1710.09608-1-15-9': 'Our numerical results provide some simple scaling relations between the amplitude of the driver and its primary frequency component as a function of [MATH].', '1710.09608-1-15-10': 'A comparison of the bifurcation diagrams of this equation obtained for [MATH] and finite [MATH] values then provides a good qualitative understanding of the changes taking place in the existence domain of the AMCs due to the presence of the inactive oscillators.', '1710.09608-1-15-11': 'AMCs are a generalized class of chimera states in which both amplitude and phase variations of the oscillators are retained and their existence is not constrained by the need to have non-local forms of coupling.', '1710.09608-1-15-12': 'Our findings can therefore have a wider applicability and be practically relevant for such states in biological or physical systems where aging can diminish the functional abilities of component parts.'}

{'1710.09608-2-0-0': 'The emergence and nature of amplitude mediated chimera states, spatio-temporal patterns of co-existing coherent and incoherent regions, are investigated for a globally coupled system of active and inactive Ginzburg-Landau oscillators.', '1710.09608-2-0-1': 'The existence domain of such states is found to shrink and shift in parametric space as the fraction of inactive oscillators is increased.', '1710.09608-2-0-2': 'The role of inactive oscillators is found to be two fold - they get activated to form a separate region of coherent oscillations and in addition decrease the common collective frequency of the coherent regions by their presence.', '1710.09608-2-0-3': 'The dynamical origin of these effects is delineated through a bifurcation analysis of a reduced model system that is based on a mean field approximation.', '1710.09608-2-0-4': 'Our results may have practical implications for the robustness of such states in biological or physical systems where age related deterioration in the functionality of components can occur.', '1710.09608-2-1-0': 'The chimera state, a novel spatio-temporal pattern of co-existing coherent and incoherent regions, was originally discovered as a collective state of a simple model system of identical phase oscillators that are non-locally coupled to each other.', '1710.09608-2-1-1': 'Since this original discovery, chimera states have been shown to occur in a wide variety of systems and have been the subject of intense theoretical and experimental studies.', '1710.09608-2-1-2': 'They serve as a useful paradigm for describing similar collective phenomena observed in many natural systems such as the variety of metastable collective states of the cortical neurons in the brain or uni-hemispheric sleep in certain birds and mammals during which one half of the brain is synchronized while the other half is in a de-synchronized state.', '1710.09608-2-1-3': 'An important question to ask is what happens to such states when some of the components of the system lose their functionality and die, or, in the context of the oscillator model, when some of the oscillators cease to oscillate.', '1710.09608-2-1-4': 'In our present work, we examine this issue by investigating the dynamics of a system consisting of a mix of active (oscillating) and inactive (non-oscillating) Ginzburg-Landau oscillators that are globally coupled to each other.', '1710.09608-2-1-5': 'The chimera states of this system, known as Amplitude Mediated Chimeras (AMCs) are a more generalized form of the original phase oscillator chimeras and display both amplitude and phase variations.', '1710.09608-2-1-6': 'We find that the inactive oscillators influence the AMCs in several distinct ways.', '1710.09608-2-1-7': 'The coupling with the rest of the active oscillators revives their oscillatory properties and they become a part of the AMC as a separate coherent cluster thereby modulating the structure of the AMC.', '1710.09608-2-1-8': 'Their presence also reduces the overall frequency of the coherent group of oscillators.', '1710.09608-2-1-9': 'Finally, they shrink the existence region of the AMCs in the parametric space of the system.', '1710.09608-2-1-10': 'A remarkable finding is that the AMCs continue to exist even in the presence of a very large fraction (about 90%) of inactive oscillators which suggests that they are very robust against aging effects.', '1710.09608-2-1-11': 'Our findings can be practically relevant for such states in biological or physical systems where aging can diminish the functional abilities of component parts.', '1710.09608-2-2-0': '# Introduction', '1710.09608-2-3-0': 'The chimera state, a novel collective phenomenon observed in coupled oscillator systems, that spontaneously emerges as a spatio-temporal pattern of co-existing synchronous and asynchronous groups of oscillators, has attracted a great deal of attention in recent years [CITATION].', '1710.09608-2-3-1': 'First observed and identified by Kuramoto and Battogtokh[CITATION] for a model system of identical phase oscillators that are non-locally coupled, such states have now been shown to occur in a wide variety of systems [CITATION] and under less restrictive conditions than previously thought of [CITATION].', '1710.09608-2-3-2': 'Theoretical and numerical studies have established the existence of chimera states in neuronal models [CITATION], in systems with non-identical oscillators [CITATION], time delay coupled systems [CITATION] and globally coupled systems that retain the amplitude dynamics of the oscillators [CITATION].', '1710.09608-2-3-3': 'Chimera states have also been observed experimentally in chemical [CITATION], optical [CITATION], mechanical [CITATION], electronic [CITATION] and electro-chemical [CITATION] oscillator systems.', '1710.09608-2-3-4': 'Furthermore, chimera states have been associated with some natural phenomena such as unihemispheric sleep [CITATION] in certain birds and mammals during which one half of the brain is synchronized while the other half is in a de-synchronized state [CITATION].', '1710.09608-2-3-5': 'In fact, the electrical activity of the brain resulting in the collective dynamics of the cortical neurons provides a rich canvas for the application of chimera states and is the subject of many present day studies [CITATION].', '1710.09608-2-3-6': 'It is believed that a prominent feature of the dynamics of the brain is the generation of a multitude of meta-stable chimera states that it keeps switching between.', '1710.09608-2-3-7': 'Such a process, presumably, is at the heart of our ability to respond to different stimuli and to much of our learning behavior.', '1710.09608-2-3-8': 'Chimera states are thus vital to the functioning of the brain and it is important therefore to investigate their existence conditions and robustness to changes in the system environment.', '1710.09608-2-3-9': 'An interesting question to ask is how the formation of chimera states can be influenced by the loss of functionality of some of the constituent components of the system.', '1710.09608-2-3-10': 'In the case of the brain it could be the damage suffered by some of the neuronal components due to aging or disease [CITATION].', '1710.09608-2-3-11': 'For a model system of coupled oscillators this can be the loss of oscillatory behavior of some of the oscillators.', '1710.09608-2-3-12': 'In this paper we address this question by studying the existence and characteristics of the amplitude mediated chimera (AMC) states in an ensemble of globally coupled Complex Ginzburg-Landau (CGL) oscillators some of which are in a non-oscillatory (inactive) state.', '1710.09608-2-3-13': 'The coupled set of CGL oscillators display a much richer dynamics [CITATION] than the coupled phase oscillator systems on which many past studies on chimeras have been carried out [CITATION].', '1710.09608-2-3-14': 'For a set of locally coupled CGL equations (corresponding to the continuum limit) Nicolaou et al [CITATION] have shown the existence of a chimera state consisting of a coherent domain of a frozen spiral structure and an incoherent domain of amplitude turbulence.', '1710.09608-2-3-15': 'The nature of the incoherent state in this case is more akin to turbulent patches observed in fluid systems and distinctly different from the incoherent behaviour obtained in non-locally coupled discrete systems [CITATION].', '1710.09608-2-3-16': 'For globally coupled systems, chimera states with amplitude variations have been obtained for a set of Stuart-Landau oscillators by Schmidt and Krischer [CITATION].', '1710.09608-2-3-17': 'However the coupling they use is of a nonlinear nature and the chimera states emerge as a result of a clustering mechanism.', '1710.09608-2-3-18': 'The AMC states, on the other hand, do not require any nonlinear coupling and exist in a system of linearly coupled CGL equations.', '1710.09608-2-3-19': 'The conditions governing the emergence of the AMCs are thus free of the topological and coupling constraints of the classical phase oscillator chimeras and may therefore have wider practical applications.', '1710.09608-2-4-0': 'Past studies on the robustness of the collective states of a population of coupled oscillators that are a mix of oscillatory and non-oscillatory (inactive) elements have been restricted to the synchronous state [CITATION].', '1710.09608-2-4-1': 'Our work extends such an analysis to the emergent dynamics of the amplitude mediated chimera state.', '1710.09608-2-4-2': 'We find that the presence of inactive elements in the system can significantly impact the existence domain and the nature of the AMCs.', '1710.09608-2-4-3': 'The existence domain of the AMC states is found to shrink and shift in parametric space as the fraction of inactive oscillators is increased in the system.', '1710.09608-2-4-4': 'Under the influence of the coupling the inactive oscillators experience a revival and turn oscillatory to form another separate coherent region.', '1710.09608-2-4-5': 'They also decrease the collective frequency of the coherent regions.', '1710.09608-2-4-6': 'These results are established through extensive numerical simulations of the coupled system and by a systematic comparison with past results obtained in the absence of the inactive oscillators.', '1710.09608-2-4-7': 'To provide a deeper understanding of the numerical results and to trace the dynamical origins of these changes we also present a bifurcation analysis of a reduced model system comprised of two single driven oscillators with a common forcing term obtained from a mean field approximation.', '1710.09608-2-5-0': 'The paper is organized as follows.', '1710.09608-2-5-1': 'In the next section (section II) we describe the full set of model equations and summarize the past results on the AMC states obtained from these equations in the absence of any inactive oscillators.', '1710.09608-2-5-2': 'Section III details our numerical simulation results obtained for various fractions of inactive oscillators and discusses the consequent modifications in the existence regions and characteristics of the AMC states.', '1710.09608-2-5-3': 'In section IV we provide some analytic results from a reduced model system based on a mean field theory to explain the findings of section III.', '1710.09608-2-5-4': 'Section V gives a brief summary and some concluding remarks on our results.', '1710.09608-2-6-0': '# Model Equations', '1710.09608-2-7-0': 'We consider a system of globally coupled complex Ginzburg-Landau type oscillators governed by the following set of equations [EQUATION] where, [MATH] is the complex amplitude of the [MATH] oscillator, [MATH] is the mean field, overdot represents a differentiation w.r.t. time ([MATH]), [MATH] are real constants, [MATH] is the total number of oscillators and [MATH] is a parameter specifying the distance from the Hopf bifurcation point.', '1710.09608-2-7-1': 'In the absence of coupling, the [MATH] oscillator exhibits a periodic oscillation if [MATH] while if [MATH] then, the [MATH] oscillator settles down to the fixed point [MATH] and does not oscillate.', '1710.09608-2-7-2': 'We therefore divide up the system of oscillators into two subsets: the oscillators with [MATH] form one population of oscillators with [MATH] and represent inactive oscillators, while the oscillators with [MATH] have [MATH] and represent active oscillators.', '1710.09608-2-7-3': 'Here [MATH]) represents the fraction of inactive oscillators in the system.', '1710.09608-2-7-4': 'We assume that our system size is sufficiently large that the ratio [MATH] can be treated as a continuous variable [CITATION].', '1710.09608-2-7-5': 'For simplicity we also assume, [MATH] and [MATH], where both [MATH] and [MATH] are positive constants.', '1710.09608-2-7-6': 'The set of equations ([REF]), in the absence of the inactive oscillators ([MATH]) has been extensively studied in the past [CITATION] and shown to possess a variety of collective states including synchronous states, splay states, single and multi-cluster states, chaotic states and more recently, also, amplitude mediated chimera states (AMCs) [CITATION].', '1710.09608-2-7-7': 'The AMC states occur in a restricted region of the ([MATH]) parameter space and are often co-existent with stable synchronous states [CITATION].', '1710.09608-2-7-8': 'Our objective in this paper is to study the effect of inactive elements on the nature of the AMCs and of modifications if any of their existence region.', '1710.09608-2-7-9': 'Accordingly, we carry out an extensive numerical exploration of the set of equations ([REF]) in the relevant parametric domain of ([MATH]) space with [MATH] and compare them to previous results [CITATION] obtained in the absence of inactive oscillators.', '1710.09608-2-8-0': '# Amplitude Mediated Chimera States', '1710.09608-2-9-0': 'In Fig. ([REF]) we display a snapshot of the time evolution of a typical AMC state obtained by a numerical solution of Eq. ([REF]) for [MATH], [MATH] and [MATH].', '1710.09608-2-9-1': 'We start with a state where the oscillators are uniformly distributed over a ring i.e. [EQUATION]', '1710.09608-2-9-2': 'However the existence and formation of the AMC states are found to be independent of the initial conditions as has also been previously shown [CITATION].', '1710.09608-2-9-3': 'The figure shows the distribution of the [MATH] oscillators in the complex plane of ([MATH]).', '1710.09608-2-9-4': 'We observe the typical signature of an AMC in the form of a string like object representing the incoherent oscillators and a cluster (represented by a black filled square) marking the coherent oscillators which move together in the complex plane.', '1710.09608-2-9-5': 'The new feature compared to the standard AMC state [CITATION] is the presence of another cluster (marked by a red triangle) that represents the dynamics of the dead oscillators which are no longer inactive now but have acquired a frequency and form another coherent region in the AMC.', '1710.09608-2-9-6': 'These oscillators have a lower amplitude than the originally active oscillators but oscillate at the same common collective frequency as them.', '1710.09608-2-9-7': 'Thus the inactive components of the system experience a revival due to the global coupling and modulate the coherent portion of the AMC profile.', '1710.09608-2-10-0': 'This can be seen more clearly in the snapshots of the profiles of [MATH] and phase [MATH] shown in Figs. ([REF]) and ([REF]) respectively for [MATH], [MATH] and [MATH].', '1710.09608-2-10-1': 'The solid line (red) close to [MATH] in Fig. ([REF]) marks the coherent region arising from the initially inactive oscillators.', '1710.09608-2-10-2': 'They have a smaller amplitude of oscillation than the initially active ones that form the coherent region shown by the other (blue) solid line.', '1710.09608-2-10-3': 'The scattered dots show the incoherent region whose oscillators drift at different frequencies.', '1710.09608-2-10-4': 'The oscillator phases of the three regions are shown in Fig. ([REF]) where one observes that the phases of the oscillators in each coherent region remain the same and there is a finite phase difference between the two regions.', '1710.09608-2-10-5': 'The incoherent regions have a random distribution of phases among the oscillators.', '1710.09608-2-11-0': 'We have next carried out extensive numerical explorations to determine the extent and location of the parametric domain where such AMCs can occur in order to determine the changes if any from the existence region for [MATH].', '1710.09608-2-11-1': 'Our results are shown in Fig. ([REF]) for several different values of [MATH] where the different curves mark the outer boundaries of the existence region of AMCs for particular values of [MATH].', '1710.09608-2-11-2': 'We find that, for a fixed value of [MATH], as the value of [MATH] is increased the parametric domain of the existence region of AMC shrinks and also shifts upward and away from the [MATH] region towards a higher value of [MATH].', '1710.09608-2-11-3': 'The area of this region asymptotically goes to zero as [MATH].', '1710.09608-2-11-4': 'Note that there is a finite parametric domain of existence even for a [MATH] value that is as large as [MATH] indicating that AMCs can occur even in the presence of a very large number of inactive oscillators and thus are very robust to aging related changes in the system environment.', '1710.09608-2-11-5': 'We have also found that the revival is dependent on the value of the coupling constant [MATH] as well as the parameters [MATH] and [MATH].', '1710.09608-2-12-0': '# Mean Field Theory', '1710.09608-2-13-0': 'In order to gain a deeper understanding of the dynamical origin of our numerical results we now analyse a reduced model that is a generalization of a similar system that has been employed in the past to study the chaotic and AMC states of Eq. ([REF]) in the absence of initially inactive oscillators [CITATION].', '1710.09608-2-13-1': 'We define appropriate mean field parameters [MATH] for the initially inactive oscillator population and [MATH] for the active population, as, [EQUATION] where [MATH] and [MATH] are the amplitudes of these mean fields and [MATH] and [MATH] are their mean frequencies.', '1710.09608-2-13-2': 'Note that the mean field for the entire system ([REF]) can also be expressed as, [EQUATION] with [EQUATION]', '1710.09608-2-13-3': 'The nature of [MATH], [MATH] and [MATH] can be seen from our numerical simulation results shown in Fig. ([REF]) where we have plotted the time evolution of [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1710.09608-2-13-4': 'We observe that the time variations of [MATH], [MATH] and [MATH] are nearly periodic and identical for the active, initially inactive and the full set of oscillators.', '1710.09608-2-13-5': 'A power spectrum plot of [MATH] given in Fig. ([REF]) further shows that the periodicity is primarily around a single frequency, [MATH], as denoted by the sharp peak in the power spectrum.', '1710.09608-2-13-6': 'Note also that this central frequency varies as a function of [MATH] and decreases as [MATH] increases.', '1710.09608-2-13-7': 'The amplitudes [MATH] and [MATH] evolve on a slower time scale and are nearly constant with small fluctuations around a mean value.', '1710.09608-2-13-8': 'The mean values also decrease as a function of [MATH] and this is shown in Fig. ([REF]) where the mean value of [MATH] is plotted against [MATH].', '1710.09608-2-13-9': 'The decrease of [MATH] with [MATH] is displayed in Fig. ([REF]).', '1710.09608-2-14-0': 'The near constancy of the amplitudes of the mean fields and the existence of a common dominant single frequency of oscillation permits the development of a simple model system, along the lines of Ref. [CITATION], in terms of the driven dynamics of representative single oscillators.', '1710.09608-2-15-0': 'We therefore define two single oscillator variables [MATH] and [MATH] to represent the dynamics of any one of the initially inactive oscillators and any one of the initially active oscillators, respectively.', '1710.09608-2-15-1': 'We take, [EQUATION]', '1710.09608-2-15-2': 'Using ([REF]) and ([REF]) in ([REF]) and re-scaling time as [EQUATION] we get two driven single oscillator equations, [EQUATION] with [EQUATION]', '1710.09608-2-15-3': 'The term [MATH] represents the mean field contribution of the entire set of oscillators and is a common driver for members of each sub-population of the oscillators.', '1710.09608-2-15-4': 'The basic difference in the dynamics of the two populations arises from the sign of the [MATH] term, namely, [MATH] for the initially inactive population and [MATH] for the active oscillators.', '1710.09608-2-15-5': 'We have taken [MATH] to agree with our numerical simulations.', '1710.09608-2-15-6': 'In the relations [REF] and [REF], [MATH] and [MATH] are now functions of [MATH] since [MATH] and [MATH] vary with [MATH] as seen from Figs. ([REF]) and ([REF]) as well as from Figs. ([REF]) and ([REF]).', '1710.09608-2-16-0': 'We can now try to understand the dynamical origin of the behaviour of the existence region of the AMC as a function of [MATH] by analyzing the two reduced equations Eq.([REF] and Eq.([REF]).', '1710.09608-2-16-1': 'For [MATH], Eq.([REF]) has been shown [CITATION] to have a rich bifurcation diagram in the [MATH] space as shown in Fig. ([REF]) where the red (solid) line is the Hopf bifurcation line and the blue (dotted) line represents the saddle-node bifurcation line.', '1710.09608-2-16-2': 'The thin black solid line between region II and V is the homoclinic bifurcation line.', '1710.09608-2-16-3': 'The big red circle and the black square indicate the Takens Bogdanov bifurcation point and the codimension-two point and are labeled as TB and G respectively.', '1710.09608-2-16-4': 'The various regions, marked as I-V, are characterized by the existence of a single or a combination of nodes, saddle points, attractive limit cycles, stable spirals and unstable spirals.', '1710.09608-2-16-5': 'The region between the red solid and the blue dotted lines represents the probable regime of AMC states.', '1710.09608-2-16-6': 'As discussed previously [CITATION] the AMC states arise from the coexistence of a stable node and a limit cycle or a spiral attractor close to the saddle-node curve.', '1710.09608-2-16-7': 'The fluctuations in the amplitude of the mean field then drive the oscillators towards these equilibrium points with those that go to the node constituting the coherent part of the AMC while those that populate the limit cycle or the stable spiral forming the incoherent part of the AMC.', '1710.09608-2-16-8': 'The distribution of the oscillators among these two sub-populations depends on the initial conditions and the kicks in phase space that they receive from the amplitude fluctuations.', '1710.09608-2-16-9': 'The location of these [MATH] AMCs are marked by the open and filled circles in Fig. ([REF]).', '1710.09608-2-16-10': 'With the increase of [MATH], [MATH] decreases, causing the determinant of the Jacobian ([MATH]) of Eq. ([REF]) to monotonically decrease while keeping the trace of the Jacobian ([MATH]) to remain unchanged.', '1710.09608-2-16-11': 'This results in the loss of a spiral into a node leading the AMC to collapse into two or three coherent cluster states.', '1710.09608-2-16-12': 'Also, as [MATH] increases, [MATH] decreases and [MATH] decreases.', '1710.09608-2-16-13': 'This leads to the loss of a node and the generation of a spiral causing the AMC states near the saddle-node bifurcation line, for [MATH], to evolve into chaotic states.', '1710.09608-2-16-14': 'The above two reasons account for the shrinkage of the existence region of the AMC with the increase of [MATH].', '1710.09608-2-16-15': 'Furthermore due to the re-scaling of [MATH] and [MATH] for a given value of [MATH], the bifurcation plot also shifts in the [MATH] space as shown in Fig. ([REF]) for [MATH].', '1710.09608-2-16-16': 'For [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH], we find [MATH] and [MATH] while for [MATH] we get [MATH] and [MATH].', '1710.09608-2-16-17': 'This shift leads to a relocation of the position of the new AMC state as indicated by the black arrow in the figure.', '1710.09608-2-16-18': 'The amount of shift can also be estimated by using relations ([REF] - [REF]).', '1710.09608-2-16-19': 'In Fig ([REF]) we show such shifts for various values of [MATH] by the directions and lengths of arrows originating from various points of the bifurcation diagram for [MATH].', '1710.09608-2-17-0': 'We can also use expressions ([REF]) and ([REF]) to understand the shift in the existence domain of the AMC in the [MATH] space, as shown in Fig. ([REF]), by a simple approximate analysis for small values of [MATH].', '1710.09608-2-17-1': 'We seek the shifts in the values of [MATH] and [MATH] for which a given AMC for [MATH] will still remain an AMC for a finite value of [MATH].', '1710.09608-2-17-2': 'This can be obtained by doing a linear perturbation analysis around the values of [MATH],[MATH], [MATH] and [MATH] for [MATH] in expressions ([REF]) and ([REF]).', '1710.09608-2-17-3': 'Writing, [EQUATION] (where the subscript [MATH] labels values at [MATH] and the terms with [MATH] represent small perturbations) substituting in ([REF]) and ([REF]) and retaining only the linear terms of the perturbed quantities, we get, [EQUATION]', '1710.09608-2-17-4': 'The above equations can be solved for [MATH] and [MATH] using the numerically obtained values for [MATH], [MATH] for a chosen small value of [MATH].', '1710.09608-2-17-5': 'For [MATH] we have [MATH] and [MATH].', '1710.09608-2-17-6': 'Using the above values at [MATH] and [MATH] we get [MATH] and [MATH].', '1710.09608-2-17-7': 'These shifts in the values of [MATH] and [MATH] are depicted by an arrow in Fig. ([REF]) and indicate the shift in the location of an AMC in the [MATH] space.', '1710.09608-2-17-8': 'The direction of the shift agrees quite well with the observed shift in the domain of the [MATH] AMC compared to the domain with [MATH].', '1710.09608-2-18-0': 'We next turn to an analysis of Eq.([REF]) to understand the behavior of the sub-population of initially inactive oscillators.', '1710.09608-2-18-1': 'In contrast to Eq.([REF]), this equation has a very simple topological structure in that it only admits a stable fixed point which corresponds to a periodic motion (with frequency [MATH]) of the corresponding variable [MATH].', '1710.09608-2-18-2': 'This explains the existence of the coherent region marked in red (around [MATH]) shown in Fig. ([REF]).', '1710.09608-2-18-3': 'In Fig. [REF](a) we plot the evolution dynamics of Eq. ([REF]) in the phase space of [MATH] for [MATH], [MATH], [MATH] and [MATH] with [MATH] and [MATH].', '1710.09608-2-18-4': 'The location of the fixed point at [MATH] corresponds to [MATH], which corresponds to the left coherent cluster shown in Fig. ([REF]).', '1710.09608-2-18-5': 'For a comparison we also show in Fig. [REF](b) a corresponding phase diagram for Eq. ([REF]) for the same set of parameters as Fig. [REF](a).', '1710.09608-2-18-6': 'We see the existence of three fixed points - one stable (marked with a [MATH] symbol) and two unstable (marked with a [MATH] symbol) and a limit cycle - the ingredients for the creation of an AMC.', '1710.09608-2-18-7': 'Thus the combined dynamics of the two model equations ([REF]) and ([REF]) provide a composite picture of the existence domain and characteristic features of the AMC in the presence of a population of inactive oscillators.', '1710.09608-2-19-0': '# Summary and Discussion', '1710.09608-2-20-0': 'To summarize, we have studied the influence of a population of inactive oscillators on the formation and dynamical features of amplitude mediated chimera states in an ensemble of globally coupled Ginzburg-Landau oscillators.', '1710.09608-2-20-1': 'From our numerical investigations we find that the inactive oscillators influence the AMCs in several distinct ways.', '1710.09608-2-20-2': 'The coupling with the rest of the active oscillators revives their oscillatory properties and they become a part of the AMC as a separate coherent cluster thereby modulating the structure of the AMC.', '1710.09608-2-20-3': 'Their presence also reduces the overall frequency of the coherent group of oscillators.', '1710.09608-2-20-4': 'Finally they shrink the existence region of the AMCs in the parametric space of the coupling strength [MATH] and the constant [MATH] (where [MATH] is the imaginary component of the coupling constant).', '1710.09608-2-20-5': 'This region continously shrinks and shifts in this parametric domain as a function of [MATH].', '1710.09608-2-20-6': 'Remarkably, the AMCs continue to exist (albeit in a very small parametric domain) even for [MATH] as large as [MATH] which is indicative of their robustness to aging effects in the system.', '1710.09608-2-20-7': 'Our numerical results can be well understood from an analytic study of a reduced model that is derived from a mean field theory and consists of two driven nonlinear oscillators that are representative of typical members of the sub-populations of initially inactive oscillators and the active oscillators.', '1710.09608-2-20-8': 'The driving term for both these single oscillators equations is the mean field arising from the global coupling of all the oscillators.', '1710.09608-2-20-9': 'A bifurcation analysis of both these model equations provides a good qualitative understanding of the changes taking place in the existence domain of the AMCs due to the presence of the inactive oscillators.', '1710.09608-2-20-10': 'AMCs are a generalized class of chimera states in which both amplitude and phase variations of the oscillators are retained and their existence is not constrained by the need to have non-local forms of coupling.', '1710.09608-2-20-11': 'Our findings can therefore have a wider applicability and be practically relevant for such states in biological or physical systems where aging can diminish the functional abilities of component parts.'}

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[]

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[]

['1710.09608-1-12-2', '1710.09608-2-15-1', '1710.09608-2-16-16']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1710.09608

hep-ph-0612175

{'hep-ph-0612175-1-0-0': 'Unitarity cuts of enhanced pomeron diagrams are analyzed in the framework of the old Reggeon Field Theory.', 'hep-ph-0612175-1-0-1': 'The complete set of cut non-loop enhanced graphs is obtained.', 'hep-ph-0612175-1-0-2': 'Important cancellations between contributions of certain sub-classes of cut diagrams are observed.', 'hep-ph-0612175-1-1-0': '# Introduction', 'hep-ph-0612175-1-2-0': 'Even nowadays, forty years after the Reggeon Field Theory (RFT) [CITATION] has been proposed, it attracts attention of researches.', 'hep-ph-0612175-1-2-1': 'This is partly due to the fact that a number of important results of the old RFT remain also valid in the perturbative BFKL pomeron calculus [CITATION].', 'hep-ph-0612175-1-2-2': 'Thus, RFT remains a testing laboratory for novel approaches, prior to their realization within more complicated BFKL framework.', 'hep-ph-0612175-1-2-3': 'On the other hand, a perturbative treatment of peripheral hadronic collisions still remains a challenge, the processes being dominated by "soft" parton physics.', 'hep-ph-0612175-1-2-4': 'Hence, numerous Monte Carlo (MC) generators employ pomeron phenomenology [CITATION] to describe the structure of "underlying" hadronic events, which proved to be a successful approach (see, e.g., [CITATION]).', 'hep-ph-0612175-1-3-0': 'Yet one usually restricts himself with the comparatively simple quasi-eikonal scheme, where hadron-hadron scattering amplitude is described by diagrams of Fig. [REF],', 'hep-ph-0612175-1-4-0': 'corresponding to independent pomeron exchanges between the two hadrons, and can be expressed via the so-called pomeron quasi-eikonal [MATH] as [CITATION][EQUATION] with [MATH], [MATH] and [MATH] being c.m. energy squared and impact parameter; [MATH] - the shower enhancement coefficient.', 'hep-ph-0612175-1-4-1': 'A small imaginary part of [MATH] can be neglected in the high energy limit.', 'hep-ph-0612175-1-5-0': 'However, at high energies one has to account for the contributions of enhanced pomeron diagrams, corresponding to pomeron-pomeron interactions [CITATION].', 'hep-ph-0612175-1-5-1': 'Thus, in addition to simple pomeron exchanges of Fig. [REF], one has to consider multiple exchanges of coupled enhanced graphs.', 'hep-ph-0612175-1-5-2': 'The elastic scattering amplitude can still be written in the usual quasi-eikonal form [EQUATION] where for the contribution of coupled enhanced graphs [MATH] one obtains the representation', 'hep-ph-0612175-1-6-0': 'of Fig. [REF] [CITATION].', 'hep-ph-0612175-1-7-0': 'Assuming [MATH]-meson dominance of multi-pomeron vertices this gives [EQUATION] where [MATH] is related to the triple-pomeron coupling [MATH] as [MATH], and we used the abbreviations [MATH], [MATH], [MATH].', 'hep-ph-0612175-1-7-1': 'The contribution [MATH] corresponds to pomeron "nets", exchanged between hadrons [MATH] and [MATH], which are generated starting from a given vertex [MATH], [MATH] are rapidity and impact parameter distances between hadron [MATH] and this vertex) according to the recursive equation of Fig. [REF]: [EQUATION]', 'hep-ph-0612175-1-7-2': 'In contrast to the usual "fan" diagram equation (which can be obtained setting [MATH] in Fig. [REF]), the "net fan" contribution [MATH] accounts for absorptive corrections due to re-scatterings both on the projectile and on the target hadrons.', 'hep-ph-0612175-1-7-3': 'The pomeron connected to the initial vertex [MATH] in Fig. [REF] will be referred as the "fan handle".', 'hep-ph-0612175-1-8-0': 'The knowledge of the elastic scattering amplitude allows one to calculate total and elastic cross sections.', 'hep-ph-0612175-1-8-1': 'However, to obtain partial weights of particular inelastic final states, one has to study unitarity cuts of elastic scattering diagrams and to re-sum the corresponding contributions to all orders.', 'hep-ph-0612175-1-8-2': 'In the following, we shall apply the AGK cutting rules [CITATION] to "net fan" graphs of Fig. [REF].', 'hep-ph-0612175-1-8-3': 'Collecting contributions of cuts of certain topologies, we shall use them as building blocks to obtain the complete set of cut diagrams, corresponding to the full discontinuity of the graphs of Fig. [REF].', 'hep-ph-0612175-1-9-0': '# Unitarity cuts of "net fans"', 'hep-ph-0612175-1-10-0': 'It is convenient to separate different unitarity cuts of "net fan" graphs in two classes: in the first sub-set cut pomerons form "fan"-like structures, some examples shown in Fig. [REF] (a), (b);', 'hep-ph-0612175-1-11-0': 'in the diagrams of the second kind some cut pomerons are connected to each other in a "zigzag" way, such that pomeron end rapidities are arranged as [MATH], see Fig. [REF] (c), (d).', 'hep-ph-0612175-1-12-0': 'Let us consider the first class and obtain separately both the total contribution of "fan"-like cuts [MATH] and the part of it, formed by diagrams with the "handle" of the "fan" being uncut, an example shown in Fig. [REF] (b), [MATH].', 'hep-ph-0612175-1-12-1': 'Applying AGK cutting rules to the graphs of Fig. [REF] and collecting contributions of cuts of desirable structures we obtain for [MATH], [MATH] the representations of Figs. [REF] and [REF],', 'hep-ph-0612175-1-13-0': 'which gives [EQUATION]', 'hep-ph-0612175-1-13-1': 'Here the omitted arguments of the eikonals in the integrands in ([REF]-[REF]) read [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0612175-1-14-0': 'The first diagram in the r.h.s. of Fig. [REF] is obtained by cutting the single pomeron exchanged between hadron [MATH] and the vertex [MATH] in the r.h.s. of Fig. [REF], whereas the other ones come from cutting the 2nd graph in the r.h.s. of Fig. [REF] in such a way that all cut pomerons are arranged in a "fan"-like structure and the cut plane passes through the "handle" of the "fan".', 'hep-ph-0612175-1-14-1': 'In graph (b) the vertex [MATH] couples together [MATH] cut projectile "net fans", each one characterized by a "fan"-like structure of cuts, and any numbers [MATH] of uncut projectile and target "net fans".', 'hep-ph-0612175-1-14-2': 'Here one has to subtract pomeron self-coupling contribution ([MATH], [MATH]) - graph (c), as well as the contributions of graphs (d) and (e), where in all [MATH] cut projectile "net fans", connected to the vertex [MATH], the "handles" of the "fans" remain uncut and all these "handles" and all the [MATH] uncut projectile "net fans" are situated on the same side of the cut plane.', 'hep-ph-0612175-1-14-3': 'Finally, in graph (f) the cut plane passes between [MATH] uncut projectile "net fans", with at least one remained on either side of the cut.', 'hep-ph-0612175-1-15-0': 'In the recursive representation of Fig. [REF] for the contribution [MATH] the graphs (a), (b), (c) in the r.h.s. of the Figure are similar to the diagrams (b), (d), (f) of Fig. [REF] correspondingly, with the difference that the "handle" of the "fan" is now uncut.', 'hep-ph-0612175-1-15-1': 'Therefore, there are [MATH] uncut target "net fans" connected to the vertex [MATH] in such a way that at least one of them is positioned on the opposite side of the cut plane with respect to the "handle" pomeron.', 'hep-ph-0612175-1-15-2': 'On the other hand, one has to add graph (d), where the vertex [MATH] couples together [MATH] projectile "net fans", which are cut in a "fan"-like way and have their "handles" uncut and positioned on the same side of the cut plane, together with any numbers [MATH] of projectile and [MATH] of target uncut "net fans", such that the vertex remains uncut.', 'hep-ph-0612175-1-15-3': 'Here one has to subtract the pomeron self-coupling ([MATH], [MATH]) - graph (e).', 'hep-ph-0612175-1-16-0': 'Adding ([REF]) to ([REF]), we obtain [EQUATION] with the solution (c.f. ([REF])) [EQUATION]', 'hep-ph-0612175-1-16-1': 'To investigate "zigzag"-like cuts of "net fan" graphs, the examples shown in Fig. [REF] (c) and (d), we introduce [MATH]-th order cut "net fan" contributions [MATH]), which, in addition to the above-considered "fan"-like cut diagrams, contain also ones with up to [MATH] cut pomerons connected to each other in a "zigzag" way, i.e., with pomeron end rapidities being arranged as [MATH].', 'hep-ph-0612175-1-16-2': 'For example, the graphs of Fig. [REF] (c) and (d) belong correspondingly to the 2nd and 3rd order cut "net fan" contributions.', 'hep-ph-0612175-1-16-3': 'As before, we consider two subsamples of the diagrams, with the "handles" of the "fans" being cut, [MATH], and uncut, [MATH], which leads us to the recursive equations of Figs. [REF] and [REF].', 'hep-ph-0612175-1-17-0': 'Compared to the ones of Figs. [REF] and [REF], they contain additional graphs, Fig. [REF] (g)-(i) and Fig. [REF] (f)-(m), where the vertex [MATH] is coupled to [MATH] cut target "net fans" of [MATH]-th order (we set [MATH], [MATH]).', 'hep-ph-0612175-1-17-1': 'Combining the contributions of the graphs of Figs. [REF] and [REF], we obtain [EQUATION] with the solution [EQUATION]', 'hep-ph-0612175-1-17-2': 'Thus, for the summary contribution of all cuts of "net fan" graphs of Fig. [REF] we obtain [MATH], as it should be.', 'hep-ph-0612175-1-17-3': 'On the other hand, contributions of various "zigzag" cuts precisely cancel each other [EQUATION]', 'hep-ph-0612175-1-17-4': 'One can obtain an alternative representation for [MATH], [MATH], as shown in Figs. [REF] and [REF],', 'hep-ph-0612175-1-18-0': 'applying ([REF]-[REF]) (correspondingly Figs. [REF] and [REF]) recursively to generate any number of vertices, connected to uncut projectile and target "net fans", along the "handle" of the "fan".', 'hep-ph-0612175-1-18-1': 'The broken pomeron lines in Figs. [REF] and [REF] correspond to [MATH]-channel sequences of cut and uncut pomerons which are separated by vertices connected to uncut projectile and target "net fans".', 'hep-ph-0612175-1-18-2': 'In particular, the contributions [MATH] and [MATH] of the first graph in the r.h.s. of Figs. [REF] and [REF] correspondingly (the index [MATH] indicates whether the downmost (uppermost) pomeron in the sequence is cut, [MATH]), or uncut, [MATH])) are defined as (c.f. ([REF]-[REF])) [EQUATION]', 'hep-ph-0612175-1-18-3': 'Similarly, we can obtain a recursive representation for [MATH]-th order "zigzag"-cut contributions [MATH], [MATH], applying recursively the relations of Figs. [REF] and [REF] to generate any number of intermediate vertices along the "fan" "handle", which are connected to uncut and [MATH]-th order cut projectile and target "net fans", until we end up with the vertex [MATH] which either couples together [MATH]-th order projectile "zigzag"-cut contributions and any numbers of uncut and [MATH]-th order cut projectile and target "net fans", or is coupled to [MATH]-th order target "zigzag"-cut contributions (in addition, to any numbers of uncut and [MATH]-th order cut target "net fans") and to uncut and [MATH]-th order cut projectile "net fans".', 'hep-ph-0612175-1-18-4': 'The corresponding relation for [MATH] is shown in Fig. [REF],', 'hep-ph-0612175-1-19-0': 'the one for [MATH] looks similarly (c.f. Figs. [REF] and [REF]).', 'hep-ph-0612175-1-20-0': '# Cut enhanced diagrams', 'hep-ph-0612175-1-21-0': 'We are going to derive the full set of cut diagrams corresponding to [MATH]-channel discontinuity of elastic scattering contributions of Fig. [REF].', 'hep-ph-0612175-1-21-1': 'Let us start with cut graphs characterized by "tree"-like structure of cut pomerons, which can be constructed coupling any numbers [MATH] of "fan"-like cut projectile and target "net fans" in one vertex.', 'hep-ph-0612175-1-22-0': 'First we consider the case of [MATH], which leads us to the set of graphs of Fig. [REF],', 'hep-ph-0612175-1-23-0': 'where we do not have any double counting of the same contributions.', 'hep-ph-0612175-1-23-1': 'For example, the graphs Fig. [REF] (a)-(e) have a single vertex [MATH], which couples together [MATH] projectile and [MATH] target cut "fans".', 'hep-ph-0612175-1-23-2': 'Correspondingly, different structures of cuts inside the "fans" and different topologies of uncut "fans" result in different diagrams.', 'hep-ph-0612175-1-23-3': 'For the combined contribution of all the graphs of Fig. [REF] we obtain, using ([REF]) [EQUATION] where [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0612175-1-24-0': 'Now we come to the case of [MATH] and [MATH], which results in the diagrams of Fig. [REF].', 'hep-ph-0612175-1-25-0': 'Similarly to the above, we obtain [EQUATION]', 'hep-ph-0612175-1-25-1': 'Finally we consider the case of [MATH] and [MATH], which can be obtained reversing the graphs of Fig. [REF] upside-down.', 'hep-ph-0612175-1-25-2': 'There we have to correct for double counting of the same contributions.', 'hep-ph-0612175-1-25-3': 'For example, considering the first diagram of Fig. [REF] being reversed upside-down and expanding its projectile cut "fan" using the relations of Figs. [REF] and [REF], we obtain the set of graphs of Fig. [REF].', 'hep-ph-0612175-1-26-0': 'Clearly, the third diagram in the r.h.s. of Fig. [REF], being symmetric with respect to the projectile and the target, will appear in a similar expansion of the first graph of Fig. [REF].', 'hep-ph-0612175-1-26-1': 'On the other hand, all the other graphs in the r.h.s. of Fig. [REF], except the first two, find their duplicates in the expansions of other diagrams of Fig. [REF].', 'hep-ph-0612175-1-26-2': 'Thus, the only new contributions are the ones of Fig. [REF] (a)-(g).', 'hep-ph-0612175-1-27-0': 'In addition, we have to include the graphs (h)-(j) of the Figure, which correspond to a [MATH]-channel sequence of [MATH] cut and uncut pomerons which are separated by vertices connected to uncut projectile and target "net fans", with the downmost and the uppermost pomerons in the sequence being cut.', 'hep-ph-0612175-1-27-1': 'The contribution of the graphs of Fig. [REF] is [EQUATION]', 'hep-ph-0612175-1-27-2': 'Adding ([REF]-[REF]) together and using ([REF]), ([REF]), ([REF]), ([REF]-[REF]), we can obtain [EQUATION]', 'hep-ph-0612175-1-27-3': 'However, the unitarity requires the sum of all cuts of the diagrams of Fig. [REF] to be equal to twice the imaginary part of the elastic scattering contribution, i.e. to [MATH].', 'hep-ph-0612175-1-27-4': 'Thus, the contributions of all cuts of non-tree ("zigzag") type should precisely cancel each other.', 'hep-ph-0612175-1-27-5': 'To verify that, we can construct the complete set of corresponding cut diagrams replacing in Figs. [REF] and [REF] some contributions [MATH] and [MATH] and [MATH] ) by [MATH] and [MATH] and [MATH]), whereas the others - by [MATH] and [MATH] and [MATH]), starting from [MATH], etc.', 'hep-ph-0612175-1-27-6': 'Using the representation of Fig. [REF] for [MATH] (similarly for [MATH]) to correct for double counts in the same way as above for "tree"-like diagrams, we obtain the set of graphs of Fig. [REF].', 'hep-ph-0612175-1-28-0': 'There, the diagrams (a)-(c) contain [MATH]-th order projectile "zigzag"-cut contributions; this gives a factor [MATH], which is equal zero due to ([REF]).', 'hep-ph-0612175-1-28-1': 'Similarly, the graphs (i)-(k) have [MATH]-th order target "zigzag"-cut "fans", which gives [MATH].', 'hep-ph-0612175-1-28-2': 'The contributions of the graphs (e) and (f) are equal up to a sign and cancel each other; the same applies to the diagrams (m) and (n).', 'hep-ph-0612175-1-28-3': 'Finally, the graphs (d), (g), (h) give together [EQUATION] where the expression in the square brackets vanishes due to ([REF]).', 'hep-ph-0612175-1-28-4': 'Similarly one demonstrates the cancellation for the graphs (l), (o), and (p).', 'hep-ph-0612175-1-28-5': 'This complets the proof of [MATH]-channel unitarity of the approach.', 'hep-ph-0612175-1-28-6': 'It is worth stressing that we obtained a cancellation for the contributions of non-"tree" type cut diagrams of Fig. [REF] to the total cross section, not to inclusive particle spectra; such graphs have to be taken into consideration in inelastic event generation procedures.', 'hep-ph-0612175-1-29-0': 'In conclusion, we derived the complete set of cut non-loop enhanced diagrams, as given in Figs. [REF], [REF], [REF], and [REF].', 'hep-ph-0612175-1-29-1': 'The obtained results open the way for a consistent implementation of the RFT in hadronic MC models.', 'hep-ph-0612175-1-29-2': 'Details of the corresponding procedure will be discussed elsewhere.', 'hep-ph-0612175-1-29-3': 'On the other hand, as already discussed in [CITATION], the scheme can be applied for calculations of diffraction cross sections and of rapidity gap survival probabilities.', 'hep-ph-0612175-1-29-4': 'It is noteworthy that current analysis does not depend on a particular parameterization for the pomeron exchange amplitude and can be extended for a phenomenological description of "hard" processes [CITATION].'}

{'hep-ph-0612175-2-0-0': 'Unitarity cuts of enhanced Pomeron diagrams are analyzed in the framework of the Reggeon Field Theory.', 'hep-ph-0612175-2-0-1': 'Assuming the validity of the Abramovskii-Gribov-Kancheli cutting rules, we derive a complete set of cut non-loop enhanced graphs and observe important cancellations between certain sub-classes of the latter.', 'hep-ph-0612175-2-0-2': 'We demonstrate also how the present method can be generalized to take into consideration Pomeron loop contributions.', 'hep-ph-0612175-2-1-0': '# Introduction', 'hep-ph-0612175-2-2-0': 'Even nowadays, forty years after the Reggeon Field Theory (RFT) [CITATION] has been proposed, it is widely applied for the description of high energy hadronic and nuclear interactions.', 'hep-ph-0612175-2-2-1': 'Partly, this is due to the fact that a number of important results of the old RFT remain also valid in the perturbative BFKL Pomeron calculus [CITATION].', 'hep-ph-0612175-2-2-2': 'Thus, RFT remains a testing laboratory for novel approaches, prior to their realization within more complicated BFKL framework.', 'hep-ph-0612175-2-2-3': 'On the other hand, a perturbative treatment of peripheral hadronic collisions still remains a challenge, the processes being dominated by "soft" parton physics.', 'hep-ph-0612175-2-2-4': 'Hence, when describing the high energy behavior of total and diffractive hadronic cross sections, calculating probabilities of large rapidity gap survival (RGS) in hadronic final states, or developing general purpose Monte Carlo (MC) generators, one applies the Pomeron phenomenology [CITATION].', 'hep-ph-0612175-2-3-0': 'Nevertheless, in MC applications one usually restricts himself with the comparatively simple multi-channel eikonal scheme, where elastic scattering amplitude is described by diagrams of Fig. [REF],', 'hep-ph-0612175-2-4-0': 'corresponding to independent Pomeron exchanges between the two hadrons, and can be expressed via the Pomeron eikonal [MATH] as [CITATION] [EQUATION] with [MATH], [MATH] and [MATH] being c.m. energy squared and impact parameter for the scattering.', 'hep-ph-0612175-2-4-1': 'A small imaginary part of [MATH] can be neglected in the high energy limit.', 'hep-ph-0612175-2-4-2': 'Here [MATH] and [MATH] are correspondingly relative weights and relative strengths of diffraction eigenstates [MATH] of hadron [MATH] in the multi-component scattering scheme [CITATION]: [EQUATION] with [MATH], [MATH].', 'hep-ph-0612175-2-5-0': 'The optical theorem allows one to obtain immediately total hadron-hadron cross section: [EQUATION]', 'hep-ph-0612175-2-5-1': 'On the other hand, in order to derive partial cross sections for various hadronic final states, one applies the Abramovskii-Gribov-Kancheli (AGK) cutting procedure [CITATION] to obtain asymptotically non-negligible unitarity cuts of the elastic scattering diagrams of Fig. [REF].', 'hep-ph-0612175-2-5-2': 'Combining together contributions of cuts of certain topologies, one can identify them with partial contributions of particular final states.', 'hep-ph-0612175-2-6-0': 'For example, the so-called topological cross sections, corresponding to the interaction being composed of [MATH] "elementary" particle production processes, are given by the contributions of of graphs in Fig. [REF] (left),', 'hep-ph-0612175-2-7-0': 'with precisely [MATH] Pomerons being cut, and with any number [MATH] of uncut ones [CITATION]: [EQUATION]', 'hep-ph-0612175-2-7-1': 'On the other hand, requiring the cut plane to pass between uncut Pomerons, with at least one on either side of the cut, as shown in Fig. [REF] (right), one obtains [EQUATION] which can be further split into elastic and diffraction dissociation cross sections [CITATION].', 'hep-ph-0612175-2-7-2': 'One can easily verify that the sum of ([REF]) and ([REF]) satisfies the [MATH]-channel unitarity relation: [EQUATION]', 'hep-ph-0612175-2-7-3': 'However, the above-described scheme can account only for low mass inelastic excitations of the projectile and target hadrons, where the integration over the masses of those inelastic intermediate states can be performed irrespective the total c. m. energy [MATH] for the scattering [CITATION].', 'hep-ph-0612175-2-7-4': 'To treat high mass diffraction, one has to generalize the scheme, including the contributions of enhanced Pomeron diagrams, i.e. to take Pomeron-Pomeron interactions into account [CITATION].', 'hep-ph-0612175-2-7-5': 'Moreover, such enhanced diagrams provide important absorptive corrections to the cross sections ([REF]-[REF]) and generate new final states of complicated topologies [CITATION].', 'hep-ph-0612175-2-7-6': 'For example, cutting the simplest triple-Pomeron diagram of Fig. [REF] (a),', 'hep-ph-0612175-2-8-0': 'one obtains the projectile high mass diffraction contribution of Fig. [REF] (b), a screening correction to one cut Pomeron process (Fig. [REF] (c)), and a new "cut Pomeron fusion" process of Fig. [REF] (d).', 'hep-ph-0612175-2-8-1': 'With increasing energy, more complicated enhanced diagrams, with numerous multi-Pomeron vertices, become important.', 'hep-ph-0612175-2-8-2': 'Thus, to obtain meaningful expressions for the contributions of various hadronic final states, one has to perform a re-summation of the whole series of the corresponding cut diagrams.', 'hep-ph-0612175-2-9-0': 'A general procedure for the re-summation of enhanced diagram contributions to the elastic scattering amplitude has been proposed in [CITATION].', 'hep-ph-0612175-2-9-1': 'The goal of the present work is to apply the method for the re-summation of cut diagram contributions and to obtain a full set of the AGK-based unitarity cuts for the considered class of uncut enhanced Pomeron graphs.', 'hep-ph-0612175-2-9-2': 'Mainly, we deal below with diagrams of "net" type, i.e. with arbitrary enhanced diagrams which do not contain Pomeron "loops" (multi-Pomeron vertices connected to each other by two or more Pomerons), although we shall demonstrate how the method can be generalized to include rather general Pomeron loop contributions.', 'hep-ph-0612175-2-9-3': 'An analysis of the structure of final states corresponding to various unitarity cuts and an implementation of the approach in a hadronic MC generator is discussed elsewhere [CITATION].', 'hep-ph-0612175-2-10-0': 'The paper is organized as follows.', 'hep-ph-0612175-2-10-1': 'In Section 2 we remind the basic results of the earlier works [CITATION] on the re-summation of uncut enhanced diagrams.', 'hep-ph-0612175-2-10-2': 'In Section 3 we analyze unitarity cuts of certain sub-graphs of general net diagrams (so-called "net fans") and perform a re-summation of the cuts characterized by certain topologies of cut Pomerons.', 'hep-ph-0612175-2-10-3': 'Next, in Section 4 we use those re-summed contributions as building blocks in the construction of the full set of cut non-loop enhanced diagrams, corresponding to the full discontinuity of the elastic scattering contributions of Section 2.', 'hep-ph-0612175-2-10-4': 'Finally, in Section 5 we outline a generalization of the present scheme to include Pomeron loop contributions.', 'hep-ph-0612175-2-10-5': 'We conclude in Section 6.', 'hep-ph-0612175-2-11-0': '# Uncut enhanced diagrams', 'hep-ph-0612175-2-12-0': 'Taking Pomeron-Pomeron interactions into account, one has to consider multiple exchanges of coupled enhanced graphs, in addition to simple Pomeron exchanges of Fig. [REF].', 'hep-ph-0612175-2-12-1': 'The elastic scattering amplitude can still be written in the usual multi-channel eikonal form (c.f. ([REF])): [EQUATION] where [MATH] stays for the eikonal contribution of irreducible enhanced graphs exchanged between the diffraction eigenstates [MATH] and [MATH] of hadrons [MATH] and [MATH].', 'hep-ph-0612175-2-12-2': 'Restricting oneself with non-loop enhanced diagrams, one can express [MATH] via the contributions [MATH] of sub-graphs of certain structure, so-called "net fans", as shown in Fig. [REF] [CITATION].', 'hep-ph-0612175-2-13-0': 'Here the "net fan" contributions [MATH] are defined by the recursive equation of Fig. [REF].', 'hep-ph-0612175-2-14-0': 'In particular, assuming eikonal structure of the vertices for the transition of [MATH] into [MATH] Pomerons [CITATION]: [EQUATION] with [MATH] being the triple-Pomeron constant, the representations of Figs. [REF], [REF] yield [CITATION]: [EQUATION] where [MATH] is the eikonal for a Pomeron exchange between hadron [MATH] and the vertex [MATH], [MATH] and [MATH] being rapidity and impact parameter distances between hadron [MATH] and that vertex, whereas the eikonal [MATH] corresponds to a Pomeron exchange between the vertices [MATH] and [MATH].', 'hep-ph-0612175-2-14-1': 'In ([REF]) we used the abbreviations [MATH], [MATH], [MATH].', 'hep-ph-0612175-2-15-0': 'The nickname "net fan" for the contribution [MATH] is because the Schwinger-Dyson equation of Fig. [REF] generates Pomeron nets exchanged between hadrons [MATH] and [MATH], starting from a given vertex [MATH], some examples shown in Fig. [REF],', 'hep-ph-0612175-2-16-0': 'and because this equation is formally similar to the usual fan diagram equation.', 'hep-ph-0612175-2-16-1': 'The latter can be recovered setting [MATH] in Fig. [REF].', 'hep-ph-0612175-2-16-2': 'In that case ([REF]) reduces to [EQUATION]', 'hep-ph-0612175-2-16-3': 'In contrast to the fan contribution [MATH], which can be associated with parton density of a free hadron [MATH] [CITATION], the "net fan" equation of Fig. [REF] accounts for absorptive corrections due to the re-scattering on the partner hadron [MATH] and corresponds to parton momentum and impact parameter distribution which is probed during the interaction [CITATION].', 'hep-ph-0612175-2-16-4': 'In the following, the Pomeron connected to the initial vertex [MATH] in Fig. [REF] will be referred to as the "fan handle".', 'hep-ph-0612175-2-17-0': 'In the representation of Fig. [REF] for enhanced diagram contribution to the elastic scattering amplitude, the first graph in the r.h.s. corresponds to any number [MATH] of projectile "net fans" [MATH] and any number [MATH] of target ones [MATH], [MATH], which are coupled together in some "central" vertex, whereas the second graph in the Figure is the double counting correction.', 'hep-ph-0612175-2-17-1': 'Any diagram with [MATH] multi-Pomeron vertices is generated [MATH] times by the first graph in the r.h.s. of Fig. [REF] (as there are [MATH] choices for the "central" vertex), from which [MATH] contributions are subtracted by the second graph.', 'hep-ph-0612175-2-18-0': '# Unitarity cuts of "net fans"', 'hep-ph-0612175-2-19-0': 'Before considering the cuts of the elastic scattering graphs of Fig. [REF], let us apply the AGK cutting procedure to the "net fan" contributions of Fig. [REF].', 'hep-ph-0612175-2-19-1': 'It is convenient to separate various unitarity cuts of "net fan" graphs in two classes: in the first sub-set cut Pomerons form fan-like structures, some examples shown in Fig. [REF] (a), (b);', 'hep-ph-0612175-2-20-0': 'in the diagrams of the second kind some cut Pomerons are connected to each other in a zigzag way, such that Pomeron end rapidities are arranged as [MATH], see Fig. [REF] (c), (d).', 'hep-ph-0612175-2-21-0': 'Let us consider the first class and obtain separately both the total contribution of fan-like cuts [MATH] and the part of it, formed by diagrams with the handle of the fan being uncut, an example shown in Fig. [REF] (b), [MATH].', 'hep-ph-0612175-2-21-1': 'Applying AGK cutting rules to the graphs of Fig. [REF] and collecting contributions of cuts of desirable structures we obtain for [MATH], [MATH] the representations of Figs. [REF] and [REF],', 'hep-ph-0612175-2-22-0': 'which gives [EQUATION]', 'hep-ph-0612175-2-22-1': 'Here the omitted indices and arguments of the eikonals in the integrands in ([REF]-[REF]) read [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0612175-2-23-0': 'The first diagram in the r.h.s. of Fig. [REF] is obtained by cutting the single Pomeron exchanged between hadron [MATH] and the vertex [MATH] in the r.h.s. of Fig. [REF], whereas the other ones come from cutting the 2nd graph in the r.h.s. of Fig. [REF] in such a way that all cut Pomerons are arranged in a fan-like structure and the cut plane passes through the handle Pomeron.', 'hep-ph-0612175-2-23-1': 'In graph (b) the vertex [MATH] couples together [MATH] cut projectile "net fans", each one characterized by a fan-like structure of cuts, and any numbers [MATH] of uncut projectile and target "net fans".', 'hep-ph-0612175-2-23-2': 'Here one has to subtract the Pomeron self-coupling contribution ([MATH], [MATH]) - graph (c), as well as the contributions of graphs (d) and (e), where in all [MATH] cut projectile "net fans", connected to the vertex [MATH], the handle Pomerons remain uncut and all these handle Pomerons and all the [MATH] uncut projectile "net fans" are situated on the same side of the cut plane.', 'hep-ph-0612175-2-23-3': 'Finally, in graph (f) the cut plane passes between [MATH] uncut projectile "net fans", with at least one remained on either side of the cut.', 'hep-ph-0612175-2-24-0': 'In the recursive representation of Fig. [REF] for the contribution [MATH], the graphs (a), (b), (c) in the r.h.s. of the Figure are similar to the diagrams (b), (d), (f) of Fig. [REF] correspondingly, with the difference that the handle of the fan is now uncut.', 'hep-ph-0612175-2-24-1': 'Therefore, there are [MATH] uncut target "net fans" connected to the vertex [MATH] in such a way that at least one of them is positioned on the opposite side of the cut plane with respect to the handle Pomeron.', 'hep-ph-0612175-2-24-2': 'On the other hand, one has to add graph (d), where the vertex [MATH] couples together [MATH] projectile "net fans", which are cut in a fan-like way and have their handle Pomerons uncut and positioned on the same side of the cut plane, together with any numbers [MATH] of projectile and [MATH] of target uncut "net fans", such that the vertex remains uncut.', 'hep-ph-0612175-2-24-3': 'Here one has to subtract the Pomeron self-coupling ([MATH], [MATH]) - graph (e).', 'hep-ph-0612175-2-25-0': 'Adding ([REF]) to ([REF]), we obtain [EQUATION] with the solution (c.f. ([REF])) [EQUATION]', 'hep-ph-0612175-2-25-1': 'To investigate zigzag-like cuts of "net fan" graphs, the examples shown in Fig. [REF] (c) and (d), we introduce [MATH]-th order cut "net fan" contributions [MATH], [MATH], which in addition to the above-considered fan-like cut diagrams contain also ones with up to [MATH] cut Pomerons connected to each other in a zigzag way, i.e., with Pomeron end rapidities being arranged as [MATH].', 'hep-ph-0612175-2-25-2': 'For example, the graphs of Fig. [REF] (c) and (d) belong correspondingly to the 2nd and 3rd order cut "net fan" contributions.', 'hep-ph-0612175-2-25-3': 'As before, we consider two subsamples of the diagrams, with the handle Pomerons being cut, [MATH], and uncut, [MATH], which leads us to the recursive equations of Figs. [REF] and [REF] respectively.', 'hep-ph-0612175-2-26-0': 'Compared to the ones of Figs. [REF] and [REF], they contain additional graphs, Fig. [REF] (g)-(i) and Fig. [REF] (f)-(m), where the vertex [MATH] is coupled to [MATH] cut target "net fans" of [MATH]-th order (we set [MATH], [MATH]).', 'hep-ph-0612175-2-26-1': 'Thus, we obtain [EQUATION]', 'hep-ph-0612175-2-26-2': 'Adding ([REF]) to ([REF]), we obtain [EQUATION] with the solution (c.f. ([REF])) [EQUATION]', 'hep-ph-0612175-2-26-3': 'Thus, for the summary contribution of all cuts of "net fan" graphs of Fig. [REF] we obtain [MATH], as it should be.', 'hep-ph-0612175-2-26-4': 'On the other hand, contributions of various zigzag-like cuts precisely cancel each other [EQUATION]', 'hep-ph-0612175-2-26-5': 'Making use of ([REF]), we can re-write ([REF]) as [EQUATION] and obtain (c.f. ([REF])) [EQUATION] i.e. the summary contribution of all AGK cuts of "net fan" graphs, with the cut plane passing through the handle Pomeron, satisfies the usual fan diagram equation ([REF]), being independent on re-scatterings on the partner hadron.', 'hep-ph-0612175-2-27-0': 'One can obtain an alternative representation for [MATH], [MATH], as shown in Figs. [REF] and [REF],', 'hep-ph-0612175-2-28-0': 'applying ([REF]-[REF]) (correspondingly Figs. [REF] and [REF]) recursively to generate any number of vertices, connected to uncut projectile and target "net fans", along the handle of the fan.', 'hep-ph-0612175-2-28-1': 'The broken Pomeron lines in Figs. [REF] and [REF] correspond to [MATH]-channel sequences of cut and uncut Pomerons, which are separated by vertices connected to uncut projectile and target "net fans"; the corresponding contributions are defined via recursive representations of Fig. [REF].', 'hep-ph-0612175-2-29-0': 'In particular, the contributions [MATH] and [MATH] of the first graphs in the r.h.s. of Figs. [REF] and [REF] respectively (the index [MATH] indicates whether the downmost (uppermost) Pomeron in the sequence is cut, [MATH]), or uncut, [MATH])) are defined as (c.f. ([REF]-[REF])) [EQUATION]', 'hep-ph-0612175-2-29-1': 'Similarly, we can obtain a recursive representation for [MATH]-th order zigzag-like cut "net fans" [MATH], [MATH], applying recursively the relations ([REF]-[REF]) (Figs. [REF] and [REF]) to generate any number of intermediate vertices along the handle Pomeron, which are connected to uncut and [MATH]-th order cut projectile and target "net fans", until we end up with the vertex [MATH], which either couples together [MATH]-th order projectile zigzag-like cut contributions and any numbers of uncut and [MATH]-th order cut projectile and target "net fans", or is coupled to [MATH]-th order target zigzag-like cut contributions (in addition, to any numbers of uncut and [MATH]-th order cut target "net fans") and to uncut and [MATH]-th order cut projectile "net fans".', 'hep-ph-0612175-2-29-2': 'The corresponding relation for [MATH] is shown in Fig. [REF],', 'hep-ph-0612175-2-30-0': 'the one for [MATH] looks similarly (c.f. Figs. [REF] and [REF]).', 'hep-ph-0612175-2-31-0': '# Cut enhanced diagrams', 'hep-ph-0612175-2-32-0': 'We are going to derive the complete set of cut diagrams corresponding to [MATH]-channel discontinuity of elastic scattering contributions of Fig. [REF].', 'hep-ph-0612175-2-32-1': 'Let us start with cut graphs characterized by a tree-like structure of cut Pomerons, which can be constructed coupling any numbers [MATH] of fan-like cut projectile and target "net fans" in one vertex.', 'hep-ph-0612175-2-33-0': 'First we consider the case of [MATH], which leads us to the set of graphs of Fig. [REF],', 'hep-ph-0612175-2-34-0': 'where we do not have any double counting of the same contributions.', 'hep-ph-0612175-2-34-1': 'For example, the graphs Fig. [REF] (a)-(e) have a single vertex [MATH], which couples together [MATH] projectile and [MATH] target fan-like cut "net fans".', 'hep-ph-0612175-2-34-2': 'Correspondingly, different structures of cut "net fans" and different topologies of the uncut ones result in different diagrams.', 'hep-ph-0612175-2-34-3': 'For the combined contribution of all the graphs of Fig. [REF] we obtain, using ([REF]), [EQUATION] where we use the abbreviations [MATH], [MATH], [MATH], [MATH].', 'hep-ph-0612175-2-35-0': 'Now we come to the case of [MATH] and [MATH], which results in the diagrams of Fig. [REF].', 'hep-ph-0612175-2-36-0': 'Similarly to the above, we obtain [EQUATION]', 'hep-ph-0612175-2-36-1': 'Finally we consider the case of [MATH] and [MATH], which can be obtained reversing the graphs of Fig. [REF] upside-down.', 'hep-ph-0612175-2-36-2': 'There we have to correct for double counting of the same contributions.', 'hep-ph-0612175-2-36-3': 'For example, considering the first diagram of Fig. [REF] being reversed upside-down and expanding its projectile fan-like cut "net fan" using the relations of Figs. [REF] and [REF], we obtain the set of graphs of Fig. [REF].', 'hep-ph-0612175-2-37-0': 'Clearly, the third diagram in the r.h.s. of Fig. [REF], being symmetric with respect to the projectile and the target, will appear in a similar expansion of the first graph of Fig. [REF].', 'hep-ph-0612175-2-37-1': 'On the other hand, all the other graphs in the r.h.s. of Fig. [REF], except the first two, find their duplicates in the expansions of other diagrams of Fig. [REF].', 'hep-ph-0612175-2-37-2': 'Thus, the only new contributions are the ones of Fig. [REF] (a)-(g).', 'hep-ph-0612175-2-38-0': 'In addition, we have to include the graphs (h)-(j) of the Figure, which correspond to a [MATH]-channel sequence of [MATH] cut and uncut Pomerons which are separated by vertices connected to uncut projectile and target "net fans", with the downmost and the uppermost Pomerons in the sequence being cut.', 'hep-ph-0612175-2-38-1': 'The contribution of the graphs of Fig. [REF] is [EQUATION]', 'hep-ph-0612175-2-38-2': 'Adding ([REF]-[REF]) together and using ([REF]), ([REF]), ([REF]), ([REF]-[REF]), we can obtain [EQUATION]', 'hep-ph-0612175-2-38-3': 'However, the unitarity requires the sum of all the cuts of the diagrams of Fig. [REF] to be equal to twice the imaginary part of the elastic scattering contribution, i.e. to [MATH].', 'hep-ph-0612175-2-38-4': 'Thus, the contributions of all cuts of non-tree (zigzag) type should precisely cancel each other.', 'hep-ph-0612175-2-38-5': 'To verify that, we can construct the complete set of corresponding cut diagrams replacing in Figs. [REF] and [REF] some contributions [MATH] and [MATH] and [MATH] ) by [MATH] and [MATH] and [MATH]), whereas the others - by [MATH] and [MATH] and [MATH]), starting from [MATH], etc.', 'hep-ph-0612175-2-38-6': 'Using the representation of Fig. [REF] for [MATH] (similarly for [MATH]) to correct for double counts in the same way as above for the tree-like cut diagrams, we obtain the set of graphs of Fig. [REF].', 'hep-ph-0612175-2-39-0': 'There, the diagrams (a)-(c) contain [MATH]-th order projectile zigzag-like cut "net fans"; this gives a factor [MATH], which is equal zero due to ([REF]).', 'hep-ph-0612175-2-39-1': 'Similarly, the graphs (i)-(k) have [MATH]-th order target zigzag-like cut "net fans", which gives [MATH].', 'hep-ph-0612175-2-39-2': 'The contributions of the graphs (e) and (f) are equal up to a sign and cancel each other; the same applies to the diagrams (m) and (n).', 'hep-ph-0612175-2-39-3': 'Finally, the graphs (d), (g), (h) give together [EQUATION] where the expression in the square brackets vanishes due to ([REF]).', 'hep-ph-0612175-2-39-4': 'Similarly one demonstrates the cancellation for the graphs (l), (o), and (p).', 'hep-ph-0612175-2-39-5': 'This completes the proof of the [MATH]-channel unitarity of the approach.', 'hep-ph-0612175-2-39-6': 'It is worth stressing that we obtained a cancellation for the contributions of non-tree type cut diagrams of Fig. [REF] to the total cross section, not to inclusive particle spectra; such graphs have to be taken into consideration in inelastic event generation procedures.', 'hep-ph-0612175-2-40-0': 'It is noteworthy that the [MATH]-channel unitarity is still violated in the described scheme in certain parts of the kinematic space, which is the price for neglecting Pomeron loop contributions.', 'hep-ph-0612175-2-40-1': 'For example, one obtains here a negative contribution for double high mass diffraction (central rapidity gap) cross section [MATH].', 'hep-ph-0612175-2-40-2': 'Indeed, dominant contribution to the process comes from hadron-hadron scattering at relatively large impact parameters, where the RGS probability is not too small, and, due to the smallness of the triple-Pomeron coupling [MATH], is given by the graphs of Fig. [REF] (left),', 'hep-ph-0612175-2-41-0': 'with only two multi-Pomeron vertices.', 'hep-ph-0612175-2-41-1': 'Thus, for [MATH] one obtains [EQUATION] where [MATH] is the (positively defined) RGS factor, i.e. the probability that additional re-scattering processes produce no secondary particles in the rapidity interval [MATH].', 'hep-ph-0612175-2-42-0': '# Pomeron loops', 'hep-ph-0612175-2-43-0': 'The above-described procedure can be easily generalized to include simple loop contributions, replacing single Pomerons connecting neighboring "cells" of Pomeron "nets" by [MATH]-channel sequences of Pomerons and Pomeron loops.', 'hep-ph-0612175-2-43-1': 'To this end, one can modify the definition ([REF]) of the "net fan" contributions [MATH], as shown in Fig. [REF],', 'hep-ph-0612175-2-44-0': 'i.e. [EQUATION] where the contributions [MATH] and [MATH] of Pomeron loop sequences, exchanged between hadron [MATH] and the vertex [MATH], respectively, between the vertices [MATH] and [MATH], are defined via the recursive representations of Figs. [REF] and [REF]:', 'hep-ph-0612175-2-45-0': '[EQUATION]', 'hep-ph-0612175-2-45-1': 'Here [MATH] and [MATH] are the contributions of such Pomeron loop sequences (exchanged between hadron [MATH] and the vertex [MATH], respectively, between the vertices [MATH] and [MATH]), which start from a single Pomeron connected to the vertex [MATH], as shown in Figs. [REF], [REF].', 'hep-ph-0612175-2-46-0': 'Redefining in a similar way the fan diagram equation ([REF]), one can literally repeat the analysis of Ref. [CITATION] and obtain the contribution of arbitrary Pomeron nets, with neighboring "cells" being connected by [MATH]-channel loop sequences, in the form of Eq. ([REF]), with the eikonal [MATH] being replaced by the corresponding loop sequence contribution [MATH], as depicted in Fig. [REF] (a), (b).', 'hep-ph-0612175-2-47-0': 'In addition, one has to consider an exchange of a single [MATH]-channel loop sequence between hadrons [MATH] and [MATH], as shown in Fig. [REF] (c), (d), such that each of the two hadrons is coupled to a single Pomeron only.', 'hep-ph-0612175-2-47-1': 'The complete eikonal contribution for the considered class of enhanced diagrams is therefore [EQUATION] where we use the same abbreviations as in ([REF]).', 'hep-ph-0612175-2-48-0': 'The analysis of unitarity cuts of the generalized scheme proceeds similarly to the one described in Sections [REF] and [REF].', 'hep-ph-0612175-2-48-1': 'For the contribution of fan-like cuts [MATH], [MATH] of "net fan" graphs of Fig. [REF] one obtains the representations of Fig. [REF]', 'hep-ph-0612175-2-49-0': '(c.f. Figs. [REF], [REF]), i.e. [EQUATION] where the contributions of cut loop sequences [MATH], [MATH], [MATH], [MATH] satisfy the recursive equations of Figs. [REF] and [REF]', 'hep-ph-0612175-2-50-0': 'respectively.', 'hep-ph-0612175-2-50-1': 'Clearly, one has [EQUATION]', 'hep-ph-0612175-2-50-2': 'Thus, repeating literally the reasoning of Sections [REF] and [REF], for the complete set of tree-like AGK cuts of the graphs of Fig. [REF] (a), (b) we can obtain the representation of Figs. [REF], [REF], and [REF], with the uncut and fan-like cut "net fans" being defined now as in Figs. [REF] and [REF] respectively, with the single cut Pomeron contribution [MATH] in Fig. [REF] (i), (j) being replaced by the one of the cut loop sequence [MATH], and with the t-channel sequences of cut and uncut Pomerons [MATH] and [MATH] (depicted as broken Pomeron lines in Fig. [REF]) being replaced by the ones of loops [MATH], [MATH].', 'hep-ph-0612175-2-50-3': 'For the latter one obtains, similarly to ([REF]-[REF]), [EQUATION]', 'hep-ph-0612175-2-50-4': 'In a similar way one generalizes the definition of the [MATH]-th order cut "net fan" contributions [MATH] and obtains the representation of Fig. [REF] for the complete set of zigzag-like cuts of the diagrams of Fig. [REF] (a), (b).', 'hep-ph-0612175-2-50-5': 'Finally, for the graphs of Fig. [REF] (c), (d) the cutting procedure is trivial, yielding a convolution of the cut loop sequences [MATH] and [MATH] of Fig. [REF] with the cut Pomeron eikonal [MATH]: [EQUATION]', 'hep-ph-0612175-2-50-6': 'The described generalization of the scheme appears to be sufficient to cure the above-mentioned problems with the violation of the [MATH]-channel unitarity in certain kinematic regions.', 'hep-ph-0612175-2-50-7': 'In the considered case of double high mass diffraction, in addition to the graph of Fig. [REF] (left) one obtains now the loop diagram of Fig. [REF] (right), such that the summary contribution becomes [EQUATION]', 'hep-ph-0612175-2-50-8': 'In the region of large impact parameters, which gives the dominant contribution to ([REF]), either [MATH] or/and [MATH] is small.', 'hep-ph-0612175-2-50-9': 'Thus, the expression in the square brackets reduces to [EQUATION] and assures a positive result for [MATH].', 'hep-ph-0612175-2-50-10': 'A systematic analysis of hadronic final states, obtained in the described scheme, will be presented elsewhere [CITATION].', 'hep-ph-0612175-2-51-0': '# Conclusions', 'hep-ph-0612175-2-52-0': 'We proposed here a method for a re-summation of the full set of AGK-based unitarity cuts of a very general class of net-like enhanced Pomeron diagrams.', 'hep-ph-0612175-2-52-1': 'This is the principal novelty of the present analysis compared to other related works [CITATION], which have been restricted to investigations of contributions of particular, notably diffractive, final states only.', 'hep-ph-0612175-2-52-2': 'Though the main derivation has been performed for the class of non-loop net-like diagrams, we have demonstrated that the method can be trivially generalized to include Pomeron loop contributions.', 'hep-ph-0612175-2-52-3': 'In the latter case, one simply considers neighboring cells of Pomeron nets to be connected by (cut or uncut) [MATH]-channel sequences of Pomerons and Pomeron loops, rather than by single Pomeron exchanges; the same applies for the connections between initial hadrons and the correspondingly neighboring net cells.', 'hep-ph-0612175-2-52-4': 'In a similar way one can include more general loop contributions [CITATION].', 'hep-ph-0612175-2-53-0': 'Although the obtained expressions for the contributions of cut enhanced diagrams are based on a particular eikonal ansatz ([REF]) for multi-Pomeron vertices, the corresponding diagrammatic representations, e.g. of Figs. [REF], [REF], [REF], [REF], [REF], [REF], [REF], [REF], are of more general character and remain applicable for arbitrary parameterizations of multi-Pomeron vertices.', 'hep-ph-0612175-2-54-0': 'It is noteworthy that current analysis does not depend on a particular parameterization for the Pomeron exchange amplitude and can be extended for a phenomenological description of "hard" partonic processes [CITATION].', 'hep-ph-0612175-2-54-1': 'In principle, the proposed method can be also applied in the perturbative BFKL Pomeron framework.', 'hep-ph-0612175-2-54-2': 'However, one should keep in mind that the principal assumption of the present analysis was that the AGK cutting rules remain valid, in particular, that multi-Pomeron vertices remain unmodified by the cutting procedure.', 'hep-ph-0612175-2-54-3': 'The fact that the AGK rules are not proven in QCD, with some deviations from the AGK prescriptions already reported in literature [CITATION], implies that the method may have to be significantly modified, when employed in the BFKL Pomeron calculus.', 'hep-ph-0612175-2-54-4': 'On the other hand, recent investigations indicate that the AGK picture still remains a reasonably good approximation in the pQCD framework [CITATION].', 'hep-ph-0612175-2-55-0': 'The obtained results open the way for a consistent implementation of the RFT in hadronic MC models.', 'hep-ph-0612175-2-55-1': 'Details of the corresponding procedure will be discussed elsewhere [CITATION].', 'hep-ph-0612175-2-55-2': 'On the other hand, the scheme can be applied for calculations of total and diffractive hadronic cross sections and of rapidity gap survival probabilities, a preliminary analysis already reported in [CITATION].'}

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[]

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[['hep-ph-0612175-1-8-2', 'hep-ph-0612175-2-19-0']]

['hep-ph-0612175-1-3-0', 'hep-ph-0612175-1-6-0', 'hep-ph-0612175-1-12-1', 'hep-ph-0612175-1-13-0', 'hep-ph-0612175-1-13-1', 'hep-ph-0612175-1-16-0', 'hep-ph-0612175-1-17-4', 'hep-ph-0612175-1-18-4', 'hep-ph-0612175-1-22-0', 'hep-ph-0612175-1-27-2', 'hep-ph-0612175-2-3-0', 'hep-ph-0612175-2-6-0', 'hep-ph-0612175-2-7-6', 'hep-ph-0612175-2-15-0', 'hep-ph-0612175-2-21-1', 'hep-ph-0612175-2-22-0', 'hep-ph-0612175-2-22-1', 'hep-ph-0612175-2-25-0', 'hep-ph-0612175-2-26-2', 'hep-ph-0612175-2-27-0', 'hep-ph-0612175-2-29-2', 'hep-ph-0612175-2-33-0', 'hep-ph-0612175-2-38-2', 'hep-ph-0612175-2-40-2', 'hep-ph-0612175-2-43-1', 'hep-ph-0612175-2-44-0', 'hep-ph-0612175-2-45-0', 'hep-ph-0612175-2-49-0', 'hep-ph-0612175-2-50-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0612175

1209.3198

{'1209.3198-1-0-0': 'The statistical evidence (or marginal likelihood) is a key quantity in Bayesian statistics, allowing one to assess the probability of the data given the model under investigation.', '1209.3198-1-0-1': 'This paper focuses on refining the power posterior approach to improve estimation of the evidence.', '1209.3198-1-0-2': 'The power posterior method involves transitioning from the prior to the posterior by powering the likelihood by a temperature variable.', '1209.3198-1-0-3': 'In common with other tempering algorithms, the power posterior involves some degree of tuning, and this paper addresses this issue.', '1209.3198-1-0-4': 'The main contributions of this article are twofold - we present a result from the numerical analysis literature which can reduce the bias in the estimate of the evidence by addressing the error arising from numerically integrating across the temperature.', '1209.3198-1-0-5': 'We also address the choice of temperature ladder, and present an adaptive algorithm which gives excellent performance in the examples considered here.', '1209.3198-1-0-6': 'A key practical point is that both of these innovations incur virtually no extra cost.', '1209.3198-1-0-7': 'Keywords: Marginal likelihood, Markov chain Monte Carlo, Power posteriors, Statistical evidence, Tempering, Thermodynamic integration.', '1209.3198-1-1-0': '# Introduction', '1209.3198-1-2-0': 'The statistical evidence (sometimes called the marginal likelihood or integrated likelihood) is a vital quantity in Bayesian statistics for the comparison of models, [MATH].', '1209.3198-1-2-1': 'Under the Bayesian paradigm we consider the posterior distribution [EQUATION] for data [MATH] and parameters [MATH] within model [MATH], where [MATH] denotes the prior distribution for parameters within model [MATH] and where [MATH] denotes the prior model probability.', '1209.3198-1-2-2': 'The evidence for data [MATH] given model [MATH] arises as the normalising constant of the posterior distribution within model [MATH], [EQUATION] and thus results from integrating the un-normalised posterior across the [MATH] parameter space, [EQUATION]', '1209.3198-1-2-3': 'This of course assumes that the prior distribution for [MATH] is proper.', '1209.3198-1-2-4': 'The marginal likelihood is often then used to calculate Bayes factors when one wants to compare two competing models, [MATH] and [MATH], [EQUATION]', '1209.3198-1-2-5': 'Here, [MATH] is the posterior probability for model [MATH] and it can be evaluated, using the evidence for each of the collection of models under consideration, [EQUATION]', '1209.3198-1-2-6': 'Estimation of the evidence is a non-trivial task for most statistical models and there has been considerable effort in the literature to find algorithms and methods for this purpose.', '1209.3198-1-2-7': "Laplace's method (Tierney and Kadane, 1986) is an early approach and very widely used.", '1209.3198-1-2-8': "Other notable and popular approaches include Chib's method (Chib 1995), annealed importance sampling (Neal 2001), nested sampling (Skilling 2006), bridge sampling (Meng and Wong, 1996) and power posteriors (Friel and Pettitt, 1998) which is the focus of this paper.", '1209.3198-1-2-9': 'For a recent review and perspective on these and other methods, see Friel and Wyse (2012).', '1209.3198-1-3-0': 'This paper is organised as follows.', '1209.3198-1-3-1': 'Section [REF] outlines the power posterior method, and the approach we propose to improve estimation of the evidence.', '1209.3198-1-3-2': 'Section [REF] illustrates the potential gain from implementing the methodology which we propose.', '1209.3198-1-3-3': 'We offer some conclusions in Section [REF].', '1209.3198-1-4-0': '# The power posterior approach', '1209.3198-1-5-0': 'In what follows we will drop the explicit conditioning on model [MATH] for notational simplicity.', '1209.3198-1-5-1': 'We follow the notation of Friel and Pettitt (2008) and denote the power posterior by [EQUATION] where [MATH] is thought of as a temperature, which has the effect of tempering the likelihood, whereby at the extreme ends of the temperature range, [MATH] and [MATH] correspond to the prior and posterior, respectively.', '1209.3198-1-5-2': 'The power posterior estimator for the evidence relies on noting that [EQUATION]', '1209.3198-1-5-3': 'As a result [EQUATION] which is the log of the desired marginal likelihood.', '1209.3198-1-6-0': 'In practice the temperature range is discretised as [MATH] to form an estimator based on ([REF]).', '1209.3198-1-6-1': 'For each [MATH], a sample from [MATH] can be used to estimate [MATH].', '1209.3198-1-6-2': 'Finally, a trapezoidal rule is used to approximate [EQUATION]', '1209.3198-1-6-3': 'Discretising [MATH] introduces an approximation into this method and the two goals of this paper are to reduce the bias in the power posterior estimation method due to the approximation and also to find an adaptive method for choosing the temperature rungs required.', '1209.3198-1-6-4': 'For both of these we will exploit the fact that the gradient of the expected log deviance curve equals its variance, as we now outline.', '1209.3198-1-7-0': 'Differentiating [MATH] with respect to [MATH] yields [EQUATION] where [MATH] denotes the variance of the log deviance at temperature [MATH].', '1209.3198-1-8-0': '## Reducing the bias by improving the numerical integration', '1209.3198-1-9-0': 'Equation ([REF]) immediately provides two useful pieces of information.', '1209.3198-1-9-1': 'First, the curve which we wish to integrate numerically is (strictly) increasing.', '1209.3198-1-9-2': 'Secondly, we can improve upon the standard trapezium rule used to numerically integrate the expected log deviance by incorporating derivative information at virtually no extra computational cost (the cost merely of calculating the variance of a set of simulations for fixed [MATH]).', '1209.3198-1-9-3': 'We do this by using the corrected trapezium rule which comes from an error analysis of the standard trapezium rule, see for example Atkinson and Han (2004), Section 5.2; when integrating a function [MATH] between points [MATH] and [MATH] [EQUATION] where [MATH] is some point in [MATH].', '1209.3198-1-9-4': 'The first term of the right hand side of this equation is the usual trapezium rule and the second can be approximated using [EQUATION]', '1209.3198-1-9-5': 'This latter form motivates the corrected trapezium rule which for unequally spaced x-axis points, taken together with the information derived above regarding the derivative of the log deviance gives [EQUATION] where both the expectations [MATH] and variances [MATH] are to be estimated using MCMC runs at a number of values of [MATH].', '1209.3198-1-10-0': '## Adaptive choice of the temperature placement', '1209.3198-1-11-0': 'The next question which arises is how to choose the [MATH] between [MATH] and [MATH].', '1209.3198-1-11-1': 'Friel and Pettitt (2008) find that setting [MATH] performs well.', '1209.3198-1-11-2': 'We refer to this as the powered fraction (PF) schedule.', '1209.3198-1-11-3': "Lartillot and Philippe (2006) discuss very similar ideas in the phylogenetics literature, although using Simpson's rule for the numerical integration; they use equally spaced temperatures between 0 and 1.", '1209.3198-1-12-0': 'Here we will only consider the discretisation error associated with the numerical integration, rather than the stochastic error arising with sampling from the different [MATH].', '1209.3198-1-12-1': 'Calderhead and Girolami (2009) show that this discretisation error depends upon the Kullback-Liebler distance between successive [MATH].', '1209.3198-1-12-2': 'Lefebvre, Steele and Vandal (2010) also consider a symmetrised Kullback-Liebler divergence in picking optimal schedules for path sampling.', '1209.3198-1-12-3': 'At first glance the Kullback-Liebler distance does not seem a particularly tractable quantity to manipulate.', '1209.3198-1-12-4': 'However, these papers and Behrens, Friel and Hurn (2012) all note that, in the notation of this paper, [EQUATION] where [MATH] denotes the Kullback-Liebler distance and [EQUATION] [MATH] can be interpreted graphically as the sum of the rectangular areas between a lower and an upper approximation to the integral of [MATH] between [MATH] and [MATH].', '1209.3198-1-12-5': 'Behrens, Friel and Hurn (2012) use minimising [MATH] as a rationale for choosing the temperatures in tempered transitions.', '1209.3198-1-12-6': 'We propose to use the same target in selecting the [MATH] for power posteriors.', '1209.3198-1-12-7': 'However, unlike in tempered transitions where the tuning forms a small part of the overall computational load, here the cost is almost exclusively the estimation of [MATH].', '1209.3198-1-12-8': 'We propose the following scheme: Initialise a set of [MATH] using the geometric placing including 0 and 1 (we will see in later examples why a reasonable starting point is necessary) where [MATH] is a small proportion of the proposed total number of rungs [MATH].', '1209.3198-1-12-9': 'These [MATH] contribute [MATH] terms [MATH] which sum to give [MATH].', '1209.3198-1-12-10': 'Identify the largest of these terms and locate the next point in the corresponding interval, say [MATH].', '1209.3198-1-12-11': 'Since we do not want to use computational resources in performing a search for the optimal location of the new [MATH] (there is no analytic solution), we follow a low cost route using the estimated gradients/variances at [MATH] and [MATH].', '1209.3198-1-12-12': 'If the estimated gradient at [MATH] is denoted by [MATH] and that at [MATH] by [MATH], we set the new point to be [EQUATION]', '1209.3198-1-12-13': 'This scheme will almost certainly not identify the optimal placing of the [MATH] rungs.', '1209.3198-1-12-14': 'However it is quick, cheap and intuitively reasonable.', '1209.3198-1-12-15': '(In practice, Monte Carlo error can mean that the function is not increasing and so the criterion is changed to picking the interval with the largest absolute contribution to [MATH].)', '1209.3198-1-13-0': '# Examples', '1209.3198-1-14-0': 'We present three examples which illustrate the gains that arise from employing the methods developed here.', '1209.3198-1-14-1': 'The first example is a non-nested linear regression comparison for which the marginal likelihoods can be calculated analytically.', '1209.3198-1-14-2': 'Example 2 is a larger problem, choosing between two logistic regression models, for which an analytic solution is not possible.', '1209.3198-1-14-3': 'These first two examples were included in the review paper by Friel and Wyse (2012) where the performance of power posteriors was compared to other existing methods.', '1209.3198-1-14-4': 'The final example is by far the largest and exhibits the most interestingly shaped [MATH].', '1209.3198-1-15-0': '## Example 1: Radiata pine', '1209.3198-1-16-0': 'The first example compares two linear regression models for the Radiata pine data originally in Williams (1959).', '1209.3198-1-16-1': 'The response variable here is the maximum compression strength parallel to the grain, [MATH], while the predictors are density, [MATH], or density adjusted for resin content, [MATH], for [MATH] specimens of radiata pine.', '1209.3198-1-16-2': 'Two possible Gaussian linear regression models are considered; [EQUATION]', '1209.3198-1-16-3': 'Priors are chosen to match the analyses of Friel and Wyse (2012) (baring a notational factor of 2).', '1209.3198-1-16-4': 'The regression parameters [MATH] and [MATH] are taken to be Normally distributed with mean [MATH] and precision [MATH] and [MATH] respectively where [MATH].', '1209.3198-1-16-5': 'The values of [MATH] and [MATH] were fixed to be 0.06 and 6.', '1209.3198-1-16-6': 'A gamma prior with shape [MATH] and rate [MATH] was assumed for both [MATH] and [MATH].', '1209.3198-1-17-0': 'Following the comparisons of Friel and Wyse (2012), we consider estimating the evidence using 10, 20, 50, 100 or 200 rungs in the tempering scheme.', '1209.3198-1-17-1': 'The parameters at all levels are updated using the Gibbs sampler.', '1209.3198-1-17-2': 'For this example, both the adaptive and the PF spacings use 20000 iterations at each rung, discarding the first fifth of these as burn in.', '1209.3198-1-17-3': 'Figure [REF] shows the expected log deviance curves for the two models using 200 rungs, their shapes suggesting that PF spacing might perform competitively (Behrens, Friel and Hurn (2012) show that a scheme where [MATH] is a constant for [MATH] minimises [MATH] when the integrand takes the form [MATH] for some constants [MATH] and [MATH]).', '1209.3198-1-18-0': 'Figure [REF] shows the upper and lower bounds of the evidence (in black), the uncorrected estimate (in red) and the corrected estimate (in blue) all for model 1 as the number of rungs increases.', '1209.3198-1-18-1': 'The PF spacing results are denoted by solid lines and the adaptive spacing results by dashed lines.', '1209.3198-1-18-2': 'The true value of the evidence is known for this example and is marked by a horizontal line.', '1209.3198-1-18-3': 'As the vertical scale differs quite significantly between [MATH] and [MATH], the figure is split into two plots, small numbers of rungs (where the upper and lower bounds are not tight) and large numbers of rungs.', '1209.3198-1-18-4': 'The adaptive temperature placement is initialised using the 10 rung PF placement.', '1209.3198-1-18-5': 'Since for the adaptive spacing, increasing the number of rungs by one requires only one additional set of MCMC iterations at the new temperature, there is an averaging effect and the dashed lines appear smoother than the solid ones (where for an increase of one rung, all temperatures apart from [MATH] and [MATH] change and so the estimates at successive rungs are independent of one another).', '1209.3198-1-19-0': 'From this figure, it appears that the corrected estimates converge faster towards the true value than do the uncorrected ones initially.', '1209.3198-1-19-1': 'By construction, the adaptive and PF schedules coincide at 10 rungs.', '1209.3198-1-19-2': 'Immediately after that, the adaptive schedule initially provides wider bounds on the evidence but after approximately 25 rungs, the bounds are consistently narrower.', '1209.3198-1-20-0': 'To quantify these observations, the bias is estimated as per the approach of Friel and Wyse (2012), performing 50 replicates at 10, 20, 50, 100 and 200 rungs using 10000 iterations of which the first fifth are discarded as burn in.', '1209.3198-1-20-1': 'The average and standard deviation of the 50 biases are given in Table [REF].', '1209.3198-1-21-0': 'As might be expected from the concave shape of the log deviance curves, the uncorrected integrals tend to underestimate the evidence, giving negative biases which decrease as the number of rungs increases.', '1209.3198-1-21-1': 'To visualise the effect of both the correction and the adaptive placing of temperatures, Figure [REF] plots the 50 observed biases separately for each number of rungs and under the two spacings with or without correction.', '1209.3198-1-21-2': 'Correction is particularly effective when smaller numbers of rungs are being used.', '1209.3198-1-21-3': 'The final panel illustrates the effect of the correction when using 100 rungs by plotting the corrected points against their uncorrected values (the line [MATH] is shown in red, corresponding to no effect of correction).', '1209.3198-1-21-4': 'There is an interesting difference between the adaptive and the PF versions, less correction is needed for the adaptive schedule, that is, it is doing a better job of the numerical integration.', '1209.3198-1-22-0': 'Given the good reductions in bias seen in Table [REF], it is important to ask how much extra time is required.', '1209.3198-1-22-1': 'To assess this, a total of 10 runs for model 1 using 20000 iterations and 200 temperatures were timed.', '1209.3198-1-22-2': 'Four versions of the algorithm were considered, corresponding to Table [REF].', '1209.3198-1-22-3': 'All the coding was in R and times are given relative to the PF non-corrected version: [EQUATION]', '1209.3198-1-22-4': 'The adaptive selection of temperatures and the correction term in the numerical integration come at negligible computational cost.', '1209.3198-1-22-5': 'Given the reductions in bias achievable by the correction in particular, there is no reason at all not to adopt this modification.', '1209.3198-1-23-0': '## Example 2: Pima indians', '1209.3198-1-24-0': 'We turn next to the Pima Indian example considered by Friel and Wyse (2012), originally described by Smith et al (1988) .', '1209.3198-1-24-1': 'These data record diabetes incidence and possible disease indicators for [MATH] Pima Indian women aged over 20.', '1209.3198-1-24-2': 'The seven possible disease indicators are the number of pregnancies (NP), plasma glucose concentration (PGC), diastolic blood pressure (BP), triceps skin fold thickness (TST), body mass index (BMI), diabetes pedigree function (DP) and age (AGE), with all these covariates standardised.', '1209.3198-1-25-0': "The model assumed for the observed diabetes incidence, [MATH], is [EQUATION] where [MATH] is the probability of incidence for person [MATH], and [MATH] is related to the [MATH] person's covariates and a constant term, denoted by [MATH], and the parameters, [MATH], by [EQUATION] where [MATH] is the number of explanatory variables.", '1209.3198-1-25-1': 'An independent multivariate Gaussian prior is assumed for [MATH], with mean zero and non-informative precision of [MATH], so that [EQUATION]', '1209.3198-1-25-2': 'There are 129 potential models ([MATH] models with covariates plus a model with only a constant term).', '1209.3198-1-25-3': 'A long reversible jump run (Green, 1995) revealed the two models with the highest posterior probability:', '1209.3198-1-26-0': '[MATH]( 0.01/t , 1/)[MATH]t[MATH][MATH]-257.2342[MATH]-259.8519[MATH]-257.2588[MATH]-259.8906[MATH]y = y_1,,y_82[MATH]z = z_1,,z_82[MATH]z_i=j[MATH]y_i[MATH]j^th[MATH]k[MATH]y_i[MATH]k=1[MATH]k=7[MATH]k=1[MATH]t[MATH]k[MATH]k=1[MATH]k=3[MATH]k=1[MATH]k=3[MATH]t_i[MATH]t=0[MATH]t=0[MATH]t_i[MATH]t=0[MATH]t=1[MATH]k=1[MATH]k=7[MATH]k=3[MATH]k=4, 5, 6[MATH]k=3[MATH]k=7[MATH]n[MATH]n[MATH]t_i[MATH]n[MATH]E_y,t(p(y))[MATH]t 0[MATH]t 0[MATH]t[MATH]p_t(y)[MATH]t=1[MATH]'}

{'1209.3198-2-0-0': 'The statistical evidence (or marginal likelihood) is a key quantity in Bayesian statistics, allowing one to assess the probability of the data given the model under investigation.', '1209.3198-2-0-1': 'This paper focuses on refining the power posterior approach to improve estimation of the evidence.', '1209.3198-2-0-2': 'The power posterior method involves transitioning from the prior to the posterior by powering the likelihood by an inverse temperature.', '1209.3198-2-0-3': 'In common with other tempering algorithms, the power posterior involves some degree of tuning.', '1209.3198-2-0-4': 'The main contributions of this article are twofold - we present a result from the numerical analysis literature which can reduce the bias in the estimate of the evidence by addressing the error arising from numerically integrating across the inverse temperatures.', '1209.3198-2-0-5': 'We also tackle the selection of the inverse temperature ladder, applying this approach additionally to the Stepping Stone sampler estimation of evidence.', '1209.3198-2-0-6': 'A key practical point is that both of these innovations incur virtually no extra cost.', '1209.3198-2-0-7': 'Keywords: Marginal likelihood, Markov chain Monte Carlo, Power posteriors, Statistical evidence, Stepping Stone sampler, Tempering, Thermodynamic integration.', '1209.3198-2-1-0': '# Introduction', '1209.3198-2-2-0': 'The statistical evidence (sometimes called the marginal likelihood or integrated likelihood) is a vital quantity in Bayesian statistics for the comparison of models, [MATH].', '1209.3198-2-2-1': 'Under the Bayesian paradigm we consider the posterior distribution [EQUATION] for data [MATH] and parameters [MATH] within model [MATH], where [MATH] denotes the prior distribution for parameters within model [MATH] and where [MATH] denotes the prior model probability.', '1209.3198-2-2-2': 'The evidence for data [MATH] given model [MATH] arises as the normalising constant of the posterior distribution within model [MATH], [EQUATION] and thus results from integrating the un-normalised posterior across the [MATH] parameter space, [EQUATION]', '1209.3198-2-2-3': 'This of course assumes that the prior distribution for [MATH] is proper.', '1209.3198-2-2-4': 'The marginal likelihood is often then used to calculate Bayes factors when one wants to compare two competing models, [MATH] and [MATH], [EQUATION]', '1209.3198-2-2-5': 'Here, [MATH] is the posterior probability for model [MATH] and it can be evaluated, using the evidence for each of the collection of models under consideration, [EQUATION]', '1209.3198-2-2-6': 'Estimation of the evidence is a non-trivial task for most statistical models and there has been considerable effort in the literature to find algorithms and methods for this purpose.', '1209.3198-2-2-7': "Laplace's method (Tierney and Kadane, 1986) is an early approach and very widely used.", '1209.3198-2-2-8': "Other notable and popular approaches include Chib's method (Chib 1995), annealed importance sampling (Neal 2001), nested sampling (Skilling 2006), bridge sampling (Meng and Wong, 1996) and power posteriors (Friel and Pettitt, 2008) which is the focus of this paper.", '1209.3198-2-2-9': 'For a recent review and perspective on these and other methods, see Friel and Wyse (2012).', '1209.3198-2-3-0': 'This paper is organised as follows.', '1209.3198-2-3-1': 'Section [REF] outlines the power posterior method, and the approach we propose to improve estimation of the evidence.', '1209.3198-2-3-2': 'Section [REF] illustrates the potential gain from implementing the methodology which we propose.', '1209.3198-2-3-3': 'We offer some conclusions in Section [REF].', '1209.3198-2-4-0': '# The power posterior approach', '1209.3198-2-5-0': 'In what follows we will drop the explicit conditioning on model [MATH] for notational simplicity.', '1209.3198-2-5-1': 'We follow the notation of Friel and Pettitt (2008) and denote the power posterior by [EQUATION] where [MATH] is thought of as an inverse temperature, which has the effect of tempering the likelihood, whereby at the extreme ends of the inverse temperature range, [MATH] and [MATH] correspond to the prior and posterior, respectively.', '1209.3198-2-5-2': 'The power posterior estimator for the evidence relies on noting that [EQUATION]', '1209.3198-2-5-3': 'As a result [EQUATION] which is the log of the desired marginal likelihood.', '1209.3198-2-6-0': 'In practice the inverse temperature range is discretised as [MATH] to form an estimator based on ([REF]).', '1209.3198-2-6-1': 'For each [MATH], a sample from [MATH] can be used to estimate [MATH].', '1209.3198-2-6-2': 'Finally, a trapezoidal rule is used to approximate [EQUATION]', '1209.3198-2-6-3': 'Discretising [MATH] introduces an approximation into this method and the two goals of this paper are to reduce the bias in the power posterior estimation method due to the approximation and also to find an adaptive method for choosing the inverse temperatures (which we also refer to as rungs, following a common analogy of an inverse temperature ladder between prior and posterior).', '1209.3198-2-6-4': 'For both of these we will exploit the fact that the gradient of the expected log deviance curve equals its variance, as we now outline.', '1209.3198-2-7-0': 'Differentiating [MATH] with respect to [MATH] yields [EQUATION] where [MATH] denotes the variance of the log deviance at inverse temperature [MATH].', '1209.3198-2-8-0': '## Reducing the bias by improving the numerical integration', '1209.3198-2-9-0': 'Equation ([REF]) immediately provides two useful pieces of information.', '1209.3198-2-9-1': 'First, the curve which we wish to integrate numerically is (strictly) increasing.', '1209.3198-2-9-2': 'Secondly, we can improve upon the standard trapezium rule used to numerically integrate the expected log deviance by incorporating derivative information at virtually no extra computational cost (the cost merely of calculating the variance of a set of simulations for fixed [MATH]).', '1209.3198-2-9-3': 'We do this by using the corrected trapezium rule which comes from an error analysis of the standard trapezium rule, see for example Atkinson and Han (2004), Section 5.2; when integrating a function [MATH] between points [MATH] and [MATH] [EQUATION] where [MATH] is some point in [MATH].', '1209.3198-2-9-4': 'The first term of the right hand side of this equation is the usual trapezium rule and the second can be approximated using [EQUATION]', '1209.3198-2-9-5': 'This latter form motivates the corrected trapezium rule which for unequally spaced x-axis points, taken together with the information derived above regarding the derivative of the log deviance gives [EQUATION] where both the expectations [MATH] and variances [MATH] are to be estimated using MCMC runs at a number of values of [MATH].', '1209.3198-2-9-6': 'We will refer to the algorithm incorporating this correction as the modified power posterior.', '1209.3198-2-10-0': '## Adaptive choice of the inverse temperature placement', '1209.3198-2-11-0': 'The next question which arises is how to choose the [MATH] between [MATH] and [MATH].', '1209.3198-2-11-1': 'Friel and Pettitt (2008) find that setting [MATH] performs well.', '1209.3198-2-11-2': 'We refer to this as the powered fraction (PF) schedule.', '1209.3198-2-11-3': "Lartillot and Philippe (2006) discuss very similar ideas in the phylogenetics literature, although using Simpson's rule for the numerical integration; they use equally spaced inverse temperatures between 0 and 1.", '1209.3198-2-12-0': 'Here we will only consider the discretisation error associated with the numerical integration, rather than the stochastic error arising with sampling from the different [MATH].', '1209.3198-2-12-1': 'Calderhead and Girolami (2009) show that this discretisation error depends upon the Kullback-Leibler distance between successive [MATH].', '1209.3198-2-12-2': 'Lefebvre, Steele and Vandal (2010) also consider a symmetrised Kullback-Leibler divergence in picking optimal schedules for path sampling.', '1209.3198-2-12-3': 'At first glance the Kullback-Leibler distance does not seem a particularly tractable quantity to manipulate.', '1209.3198-2-12-4': 'However, these papers and Behrens, Friel and Hurn (2012) all note that, in the notation of this paper, [EQUATION] where [MATH] denotes the Kullback-Leibler distance and [EQUATION] [MATH] can be interpreted graphically as the sum of the rectangular areas between a lower and an upper approximation to the integral of [MATH] between [MATH] and [MATH].', '1209.3198-2-12-5': 'Behrens, Friel and Hurn (2012) use minimising [MATH] as a rationale for choosing the inverse temperatures in tempered transitions.', '1209.3198-2-12-6': 'We propose to use the same target in selecting the [MATH] for power posteriors.', '1209.3198-2-12-7': 'However, unlike in tempered transitions where the tuning forms a small part of the overall computational load, here the cost is almost exclusively the estimation of [MATH] and its gradient.', '1209.3198-2-13-0': 'We propose the following approach.', '1209.3198-2-13-1': 'Begin by estimating the expected log deviance and its gradient at [MATH] then [MATH] (both points which are needed in all possible schemes).', '1209.3198-2-13-2': 'Where to site the next [MATH]?', '1209.3198-2-13-3': 'There is no analytic solution without knowing the curve and we do not want to use computational resources in performing a search for the optimal location.', '1209.3198-2-13-4': 'Instead we find the intersection of the two straight lines defined by our current knowledge of the curve: If the estimated function and gradient at [MATH] are denoted by [MATH] and [MATH] respectively, and those at [MATH] by [MATH] and [MATH], we set the new point to be [EQUATION]', '1209.3198-2-13-5': 'If this intersecting point is outside the interval [MATH], it suggests there is some sort of inflection within the interval and instead we use a simple weighted average.', '1209.3198-2-13-6': '[EQUATION]', '1209.3198-2-13-7': 'This now gives us two rectangular contributions to [MATH].', '1209.3198-2-13-8': 'We identify the larger of these terms and locate the next point in the corresponding interval, and so on iteratively.', '1209.3198-2-13-9': 'This scheme will almost certainly not identify the optimal placing of the [MATH] interior rungs.', '1209.3198-2-13-10': 'However it is quick, cheap and intuitively reasonable.', '1209.3198-2-13-11': '(In practice, Monte Carlo error can mean that the function is not increasing and so the criterion is changed to picking the interval with the largest absolute contribution to [MATH].', '1209.3198-2-13-12': 'In the event of picking such an interval, we set the new [MATH] to be the midpoint of the interval reflecting the significant levels of uncertainty.)', '1209.3198-2-14-0': '## Computational details', '1209.3198-2-15-0': 'We use sequential runs of MCMC, beginning with sampling from the posterior, [MATH].', '1209.3198-2-15-1': 'From each MCMC run we store the estimated expected log deviance, its estimated gradient and the values at the last iteration of the run.', '1209.3198-2-15-2': 'These final values are used as starting points for runs at a smaller value of [MATH].', '1209.3198-2-15-3': 'In the case of the power fraction placements where the inverse temperatures are deterministic, this means that the run at [MATH] is initialised with the values from [MATH].', '1209.3198-2-15-4': 'In the adaptive placements, the run at each new [MATH] is initialised with the values of the currently closest larger [MATH].', '1209.3198-2-16-0': '# Examples', '1209.3198-2-17-0': 'We present three examples, the first two of which were included in the review paper by Friel and Wyse (2012) where the performance of power posteriors was compared to some other existing methods.', '1209.3198-2-17-1': 'The first is a non-nested linear regression comparison for which the marginal likelihoods can be calculated analytically.', '1209.3198-2-17-2': 'Here our experiments concentrate on the effects of the modified integration rule and the adaptive placement of the inverse temperatures.', '1209.3198-2-17-3': 'Example 2 is a larger problem, choosing between two logistic regression models, for which an analytic solution is not possible; we use this example to compare the improved algorithm with the Stepping Stone sampling approach of Xie et al (2011).', '1209.3198-2-17-4': 'The final example is by far the largest and exhibits the most interestingly shaped [MATH], the focus here is on the stability of the approach to poor estimation of the gradient.', '1209.3198-2-18-0': '## Example 1: Radiata pine', '1209.3198-2-19-0': 'The first example compares two linear regression models for the Radiata pine data originally in Williams (1959).', '1209.3198-2-19-1': 'The response variable here is the maximum compression strength parallel to the grain, [MATH], while the predictors are density, [MATH], or density adjusted for resin content, [MATH], for [MATH] specimens of radiata pine.', '1209.3198-2-19-2': 'Two possible Gaussian linear regression models are considered; [EQUATION]', '1209.3198-2-19-3': 'Priors are chosen to match the analyses of Friel and Wyse (2012) (baring a notational factor of 2).', '1209.3198-2-19-4': 'The regression parameters [MATH] and [MATH] were taken to be Normally distributed with mean [MATH] and precision [MATH] and [MATH] respectively where [MATH].', '1209.3198-2-19-5': 'The values of [MATH] and [MATH] were fixed to be 0.06 and 6.', '1209.3198-2-19-6': 'A gamma prior with shape [MATH] and rate [MATH] was assumed for both [MATH] and [MATH].', '1209.3198-2-20-0': 'We consider estimating the evidence using [MATH], 20, 50 or 100 rungs in the tempering scheme where the rungs are chosen either according to the powered fraction (PF) rule or adaptively following the heuristic in Section [REF].', '1209.3198-2-20-1': 'The parameters at all levels are updated using the Gibbs sampler, with 10000 iterations at each rung, discarding the first fifth of these as burn in.', '1209.3198-2-20-2': 'To quantify the performances, the bias, standard deviation and Root Mean Square Error (RMSE) are estimated by performing 100 replicates of the four schemes at 10, 20, 50 and 100 rungs.', '1209.3198-2-20-3': 'The results are given in Table [REF].', '1209.3198-2-20-4': 'Comparing the first column with the second, and the third with the fourth, we can see that the modified power posterior approach dominates the usual one in all cases.', '1209.3198-2-20-5': 'For small numbers of rungs, where the RMSE is dominated by the bias term, this effect is particularly dramatic.', '1209.3198-2-20-6': 'The standard error is not greatly affected by the choice of scheme, although in this example the smallest values all occur for modified schemes.', '1209.3198-2-20-7': 'As the number of rungs increases, the RMSE becomes more affected by the standard error than the bias.', '1209.3198-2-21-0': 'Comparing the first column of Table [REF] with the third, we can isolate the effect of the adaptive placement.', '1209.3198-2-21-1': 'This is also most noticeable at small numbers of rungs, perhaps unsurprisingly given that the placing of an individual [MATH] makes less difference as their total number increases.', '1209.3198-2-21-2': 'To help to visualise the effect of both the correction and the adaptive placing of inverse temperatures, Figure [REF] plots the 100 observed biases pairwise for the standard vs modified power posterior.', '1209.3198-2-21-3': 'Points are plotted for both the PF spacing and the adaptive spacings, and for each number of rungs.', '1209.3198-2-21-4': 'How much change occurs for each standard power posterior under the additive modification given by Equation ([REF]) can be seen as the vertical distance between the point and the red [MATH] line.', '1209.3198-2-21-5': 'A few points are immediately clear.', '1209.3198-2-21-6': 'Firstly the correction is fairly well estimated (the sets of points are roughly parallel to the [MATH] line, even in the [MATH] case; this is a function of the MCMC run lengths).', '1209.3198-2-21-7': 'Second, there is an interesting difference between the adaptive and the PF versions, less correction is needed for the adaptive schedule, that is, it is already doing a better job of linearly approximating the function for the numerical integration.', '1209.3198-2-21-8': 'Thirdly, as was already observed in Table [REF], as the number of rungs increases, the differences between the two types of spacings become less pronounced.', '1209.3198-2-22-0': 'Given the good reductions in bias seen in Table [REF], it is important to ask how much extra time is required.', '1209.3198-2-22-1': 'To assess this, a total of 20 runs for Model 1 using 10000 iterations and 100 inverse temperatures were timed.', '1209.3198-2-22-2': 'Four versions of the algorithm were considered, corresponding to Table [REF].', '1209.3198-2-22-3': 'All the coding was in R and times are given relative to the PF standard power posterior version: [EQUATION]', '1209.3198-2-22-4': 'What we see is that the adaptive selection of inverse temperatures and the correction term in the numerical integration come at an acceptably small computational cost.', '1209.3198-2-22-5': 'Given this, our recommendation would be to use the modified integration rule when employing power posteriors.', '1209.3198-2-22-6': 'The benefits of the adaptive placement are less dramatic except at small [MATH] but as it provides a completely automatic way to choose the inverse temperatures for very little cost, we would also recommend it.', '1209.3198-2-23-0': '## Example 2: Pima indians', '1209.3198-2-24-0': 'We turn next to the Pima Indian example considered by Friel and Wyse (2012), originally described by Smith et al (1988) .', '1209.3198-2-24-1': 'These data record diabetes incidence and possible disease indicators for [MATH] Pima Indian women aged over 20.', '1209.3198-2-24-2': 'The seven possible disease indicators are the number of pregnancies (NP), plasma glucose concentration (PGC), diastolic blood pressure (BP), triceps skin fold thickness (TST), body mass index (BMI), diabetes pedigree function (DP) and age (AGE), with all these covariates standardised.', '1209.3198-2-25-0': "The model assumed for the observed diabetes incidence, [MATH], is [EQUATION] where [MATH] is the probability of incidence for person [MATH], and [MATH] is related to the [MATH] person's covariates and a constant term, denoted by [MATH], and the parameters, [MATH], by [EQUATION] where [MATH] is the number of explanatory variables.", '1209.3198-2-25-1': 'An independent multivariate Gaussian prior is assumed for [MATH], with mean zero and non-informative precision of [MATH], so that [EQUATION]', '1209.3198-2-25-2': 'There are 129 potential models ([MATH] models with covariates plus a model with only a constant term).', '1209.3198-2-25-3': 'A long reversible jump run (Green, 1995) revealed the two models with the highest posterior probability:', '1209.3198-2-26-0': '[MATH]( 0.01/t , 1/)[MATH]t[MATH][MATH]n=50[MATH]-257.2342[MATH]-259.8519[MATH]-257.2588[MATH]-259.8906[MATH]t_0=0[MATH]t_n-1[MATH]r_k=z(yt_k+1)/z(yt_k)[MATH]z(yt)[MATH]t_k[MATH]m[MATH]^(i)[MATH]p_t_k( y)[MATH]z(y t_n=1)[MATH]n[MATH]_k=0^n-1 r_k[MATH]t_n=1[MATH]t_n-1[MATH]n[MATH]t_i[MATH]n=10[MATH]n[MATH]n[MATH]y = y_1,,y_82[MATH]z = z_1,,z_82[MATH]z_i=j[MATH]y_i[MATH]j^th[MATH]k[MATH]y_i[MATH]k=3[MATH]k=4[MATH]t=0[MATH]n[MATH]k=3[MATH][-229.8626,-227.8737][MATH]k=4[MATH][-230.5911,-228.0902][MATH]k=3[MATH]k=4[MATH]t_i[MATH]t=0[MATH]n 100[MATH]n=200[MATH]n[MATH]t_i[MATH]n[MATH]E_y,t(p(y))[MATH]t 0[MATH]t 1[MATH]t[MATH]p_t(y)[MATH]t=1[MATH]'}

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'1209.3198-3-15-2'], ['1209.3198-2-15-3', '1209.3198-3-15-3'], ['1209.3198-2-15-4', '1209.3198-3-15-4'], ['1209.3198-2-9-0', '1209.3198-3-9-0'], ['1209.3198-2-9-1', '1209.3198-3-9-1'], ['1209.3198-2-9-2', '1209.3198-3-9-2'], ['1209.3198-2-9-3', '1209.3198-3-9-3'], ['1209.3198-2-9-4', '1209.3198-3-9-4'], ['1209.3198-2-9-5', '1209.3198-3-9-5'], ['1209.3198-2-9-6', '1209.3198-3-9-6'], ['1209.3198-1-12-0', '1209.3198-2-12-0'], ['1209.3198-1-12-6', '1209.3198-2-12-6'], ['1209.3198-1-12-14', '1209.3198-2-13-10'], ['1209.3198-1-14-1', '1209.3198-2-17-1'], ['1209.3198-1-14-3', '1209.3198-2-17-0'], ['1209.3198-1-5-1', '1209.3198-2-5-1'], ['1209.3198-1-0-4', '1209.3198-2-0-4'], ['1209.3198-1-0-7', '1209.3198-2-0-7'], ['1209.3198-1-7-0', '1209.3198-2-7-0'], ['1209.3198-1-22-3', '1209.3198-2-22-3'], ['1209.3198-1-6-0', '1209.3198-2-6-0'], ['1209.3198-1-16-4', '1209.3198-2-19-4'], ['1209.3198-1-2-8', '1209.3198-2-2-8'], ['1209.3198-1-21-4', '1209.3198-2-21-7'], ['1209.3198-1-11-3', 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'1209.3198-3-15-2'], ['1209.3198-2-15-3', '1209.3198-3-15-3'], ['1209.3198-2-15-4', '1209.3198-3-15-4'], ['1209.3198-2-9-0', '1209.3198-3-9-0'], ['1209.3198-2-9-1', '1209.3198-3-9-1'], ['1209.3198-2-9-2', '1209.3198-3-9-2'], ['1209.3198-2-9-3', '1209.3198-3-9-3'], ['1209.3198-2-9-4', '1209.3198-3-9-4'], ['1209.3198-2-9-5', '1209.3198-3-9-5'], ['1209.3198-2-9-6', '1209.3198-3-9-6'], ['1209.3198-1-12-0', '1209.3198-2-12-0'], ['1209.3198-1-12-6', '1209.3198-2-12-6'], ['1209.3198-1-12-14', '1209.3198-2-13-10']]

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[]

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[]

['1209.3198-1-25-3', '1209.3198-1-26-0', '1209.3198-2-13-6', '1209.3198-2-25-3', '1209.3198-2-26-0', '1209.3198-3-13-6', '1209.3198-3-25-2', '1209.3198-3-26-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1209.3198

{'1209.3198-3-0-0': 'The statistical evidence (or marginal likelihood) is a key quantity in Bayesian statistics, allowing one to assess the probability of the data given the model under investigation.', '1209.3198-3-0-1': 'This paper focuses on refining the power posterior approach to improve estimation of the evidence.', '1209.3198-3-0-2': 'The power posterior method involves transitioning from the prior to the posterior by powering the likelihood by an inverse temperature.', '1209.3198-3-0-3': 'In common with other tempering algorithms, the power posterior involves some degree of tuning.', '1209.3198-3-0-4': 'The main contributions of this article are twofold - we present a result from the numerical analysis literature which can reduce the bias in the estimate of the evidence by addressing the error arising from numerically integrating across the inverse temperatures.', '1209.3198-3-0-5': 'We also tackle the selection of the inverse temperature ladder, applying this approach additionally to the Stepping Stone sampler estimation of evidence.', '1209.3198-3-0-6': 'A key practical point is that both of these innovations incur virtually no extra cost.', '1209.3198-3-0-7': 'Keywords: Marginal likelihood, Markov chain Monte Carlo, Power posteriors, Statistical evidence, Stepping Stone sampler, Tempering, Thermodynamic integration.', '1209.3198-3-1-0': '# Introduction', '1209.3198-3-2-0': 'The statistical evidence (sometimes called the marginal likelihood or integrated likelihood) is a vital quantity in Bayesian statistics for the comparison of models, [MATH].', '1209.3198-3-2-1': 'Under the Bayesian paradigm we consider the posterior distribution [EQUATION] for data [MATH] and parameters [MATH] within model [MATH], where [MATH] denotes the prior distribution for parameters within model [MATH] and where [MATH] denotes the prior model probability.', '1209.3198-3-2-2': 'The evidence for data [MATH] given model [MATH] arises as the normalising constant of the posterior distribution within model [MATH], [EQUATION] and thus results from integrating the un-normalised posterior across the [MATH] parameter space, [EQUATION]', '1209.3198-3-2-3': 'This of course assumes that the prior distribution for [MATH] is proper.', '1209.3198-3-2-4': 'The marginal likelihood is often then used to calculate Bayes factors when one wants to compare two competing models, [MATH] and [MATH], [EQUATION]', '1209.3198-3-2-5': 'Here, [MATH] is the posterior probability for model [MATH] and it can be evaluated, using the evidence for each of the collection of models under consideration, [EQUATION]', '1209.3198-3-2-6': 'Estimation of the evidence is a non-trivial task for most statistical models and there has been considerable effort in the literature to find algorithms and methods for this purpose.', '1209.3198-3-2-7': "Laplace's method (Tierney and Kadane, 1986) is an early approach and very widely used.", '1209.3198-3-2-8': "Other notable and popular approaches include Chib's method (Chib 1995), annealed importance sampling (Neal 2001), nested sampling (Skilling 2006), bridge sampling (Meng and Wong, 1996) and power posteriors (Friel and Pettitt, 2008) which is the focus of this paper.", '1209.3198-3-2-9': 'For a recent review and perspective on these and other methods, see Friel and Wyse (2012).', '1209.3198-3-3-0': 'This paper is organised as follows.', '1209.3198-3-3-1': 'Section [REF] outlines the power posterior method, and the approach we propose to improve estimation of the evidence.', '1209.3198-3-3-2': 'Section [REF] illustrates the potential gain from implementing the methodology which we propose.', '1209.3198-3-3-3': 'We offer some conclusions in Section [REF].', '1209.3198-3-4-0': '# The power posterior approach', '1209.3198-3-5-0': 'In what follows we will drop the explicit conditioning on model [MATH] for notational simplicity.', '1209.3198-3-5-1': 'We follow the notation of Friel and Pettitt (2008) and denote the power posterior by [EQUATION] where [MATH] is thought of as an inverse temperature, which has the effect of tempering the likelihood, whereby at the extreme ends of the inverse temperature range, [MATH] and [MATH] correspond to the prior and posterior, respectively.', '1209.3198-3-5-2': 'The power posterior estimator for the evidence relies on noting that [EQUATION]', '1209.3198-3-5-3': 'As a result [EQUATION] which is the log of the desired marginal likelihood.', '1209.3198-3-6-0': 'In practice the inverse temperature range is discretised as [MATH] to form an estimator based on ([REF]).', '1209.3198-3-6-1': 'For each [MATH], a sample from [MATH] can be used to estimate [MATH].', '1209.3198-3-6-2': 'Finally, a trapezoidal rule is used to approximate [EQUATION]', '1209.3198-3-6-3': 'Discretising [MATH] introduces an approximation into this method and the two goals of this paper are to reduce the bias in the power posterior estimation method due to the approximation and also to find an adaptive method for choosing the inverse temperatures (which we also refer to as rungs, following a common analogy of an inverse temperature ladder between prior and posterior).', '1209.3198-3-6-4': 'For both of these we will exploit the fact that the gradient of the expected log deviance curve equals its variance, as we now outline.', '1209.3198-3-7-0': 'Differentiating [MATH] with respect to [MATH] yields [EQUATION] where [MATH] denotes the variance of the log deviance at inverse temperature [MATH].', '1209.3198-3-8-0': '## Reducing the bias by improving the numerical integration', '1209.3198-3-9-0': 'Equation ([REF]) immediately provides two useful pieces of information.', '1209.3198-3-9-1': 'First, the curve which we wish to integrate numerically is (strictly) increasing.', '1209.3198-3-9-2': 'Secondly, we can improve upon the standard trapezium rule used to numerically integrate the expected log deviance by incorporating derivative information at virtually no extra computational cost (the cost merely of calculating the variance of a set of simulations for fixed [MATH]).', '1209.3198-3-9-3': 'We do this by using the corrected trapezium rule which comes from an error analysis of the standard trapezium rule, see for example Atkinson and Han (2004), Section 5.2; when integrating a function [MATH] between points [MATH] and [MATH] [EQUATION] where [MATH] is some point in [MATH].', '1209.3198-3-9-4': 'The first term of the right hand side of this equation is the usual trapezium rule and the second can be approximated using [EQUATION]', '1209.3198-3-9-5': 'This latter form motivates the corrected trapezium rule which for unequally spaced x-axis points, taken together with the information derived above regarding the derivative of the log deviance gives [EQUATION] where both the expectations [MATH] and variances [MATH] are to be estimated using MCMC runs at a number of values of [MATH].', '1209.3198-3-9-6': 'We will refer to the algorithm incorporating this correction as the modified power posterior.', '1209.3198-3-10-0': '## Adaptive choice of the inverse temperature placement', '1209.3198-3-11-0': 'The next question which arises is how to choose the [MATH] between [MATH] and [MATH].', '1209.3198-3-11-1': 'Friel and Pettitt (2008) find that setting [MATH] performs well.', '1209.3198-3-11-2': 'We refer to this as the powered fraction (PF) schedule.', '1209.3198-3-11-3': "Lartillot and Philippe (2006) discuss very similar ideas in the phylogenetics literature, although using Simpson's rule for the numerical integration; they use equally spaced inverse temperatures between 0 and 1.", '1209.3198-3-12-0': 'Here we will only consider the discretisation error associated with the numerical integration, rather than the stochastic error arising with sampling from the different [MATH].', '1209.3198-3-12-1': 'Calderhead and Girolami (2009) show that this discretisation error depends upon the Kullback-Leibler distance between successive [MATH].', '1209.3198-3-12-2': 'Lefebvre, Steele and Vandal (2010) also consider a symmetrised Kullback-Leibler divergence in picking optimal schedules for path sampling.', '1209.3198-3-12-3': 'At first glance the Kullback-Leibler distance does not seem a particularly tractable quantity to manipulate.', '1209.3198-3-12-4': 'However, these papers and Behrens, Friel and Hurn (2012) all note that, in the notation of this paper, [EQUATION] where [MATH] denotes the Kullback-Leibler distance and [EQUATION] [MATH] can be interpreted graphically as the sum of the rectangular areas between a lower and an upper approximation to the integral of [MATH] between [MATH] and [MATH].', '1209.3198-3-12-5': 'Behrens, Friel and Hurn (2012) use minimising [MATH] as a rationale for choosing the inverse temperatures in tempered transitions.', '1209.3198-3-12-6': 'We propose to use the same target in selecting the [MATH] for power posteriors.', '1209.3198-3-12-7': 'However, unlike in tempered transitions where the tuning forms a small part of the overall computational load, here the cost is almost exclusively the estimation of [MATH] and its gradient.', '1209.3198-3-13-0': 'We propose the following approach.', '1209.3198-3-13-1': 'Begin by estimating the expected log deviance and its gradient at [MATH] then [MATH] (both points which are needed in all possible schemes).', '1209.3198-3-13-2': 'Where to site the next [MATH]?', '1209.3198-3-13-3': 'There is no analytic solution without knowing the curve and we do not want to use computational resources in performing a search for the optimal location.', '1209.3198-3-13-4': 'Instead we find the intersection of the two straight lines defined by our current knowledge of the curve: If the estimated function and gradient at [MATH] are denoted by [MATH] and [MATH] respectively, and those at [MATH] by [MATH] and [MATH], we set the new point to be [EQUATION]', '1209.3198-3-13-5': 'If this intersecting point is outside the interval [MATH], it suggests there is some sort of inflection within the interval and instead we use a simple weighted average.', '1209.3198-3-13-6': '[EQUATION]', '1209.3198-3-13-7': 'This now gives us two rectangular contributions to [MATH].', '1209.3198-3-13-8': 'We identify the larger of these terms and locate the next point in the corresponding interval, and so on iteratively.', '1209.3198-3-13-9': 'This scheme will almost certainly not identify the optimal placing of the [MATH] interior rungs.', '1209.3198-3-13-10': 'However it is quick, cheap and intuitively reasonable.', '1209.3198-3-13-11': '(In practice, Monte Carlo error can mean that the function is not increasing and so the criterion is changed to picking the interval with the largest absolute contribution to [MATH].', '1209.3198-3-13-12': 'In the event of picking such an interval, we set the new [MATH] to be the midpoint of the interval reflecting the significant levels of uncertainty.)', '1209.3198-3-14-0': '## Computational details', '1209.3198-3-15-0': 'We use sequential runs of MCMC, beginning with sampling from the posterior, [MATH].', '1209.3198-3-15-1': 'From each MCMC run we store the estimated expected log deviance, its estimated gradient and the values at the last iteration of the run.', '1209.3198-3-15-2': 'These final values are used as starting points for runs at a smaller value of [MATH].', '1209.3198-3-15-3': 'In the case of the power fraction placements where the inverse temperatures are deterministic, this means that the run at [MATH] is initialised with the values from [MATH].', '1209.3198-3-15-4': 'In the adaptive placements, the run at each new [MATH] is initialised with the values of the currently closest larger [MATH].', '1209.3198-3-16-0': '# Examples', '1209.3198-3-17-0': 'We present three examples, the first two of which were included in the review paper by Friel and Wyse (2012) where the performance of power posteriors was compared to some other existing methods.', '1209.3198-3-17-1': 'The first is a non-nested linear regression comparison for which the marginal likelihoods can be calculated analytically.', '1209.3198-3-17-2': 'Here our experiments concentrate on the effects of the modified integration rule and the adaptive placement of the inverse temperatures.', '1209.3198-3-17-3': 'Example 2 is a larger problem, choosing between two logistic regression models, for which an analytic solution is not possible; we use this example to compare the improved algorithm with the Stepping Stone sampling approach of Xie et al (2011).', '1209.3198-3-17-4': 'The final example is by far the largest and exhibits the most interestingly shaped [MATH], the focus here is on the stability of the approach to poor estimation of the gradient.', '1209.3198-3-18-0': '## Example 1: Radiata pine', '1209.3198-3-19-0': 'The first example compares two linear regression models for the Radiata pine data originally in Williams (1959).', '1209.3198-3-19-1': 'The response variable here is the maximum compression strength parallel to the grain, [MATH], while the predictors are density, [MATH], or density adjusted for resin content, [MATH], for [MATH] specimens of radiata pine.', '1209.3198-3-19-2': 'Two possible Gaussian linear regression models are considered; [EQUATION]', '1209.3198-3-19-3': 'Priors are chosen to match the analyses of Friel and Wyse (2012) (baring a notational factor of 2).', '1209.3198-3-19-4': 'The regression parameters [MATH] and [MATH] were taken to be Normally distributed with mean [MATH] and precision [MATH] and [MATH] respectively where [MATH].', '1209.3198-3-19-5': 'The values of [MATH] and [MATH] were fixed to be 0.06 and 6.', '1209.3198-3-19-6': 'A gamma prior with shape [MATH] and rate [MATH] was assumed for both [MATH] and [MATH].', '1209.3198-3-20-0': 'We consider estimating the evidence using [MATH], 20, 50 or 100 rungs in the tempering scheme where the rungs are chosen either according to the powered fraction (PF) rule or adaptively following the heuristic in Section [REF].', '1209.3198-3-20-1': 'The parameters at all levels are updated using the Gibbs sampler, with 10000 iterations at each rung, discarding the first fifth of these as burn in.', '1209.3198-3-20-2': 'To quantify the performances, the bias, standard deviation and Root Mean Square Error (RMSE) are estimated by performing 100 replicates of the four schemes at 10, 20, 50 and 100 rungs.', '1209.3198-3-20-3': 'The results are given in Table [REF].', '1209.3198-3-20-4': 'Comparing the first column with the second, and the third with the fourth, we can see that the modified power posterior approach dominates the usual one in all cases.', '1209.3198-3-20-5': 'For small numbers of rungs, where the RMSE is dominated by the bias term, this effect is particularly dramatic.', '1209.3198-3-20-6': 'The standard error is not greatly affected by the choice of scheme, although in this example the smallest values all occur for modified schemes.', '1209.3198-3-20-7': 'As the number of rungs increases, the RMSE becomes more affected by the standard error than the bias.', '1209.3198-3-21-0': 'Comparing the first column of Table [REF] with the third, we can isolate the effect of the adaptive placement.', '1209.3198-3-21-1': 'This is also most noticeable at small numbers of rungs, perhaps unsurprisingly given that the placing of an individual [MATH] makes less difference as their total number increases.', '1209.3198-3-21-2': 'To help to visualise the effect of both the correction and the adaptive placing of inverse temperatures, Figure [REF] plots the 100 observed biases pairwise for the standard vs modified power posterior.', '1209.3198-3-21-3': 'Points are plotted for both the PF spacing and the adaptive spacings, and for each number of rungs.', '1209.3198-3-21-4': 'How much change occurs for each standard power posterior under the additive modification given by Equation ([REF]) can be seen as the vertical distance between the point and the red [MATH] line.', '1209.3198-3-21-5': 'A few points are immediately clear.', '1209.3198-3-21-6': 'Firstly the correction is fairly well estimated (the sets of points are roughly parallel to the [MATH] line, even in the [MATH] case; this is a function of the MCMC run lengths).', '1209.3198-3-21-7': 'Second, there is an interesting difference between the adaptive and the PF versions, less correction is needed for the adaptive schedule, that is, it is already doing a better job of linearly approximating the function for the numerical integration.', '1209.3198-3-21-8': 'Thirdly, as was already observed in Table [REF], as the number of rungs increases, the differences between the two types of spacings become less pronounced.', '1209.3198-3-22-0': 'Given the good reductions in bias seen in Table [REF], it is important to ask how much extra time is required.', '1209.3198-3-22-1': 'To assess this, a total of 20 runs for Model 1 using 10000 iterations and 100 inverse temperatures were timed.', '1209.3198-3-22-2': 'Four versions of the algorithm were considered, corresponding to Table [REF].', '1209.3198-3-22-3': 'All the coding was in R and times are given relative to the PF standard power posterior version: [EQUATION]', '1209.3198-3-22-4': 'What we see is that the adaptive selection of inverse temperatures and the correction term in the numerical integration come at an acceptably small computational cost.', '1209.3198-3-22-5': 'Given this, our recommendation would be to use the modified integration rule when employing power posteriors.', '1209.3198-3-22-6': 'The benefits of the adaptive placement are less dramatic except at small [MATH] but as it provides a completely automatic way to choose the inverse temperatures for very little cost, we would also recommend it.', '1209.3198-3-23-0': '## Example 2: Pima indians', '1209.3198-3-24-0': 'We turn next to the Pima Indian example considered by Friel and Wyse (2012), originally described by Smith et al (1988) .', '1209.3198-3-24-1': 'These data record diabetes incidence and possible disease indicators for [MATH] Pima Indian women aged over 20.', '1209.3198-3-24-2': 'The seven possible disease indicators are the number of pregnancies (NP), plasma glucose concentration (PGC), diastolic blood pressure (BP), triceps skin fold thickness (TST), body mass index (BMI), diabetes pedigree function (DP) and age (AGE), with all these covariates standardised.', '1209.3198-3-25-0': "The model assumed for the observed diabetes incidence, [MATH], is [EQUATION] where [MATH] is the probability of incidence for person [MATH], and [MATH] is related to the [MATH] person's covariates and a constant term, denoted by [MATH], and the parameters, [MATH], by [EQUATION] where [MATH] is the number of explanatory variables.", '1209.3198-3-25-1': 'An independent multivariate Gaussian prior is assumed for [MATH], with mean zero and non-informative precision of [MATH], so that [EQUATION]', '1209.3198-3-25-2': 'A long reversible jump run (Green, 1995) revealed the two models with the highest posterior probability:', '1209.3198-3-26-0': '[MATH]( 0.01/t , 1/)[MATH]t[MATH][MATH]n=50[MATH]-257.2342[MATH]-259.8519[MATH]-257.2588[MATH]-259.8906[MATH]t_0=0[MATH]t_n-1[MATH]r_k=z(yt_k+1)/z(yt_k)[MATH]z(yt)[MATH]t_k[MATH]m[MATH]^(i)[MATH]p_t_k( y)[MATH]z(y t_n=1)[MATH]n[MATH]_k=0^n-1 r_k[MATH]t_n=1[MATH]t_n-1[MATH]n[MATH]t_i[MATH]n=10[MATH]n[MATH]n[MATH]y = y_1,,y_82[MATH]z = z_1,,z_82[MATH]z_i=j[MATH]y_i[MATH]j^th[MATH]k[MATH]y_i[MATH]k=3[MATH]k=4[MATH]t=0[MATH]n[MATH]k=3[MATH][-229.8626,-227.8737][MATH]k=4[MATH][-230.5911,-228.0902][MATH]k=3[MATH]k=4[MATH]t_i[MATH]t=0[MATH]n 100[MATH]n=200[MATH]n[MATH]t_i[MATH]n[MATH]E_y,t(p(y))[MATH]t 0[MATH]t 1[MATH]t[MATH]p_t(y)[MATH]t=1[MATH]'}

1510.08849

{'1510.08849-1-0-0': 'Hard X-ray observations are crucial to study the non-thermal jet emission from high-redshift, powerful blazars.', '1510.08849-1-0-1': 'We observed two bright [MATH] flat spectrum radio quasars (FSRQs) in hard X-rays to explore the details of their relativistic jets and their possible variability.', '1510.08849-1-0-2': 'S5 0014+81 (at [MATH]) and B0222+185 (at [MATH]) have been observed twice by the Nuclear Spectroscopic Telescope Array (NuSTAR) simultaneously with Swift/XRT, showing different variability behaviours.', '1510.08849-1-0-3': 'We found that NuSTAR is instrumental to explore the variability of powerful high-redshift blazars, even when no [MATH]-ray emission is detected.', '1510.08849-1-0-4': 'The two sources have proven to have respectively the most luminous accretion disk and the most powerful jet among known blazars.', '1510.08849-1-0-5': 'They are located at the extreme end of the jet-accretion disk relation previously found for [MATH]-ray detected blazars.', '1510.08849-1-1-0': '# Introduction', '1510.08849-1-2-0': 'Blazars are active galactic nuclei (AGN) with their broad-band emission dominated by the relativistic jet, oriented close to our line of sight.', '1510.08849-1-2-1': 'The two humps that characterise their spectral energy distribution (SED) are the signature of this relativistically beamed emission.', '1510.08849-1-2-2': 'They are attributed to synchrotron (at lower frequencies) and inverse Compton (at high frequencies) processes, and peak in the sub-millimeter and X- to [MATH]-ray, respectively.', '1510.08849-1-2-3': 'The electron population involved in the inverse Compton emission is thought to interact either with the synchrotron photons involved in the lower-frequency emission, or with photons coming from structures external to the relativistic jet (synchrotron self-Compton, SSC, or external Compton, EC, emissions, respectively).', '1510.08849-1-2-4': 'The latter is likely the primary process in sources that present a pronounced dominance of the higher frequencies hump over the synchrotron one.', '1510.08849-1-2-5': 'This usually happens in the most powerful blazars, i.e. the flat-spectrum radio quasars (FSRQs).', '1510.08849-1-2-6': 'These sources are thought to have more sources of seed photons for an EC process (i.e. accretion disk, broad line region, torus), compared to the BL Lacertae objects (BL Lacs) that have weak or absent broad lines and no accretion or torus emission (see e.g. Chiaberge, Capetti Celotti 1999; Ghisellini et al. 2011, Sbarrato et al. 2012).', '1510.08849-1-3-0': 'The most immediate signature of the blazar nature of an AGN is its emission in the [MATH]-rays.', '1510.08849-1-3-1': 'The high-energy hump in the blazar SED, in fact, usually peaks at tens or hundreds of MeV, and therefore it can be easily observed with [MATH]-ray telescopes, such as the Large Area Telescope (LAT) onboard the Fermi Gamma-Ray Space Telescope (Atwood et al. 2009).', '1510.08849-1-3-2': 'Fermi/LAT provides an all-sky survey in the [MATH]-rays, that ensures a clear classification of the blazar candidates.', '1510.08849-1-3-3': 'Nevertheless, at higher redshifts, Fermi/LAT is less efficient in detecting blazars, even those with a very large bolometric luminosity.', '1510.08849-1-3-4': 'This is because the most powerful blazars have their high energy peak at [MATH]MeV energies or below, and this peak is seen redshifted.', '1510.08849-1-3-5': 'This is the reason why the fraction of high redshift blazars (i.e. at [MATH]) detected in the hard X-rays by the Burst Alert Telescope (BAT) onboard the Swift satellite is much larger than for Fermi/LAT (see Ajello et al. 2009, Ghisellini et al 2010a).', '1510.08849-1-4-0': 'Indeed, blazars observed so far show a trend: the humps in the SEDs of the more powerful ones peak at lower frequencies as compared to less powerful blazars.', '1510.08849-1-4-1': 'This trend is known as the "blazar sequence" (Fossati et al. 1998, but see also Giommi et al. 2012).', '1510.08849-1-4-2': 'As a result, it is likely that the most powerful and distant blazars cannot be detected by Fermi/LAT.', '1510.08849-1-4-3': 'Hard X-ray instruments, instead, like Swift/BAT and now the Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison et al. 2013), are the most suitable instruments now available to investigate jet emission in the most powerful blazars at [MATH].', '1510.08849-1-5-0': 'In this paper we report on observations of S5 0014+81 (at [MATH]) and B0222+185 ([MATH]) by NuSTAR.', '1510.08849-1-5-1': 'These two blazars have been previously detected in the 3-year all-sky survey of Swift/BAT (Ajello et al. 2009, see also Ajello et al. 2012 and Baumgartner et al. 2013), and are amongst the most powerful blazars ever observed.', '1510.08849-1-5-2': 'As with other powerful and high-redshift FSRQs, their optical flux shows contributions due to thermal emission from the accretion disk, particularly prominent in S5 0014+813, whose luminosity reaches [MATH] erg s[MATH] (Ghisellini et al. 2010a).', '1510.08849-1-5-3': 'For both sources, the Swift/BAT spectrum together with the Fermi/LAT upper limit already constrained the location of the high energy peak, but with a relatively large uncertainty given the poor spectral slope determination of Swift/BAT.', '1510.08849-1-5-4': 'This motivated the NuSTAR observations.', '1510.08849-1-6-0': 'In this work, we adopt a flat cosmology with [MATH] km s[MATH] Mpc[MATH] and [MATH].', '1510.08849-1-7-0': '# Observations and Data Analysis', '1510.08849-1-8-0': '## NuSTAR observations', '1510.08849-1-9-0': 'The NuSTAR satellite observed S5 0014+81 on 2014 December 21 (obsID 60001098002) and on 2015 January 23 (obsID 60001098004) for total net exposure times of 31.0 ks and 36.4 ks, respectively.', '1510.08849-1-9-1': 'B0222+185 was observed by NuSTAR on 2014 December 24 (obsID 60001101002) and on 2015 January 18 (obsID 60001101004).', '1510.08849-1-9-2': 'The total net exposure times were 32.0 ks and 37.4 ks, respectively.', '1510.08849-1-10-0': 'The Focal Plane Module A (FPMA) and Focal Plane Module B (FPMB) data sets were first processed with the NuSTARDAS software package (v.1.4.1) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (Caltech, USA).', '1510.08849-1-10-1': 'Event files were calibrated and cleaned with standard filtering criteria using the nupipeline task (version 20150316) of the NuSTAR CALDB.', '1510.08849-1-11-0': 'The two sources were well detected in the NuSTAR 3-79 keV energy band.', '1510.08849-1-11-1': 'In both cases the FPMA and FPMB spectra of the target were extracted from the cleaned event files using a circle of 20 pixel ([MATH] arcsec) radius, while the background was extracted from nearby circular regions of 40 pixel radius.', '1510.08849-1-11-2': 'The ancillary response files were generated with the numkarf task, applying corrections for the PSF losses, exposure maps and vignetting.', '1510.08849-1-11-3': 'All spectra were binned to ensure a minimum of 30 counts per bin.', '1510.08849-1-12-0': '## Swift observations', '1510.08849-1-13-0': 'The Swift satellite observed S5 0014+81 on 2014 December 21 (obsIDs 00080003001) and on 2015 January 23 (obsID 00080003002) while B0222+185 was observed on 2014 December 24 (obsID 00080243001) and on 2015 January 18 (00080243002).', '1510.08849-1-13-1': 'The total net exposure times were 6.5 ks (December 2014) and 6.6 ks (January 2015) for S5 0014+81 and 4.9 ks (December 2014) and 5.1 ks (January 2015) for B0222+185.', '1510.08849-1-14-0': '### XRT observations', '1510.08849-1-15-0': 'Swift/XRT (Burrows et al. 2005) observations were carried out using the Photon Counting (PC) CCD readout mode and in the four observations the sources were well detected in the 0.3-10 keV XRT energy band.', '1510.08849-1-15-1': 'The XRT data sets were first processed with the XRTDAS software package (v.3.0.0) developed at the ASI Science Data Center (ASDC) and distributed by HEASARC within the HEASoft package (v. 6.16).', '1510.08849-1-15-2': 'In particular, event files were calibrated and cleaned with standard filtering criteria with the xrtpipeline task using the calibration files available in the version 20140709 of the Swift/XRT CALDB.', '1510.08849-1-16-0': 'The energy spectra were then extracted from the calibrated and cleaned event files.', '1510.08849-1-16-1': 'Events for the source spectral analysis were selected within a circle of 20 pixel ([MATH] arcsec) radius, enclosing about 90% of the PSF, while the background was extracted from a nearby circular region of 80 pixel radius.', '1510.08849-1-16-2': 'The ancillary response files were generated with the xrtmkarf task, applying corrections for the PSF losses and CCD defects using the cumulative exposure map.', '1510.08849-1-16-3': 'The source spectra were binned to ensure a minimum of 30 counts per bin.', '1510.08849-1-17-0': '### UVOT observations', '1510.08849-1-18-0': 'Swift/UVOT (Roming et al. 2005) observations were performed with all six optical and UV lenticular filters (namely [MATH]).', '1510.08849-1-18-1': 'We performed aperture photometry for all filters in all the observations using the standard UVOT software distributed within the HEAsoft package (version 6.16) and the calibration included in the latest release of the CALDB.', '1510.08849-1-18-2': 'Counts were extracted from apertures of 5 arcsec radius for all filters and converted to fluxes using the standard zero points (Poole et al. 2008).', '1510.08849-1-18-3': 'The fluxes were then de-reddened using the appropriate values of [MATH] taken from Schlegel et al. (1998) and Schlafly et al. (2011) with [MATH] ratios calculated for UVOT filters using the mean Galactic interstellar extinction curve from Fitzpatrick (1999).', '1510.08849-1-18-4': 'No variability was detected within single exposures in any filter.', '1510.08849-1-18-5': 'The processing results were carefully verified, checking for possible contaminations from nearby objects within the source apertures and from objects falling within the background apertures.', '1510.08849-1-19-0': '## X-ray spectral analysis', '1510.08849-1-20-0': 'The spectral analysis of the December 2014 and January 2015 NuSTAR and Swift/XRT simultaneous observations of S5 0014+81 and B0222+185 were performed using the XSPEC package.', '1510.08849-1-20-1': 'In all four observations a broken power-law model with an absorption hydrogen-equivalent column density fixed to the Galactic value of [MATH] cm[MATH] (S5 0014+81) and [MATH] cm[MATH] (B0222+185) (Kalberla et al. 2005) was found to provide a good description of the observed spectra in the 0.3-79 keV energy band.', '1510.08849-1-20-2': 'The inter-calibration factors between the three instruments (NuSTAR/FPMA, NuSTAR/FPMB and Swift/XRT) were taken into account adding a multiplicative constant (kept to 1 for NuSTAR/FPMA) to the spectral model.', '1510.08849-1-20-3': 'We found values in the 2% range for NuSTAR/FPMB and in the 10% range for Swift/XRT which are consistent with the cross-calibration uncertainties for the instruments (Madsen et al. 2015).', '1510.08849-1-20-4': 'The results of the spectral fits are shown in Table [REF], and Figure [REF] shows the X-ray spectra of the two sources.', '1510.08849-1-21-0': 'We also tried a simpler spectral model, namely a single power-law with an absorption hydrogen-equivalent column density fixed to the Galactic value.', '1510.08849-1-21-1': 'However, for all four observations we found significantly higher [MATH] values, thus confirming the existence of an intrinsic spectral curvature within the observed 0.3-79 keV energy band.', '1510.08849-1-22-0': '# Black hole Mass Estimate', '1510.08849-1-23-0': 'The black hole mass of a blazar is an important feature to characterise it.', '1510.08849-1-23-1': 'When optical or infrared spectra are not available, the virial mass estimate method cannot be applied.', '1510.08849-1-23-2': 'At high redshift, the available lines to apply such a method are also less reliable.', '1510.08849-1-23-3': 'With a good photometric coverage of the UV-optical-IR band, instead, the accretion disk emission can be fitted, and the black hole mass can be estimated from this fitting process (Calderone et al. 2013).', '1510.08849-1-24-0': 'The Swift/UVOT photometric data, along with archival data from the ASI Science Data Center (ASDC) allow us to cover the optical-UV band of the SEDs.', '1510.08849-1-24-1': 'The first feature to notice in the IR-optical-UV SEDs of the two blazars is the absorption at frequencies higher than [MATH] rest frame.', '1510.08849-1-24-2': 'This is the Ly[MATH] forest, due to intervening clouds absorbing hydrogen Ly[MATH] photons, and affects wavelengths shortward of 1216 .', '1510.08849-1-24-3': 'The bluer UVOT bands fall in this frequency range in the case of S5 0014+318 and B0222+185, and in fact only upper limits could be derived for those photometric bands.', '1510.08849-1-24-4': 'At frequencies lower than this prominent absorption feature, though, a peak in the SEDs is clearly visible.', '1510.08849-1-24-5': 'Below this peak, the optical flux decreases with frequency, suggesting a power-law trend, especially in S5 0014+813.', '1510.08849-1-24-6': 'This is the clear signature of an accretion disk, which can be fitted with a simple Shakura-Sunyaev model (Shakura Sunyaev 1973; Calderone et al. 2013; Sbarrato et al. 2013).', '1510.08849-1-24-7': 'At lower frequencies, another feature is evident from the IR-optical-UV SEDs of these two sources: the WISE (Wright et al. 2010) IR bands show an increase of the flux, that breaks the power-law-like trend in the optical.', '1510.08849-1-24-8': 'This is likely the signature of the IR emission from a dusty torus around the nucleus.', '1510.08849-1-25-0': 'With these premises, a reliable way to estimate the black hole mass of S5 0014+318 and B0222+185 is to fit their IR-optical-UV SEDs with a simple model of accretion disk emission.', '1510.08849-1-25-1': 'We applied the radiatively efficient, geometrically thin, optically thick Shakura Sunyaev (1973) model.', '1510.08849-1-25-2': 'Assuming a standard radiative efficiency [MATH], only two free parameters are left to be fitted: the accretion rate [MATH], that can be traced by the intrinsic disk luminosity [MATH], and the black hole mass [MATH] itself.', '1510.08849-1-25-3': 'We can constrain the overall disk luminosity thanks to the visibility of the peak of the disk emission, with some consideration regarding its anisotropic properties (as throughly explained by Calderone et al. 2013):', '1510.08849-1-26-0': 'according to the Shakura-Sunyaev model, the peak luminosity [MATH] corresponds to half the total observed luminosity [MATH].', '1510.08849-1-27-0': 'the observed luminosity depends on the viewing angle of the accretion disk: [EQUATION] where [MATH] is the intrinsic total luminosity emitted by the accretion disk.', '1510.08849-1-27-1': 'In the case of a blazar [MATH] since we see the accretion disk face-on.', '1510.08849-1-28-0': 'We can therefore derive the intrinsic total luminosity from the peak luminosity of our sources: [EQUATION]', '1510.08849-1-28-1': 'This means that in the case of these two sources, [MATH] is constrained by observations, and only [MATH] is left as a free parameter to be derived with the IR-optical-UV SED fitting.', '1510.08849-1-29-0': 'We find that both sources have large black hole masses and are fast accreting, even if not super-Eddington.', '1510.08849-1-29-1': 'We derive [MATH] and [MATH] for B0222+185 and [MATH] and [MATH]erg s[MATH] for S5 0014+813.', '1510.08849-1-29-2': 'These values are significantly smaller than what was derived in Ghisellini et al. (2009 and 2010a).', '1510.08849-1-29-3': 'For S5 0014+813 the reason is due to i) the better coverage of the IR band achieved with WISE data and ii) neglecting the optical data taken from Bechtold et al. (1994).', '1510.08849-1-29-4': 'We now prefer to discard those data because the derivation of flux and luminosities are not sufficiently clear in that paper.', '1510.08849-1-30-0': 'For B0222+185, the smaller values of [MATH] and [MATH] are due to the new infrared data (not available in the previous work), that now help more accurately constrain the peak frequency of the disk emission.', '1510.08849-1-31-0': '# Modelling the broad-band SED', '1510.08849-1-32-0': 'Figure [REF] shows that both S5 0014+318 and B0222+185 have overall SEDs characterised by a prominent high-energy component, that along with the characteristic flat and intense radio luminosity is attributed to non-thermal emission from a relativistic jet.', '1510.08849-1-32-1': 'In the IR-optical-UV range of both sources, the SEDs are dominated by thermal emission attributed to the accretion disk, as discussed in Section [REF].', '1510.08849-1-33-0': 'Not being detected in the [MATH]-rays by Fermi/LAT, X-ray data are necessary to study the non-thermal high-energy emission of S5 0014+318 and B0222+185.', '1510.08849-1-33-1': 'Specifically, NuSTAR data are crucial for understanding the X-ray spectral profile and possible variability in this kind of high-redshift source, as can be seen in Figure [REF].', '1510.08849-1-33-2': 'X-ray data contribute significantly to the modelling of the broad-band SEDs of the two sources (Figure [REF]).', '1510.08849-1-34-0': 'To interpret the SEDs of the two sources, we used a leptonic one-zone emitting model, fully described in Ghisellini Tavecchio (2009).', '1510.08849-1-34-1': 'We refer to the original paper for details, providing here only a very brief description of the most important features of the models.', '1510.08849-1-34-2': 'The emitting source is assumed to be a spherical region in which relativistic electrons emit by synchrotron and inverse Compton processes.', '1510.08849-1-34-3': 'This homogeneous spherical blob is assumed to be located at a distance [MATH] from the central black hole, moving with a bulk Lorentz factor [MATH] at an angle [MATH] from our line-of-sight.', '1510.08849-1-34-4': 'Relativistic electrons are injected throughout the source, with a power [MATH] as measured in the comoving frame.', '1510.08849-1-34-5': 'The energy distribution [MATH] of the injected electrons is a smoothly broken power law with slopes [MATH] and [MATH] (defined as [MATH]) below and above the random Lorentz factor [MATH].', '1510.08849-1-34-6': 'Note that, even if [MATH] is a broken power law, [MATH] derived through the continuity equation maintains a break, albeit smoother than the injected broken power law.', '1510.08849-1-34-7': 'This produces a gently curved spectrum, as shown in Figure 1.', '1510.08849-1-34-8': 'The broad line region is located at a distance [MATH] from the black hole, while the infrared emitting torus is at [MATH].', '1510.08849-1-34-9': '[MATH] is the accretion disk luminosity in units of [MATH], and it is derived as in Section [REF], together with the central black hole mass.', '1510.08849-1-34-10': 'The values of the parameters adopted for the models are reported in Table [REF].', '1510.08849-1-35-0': 'Table [REF] reports the different forms of the power carried by the jet: the power [MATH] spent in producing the radiation we observe, the Poynting flux [MATH], the power associated to the bulk motion of relativistic electrons ([MATH]) and cold protons ([MATH]), assuming one proton per relativistic electron.', '1510.08849-1-35-1': 'This assumption is consistent with independent results on blazar and GRB jets by Nemmen et al. (2012).', '1510.08849-1-35-2': 'They found that the total jet power for both classes is ten times the radiative power [MATH], i.e. similar to what we find in this work (see Table [REF]).', '1510.08849-1-35-3': 'Different proton-to-relativistic electron ratios were explored by Sikora Madejski (2000), who found that the relativistic pairs must be less than 10-20 per proton.', '1510.08849-1-35-4': 'With this combination, the total jet power can result equal to or even less than the radiative power, that instead is only a part of the total power carried by the jet, and hence should be a lower limit to the total [MATH] (Ghisellini 2012; Ghisellini et al. 2014).', '1510.08849-1-35-5': 'Therefore, assuming one proton per relativistic electron is reasonable to explain the observed jet features and its physics.', '1510.08849-1-36-0': 'The emitting regions of both sources are located within the broad line region (and the infrared torus).', '1510.08849-1-36-1': 'In this way, the energy density of photons from the broad line region feed the inverse Compton process, together with photons from the torus.', '1510.08849-1-36-2': 'The inverse Compton process is dominated by external Compton instead of synchrotron-self Compton, as expected in FSRQs.', '1510.08849-1-36-3': 'Tagliaferri et al. (2015) obtained the same result for two other [MATH] blazars observed by NuSTAR: the emitting regions of both S5 0836+710 and PKS 2149-306 are located between the BLR and IR torus.', '1510.08849-1-36-4': 'Their results were obtained through SED fitting, and were also confirmed on the basis of the variability timescales obtained with two NuSTAR observations per source.', '1510.08849-1-37-0': '# Discussion', '1510.08849-1-38-0': 'Blazars are characterised by their prominent relativistically boosted jet emission.', '1510.08849-1-38-1': 'They usually show prominent high-energy emission, which results in high [MATH]-ray luminosities, well detected by instruments like Fermi/LAT.', '1510.08849-1-38-2': 'In some cases, though, blazars are not detected in such energy bands.', '1510.08849-1-38-3': 'This is the case for S5 0014+318 and B0222+185.', '1510.08849-1-38-4': 'Even if lacking a high-energy detection, the Compton bump can be observed in the X-ray frequency range, but a detection in the soft X-rays usually is not enough to determine the relativistic jet features of a blazar, nor its orientation.', '1510.08849-1-38-5': 'S5 0014+318 and B0222+185 were detected by Swift/BAT, but these data were not precise enough to derive exact estimates of the bulk Lorentz factors and viewing angles.', '1510.08849-1-38-6': 'Figures [REF] and [REF] show that the Swift/BAT data do not have enough precision to constrain the hard X-ray slope.', '1510.08849-1-38-7': 'NuSTAR, however, provides a broad-band, precise measurement for both sources, confirming that both blazars are seen at small viewing angles, i.e. their jets are directed along our line of sight.', '1510.08849-1-38-8': 'Both sources also host massive central black holes (both with [MATH]).', '1510.08849-1-39-0': '## Variability', '1510.08849-1-40-0': 'Looking in detail at the broad-band SEDs of these two objects, we see some interesting differences.', '1510.08849-1-40-1': 'Both sources show the two humps, i.e. the signature of aligned jet emission, while in the IR-optical-UV band, the accretion disk emission dominates over the non-thermal jet emission.', '1510.08849-1-40-2': 'Comparing the two panels of Figure [REF], it can be noticed that S5 0014+318 and B0222+185 also show different variability behaviours.', '1510.08849-1-40-3': 'They were both observed in two epochs separated by [MATH] month.', '1510.08849-1-40-4': 'S5 0014+318 does not show flux or spectral variation between the two observations, and the new data are consistent with archival data.', '1510.08849-1-40-5': 'Only the Swift/BAT detection could suggest a different state of the source, but due to the large uncertainty we cannot draw any strong conclusion.', '1510.08849-1-41-0': 'B0222+185, instead, shows a clear variation of the X-ray flux between the two Swift/XRT + NuSTAR observations: in December 2014 the source was in a higher state, compared to both January 2015 and archival data.', '1510.08849-1-41-1': 'The right panel of Figure [REF] shows the peak of the high-energy hump shifted towards higher frequencies in the higher state.', '1510.08849-1-41-2': 'This is opposite to the general trend displayed by the blazar sequence.', '1510.08849-1-42-0': 'Such behaviour is not uncommon in rapidly varying FSRQs: although they follow the blazar sequence when considering different sources, an individual object can behave opposite to the sequence itself while varying.', '1510.08849-1-42-1': 'According to the model shown in Figure [REF], the B0222+185 variability can be described by a variation in the injected power (see second and third lines of Table [REF]), accompanied by [MATH] increasing in the high state.', '1510.08849-1-43-0': 'Another remarkable example of this kind of variation, very similar to the one showed by B0222+185, but much more pronounced, has recently been seen for S5 0836+710 (Ciprini et al. 2015; Vercellone et al. 2015; Giroletti et al. 2015) during its August 2015 [MATH]-ray flare, which triggered observations at X-ray and radio frequencies.', '1510.08849-1-43-1': 'The amplitude of the flux variation was huge in the Fermi/LAT band (factor 65 greater than the average flux reported in the third Fermi/LAT catalog of Acero et al. 2015) and rather modest in the high energy part of the Swift/XRT band.', '1510.08849-1-43-2': 'This implies that the X-ray flux had to change more at larger energies, to connect to the enhanced [MATH]-ray flux, and that the peak frequency of the high energy hump must be "bluer" than what was displayed during the NuSTAR observation described in Tagliaferri et al. (2015).', '1510.08849-1-43-3': 'In other words: if NuSTAR had followed the August 2015 flare of S5 0836+710, it likely would have detected a clear flux and spectral variation, leading to a predicted shift of the high energy hump towards higher frequencies even in the absence of [MATH]-ray data.', '1510.08849-1-43-4': 'The right panel of Figure [REF] shows that Swift/XRT would not have been able to discriminate between the two states of B0222+185, while NuSTAR distinguishes them clearly.', '1510.08849-1-43-5': 'We conclude that:', '1510.08849-1-44-0': '## Jets and accretion of the two most powerful blazars', '1510.08849-1-45-0': 'We now aim to frame S5 0014+318 and B0222+185 within the larger blazar picture.', '1510.08849-1-45-1': 'We consider them in the jet-accretion correlation scenario.', '1510.08849-1-45-2': 'Ghisellini et al. (2014) found that in blazars the jet power not only correlates with the accretion power, but it is even larger.', '1510.08849-1-45-3': 'This suggests that accretion is strongly related to jet power, implying a role in jet production.', '1510.08849-1-45-4': 'At the same time, the fact that jet power is larger than accretion power tells us that some other process must play a role in the jet launch and acceleration.', '1510.08849-1-45-5': 'Black hole spin is the best candidate to play such a role.', '1510.08849-1-45-6': 'This result was obtained by studying a sample of Fermi-detected blazars, for which Shaw et al. (2012; 2013) obtained optical spectra.', '1510.08849-1-45-7': 'Ghisellini et al. (2014) selected all the objects with broad emission lines, in order to have a proxy of accretion luminosity, and compared jet and accretion power for the 226 blazars in this sample.', '1510.08849-1-45-8': 'However, this sample did not include the most extreme blazars known, leaving open the questions:', '1510.08849-1-46-0': 'how does the jet-accretion relation look in the case of the most powerful blazars?', '1510.08849-1-47-0': 'Does the power balance change when accretion or jet emission are extreme?', '1510.08849-1-47-1': 'These questions will guide us in the following discussion.', '1510.08849-1-48-0': 'First we add to the original blazar sample the sources expected to be the most powerful.', '1510.08849-1-48-1': 'To this aim we select the [MATH] blazars detected by Swift/BAT and all known high-redshift ([MATH]) blazars.', '1510.08849-1-48-2': 'The BAT sensitivity limit is not very deep, and at high redshift it can detect mainly the most powerful sources, whose high-energy components peak in the [MATH]MeV range.', '1510.08849-1-48-3': 'BAT detected 10 [MATH] blazars, including S5 0014+318 and B0222+185, that we add to the blazar sample of Ghisellini et al. (2014).', '1510.08849-1-48-4': 'We also include all the known blazars at [MATH], as listed in Ghisellini et al. (2015).', '1510.08849-1-48-5': 'Being the highest redshift blazars currently known, they are expected to be among the most powerful blazars.', '1510.08849-1-48-6': 'They are not present in the BAT blazar catalog because their distance makes their hard X-ray flux too weak for a detection with BAT.', '1510.08849-1-48-7': 'Since most of them were selected starting from optical catalogs, they are likely very powerful in accretion luminosity.', '1510.08849-1-49-0': 'Figure [REF] shows how these samples are located in the overall jet-accretion relation, along with S5 0014+318 and B0222+185.', '1510.08849-1-49-1': 'The total jet power (calculated as the sum of different jet power components listed in Table [REF]) is plotted as a function of the disk luminosity.', '1510.08849-1-49-2': 'The grey stripes show the best fit of the sample by Ghisellini et al. (2014).', '1510.08849-1-49-3': 'Note that the powerful blazars we add in this work are all located within the 2[MATH] dispersion of the previous correlation.', '1510.08849-1-49-4': 'This means that they still follow the jet-accretion relation found by Ghisellini et al. (2014), even if they are among the most powerful sources in the set.', '1510.08849-1-50-0': 'S5 0014+318 and B0222+185 can be considered the two most extreme sources: respectively, they are the blazars with the most luminous disk and the most powerful jet.', '1510.08849-1-50-1': 'Still, they are close enough to the known jet-accretion correlation, to be less than 2[MATH] from the Ghisellini et al. (2014) result.', '1510.08849-1-50-2': 'Thus we conclude that even the most powerful blazars follow the same jet-accretion relation as the [MATH]-ray detected bulk sample.', '1510.08849-1-50-3': 'The second interesting conclusion that we can draw from this comparison is that NuSTAR is once again confirmed to be the most suitable telescope to study the most powerful blazars in our Universe.', '1510.08849-1-51-0': '# Conclusions', '1510.08849-1-52-0': 'The simultaneous X-ray observations of S5 0014+318 and B0222+185 performed with Swift/XRT and NuSTAR gave us an interesting insight into the jet emissions of these two sources.', '1510.08849-1-52-1': 'We confirmed their blazar nature, with a refined estimate of their bulk Lorentz factors and viewing angles, supported by more precise sets of parameters (Tables [REF] and [REF]).', '1510.08849-1-52-2': 'The accretion disk fitting to a more complete data set gave us the possibility to refine our previous estimates of the black hole mass and accretion luminosity of these two sources, implying fast accreting objects with extreme masses of [MATH].', '1510.08849-1-53-0': 'The overall SED modelling allowed us to estimate the jet power and accretion disk luminosity, allowing a comparison of these two sources with the overall blazar jet-accretion relation.', '1510.08849-1-53-1': 'The two sources are among the most powerful blazars known, and they populate the highest disk luminosity and jet power part of the jet-accretion correlation (Ghisellini et al. 2014; Figure [REF]).', '1510.08849-1-53-2': 'It is remarkable that a sample formed by the most powerful blazars known is still within [MATH] from the correlation derived from a sample of blazars whose [MATH]-ray flux was averaged over two years.', '1510.08849-1-53-3': 'The mechanisms governing the jet formation and evolution in the most extreme sources must not be different from the processes powering the more moderate objects.', '1510.08849-1-53-4': 'S5 0014+318 and B0222+185 themselves are consistent with the relation derived from the [MATH]-ray blazars, even if they are the blazars with the most luminous accretion disk and the most powerful jet, respectively.', '1510.08849-1-54-0': 'We found a different variability behaviour between the two sources: while S5 0014+318 did not vary in the two observation epochs separated by [MATH] month, B0222+185 shows a clear variation, with an amplitude larger at larger frequencies.', '1510.08849-1-54-1': 'This last feature is the main reason why NuSTAR is such a crucial instrument to study high-redshift and powerful blazars.', '1510.08849-1-54-2': 'A soft X-ray telescope alone could not see a variability event like the one shown by B0222+185: NuSTAR instead observes at frequencies where the amplitude of a flaring activity is large enough to be seen.', '1510.08849-1-54-3': 'Were we lacking the [MATH]-ray signature of a flare, NuSTAR could do the job.'}

{'1510.08849-2-0-0': 'Hard X-ray observations are crucial to study the non-thermal jet emission from high-redshift, powerful blazars.', '1510.08849-2-0-1': 'We observed two bright [MATH] flat spectrum radio quasars (FSRQs) in hard X-rays to explore the details of their relativistic jets and their possible variability.', '1510.08849-2-0-2': 'S5 0014+81 (at [MATH]) and B0222+185 (at [MATH]) have been observed twice by the Nuclear Spectroscopic Telescope Array (NuSTAR) simultaneously with Swift/XRT, showing different variability behaviours.', '1510.08849-2-0-3': 'We found that NuSTAR is instrumental to explore the variability of powerful high-redshift blazars, even when no [MATH]-ray emission is detected.', '1510.08849-2-0-4': 'The two sources have proven to have respectively the most luminous accretion disk and the most powerful jet among known blazars.', '1510.08849-2-0-5': 'Thanks to these properties, they are located at the extreme end of the jet-accretion disk relation previously found for [MATH]-ray detected blazars, to which they are consistent.', '1510.08849-2-1-0': '# Introduction', '1510.08849-2-2-0': 'Blazars are active galactic nuclei (AGN) with their broad-band emission dominated by the relativistic jet, oriented close to our line of sight.', '1510.08849-2-2-1': 'The two humps that characterise their spectral energy distribution (SED) are the signature of this relativistically beamed emission.', '1510.08849-2-2-2': 'They are attributed to synchrotron (at low frequencies) and inverse Compton (at high frequencies) processes, and in the radio/sub-millimeter and X- /[MATH]-ray, respectively.', '1510.08849-2-2-3': 'The electron population involved in the inverse Compton emission is thought to interact either with the synchrotron photons involved in the low-frequency emission, or with photons coming from structures external to the relativistic jet (synchrotron self-Compton, SSC, or external Compton, EC, emissions, respectively).', '1510.08849-2-2-4': 'The latter is likely the primary process in sources that present a pronounced dominance of the higher frequencies hump over the synchrotron one.', '1510.08849-2-2-5': 'This usually happens in the most powerful blazars, i.e. the flat-spectrum radio quasars (FSRQs).', '1510.08849-2-2-6': 'These sources are thought to have more sources of seed photons for an EC process (i.e. accretion disk, broad line region, torus), compared to the BL Lacertae objects (BL Lacs) that have weak or absent broad lines and no accretion or torus emission (see e.g. Chiaberge, Capetti Celotti 1999; Ghisellini et al. 2011, Sbarrato et al. 2012).', '1510.08849-2-3-0': 'The most immediate signature of the blazar nature of an AGN is its emission in the [MATH]-rays.', '1510.08849-2-3-1': 'The high-energy hump in very powerful blazar SED, in particular, usually peaks in the MeV-GeV range, and therefore it can be easily observed with [MATH]-ray telescopes, such as the Large Area Telescope (LAT) onboard the Fermi Gamma-Ray Space Telescope (Atwood et al. 2009).', '1510.08849-2-3-2': 'The Fermi/LAT team built an all-sky [MATH]-ray catalog, providing a clear classification of all the sources included in the survey, through multifrequency studies.', '1510.08849-2-3-3': 'This provides a complete, all-sky blazar catalog (Ackermann et al. 2015).', '1510.08849-2-3-4': 'Nevertheless, at higher redshifts, Fermi/LAT is less efficient in detecting blazars, even those with a very large bolometric luminosity.', '1510.08849-2-3-5': 'This is because the most powerful blazars have their high energy peak at [MATH]MeV energies or below, and this peak is seen redshifted.', '1510.08849-2-3-6': 'This is the reason why the fraction of high redshift blazars (i.e. at [MATH]) detected in the hard X-rays by the Burst Alert Telescope (BAT) onboard the Swift satellite is much larger than for Fermi/LAT (see Ajello et al. 2009, Ghisellini et al 2010a).', '1510.08849-2-4-0': 'Indeed, blazars observed so far show a trend: the humps in the SEDs of the more powerful ones peak at lower frequencies as compared to less powerful blazars.', '1510.08849-2-4-1': 'This trend is known as the "blazar sequence" (Fossati et al. 1998).', '1510.08849-2-4-2': 'Although the original concept of "blazar sequence" finds confirmation through Fermi blazar data (i.e. Ajello et al. 2015; Ackermann et al. 2015), there is some dispute about its reality (see e.g. the reviews by Padovani 2007 and Ghisellini et al. 2008).', '1510.08849-2-4-3': 'For instance, Giommi et al. (2012) proposed what they called a "simplified blazar scenario", in which they postulate that the shape of the SED of blazars is uncorrelated with their luminosity.', '1510.08849-2-4-4': 'Then they assume a given probability for the different blazar shape: there are more blazar with low-frequency peaks than high-frequency ones.', '1510.08849-2-4-5': 'Taking into account the observation constraints and the limiting fluxes of the current blazar surveys, they can reproduce what is observed.', '1510.08849-2-4-6': 'This should be taken as a test that both the "blazar sequence" and the "simplified scenario" pass, not as a proof that the blazar sequence is wrong.', '1510.08849-2-4-7': 'Both frameworks can describe the considered existing data.', '1510.08849-2-4-8': 'On the other hand, the blazar sequence found an easy physical explanation in terms of radiative cooling (Ghisellini et al. 1998), while the simplified scenario is based on the assumed SED distribution, that has no physical explanation (yet).', '1510.08849-2-4-9': 'Whether the blazar sequence is intrinsic or only a selection effect, the most powerful and distant blazars are hardly detected by Fermi/LAT, their inverse Compton emission peak being shifted towards the (observed) MeV band (Ghisellini et al. 2010).', '1510.08849-2-4-10': 'Hard X-ray instruments, instead, like Swift/BAT and now the Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison et al. 2013), are the most suitable instruments now available to investigate jet emission in the most powerful blazars at [MATH].', '1510.08849-2-5-0': 'In this paper we report on observations of S5 0014+81 ([MATH], [MATH]) and B0222+185 ([MATH], [MATH]) by NuSTAR.', '1510.08849-2-5-1': 'These two blazars have been previously detected in the 3-year all-sky survey of Swift/BAT (Ajello et al. 2009, see also Ajello et al. 2012 and Baumgartner et al. 2013), and are amongst the most powerful blazars ever observed.', '1510.08849-2-5-2': 'As with other powerful and high-redshift FSRQs, their optical flux shows contributions due to thermal emission from the accretion disk, particularly prominent in S5 0014+813, whose luminosity reaches [MATH] erg s[MATH] (Ghisellini et al. 2010a).', '1510.08849-2-5-3': 'For both sources, the Swift/BAT spectrum together with the Fermi/LAT upper limit already constrained the location of the high energy peak, but with a relatively large uncertainty given the poor spectral slope determination of Swift/BAT.', '1510.08849-2-5-4': 'This motivated the NuSTAR observations.', '1510.08849-2-6-0': 'In this work, we adopt a flat cosmology with [MATH] km s[MATH] Mpc[MATH] and [MATH], as found by Planck Collaboration XIII (2015).', '1510.08849-2-7-0': '# Observations and Data Analysis', '1510.08849-2-8-0': 'We performed simultaneous NuSTAR and Swift observations in two observing periods for each source.', '1510.08849-2-8-1': 'This section describes the data analysis performed on these new X-ray and optical-UV data.', '1510.08849-2-8-2': 'Along with the new data sets, we consider archival data for the overall SED modeling (see [REF] and [REF]).', '1510.08849-2-8-3': 'Specifically, S5 0014+813 radio and IR data are from Ghisellini et al. (2009), integrated with new IR photometry from the Wide-field Infrared Explorer (WISE; Wright et al. 2010).', '1510.08849-2-8-4': 'In the case of B0222+185, archival data were all retrieved through the ASI Science Data Center (ASDC).', '1510.08849-2-8-5': 'Both sources have not been detected by Fermi/LAT, but we obtained some information from this lack of detection with an "upper limit" on their [MATH]-ray fluxes.', '1510.08849-2-8-6': 'The sensitivity limit at 5[MATH] estimated on 5 years of observations gives a good constrain on the high-energy emission of S5 0014+813 and B0222+185.', '1510.08849-2-9-0': '## NuSTAR observations', '1510.08849-2-10-0': 'The NuSTAR satellite observed S5 0014+81 on 2014 December 21 (obsID 60001098002) and on 2015 January 23 (obsID 60001098004) for total net exposure times of 31.0 ks and 36.4 ks, respectively.', '1510.08849-2-10-1': 'B0222+185 was observed by NuSTAR on 2014 December 24 (obsID 60001101002) and on 2015 January 18 (obsID 60001101004).', '1510.08849-2-10-2': 'The total net exposure times were 32.0 ks and 37.4 ks, respectively.', '1510.08849-2-11-0': 'The Focal Plane Module A (FPMA) and Focal Plane Module B (FPMB) data sets were first processed with the NuSTARDAS software package (v.1.4.1) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (Caltech, USA).', '1510.08849-2-11-1': 'Event files were calibrated and cleaned with standard filtering criteria using the nupipeline task (version 20150316) of the NuSTAR CALDB.', '1510.08849-2-12-0': 'The two sources were well detected in the NuSTAR 3-79 keV energy band.', '1510.08849-2-12-1': 'In both cases the FPMA and FPMB spectra of the target were extracted from the cleaned event files using a circle of 20 pixel ([MATH] arcsec) radius, while the background was extracted from nearby circular regions of 40 pixel radius.', '1510.08849-2-12-2': 'The ancillary response files were generated with the numkarf task, applying corrections for the PSF losses, exposure maps and vignetting.', '1510.08849-2-12-3': 'All spectra were binned to ensure a minimum of 30 counts per bin.', '1510.08849-2-13-0': '## Swift observations', '1510.08849-2-14-0': 'The Swift satellite observed S5 0014+81 on 2014 December 21 (obsID 00080003001) and on 2015 January 23 (obsID 00080003002) while B0222+185 was observed on 2014 December 24 (obsID 00080243001) and on 2015 January 18 (obsID 00080243002).', '1510.08849-2-14-1': 'The total net exposure times were 6.5 ks (December 2014) and 6.6 ks (January 2015) for S5 0014+81 and 4.9 ks (December 2014) and 5.1 ks (January 2015) for B0222+185.', '1510.08849-2-15-0': '### XRT observations', '1510.08849-2-16-0': 'Swift/XRT (Burrows et al. 2005) observations were carried out using the Photon Counting (PC) CCD readout mode and in the four observations the sources were well detected in the 0.3-10 keV XRT energy band.', '1510.08849-2-16-1': 'The XRT data sets were first processed with the XRTDAS software package (v.3.0.0) developed at the ASI Science Data Center (ASDC) and distributed by HEASARC within the HEASoft package (v. 6.16).', '1510.08849-2-16-2': 'In particular, event files were calibrated and cleaned with standard filtering criteria with the xrtpipeline task using the calibration files available in the version 20140709 of the Swift/XRT CALDB.', '1510.08849-2-17-0': 'The energy spectra were then extracted from the calibrated and cleaned event files.', '1510.08849-2-17-1': 'Events for the source spectral analysis were selected within a circle of 20 pixel ([MATH] arcsec) radius, enclosing about 90% of the PSF, while the background was extracted from a nearby circular region of 80 pixel radius.', '1510.08849-2-17-2': 'The ancillary response files were generated with the xrtmkarf task, applying corrections for the PSF losses and CCD defects using the cumulative exposure map.', '1510.08849-2-17-3': 'The source spectra were binned to ensure a minimum of 30 counts per bin.', '1510.08849-2-18-0': '### UVOT observations', '1510.08849-2-19-0': 'Swift/UVOT (Roming et al. 2005) observations were performed with all six optical and UV lenticular filters (namely [MATH]).', '1510.08849-2-19-1': 'We performed aperture photometry for all filters in all the observations using the standard UVOT software distributed within the HEAsoft package (version 6.16) and the calibration included in the latest release of the CALDB.', '1510.08849-2-19-2': 'Counts were extracted from apertures of 5 arcsec radius for all filters and converted to fluxes using the standard zero points (Poole et al. 2008).', '1510.08849-2-19-3': 'The fluxes were then de-reddened using the appropriate values of [MATH] taken from Schlegel et al. (1998) and Schlafly et al. (2011) with [MATH] ratios calculated for UVOT filters using the mean Galactic interstellar extinction curve from Fitzpatrick (1999).', '1510.08849-2-19-4': 'No variability was detected within single exposures in any filter.', '1510.08849-2-19-5': 'The processing results were carefully verified, checking for possible contaminations from nearby objects within the source apertures and from objects falling within the background apertures.', '1510.08849-2-20-0': '## X-ray spectral analysis', '1510.08849-2-21-0': 'The spectral analysis of the December 2014 and January 2015 NuSTAR and Swift/XRT simultaneous observations of S5 0014+81 and B0222+185 were performed using the XSPEC package.', '1510.08849-2-21-1': 'In all four observations a broken power-law model with an absorption hydrogen-equivalent column density fixed to the Galactic value of [MATH] cm[MATH] (S5 0014+81) and [MATH] cm[MATH] (B0222+185) (Kalberla et al. 2005) was found to provide a good description of the observed spectra in the 0.3-79 keV energy band.', '1510.08849-2-21-2': 'The inter-calibration factors between the three instruments (NuSTAR/FPMA, NuSTAR/FPMB and Swift/XRT) were taken into account adding a multiplicative constant (kept to 1 for NuSTAR/FPMA) to the spectral model.', '1510.08849-2-21-3': 'We found values in the 2% range for NuSTAR/FPMB and in the 10% range for Swift/XRT which are consistent with the cross-calibration uncertainties for the instruments (Madsen et al. 2015).', '1510.08849-2-21-4': 'The results of the spectral fits are shown in Table [REF], and Figure [REF] shows the X-ray spectra of the two sources.', '1510.08849-2-22-0': 'These results describe intrinsic broken power-laws, not consistent with absorption.', '1510.08849-2-22-1': 'We tested different spectral models, namely single power-laws with an absorption hydrogen-equivalent column density fixed to the Galactic value or left free to vary.', '1510.08849-2-22-2': 'These were possibilities explored by other authors also for these sources (e.g. Page et al. 2005; Piconcelli Guainazzi 2005; Tavecchio et al. 2007; Eitan Behar 2013).', '1510.08849-2-22-3': 'When we left [MATH] free to vary, the inter-calibration factor between Swift/XRT and NuSTAR/FPMA-FMPB is no more consistent with the cross-calibration uncertainties for the instruments, with values that differ of more than 25%.', '1510.08849-2-22-4': 'The [MATH] associated with this option is significantly higher than the broken power-law option, for all the observations.', '1510.08849-2-22-5': 'We performed a second test by fixing the absorption column density at the Galactic value, in the case of a single power-law.', '1510.08849-2-22-6': 'The inter-calibration factor in this case is even less consistent with the acceptable cross-calibration uncertainties (i.e. [MATH]), along with higher [MATH] values.', '1510.08849-2-22-7': 'The results from these tests confirm the existence of an intrinsic spectral curvature within the observed 0.3-79 keV energy band.', '1510.08849-2-23-0': '# Black hole Mass Estimate', '1510.08849-2-24-0': 'The black hole mass of a blazar is an important feature to characterise it.', '1510.08849-2-24-1': 'When optical or infrared spectra are not available, the virial mass estimate method cannot be applied.', '1510.08849-2-24-2': 'At high redshift, the available lines to apply such a method are also less reliable.', '1510.08849-2-24-3': 'With a good photometric coverage of the UV-optical-IR band, instead, the accretion disk emission can be fitted, and the black hole mass can be estimated from this fitting process (Calderone et al. 2013).', '1510.08849-2-25-0': 'Figure [REF] shows the optical-UV band of the SEDs, along with Swift/UVOT photometric data, archival data from the ASI Science Data Center (ASDC), that cover well the whole band.', '1510.08849-2-25-1': 'First we need to point out that at frequencies higher than [MATH] rest frame a prominent absorption feature is usually present, i.e. the Ly[MATH] forest, due to intervening clouds absorbing hydrogen Ly[MATH] photons at wavelengths shortward of 1216 .', '1510.08849-2-25-2': 'We corrected for the absorption following Ghisellini et al. (2010).', '1510.08849-2-25-3': 'However, the distribution of intervening clouds varies randomly along every line-of-sight.', '1510.08849-2-25-4': 'Only an average correction can be done, and it is not sufficiently reliable when applied on single sources.', '1510.08849-2-25-5': 'For this reason, we do not consider the data point at [MATH] in our modeling.', '1510.08849-2-25-6': 'The bluer UVOT bands fall in this frequency range in the case of S5 0014+318 and B0222+185.', '1510.08849-2-25-7': 'At frequencies lower than this prominent absorption feature, though, a peak in the SEDs is clearly visible.', '1510.08849-2-25-8': 'Below this peak, the optical flux decreases with frequency, suggesting a power-law trend, especially in S5 0014+813.', '1510.08849-2-25-9': 'This is the clear signature of an accretion disk, which can be fitted with a simple Shakura-Sunyaev model (Shakura Sunyaev 1973; Calderone et al. 2013; Sbarrato et al. 2013).', '1510.08849-2-26-0': 'At lower frequencies, another feature is evident from the IR-optical-UV SEDs of these two sources: the WISE IR bands show an increase of the flux, that breaks the power-law-like trend in the optical.', '1510.08849-2-26-1': 'This is likely the signature of the IR emission from a dusty torus around the nucleus.', '1510.08849-2-26-2': 'Such a steep IR spectrum, in fact, could not be produced by synchrotron emission: either a self-absorption frequency larger that [MATH]Hz or a steep thin synchrotron spectrum ending with an exponential cut would be needed to justify such a spectral profile.', '1510.08849-2-26-3': 'Both options would show up with prominent signatures in the high-energy emission of the sources, that we do not observe.', '1510.08849-2-26-4': 'Figure [REF] shows that our models do not perfectly reproduce far-IR data.', '1510.08849-2-26-5': 'In fact our model over-simplifies the torus emission: we describe it as a black body emission, while there is evidence (see e.g. Calderone, Sbarrato Ghisellini 2012) that it is best represented as a multi-temperature structure.', '1510.08849-2-26-6': 'The hottest part, closer to the accretion disk, has likely a temperature [MATH]K, i.e. of the order of the dust sublimation temperature.', '1510.08849-2-26-7': 'Our far-IR data, in fact, show an increase in flux at frequencies [MATH]Hz, that roughly corresponds to these range of temperatures.', '1510.08849-2-27-0': 'With these premises, a reliable way to estimate the black hole mass of S5 0014+318 and B0222+185 is to fit their IR-optical-UV SEDs with a simple model of accretion disk emission.', '1510.08849-2-27-1': 'We applied the radiatively efficient, geometrically thin, optically thick Shakura Sunyaev (1973) model.', '1510.08849-2-27-2': 'Assuming a standard radiative efficiency [MATH], only two free parameters are left to be fitted: the accretion rate [MATH], that can be traced by the intrinsic disk luminosity [MATH], and the black hole mass [MATH] itself.', '1510.08849-2-27-3': 'In the case of S5 0014+318, we can constrain the overall disk luminosity thanks to the visibility of the peak of the disk emission, with some consideration regarding its anisotropic properties (as thoroughly explained by Calderone et al. 2013):', '1510.08849-2-28-0': 'according to the Shakura-Sunyaev model, the peak luminosity [MATH] corresponds to half the total observed luminosity [MATH].', '1510.08849-2-29-0': 'the observed luminosity depends on the viewing angle of the accretion disk: [EQUATION] where [MATH] is the intrinsic total luminosity emitted by the accretion disk.', '1510.08849-2-29-1': 'In the case of a blazar [MATH] since we see the accretion disk face-on.', '1510.08849-2-30-0': 'We can therefore derive the intrinsic total luminosity from the peak luminosity of our sources: [EQUATION]', '1510.08849-2-30-1': 'This means that in the case of S5 0014+318, [MATH] is constrained by observations (i.e. 2MASS and UVOT data), and only [MATH] is left as a free parameter to be derived with the IR-optical-UV SED fitting.', '1510.08849-2-31-0': 'We find that both sources have large black hole masses and are fast accreting, even if not super-Eddington.', '1510.08849-2-31-1': 'We derive [MATH] and [MATH] for B0222+185 and [MATH] and [MATH]erg s[MATH] for S5 0014+813.', '1510.08849-2-31-2': 'These values are significantly smaller than what was derived in Ghisellini et al. (2009 and 2010a).', '1510.08849-2-31-3': 'For S5 0014+813 the reason is due to i) the better coverage of the IR band achieved with WISE data and ii) neglecting the optical data taken from Bechtold et al. (1994).', '1510.08849-2-31-4': 'We now prefer to discard those data because the derivation of flux and luminosities are not sufficiently clear in that paper.', '1510.08849-2-31-5': 'The confidence range of S5 0014+813 mass ([MATH]) is indicated by the dashed lines in the left panel of Fig. [REF].', '1510.08849-2-31-6': 'Note that it strongly depends on data quality.', '1510.08849-2-31-7': 'In this case, the range is rather narrow because of very good data.', '1510.08849-2-31-8': 'More precise data would lead to even more refined estimates.', '1510.08849-2-31-9': 'A lower limit on the mass is anyway fixed by the strong constraint given by the Eddington limit.', '1510.08849-2-32-0': 'For B0222+185, the smaller values of [MATH] and [MATH] are due to the new infrared data (not available in the previous work), that now help in roughly constraining the peak frequency of the disk emission.', '1510.08849-2-32-1': 'The data coverage in this case is not enough to constrain the peak frequency luminosity, therefore the estimate on the black hole mass is less constrained.', '1510.08849-2-32-2': 'This should be taken as an indication of [MATH], not as a best fit.', '1510.08849-2-33-0': 'A side result of [MATH] and [MATH] studies is an estimate of the broad line region covering factor with respect to the accretion disc [MATH].', '1510.08849-2-33-1': 'The BLR is thought to reprocess a fraction [MATH] of the radiation emitted from the disk, thus usually a standard value [MATH] is used.', '1510.08849-2-33-2': 'When BLR and disk luminosities are obtained independently, the BLR covering factor can be derived, and this is the case.', '1510.08849-2-33-3': 'Cao Jiang (1999) derived [MATH] for S5 0014+813.', '1510.08849-2-33-4': 'By comparing it with our result, we obtain a covering factor [MATH].', '1510.08849-2-34-0': '# Modelling the broad-band SED', '1510.08849-2-35-0': 'Figure [REF] shows that both S5 0014+318 and B0222+185 have overall SEDs characterised by a prominent high-energy component, that along with the characteristic flat and intense radio luminosity is attributed to non-thermal emission from a relativistic jet.', '1510.08849-2-35-1': 'In the IR-optical-UV range of both sources, the SEDs are dominated by thermal emission attributed to the accretion disk, as discussed in Section [REF].', '1510.08849-2-36-0': 'Not being detected in the [MATH]-rays by Fermi/LAT, X-ray data are necessary to study the non-thermal high-energy emission of S5 0014+318 and B0222+185.', '1510.08849-2-36-1': 'Specifically, NuSTAR data are crucial for understanding the X-ray spectral profile and possible variability in this kind of high-redshift source, as can be seen in Figure [REF].', '1510.08849-2-36-2': 'X-ray data contribute significantly to the modelling of the broad-band SEDs of the two sources (Figure [REF]).', '1510.08849-2-37-0': 'To interpret the SEDs of the two sources, we used a leptonic one-zone emitting model, fully described in Ghisellini Tavecchio (2009).', '1510.08849-2-37-1': 'We refer to the original paper for details, providing here only a very brief description of the most important features of the models.', '1510.08849-2-37-2': 'The emitting source is assumed to be a spherical region in which relativistic electrons emit by synchrotron and inverse Compton processes.', '1510.08849-2-37-3': 'This homogeneous spherical blob is assumed to be located at a distance [MATH] from the central black hole, moving with a bulk Lorentz factor [MATH] at an angle [MATH] from our line-of-sight.', '1510.08849-2-37-4': 'Relativistic electrons are injected throughout the source, with a power [MATH] as measured in the comoving frame.', '1510.08849-2-37-5': 'The energy distribution [MATH] of the injected electrons is a smoothly broken power law with slopes [MATH] and [MATH] (defined as [MATH]) below and above the random Lorentz factor [MATH].', '1510.08849-2-37-6': 'Note that, even if [MATH] is a broken power law, the particle energy distribution [MATH] derived through the continuity equation maintains a break, albeit smoother than the injected broken power law.', '1510.08849-2-37-7': 'This produces a gently curved spectrum, as shown in Figure 1.', '1510.08849-2-37-8': 'The broad line region is located at a distance [MATH] from the black hole, while the infrared emitting torus is at [MATH].', '1510.08849-2-37-9': '[MATH] is the accretion disk luminosity in units of [MATH], and it is derived as in Section [REF], together with the central black hole mass.', '1510.08849-2-37-10': 'The values of the parameters adopted for the models are reported in Table [REF].', '1510.08849-2-37-11': 'Note that the model we apply is very sensitive to changes in the derived parameters.', '1510.08849-2-37-12': 'The emission profile and intensity reproduced by the SED fitting change significantly even after small parameter variations, as shown for small variations in the viewing angle in Fig. 3 of Sbarrato et al. (2015).', '1510.08849-2-38-0': 'Table [REF] reports the different forms of the power carried by the jet: the power [MATH] spent in producing the radiation we observe, the Poynting flux [MATH], the power associated to the bulk motion of relativistic electrons ([MATH]) and cold protons ([MATH]), assuming one proton per relativistic electron.', '1510.08849-2-38-1': 'This assumption is consistent with independent results on blazar and GRB jets by Nemmen et al. (2012).', '1510.08849-2-38-2': 'They found that the total jet power for both classes is ten times the radiative power [MATH], i.e. similar to what we find in this work (see Table [REF]).', '1510.08849-2-38-3': 'Different proton-to-relativistic electron ratios were explored by Sikora Madejski (2000), who found that the relativistic pairs must be less than 10-20 per proton.', '1510.08849-2-38-4': 'With this combination, the total jet power can result equal to or even less than the radiative power, that instead is only a part of the total power carried by the jet, and hence should be a lower limit to the total [MATH] (Ghisellini 2012; Ghisellini et al. 2014).', '1510.08849-2-38-5': 'Therefore, assuming one proton per relativistic electron is reasonable to explain the observed jet features and its physics.', '1510.08849-2-39-0': 'The emitting regions of both sources are located within the broad line region (and the infrared torus).', '1510.08849-2-39-1': 'In this way, the energy density of photons from the broad line region feed the inverse Compton process, together with photons from the torus.', '1510.08849-2-39-2': 'The inverse Compton process is dominated by external Compton instead of synchrotron-self Compton, as expected in FSRQs.', '1510.08849-2-39-3': 'Tagliaferri et al. (2015) obtained the same result for two other [MATH] blazars observed by NuSTAR: the emitting regions of both S5 0836+710 and PKS 2149-306 are located between the BLR and IR torus.', '1510.08849-2-39-4': 'Their results were obtained through SED fitting, and were also confirmed on the basis of the variability timescales obtained with two NuSTAR observations per source.', '1510.08849-2-40-0': '# Discussion', '1510.08849-2-41-0': 'Blazars are characterised by their prominent relativistically boosted jet emission.', '1510.08849-2-41-1': 'They usually show prominent high-energy emission, which results in high [MATH]-ray luminosities, well detected by instruments like Fermi/LAT.', '1510.08849-2-41-2': 'In some cases, though, blazars are not detected in such energy bands.', '1510.08849-2-41-3': 'This is the case for S5 0014+318 and B0222+185.', '1510.08849-2-41-4': 'Even if lacking a high-energy detection, the Compton bump can be observed in the X-ray frequency range, but a detection in the soft X-rays usually is not enough to determine the relativistic jet features of a blazar, nor its orientation.', '1510.08849-2-41-5': 'S5 0014+318 and B0222+185 were detected by Swift/BAT, but these data were not precise enough to derive exact estimates of the bulk Lorentz factors and viewing angles.', '1510.08849-2-41-6': 'Figures [REF] and [REF] show that the Swift/BAT data do not have enough precision to constrain the hard X-ray slope.', '1510.08849-2-41-7': 'NuSTAR, however, provides a broad-band, precise measurement for both sources, confirming that both blazars are seen at small viewing angles, i.e. their jets are directed along our line of sight.', '1510.08849-2-41-8': 'Both sources also host massive central black holes (both with [MATH]).', '1510.08849-2-42-0': '## Variability', '1510.08849-2-43-0': 'Looking in detail at the broad-band SEDs of these two objects, we see some interesting differences.', '1510.08849-2-43-1': 'Both sources show the two humps, i.e. the signature of aligned jet emission, while in the IR-optical-UV band, the accretion disk emission dominates over the non-thermal jet emission.', '1510.08849-2-43-2': 'Comparing the two panels of Figure [REF], it can be noticed that S5 0014+318 and B0222+185 also show different variability behaviours.', '1510.08849-2-43-3': 'They were both observed in two epochs separated by [MATH] month.', '1510.08849-2-43-4': 'S5 0014+318 does not show flux or spectral variation between the two observations, and the new data are consistent with archival data.', '1510.08849-2-43-5': 'Only the Swift/BAT detection could suggest a different state of the source, but due to the large uncertainty we cannot draw any strong conclusion.', '1510.08849-2-44-0': 'B0222+185, instead, shows a clear variation of the X-ray flux between the two Swift/XRT + NuSTAR observations: in December 2014 the source was in a higher state, compared to both January 2015 and archival data.', '1510.08849-2-44-1': 'The different hard X-ray spectra showed in the two B0222+185 observations (see right panel of Figure [REF]) suggest that the peak of the high-energy hump is at the same or higher frequency when in the higher state, compared to the lower state.', '1510.08849-2-44-2': 'This would be opposite to the general trend displayed by the blazar sequence, but missing a higher-frequency detection, this speculation is not conclusive.', '1510.08849-2-44-3': 'If real, such behaviour would not be uncommon in rapidly varying FSRQs: although they follow the blazar sequence when considering different sources, an individual object can behave opposite to the sequence itself while varying.', '1510.08849-2-44-4': 'According to the model shown in Figure [REF], B0222+185 variability can be described by a variation in the injected power (see second and third lines of Table [REF]), accompanied by [MATH] increasing in the high state.', '1510.08849-2-45-0': 'Another remarkable example of this kind of variation, very similar to the one showed by B0222+185, but much more pronounced, has recently been seen for S5 0836+710 (Ciprini et al. 2015; Vercellone et al. 2015; Giroletti et al. 2015) during its August 2015 [MATH]-ray flare, which triggered observations at X-ray and radio frequencies.', '1510.08849-2-45-1': 'The amplitude of the flux variation was huge in the Fermi/LAT band (factor 65 greater than the average flux reported in the third Fermi/LAT catalog of Acero et al. 2015) and rather modest in the high energy part of the Swift/XRT band.', '1510.08849-2-45-2': 'This implies that the X-ray flux had to change more at larger energies, to connect to the enhanced [MATH]-ray flux, and that the peak frequency of the high energy hump must be "bluer" than what was displayed during the NuSTAR observation described in Tagliaferri et al. (2015).', '1510.08849-2-45-3': 'In other words: if NuSTAR had followed the August 2015 flare of S5 0836+710, it likely would have detected a clear flux and spectral variation, leading to a predicted shift of the high energy hump towards higher frequencies even in the absence of [MATH]-ray data.', '1510.08849-2-45-4': 'The right panel of Figure [REF] shows that Swift/XRT would not have been able to discriminate between the two states of B0222+185, while NuSTAR distinguishes them clearly.', '1510.08849-2-45-5': 'We conclude that:', '1510.08849-2-46-0': '## Jets and accretion of the two most powerful blazars', '1510.08849-2-47-0': 'We now aim to frame S5 0014+318 and B0222+185 within the larger blazar picture.', '1510.08849-2-47-1': 'We consider them in the jet-accretion correlation scenario.', '1510.08849-2-47-2': 'Ghisellini et al. (2014) found that in blazars the jet power not only correlates with the accretion power, but it is even larger.', '1510.08849-2-47-3': 'This suggests that accretion is strongly related to jet power, implying a role in jet production.', '1510.08849-2-47-4': 'At the same time, the fact that jet power is larger than accretion power tells us that some other process must play a role in the jet launch and acceleration.', '1510.08849-2-47-5': 'Black hole spin is the best candidate to play such a role.', '1510.08849-2-47-6': 'This result was obtained by studying a sample of Fermi-detected blazars, for which Shaw et al. (2012; 2013) obtained optical spectra.', '1510.08849-2-47-7': 'Ghisellini et al. (2014) selected all the objects with broad emission lines, in order to have a proxy of accretion luminosity, and compared jet and accretion power for the 226 blazars in this sample.', '1510.08849-2-47-8': 'However, this sample did not include the most extreme blazars known, leaving open the questions:', '1510.08849-2-48-0': 'how does the jet-accretion relation look in the case of the most powerful blazars?', '1510.08849-2-49-0': 'Does the power balance change when accretion or jet emission are extreme?', '1510.08849-2-49-1': 'These questions will guide us in the following discussion.', '1510.08849-2-50-0': 'First we add to the original blazar sample the sources expected to be the most powerful.', '1510.08849-2-50-1': 'To this aim we select the [MATH] blazars detected by Swift/BAT and all known high-redshift ([MATH]) blazars.', '1510.08849-2-50-2': 'The BAT sensitivity limit is not very deep, and at high redshift it can detect mainly the most powerful sources, whose high-energy components peak in the [MATH]MeV range.', '1510.08849-2-50-3': 'BAT detected 10 [MATH] blazars, including S5 0014+318 and B0222+185, that we add to the blazar sample of Ghisellini et al. (2014).', '1510.08849-2-50-4': 'We also include all the known blazars at [MATH], as listed in Ghisellini et al. (2015).', '1510.08849-2-50-5': 'Being the highest redshift blazars currently known, they are expected to be among the most powerful blazars.', '1510.08849-2-50-6': 'They are not present in the BAT blazar catalog because their distance makes their hard X-ray flux too weak for a detection with BAT.', '1510.08849-2-50-7': 'Since most of them were selected starting from optical catalogs, they are likely very powerful in accretion luminosity.', '1510.08849-2-51-0': 'Figure [REF] shows how these samples are located in the overall jet-accretion relation, along with S5 0014+318 and B0222+185.', '1510.08849-2-51-1': 'The total jet power (calculated as the sum of different jet power components listed in Table [REF]) is plotted as a function of the disk luminosity.', '1510.08849-2-51-2': 'The grey stripes show the best fit of the sample by Ghisellini et al. (2014).', '1510.08849-2-51-3': 'Note that the powerful blazars we add in this work are all located within the 2[MATH] dispersion of the previous correlation.', '1510.08849-2-51-4': 'This means that they still follow the jet-accretion relation found by Ghisellini et al. (2014), even if they are among the most powerful sources in the set.', '1510.08849-2-52-0': 'S5 0014+318 and B0222+185 can be considered the two most extreme sources: respectively, they are the blazars with the most luminous disk and the most powerful jet.', '1510.08849-2-52-1': 'Still, they are close enough to the known jet-accretion correlation, to be less than 2[MATH] from the Ghisellini et al. (2014) result.', '1510.08849-2-52-2': 'Thus we conclude that even the most powerful blazars follow the same jet-accretion relation as the [MATH]-ray detected bulk sample.', '1510.08849-2-52-3': 'The second interesting conclusion that we can draw from this comparison is that NuSTAR is once again confirmed to be the most suitable telescope to study the most powerful blazars in our Universe.', '1510.08849-2-53-0': '# Conclusions', '1510.08849-2-54-0': 'The simultaneous X-ray observations of S5 0014+318 and B0222+185 performed with Swift/XRT and NuSTAR gave us an interesting insight into the jet emissions of these two sources.', '1510.08849-2-54-1': 'We confirmed their blazar nature, with a refined estimate of their bulk Lorentz factors and viewing angles, supported by more precise sets of parameters (Tables [REF] and [REF]).', '1510.08849-2-54-2': 'The accretion disk fitting to a more complete data set gave us the possibility to refine our previous estimates of the black hole mass and accretion luminosity of these two sources, implying fast accreting objects with extreme masses of [MATH].', '1510.08849-2-55-0': 'The overall SED modelling allowed us to estimate the jet power and accretion disk luminosity, allowing a comparison of these two sources with the overall blazar jet-accretion relation.', '1510.08849-2-55-1': 'The two sources are among the most powerful blazars known, and they populate the highest disk luminosity and jet power part of the jet-accretion correlation (Ghisellini et al. 2014; Figure [REF]).', '1510.08849-2-55-2': 'It is remarkable that a sample formed by the most powerful blazars known is still within [MATH] from the correlation derived from a sample of blazars whose [MATH]-ray flux was averaged over two years.', '1510.08849-2-55-3': 'The mechanisms governing the jet formation and evolution in the most extreme sources must not be different from the processes powering the more moderate objects.', '1510.08849-2-55-4': 'S5 0014+318 and B0222+185 themselves are consistent with the relation derived from the [MATH]-ray blazars, even if they are the blazars with the most luminous accretion disk and the most powerful jet, respectively.', '1510.08849-2-56-0': 'We found a different variability behaviour between the two sources: while S5 0014+318 did not vary in the two observation epochs separated by [MATH] month, B0222+185 shows a clear variation, with an amplitude larger at larger frequencies.', '1510.08849-2-56-1': 'This last feature is the main reason why NuSTAR is such a crucial instrument to study high-redshift and powerful blazars.', '1510.08849-2-56-2': 'A soft X-ray telescope alone could not see a variability event like the one shown by B0222+185: NuSTAR instead observes at frequencies where the amplitude of a flaring activity is large enough to be seen.', '1510.08849-2-56-3': 'Were we lacking the [MATH]-ray signature of a flare, NuSTAR could do the job.'}

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'1510.08849-2-38-5'], ['1510.08849-1-2-0', '1510.08849-2-2-0'], ['1510.08849-1-2-1', '1510.08849-2-2-1'], ['1510.08849-1-2-4', '1510.08849-2-2-4'], ['1510.08849-1-2-5', '1510.08849-2-2-5'], ['1510.08849-1-2-6', '1510.08849-2-2-6'], ['1510.08849-1-16-0', '1510.08849-2-17-0'], ['1510.08849-1-16-1', '1510.08849-2-17-1'], ['1510.08849-1-16-2', '1510.08849-2-17-2'], ['1510.08849-1-16-3', '1510.08849-2-17-3'], ['1510.08849-1-24-4', '1510.08849-2-25-7'], ['1510.08849-1-24-5', '1510.08849-2-25-8'], ['1510.08849-1-24-6', '1510.08849-2-25-9'], ['1510.08849-1-24-8', '1510.08849-2-26-1'], ['1510.08849-1-41-0', '1510.08849-2-44-0']]

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[]

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[]

['1510.08849-1-5-0', '1510.08849-1-9-0', '1510.08849-1-9-1', '1510.08849-1-13-0', '1510.08849-1-13-1', '1510.08849-1-25-3', '1510.08849-1-29-1', '1510.08849-1-38-3', '1510.08849-1-43-5', '1510.08849-1-45-8', '1510.08849-2-5-0', '1510.08849-2-10-0', '1510.08849-2-10-1', '1510.08849-2-14-0', '1510.08849-2-14-1', '1510.08849-2-27-3', '1510.08849-2-31-1', '1510.08849-2-33-3', '1510.08849-2-41-3', '1510.08849-2-45-5', '1510.08849-2-47-8']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1510.08849

1610.05667

{'1610.05667-1-0-0': 'A plastic scintillator bar with dimensions 3 m [MATH] 2.5 cm [MATH] 11 cm was exposed to a focused muon beam to study its light yield and timing characteristics as a function of position and angle of incidence.', '1610.05667-1-0-1': 'The scintillating light was read out at both ends by photomultiplier tubes whose pulse shapes were recorded by waveform digitizers.', '1610.05667-1-0-2': 'Results obtained with the WAVECATCHER and SAMPIC digitizers are analyzed and compared.', '1610.05667-1-0-3': 'A comprehensive discussion of the various factors affecting the timing resolution is presented.', '1610.05667-1-0-4': 'Prospects for applications of plastic scintillator technology in large-scale particle physics detectors with timing resolution around 100 ps are provided in light of the results.', '1610.05667-1-1-0': '# Introduction', '1610.05667-1-2-0': 'Plastic scintillator detectors have been extensively used in particle physics experiments since decades.', '1610.05667-1-2-1': 'In large-scale experiments, they typically represent an array of bars covering a large surface which can provide a fast trigger signal or particle identification using time-of-flight.', '1610.05667-1-2-2': 'Depending on the bar dimensions, scintillator type, and light readout sensor the time resolution for such detectors typically ranges from 50 ps (50 cm bars of the ToF system of MICE [CITATION]) to 350 ps (6.8 m bars of the ToF system of OPAL [CITATION]).', '1610.05667-1-3-0': 'As a matter of actual practice, counters which are made of a bulk scintillator do not exceed 3 m in length.', '1610.05667-1-3-1': 'This restriction comes naturally from a signal attenuation and an uncertainty related to the dispersion of photon path lengths which gets dominant for long counters.', '1610.05667-1-3-2': 'Moreover this uncertainty grows exponentially with a decrease of the thickness of a bar [CITATION].', '1610.05667-1-3-3': 'It makes a bar cross section close to a square shape advantageous in detectors [CITATION].', '1610.05667-1-3-4': 'However, when a detector covers a large surface, for reasons of economy, the thickness of a counter along the beam is often chosen to be significantly smaller than its width.', '1610.05667-1-3-5': 'In this case the thickness becomes a limiting factor for the precision time measurement.', '1610.05667-1-3-6': 'Recent examples of detectors using this type of bars are the trigger hodoscopes system in COMPASS [CITATION] and the NA61/SHINE ToF detector [CITATION].', '1610.05667-1-3-7': 'Another example of a detector which combines the requirements of a large covered surface and an excellent time resolution is the timing detector of the proposed SHiP experiment at the CERN SPS [CITATION].', '1610.05667-1-3-8': 'To efficiently distinguish between vertices from random muon crossings and genuine particle decays, the SHiP timing detector needs to cover a 6 m [MATH] 12 m area with a time resolution of 100 ps or better [CITATION] at an affordable price, which is a challenge.', '1610.05667-1-3-9': 'One option proposed in the SHiP technical proposal is an array of 3 m long plastic scintillator bars read out by photomultiplier tubes (PMTs) [CITATION].', '1610.05667-1-3-10': 'Another feature of SHiP is a software trigger running on an online computer farm, thus favoring the use of a DAQ electronics which has the particularity to tolerate relatively high event rates and at the same time allow for each channel to operate in a self-triggering mode.', '1610.05667-1-4-0': 'Novel types of acquisition electronics which perform a waveform sampling using a switched capacitor array (SCA) are employed since recently in particle physics experiments.', '1610.05667-1-4-1': 'The use of an analog memory which is added in parallel with a delay line allows to sample an analog signal at very high rate.', '1610.05667-1-4-2': 'In addition, having the waveform recorded, one can extract various kinds of information like baseline, amplitude, charge and time.', '1610.05667-1-4-3': 'The measurements presented in this article with a 3 m bar were made with the two acquisition modules WAVECATCHER [CITATION] and SAMPIC [CITATION].', '1610.05667-1-4-4': 'Both digitizers were used at the 3.2 GS/s sampling rate.', '1610.05667-1-4-5': 'The circular buffer of WAVECATCHER contains 1024 cells what makes possible to cover a 320 ns time window, thus to record the full signal coming from PMT.', '1610.05667-1-4-6': 'The WAVECATCHER measurements are therefore used as a reference.', '1610.05667-1-4-7': 'The SAMPIC, on the other hand, has a short buffer of 64 cells for a 20 ns window.', '1610.05667-1-4-8': 'Combined with an on-chip ADC, this allows though to work with rates two orders of magnitude higher than WAVECATCHER, making it attractive for applications such as the SHiP timing detector.', '1610.05667-1-4-9': 'The test-bench used here can thus be considered as a prototype for the conservative design of the timing detector of the SHiP experiment described in the technical proposal [CITATION].', '1610.05667-1-5-0': 'The article is organized as follows.', '1610.05667-1-5-1': 'The experimental setup is described in Section [REF].', '1610.05667-1-5-2': 'Section [REF] provides a detailed description of the DAQ electronics.', '1610.05667-1-5-3': 'The analysis procedure is presented in Section [REF].', '1610.05667-1-5-4': 'The results of the measurements are discussed in Section [REF].', '1610.05667-1-5-5': 'Finally, a summary is given in Section [REF].', '1610.05667-1-6-0': '# Experimental setup', '1610.05667-1-7-0': 'We present results of test-beam measurements which took place at the CERN PS in June 2016.', '1610.05667-1-7-1': 'The layout of the setup is shown in Fig. [REF].', '1610.05667-1-7-2': 'The coordinate system is chosen such that the [MATH] axis is directed along the beam, the [MATH] axis is along the bar and the [MATH] axis is directed vertically in such a way that the coordinate system is right-handed.', '1610.05667-1-7-3': 'The origin of the system is at the left side of the bar, in the center of the cross section.', '1610.05667-1-8-0': '## Plastic bar and PMTs', '1610.05667-1-9-0': 'The scintillator counter has been purchased from the SCIONIX Radiation Detector Crystals company [CITATION].', '1610.05667-1-9-1': 'The bar length is 3 m and its transverse cross section is 2.5 cm [MATH] 11 cm.', '1610.05667-1-9-2': 'The two larger surfaces of the bar (3 m [MATH] 11 cm) were in contact with a casting form and had no other preparation.', '1610.05667-1-9-3': 'The four other surfaces were diamond milled.', '1610.05667-1-9-4': 'The choice of plastic was primarily driven by the length of the bar: EJ-200 provides an optimal combination of an optical attenuation length, fast timing and high light output.', '1610.05667-1-9-5': 'The properties of EJ-200 quoted by producer are: a rise time of 0.9 ns; a decay time of 2.1 ns; a bulk attenuation length of 4 m; and a refraction index of 1.58.', '1610.05667-1-9-6': 'The peak in the emission spectrum resides in the violet region of the visible spectrum.', '1610.05667-1-9-7': 'As shown in Fig. [REF], this spectrum is compatible with the sensitivity region of the PMT and the reflection efficiency of an aluminum foil which was used to wrap the bar.', '1610.05667-1-10-0': 'The bar is attached via tapered light guides to 2" phototubes on both ends.', '1610.05667-1-10-1': 'The fast Hamamatsu R13089-10 PMT [CITATION] is chosen because of its good time resolution and moderate cost.', '1610.05667-1-10-2': 'It has a linear-focused dynode structure with 8 stages and a typical anode gain value [MATH].', '1610.05667-1-10-3': 'Its signal output was coupled directly to the acquisition module.', '1610.05667-1-10-4': 'This results in a signal amplitude in the range [MATH] mV at one MIP which fits perfectly the dynamic range of the acquisition modules.', '1610.05667-1-10-5': 'The quantum efficiency of the photocathode as given by the manufacturer is 25% at the peak emission of the scintillator (see Fig. [REF]).', '1610.05667-1-10-6': 'Parameters relevant for the precision time measurements are a rise time of 2 ns and a transit time spread of 230 ps.', '1610.05667-1-11-0': 'We did not use an optical coupling to connect the photocathode of the PMT to the light guide.', '1610.05667-1-11-1': 'It makes possible a thin air gap between them which may reduce the amount of photons at large angles due to total internal reflection.', '1610.05667-1-11-2': 'Furthermore, the cross-sectional area of the bar is larger than the area of the photocathode by about 34%.', '1610.05667-1-11-3': 'Due to phase-space conservation of the photon flux the light output should be reduced by about the same amount, for the case when an interaction took place in the proximity of the PMT.', '1610.05667-1-11-4': 'Both effects described in this paragraph can contribute to light reflection thus enhancing the signal in the PMT at the opposite side of the bar.', '1610.05667-1-12-0': 'The bar and PMTs were fixed to an aluminum frame which could be moved vertically and horizontally with respect to the beam.', '1610.05667-1-13-0': '## Beam and trigger system', '1610.05667-1-14-0': 'Measurements were carried out using a 10 GeV/[MATH] muon beam produced by interactions of 14 GeV/[MATH] protons from the CERN PS accelerator with closed shutters at the T9 beamline of the East Hall.', '1610.05667-1-15-0': 'The trigger was formed by the coincidence of signals from two beam counters installed 50 cm up- and downstream with respect to the bar under test as shown in Fig. [REF].', '1610.05667-1-15-1': 'The counters are shaped as cubes with 2 cm sides made of a fast EJ-228 scintillator with rise and decay constants 0.5 ns and 1.4 ns, respectively.', '1610.05667-1-15-2': 'They were coupled to 1" PMTs (Philips Xp2972) from two sides via 5 cm long light guides.', '1610.05667-1-16-0': 'The trigger time is calculated as an average of the measurements of all four trigger PMTs.', '1610.05667-1-16-1': 'This time is used as a reference for the measurement of the counter under test.', '1610.05667-1-16-2': 'The resolution of the trigger system is determined by subtracting time measurements of up- and downstream counters.', '1610.05667-1-16-3': 'It is found to be 40 ps rms.', '1610.05667-1-16-4': 'This value and the uncertainty associated with a finite size of the beam counters are subtracted in quadrature from the resolution obtained with the counter under test.', '1610.05667-1-17-0': '# DAQ electronics', '1610.05667-1-18-0': 'The major design criterion for the DAQ system is an internal time resolution which has to be much better than the expected resolution of the scintillator counter.', '1610.05667-1-18-1': 'The chosen electronics modules WAVECATCHER [CITATION] and SAMPIC [CITATION] are based on waveform digitizer ASICs which have been developed by LAL and IRFU teams since 1992.', '1610.05667-1-18-2': 'The technology employs a circular buffer, based on arrays of switched capacitors (SCA) which record an analog signal at very high rate.', '1610.05667-1-18-3': 'The time information is extracted with an interpolation of samples in the leading edge of the signal which permits reaching a timing accuracy of about 5 ps rms.', '1610.05667-1-18-4': 'Fast detectors for time-of-flight measurements thus represent a natural target of these digitizers.', '1610.05667-1-18-5': 'In addition they can be used for the pile-up rejection.', '1610.05667-1-19-0': 'In both modules, common clock and trigger signals can be provided externally.', '1610.05667-1-19-1': 'Besides that, each channel also integrates a discriminator that can trigger individually or participate in a more complex trigger.', '1610.05667-1-19-2': 'This capability makes these digitizer modules very suitable for a test bench experiment since an extra electronics for trigger is not required.', '1610.05667-1-19-3': 'The digitizers combine in a single module the functions of TDC and ADC, and permit performing a digital CFD which, in case of WAVECATCHER, is a part of the firmware.', '1610.05667-1-20-0': 'The desktop versions of the acquisition modules have been used.', '1610.05667-1-20-1': 'They house the USB2 interface which permits a connection to PC with a 480 Mbit/s speed.', '1610.05667-1-20-2': 'The acquisition softwares run on PC under Windows saving data directly on disk.', '1610.05667-1-21-0': 'The analog signal sampling in WAVECATCHER is based on the SAMLONG SCA developed since 2010 [CITATION].', '1610.05667-1-21-1': 'The chip was designed with the AMS CMOS 0.35-[MATH]m technology and houses two fully differential channels.', '1610.05667-1-21-2': 'The bandwidth of 500 MHz suits well for detection of very short pulses.', '1610.05667-1-21-3': 'An analog signal is sampled at a rate which can be set between 0.4 and 3.2 GS/s.', '1610.05667-1-21-4': 'Voltages stored in capacitors are further digitized with commercial ADCs at a much lower rate (10 to 20 MHz).', '1610.05667-1-21-5': 'This results in an overall readout dead time of about 120 [MATH]s for the full sampling depth of the circular buffer which comprises 1024 cells.', '1610.05667-1-21-6': 'However one can define for the readout an interval of interest which can be a subset of the whole channel.', '1610.05667-1-21-7': 'The board is DC-coupled with dynamic range of 2.5 V and adjustable offsets are coded over 16 bits.', '1610.05667-1-21-8': 'In the acquisition of data presented in this article the desktop module with 8 channels running at 3.2 GS/s has been used.', '1610.05667-1-22-0': 'The SAMPIC acquisition module is based on the cognominal ASIC designed to be the first TDC directly working on analog signals [CITATION].', '1610.05667-1-22-1': 'SAMPIC was developed in the frame of the RD project aiming at a Waveform and Time to Digital Converter (WTDC) multichannel chip.', '1610.05667-1-22-2': 'It was initially intended to address needs for high precision timing detectors (5 ps rms) required by ATLAS AFP and SuperB FTOF.', '1610.05667-1-23-0': 'SAMPIC is a 16-channel ASIC designed with the AMS CMOS 0.18-[MATH]m technology.', '1610.05667-1-23-1': 'Each channel associates a DLL-based TDC providing a raw time with a 64-cell ultra-fast analog memory sampling up to 10.2 GS/s thus assuring the high resolution timing information.', '1610.05667-1-23-2': 'Analog data are digitized by an on-chip ADC (8 to 11 bits).', '1610.05667-1-23-3': 'The relatively small sampling depth allows for a low deadtime around 1.5 [MATH]s when using the ADC in the 11-bit mode and of the order of 0.2 [MATH]s in the 8-bit mode.', '1610.05667-1-23-4': 'The chip has been designed to offer a signal bandwidth of 1.5 GHz and a usable dynamic range of 1 V.', '1610.05667-1-24-0': 'In the work presented in this paper SAMPIC was exploited at the sampling rate 3.2 GS/s.', '1610.05667-1-24-1': 'The typical length of a signal in the scintillator counter is 50 ns, thus SAMPIC would not be able to record the full signal waveform but the sampling depth of 20 ns is enough to determine the baseline and cover the front edge of the signal.', '1610.05667-1-24-2': 'The sampling rate, in general, can be lowered to 1.6 GS/s thus increasing the time window by a factor of two.', '1610.05667-1-25-0': 'Each channel of SAMPIC integrates a discriminator which, contrary to WAVECATCHER, can trigger itself independently of other channels.', '1610.05667-1-25-1': 'This mode was used in the data taking.', '1610.05667-1-25-2': 'Coincidence of reference counters has been checked offline and selects about 1% of recorded events.', '1610.05667-1-25-3': 'The rest of data represents a muon pile-up and cosmics.', '1610.05667-1-26-0': '# Analysis', '1610.05667-1-27-0': 'The time resolution of the counter is studied using either WAVECATCHER or SAMPIC as a function of the position of the signal in the bar (both longitudinally and transversely) and angle of incidence of the particle trajectory (with respect to both horizontal and vertical planes).', '1610.05667-1-27-1': 'The counter was exposed to the muon beam to collect about two thousand triggers for every point, after which the bar was manually either shifted or rotated to the next position.', '1610.05667-1-27-2': 'Measurements were done for 15 points along the [MATH]-axis of the bar .', '1610.05667-1-27-3': 'For each of these, two measurements were made, one at [MATH] cm, and one at the edge, [MATH] cm.', '1610.05667-1-27-4': 'Rotations were performed only for a beam position at the center of the bar, with 8 different angles between 40[MATH] and 90[MATH] in the horizontal plane and three different angles between 60[MATH] and 90[MATH] in the vertical plane.', '1610.05667-1-28-0': 'There are several techniques for the extraction of time from the sampled waveforms [CITATION].', '1610.05667-1-28-1': 'However it was shown that results obtained with the digital constant fraction discrimination (dCFD) technique provides the best time resolution [CITATION].', '1610.05667-1-28-2': 'In the dCFD approach the signal time is determined by the crossing point of the interpolated digitized signal at the threshold which, in turn, is a constant fraction of the pulse amplitude.', '1610.05667-1-29-0': 'The variance of time measurements can be expressed as [CITATION] [EQUATION] where [MATH] is the variance of the measured voltage and [MATH] is the slope of the rising edge of a signal.', '1610.05667-1-29-1': '[MATH] is an uncertainty due to the limited precision of measurements (for instance the TDC jitter) which is independent of a signal.', '1610.05667-1-30-0': 'The waveforms of a typical signal detected by PMT1 and PMT2 at [MATH] cm and recorded by WAVECATCHER are shown in Fig. [REF].', '1610.05667-1-30-1': 'Two methods are applied to determine the crossing point of dCFD.', '1610.05667-1-30-2': 'In the first method the linear interpolation between two neighboring samples is utilized.', '1610.05667-1-30-3': 'In the second method an analytic function is used to fit the part of the waveform which includes eight consecutive samples.', '1610.05667-1-30-4': 'The second method can give an advantage in case of a long propagation of a signal along the bar, leading to a smaller number of detected photons.', '1610.05667-1-30-5': 'Indeed, if [MATH] is dominated by statistical fluctuations the fit can improve the precision.', '1610.05667-1-30-6': 'If the signal propagation distance is short and, in turn, the statistical contribution to [MATH] is smaller the correlation effect between samples [CITATION] negates the improvement due to the fit, leading to no significant difference between the two methods.', '1610.05667-1-31-0': 'The dCFD fraction should be chosen as a compromise between increasing fluctuations of the rising amplitude and improving steepness of slope (see Eq. [REF]).', '1610.05667-1-31-1': 'To determine the optimal fraction a scan is performed in the interval [MATH].', '1610.05667-1-31-2': 'The time resolution of the counter as viewed by PMT1 as a function of the dCFD fraction is shown on Fig. [REF].', '1610.05667-1-31-3': 'The value depends weakly on the distance between the interaction point and the PMT.', '1610.05667-1-31-4': 'For the analysis presented in this work 0.14 is chosen.', '1610.05667-1-31-5': 'This fraction is indicated by crosses on top of the graphs in Fig. [REF].', '1610.05667-1-32-0': '# Results', '1610.05667-1-33-0': 'The dependency of the measured time versus position of the crossing point along the bar as viewed by both PMTs is shown in Fig. [REF].', '1610.05667-1-33-1': 'The graphs are approximated by a polynomial of the first order.', '1610.05667-1-33-2': 'The line slope represents the effective speed of light along the [MATH]-axis of the bar, which is 16.1 cm/ns.', '1610.05667-1-33-3': 'Using the refraction index of the plastic one can convert this value to the effective average reflection angle [MATH].', '1610.05667-1-33-4': 'Yet, observant eyes would reveal a systematic trend in the position of the points with respect to the linear function.', '1610.05667-1-33-5': 'In order to understand it we calculate the velocity and the reflection angle for every point with respect to the first measured point.', '1610.05667-1-33-6': 'Results are shown in Figs [REF] and [REF].', '1610.05667-1-33-7': 'The velocity of the signal propagation in the proximity of the PMT is 14 ns/cm ([MATH]).', '1610.05667-1-33-8': 'It increases up to 16 ns/cm ([MATH]) at [MATH] cm and stays rather unchanged till the end of the bar.', '1610.05667-1-33-9': 'This behavior results from a combination of multiple effects: contribution of refractive photons which, as opposed to reflective photons, enlarge the rising edge of a signal [CITATION]; broadening of the signal due to propagation; and walk effect which however is small since CFD technique has been applied.', '1610.05667-1-34-0': 'Time resolution was obtained in the gaussian fit of a distribution of the PMT1 and PMT2 time with the trigger time [MATH] subtracted.', '1610.05667-1-34-1': 'The time resolution as a function of longitudinal distance at [MATH] cm obtained in measurements with WAVECATCHER is shown in the left panel of Fig. [REF].', '1610.05667-1-34-2': 'The time of dCFD has been extracted in the fit of a waveform as described in the previous section.', '1610.05667-1-34-3': 'An advantage of this method is demonstrated in Fig. [REF] which shows the ratio of resolutions obtained in the dCFD analysis of the waveform by a fit and by a linear interpolation of samples.', '1610.05667-1-34-4': 'The former provides better precision by 2% in the proximity of PMT, however an advantage becomes obvious in the outer side of the bar where the precision improves by 8%.', '1610.05667-1-34-5': 'The contribution from the reference counters is subtracted in quadrature.', '1610.05667-1-35-0': 'The time resolution of an individual PMT evolves from 80 ps for the crossing point near the phototube to 320 ps for the light propagation along the 280 cm distance.', '1610.05667-1-35-1': 'An improvement of the resolution, in case of the crossing point approaching the end of the bar at 300 cm, is likely an effect of a signal strengthening due to the reflection (see Sec.[REF]).', '1610.05667-1-35-2': 'Similar effect was observed in Ref.[CITATION].', '1610.05667-1-35-3': 'The distribution is fitted by an analytic function (sum of two exponential functions and a constant) which is shown by a solid curve in Fig. [REF].', '1610.05667-1-35-4': 'The physics motivated analysis of the distribution will be presented in Sec. [REF].', '1610.05667-1-36-0': 'In case when an information on the position of the crossing point was not used, the time of the interaction was calculated as a simple average [EQUATION] where [MATH] and [MATH] are the time measured by PMT1 or PMT2, respectively, with [MATH] subtracted.', '1610.05667-1-36-1': 'However in a real experiment the crossing point can be well determined by other precision detectors.', '1610.05667-1-36-2': 'In our case the interaction point is defined by the position of the reference counters.', '1610.05667-1-36-3': 'Therefore one can weight the time measured by PMT1 or PMT2 according to their uncertainties [MATH] and [MATH] [EQUATION]', '1610.05667-1-36-4': 'The counter time resolutions extracted in the gaussian fit of the distribution obtained with Eq. ([REF]) and Eq. ([REF]) are shown in Fig. [REF] by open and full diamond symbols, respectively.', '1610.05667-1-36-5': 'Both approaches provide a precision of 133 ps in the central region of the bar.', '1610.05667-1-36-6': 'On the other hand, in the outer parts of the bar the phototube which is closer to the crossing point assures a substantially better resolution.', '1610.05667-1-36-7': 'In this region the weighted average provides a significant advantage, with a time resolution of 80 ps, while the resolution of the simple average is around 160 ps.', '1610.05667-1-36-8': 'Therefore, the intrinsic time resolution of the counter can be regarded as being 150 ps for the simple average and 100 ps for the weighted average approach over the entire counter length of 3 m.', '1610.05667-1-37-0': 'The time resolution as a function of longitudinal distance at [MATH] cm obtained in measurements with SAMPIC is shown in the right panel of Fig. [REF].', '1610.05667-1-37-1': 'The time resolution of the counter when measured by a single PMT varies from 80 ps to 340 ps.', '1610.05667-1-37-2': 'The resolution in the center of the bar, when calculated as an average for two PMTs, is found to be 140 ps.', '1610.05667-1-37-3': 'The slight differences in the values obtained with SAMPIC and with WAVECATCHER could be, in general, a consequence of a shorter waveform recorded.', '1610.05667-1-37-4': 'The ratio of the two is shown in Fig. [REF].', '1610.05667-1-38-0': '## Resolution vs transverse coordinate', '1610.05667-1-39-0': 'The time resolution of the counter was studied as a function of [MATH].', '1610.05667-1-39-1': 'In addition to the scan at [MATH] cm, which is described above, a second scan with the position of trigger counters shifted by [MATH] cm upward was performed.', '1610.05667-1-40-0': 'The ratio of time resolutions obtained in scans at [MATH] cm and [MATH] cm for PMT2 is shown in Fig. [REF].', '1610.05667-1-40-1': 'A fit with a constant function gives a value compatible with 1, showing that the shift does not alter the time resolution.', '1610.05667-1-41-0': '## Phenomenological analysis', '1610.05667-1-42-0': 'In order to quantify the effects discussed above, a phenomenological analysis is performed aiming to describe the resolution [MATH] in terms of different contributions.', '1610.05667-1-43-0': 'The time resolution as a function of distance [MATH] can be decomposed as [CITATION] [EQUATION] where the parameter [MATH] represents a contribution from the emission time of the scintillator and the time jitter of the PMT; [MATH] accounts for a time spread due to the light transmission; and [MATH] is the contribution which is independent of the light strength, such as readout electronics noise.', '1610.05667-1-43-1': 'The number of observed photoelectrons [MATH] depends on distance and can be directly extracted in the experiment.', '1610.05667-1-44-0': 'The signal amplitude is shown in Fig. [REF] as a function of distance.', '1610.05667-1-44-1': 'One can distinguish three regions, motivating a fit with a sum of three exponential functions.', '1610.05667-1-44-2': 'The distribution falls down quickly along the first 20 cm.', '1610.05667-1-44-3': 'It could be explained by the contribution of light reflected from the aluminum foil which weakens quickly with distance since in every reflection about 40% of the light is lost (see Fig. [REF]).', '1610.05667-1-44-4': 'Photons which undergo the total internal reflection travel a much longer distance along the bar.', '1610.05667-1-44-5': 'An angle between their trajectories and the surface of the bar has to be smaller than [MATH] (see Fig. [REF]).', '1610.05667-1-44-6': 'They make the principal contribution to the light transmission in the bar.', '1610.05667-1-44-7': 'When fitting by an exponent one obtains the effective attenuation length [MATH] cm.', '1610.05667-1-44-8': 'Finally, when an interaction takes place at the far end of the bar reflection effects give an additional contribution to the strength of the signal.', '1610.05667-1-45-0': 'In order to convert the amplitude into the number of photoelectrons [MATH], we assume that they are proportional.', '1610.05667-1-45-1': 'Thus the asymmetry of amplitudes of signals detected in PMT1 and PMT2 would be equal to the asymmetry of the number of photoelectrons.', '1610.05667-1-45-2': 'The asymmetry was calculated on an event-by-event basis.', '1610.05667-1-45-3': 'Its distribution exhibits a Gaussian shape and its spread provides the total number of photoelectrons detected in both phototubes [EQUATION].', '1610.05667-1-45-4': 'When calculated in the center of the bar, the procedure results in [MATH].', '1610.05667-1-45-5': 'This value is used to convert the distribution of amplitude to that of photoelectrons on which the fit of the time resolution [MATH] with the function of Eq. ([REF]) is performed.', '1610.05667-1-46-0': 'The distribution of the time resolution for PMT1 overlaid with the fitting function is shown in Fig. [REF].', '1610.05667-1-46-1': 'At short distance the resolution is driven by the photoemission properties of the plastic.', '1610.05667-1-46-2': 'The value obtained in the fit [MATH] ns is compatible with the time constants of EJ-200 quoted in Section [REF].', '1610.05667-1-46-3': 'The PMT also contributes to this uncertainty via the transit time spread but at a much lower level.', '1610.05667-1-46-4': 'In the middle of the bar the contribution of the path length dispersion gets equal in size with the plastic uncertainty.', '1610.05667-1-46-5': 'This contribution is linear in distance and it enlarges the uncertainty as [MATH] ns/m.', '1610.05667-1-46-6': 'The last term which reflects the contribution from the readout electronics is negligible, [MATH] ns.', '1610.05667-1-47-0': 'The fit is able to describe reasonably the time resolution except for one point at the largest distance to the PMT.', '1610.05667-1-47-1': 'This can be due to focusing effects of the reflected light from the nearest lightguide and PMT (as mentioned in Section [REF]) leading to a narrower time spread, which is not accounted for by the model.', '1610.05667-1-48-0': '## Resolution vs incident angle', '1610.05667-1-49-0': 'We also studied the behavior of [MATH] as a function of the angle between the surface of the bar and the beam trajectory.', '1610.05667-1-49-1': 'To do so the counter was rotated in either the horizontal or the vertical plane.', '1610.05667-1-49-2': 'The time was calculated as an average between measurements of the two PMTs at the center of the bar.', '1610.05667-1-49-3': 'Results are presented in Fig. [REF].', '1610.05667-1-50-0': 'There are two effects which can be discussed in this respect.', '1610.05667-1-50-1': 'The track length inside the plastic gets longer in case of the oblique incidence, thus the number of emitted photons increases proportionally and the time resolution improves as a square root of this number.', '1610.05667-1-50-2': 'Indeed the measurements follow a [MATH] behavior for a rotation in the vertical plane.', '1610.05667-1-50-3': 'In case of a rotation in the horizontal plane, an additional uncertainty comes from the fact that the entrance and exit points of a track are located at different distances from the PMT.', '1610.05667-1-50-4': 'As it can be seen in Fig. [REF], the resolution does not change in the interval from 50[MATH] to 90[MATH], which is in general agreement with observation presented in Refs[CITATION].', '1610.05667-1-50-5': 'However at smaller angles the resolution improves.', '1610.05667-1-51-0': '# Conclusions', '1610.05667-1-52-0': 'The timing properties of a plastic scintillator counter with dimensions 3 m [MATH] 11 cm [MATH] 2.5 cm were studied using the test-beam facility of the East Area of the CERN PS.', '1610.05667-1-52-1': 'Two waveform digitizers, WAVECATCHER and SAMPIC, were exploited as a DAQ electronics.', '1610.05667-1-53-0': 'The time resolution of the counter when measured by a single PMT varies from 80 ps to 320 ps and from 80 ps to 340 ps in case of WAVECATCHER and SAMPIC, respectively and its behavior as a function of distance can be reasonably modeled.', '1610.05667-1-53-1': 'The resolution in the center of the bar, when calculated as the average of the two PMTs, is found to be 133 ps and 140 ps for the two DAQ modules, respectively.', '1610.05667-1-53-2': 'Results of measurements along the central line ([MATH] cm) of the bar and shifted transversely by 4 cm show no visible difference in the time resolution.', '1610.05667-1-53-3': 'The behavior of the time resolution versus incident angle in horizontal and vertical planes was also studied and the difference between the two is understood.', '1610.05667-1-54-0': 'These results confirm that the use of long scintillator counters can provide a time resolution of approximately 100 ps for a large-scale detector used in particle physics experiments.', '1610.05667-1-54-1': 'The SAMPIC waveform digitizer appears to be adequate for such applications, especially if a self-triggering capacity and a tolerance for high signal rates is desired.'}

{'1610.05667-2-0-0': 'A plastic scintillator bar with dimensions 300 cm [MATH] 2.5 cm [MATH] 11 cm was exposed to a focused muon beam to study its light yield and timing characteristics as a function of position and angle of incidence.', '1610.05667-2-0-1': 'The scintillating light was read out at both ends by photomultiplier tubes whose pulse shapes were recorded by waveform digitizers.', '1610.05667-2-0-2': 'Results obtained with the WAVECATCHER and SAMPIC digitizers are analyzed and compared.', '1610.05667-2-0-3': 'A discussion of the various factors affecting the timing resolution is presented.', '1610.05667-2-0-4': 'Prospects for applications of plastic scintillator technology in large-scale particle physics detectors with timing resolution around 100 ps are provided in light of the results.', '1610.05667-2-1-0': '# Introduction', '1610.05667-2-2-0': 'Plastic scintillator detectors have been extensively used in particle physics experiments for decades.', '1610.05667-2-2-1': 'In large-scale experiments, they are typically arranged as an array covering a large surface which can provide a fast trigger signal or particle identification using the time-of-flight (ToF) technique.', '1610.05667-2-2-2': 'Depending on the bar dimensions, scintillator type and light readout sensor, the time resolution for such detectors typically ranges from 50 ps (0.5 m bars of the ToF system of MICE [CITATION]) to 350 ps (6.8 m bars of the ToF system of OPAL [CITATION]).', '1610.05667-2-3-0': 'In practice, bars which are made of a bulk scintillator do not exceed 3 m in length.', '1610.05667-2-3-1': 'This restriction comes naturally from light attenuation within the plastic and an uncertainty related to the dispersion of photon path lengths which becomes dominant for long bars.', '1610.05667-2-3-2': 'Moreover, this uncertainty grows exponentially with decreasing bar thickness [CITATION].', '1610.05667-2-3-3': 'It makes a bar cross section close to a square shape advantageous in detectors [CITATION].', '1610.05667-2-3-4': 'However, when a detector covers a large surface, for reasons of economy, the bar thickness along the beam is often chosen to be significantly smaller than its width.', '1610.05667-2-3-5': 'In this case the thickness becomes a limiting factor for the precision of the time measurement.', '1610.05667-2-3-6': 'Recent examples of detectors using this type of bars are the trigger hodoscopes system in COMPASS [CITATION] and in the NA61/SHINE ToF detector [CITATION].', '1610.05667-2-4-0': 'Another example of a detector which combines the requirements of a large covered surface and an excellent time resolution is the timing detector of the proposed SHiP experiment at the CERN SPS [CITATION].', '1610.05667-2-4-1': 'To efficiently distinguish between vertices from random muon crossings and genuine particle decays, the SHiP timing detector needs to cover a 6 m [MATH] 12 m area with a time resolution of 100 ps or better [CITATION] at an affordable price, which is a challenge.', '1610.05667-2-4-2': 'One option considered in the SHiP technical proposal is an array of 3 m long plastic scintillator bars with the light collected by photomultiplier tubes (PMTs) [CITATION].', '1610.05667-2-4-3': 'Another feature of SHiP is a software trigger running on an online computer farm, thus favoring the use of a DAQ electronics which has the particularity to tolerate relatively high event rates and at the same time allow for each channel to operate in a self-triggering mode.', '1610.05667-2-5-0': 'Novel types of acquisition electronics which perform waveform sampling using a switched capacitor array (SCA) have only recently been employed in particle physics experiments [CITATION].', '1610.05667-2-5-1': 'The use of an analogue memory which is added in parallel with a delay line allows for analog signal sampling at a very high rate.', '1610.05667-2-5-2': 'In addition, having the waveform recorded, one can extract various kinds of information such as baseline, amplitude, charge and time.', '1610.05667-2-5-3': 'The measurements presented in this article with a 3 m bar were made with the two acquisition modules WAVECATCHER [CITATION] and SAMPIC [CITATION].', '1610.05667-2-5-4': 'The latter is proposed for the data acquisition system for the SHiP timing detector.', '1610.05667-2-5-5': 'The test-bench used here can thus be considered as a prototype for the design of the timing detector of the SHiP experiment described in the technical proposal [CITATION].', '1610.05667-2-6-0': 'The article is organized as follows.', '1610.05667-2-6-1': 'The experimental setup is described in Section [REF].', '1610.05667-2-6-2': 'Section [REF] provides a detailed description of the DAQ electronics.', '1610.05667-2-6-3': 'The analysis procedure is presented in Section [REF].', '1610.05667-2-6-4': 'The results of the measurements are discussed in Section [REF].', '1610.05667-2-6-5': 'Finally, a summary is given in Section [REF].', '1610.05667-2-7-0': '# Experimental setup', '1610.05667-2-8-0': 'We present results of test-beam measurements which took place at the CERN PS in June 2016.', '1610.05667-2-8-1': 'The layout of the setup is shown in Fig. [REF].', '1610.05667-2-8-2': 'The coordinate system is chosen such that the [MATH] axis is directed along the beam, the [MATH] axis is along the bar and the [MATH] axis is directed vertically in such a way that the coordinate system is right-handed.', '1610.05667-2-8-3': 'The origin of the system is at the left side of the bar, in the center of the [MATH] cross section.', '1610.05667-2-9-0': '## Plastic bar and PMTs', '1610.05667-2-10-0': 'The scintillator bar was purchased from the SCIONIX Radiation Detector Crystals company [CITATION].', '1610.05667-2-10-1': 'The bar length is 300 cm and its transverse cross section is 2.5 cm [MATH] 11 cm.', '1610.05667-2-10-2': 'The two larger surfaces of the bar (300 cm [MATH] 11 cm) were in contact with a casting form and had no other preparation.', '1610.05667-2-10-3': 'The four other surfaces were diamond milled.', '1610.05667-2-10-4': 'The choice of plastic was primarily driven by the length of the bar: EJ-200 provides an optimal combination of a suitable optical attenuation length, fast timing and high light output.', '1610.05667-2-10-5': 'The properties of EJ-200 quoted by the producer are: a rise time of 0.9 ns; a decay time of 2.1 ns; a bulk attenuation length of 4 m; and a refraction index of 1.58.', '1610.05667-2-10-6': 'The peak in the emission spectrum resides in the violet region of the visible spectrum.', '1610.05667-2-10-7': 'As shown in Fig. [REF], this spectrum is compatible with the sensitivity region of the PMT and the reflection efficiency of an aluminum foil which was used to wrap the bar.', '1610.05667-2-11-0': 'The bar is attached via tapered light guides to 2" phototubes on both ends.', '1610.05667-2-11-1': 'The fast Hamamatsu R13089-10 PMT [CITATION] is chosen because of its good time resolution and moderate cost.', '1610.05667-2-11-2': 'It has a linear-focused dynode structure with 8 stages and a typical anode gain value of [MATH].', '1610.05667-2-11-3': 'The voltage divider optimized for timing applications was provided by the company.', '1610.05667-2-11-4': 'The PMT output signal was coupled directly to the acquisition module.', '1610.05667-2-11-5': 'This results in a signal amplitude in the range [MATH] mV (given for the most probable value) which fits perfectly the dynamic range of the acquisition modules.', '1610.05667-2-11-6': 'The quantum efficiency of the photocathode as given by the manufacturer is 25% at the emission peak for the scintillator (see Fig. [REF]).', '1610.05667-2-11-7': 'Parameters relevant for the precision time measurements are a rise time of 2 ns and a transit time spread of 230 ps.', '1610.05667-2-12-0': 'The phototubes were pressed towards the light guides.', '1610.05667-2-12-1': 'The probable presence of air gaps between the photocathode and plastic may however reduce the amount of photons at large angles due to total internal reflection.', '1610.05667-2-12-2': 'Also, the cross-sectional area of the bar is larger than the area of the photocathode by about 34%.', '1610.05667-2-12-3': 'Due to phase-space conservation of the photon flux the light output should be reduced by about the same amount in the case where an interaction took place in the proximity of the PMT.', '1610.05667-2-13-0': 'The bar and PMTs were fixed to an aluminum frame which could be moved vertically and horizontally with respect to the beam.', '1610.05667-2-14-0': '## Beam and trigger system', '1610.05667-2-15-0': 'Measurements were carried out using a 10 GeV/[MATH] muon beam produced by interactions of 24 GeV/[MATH] protons from the CERN PS accelerator with closed shutters at the T9 beam line of the East Hall.', '1610.05667-2-16-0': 'The trigger was formed by the coincidence of signals from two beam counters installed 50 cm up- and downstream with respect to the bar under test as shown in Fig. [REF].', '1610.05667-2-16-1': 'The counters are shaped as cubes with 2 cm sides made of a fast EJ-228 scintillator with rise and decay constants 0.5 ns and 1.4 ns, respectively.', '1610.05667-2-16-2': 'They were coupled to 1" PMTs (Philips Xp2972) from two sides via 5 cm long light guides.', '1610.05667-2-17-0': 'The trigger time is calculated as an average of the measurements of all four trigger PMTs.', '1610.05667-2-17-1': 'This time is used as a reference for the measurement of the counter under test.', '1610.05667-2-17-2': 'The resolution of the trigger system is derived from the width of a distribution of the time difference between the time measurements by up- and downstream counters.', '1610.05667-2-17-3': 'It is found to be 40 ps.', '1610.05667-2-17-4': 'Another contribution to the trigger time resolution is associated with a finite size of the beam counters.', '1610.05667-2-17-5': 'It was estimated to be 36 ps assuming a uniform distribution of the beam within the counter area.', '1610.05667-2-17-6': 'Both contributions are further subtracted in quadrature from the uncertainty obtained with the counter under test.', '1610.05667-2-18-0': '# DAQ electronics', '1610.05667-2-19-0': 'The major design criterion for the DAQ system is an internal time resolution which has to be much better than the expected resolution of the scintillator counter.', '1610.05667-2-19-1': 'The chosen electronics modules WAVECATCHER [CITATION] and SAMPIC [CITATION] are based on waveform digitizer ASICs which have been developed by LAL and IRFU teams since 1992.', '1610.05667-2-19-2': 'The technology employs a circular buffer, based on arrays of switched capacitors (SCA) which record an analogue signal at very high rate.', '1610.05667-2-19-3': 'The time information is extracted with an interpolation of samples in the leading edge of the signal which permits reaching a timing accuracy of about 5 ps.', '1610.05667-2-19-4': 'Fast detectors for ToF measurements thus represent a natural target for these digitizers.', '1610.05667-2-20-0': 'In both modules, common clock and trigger signals can be provided externally.', '1610.05667-2-20-1': 'Besides that, each channel also integrates a discriminator that can trigger individually or participate in a more complex trigger.', '1610.05667-2-20-2': 'This capability makes these digitizer modules very suitable for a test bench experiment since no extra electronics are required for the trigger.', '1610.05667-2-20-3': 'The digitizers combine in a single module the functions of TDC and ADC, and allow to perform a digital constant fraction discrimination (dCFD) which, in the case of the WAVECATCHER, is a part of the firmware.', '1610.05667-2-21-0': 'The desktop versions of the acquisition modules were used.', '1610.05667-2-21-1': 'They house the USB2 interface which permits a connection to PC with 480 Mbit/s.', '1610.05667-2-21-2': 'The acquisition software runs under Windows saving data directly on disk.', '1610.05667-2-22-0': 'The analogue signal sampling in WAVECATCHER is based on the SAMLONG SCA [CITATION].', '1610.05667-2-22-1': 'The chip was designed with the AMS CMOS 0.35-[MATH]m technology and houses two fully differential channels.', '1610.05667-2-22-2': 'The bandwidth of 500 MHz is well suited for detection of very short pulses.', '1610.05667-2-22-3': 'An analogue signal is sampled at a rate which can be set between 0.4 and 3.2 GS/s.', '1610.05667-2-22-4': 'Voltages stored in capacitors are further digitized with commercial ADCs at a much lower rate (10 to 20 MHz).', '1610.05667-2-22-5': 'This results in an overall readout dead time of about 120 [MATH]s for the full sampling depth of the circular buffer which comprises 1024 cells.', '1610.05667-2-22-6': 'However one can define for the readout an interval of interest which can be a subset of the whole buffer.', '1610.05667-2-22-7': 'The board is DC-coupled with a dynamic range of 2.5 V and adjustable offsets are coded over 16 bits.', '1610.05667-2-22-8': 'In the acquisition of data presented here the desktop module with 8 channels running at 3.2 GS/s (buffer depth 320 ns) has been used.', '1610.05667-2-23-0': 'The SAMPIC acquisition module is based on the cognominal ASIC designed to be the first TDC directly working on analogue signals [CITATION].', '1610.05667-2-23-1': 'SAMPIC was developed in the frame of the RD project aiming at a Waveform and Time to Digital Converter (WTDC) multi-channel chip.', '1610.05667-2-23-2': 'It was initially intended to address needs for high precision timing detectors (5 ps) required by ATLAS AFP and SuperB FTOF.', '1610.05667-2-24-0': 'SAMPIC is a 16-channel ASIC designed with the AMS CMOS 0.18-[MATH]m technology.', '1610.05667-2-24-1': 'Each channel associates a DLL-based TDC providing a raw time with a 64-cell ultra-fast analogue memory sampling up to 10.2 GS/s thus assuring high-resolution timing information.', '1610.05667-2-24-2': 'Analogue data are digitized by an on-chip ADC (8 to 11 bits).', '1610.05667-2-24-3': 'The relatively small sampling depth allows for a low dead time, around 1.5 [MATH]s when using the ADC in the 11-bit mode and of the order of 0.2 [MATH]s in the 8-bit mode.', '1610.05667-2-24-4': 'The chip is designed to offer a signal bandwidth of 1.5 GHz and a usable dynamic range of 1 V.', '1610.05667-2-25-0': 'In the work presented here SAMPIC was exploited at the 3.2 GS/s sampling rate.', '1610.05667-2-25-1': 'The typical width of a signal in the scintillator counter is 25 ns.', '1610.05667-2-25-2': 'Thus, SAMPIC is generally not able to record the full signal waveform, but the sampling depth of 20 ns is enough to determine the baseline and cover the front edge of the signal.', '1610.05667-2-25-3': 'The sampling rate can in principle be lowered to 1.6 GS/s which would increase the time window by a factor of two.', '1610.05667-2-26-0': 'Each SAMPIC channel integrates a discriminator which, contrary to WAVECATCHER, can trigger itself independently of other channels.', '1610.05667-2-26-1': 'This mode was used in the data taking.', '1610.05667-2-26-2': 'Coincidence of reference counters has been checked offline and selects about 1% of recorded events.', '1610.05667-2-26-3': 'The rest of the data represent interactions from muon pile-up and cosmics.', '1610.05667-2-27-0': '# Analysis', '1610.05667-2-28-0': 'The time resolution of the counter is studied using either WAVECATCHER or SAMPIC as a function of the position of a charged-particle interaction either along [MATH] or [MATH] axis and angle of incidence of the particle trajectory.', '1610.05667-2-28-1': 'The counter was exposed to the muon beam to collect about two thousand triggers for every point, after which the bar was manually either shifted or rotated to the next position.', '1610.05667-2-28-2': 'Measurements were done for 15 points along the [MATH] axis of the bar.', '1610.05667-2-28-3': 'For each of these, two measurements were made: one at the center [MATH] cm and another at the edge [MATH] cm.', '1610.05667-2-28-4': 'Rotations were performed only for a beam position at the center of the bar, with 8 different angles between 40[MATH] and 90[MATH] in the horizontal [MATH] plane and three different angles between 60[MATH] and 90[MATH] in the vertical [MATH] plane.', '1610.05667-2-29-0': 'In the analysis, the time [MATH] corresponding to either PMT1 or PMT2 is the measured time subtracted from the reference time (defined as the mean value of the times registered by the two trigger counters).', '1610.05667-2-29-1': 'Examples of time spectra as measured for different positions along [MATH] are shown in Fig. [REF].', '1610.05667-2-29-2': 'The time spectra can be reasonably approximated by Gaussian functions.', '1610.05667-2-29-3': 'For each position, the variance and the mean of the function were used to obtain the time resolution and the peak position of the distribution.', '1610.05667-2-30-0': 'There are several techniques for the extraction of time from the sampled waveforms [CITATION].', '1610.05667-2-30-1': 'However it was shown that results obtained with a digital constant fraction discrimination technique typically show a better time resolution [CITATION].', '1610.05667-2-30-2': 'In the dCFD approach the signal time is determined by the crossing point of the interpolated digitized signal at the threshold which, in turn, is a constant fraction of the pulse amplitude.', '1610.05667-2-31-0': 'The variance of time measurements can be expressed as [CITATION] [EQUATION] where [MATH] is a variance of the measured voltage and [MATH] is a slope of the rising edge of a signal.', '1610.05667-2-31-1': '[MATH] is an uncertainty due to the limited precision of measurements (for instance the TDC jitter) which is independent of a signal.', '1610.05667-2-32-0': 'Waveforms of typical signals detected by PMT1 and PMT2 at [MATH] cm and recorded by the WAVECATCHER are shown in Fig. [REF].', '1610.05667-2-32-1': 'Two methods are applied to determine the crossing point of dCFD.', '1610.05667-2-32-2': 'In the first method a linear interpolation between two neighboring samples is utilized.', '1610.05667-2-32-3': 'In the second method a Gaussian function is used to approximate a part of the waveform which includes eight consecutive samples.', '1610.05667-2-32-4': 'The second method can give an advantage in the case of a smaller number of detected photons resulting from long signal propagation along the bar.', '1610.05667-2-32-5': 'Indeed, if [MATH] is dominated by statistical fluctuations the fit can improve the precision.', '1610.05667-2-32-6': 'On the other hand, if the signal propagation distance is short and, in turn, the statistical contribution to [MATH] becomes less significant, the correlation effect between samples [CITATION] negates the improvement due to the fit, leading to no significant difference between the two methods.', '1610.05667-2-33-0': 'The value of the dCFD fraction should be chosen as a compromise between growing fluctuations [MATH] of the rising amplitude and an improving steepness of the slope [MATH] (see Eq. [REF]).', '1610.05667-2-33-1': 'A scan is performed in the interval 0.04 [MATH] 0.7 to determine the optimal fraction.', '1610.05667-2-33-2': 'The time resolution of the counter as viewed by PMT1 as a function of the dCFD fraction is shown on Fig. [REF].', '1610.05667-2-33-3': 'The optimal value depends weakly on the distance between the interaction point and the PMT.', '1610.05667-2-33-4': 'For the analysis presented in this work 0.14 is chosen.', '1610.05667-2-33-5': 'This fraction is indicated by crosses on top of the graphs in Fig. [REF].', '1610.05667-2-34-0': '# Results', '1610.05667-2-35-0': 'The dependency of the measured time versus position of the crossing point along the bar as viewed by both PMTs is shown in Fig. [REF].', '1610.05667-2-35-1': 'The graphs are approximated by linear functions whose slopes represent the effective average speed of light along the [MATH] axis, which is found to be [MATH] cm/ns.', '1610.05667-2-35-2': 'Using the refraction index of the plastic one can convert this value into the effective average reflection angle: [MATH].', '1610.05667-2-36-0': 'As one can see in Fig. [REF], points lie above the lines for interactions taking place nearby a PMT ([MATH] ns) and they go below for interactions which are closer to the bar center.', '1610.05667-2-36-1': 'In order to understand this behavior we calculate [MATH] and [MATH] as a function of distance.', '1610.05667-2-36-2': 'Results are shown in Fig. [REF].', '1610.05667-2-36-3': 'The value of [MATH] in the proximity of the PMT is 14 ns/cm ([MATH]).', '1610.05667-2-36-4': 'It increases up to 16 ns/cm ([MATH]) at [MATH] cm and stays rather unchanged until the end of the bar.', '1610.05667-2-36-5': 'Presumably this effect can be explained by the fact that photons at larger angles with respect to the surface of the bar have to travel longer distances before reaching the PMT, and are therefore more strongly affected by attenuation.', '1610.05667-2-37-0': 'The time resolution as a function of longitudinal distance at [MATH] cm obtained in measurements with WAVECATCHER is shown in the left panel of Fig. [REF].', '1610.05667-2-37-1': 'The time of dCFD was extracted from the fit to the waveform as described in the previous section.', '1610.05667-2-37-2': 'The advantage of this method is demonstrated in Fig. [REF] which shows the ratio between the resolution obtained with the fit method and the resolution obtained using a linear interpolation.', '1610.05667-2-37-3': 'The former provides better precision by 2% in the proximity of the PMT, and the improvement reaches 8% for the case of an interaction taking place at the far end of the bar.', '1610.05667-2-38-0': 'The time resolution of an individual PMT evolves from 80 ps for the crossing point near the phototube to 320 ps for the light propagation along the 280 cm distance.', '1610.05667-2-38-1': 'A slight improvement of the resolution is observed in case of the crossing point being at the proximity of 300 cm.', '1610.05667-2-38-2': 'This could possibly be an effect of light reflected backwards.', '1610.05667-2-38-3': 'A similar effect was observed in Ref. [CITATION].', '1610.05667-2-38-4': 'The distribution is approximated by an analytic function (sum of two exponential functions and a constant) which is shown by a solid curve in Fig. [REF].', '1610.05667-2-39-0': 'Measurements of time done by the two PMTs on both ends of the bar can be combined in two different ways.', '1610.05667-2-39-1': 'In the case where the position of the crossing point between the particle trajectory and the bar is unknown, the time of the interaction can only be calculated as a simple average [EQUATION] where [MATH] and [MATH] are the times measured by PMT1 and PMT2, respectively.', '1610.05667-2-39-2': 'On the other hand, in a real experiment the crossing point can be well determined by other precision tracking detectors.', '1610.05667-2-39-3': 'In our case the interaction point is defined by the position of the reference trigger counters.', '1610.05667-2-39-4': 'Therefore one can weight the measurements of PMT1 and PMT2 according to their uncertainties [MATH] and [MATH] [EQUATION]', '1610.05667-2-39-5': 'The time resolutions extracted from fits of the distributions obtained with Eq. ([REF]) and Eq. ([REF]) are shown in Fig. [REF] by open and full diamond symbols, respectively.', '1610.05667-2-39-6': 'Both approaches provide 133 ps resolution in the central region of the bar.', '1610.05667-2-39-7': 'On the other hand, in the outer parts of the bar the phototube which is closer to the crossing point makes possible a substantially better resolution.', '1610.05667-2-39-8': 'In this region the weighted average provides a significant advantage, with a time resolution of 80 ps, while the resolution of the simple average is around 160 ps.', '1610.05667-2-39-9': 'Therefore, the time resolution of the bar can be regarded as being 150 ps for the simple average and 100 ps for the weighted average approach over the entire bar length of 3 m.', '1610.05667-2-40-0': 'The time resolution as a function of longitudinal distance at [MATH] cm obtained in measurements with SAMPIC is shown in the right panel of Fig. [REF].', '1610.05667-2-40-1': 'The time resolution of the bar when measured by a single PMT varies from 80 ps to 340 ps.', '1610.05667-2-40-2': 'The resolution in the center of the bar, when calculated as an average for two PMTs, is found to be 140 ps.', '1610.05667-2-40-3': 'The slight differences in the values obtained with SAMPIC and with WAVECATCHER are mostly a consequence of a shorter waveform recorded by SAMPIC.', '1610.05667-2-40-4': 'The ratio of the two is shown in Fig. [REF].', '1610.05667-2-41-0': '## Resolution vs transverse coordinate', '1610.05667-2-42-0': 'The time resolution of the bar was studied as a function of [MATH], using a second scan with the position of the bar shifted by [MATH] cm upward.', '1610.05667-2-42-1': 'The ratio of the time resolutions obtained in scans at [MATH] cm and [MATH] cm is shown in Fig. [REF].', '1610.05667-2-42-2': 'A fit with a constant function gives a value compatible with 1, showing that the shift does not alter the time resolution.', '1610.05667-2-43-0': '## Phenomenological analysis', '1610.05667-2-44-0': 'In order to quantify the effects discussed at the beginning of this section, a phenomenological analysis is performed to describe the resolution [MATH] in terms of physics-motivated contributions.', '1610.05667-2-45-0': 'The time resolution as a function of the light propagation distance [MATH] can be decomposed as [CITATION] [EQUATION] where the parameter [MATH] represents a contribution from a spread of the photon emission time of the scintillator and the time jitter of the PMT; [MATH] accounts for a time spread due to the light transmission; and [MATH] is the contribution which is independent of the light strength, such as a noise of the readout electronics.', '1610.05667-2-45-1': 'The number of observed photoelectrons [MATH] depends on distance and can be directly extracted in the experiment.', '1610.05667-2-46-0': 'The most probable value of the pulse height spectrum is shown in Fig. [REF] as a function of the light propagation distance.', '1610.05667-2-46-1': 'One can distinguish several regions.', '1610.05667-2-46-2': 'The distribution falls down more steeply for the first 20 cm.', '1610.05667-2-46-3': 'It could be explained by a strong dependence of the attenuation length on the photon wavelength as shown in Fig. [REF], i.e. the contribution of short-wavelength photons is significant for the short propagation of light and rapidly diminishes at larger distances.', '1610.05667-2-46-4': 'The behavior of the light transmission at larger distance is mainly dictated by photons which undergo total internal reflection, traveling therefore a much longer distance inside the plastic.', '1610.05667-2-46-5': 'For total reflection, the angle between the photon trajectory and the surface of the bar has to be smaller than [MATH], where [MATH] is the refraction index of the plastic.', '1610.05667-2-46-6': 'When fitting by a sum of two exponents one obtains the effective attenuation length [MATH] cm for this region.', '1610.05667-2-46-7': 'For the last 50 cm of the bar the attenuation is not observed.', '1610.05667-2-47-0': 'In order to estimate the number of photoelectrons collected by phototubes the procedure described in Ref. [CITATION] is applied.', '1610.05667-2-47-1': 'The number of detected photoelectrons is assumed to be proportional to the signal amplitude.', '1610.05667-2-47-2': 'The spread of the asymmetry of amplitudes of PMT1 and PMT2 signals provides a lower estimate for the number of photoelectrons.', '1610.05667-2-47-3': 'The procedure results in [MATH] at the center of the bar.', '1610.05667-2-47-4': 'This value is used to rescale the amplitude distribution to that of the number of photoelectrons which, in turn, is used in the fit of the time resolution [MATH] with the function of Eq. ([REF]).', '1610.05667-2-48-0': 'The distribution of the time resolution for PMT1 overlaid with the fitting function is shown in Fig. [REF].', '1610.05667-2-48-1': 'At short distance the resolution is driven by the photoemission properties of the plastic.', '1610.05667-2-48-2': 'The PMT also contributes to this uncertainty via the transit time spread but at a much lower level.', '1610.05667-2-48-3': 'The value obtained from the fit is [MATH] ns.', '1610.05667-2-48-4': 'In the middle of the bar the contribution of the path length dispersion becomes equal in size with the photoemission spread.', '1610.05667-2-48-5': 'This contribution is linear in distance and it enlarges the uncertainty as [MATH] ns/m.', '1610.05667-2-48-6': 'The last term which reflects the contribution from the readout electronics, [MATH], is not significant.', '1610.05667-2-48-7': 'The fitting function is able to approximate reasonably the time resolution except as the point at the far end of the bar.', '1610.05667-2-48-8': 'This improvement of the resolution is likely to be due to the presence of reflected light and is not described by Eq. ([REF]).', '1610.05667-2-49-0': '## Resolution vs incident angle', '1610.05667-2-50-0': 'The behavior of [MATH] as a function of the angle between the surface of the bar and the beam trajectory is also studied.', '1610.05667-2-50-1': 'To do so the bar was rotated in either the horizontal [MATH] or the vertical [MATH] plane.', '1610.05667-2-50-2': 'The time was calculated as an average between measurements of the two PMTs at the center of the bar.', '1610.05667-2-50-3': 'Results are presented in Fig. [REF].', '1610.05667-2-51-0': 'There are two effects which can be discussed in this respect.', '1610.05667-2-51-1': 'The track length inside the bar gets longer in case of the oblique incidence, thus the number of emitted photons increases proportionally and the time resolution improves as a square root of this number.', '1610.05667-2-51-2': 'Indeed the measurements follow a [MATH] behavior for the rotation in the vertical plane.', '1610.05667-2-51-3': 'In case of the rotation in the horizontal plane, an additional uncertainty comes from the fact that the entrance and exit points of a track are located at different distances from the PMT.', '1610.05667-2-51-4': 'It increases the rise time of a signal, thus worsening the time resolution.', '1610.05667-2-51-5': 'As it can be seen in Fig. [REF], the resolution does not change in the interval from 50[MATH] to 90[MATH], which presumably indicates a compensation between the increase of the number of photons and the smearing of the signal rising edge.', '1610.05667-2-51-6': 'This observation is in general agreement with results presented in Refs [CITATION].', '1610.05667-2-51-7': 'The resolution improves for angles smaller than 50[MATH].', '1610.05667-2-51-8': 'In this region a measurable fraction of Cherenkov photons can reach one of PMTs without reflections.', '1610.05667-2-51-9': 'The number of Cherenkov photons increases for smaller values of angle which presumably explains the improved time resolution.', '1610.05667-2-52-0': '# Conclusions', '1610.05667-2-53-0': 'The timing properties of a plastic scintillator counter with dimensions 300 cm [MATH] 11 cm [MATH] 2.5 cm were studied in detail using the test-beam facility of the East Area of the CERN PS.', '1610.05667-2-53-1': 'Two waveform digitizer modules, WAVECATCHER and SAMPIC, were exploited for the data acquisition.', '1610.05667-2-54-0': 'The time resolution of the counter when measured by a single PMT varies from 80 ps to 320 ps and from 80 ps to 340 ps in case of WAVECATCHER and SAMPIC, respectively.', '1610.05667-2-54-1': 'The resolution in the center of the bar, when calculated as an average of the two PMTs, is found to be 133 ps and 140 ps for the two DAQ modules, respectively.', '1610.05667-2-54-2': 'The behavior of the resolution as a function of the light propagation distance was successfully parametrized with a physics-motivated model.', '1610.05667-2-54-3': 'Measurements along the central line of the bar show the same time resolution as those displaced by 4 cm from the central line.', '1610.05667-2-54-4': 'The behavior of the time resolution versus incident angle for rotations in horizontal and vertical planes was also discussed and understood.', '1610.05667-2-55-0': 'These results confirm that the use of long scintillator counters can provide a time resolution of approximately 100 ps for a large-scale detector used in particle physics experiments.', '1610.05667-2-55-1': 'The SAMPIC waveform digitizer appears to be adequate for such applications, especially if a self-triggering capacity and a tolerance for high signal rates is desired.'}

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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1610.05667

1503.00945

{'1503.00945-1-0-0': 'We prove local smoothing estimates for the massless 3D Dirac equation with a Coulomb potential.', '1503.00945-1-0-1': 'Our strategy is inspired by [CITATION] and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.', '1503.00945-1-1-0': '# Introduction and generalities', '1503.00945-1-2-0': 'The 3D massless Dirac equation with an electric Coulomb potential reads as [EQUATION] where [MATH] is the 3D massless Dirac operator, i.e. [EQUATION] and the [MATH] Dirac matrices can be written as [EQUATION] in terms of the Pauli matrices [EQUATION]', '1503.00945-1-2-1': 'The Dirac equation represents one of the most important and investigated models in quantum mechanics, and it has been widely and successfully used in physics to describe relativistic particles having a spin-[MATH] (see e.g. [CITATION], [CITATION]).', '1503.00945-1-2-2': 'From a mathematical viewpoint, there have been many results in the recent years on the stationary (massive) Dirac equation (we refer to the recent survey [CITATION]).', '1503.00945-1-2-3': 'On the other hand, the state-of-the-art understanding of its dynamics is still far from being satisfactory.', '1503.00945-1-3-0': 'The [MATH] matrices satisfy the following anticommutation relations [EQUATION] which imply that [MATH] (we recall that the spectrum of [MATH] is the whole line [MATH]); in particular, this means that [EQUATION] which strictly links the dynamics of the free massless Dirac equation to a system of wave equations.', '1503.00945-1-3-1': 'Therefore, dynamical properties for the free Dirac equation can easily be derived from their wave counterparts: dispersive properties of the flow are thus mainly encoded in the celebrated family of Strichartz estimates, which are given by [EQUATION] where the exponents [MATH] are wave admissible, i.e. satisfy [EQUATION] and [MATH] represents the propagator for the solutions to [REF].', '1503.00945-1-3-2': 'We stress the fact that the estimate corresponding to the couple [MATH], the so called endpoint, that is [EQUATION] is known to fail as the corresponding one for the 3D wave flow.', '1503.00945-1-3-3': 'In [CITATION] the authors proved a refined version of this estimate involving angular spaces, and this allowed them to prove well-posedness for the celebrated cubic nonlinear Dirac equation assuming initial data in [MATH] with slight additional regularity on the angular variable.', '1503.00945-1-3-4': 'The problem was afterwards completely solved in [CITATION].', '1503.00945-1-4-0': 'Another family of a priori estimates encoding informations about dispersion are the so called local-smoothing type estimates, for which different forms are available: to the best of our knowledge the most general one in this setting is the following [EQUATION] where [MATH], [MATH].', '1503.00945-1-4-1': '(Here and in the following we shall use the notation [MATH] for the homogeneous Sobolev space with the norm [MATH] where [MATH], and [MATH] for the mixed space-time Strichartz spaces).', '1503.00945-1-4-2': 'Both estimates [REF] and [REF] can be found in [CITATION] (see also [CITATION], [CITATION]).', '1503.00945-1-5-0': 'In recent years a lot of effort has been spent in order to investigate what happens to dispersive flows when potential perturbations come into play.', '1503.00945-1-5-1': 'In [CITATION] the authors proved that both [REF] and [REF] hold for the flow [MATH] under the assumption [EQUATION] for some [MATH] sufficiently small, which by the way is a sufficient condition to guarantee that the operator [MATH] is selfadjoint (see [CITATION]).', '1503.00945-1-5-2': 'Similar results are available for magnetic potentials as well (see e.g. [CITATION], [CITATION]).', '1503.00945-1-5-3': 'In [CITATION], [CITATION] the authors generalized the angular endpoint estimates proved in [CITATION] to small potential perturbations by introducing some new mixed Strichartz-smoothing estimates.', '1503.00945-1-6-0': 'At the same time, some effort has been spent in order to find examples of potentials such that the correspoding flow does not disperse (in the sense we have discussed above).', '1503.00945-1-6-1': 'This problem has been tackled for the magnetic Dirac equation in [CITATION], essentially showing that for vector potentials of the form [MATH] with [MATH] most of the mass of the solution is localized around a non-dispersive function.', '1503.00945-1-6-2': 'Other previous results in this direction include [CITATION] for the Schrodinger equation.', '1503.00945-1-6-3': 'These arguments suggest that, heuristically, the degree of homogeneity of the operator works as a threshold for the validity of dispersive estimates: therefore, for the Dirac equation, the Coulomb potential is to be thought of as a critical case (we stress the fact that the restriction to the massless case seems unavoidable now).', '1503.00945-1-6-4': 'In the very last years the problem of understanding dispersive estimates for scaling invariant potentials has been approached for other dynamics: in [CITATION] and [CITATION] the authors have proved indeed that Morawetz and Strichartz estimates can be recovered for both the Schrodinger and wave equations with inverse square potentials, and in [CITATION] the [MATH] time decay estimate is proved in the electromagnetic case.', '1503.00945-1-7-0': 'The aim of this paper is to try to adapt the techniques of [CITATION] to the massless Dirac Coulomb model, and to prove local smoothing estimates for solutions to [REF].', '1503.00945-1-7-1': 'The strategy of proof of [CITATION] is fairly straightforward: the basic idea is to use a sphErical harmonics decomposition to reduce the equation to a radial one and then rely on the Hankel transform to diagonalize it.', '1503.00945-1-7-2': 'Several problems arise when trying to adapt this strategy to the Dirac setting.', '1503.00945-1-7-3': 'The first difficulty is given by the fact that the Dirac operator does not preserve radiality, and therefore one needs a more complex, but still classical, sphErical harmonics decomposition related to the [MATH] group action (see [CITATION]).', '1503.00945-1-7-4': 'Another issue is that we cannot use the Hankel transform, and we have to build up a relativistic analogue.', '1503.00945-1-7-5': 'To do this, we need to study the continuous spectrum of the Dirac-Coulomb model.', '1503.00945-1-7-6': 'The corresponding (approximated) eigenfunctions are well known (see for instance [CITATION]), but their rich structure will raise some additional technical difficulties.', '1503.00945-1-7-7': 'We stress the fact that, to the best of our knowledge, this is the first result concerning dispersive properties of Dirac-Coulomb, an operator which plays a fundamental role in relativistic quantum chemistry.', '1503.00945-1-7-8': 'We hope that our result will be useful in the study of the dynamics of nonlinear models involving, for instance, several relativistic electrons in a molecule.', '1503.00945-1-8-0': 'Before stating our main Theorem, we introduce some notations that will be used throughout the paper.', '1503.00945-1-9-0': 'Notation We will indicate with [MATH] and [MATH], respectively, the standard Lebesgue and Sobolev spaces.', '1503.00945-1-9-1': 'The Dirac operator with the Coulomb potential will be [EQUATION] we will need the fractional powers of this operator as well, that we will denote with [MATH], that will be precisely defined the next section (see Proposition [REF]).', '1503.00945-1-9-2': 'With [MATH] we will denote the operator [EQUATION] and with a little abuse of notation we will use the same symbol to indicate the operators which are pointwise equal for all times, [EQUATION]', '1503.00945-1-9-3': 'Moreover, for any [MATH] for [MATH] we will denote with [MATH] the direct sum [EQUATION] where the [MATH] spaces are defined in details in the next section.', '1503.00945-1-10-0': 'We are now ready to state our main result.', '1503.00945-1-11-0': 'Let [MATH] be solution of [REF] and let [MATH] for [MATH].', '1503.00945-1-11-1': 'Then for any [MATH] and any [MATH] there exists a constant [MATH] such that the following estimate holds [EQUATION]', '1503.00945-1-11-2': 'The essential self-adjointness of the Dirac operator with a Coulomb potential is in fact guaranteed if [MATH].', '1503.00945-1-11-3': 'On the other hand, in the interval [MATH] it is anyway possible to build a distinguished selfadjoint extension.', '1503.00945-1-11-4': 'We refer to [CITATION] and references therein for a detailed discussion of the topic in a more general setting.', '1503.00945-1-11-5': 'We stress here the fact that, for [MATH] approaching [MATH], the range of admissible exponents when [MATH] for our estimate [REF] shrinks to zero.', '1503.00945-1-12-0': "The endpoint case [MATH] in [REF] would permit to recover with a standard argument the full set of Strichartz estimates for the perturbed flow (see e.g. [CITATION]); even though we don't have a concrete counterexample, this estimate seems to fail.", '1503.00945-1-12-1': 'Forthcoming remark [REF] gives more details on the issue.', '1503.00945-1-13-0': 'The plan of the paper is the following: in section [REF] we build the setup, reviewing the theory of partial wave subspaces, the spectrum of the Dirac-Coulomb operator and defining our analogue of the Hankel transform, while section [REF] is devoted to the proof of Theorem [REF].', '1503.00945-1-13-1': 'An appendix containing some generalities on special functions is included at the end.'}

{'1503.00945-2-0-0': 'We prove local smoothing estimates for the massless Dirac equation with a Coulomb potential in [MATH] and [MATH] dimensions.', '1503.00945-2-0-1': 'Our strategy is inspired by [CITATION] and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.', '1503.00945-2-1-0': '# Introduction and generalities', '1503.00945-2-2-0': 'The massless Dirac equation with an electric Coulomb potential reads [EQUATION] where the massless Dirac operator [MATH] is defined in terms of the Pauli matrices [EQUATION] as [EQUATION] (we denote [MATH] and [MATH]) in dimension [MATH] with [MATH], and [EQUATION] where the [MATH] Dirac matrices are given by [EQUATION] in dimension [MATH], with [MATH].', '1503.00945-2-3-0': 'The Dirac equation is widely used in physics to describe relativistic particles of spin [MATH] (see e.g. [CITATION], [CITATION]).', '1503.00945-2-3-1': 'In particular, the [MATH] massless equation is used as a model for charge carriers in graphene, a layer of carbon atoms arranged in a honeycomb lattice (see e.g. the physics survey [CITATION] and the mathematical papers [CITATION], [CITATION], [CITATION]).', '1503.00945-2-3-2': 'Note that there have been many rigorous results in the recent years on the stationary (massive) Dirac equation, essentially in dimension 3 (see e.g. [CITATION] for references).', '1503.00945-2-3-3': 'Comparatively, its dynamics has been less investigated up to now.', '1503.00945-2-4-0': 'The [MATH] and [MATH] matrices were introduced in view of making the Dirac operator a square root of the Laplace operator: therefore, they satisfy by construction the following anticommutating relations [EQUATION] (we recall that the spectrum of [MATH] is the whole line [MATH]).', '1503.00945-2-4-1': 'These conditions ensure that [EQUATION] which strictly links the dynamics of the free massless Dirac equation to a system of [MATH] decoupled wave equations.', '1503.00945-2-4-2': 'Therefore, dynamical properties for the free Dirac equation can directly be derived from their wave counterparts: dispersive properties of the flow can be mainly encoded in the celebrated family of Strichartz estimates, which are given by [EQUATION] where the exponents [MATH] are wave admissible, i.e. satisfy [EQUATION] and [MATH] represents the propagator for the solutions to [REF] with [MATH] for [MATH].', '1503.00945-2-4-3': 'We stress the fact that the estimate corresponding to the couple [MATH] in dimension [MATH], the so called endpoint, that is [EQUATION] is known to fail as the corresponding one for the 3D wave flow.', '1503.00945-2-4-4': 'In [CITATION] a refined version of this estimate involving angular spaces was given.', '1503.00945-2-4-5': 'As a consequence, the well-posedness of the celebrated cubic nonlinear Dirac equation was established, assuming initial data in [MATH] with slight additional regularity on the angular variable.', '1503.00945-2-4-6': 'The problem was afterwards completely solved in [CITATION].', '1503.00945-2-4-7': 'See also [CITATION] for the [MATH] case.', '1503.00945-2-5-0': 'Another family of a priori estimates encoding informations about dispersion is given by the so called local-smoothing type estimates, which turn to be particularly useful to handle potential-type perturbations.', '1503.00945-2-5-1': 'Different versions of these estimates are available: to the best of our knowledge the most general one in this setting is [EQUATION] where [MATH], [MATH].', '1503.00945-2-5-2': 'Both [REF] and [REF] can be found in [CITATION] (see also [CITATION], [CITATION]).', '1503.00945-2-5-3': 'We are not aware of any corresponding result for the [MATH] case; nevertheless, the strategy developed in [CITATION] can be suitably adapted to this context to obtain similar results.', '1503.00945-2-6-0': 'In recent years a lot of effort has been spent in order to investigate what happens to dispersive flows when potential perturbations come into play.', '1503.00945-2-6-1': 'In [CITATION] both [REF] and [REF] were proved for the flow [MATH] under the assumption [EQUATION] for some [MATH] sufficiently small, which by the way is a sufficient condition to guarantee that the operator [MATH] is selfadjoint (see [CITATION]).', '1503.00945-2-6-2': 'Similar results are available for magnetic potentials as well (see e.g. [CITATION], [CITATION]).', '1503.00945-2-6-3': 'The angular endpoint estimates proved in [CITATION] have been generalized in [CITATION], [CITATION] to small potential perturbations, by introducing some new mixed Strichartz-smoothing estimates.', '1503.00945-2-6-4': 'At the same time, some effort has been spent in order to find examples of potentials such that the correspoding flows do not disperse (in the sense discussed above).', '1503.00945-2-6-5': 'This problem has been tackled for the magnetic Dirac equation in [CITATION] in [MATH], essentially showing that for vector potentials of the form [MATH] with [MATH] most of the mass of the solution is localized around a non-dispersive function.', '1503.00945-2-6-6': 'These arguments suggest that, heuristically, the degree of homogeneity of the operator works as a threshold for the validity of dispersive estimates: therefore, for the Dirac equation, the Coulomb potential is to be thought of as a critical case (we stress the fact that the restriction to the massless case seems unavoidable now).', '1503.00945-2-6-7': 'In the very last years the problem of understanding dispersive estimates for scaling invariant potentials has been approached for other dynamics: in [CITATION] and [CITATION] the authors have proved indeed that Morawetz and Strichartz estimates can be recovered for both the Schrodinger and wave equations with inverse square potentials, and in [CITATION] the [MATH] time decay estimate is proved in the electromagnetic case.', '1503.00945-2-7-0': 'The aim of this paper is to adapt the techniques of [CITATION] to the massless Dirac Coulomb equation, and thus to prove local smoothing estimates for solutions to [REF].', '1503.00945-2-7-1': 'The strategy of proof of [CITATION] is fairly straightforward: the basic idea is to use a spherical harmonics decomposition to reduce the equation to a radial one and then rely on the Hankel transform to diagonalize it.', '1503.00945-2-7-2': 'Several problems arise when trying to adapt these ideas to the Dirac setting.', '1503.00945-2-7-3': 'The first difficulty is given by the fact that the Dirac operator does not preserve radiality, and therefore one needs a more complex, but still classical, spherical harmonics decomposition related to the [MATH] group action (see [CITATION], section 4.6).', '1503.00945-2-7-4': 'Another issue is that we cannot use the Hankel transform, and we have to build up a relativistic analogue.', '1503.00945-2-7-5': 'To do this, we need to study the continuous spectrum of the Dirac-Coulomb model, both in [MATH] and [MATH] dimensions.', '1503.00945-2-7-6': 'The corresponding generalized eigenfunctions are well known (see for instance [CITATION] and [CITATION]), but their rich structure will raise some additional technical difficulties.', '1503.00945-2-7-7': 'We stress the fact that, to the best of our knowledge, this is the first result concerning dispersive properties of Dirac-Coulomb, an operator which plays a fundamental role in relativistic quantum chemistry.', '1503.00945-2-7-8': 'We hope that our result will be useful in the study of the dynamics of nonlinear models involving, for instance, several relativistic electrons in a molecule.', '1503.00945-2-8-0': 'Before stating our main result, we introduce some notations that will be used throughout the paper.', '1503.00945-2-9-0': 'Notations.', '1503.00945-2-10-0': 'We shall use the standard notation [MATH] for the homogeneous Sobolev space with the norm [MATH] where [MATH], and [MATH] for the mixed space-time Strichartz spaces.', '1503.00945-2-10-1': 'We denote with [MATH] the operator [EQUATION] and with a little abuse of notation we will use the same symbol to indicate the operators which are pointwise equal for all times, [EQUATION]', '1503.00945-2-10-2': 'We introduce the following sets of indices [EQUATION]', '1503.00945-2-10-3': 'As most of the forthcoming objects will be significantly different in [MATH] or [MATH], we will often use the apex [MATH] to distinguish the dimensions; this will not have to be confused with standard powers, but interpretation will be clear from time to time.', '1503.00945-2-10-4': 'For the definition of the angular spaces [MATH], [MATH] we refer to forthcoming Proposition [REF] and subsequent Remark [REF].', '1503.00945-2-11-0': 'We are now ready to state our main Theorem.', '1503.00945-2-12-0': 'Let [MATH], [MATH], and let [MATH].', '1503.00945-2-12-1': 'Let [MATH] be a solution of [REF].', '1503.00945-2-12-2': 'Then, for any [EQUATION] and any [MATH] there exists a constant [MATH] such that the following estimate holds [EQUATION]', '1503.00945-2-12-3': 'The dependence on the dimension of [MATH], which will be neglected in the rest of the paper, has been added in the statement of the Theorem to allow us to remark the different ranges of admissible charges in dimensions [MATH] and [MATH].', '1503.00945-2-12-4': 'These ranges are in fact imposed by the condition on the Dirac Coulomb operator to be self-adjoint, which is indeed guaranteed in those intervals (see Remark [REF]).', '1503.00945-2-12-5': 'We stress also the fact that for [MATH] approaching the endpoints, respectively [MATH] in [MATH] and [MATH] in [MATH], the range of admissible exponents for the lowest value of [MATH] in estimate [REF] shrinks to zero.', '1503.00945-2-13-0': 'It seems to be possible to generalize our proof of Theorem [REF] to any space dimension [MATH].', '1503.00945-2-13-1': 'As it will be clear, the key ingredients are given indeed by the construction of a suitable integral operator which transforms the problem into an ODE, and then by the careful analysis of some interaction integrals between generalized eigenstates of the Dirac-Coulomb equation that naturally appear.', '1503.00945-2-13-2': 'Therefore, the crucial step consists in finding the generalized eigenstates of the Dirac-Coulomb equation: this has been performed in [CITATION] in higher space dimension [MATH] by generalizing the spherical wave decomposition (see Proposition [REF]), i.e. by carefully analyzing the symmetries of the [MATH] group.', '1503.00945-2-13-3': 'Anyway, we prefer not to deal with this problem here both because technicalities would lead us too far and because of the scarce physical interest of it.', '1503.00945-2-14-0': "The endpoint case [MATH] in [REF] would permit to recover with a standard argument the full set of Strichartz estimates for the perturbed flow (see e.g. [CITATION]); even though we don't have a concrete counterexample, this estimate seems to fail.", '1503.00945-2-14-1': 'Forthcoming remark [REF] gives more details on the issue.', '1503.00945-2-15-0': 'In order to have a unitary flow for equation [REF] one needs the operator [MATH] to be selfadjoint; many papers have been devoted to the study of this property (see e.g. [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION]) in the [MATH] setting (note that the methods introduced in these papers are quite general and can be adapted to other dimensions).', '1503.00945-2-15-1': 'In [MATH] the essential self-adjointness is in fact guaranteed if [MATH], while in the interval [MATH] it is still possible to build a distinguished selfadjoint extension.', '1503.00945-2-15-2': 'In the [MATH] case, as soon as [MATH] is nonzero, the Dirac-Coulomb operator [MATH] defined on [MATH] is not essentially selfadjoint (this is an immediate consequence of Theorem 4.16 in [CITATION]).', '1503.00945-2-15-3': 'But C. Warmt ([CITATION], Satz 2.2.6) recently proved the existence of a distinguished self-adjoint extension if [MATH].', '1503.00945-2-15-4': 'Therefore, in the range of [MATH] for which we prove our Theorem [REF], the operator [MATH] is self-adjoint (in the sense just discussed).', '1503.00945-2-16-0': 'The plan of the paper is the following: in section [REF] we build the setup, reviewing the theory of partial wave subspaces and the spectrum of the Dirac-Coulomb operator in [MATH] and [MATH]D; then we define our analogue of the Hankel transform used in [CITATION].', '1503.00945-2-16-1': 'Section [REF] is devoted to the proof of Theorem [REF].', '1503.00945-2-16-2': "A short appendix containing some generalities on special functions is included at the end for the reader's convenience."}

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[]

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[]

['1503.00945-1-10-0', '1503.00945-2-9-0', '1503.00945-2-11-0', '1503.00945-2-12-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1503.00945

0807.0605

{'0807.0605-1-0-0': 'In this talk we present recent next-to-leading order results relevant for LHC phenomenology obtained with the GOLEM method.', '0807.0605-1-0-1': 'After reviewing the status of this Feynman diagrammatic approach for multi-leg one-loop calculations we discuss three applications: the loop-induced process [MATH] and the virtual corrections to the five and six point processes [MATH] and [MATH].', '0807.0605-1-0-2': 'We demonstrate that our method leads to representations of such amplitudes which allow for efficient phase space integration.', '0807.0605-1-0-3': 'In this context we propose a reweighting technique of the leading order unweighted events by local K-factors.', '0807.0605-1-1-0': '# Introduction', '0807.0605-1-2-0': 'The Large Hadron Collider will explore our understanding of fundamental interactions in the multi-TeV range.', '0807.0605-1-2-1': 'Apart from testing the Higgs mechanism which is the final cornerstone of the Standard Model (SM), also a plethora of extensions of the SM will be put under scrutiny.', '0807.0605-1-2-2': 'Whatever final states will be detected, the initial state will always consist of QCD partons.', '0807.0605-1-2-3': 'The perturbative description of such processes is necessarily plagued by renormalisation and factorisation scale uncertainties.', '0807.0605-1-2-4': 'Only if next-to-leading order (NLO) corrections are included the logarithmic dependence on these scales is tamed and one arrives at sufficiently reliable predictions for various signal and background predictions.', '0807.0605-1-2-5': 'For a discussion of what remains to be done see [CITATION].', '0807.0605-1-3-0': 'For a full next-to-leading order evaluation of an N-point process one has to combine virtual corrections with real-emission corrections using some infrared subtraction method.', '0807.0605-1-3-1': 'The tree-like processes can be evaluated by using standard leading order tools.', '0807.0605-1-3-2': 'Meanwhile, automated ways to deal with the IR subtractions based on the Catani-Seymour dipole approach [CITATION] are also on the market [CITATION].', '0807.0605-1-3-3': 'The evaluation of the one-loop contribution for N-point processes is not yet at this level of automation, although new ideas emerged and a lot of progress has been made recently in various directions [CITATION].', '0807.0605-1-4-0': '# The GOLEM method', '0807.0605-1-5-0': 'The aim of our collaboration is to provide a tool which allows for a numerically stable evaluation of multi-leg one-loop amplitudes: GOLEM.', '0807.0605-1-5-1': 'It is based on the method described in [CITATION].', '0807.0605-1-5-2': 'The approach relies on the evaluation of Feynman diagrams and the reduction of tensor integrals using a form factor approach.', '0807.0605-1-5-3': 'The form factors can be evaluated in various ways as outlined below.', '0807.0605-1-6-0': 'We organize the evaluation of a one-loop amplitude as follows:', '0807.0605-1-7-0': 'At this point two independent set-ups exist.', '0807.0605-1-7-1': 'Firstly, a completely symbolic reduction to standard scalar integrals with up to four external legs can be obtained using FORM [CITATION] and MAPLE.', '0807.0605-1-7-2': '[EQUATION]', '0807.0605-1-7-3': 'The respective coefficients are rational polynomials in Mandelstam variables.', '0807.0605-1-7-4': 'The extraction of the rational part, [MATH], of the amplitude is transparent [CITATION].', '0807.0605-1-7-5': 'As long as no efficient tools for the manipulation of multivariate rational polynomials are available, interactive user input is needed to produce sufficiently compact amplitude expressions in the purely symbolic approach.', '0807.0605-1-7-6': 'Secondly, apart from the symbolic approach, we provide a numerical tensor reduction.', '0807.0605-1-7-7': 'Schematically the amplitude is expressed in terms of form factors which resemble Feynman parameter integrals with Feynman parameters in the numerator.', '0807.0605-1-7-8': '[EQUATION]', '0807.0605-1-7-9': 'These form factors are implemented in a FORTAN90 code and can be evaluated by numerical reduction and also by using one-dimensional integral representations.', '0807.0605-1-7-10': 'The form factors were designed to avoid the occurrence of so-called Gram determinants which usually hamper a numerically stable evaluation of large Feynman diagrammatic expressions.', '0807.0605-1-7-11': 'Our method has been successfully applied to various calculations with up to six point functions [CITATION].', '0807.0605-1-8-0': '# Applications for LHC phenomenology', '0807.0605-1-9-0': 'We discuss now recent evaluations of three loop amplitudes which are relevant in the context of Higgs searches at the LHC.', '0807.0605-1-10-0': '## The process [MATH]', '0807.0605-1-11-0': 'In [CITATION] we have shown that the gluon induced production of charged vector boson pairs accounts for about 30 percent of the background to Higgs searches in that channel after cuts.', '0807.0605-1-11-1': 'Although a similar calculation for neutral vector bosons has been performed a long time ago [CITATION], no public code is available which motivated us to redo this calculation using our method.', '0807.0605-1-12-0': 'As only 4-point functions are present, a symbolic expression for the amplitude could be obtained, where numerically dangerous denominators have been cancelled algebraically.', '0807.0605-1-12-1': 'The expressions are implemented in a flexible computer program GG2ZZ [CITATION] which also contains the photonic contributions.', '0807.0605-1-12-2': 'The size of the gluon contribution to the [MATH] cross section in relation to the quark induced part is included in the following table:', '0807.0605-1-13-0': 'For the numerical results we use the following set of input parameters: [MATH] GeV, [MATH] GeV, [MATH] GeV[MATH], [MATH] GeV.', '0807.0605-1-13-1': 'The electromagnetic coupling is defined in the [MATH] scheme.', '0807.0605-1-13-2': 'The pp cross sections are calculated at [MATH] TeV employing the CTEQ6L1 and CTEQ6M parton distribution functions at tree- and loop-level, for more details see [CITATION].', '0807.0605-1-13-3': 'Applying standard cuts: [MATH] GeV,75 GeV[MATH] 105 GeV, [MATH] GeV, we find that the gluon contribution accounts for 14% to the total [MATH] process.', '0807.0605-1-13-4': 'The [MATH] contribution was evaluated using MCFM [CITATION].', '0807.0605-1-14-0': 'The effect of the photon contribution can be seen best in the invariant mass distribution of the 4 leptons in Fig [REF].', '0807.0605-1-15-0': 'Between the one- and two-Z threshold the interference effects are sizable.', '0807.0605-1-16-0': '## The process [MATH]jet', '0807.0605-1-17-0': 'This process is of relevance in the context of Higgs searches in the Higgs plus jet channel.', '0807.0605-1-17-1': 'We have obtained analytic expressions for all 36 partonic one-loop helicity amplitudes [MATH] which contribute to this process.', '0807.0605-1-17-2': 'The colour structure is simple, one finds three different colour structures.', '0807.0605-1-17-3': 'By applying projection operators to each Feynman diagram, reducible scalar products between the loop momentum and external momenta can be expressed by inverse propagators and cancelled.', '0807.0605-1-17-4': 'In this way only rank one 5-point functions remain, together with 2-, 3- and 4-point tensor integrals.', '0807.0605-1-17-5': "We use the 'tHooft-Veltman scheme which needs an accompanying prescription for [MATH].", '0807.0605-1-17-6': 'By splitting the [MATH]-algebra and the loop momentum in a 4- and (n-4)-dimensional part we have to add a finite counterterm [MATH] to the axial coupling to guarantee that the axial and vector part of the vector boson renormalise in the same way [CITATION].', '0807.0605-1-17-7': 'After UV renormalization, only IR poles remain.', '0807.0605-1-17-8': 'A finite expression can be obtained by adding the insertion operator, [MATH] [CITATION], to the result.', '0807.0605-1-17-9': 'We have integrated the resulting expression for the LHC energy over the phase space using the following cuts: [MATH] GeV, [MATH], and find: [EQUATION]', '0807.0605-1-17-10': 'Here [MATH] GeV, [MATH] GeV, [MATH] GeV[MATH] are used.', '0807.0605-1-17-11': '[MATH] is evaluated in the [MATH] scheme and we used [MATH].', '0807.0605-1-17-12': 'For the LO result we use the CTEQ6L1 and for NLO the CTEQ6M parton distribution functions.', '0807.0605-1-17-13': '[MATH] is evaluated using the LHAPDF routines.', '0807.0605-1-17-14': 'Note that the number does not include the closed fermion loop contribution to this process as it turned out to be numerically irrelevant.', '0807.0605-1-18-0': 'We compare the scale dependence of the LO term and the NLO virtual part in Fig [REF].', '0807.0605-1-19-0': 'The figures indicate that the inclusion of the virtual corrections indeed stabilise the prediction for scales around [MATH].', '0807.0605-1-19-1': 'Note that the real emission part of the NLO correction will add another [MATH] dependent term which stems from the initial state singularities.', '0807.0605-1-19-2': 'As new colour channels are present at NLO one expects actually a deterioration of the scale dependence, as has been observed in the [MATH]jet case [CITATION].', '0807.0605-1-20-0': 'Results for this related process have been presented by two other groups already [CITATION].', '0807.0605-1-20-1': 'We have also evaluated this process and compared our evaluation with both groups for single phase space points.', '0807.0605-1-20-2': 'Perfect agreement was found [CITATION].', '0807.0605-1-21-0': '## The process [MATH]', '0807.0605-1-22-0': 'Signatures of beyond SM processes contain typically leptons, jets and missing energy and one easily reaches large numbers of final state partons at the LHC.', '0807.0605-1-22-1': 'Up to now no complete 6-point process, which is related to a [MATH] kinematics, is evaluated at next-to-leading order in [MATH].', '0807.0605-1-22-2': 'An example for a 6-point process emerges in two Higgs doublet models, which lead in certain parameter regions predominantly to 4 b-jets in the final state.', '0807.0605-1-22-3': 'To understand the related background in detail, the process [MATH] has to be known at NLO.', '0807.0605-1-22-4': 'Note that at the LHC one can safely neglect the bottom mass, if realistic [MATH] and [MATH] separation cuts are applied.', '0807.0605-1-22-5': 'The partonic amplitudes [MATH] and [MATH] are sufficient to predict this cross section in the massless limit.', '0807.0605-1-22-6': 'We report here on the successful evaluation of the virtual part of the first of these two amplitudes.', '0807.0605-1-23-0': 'In detail, the [MATH] amplitude can be written in terms of six colour structures.', '0807.0605-1-23-1': 'Two independent helicity amplitudes, [MATH] and [MATH], are needed.', '0807.0605-1-23-2': 'They were evaluated in two completely independent ways.', '0807.0605-1-23-3': 'In one evaluation a fully symbolic reduction to scalar integrals was performed, in the other one each Feynman diagram was mapped to a form factor representation and translated into a FORTRAN90 code.', '0807.0605-1-23-4': 'Both calculations are highly automated such that the evaluation of other processes only needs the respective Feynman diagrammatic input and the specification of colour and helicity projections.', '0807.0605-1-23-5': 'In the given case one has to evaluate 25 pentagon and 8 hexagon diagrams.', '0807.0605-1-23-6': 'After UV renormalization and adding the IR insertion operator all poles in [MATH] cancel and one is left with a finite expression which can be evaluated numerically.', '0807.0605-1-23-7': 'The FORTRAN90 code was organized such that the reevaluation of algebraic terms was avoided by recursive organization of the expressions and caching.', '0807.0605-1-23-8': 'The evaluation time for one phase space point of the full amplitude, summed over helicities and colour is about 0.8 seconds on a 3.2 GHz Intel Pentium 4 processor.', '0807.0605-1-23-9': 'As the integration over phase space can be trivially parallelised this is sufficiently fast in what concerns the evaluation of distributions.', '0807.0605-1-24-0': 'As the evaluation of the amplitude needs a large number of numerical operations, one typically observes numerical problems in parts of the phase space where denominators become small and form factors, respectively scalar integrals, are not linearly independent anymore.', '0807.0605-1-24-1': 'If one integrates directly the LO plus finite virtual corrections over the phase space, adaptive numerical integrators tend to sample phase space points in these critical phase space regions.', '0807.0605-1-24-2': 'This happens if the induced variations influence the result at the order of the accuracy goal.', '0807.0605-1-24-3': 'To avoid this kind of destabilisation we have applied the following method for integrating the virtual NLO corrections.', '0807.0605-1-25-0': 'We first evaluate the LO contribution over the target phase space [EQUATION] and generate unweighted events [MATH].', '0807.0605-1-25-1': 'The latter are related to a parameter transformation [MATH] on phase space such that the new variables have constant density [MATH].', '0807.0605-1-25-2': 'Any observable [MATH] can be estimated by distributing the unweighted events, [MATH], into the respective bins.', '0807.0605-1-25-3': '[EQUATION]', '0807.0605-1-25-4': 'An estimate of the LO plus virtual corrections can now be obtained using the same set of events.', '0807.0605-1-25-5': 'The relation [EQUATION] where [MATH] is a local K-factor, implies [EQUATION] which is a simple reweighing of a LO event sample.', '0807.0605-1-25-6': 'For this purpose the LO events should be evaluated with NLO pdfs.', '0807.0605-1-25-7': 'In this way no integration over the finite virtual corrections is needed, one simply has to evaluate the virtual corrections for each unweighted event which belongs to a specified observable.', '0807.0605-1-25-8': 'Of course it still can happen that this evaluation is plagued by numerical problems but it does not negatively affect the sampling of test points in integration methods.', '0807.0605-1-25-9': 'This method leads to a good estimate if the virtual corrections are sufficiently close to the LO distribution, such that the unweighted events are also representative for the LO+virtual differential cross section.', '0807.0605-1-25-10': 'Note that this has to be fulfilled for any observable, for which perturbation theory is meaningful in the first place.', '0807.0605-1-26-0': 'To illustrate this method we show in Fig. [REF] the effect of the virtual contribution on the distribution of the leading jet, i.e. the jet with the highest energy.', '0807.0605-1-26-1': 'To define the LO 4-jet observable we use the cuts: [MATH] GeV, [MATH] and [MATH].', '0807.0605-1-26-2': 'The LO cross section and the corresponding unweighted events were evaluated using WHIZARD [CITATION] with CTEQ6M pdfs and the scale choice [MATH], [MATH] GeV and one-loop running for [MATH].', '0807.0605-1-26-3': 'The LO + finite virtual contribution was evaluated as described above.', '0807.0605-1-26-4': 'For the LO and LO+virtual contribution we obtain [EQUATION]', '0807.0605-1-26-5': 'The histograms in Fig. [REF] are filled with 200,000 unweighted events.', '0807.0605-1-26-6': 'When evaluating the local K-factors, less than 1% of all points showed an indication of numerical instability.', '0807.0605-1-26-7': 'These critical points where simply reevaluated by using the quadruple precision version of our code.', '0807.0605-1-26-8': 'A similar evaluation of the [MATH] amplitude and the combination with the real emission corrections is in progress.', '0807.0605-1-27-0': '# Conclusion', '0807.0605-1-28-0': 'In this talk we have presented recent results of the GOLEM collaboration.', '0807.0605-1-28-1': 'The implementation of our method to evaluate Feynman diagrammatic representations of amplitudes in symbolic/numerical computer programs has been completed in the context of one-loop amplitude evaluations relevant for the LHC.', '0807.0605-1-28-2': 'Here we presented results for the process [MATH], and the virtual corrections to [MATH] and [MATH].', '0807.0605-1-28-3': 'We have proposed a new indirect integration method for virtual corrections which is based on the evaluation of local K-factors for unweighted events defined by the LO cross section.', '0807.0605-1-28-4': 'We conclude that our method is numerically efficient and can provide predictions for multi-leg one loop processes at TeV colliders.'}

{'0807.0605-2-0-0': 'In this talk we present recent next-to-leading order results relevant for LHC phenomenology obtained with the GOLEM method.', '0807.0605-2-0-1': 'After reviewing the status of this Feynman diagrammatic approach for multi-leg one-loop calculations we discuss three applications: the loop-induced process [MATH] and the virtual corrections to the five and six point processes [MATH] and [MATH].', '0807.0605-2-0-2': 'We demonstrate that our method leads to representations of such amplitudes which allow for efficient phase space integration.', '0807.0605-2-0-3': 'In this context we propose a reweighting technique of the leading order unweighted events by local K-factors.', '0807.0605-2-1-0': '# Introduction', '0807.0605-2-2-0': 'The Large Hadron Collider will explore our understanding of fundamental interactions in the multi-TeV range.', '0807.0605-2-2-1': 'Apart from testing the Higgs mechanism which is the final cornerstone of the Standard Model (SM), also a plethora of extensions of the SM will be put under scrutiny.', '0807.0605-2-2-2': 'Whatever final states will be detected, the initial state will always consist of QCD partons.', '0807.0605-2-2-3': 'The perturbative description of such processes is necessarily plagued by renormalisation and factorisation scale uncertainties.', '0807.0605-2-2-4': 'Only if next-to-leading order (NLO) corrections are included the logarithmic dependence on these scales is tamed and one arrives at sufficiently reliable predictions for various signal and background processes.', '0807.0605-2-2-5': 'For a discussion of what remains to be done see [CITATION].', '0807.0605-2-3-0': 'For a full next-to-leading order evaluation of an N-point process one has to combine virtual corrections with real-emission corrections using some infrared subtraction method.', '0807.0605-2-3-1': 'The tree-like processes can be evaluated by using standard leading order tools.', '0807.0605-2-3-2': 'Meanwhile, automated ways to deal with the IR subtractions based on the Catani-Seymour dipole approach [CITATION] are also on the market [CITATION].', '0807.0605-2-3-3': 'The evaluation of the one-loop contribution for N-point processes is not yet at this level of automation, although new ideas emerged and a lot of progress has been made recently in various directions [CITATION].', '0807.0605-2-4-0': '# The GOLEM method', '0807.0605-2-5-0': 'The aim of our collaboration is to provide a tool which allows for a numerically stable evaluation of multi-leg one-loop amplitudes: GOLEM.', '0807.0605-2-5-1': 'It is based on the method described in [CITATION].', '0807.0605-2-5-2': 'The approach relies on the evaluation of Feynman diagrams and the reduction of tensor integrals using a form factor approach.', '0807.0605-2-5-3': 'The form factors can be evaluated in various ways as outlined below.', '0807.0605-2-6-0': 'We organize the evaluation of a one-loop amplitude as follows:', '0807.0605-2-7-0': 'At this point two independent set-ups exist.', '0807.0605-2-7-1': 'Firstly, a completely symbolic reduction to standard scalar integrals with up to four external legs can be obtained using FORM [CITATION] and MAPLE.', '0807.0605-2-7-2': '[EQUATION]', '0807.0605-2-7-3': 'The respective coefficients are rational polynomials in Mandelstam variables.', '0807.0605-2-7-4': 'The extraction of the rational part, [MATH], of the amplitude can be done separately [CITATION].', '0807.0605-2-7-5': 'As long as no efficient tools for the manipulation of multivariate rational polynomials are available, interactive user input is needed to produce sufficiently compact amplitude expressions in the purely symbolic approach.', '0807.0605-2-7-6': 'Secondly, apart from the symbolic approach, we provide a numerical tensor reduction.', '0807.0605-2-7-7': 'Schematically the amplitude is expressed in terms of form factors which resemble Feynman parameter integrals with Feynman parameters in the numerator.', '0807.0605-2-7-8': '[EQUATION]', '0807.0605-2-7-9': 'These form factors are implemented in a FORTAN90 code and can be evaluated by numerical reduction and also by using one-dimensional integral representations.', '0807.0605-2-7-10': 'The form factors were designed to avoid the occurrence of so-called Gram determinants which usually hamper a numerically stable evaluation of large Feynman diagrammatic expressions.', '0807.0605-2-7-11': 'Our method has been successfully applied to various calculations with up to six point functions [CITATION].', '0807.0605-2-8-0': '# Applications for LHC phenomenology', '0807.0605-2-9-0': 'We discuss now recent evaluations of three loop amplitudes which are relevant in the context of Higgs searches at the LHC.', '0807.0605-2-10-0': '## The process [MATH]', '0807.0605-2-11-0': 'In [CITATION] it has been shown that the gluon induced production of charged vector boson pairs accounts for about 30 percent of the background to Higgs searches in that channel after cuts.', '0807.0605-2-11-1': 'Although a similar calculation for neutral vector bosons has been performed a long time ago [CITATION], no public code is available which motivated us to redo this calculation using our method.', '0807.0605-2-12-0': 'As only 4-point functions are present, a symbolic expression for the amplitude could be obtained, where numerically dangerous denominators have been cancelled algebraically.', '0807.0605-2-12-1': 'The expressions are implemented in a flexible computer program GG2ZZ [CITATION] which also contains the photonic contributions.', '0807.0605-2-12-2': 'The size of the gluon contribution to the [MATH] cross section in relation to the quark induced part is included in the following table:', '0807.0605-2-13-0': 'For the numerical results we use the following set of input parameters: [MATH] GeV, [MATH] GeV, [MATH] GeV[MATH], [MATH] GeV.', '0807.0605-2-13-1': 'The electromagnetic coupling is defined in the [MATH] scheme.', '0807.0605-2-13-2': 'The pp cross sections are calculated at [MATH] TeV employing the CTEQ6L1 and CTEQ6M parton distribution functions at tree- and loop-level, for more details see [CITATION].', '0807.0605-2-13-3': 'Applying standard cuts: [MATH] GeV,75 GeV[MATH] 105 GeV, [MATH] GeV, we find that the gluon contribution accounts for 14% to the total [MATH] process.', '0807.0605-2-13-4': 'The [MATH] contribution was evaluated using MCFM [CITATION].', '0807.0605-2-14-0': 'The effect of the photon contribution can be seen best in the invariant mass distribution of the 4 leptons in Fig [REF].', '0807.0605-2-15-0': 'Between the one- and two-Z threshold the interference effects are sizable.', '0807.0605-2-16-0': '## The process [MATH]jet', '0807.0605-2-17-0': 'This process is of relevance in the context of Higgs searches in the Higgs plus jet channel.', '0807.0605-2-17-1': 'We have obtained analytic expressions for all 36 partonic one-loop helicity amplitudes [MATH] which contribute to this process.', '0807.0605-2-17-2': 'The colour structure is simple, one finds three different colour structures.', '0807.0605-2-17-3': 'By applying projection operators to each Feynman diagram, reducible scalar products between the loop momentum and external momenta can be expressed by inverse propagators and cancelled.', '0807.0605-2-17-4': 'In this way only rank one five-point functions remain, together with two-, three- and four-point tensor integrals.', '0807.0605-2-17-5': "We use the 'tHooft-Veltman scheme which needs an accompanying prescription for [MATH].", '0807.0605-2-17-6': 'By splitting the [MATH]-algebra and the loop momentum in a 4- and (n-4)-dimensional part we have to add a finite counterterm [MATH] to the axial coupling to guarantee that the axial and vector part of the vector boson renormalise in the same way [CITATION].', '0807.0605-2-17-7': 'After UV renormalization, only IR poles remain.', '0807.0605-2-17-8': 'A finite expression can be obtained by adding the Catani-Seymour insertion operator, [MATH] [CITATION], to the result.', '0807.0605-2-17-9': 'We have integrated the resulting expression for the LHC energy over the phase space using the cut [MATH] GeV and find: [EQUATION]', '0807.0605-2-17-10': 'Here [MATH] GeV, [MATH] GeV, [MATH] GeV[MATH] are used.', '0807.0605-2-17-11': '[MATH] is evaluated in the [MATH] scheme and we used [MATH].', '0807.0605-2-17-12': 'For the LO result we use the CTEQ6L1 and for NLO the CTEQ6M parton distribution functions.', '0807.0605-2-17-13': '[MATH] is evaluated using the LHAPDF routines.', '0807.0605-2-17-14': 'Note that the number does not include the closed fermion loop contribution to this process as it turned out to be numerically irrelevant.', '0807.0605-2-18-0': 'We compare the scale dependence of the LO term and the NLO virtual part in Fig [REF].', '0807.0605-2-19-0': 'The figures indicate that the inclusion of the virtual corrections indeed stabilise the prediction for scales around [MATH].', '0807.0605-2-19-1': 'Note that the real emission part of the NLO correction will add another [MATH] dependent term which stems from the initial state singularities.', '0807.0605-2-19-2': 'As new colour channels are present at NLO one expects actually a deterioration of the scale dependence, as has been observed in the [MATH]jet case [CITATION].', '0807.0605-2-20-0': 'Results for this related process have been presented by two other groups already [CITATION].', '0807.0605-2-20-1': 'We have also evaluated this process and compared our evaluation with both groups for single phase space points.', '0807.0605-2-20-2': 'Perfect agreement was found [CITATION].', '0807.0605-2-21-0': '## The process [MATH]', '0807.0605-2-22-0': 'Signatures of beyond SM processes contain typically leptons, jets and missing energy and one easily reaches large numbers of final state partons at the LHC.', '0807.0605-2-22-1': 'Up to now no complete 6-point process, which is related to a [MATH] kinematics, is evaluated at next-to-leading order in [MATH].', '0807.0605-2-22-2': 'Note that progress in that direction has been reported for the process [MATH] at this workshop [CITATION].', '0807.0605-2-22-3': 'An example for another relevant 6-point process emerges in two Higgs doublet models, which lead in certain parameter regions predominantly to 4 b-jets in the final state.', '0807.0605-2-22-4': 'To understand the related background in detail, the process [MATH] has to be known at NLO.', '0807.0605-2-22-5': 'Note that at the LHC one can safely neglect the bottom mass, if realistic [MATH] and [MATH] separation cuts are applied.', '0807.0605-2-22-6': 'The partonic amplitudes [MATH] and [MATH] are sufficient to predict this cross section in the massless limit.', '0807.0605-2-22-7': 'We report here on the successful evaluation of the virtual part of the first of these two amplitudes.', '0807.0605-2-23-0': 'In detail, the [MATH] amplitude can be written in terms of six colour structures.', '0807.0605-2-23-1': 'Two independent helicity amplitudes, [MATH] and [MATH], are needed.', '0807.0605-2-23-2': 'They were evaluated in two completely independent ways.', '0807.0605-2-23-3': 'In one evaluation a fully symbolic reduction to scalar integrals was performed, in the other one each Feynman diagram was mapped to a form factor representation and translated into a FORTRAN90 code.', '0807.0605-2-23-4': 'Both calculations are highly automated such that the evaluation of other processes only needs the respective Feynman diagrammatic input and the specification of colour and helicity projections.', '0807.0605-2-23-5': 'In the given case one has to evaluate 25 pentagon and 8 hexagon diagrams.', '0807.0605-2-23-6': 'After UV renormalization and adding the IR insertion operator all poles in [MATH] cancel and one is left with a finite expression which can be evaluated numerically.', '0807.0605-2-23-7': 'The FORTRAN90 code was organized such that the reevaluation of algebraic terms was avoided by recursive organization of the expressions and caching.', '0807.0605-2-23-8': 'The evaluation time for one phase space point of the full amplitude, summed over helicities and colour is about 0.8 seconds on a 3.2 GHz Intel Pentium 4 processor.', '0807.0605-2-23-9': 'As the integration over phase space can be trivially parallelised this is sufficiently fast in what concerns the evaluation of distributions.', '0807.0605-2-24-0': 'As the evaluation of the amplitude needs a large number of numerical operations, one typically observes numerical problems in parts of the phase space where denominators become small and form factors, respectively scalar integrals, are not linearly independent anymore.', '0807.0605-2-24-1': 'If one integrates directly the LO plus finite virtual corrections over the phase space, adaptive numerical integrators tend to sample phase space points in these critical phase space regions.', '0807.0605-2-24-2': 'This happens if the induced variations influence the result at the order of the accuracy goal.', '0807.0605-2-24-3': 'To avoid this kind of destabilisation we have applied the following method for integrating the virtual NLO corrections.', '0807.0605-2-25-0': 'We first evaluate the LO contribution over the target phase space [EQUATION] and generate unweighted events [MATH].', '0807.0605-2-25-1': 'The latter are related to a parameter transformation [MATH] on phase space such that the new variables have constant density [MATH].', '0807.0605-2-25-2': 'Any observable [MATH] can be estimated by distributing the unweighted events, [MATH], into the respective bins.', '0807.0605-2-25-3': '[EQUATION]', '0807.0605-2-25-4': 'An estimate of the LO plus virtual corrections can now be obtained using the same set of events.', '0807.0605-2-25-5': 'The relation [EQUATION] where [MATH] is a local K-factor, implies [EQUATION] which is a simple reweighting of a LO event sample.', '0807.0605-2-25-6': 'For this purpose the LO events should be evaluated with NLO pdfs.', '0807.0605-2-25-7': 'In this way no integration over the finite virtual corrections is needed, one simply has to evaluate the virtual corrections for each unweighted event which belongs to a specified observable.', '0807.0605-2-25-8': 'Of course it still can happen that this evaluation is plagued by numerical problems but it does not negatively affect the sampling of test points in integration methods.', '0807.0605-2-25-9': 'This method leads to a good estimate if the virtual corrections are sufficiently close to the LO distribution, such that the unweighted events are also representative for the LO+virtual differential cross section.', '0807.0605-2-25-10': 'Note that this has to be fulfilled for any observable for which perturbation theory is meaningful in the first place.', '0807.0605-2-26-0': 'To illustrate this method we show in Fig. [REF] the effect of the virtual contribution on the distribution of the leading jet, i.e. the jet with the highest energy.', '0807.0605-2-26-1': 'To define the LO 4-jet observable we use the cuts: [MATH] GeV, [MATH] and [MATH].', '0807.0605-2-26-2': 'The LO cross section and the corresponding unweighted events were evaluated using WHIZARD [CITATION] with CTEQ6M pdfs and the scale choice [MATH], [MATH] GeV and one-loop running for [MATH].', '0807.0605-2-26-3': 'The LO + finite virtual contribution was evaluated as described above.', '0807.0605-2-26-4': 'For the LO and LO+virtual contribution we obtain [EQUATION]', '0807.0605-2-26-5': 'The histograms in Fig. [REF] are filled with 200,000 unweighted events.', '0807.0605-2-26-6': 'When evaluating the local K-factors, less than 1% of all points showed an indication of numerical instability.', '0807.0605-2-26-7': 'These critical points where simply reevaluated by using the quadruple precision version of our code.', '0807.0605-2-26-8': 'A similar evaluation of the [MATH] amplitude and the combination with the real emission corrections is in progress.', '0807.0605-2-27-0': '# Conclusion', '0807.0605-2-28-0': 'In this talk we have presented recent results of the GOLEM collaboration.', '0807.0605-2-28-1': 'The implementation of our method to evaluate Feynman diagrammatic representations of amplitudes in symbolic/numerical computer programs has been completed in the context of one-loop amplitude evaluations relevant for the LHC.', '0807.0605-2-28-2': 'Here we presented results for the process [MATH], and the virtual corrections to [MATH] and [MATH].', '0807.0605-2-28-3': 'We have proposed a new indirect integration method for virtual corrections which is based on the evaluation of local K-factors for unweighted events defined by the LO cross section.', '0807.0605-2-28-4': 'We conclude that our method is numerically efficient and can provide predictions for multi-leg one loop processes at TeV colliders.'}

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[]

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[]

['0807.0605-1-6-0', '0807.0605-1-7-2', '0807.0605-1-7-8', '0807.0605-1-12-2', '0807.0605-1-25-3', '0807.0605-2-6-0', '0807.0605-2-7-2', '0807.0605-2-7-8', '0807.0605-2-12-2', '0807.0605-2-25-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0807.0605

1403.7020

{'1403.7020-1-0-0': 'We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold.', '1403.7020-1-0-1': 'In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.', '1403.7020-1-1-0': "The research leading to the results contained in this paper has received funding from the European Research Council (E.R.C.) under European Union's Seventh Framework Program (FP7/2007-2013)/ ERC grant agreement No. 291060.", '1403.7020-1-2-0': '# Introduction', '1403.7020-1-3-0': 'The eta invariant was introduced by Atiyah, Patodi and Singer in [CITATION] as a correction term to an index theorem for manifolds with boundary.', '1403.7020-1-3-1': 'Consider a first order, elliptic and self-adjoint operator [MATH] on a compact manifold.', '1403.7020-1-3-2': 'Formally, the eta invariant [MATH] of this operator can be interpreted as its signature, or the difference between the number of positive and the number of negative eigenvalues of [MATH].', '1403.7020-1-3-3': 'In reality, since [MATH] has infinitely many eigenvalues of each sign this needs to be defined via regularization (see sec:Preliminaries).', '1403.7020-1-4-0': 'A key feature of the invariant [MATH], much like the signature of a matrix, is that it is [MATH] in general a continuous function of the operator [MATH].', '1403.7020-1-4-1': 'In particular consider a smooth one-parameter family of operators [MATH].', '1403.7020-1-4-2': 'The corresponding eta invariant [MATH] is then in general a discontinuous function of the parameter [MATH], making it difficult to understand how it behaves as [MATH] varies.', '1403.7020-1-4-3': 'In this paper we shall investigate how the eta invariant of such a one parameter family behaves asymptotically as the parameter gets large.', '1403.7020-1-5-0': 'More precisely, consider a compact, oriented Riemannian manifold [MATH] of odd dimension [MATH], equipped with a spin structure.', '1403.7020-1-5-1': 'Let [MATH] be the corresponding spin bundle on [MATH].', '1403.7020-1-5-2': 'Let [MATH] be a Hermitian line bundle on [MATH].', '1403.7020-1-5-3': 'Let [MATH] be a fixed unitary connection on [MATH] and let [MATH] be an imaginary one form on [MATH].', '1403.7020-1-5-4': 'This gives a family [MATH] of unitary connections on [MATH], with [MATH] being a real parameter .', '1403.7020-1-5-5': 'Each connection in this family gives a coupled Dirac operator [MATH] acting on sections of [MATH].', '1403.7020-1-5-6': 'Our first result, regarding the asymptotics of the reduced eta invariant [MATH], is the following.', '1403.7020-1-6-0': 'As [MATH], the reduced eta invariant satisfies the asymptotics [EQUATION]', '1403.7020-1-6-1': 'It is an interesting question as to what extent the little [MATH] estimate of thm:main theorem can be improved.', '1403.7020-1-7-0': 'In order to investigate this question we consider the eta invariant of such a family in the case where [MATH] is the total space of a circle bundle.', '1403.7020-1-7-1': 'In particular, we shall let [MATH] be the space of unit elements of a positive line bundle [MATH] over a complex manifold [MATH].', '1403.7020-1-7-2': 'We shall further equip [MATH] with an adiabatic family of metrics [MATH] (see sec:Eta-invariant-of).', '1403.7020-1-7-3': 'Under an appropriate choice of the family of connections, this gives the corresponding eta invariant [MATH], with now an additional dependence on the adiabatic parameter [MATH].', '1403.7020-1-7-4': 'Letting [MATH] denote the [MATH]-genus of [MATH], we now prove the following more precise formula for the eta invariant (see theorem thm:Eta invrian asmp S1 bundle) [EQUATION]', '1403.7020-1-8-0': 'From this formula we observe that [MATH], in this case, exhibits jump discontinuities at integer values of [MATH].', '1403.7020-1-8-1': 'Furthermore, the size of the jumps is growing at the rate [MATH] as [MATH] .', '1403.7020-1-8-2': 'Hence this calculation demonstrates that thm:main theorem cannot be improved beyond an [MATH] estimate on the eta invariant.', '1403.7020-1-9-0': 'The eta invariant is a non-local quantity.', '1403.7020-1-9-1': 'That is, it cannot be written as an integral over the manifold of a canonical differential from obtained from the symbol of the operator.', '1403.7020-1-9-2': 'This makes it difficult to compute the eta invariant explicitly.', '1403.7020-1-9-3': 'In the final section of this paper we give an exact formula for the eta invariant [MATH], assuming the value of the adiabatic parameter [MATH] to be small, using the adiabatic limit technique of Bismut-Cheeger, Dai and Zhang [CITATION].', '1403.7020-1-9-4': 'We refer to thm:Eta invariant explicit computation for the exact formula arising from the computation.', '1403.7020-1-9-5': 'A striking feature of this formula is that it expresses the eta invariant [MATH] in purely topological terms on the base [MATH].', '1403.7020-1-9-6': 'This generalizes a similar known computation in dimension three of Nicolaescu [CITATION].', '1403.7020-1-10-0': 'An asymptotic result of the form thm:main theorem was used by Taubes in [CITATION] in order to prove the Weinstein conjecture on the existence of Reeb orbits on three dimensional contact manifolds.', '1403.7020-1-10-1': 'Our results improve the estimates obtained therein and could lead to further information regarding Reeb orbits.', '1403.7020-1-10-2': 'The three dimensional case has been further explored, under certain hypotheses, by Tsai in [CITATION].', '1403.7020-1-11-0': 'In another direction, the asymptotics considered in this paper are closely related to the asymptotic results of Bismut-Vasserot from [CITATION].', '1403.7020-1-11-1': 'In [CITATION] the authors considered the Dolbeault Laplacian [MATH] acting on [MATH]-forms, with values in a tensor power [MATH], of the positive line bundle [MATH] considered here earlier.', '1403.7020-1-11-2': 'They then derived an asymptotic formula for the holomorphic torsion of [MATH] in the limit as [MATH].', '1403.7020-1-11-3': "The asymptotics of the heat trace of [MATH] were used in [CITATION] to prove Demailly's asymptotic Morse inequalities.", '1403.7020-1-11-4': 'This Laplacian will arise in our computations in sec:Eta-invariant-of and it would be interesting to explore this connection further.', '1403.7020-1-12-0': 'The paper is organized as follows.', '1403.7020-1-12-1': 'In sec:Preliminaries we begin with preliminary notations and facts used in the paper.', '1403.7020-1-12-2': 'In sec:Asymptotics-of-the we derive asymptotics of heat traces required in the proof of thm:main theorem.', '1403.7020-1-12-3': 'In sec:Asymptotics- we derive the asymptotics of the spectral measure of a rescaled Dirac operator and prove thm:main theorem.', '1403.7020-1-12-4': 'In sec:Eta-invariant-of we consider the eta invariant of the circle bundle.', '1403.7020-1-12-5': 'There we prove thm:Eta invrian asmp S1 bundle and give the exact computation for the eta invariant of thm:Eta invariant explicit computation.', '1403.7020-1-13-0': '# Preliminaries', '1403.7020-1-14-0': 'Consider a compact, oriented, Riemannian manifold [MATH] of odd dimension [MATH] equipped with a spin structure.', '1403.7020-1-14-1': 'Let [MATH] be the corresponding spin bundle on [MATH].', '1403.7020-1-14-2': 'Let [MATH] denote the Levi-Civita connection on [MATH].', '1403.7020-1-14-3': 'This lifts to the spin connection [MATH] on the spin bundle [MATH].', '1403.7020-1-14-4': 'We denote the Clifford multiplication endomorphism by [MATH] satisfying [EQUATION]', '1403.7020-1-14-5': 'Let [MATH] be a Hermitian line bundle on [MATH].', '1403.7020-1-14-6': 'Let [MATH] be a fixed unitary connection on [MATH] and let [MATH] be an imaginary 1-form on [MATH].', '1403.7020-1-14-7': 'This gives a family [MATH] of unitary connections on [MATH].', '1403.7020-1-14-8': 'We denote by [MATH] the tensor product connection on [MATH].', '1403.7020-1-14-9': 'Each such connection defines a coupled Dirac operator [EQUATION]', '1403.7020-1-14-10': 'Each Dirac operator [MATH] is elliptic and self-adjoint.', '1403.7020-1-14-11': 'It hence possesses a discrete spectrum of eigenvalues.', '1403.7020-1-14-12': 'Define the eta function of [MATH] by the formula [EQUATION]', '1403.7020-1-14-13': 'Here, and in the remainder of the paper, we use the convention that [MATH] denotes a multiset with each eigenvalue of [MATH] being counted with its multiplicity.', '1403.7020-1-14-14': 'The above series converges for [MATH].', '1403.7020-1-14-15': 'It was shown in [CITATION] that the eta function possesses a meromorphic continuation to the entire complex [MATH]-plane and has no pole at zero.', '1403.7020-1-14-16': 'Its value at zero is defined to be the eta invariant of the operator [MATH].', '1403.7020-1-14-17': 'By including the zero eigenvalue in eq:eta invariant definition, with an appropriate convention, we may define a variant known as the reduced eta invariant by [EQUATION]', '1403.7020-1-14-18': 'We shall henceforth denote the reduced eta invariant by the shorthand [MATH], and would like to investigate its asymptotics for large [MATH].', '1403.7020-1-14-19': 'Our results will apply equally well to the unreduced version [MATH].', '1403.7020-1-15-0': 'Let [MATH] denote the Schwartz kernel of the operator [MATH] on the product [MATH].', '1403.7020-1-15-1': 'Denote by [MATH] the pointwise trace of [MATH] along the diagonal.', '1403.7020-1-15-2': 'We may now analogously define the function [EQUATION]', '1403.7020-1-15-3': 'In [CITATION] theorem 2.6, the authors showed that for [MATH], the function [MATH] is holomorphic in [MATH] and smooth in [MATH].', '1403.7020-1-15-4': 'From eq:eta function diagonal it is clear that this is equivalent to [EQUATION]', '1403.7020-1-16-0': '# Asymptotics of the heat kernel', '1403.7020-1-17-0': 'In order to control the eta invariant we shall need to find the asymptotics for the heat traces of [MATH].', '1403.7020-1-17-1': 'We begin with an estimate on its heat kernel.', '1403.7020-1-17-2': 'We denote by [MATH] the Riemannian volume form on [MATH].', '1403.7020-1-17-3': 'All kernels will be calculated with respect to [MATH] in what follows.', '1403.7020-1-17-4': 'Let [MATH] denote the injectivity radius of [MATH].', '1403.7020-1-17-5': 'Let [MATH] denote the geodesic distance function between two given points [MATH].', '1403.7020-1-17-6': 'Define a function on [MATH] by the following formula [EQUATION].', '1403.7020-1-17-7': 'Let [MATH] denote the kernel of [MATH] for [MATH].', '1403.7020-1-17-8': 'We now have the following estimate.', '1403.7020-1-18-0': 'Let [MATH] denote the spin connection on [MATH] and [MATH] be the tensor product connection on [MATH].', '1403.7020-1-18-1': 'First observe that for fixed [MATH] the section [MATH] satisfies the heat equation [MATH].', '1403.7020-1-18-2': 'The Weitzenbock formula gives', '1403.7020-1-19-0': '[EQUATION] where [MATH] denote the curvature of [MATH] and the scalar curvature of [MATH] respectively.', '1403.7020-1-19-1': 'Using the Weitzenbock formula and the heat equation [MATH], we now see that the function [MATH] obeys the inequality [EQUATION] for some constant [MATH] independent of [MATH].', '1403.7020-1-19-2': 'Hence the function [MATH] satisfies the inequality [EQUATION]', '1403.7020-1-19-3': 'Let [MATH] denote the heat kernel [MATH] for the Laplace operator acting on functions on [MATH].', '1403.7020-1-19-4': 'Now since [MATH] and [MATH] have the same asymptotics as [MATH], an application of the maximum principle for the heat equation gives [EQUATION] for all time [MATH].', '1403.7020-1-19-5': 'Next we use the estimate [EQUATION] on the heat kernel.', '1403.7020-1-19-6': 'Equation eq:scalar heat kernel estimate follows for large time since the heat kernel is bounded [EQUATION]', '1403.7020-1-19-7': 'For small time, eq:scalar heat kernel estimate follows from the heat kernel estimate of [CITATION].', '1403.7020-1-19-8': 'The proposition now follows from eq:scalar heat kernel comp.', '1403.7020-1-19-9': 'and eq:scalar heat kernel estimate.', '1403.7020-1-20-0': "Following this we shall prove a more refined estimate on the heat kernel comparing it with Mehler's kernel.", '1403.7020-1-20-1': "We first recall the definition of the Mehler's kernel.", '1403.7020-1-20-2': 'Define an antisymmetric endomorphism [MATH] of [MATH] via [EQUATION]', '1403.7020-1-20-3': 'Let [MATH] be two points of [MATH] such that [MATH].', '1403.7020-1-20-4': 'Let [MATH] such that [MATH].', '1403.7020-1-20-5': 'Define a function on a geodesic neighborhood of the diagonal in [MATH] by [EQUATION]', '1403.7020-1-20-6': 'Now let [MATH] and [MATH] denote the projections onto the two factors of [MATH] and define a section [MATH] of [MATH], in a geodesic neighborhood of the diagonal.', '1403.7020-1-20-7': 'This restricts to [MATH] at the diagonal [MATH] and is parallel along geodesics [MATH].', '1403.7020-1-20-8': 'Consider a smooth cutoff function satisfying [EQUATION]', '1403.7020-1-20-9': "Mehler's kernel is now defined via [EQUATION]", '1403.7020-1-20-10': 'First fix a point [MATH] and a set of geodesic coordinates centered at [MATH].', '1403.7020-1-20-11': 'Now choose a basis [MATH] for [MATH] and a basis [MATH] for [MATH].', '1403.7020-1-20-12': 'Parallel transport this basis along geodesics using the connections [MATH] to obtain trivializations [MATH] and [MATH] of [MATH] and [MATH] respectively near [MATH].', '1403.7020-1-20-13': 'Now define local orthonormal sections of [MATH] via [EQUATION]', '1403.7020-1-20-14': 'The connection [MATH] can be expressed in this frame and these coordinates as [EQUATION] where each [MATH] is a Christoffel symbol of [MATH] (or [MATH] copies of it) and each [MATH] is a Christoffel symbol of the spin connection on [MATH].', '1403.7020-1-20-15': 'Since the section [MATH] is obtained via parallel transport along geodesics, the connection coefficient [MATH] maybe written in terms of the curvature [MATH] of [MATH] via [EQUATION] with the Einstein summation convention being used.', '1403.7020-1-20-16': 'The dependence of the curvature coefficients [MATH] on the parameter [MATH] is seen to be linear [MATH] despite the fact that they are expressed in the [MATH] dependent frame [MATH].', '1403.7020-1-20-17': 'This is because a gauge transformation from an [MATH] independent frame into [MATH] changes the curvature coefficient by conjugation.', '1403.7020-1-20-18': 'Since [MATH] is a line bundle this is conjugation by a function and hence does not change the coefficient.', '1403.7020-1-20-19': 'Next, using the Taylor expansion [MATH], we see that the connection [MATH] has the form [EQUATION]', '1403.7020-1-20-20': 'Here [MATH], [MATH] and [MATH] are all independent of [MATH].', '1403.7020-1-21-0': "Now using Weitzenbock's formula, we note that the operator [MATH] has the form [EQUATION]", '1403.7020-1-21-1': 'Here [MATH] and [MATH] are each smooth endomorphisms of [MATH] independent of [MATH].', '1403.7020-1-21-2': 'Since [MATH] we now have [EQUATION]', '1403.7020-1-21-3': "Note that the right hand side of eq:hdhm is zero for [MATH] since [MATH] is supported in a geodesic neighborhood of the diagonal, by eq:Mehler's kernel def.", '1403.7020-1-21-4': "From the defining equations eq:function m_t in Mehler's kernel and eq:Mehler's kernel def, Mehler's kernel is given in geodesic coordinates via [EQUATION]", '1403.7020-1-21-5': "We now differentiate eq:Mehler's kernel coordinates using eq:dssf-eq:ser to compute the right hand side of eq:hdhm.", '1403.7020-1-21-6': "By Mehler's formula, see section 4.2 in [CITATION], we have [MATH] for [MATH].", '1403.7020-1-21-7': 'Differentiating the rest, we observe that the right hand side of eq:hdhm has the form of a finite sum [EQUATION]', '1403.7020-1-21-8': "Now since the kernels [MATH] and [MATH] both have the same asymptotics as [MATH], Duhamel's principle, using eq:hdhm, gives [EQUATION]", '1403.7020-1-21-9': 'Now we substitute eq:hdsf into eq:ddhm.', '1403.7020-1-21-10': 'Following this substitution, we use the heat kernel bound eq:mest, eq:function exponential bound and the bound [EQUATION] for constants [MATH] and [MATH] independent of [MATH].', '1403.7020-1-21-11': 'These bounds can be used to estimate the right hand side of eq:ddhm by a sum of finitely many terms of the form [EQUATION] with each multi-index [MATH] above satisfying eq:iin.', '1403.7020-1-21-12': 'Finally, eq:iin and the inequalities [EQUATION] (see sec:Estimates-on-Gaussian for a proof of eq:ehi2) give eq:mhest.', '1403.7020-1-22-0': '## Bound on the trace of [MATH]', '1403.7020-1-23-0': 'We now turn to bound the pointwise [MATH].', '1403.7020-1-23-1': 'To this end, first consider the expansion for the heat kernel [MATH] given by [EQUATION]', '1403.7020-1-23-2': 'Here the coefficients [MATH] are smooth sections defined on the neighborhood [MATH] of the diagonal in [MATH].', '1403.7020-1-23-3': 'They are generated by solving a recursive system of transport equations along geodesics as in chapter 7 of [CITATION].', '1403.7020-1-23-4': 'The kernel [MATH] of [MATH] is simply [MATH].', '1403.7020-1-23-5': 'It hence has an expansion given by [EQUATION]', '1403.7020-1-23-6': 'By eq:cutoff, [MATH] is an endomorphism of the spin bundle supported in the region where [MATH].', '1403.7020-1-23-7': 'By eq:pointwise trace asymp as t->0 and eq:dtre, the pointwise trace [MATH] along the diagonal has an expansion starting with a leading term of order [MATH].', '1403.7020-1-23-8': 'Since the restriction to the diagonal of the [MATH] term in eq:dtre is zero, this implies that [EQUATION] at each point on the diagonal.', '1403.7020-1-23-9': 'To bound the trace of [MATH] we will need a lemma giving a schematic form for the coefficients [MATH].', '1403.7020-1-23-10': 'We again work in geodesic coordinates centered at a point [MATH].', '1403.7020-1-23-11': 'Each heat kernel coefficient can be written in terms of the frame eq:parallel transport frame as [EQUATION] for some functions [MATH].', '1403.7020-1-24-0': 'Each function [MATH] appearing in eq:expansion of hk coeff special frame can be written as a finite sum [EQUATION] for some functions [MATH], independent of [MATH].', '1403.7020-1-24-1': 'Moreover, each multi-index [MATH] appearing in eq:schf satisfies the inequality [EQUATION]', '1403.7020-1-24-2': 'The heat kernel coefficients [MATH] are given, as in chapter 7 of [CITATION], by the recursion [EQUATION] where [MATH] denotes the determinant of the metric on [MATH].', '1403.7020-1-24-3': 'Hence [MATH] is clearly seen to be of the form eq:schf.', '1403.7020-1-24-4': 'Equations eq:dssf-eq:ser and eq:rec2 imply that [MATH] has the form eq:schf assuming it to be true for [MATH].', '1403.7020-1-24-5': 'The lemma now follows by induction on [MATH].', '1403.7020-1-25-0': 'Following this we are ready to bound the pointwise trace [MATH].', '1403.7020-1-25-1': 'The above lemma will play an important role in the proposition below.', '1403.7020-1-26-0': 'Consider the remainder obtained after subtracting the first [MATH] terms of the kernel expansion eq:dtre [EQUATION]', '1403.7020-1-26-1': 'From eq:dtre and eq:trace of coeff.', '1403.7020-1-26-2': 'L_t vanish we see that [EQUATION] and it hence suffices to bound [MATH].', '1403.7020-1-26-3': 'We clearly have [MATH] with [MATH] being the analogous remainder in the kernel expansion for the heat trace [EQUATION]', '1403.7020-1-26-4': 'Let us denote [EQUATION]', '1403.7020-1-26-5': 'The result of applying the heat operator to eq:def remainder D(heat trace exp) is then [EQUATION]', '1403.7020-1-26-6': 'In other words, [MATH] is the sum of the terms obtained when some derivative differentiates the cutoff function [MATH] in eq:def remainder D(heat trace exp).', '1403.7020-1-26-7': "Now since [MATH] as [MATH], Duhamel's principle applied to eq:heat operator applied to L gives [EQUATION]", '1403.7020-1-26-8': 'We first bound [MATH], again working in geodesic coordinates and the frame eq:parallel transport frame.', '1403.7020-1-26-9': 'Using eq:cdic and the fact that Clifford multiplication is parallel, the Dirac operator is seen to be of the form [EQUATION] where [MATH], [MATH] and [MATH] are endomorphisms of [MATH] independent of [MATH].', '1403.7020-1-26-10': 'Using eq:expansion of hk coeff special frame,eq:formula for S_t,eq:formula for R_t and eq:dic we may write [EQUATION] in the frame eq:parallel transport frame for some coefficient functions [MATH].', '1403.7020-1-26-11': 'By eq:schf each [MATH] can be written as a finite sum [EQUATION] for some functions [MATH] independent of [MATH].', '1403.7020-1-26-12': 'Since [MATH] in a neighborhood of the diagonal, eq:formula for R_t implies each [MATH] vanishes to infinite order near the diagonal.', '1403.7020-1-26-13': 'Hence by eq:main index inequality we may assume that each multi-index [MATH] in eq:f0 sch form satisfies [MATH].', '1403.7020-1-26-14': 'We now substitute eq:expansion R orth frame and eq:f0 sch form in eq:formula F. Using this substitution along with the heat kernel bound eq:mest, we may bound [MATH] by a sum of terms of the form eq:generic form integral each satisfying [MATH].', '1403.7020-1-26-15': 'The inequalities eq:ehi1-eq:ehi2 then give the estimate [EQUATION]', '1403.7020-1-26-16': 'Next we estimate [MATH].', '1403.7020-1-26-17': 'Following eq:formula for S_t and using the cancellations in the kernel expansion, resulting from the transport equation, we see that [EQUATION]', '1403.7020-1-26-18': 'Equation eq:formula E now gives', '1403.7020-1-27-0': 'We denote by [MATH] and [MATH] the kernels obtained by replacing [MATH] in eq:rduha by [MATH] and [MATH] respectively', '1403.7020-1-28-0': 'It is clear that [EQUATION]', '1403.7020-1-28-1': 'We first bound [MATH], again working in geodesic coordinates and the frame eq:parallel transport frame.', '1403.7020-1-28-2': 'In terms of the orthonormal frame we may write [EQUATION] for some coefficient functions [MATH] and [MATH].', '1403.7020-1-28-3': 'Using eq:schf and eq:dic these can be expressed as finite sums [EQUATION] where [MATH] are functions independent of [MATH] and [MATH] are finite subsets of [MATH].', '1403.7020-1-28-4': 'Moreover, eq:main index inequality now gives [EQUATION]', '1403.7020-1-28-5': 'Again we substitute eq:dschf-eq:schematic form coeffs 2 into eq:rduha U.', '1403.7020-1-28-6': 'This substitution, the bound eq:mhest, along with the inequalities eq:ehi1-eq:ehi2 and eq:schematic form coeffs 1-eq:schematic form coeffs 2 now give the estimate [EQUATION]', '1403.7020-1-28-7': 'Next we estimate [MATH].', '1403.7020-1-28-8': 'First we use a Taylor expansion to write [EQUATION] where each of [MATH] and [MATH] is an even function in these coordinates.', '1403.7020-1-28-9': 'We now let [EQUATION] and define [EQUATION]', '1403.7020-1-28-10': 'Clearly, by eq:schematic form coeffs 1-eq:schematic form coeffs 2 and eq:Taylor expansion odd even-eq:splitting f_ab 2, [EQUATION]', '1403.7020-1-28-11': 'Next we claim that the contribution of [MATH] to [MATH] is zero.', '1403.7020-1-28-12': 'To see this, first observe that [MATH] implies [MATH] is odd by eq:S1 bar.', '1403.7020-1-28-13': 'Hence [MATH] is an odd function, using eq:splitting f_ab 1 and the fact that [MATH] is even.', '1403.7020-1-28-14': 'Hence the integral corresponding to [MATH] in eq:rduha V is zero, being the integral of an odd function in these coordinates.', '1403.7020-1-28-15': 'Similarly, we claim that the contribution of [MATH] to [MATH] is zero.', '1403.7020-1-28-16': 'This time, [MATH] implies [MATH] is even by eq:S2 bar.', '1403.7020-1-28-17': 'Hence [MATH] is an even function using eq:splitting f_ab 2.', '1403.7020-1-28-18': 'However the integral corresponding to [MATH] in eq:rduha V is still the integral of an odd function in these coordinates, because of the [MATH] term in eq:rduha V.', '1403.7020-1-29-0': 'Following this the contribution of [MATH] to [MATH] can be bounded by a finite sum of terms of the form [EQUATION]', '1403.7020-1-29-1': 'Again using the inequalities eq:ehi1-eq:ehi2, and estimating the contribution of [MATH] in similar fashion, gives the estimate [EQUATION]', '1403.7020-1-29-2': 'Following eq:uest, eq:vest and eq:E=00003DU+V we obtain the estimate [EQUATION] for constants [MATH] and [MATH] independent of [MATH].', '1403.7020-1-29-3': 'Equations eq:L=00003DE+F, eq:F estimate and eq:E est then give [EQUATION] for constants [MATH] and [MATH] independent of [MATH].', '1403.7020-1-29-4': 'The proposition now follows from eq:tr(L_t)=00003Dtr(L_t^(n-1)/2) and eq:L est.', '1403.7020-1-30-0': '# Asymptotics of the spectral measure', '1403.7020-1-31-0': 'We now consider the rescaled operator [MATH].', '1403.7020-1-31-1': 'Here we shall find the asymptotics of its spectral measure using the heat trace estimates of the previous section.', '1403.7020-1-31-2': 'We first consider the heat traces of [MATH].', '1403.7020-1-32-0': 'For any [MATH], [EQUATION] where [MATH] is as defined by eq:antisymm endomorphism A.', '1403.7020-1-32-1': 'The convergences above are uniform in compact intervals of [MATH] respectively.', '1403.7020-1-33-0': 'If [MATH] denotes the kernel of [MATH] it is clear that [MATH] after rescaling.', '1403.7020-1-33-1': 'Hence proposition prop:Heat Mehler comparison gives the estimate [EQUATION] for some constants [MATH] independent of [MATH].', '1403.7020-1-33-2': 'Hence [MATH].', '1403.7020-1-33-3': 'It now remains to compute [MATH] in order to prove eq:rescaled heat trace limit.', '1403.7020-1-33-4': "By eq:Mehler's kernel def, the highest order part in [MATH] of [MATH] is given by [EQUATION].", '1403.7020-1-33-5': 'Since [MATH] is an antisymmetric endomorphism it maybe diagonalized to give an orthonormal basis [MATH] of [MATH] with eigenvalues [MATH] respectively.', '1403.7020-1-33-6': 'We hence have [MATH] and [EQUATION]', '1403.7020-1-33-7': 'Now if [MATH], we have the commutation [EQUATION].', '1403.7020-1-33-8': 'This shows that the only traceless terms in the expansion of eq:Exponential Clifford multiplication are the constants and hence [MATH].', '1403.7020-1-33-9': 'Equation eq:rescaled heat trace limit now follows.', '1403.7020-1-33-10': 'For the second part of the theorem note that proposition prop:trace Dexp(td2) estimate gives the estimate [EQUATION] for uniform constants [MATH] independent of [MATH].', '1403.7020-1-33-11': 'From this equation, eq:rescaled tr(Dexp(tD2)) estimate follows.', '1403.7020-1-33-12': 'The uniformity of the limits eq:rescaled heat trace limit,eq:rescaled tr(Dexp(tD2)) estimate is also an easy consequence of the estimates eq:rescaled heat kernel bound and eq:rescaled trace bound.', '1403.7020-1-34-0': 'The above theorem also follows from the rescaling argument as described in section c) of [CITATION].', '1403.7020-1-35-0': 'Next we let [MATH]) denote the number of eigenvalues of [MATH] in the interval [MATH].', '1403.7020-1-35-1': 'We also use the notation [MATH] for any [MATH].', '1403.7020-1-35-2': 'We shall need the following estimates.', '1403.7020-1-36-0': 'We begin with [EQUATION] using proposition prop:Maximum Principle estimate.', '1403.7020-1-36-1': 'This gives [MATH] from whicheq:Counting function bound follows on substituting [MATH].', '1403.7020-1-37-0': 'For the second part we estimate [EQUATION]', '1403.7020-1-37-1': 'Finally to prove the third part note that eq:odd trace cancellation is true for the family of odd Schwartz functions [MATH] on account of eq:rescaled tr(Dexp(tD2)) estimate.', '1403.7020-1-37-2': 'Since the convergence in eq:rescaled tr(Dexp(tD2)) estimate is uniform it may be differentiated to obtain eq:odd trace cancellation for the odd functions [MATH].', '1403.7020-1-37-3': 'Now eq:Functional trace bound along with the fact that the span of [MATH] is dense in the space of odd Schwartz functions gives eq:odd trace cancellation.', '1403.7020-1-38-0': 'Now consider the rescaled spectral measure of [MATH] given by [EQUATION].', '1403.7020-1-38-1': 'Let [MATH] denote the space of bounded, continuous functions on [MATH] vanishing at [MATH].', '1403.7020-1-38-2': 'Consider the Banach space [MATH]S_2k[MATH][MATH]^TX[MATH]TX^1,0,TX^0,1[MATH]^TX^1,0,^TX^0,1[MATH]^TX[MATH]TX^1,0,TX^0,1[MATH]R^TX^1,0,R^TX^0,1[MATH] _i _i=2^[MATH](R^TX+2J+)^N[MATH]R^TX,2J[MATH][MATH][R^TX,J]=0[MATH]R^TX[MATH][MATH]a=(a_1,,a_k+1)N_0^k+1[MATH]a=_i=1^k+1a_i.', '1403.7020-1-38-3': "[MATH]aN_0^k+1[MATH]a=N-k[MATH]2N2k[MATH]J(R^TX+2J+)^N-1[MATH]R^TX[MATH][MATH]aN_0^k+1[MATH]a=N-1-k[MATH]N2k+N2-1k[MATH]J(R^TX+2J+)^N-2[MATH][MATH]aN_0^k+1[MATH]a=N-2-k[MATH]N2-1k[MATH](R^TZ)^N[MATH]d[MATH]e^*[MATH](R^TZ)^N[MATH]de^*J,d_1,e^*_2[MATH]^2e^*_3[MATH]P_i[MATH]i[MATH](R^TX)^N[MATH]P_0,P_1[MATH]P_2[MATH]P_0[MATH]P_1[MATH]P_1[MATH]d_1,e^*_2[MATH]^2e^*_3[MATH]de^*J[MATH]_1,_2[MATH]_3[MATH]T^HY[MATH]T^VY[MATH]P_2[MATH]P_2[MATH]d[MATH]e^*[MATH]de^*J[MATH]d_1,e^*_2[MATH]^2e^*_3[MATH]P_2^1[MATH](R^TX)^N[MATH]d_1,e^*_2[MATH]de^*J[MATH]^2e^*_3[MATH]P_2^2[MATH](R^TX)^N[MATH]_1[MATH]T^VY[MATH]T^HY[MATH]_2[MATH]T^HY[MATH]T^VY[MATH]A,B[MATH]C[MATH]R^TX,2J[MATH][MATH]d_1,e^*_2[MATH]d_1[MATH]^2e^*_3[MATH]de^*J[MATH]e^*_2[MATH]_1[MATH]_3[MATH]T^HY[MATH]T^VY[MATH]T^HY[MATH]X[MATH]g^TX[MATH]l[MATH]l[MATH]J=-[MATH]X[MATH][MATH][MATH]_Xl[MATH]LX[MATH]c_1(L)=l[MATH]L[MATH]R=2[MATH]L[MATH]Y[MATH]L[MATH]X[MATH]^0,[MATH][MATH]r=0[MATH][MATH]r^2g^TS^11l^*g^TX[MATH]2l[MATH]C_1,C_2[MATH]C_3[MATH](Y,g)[MATH]x,zY[MATH]t,t'>0[MATH](1).", '1403.7020-1-38-4': '[MATH]B= y(x,y)<i_g [MATH]B[MATH]B^c[MATH]B[MATH]_R^ne^-r^24t(4t)^n2dr=1.', '1403.7020-1-38-5': '[MATH]B^c[MATH]e^-(x,y)^24t(4t)^n2(n/2)!', '1403.7020-1-38-6': '^n/2i_g^n[MATH]_B^ch_t(x,y)dy(n/2)!', '1403.7020-1-38-7': '^n/2i_g^nvol(Y)[MATH](2).', '1403.7020-1-38-8': "[MATH]tt'.", '1403.7020-1-38-9': "[MATH](x,y)^2t+(y,z)^2t'(x,z)^22(t+t')[MATH]1t'2t+t'[MATH]e^-(y,z)^28t'1[MATH]2^nh_4(t+t')(x,z)(_Yh_2t(x,y)dy).", '1403.7020-1-38-10': '[MATH]'}

{'1403.7020-2-0-0': 'We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold.', '1403.7020-2-0-1': 'In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.', '1403.7020-2-1-0': "The research leading to the results contained in this paper has received funding from the European Research Council (E.R.C.) under European Union's Seventh Framework Program (FP7/2007-2013)/ ERC grant agreement No. 291060.", '1403.7020-2-2-0': '# Introduction', '1403.7020-2-3-0': 'The eta invariant was introduced by Atiyah, Patodi and Singer in [CITATION] as a correction term to an index theorem for manifolds with boundary.', '1403.7020-2-3-1': 'Consider a first order, elliptic and self-adjoint operator [MATH] on a compact manifold.', '1403.7020-2-3-2': 'Formally, the eta invariant [MATH] of this operator can be interpreted as its signature, or the difference between the number of positive and the number of negative eigenvalues of [MATH].', '1403.7020-2-3-3': 'In reality, since [MATH] has infinitely many eigenvalues of each sign this needs to be defined via regularization (see sec:Preliminaries).', '1403.7020-2-4-0': 'A key feature of the invariant [MATH], much like the signature of a matrix, is that it is [MATH] in general a continuous function of the operator [MATH].', '1403.7020-2-4-1': 'In particular consider a smooth one-parameter family of operators [MATH].', '1403.7020-2-4-2': 'The corresponding eta invariant [MATH] is then in general a discontinuous function of the parameter [MATH], making it difficult to understand how it behaves as [MATH] varies.', '1403.7020-2-4-3': 'In this paper we shall investigate how the eta invariant of such a one parameter family behaves asymptotically as the parameter gets large.', '1403.7020-2-5-0': 'More precisely, consider a compact, oriented Riemannian manifold [MATH] of odd dimension [MATH], equipped with a spin structure.', '1403.7020-2-5-1': 'Let [MATH] be the corresponding spin bundle on [MATH].', '1403.7020-2-5-2': 'Let [MATH] be a Hermitian line bundle on [MATH].', '1403.7020-2-5-3': 'Let [MATH] be a fixed unitary connection on [MATH] and let [MATH] be an imaginary one form on [MATH].', '1403.7020-2-5-4': 'This gives a family [MATH] of unitary connections on [MATH], with [MATH] being a real parameter .', '1403.7020-2-5-5': 'Each connection in this family gives a coupled Dirac operator [MATH] acting on sections of [MATH].', '1403.7020-2-5-6': 'Our first result, regarding the asymptotics of the reduced eta invariant [MATH], is the following.', '1403.7020-2-6-0': 'As [MATH], the reduced eta invariant satisfies the asymptotics [EQUATION]', '1403.7020-2-6-1': 'It is an interesting question as to what extent the little [MATH] estimate of thm:main theorem can be improved.', '1403.7020-2-7-0': 'In order to investigate this question we consider the eta invariant of such a family in the case where [MATH] is the total space of a circle bundle.', '1403.7020-2-7-1': 'In particular, we shall let [MATH] be the space of unit elements of a positive line bundle [MATH] over a complex manifold [MATH].', '1403.7020-2-7-2': 'We shall further equip [MATH] with an adiabatic family of metrics [MATH] (see sec:Eta-invariant-of).', '1403.7020-2-7-3': 'Under an appropriate choice of the family of connections, this gives the corresponding eta invariant [MATH], with now an additional dependence on the adiabatic parameter [MATH].', '1403.7020-2-7-4': 'Letting [MATH] denote the [MATH]-genus of [MATH], we now prove the following more precise formula for the eta invariant (see theorem thm:Eta invrian asmp S1 bundle) [EQUATION]', '1403.7020-2-8-0': 'From this formula we observe that [MATH], in this case, exhibits jump discontinuities at integer values of [MATH].', '1403.7020-2-8-1': 'Furthermore, the size of the jumps is growing at the rate [MATH] as [MATH] .', '1403.7020-2-8-2': 'Hence this calculation demonstrates that thm:main theorem cannot be improved beyond an [MATH] estimate on the eta invariant.', '1403.7020-2-9-0': 'The eta invariant is a non-local quantity.', '1403.7020-2-9-1': 'That is, it cannot be written as an integral over the manifold of a canonical differential from obtained from the symbol of the operator.', '1403.7020-2-9-2': 'This makes it difficult to compute the eta invariant explicitly.', '1403.7020-2-9-3': 'In the final section of this paper we give an exact formula for the eta invariant [MATH], assuming the value of the adiabatic parameter [MATH] to be small, using the adiabatic limit technique of Bismut-Cheeger, Dai and Zhang [CITATION].', '1403.7020-2-9-4': 'We refer to thm:Eta invariant explicit computation for the exact formula arising from the computation.', '1403.7020-2-9-5': 'A striking feature of this formula is that it expresses the eta invariant [MATH] in purely topological terms on the base [MATH].', '1403.7020-2-9-6': 'This generalizes a similar known computation in dimension three of Nicolaescu [CITATION].', '1403.7020-2-10-0': 'An asymptotic result of the form thm:main theorem was used by Taubes in [CITATION] in order to prove the Weinstein conjecture on the existence of Reeb orbits on three dimensional contact manifolds.', '1403.7020-2-10-1': 'Our results improve the estimates obtained therein and could lead to further information regarding Reeb orbits.', '1403.7020-2-10-2': 'The three dimensional case has been further explored, under certain hypotheses, by Tsai in [CITATION].', '1403.7020-2-11-0': 'In another direction, the asymptotics considered in this paper are closely related to the asymptotic results of Bismut-Vasserot from [CITATION].', '1403.7020-2-11-1': 'In [CITATION] the authors considered the Dolbeault Laplacian [MATH] acting on [MATH]-forms, with values in a tensor power [MATH], of the positive line bundle [MATH] considered here earlier.', '1403.7020-2-11-2': 'They then derived an asymptotic formula for the holomorphic torsion of [MATH] in the limit as [MATH].', '1403.7020-2-11-3': "The asymptotics of the heat trace of [MATH] were used in [CITATION] to prove Demailly's asymptotic Morse inequalities.", '1403.7020-2-11-4': 'This Laplacian will arise in our computations in sec:Eta-invariant-of and it would be interesting to explore this connection further.', '1403.7020-2-12-0': 'The paper is organized as follows.', '1403.7020-2-12-1': 'In sec:Preliminaries we begin with preliminary notations and facts used in the paper.', '1403.7020-2-12-2': 'In sec:Asymptotics-of-the we derive asymptotics of heat traces required in the proof of thm:main theorem.', '1403.7020-2-12-3': 'In sec:Asymptotics- we derive the asymptotics of the spectral measure of a rescaled Dirac operator and prove thm:main theorem.', '1403.7020-2-12-4': 'In sec:Eta-invariant-of we consider the eta invariant of the circle bundle.', '1403.7020-2-12-5': 'There we prove thm:Eta invrian asmp S1 bundle and give the exact computation for the eta invariant of thm:Eta invariant explicit computation.', '1403.7020-2-13-0': '# Preliminaries', '1403.7020-2-14-0': 'Consider a compact, oriented, Riemannian manifold [MATH] of odd dimension [MATH] equipped with a spin structure.', '1403.7020-2-14-1': 'Let [MATH] be the corresponding spin bundle on [MATH].', '1403.7020-2-14-2': 'Let [MATH] denote the Levi-Civita connection on [MATH].', '1403.7020-2-14-3': 'This lifts to the spin connection [MATH] on the spin bundle [MATH].', '1403.7020-2-14-4': 'We denote the Clifford multiplication endomorphism by [MATH] satisfying [EQUATION]', '1403.7020-2-14-5': 'Let [MATH] be a Hermitian line bundle on [MATH].', '1403.7020-2-14-6': 'Let [MATH] be a fixed unitary connection on [MATH] and let [MATH] be an imaginary 1-form on [MATH].', '1403.7020-2-14-7': 'This gives a family [MATH] of unitary connections on [MATH].', '1403.7020-2-14-8': 'We denote by [MATH] the tensor product connection on [MATH].', '1403.7020-2-14-9': 'Each such connection defines a coupled Dirac operator [EQUATION]', '1403.7020-2-14-10': 'Each Dirac operator [MATH] is elliptic and self-adjoint.', '1403.7020-2-14-11': 'It hence possesses a discrete spectrum of eigenvalues.', '1403.7020-2-14-12': 'Define the eta function of [MATH] by the formula [EQUATION]', '1403.7020-2-14-13': 'Here, and in the remainder of the paper, we use the convention that [MATH] denotes a multiset with each eigenvalue of [MATH] being counted with its multiplicity.', '1403.7020-2-14-14': 'The above series converges for [MATH].', '1403.7020-2-14-15': 'It was shown in [CITATION] that the eta function possesses a meromorphic continuation to the entire complex [MATH]-plane and has no pole at zero.', '1403.7020-2-14-16': 'Its value at zero is defined to be the eta invariant of the operator [MATH].', '1403.7020-2-14-17': 'By including the zero eigenvalue in eq:eta invariant definition, with an appropriate convention, we may define a variant known as the reduced eta invariant by [EQUATION]', '1403.7020-2-14-18': 'We shall henceforth denote the reduced eta invariant by the shorthand [MATH], and would like to investigate its asymptotics for large [MATH].', '1403.7020-2-14-19': 'Our results will apply equally well to the unreduced version [MATH].', '1403.7020-2-15-0': 'Let [MATH] denote the Schwartz kernel of the operator [MATH] on the product [MATH].', '1403.7020-2-15-1': 'Denote by [MATH] the pointwise trace of [MATH] along the diagonal.', '1403.7020-2-15-2': 'We may now analogously define the function [EQUATION]', '1403.7020-2-15-3': 'In [CITATION] theorem 2.6, the authors showed that for [MATH], the function [MATH] is holomorphic in [MATH] and smooth in [MATH].', '1403.7020-2-15-4': 'From eq:eta function diagonal it is clear that this is equivalent to [EQUATION]', '1403.7020-2-16-0': '# Asymptotics of the heat kernel', '1403.7020-2-17-0': 'In order to control the eta invariant we shall need to find the asymptotics for the heat traces of [MATH].', '1403.7020-2-17-1': 'We begin with an estimate on its heat kernel.', '1403.7020-2-17-2': 'We denote by [MATH] the Riemannian volume form on [MATH].', '1403.7020-2-17-3': 'All kernels will be calculated with respect to [MATH] in what follows.', '1403.7020-2-17-4': 'Let [MATH] denote the injectivity radius of [MATH].', '1403.7020-2-17-5': 'Let [MATH] denote the geodesic distance function between two given points [MATH].', '1403.7020-2-17-6': 'Define a function on [MATH] by the following formula [EQUATION].', '1403.7020-2-17-7': 'Let [MATH] denote the kernel of [MATH] for [MATH].', '1403.7020-2-17-8': 'We now have the following estimate.', '1403.7020-2-18-0': 'Let [MATH] denote the spin connection on [MATH] and [MATH] be the tensor product connection on [MATH].', '1403.7020-2-18-1': 'First observe that for fixed [MATH] the section [MATH] satisfies the heat equation [MATH].', '1403.7020-2-18-2': 'The Weitzenbock formula gives', '1403.7020-2-19-0': '[EQUATION] where [MATH] denote the curvature of [MATH] and the scalar curvature of [MATH] respectively.', '1403.7020-2-19-1': 'Using the Weitzenbock formula and the heat equation [MATH], we now see that the function [MATH] obeys the inequality [EQUATION] for some constant [MATH] independent of [MATH].', '1403.7020-2-19-2': 'Hence the function [MATH] satisfies the inequality [EQUATION]', '1403.7020-2-19-3': 'Let [MATH] denote the heat kernel [MATH] for the Laplace operator acting on functions on [MATH].', '1403.7020-2-19-4': 'Now since [MATH] and [MATH] have the same asymptotics as [MATH], an application of the maximum principle for the heat equation gives [EQUATION] for all time [MATH].', '1403.7020-2-19-5': 'Next we use the estimate [EQUATION] on the heat kernel.', '1403.7020-2-19-6': 'Equation eq:scalar heat kernel estimate follows for large time since the heat kernel is bounded [EQUATION]', '1403.7020-2-19-7': 'For small time, eq:scalar heat kernel estimate follows from the heat kernel estimate of [CITATION].', '1403.7020-2-19-8': 'The proposition now follows from eq:scalar heat kernel comp.', '1403.7020-2-19-9': 'and eq:scalar heat kernel estimate.', '1403.7020-2-20-0': "Following this we shall prove a more refined estimate on the heat kernel comparing it with Mehler's kernel.", '1403.7020-2-20-1': "We first recall the definition of the Mehler's kernel.", '1403.7020-2-20-2': 'Define an antisymmetric endomorphism [MATH] of [MATH] via [EQUATION]', '1403.7020-2-20-3': 'Let [MATH] be two points of [MATH] such that [MATH].', '1403.7020-2-20-4': 'Let [MATH] such that [MATH].', '1403.7020-2-20-5': 'Define a function on a geodesic neighborhood of the diagonal in [MATH] by [EQUATION]', '1403.7020-2-20-6': 'Now let [MATH] and [MATH] denote the projections onto the two factors of [MATH] and define a section [MATH] of [MATH], in a geodesic neighborhood of the diagonal.', '1403.7020-2-20-7': 'This restricts to [MATH] at the diagonal [MATH] and is parallel along geodesics [MATH].', '1403.7020-2-20-8': 'Consider a smooth cutoff function satisfying [EQUATION]', '1403.7020-2-20-9': "Mehler's kernel is now defined via [EQUATION]", '1403.7020-2-20-10': 'First fix a point [MATH] and a set of geodesic coordinates centered at [MATH].', '1403.7020-2-20-11': 'Now choose a basis [MATH] for [MATH] and a basis [MATH] for [MATH].', '1403.7020-2-20-12': 'Parallel transport this basis along geodesics using the connections [MATH] to obtain trivializations [MATH] and [MATH] of [MATH] and [MATH] respectively near [MATH].', '1403.7020-2-20-13': 'Now define local orthonormal sections of [MATH] via [EQUATION]', '1403.7020-2-20-14': 'The connection [MATH] can be expressed in this frame and these coordinates as [EQUATION] where each [MATH] is a Christoffel symbol of [MATH] (or [MATH] copies of it) and each [MATH] is a Christoffel symbol of the spin connection on [MATH].', '1403.7020-2-20-15': 'Since the section [MATH] is obtained via parallel transport along geodesics, the connection coefficient [MATH] maybe written in terms of the curvature [MATH] of [MATH] via [EQUATION] with the Einstein summation convention being used.', '1403.7020-2-20-16': 'The dependence of the curvature coefficients [MATH] on the parameter [MATH] is seen to be linear [MATH] despite the fact that they are expressed in the [MATH] dependent frame [MATH].', '1403.7020-2-20-17': 'This is because a gauge transformation from an [MATH] independent frame into [MATH] changes the curvature coefficient by conjugation.', '1403.7020-2-20-18': 'Since [MATH] is a line bundle this is conjugation by a function and hence does not change the coefficient.', '1403.7020-2-20-19': 'Next, using the Taylor expansion [MATH], we see that the connection [MATH] has the form [EQUATION]', '1403.7020-2-20-20': 'Here [MATH], [MATH] and [MATH] are all independent of [MATH].', '1403.7020-2-21-0': "Now using Weitzenbock's formula, we note that the operator [MATH] has the form [EQUATION]", '1403.7020-2-21-1': 'Here [MATH] and [MATH] are each smooth endomorphisms of [MATH] independent of [MATH].', '1403.7020-2-21-2': 'Since [MATH] we now have [EQUATION]', '1403.7020-2-21-3': "Note that the right hand side of eq:hdhm is zero for [MATH] since [MATH] is supported in a geodesic neighborhood of the diagonal, by eq:Mehler's kernel def.", '1403.7020-2-21-4': "From the defining equations eq:function m_t in Mehler's kernel and eq:Mehler's kernel def, Mehler's kernel is given in geodesic coordinates via [EQUATION]", '1403.7020-2-21-5': "We now differentiate eq:Mehler's kernel coordinates using eq:dssf-eq:ser to compute the right hand side of eq:hdhm.", '1403.7020-2-21-6': "By Mehler's formula, see section 4.2 in [CITATION], we have [MATH] for [MATH].", '1403.7020-2-21-7': 'Differentiating the rest, we observe that the right hand side of eq:hdhm has the form of a finite sum [EQUATION]', '1403.7020-2-21-8': "Now since the kernels [MATH] and [MATH] both have the same asymptotics as [MATH], Duhamel's principle, using eq:hdhm, gives [EQUATION]", '1403.7020-2-21-9': 'Now we substitute eq:hdsf into eq:ddhm.', '1403.7020-2-21-10': 'Following this substitution, we use the heat kernel bound eq:mest, eq:function exponential bound and the bound [EQUATION] for constants [MATH] and [MATH] independent of [MATH].', '1403.7020-2-21-11': 'These bounds can be used to estimate the right hand side of eq:ddhm by a sum of finitely many terms of the form [EQUATION] with each multi-index [MATH] above satisfying eq:iin.', '1403.7020-2-21-12': 'Finally, eq:iin and the inequalities [EQUATION] (see sec:Estimates-on-Gaussian for a proof of eq:ehi2) give eq:mhest.', '1403.7020-2-22-0': '## Bound on the trace of [MATH]', '1403.7020-2-23-0': 'We now turn to bound the pointwise [MATH].', '1403.7020-2-23-1': 'To this end, first consider the expansion for the heat kernel [MATH] given by [EQUATION]', '1403.7020-2-23-2': 'Here the coefficients [MATH] are smooth sections defined on the neighborhood [MATH] of the diagonal in [MATH].', '1403.7020-2-23-3': 'They are generated by solving a recursive system of transport equations along geodesics as in chapter 7 of [CITATION].', '1403.7020-2-23-4': 'The kernel [MATH] of [MATH] is simply [MATH].', '1403.7020-2-23-5': 'It hence has an expansion given by [EQUATION]', '1403.7020-2-23-6': 'By eq:cutoff, [MATH] is an endomorphism of the spin bundle supported in the region where [MATH].', '1403.7020-2-23-7': 'By eq:pointwise trace asymp as t->0 and eq:dtre, the pointwise trace [MATH] along the diagonal has an expansion starting with a leading term of order [MATH].', '1403.7020-2-23-8': 'Since the restriction to the diagonal of the [MATH] term in eq:dtre is zero, this implies that [EQUATION] at each point on the diagonal.', '1403.7020-2-23-9': 'To bound the trace of [MATH] we will need a lemma giving a schematic form for the coefficients [MATH].', '1403.7020-2-23-10': 'We again work in geodesic coordinates centered at a point [MATH].', '1403.7020-2-23-11': 'Each heat kernel coefficient can be written in terms of the frame eq:parallel transport frame as [EQUATION] for some functions [MATH].', '1403.7020-2-24-0': 'Each function [MATH] appearing in eq:expansion of hk coeff special frame can be written as a finite sum [EQUATION] for some functions [MATH], independent of [MATH].', '1403.7020-2-24-1': 'Moreover, each multi-index [MATH] appearing in eq:schf satisfies the inequality [EQUATION]', '1403.7020-2-24-2': 'The heat kernel coefficients [MATH] are given, as in chapter 7 of [CITATION], by the recursion [EQUATION] where [MATH] denotes the determinant of the metric on [MATH].', '1403.7020-2-24-3': 'Hence [MATH] is clearly seen to be of the form eq:schf.', '1403.7020-2-24-4': 'Equations eq:dssf-eq:ser and eq:rec2 imply that [MATH] has the form eq:schf assuming it to be true for [MATH].', '1403.7020-2-24-5': 'The lemma now follows by induction on [MATH].', '1403.7020-2-25-0': 'Following this we are ready to bound the pointwise trace [MATH].', '1403.7020-2-25-1': 'The above lemma will play an important role in the proposition below.', '1403.7020-2-26-0': 'Consider the remainder obtained after subtracting the first [MATH] terms of the kernel expansion eq:dtre [EQUATION]', '1403.7020-2-26-1': 'From eq:dtre and eq:trace of coeff.', '1403.7020-2-26-2': 'L_t vanish we see that [EQUATION] and it hence suffices to bound [MATH].', '1403.7020-2-26-3': 'We clearly have [MATH] with [MATH] being the analogous remainder in the kernel expansion for the heat trace [EQUATION]', '1403.7020-2-26-4': 'Let us denote [EQUATION]', '1403.7020-2-26-5': 'The result of applying the heat operator to eq:def remainder D(heat trace exp) is then [EQUATION]', '1403.7020-2-26-6': 'In other words, [MATH] is the sum of the terms obtained when some derivative differentiates the cutoff function [MATH] in eq:def remainder D(heat trace exp).', '1403.7020-2-26-7': "Now since [MATH] as [MATH], Duhamel's principle applied to eq:heat operator applied to L gives [EQUATION]", '1403.7020-2-26-8': 'We first bound [MATH], again working in geodesic coordinates and the frame eq:parallel transport frame.', '1403.7020-2-26-9': 'Using eq:cdic and the fact that Clifford multiplication is parallel, the Dirac operator is seen to be of the form [EQUATION] where [MATH], [MATH] and [MATH] are endomorphisms of [MATH] independent of [MATH].', '1403.7020-2-26-10': 'Using eq:expansion of hk coeff special frame,eq:formula for S_t,eq:formula for R_t and eq:dic we may write [EQUATION] in the frame eq:parallel transport frame for some coefficient functions [MATH].', '1403.7020-2-26-11': 'By eq:schf each [MATH] can be written as a finite sum [EQUATION] for some functions [MATH] independent of [MATH].', '1403.7020-2-26-12': 'Since [MATH] in a neighborhood of the diagonal, eq:formula for R_t implies each [MATH] vanishes to infinite order near the diagonal.', '1403.7020-2-26-13': 'Hence by eq:main index inequality we may assume that each multi-index [MATH] in eq:f0 sch form satisfies [MATH].', '1403.7020-2-26-14': 'We now substitute eq:expansion R orth frame and eq:f0 sch form in eq:formula F. Using this substitution along with the heat kernel bound eq:mest, we may bound [MATH] by a sum of terms of the form eq:generic form integral each satisfying [MATH].', '1403.7020-2-26-15': 'The inequalities eq:ehi1-eq:ehi2 then give the estimate [EQUATION]', '1403.7020-2-26-16': 'Next we estimate [MATH].', '1403.7020-2-26-17': 'Following eq:formula for S_t and using the cancellations in the kernel expansion, resulting from the transport equation, we see that [EQUATION]', '1403.7020-2-26-18': 'Equation eq:formula E now gives', '1403.7020-2-27-0': 'We denote by [MATH] and [MATH] the kernels obtained by replacing [MATH] in eq:rduha by [MATH] and [MATH] respectively', '1403.7020-2-28-0': 'It is clear that [EQUATION]', '1403.7020-2-28-1': 'We first bound [MATH], again working in geodesic coordinates and the frame eq:parallel transport frame.', '1403.7020-2-28-2': 'In terms of the orthonormal frame we may write [EQUATION] for some coefficient functions [MATH] and [MATH].', '1403.7020-2-28-3': 'Using eq:schf and eq:dic these can be expressed as finite sums [EQUATION] where [MATH] are functions independent of [MATH] and [MATH] are finite subsets of [MATH].', '1403.7020-2-28-4': 'Moreover, eq:main index inequality now gives [EQUATION]', '1403.7020-2-28-5': 'Again we substitute eq:dschf-eq:schematic form coeffs 2 into eq:rduha U.', '1403.7020-2-28-6': 'This substitution, the bound eq:mhest, along with the inequalities eq:ehi1-eq:ehi2 and eq:schematic form coeffs 1-eq:schematic form coeffs 2 now give the estimate [EQUATION]', '1403.7020-2-28-7': 'Next we estimate [MATH].', '1403.7020-2-28-8': 'First we use a Taylor expansion to write [EQUATION] where each of [MATH] and [MATH] is an even function in these coordinates.', '1403.7020-2-28-9': 'We now let [EQUATION] and define [EQUATION]', '1403.7020-2-28-10': 'Clearly, by eq:schematic form coeffs 1-eq:schematic form coeffs 2 and eq:Taylor expansion odd even-eq:splitting f_ab 2, [EQUATION]', '1403.7020-2-28-11': 'Next we claim that the contribution of [MATH] to [MATH] is zero.', '1403.7020-2-28-12': 'To see this, first observe that [MATH] implies [MATH] is odd by eq:S1 bar.', '1403.7020-2-28-13': 'Hence [MATH] is an odd function, using eq:splitting f_ab 1 and the fact that [MATH] is even.', '1403.7020-2-28-14': 'Hence the integral corresponding to [MATH] in eq:rduha V is zero, being the integral of an odd function in these coordinates.', '1403.7020-2-28-15': 'Similarly, we claim that the contribution of [MATH] to [MATH] is zero.', '1403.7020-2-28-16': 'This time, [MATH] implies [MATH] is even by eq:S2 bar.', '1403.7020-2-28-17': 'Hence [MATH] is an even function using eq:splitting f_ab 2.', '1403.7020-2-28-18': 'However the integral corresponding to [MATH] in eq:rduha V is still the integral of an odd function in these coordinates, because of the [MATH] term in eq:rduha V.', '1403.7020-2-29-0': 'Following this the contribution of [MATH] to [MATH] can be bounded by a finite sum of terms of the form [EQUATION]', '1403.7020-2-29-1': 'Again using the inequalities eq:ehi1-eq:ehi2, and estimating the contribution of [MATH] in similar fashion, gives the estimate [EQUATION]', '1403.7020-2-29-2': 'Following eq:uest, eq:vest and eq:E=00003DU+V we obtain the estimate [EQUATION] for constants [MATH] and [MATH] independent of [MATH].', '1403.7020-2-29-3': 'Equations eq:L=00003DE+F, eq:F estimate and eq:E est then give [EQUATION] for constants [MATH] and [MATH] independent of [MATH].', '1403.7020-2-29-4': 'The proposition now follows from eq:tr(L_t)=00003Dtr(L_t^(n-1)/2) and eq:L est.', '1403.7020-2-30-0': '# Asymptotics of the spectral measure', '1403.7020-2-31-0': 'We now consider the rescaled operator [MATH].', '1403.7020-2-31-1': 'Here we shall find the asymptotics of its spectral measure using the heat trace estimates of the previous section.', '1403.7020-2-31-2': 'We first consider the heat traces of [MATH].', '1403.7020-2-32-0': 'For any [MATH], [EQUATION] where [MATH] is as defined by eq:antisymm endomorphism A.', '1403.7020-2-32-1': 'The convergences above are uniform in compact intervals of [MATH] respectively.', '1403.7020-2-33-0': 'If [MATH] denotes the kernel of [MATH] it is clear that [MATH] after rescaling.', '1403.7020-2-33-1': 'Hence proposition prop:Heat Mehler comparison gives the estimate [EQUATION] for some constants [MATH] independent of [MATH].', '1403.7020-2-33-2': 'Hence [MATH].', '1403.7020-2-33-3': 'It now remains to compute [MATH] in order to prove eq:rescaled heat trace limit.', '1403.7020-2-33-4': "By eq:Mehler's kernel def, the highest order part in [MATH] of [MATH] is given by [EQUATION].", '1403.7020-2-33-5': 'Since [MATH] is an antisymmetric endomorphism it maybe diagonalized to give an orthonormal basis [MATH] of [MATH] with eigenvalues [MATH] respectively.', '1403.7020-2-33-6': 'We hence have [MATH] and [EQUATION]', '1403.7020-2-33-7': 'Now if [MATH], we have the commutation [EQUATION].', '1403.7020-2-33-8': 'This shows that the only traceless terms in the expansion of eq:Exponential Clifford multiplication are the constants and hence [MATH].', '1403.7020-2-33-9': 'Equation eq:rescaled heat trace limit now follows.', '1403.7020-2-33-10': 'For the second part of the theorem note that proposition prop:trace Dexp(td2) estimate gives the estimate [EQUATION] for uniform constants [MATH] independent of [MATH].', '1403.7020-2-33-11': 'From this equation, eq:rescaled tr(Dexp(tD2)) estimate follows.', '1403.7020-2-33-12': 'The uniformity of the limits eq:rescaled heat trace limit,eq:rescaled tr(Dexp(tD2)) estimate is also an easy consequence of the estimates eq:rescaled heat kernel bound and eq:rescaled trace bound.', '1403.7020-2-34-0': 'The above theorem also follows from the rescaling argument as described in section c) of [CITATION].', '1403.7020-2-35-0': 'Next we let [MATH]) denote the number of eigenvalues of [MATH] in the interval [MATH].', '1403.7020-2-35-1': 'We also use the notation [MATH] for any [MATH].', '1403.7020-2-35-2': 'We shall need the following estimates.', '1403.7020-2-36-0': 'We begin with [EQUATION] using proposition prop:Maximum Principle estimate.', '1403.7020-2-36-1': 'This gives [MATH] from whicheq:Counting function bound follows on substituting [MATH].', '1403.7020-2-37-0': 'For the second part we estimate [EQUATION]', '1403.7020-2-37-1': 'Finally to prove the third part note that eq:odd trace cancellation is true for the family of odd Schwartz functions [MATH] on account of eq:rescaled tr(Dexp(tD2)) estimate.', '1403.7020-2-37-2': 'Since the convergence in eq:rescaled tr(Dexp(tD2)) estimate is uniform it may be differentiated to obtain eq:odd trace cancellation for the odd functions [MATH].', '1403.7020-2-37-3': 'Now eq:Functional trace bound along with the fact that the span of [MATH] is dense in the space of odd Schwartz functions gives eq:odd trace cancellation.', '1403.7020-2-38-0': 'Now consider the rescaled spectral measure of [MATH] given by [EQUATION].', '1403.7020-2-38-1': 'Let [MATH] denote the space of bounded, continuous functions on [MATH] vanishing at [MATH].', '1403.7020-2-38-2': 'Consider the Banach space [MATH]S_2k[MATH][MATH]^TX[MATH]TX^1,0,TX^0,1[MATH]^TX^1,0,^TX^0,1[MATH]^TX[MATH]TX^1,0,TX^0,1[MATH]R^TX^1,0,R^TX^0,1[MATH] _i _i=2^[MATH](R^TX+2J+)^N[MATH]R^TX,2J[MATH][MATH][R^TX,J]=0[MATH]R^TX[MATH][MATH]a=(a_1,,a_k+1)N_0^k+1[MATH]a=_i=1^k+1a_i.', '1403.7020-2-38-3': "[MATH]aN_0^k+1[MATH]a=N-k[MATH]2N2k[MATH]J(R^TX+2J+)^N-1[MATH]R^TX[MATH][MATH]aN_0^k+1[MATH]a=N-1-k[MATH]N2k+N2-1k[MATH]J(R^TX+2J+)^N-2[MATH][MATH]aN_0^k+1[MATH]a=N-2-k[MATH]N2-1k[MATH](R^TZ)^N[MATH]d[MATH]e^*[MATH](R^TZ)^N[MATH]de^*J,d_1,e^*_2[MATH]^2e^*_3[MATH]P_i[MATH]i[MATH](R^TX)^N[MATH]P_0,P_1[MATH]P_2[MATH]P_0[MATH]P_1[MATH]P_1[MATH]d_1,e^*_2[MATH]^2e^*_3[MATH]de^*J[MATH]_1,_2[MATH]_3[MATH]T^HY[MATH]T^VY[MATH]P_2[MATH]P_2[MATH]d[MATH]e^*[MATH]de^*J[MATH]d_1,e^*_2[MATH]^2e^*_3[MATH]P_2^1[MATH](R^TX)^N[MATH]d_1,e^*_2[MATH]de^*J[MATH]^2e^*_3[MATH]P_2^2[MATH](R^TX)^N[MATH]_1[MATH]T^VY[MATH]T^HY[MATH]_2[MATH]T^HY[MATH]T^VY[MATH]A,B[MATH]C[MATH]R^TX,2J[MATH][MATH]d_1,e^*_2[MATH]d_1[MATH]^2e^*_3[MATH]de^*J[MATH]e^*_2[MATH]_1[MATH]_3[MATH]T^HY[MATH]T^VY[MATH]T^HY[MATH]X[MATH]g^TX[MATH]l[MATH]l[MATH]J=-[MATH]X[MATH][MATH][MATH]_Xl[MATH]LX[MATH]c_1(L)=l[MATH]L[MATH]R=2[MATH]L[MATH]Y[MATH]L[MATH]X[MATH]^0,[MATH][MATH]r=0[MATH][MATH]r^2g^TS^11l^*g^TX[MATH]2l[MATH]C_1,C_2[MATH]C_3[MATH](Y,g)[MATH]x,zY[MATH]t,t'>0[MATH](1).", '1403.7020-2-38-4': '[MATH]B= y(x,y)<i_g [MATH]B[MATH]B^c[MATH]B[MATH]_R^ne^-r^24t(4t)^n2dr=1.', '1403.7020-2-38-5': '[MATH]B^c[MATH]e^-(x,y)^24t(4t)^n2(n/2)!', '1403.7020-2-38-6': '^n/2i_g^n[MATH]_B^ch_t(x,y)dy(n/2)!', '1403.7020-2-38-7': '^n/2i_g^nvol(Y)[MATH](2).', '1403.7020-2-38-8': "[MATH]tt'.", '1403.7020-2-38-9': "[MATH](x,y)^2t+(y,z)^2t'(x,z)^22(t+t')[MATH]1t'2t+t'[MATH]e^-(y,z)^28t'1[MATH]2^nh_4(t+t')(x,z)(_Yh_2t(x,y)dy).", '1403.7020-2-38-10': '[MATH]'}

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'1403.7020-2-17-8'], ['1403.7020-1-25-0', '1403.7020-2-25-0'], ['1403.7020-1-25-1', '1403.7020-2-25-1'], ['1403.7020-1-34-0', '1403.7020-2-34-0'], ['1403.7020-1-10-0', '1403.7020-2-10-0'], ['1403.7020-1-10-1', '1403.7020-2-10-1'], ['1403.7020-1-10-2', '1403.7020-2-10-2'], ['1403.7020-1-4-0', '1403.7020-2-4-0'], ['1403.7020-1-4-1', '1403.7020-2-4-1'], ['1403.7020-1-4-2', '1403.7020-2-4-2'], ['1403.7020-1-4-3', '1403.7020-2-4-3'], ['1403.7020-1-38-0', '1403.7020-2-38-0'], ['1403.7020-1-38-1', '1403.7020-2-38-1'], ['1403.7020-1-28-0', '1403.7020-2-28-0'], ['1403.7020-1-28-1', '1403.7020-2-28-1'], ['1403.7020-1-28-2', '1403.7020-2-28-2'], ['1403.7020-1-28-3', '1403.7020-2-28-3'], ['1403.7020-1-28-4', '1403.7020-2-28-4'], ['1403.7020-1-28-5', '1403.7020-2-28-5'], ['1403.7020-1-28-6', '1403.7020-2-28-6'], ['1403.7020-1-28-7', '1403.7020-2-28-7'], ['1403.7020-1-28-8', '1403.7020-2-28-8'], ['1403.7020-1-28-9', '1403.7020-2-28-9'], ['1403.7020-1-28-10', '1403.7020-2-28-10'], ['1403.7020-1-28-11', '1403.7020-2-28-11'], ['1403.7020-1-28-12', '1403.7020-2-28-12'], ['1403.7020-1-28-13', '1403.7020-2-28-13'], ['1403.7020-1-28-14', '1403.7020-2-28-14'], ['1403.7020-1-28-15', '1403.7020-2-28-15'], ['1403.7020-1-28-16', '1403.7020-2-28-16'], ['1403.7020-1-28-17', '1403.7020-2-28-17'], ['1403.7020-1-28-18', '1403.7020-2-28-18'], ['1403.7020-1-5-0', '1403.7020-2-5-0'], ['1403.7020-1-5-1', '1403.7020-2-5-1'], ['1403.7020-1-5-2', '1403.7020-2-5-2'], ['1403.7020-1-5-3', '1403.7020-2-5-3'], ['1403.7020-1-5-4', '1403.7020-2-5-4'], ['1403.7020-1-5-5', '1403.7020-2-5-5'], ['1403.7020-1-5-6', '1403.7020-2-5-6'], ['1403.7020-1-18-0', '1403.7020-2-18-0'], ['1403.7020-1-18-1', '1403.7020-2-18-1'], ['1403.7020-1-18-2', '1403.7020-2-18-2'], ['1403.7020-1-23-0', '1403.7020-2-23-0'], ['1403.7020-1-23-1', '1403.7020-2-23-1'], ['1403.7020-1-23-2', '1403.7020-2-23-2'], ['1403.7020-1-23-3', '1403.7020-2-23-3'], ['1403.7020-1-23-4', '1403.7020-2-23-4'], ['1403.7020-1-23-5', '1403.7020-2-23-5'], ['1403.7020-1-23-6', '1403.7020-2-23-6'], ['1403.7020-1-23-7', '1403.7020-2-23-7'], ['1403.7020-1-23-8', '1403.7020-2-23-8'], ['1403.7020-1-23-9', '1403.7020-2-23-9'], ['1403.7020-1-23-10', '1403.7020-2-23-10'], ['1403.7020-1-23-11', '1403.7020-2-23-11'], ['1403.7020-1-37-0', '1403.7020-2-37-0'], ['1403.7020-1-37-1', '1403.7020-2-37-1'], ['1403.7020-1-37-2', '1403.7020-2-37-2'], ['1403.7020-1-37-3', '1403.7020-2-37-3']]

[]

[]

[]

[]

['1403.7020-1-29-4', '1403.7020-1-33-2', '1403.7020-1-38-2', '1403.7020-1-38-3', '1403.7020-1-38-4', '1403.7020-1-38-5', '1403.7020-1-38-6', '1403.7020-1-38-7', '1403.7020-1-38-8', '1403.7020-1-38-9', '1403.7020-1-38-10', '1403.7020-2-29-4', '1403.7020-2-33-2', '1403.7020-2-38-2', '1403.7020-2-38-3', '1403.7020-2-38-4', '1403.7020-2-38-5', '1403.7020-2-38-6', '1403.7020-2-38-7', '1403.7020-2-38-8', '1403.7020-2-38-9', '1403.7020-2-38-10']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1403.7020

0911.5228

{'0911.5228-1-0-0': 'We study the string instanton wrapping a non-trivial two-cycle in [MATH] of the type IIA string theory compactified on [MATH] superspace and find that it has twelve fermionic zero modes associated with 1/2 of the supersymmetry of the background.', '0911.5228-1-0-1': 'Being goldstinos, the instanton fermionic zero modes thus spontaneously break supersymmetry of the string worldsheet theory.', '0911.5228-1-1-0': '# Introduction', '0911.5228-1-2-0': 'M-theory compactified on [MATH] and a corresponding Type IIA superstring theory compactified on [MATH] are on the bulk side of the [MATH] holography whose boundary superconformal Chern-Simons-matter theory is assumed to provide an effective worldvolume description of a stack of multiple M2-branes [CITATION].', '0911.5228-1-2-1': 'This new [MATH] correspondence shares some features with the well studied [MATH] correspondence whose bulk theory is Type IIB superstring theory compactified on [MATH] and the boundary theory is the superconformal [MATH], [MATH] super Yang-Mills theory.', '0911.5228-1-3-0': 'However, the two examples of the [MATH] correspondence are quite different.', '0911.5228-1-3-1': 'First of all, the Type IIB superstring in [MATH] is maximally supersymmetric.', '0911.5228-1-3-2': 'Its 32 supersymmetries are part of the superisometry group [MATH].', '0911.5228-1-3-3': 'On the other hand the [MATH] solution of [MATH] supergravity has 32 supersymmetries only for [MATH], while for [MATH] the theory is invariant under 24 supersymmetries and so is its ten-dimensional counterpart, the Type IIA superstring theory in the [MATH] supergravity background whose superisometries form the supergroup [MATH] [CITATION].', '0911.5228-1-3-4': 'As a result, while the Green-Schwarz action for the [MATH] superstring amounts to a worldsheet sigma-model on the supercoset space [MATH] [CITATION], the target superspace of the Green-Schwarz action of the [MATH] superstring is not a supercoset space, though it possesses the [MATH] isometry [CITATION].', '0911.5228-1-3-5': 'Only when the superstring is extended in [MATH], its dynamics can be described by the [MATH] supercoset sigma-model [CITATION].', '0911.5228-1-3-6': 'It can be obtained from the complete Green-Schwarz action [CITATION] by partially gauge fixing kappa-symmetry in a way which puts to zero the eight worldsheet fermionic modes corresponding to the eight broken supersymmetries of the [MATH] superbackground.', '0911.5228-1-3-7': 'Such a gauge fixing is admissible only when the string moves in [MATH].', '0911.5228-1-3-8': 'This gauge is, however not admissible when the string moves entirely in the [MATH] part of the superbackground.', '0911.5228-1-3-9': 'One of the consequences of this peculiar situation is that though the subsector of the [MATH] superstring theory described by the supercoset sigma-model is classically integrable [CITATION], the explicit proof of the classical integrability of the complete [MATH] superstring still remains an open problem.', '0911.5228-1-3-10': 'This is because the complete [MATH] superspace is not a supercoset space, and the methods used to prove the integrability of the [MATH] superstring [CITATION] and the supercoset sector of the [MATH] superstring [CITATION] do not apply.', '0911.5228-1-4-0': 'Another interesting peculiarity of the [MATH] superstring, which the [MATH] superstring does not have, is the existence on [MATH] of string instantons.', '0911.5228-1-4-1': 'They are formed in the Wick rotated theory by the string worldsheet wrapping a topologically non-trivial two-cycle of [MATH].', '0911.5228-1-4-2': 'This two-cycle is a [MATH] corresponding to the closed Kahler two-form [MATH] on [MATH].', '0911.5228-1-4-3': 'As we shall show, also in the case of the string instantons on [MATH] the consistent gauge fixing of kappa-symmetry does not allow reducing the string action to the supercoset sigma model, i.e. to eliminate, by using kappa-symmetry, the eight fermionic modes corresponding to the broken supersymmetries.', '0911.5228-1-5-0': 'The main goal of this paper is to study the superstring instanton on [MATH] and analyze its fermionic zero modes.', '0911.5228-1-5-1': 'In the case of branes moving in a supersymmetric background, their fermionic equations have solutions which are associated with the Killing spinors of the background that guarantee its supersymmetries.', '0911.5228-1-5-2': 'The number of physical modes of the worldvolume Dirac operator associated with the Killing spinors is equal to the number of components of the Killing spinors which are not annihilated by the kappa-symmetry projector of the brane.', '0911.5228-1-5-3': 'If the background is maximally supersymmetric, one concludes that the number of dynamical zero modes on the brane are half the number of supersymmetries, since the rank of the kappa-symmetry projector is equal to half the number of the maximal supersymmetries of the background.', '0911.5228-1-5-4': 'When the background is not maximally supersymmetric, as the [MATH] one, the number of the brane fermion zero modes associated with unbroken supersymmetries depends on how many of them are not eliminated by the kappa-symmetry projector.', '0911.5228-1-5-5': 'The action of the kappa-symmetry projector and the corresponding number of fermionic zero modes depends on how the given brane configuration is embedded into target space.', '0911.5228-1-5-6': 'In the cases with less supersymmetries the worldvolume Dirac equation may, in general, also have solutions which are not associated with unbroken supersymmetries, but with the broken ones.', '0911.5228-1-5-7': 'Thus the analysis of the brane fermionic modes in less supersymmetric backgrounds should be made case by case (see e.g. [CITATION] for a more detailed discussion of this point).', '0911.5228-1-6-0': 'As we shall see, in the [MATH] case the string instanton on [MATH] has twelve zero modes all of which are associated with unbroken supersymmetries of the background and there are no zero modes of the fermions corresponding to broken supersymmetries.', '0911.5228-1-6-1': 'It is interesting that these twelve zero modes are divided into eight and four ones which have different geometrical and physical meaning.', '0911.5228-1-6-2': 'The eight massive fermionic zero modes are four copies of the two-component Killing spinor on [MATH] and the four other fermionic modes are two copies of massless chiral and anti-chiral fermion on [MATH] electrically coupled to the electromagnetic potential created on [MATH] by a monopole placed in the center of [MATH].', '0911.5228-1-6-3': 'The monopole potential arises as part of the [MATH] spin connection pulled-back on the instanton [MATH].', '0911.5228-1-6-4': 'This, at least formally and remotely, reminds us the peculiarity of the presence of different, light and heavy, physical worldsheet degrees of freedom in a Penrose limit of the [MATH] superstring [CITATION].', '0911.5228-1-7-0': 'The presence of the string instanton and its fermionic zero modes may generate non-perturbative corrections to the string effective action, which may affect its properties and if so should be taken into account in studying, e.g. the [MATH] correspondence.', '0911.5228-1-7-1': 'The instantons may, perhaps, contribute to the worldsheet S-matrix and/or to energies of a semiclassical string.', '0911.5228-1-7-2': 'To study these effects one needs to find a way of merging the instanton and Minkowski solutions, such as spinning strings or BMN geodesics.', '0911.5228-1-7-3': 'In addition, in the presence of the instanton fermionic zero modes, the worldsheet correlator, to be non-zero, should contain a number of fermion insertions.', '0911.5228-1-8-0': 'The paper is organized as follows.', '0911.5228-1-8-1': 'In Section [REF] we review the form of the string action in the [MATH] superbackground and truncate it to the second order in fermions.', '0911.5228-1-8-2': 'In Section [REF] we describe the bosonic part of the string instanton solution and in Section [REF] we study its fermionic zero modes.', '0911.5228-1-8-3': 'Appendix A contains a description of our conventions and notation and of the geometry of the [MATH] superspace.', '0911.5228-1-8-4': 'In Appendix B we give the explicit form of the [MATH] Fubini-Study metric, vielbeins and connection which were used for the analysis of the string fermion equations.', '0911.5228-1-9-0': '# [MATH] superstring action up to the quadratic order in fermions', '0911.5228-1-10-0': 'To simplify the study of the fermionic zero modes of the string instanton we reduce the complete superstring action of [CITATION] to the quadratic order in fermions (though, the solutions we find satisfy the complete non-linear equations of motion to all orders in fermions).', '0911.5228-1-10-1': 'Alternatively, one can use the quadratic type IIA superstring action derived in [CITATION] for a generic superbackground and substitute into it the values of the supergravity fields corresponding to the [MATH] background.', '0911.5228-1-10-2': 'As a consistency check we have performed the reduction of both of the actions.', '0911.5228-1-10-3': 'We shall see that they give the same result upon a redefinition of bosonic [MATH] coordinates of the target-superspace of [CITATION].', '0911.5228-1-10-4': 'This redefinition is required due to a particular parametrization used in [CITATION] to construct an explicit form of the [MATH] supergeometry.', '0911.5228-1-11-0': 'The Green-Schwarz superstring action in a generic type IIA supergravity background has the well known form [EQUATION] where [MATH] are the worldsheet coordinates, [MATH] is an intrinsic worldsheet metric, [MATH] are worldsheet pullbacks of target superspace vector supervielbeins and [MATH] is the pull-back of the NS-NS 2-form.', '0911.5228-1-12-0': 'The kappa-symmetry transformations of the worldsheet fields [MATH] which leave the superstring action ([REF]) invariant (provided the superbackground obeys the superspace supergravity constraints) are [EQUATION] where [MATH] is a 32-component spinor parameter, [MATH] is a spinor projection matrix with [EQUATION] where [MATH] is the induced metric on the worldsheet.', '0911.5228-1-12-1': 'The explicit form of the supervielbeins [MATH] and the NS-NS 2-form [MATH] which describe the geometry of the [MATH] superspace are given in Appendix A (see also [CITATION]).', '0911.5228-1-13-0': 'Up to second order in fermions the supervielbeins and the [MATH]-field have the following form 0 [EQUATION] [EQUATION] where [MATH] is the vacuum expectation value of the dilaton, [MATH] is the radius of the [MATH] sphere whose base is [MATH], [MATH] is the eleven-dimensional Planck length related to the string tension as follows [MATH] and [MATH] is the Chern-Simons level related to the units of [MATH] and [MATH] flux which support the [MATH] solution of Type IIA supergravity.', '0911.5228-1-13-1': 'The 32-component fermionic variable [MATH] is split by projectors [MATH] and [MATH] (see Appendix A.5) into the 24-component spinors [MATH]; [MATH]) which correspond to the 24 supersymmetries of the [MATH] solution and the 8-component spinors [MATH] which correspond to the broken supersymmetries.', '0911.5228-1-13-2': 'The index [MATH] is a spinor index of [MATH] (see Appendix A for more details).', '0911.5228-1-13-3': 'The covariant derivative [MATH] is defined as follows [EQUATION] where [MATH], [MATH] and [MATH] are, respectively the vielbein, connection and Dirac-matrices of [MATH] of radius [MATH].', '0911.5228-1-13-4': '[MATH] and [MATH] are, respectively, the vielbein and connection on [MATH] of radius [MATH] and [MATH] are [MATH] gamma-matrices of [MATH].', '0911.5228-1-13-5': '[MATH] is the RR one-form potential whose field strength is the Kahler form on [MATH], [MATH].', '0911.5228-1-13-6': 'See Appendix A for more details regarding the notation and conventions.', '0911.5228-1-14-0': 'Substituting the expressions for the vielbeins [REF] and the NS-NS two-form [REF] into the action [REF] and keeping only terms up to quadratic order in fermions we get the following action [EQUATION] where [MATH] is the worldsheet pullback of the conventional [MATH] covariant derivative.', '0911.5228-1-15-0': 'The first two lines of this action coincide with the action which one gets by reducing to [MATH] the quadratic Green-Schwarz action in a generic type IIA superbackground [CITATION] .', '0911.5228-1-15-1': 'The last term in the action appeared because of our choice of parametrization of the [MATH] superspace which allowed us to write its geometry in the simplest form.', '0911.5228-1-15-2': 'It is not hard to see that the last term in [REF] can be canceled (modulo higher order terms in fermions) by making the following shift of the bosonic coordinates [MATH] of [MATH] [EQUATION]', '0911.5228-1-15-3': 'After this field redefinition the two forms of the string action become equivalent.', '0911.5228-1-16-0': 'To study the string instantons we should perform a Wick rotation of the worldsheet and the target space in the action [REF] to Euclidean signature.', '0911.5228-1-16-1': 'The Wick rotation basically consists in replacing [MATH] and [MATH], respectively with [MATH] and [MATH], replacing [MATH] with [MATH] and taking into account that the fermions [MATH] become complex spinors, since there are no Majorana spinors in ten-dimensional Euclidean space.', '0911.5228-1-16-2': 'However, the complex conjugate spinors do not appear in the Wick rotated action and, hence, the number of the fermionic degrees of freedom formally remains the same as before the Wick rotation.', '0911.5228-1-16-3': 'Note also that the Euclidean [MATH] is defined as [MATH], where [MATH] is the Wick rotated [MATH].', '0911.5228-1-16-4': 'So [MATH] as in the case of Minkowski signature.', '0911.5228-1-17-0': 'Thus, after the redefinition [REF] and the Wick rotation the action takes the following form [EQUATION] and the kappa-symmetry matrix [MATH] gets replaced by [EQUATION]', '0911.5228-1-18-0': '# String instanton wrapping a two-sphere inside [MATH] We are interested in a string whose worldsheet wraps a topologically non-trivial two-cycle inside [MATH] and thus is a stringy counterpart of the instantons of two-dimensional [MATH] sigma-models.', '0911.5228-1-18-1': 'To be topologically non-trivial this two-cycle should have a non-zero pull-back on its worldsheet of the Kahler two-form [MATH] of [MATH].', '0911.5228-1-18-2': 'Such a two-cycle is a [MATH] subspace of [MATH].', '0911.5228-1-18-3': 'To identify it, it is convenient to consider the form of the Fubini-Study metric on [MATH] given in [CITATION] [EQUATION] where [MATH], [MATH] and [MATH], and [MATH] 0 [EQUATION] are three left-invariant one-forms on [MATH] obeying [MATH] etc. (see Appendix B for more details).', '0911.5228-1-18-4': 'Notice that with this choice of the [MATH] coordinates, [MATH] and [MATH] parameterize a two-sphere of radius [MATH].', '0911.5228-1-18-5': 'This two-sphere is topologically non-trivial and associated to the Kahler form on [MATH].', '0911.5228-1-18-6': 'The string instanton wraps this sphere.', '0911.5228-1-18-7': 'For instance, if it wraps the sphere once [MATH] and [MATH] can be identified with the string worldsheet coordinates, while all other [MATH] as well as [MATH] coordinates are worldsheet constants for the instanton solution.', '0911.5228-1-18-8': 'Thus the pullback on the string instanton of the metric [REF] of [MATH] (of radius [MATH]) is the metric of the sphere of radius [MATH] [EQUATION]', '0911.5228-1-18-9': 'In this coordinate system the [MATH] vielbein [MATH] and the spin connection [MATH] can be chosen in the form [EQUATION] and the [MATH] curvature is [EQUATION]', '0911.5228-1-19-0': '## Bosonic part of the instanton solution', '0911.5228-1-20-0': 'The bosonic part of the Wick rotated string action [REF] is [EQUATION] where [MATH] and [MATH] are the vielbeins on [MATH].', '0911.5228-1-20-1': 'To discuss the instanton solution of this [MATH] sigma model it is convenient to introduce complex coordinates both on the worldsheet and in target space (see [CITATION] for a review of instantons in two-dimensional sigma models).', '0911.5228-1-20-2': 'In the conformal gauge [MATH] and in the [MATH] coordinate system on the worldsheet the action takes the form [EQUATION]', '0911.5228-1-20-3': 'To introduce complex coordinates on the target sphere it is convenient to describe it as [MATH].', '0911.5228-1-20-4': 'The Fubini-Study metric on [MATH] is [EQUATION]', '0911.5228-1-20-5': 'If we choose [MATH] to be [EQUATION] eq. [REF] takes the form of the metric on [MATH] of radius [MATH] [EQUATION]', '0911.5228-1-20-6': 'In the [MATH], [MATH] coordinate system the string action takes the following form [EQUATION]', '0911.5228-1-20-7': 'It is now obvious that a local minimum is attained if [MATH] or [MATH], i.e. the embedding is given by a holomorphic function [MATH] for the instanton or by an anti-holomorphic function [MATH] for the anti-instanton.', '0911.5228-1-20-8': 'The remaining part of the action can be shown to be a topological invariant.', '0911.5228-1-20-9': 'What we have just reproduced is the classical instanton solution of the two-dimensional [MATH] sigma-model [CITATION] or rather its extension to [MATH] [CITATION].', '0911.5228-1-20-10': 'The only difference is that in our case the classical string solution should also satisfy the Virasoro constraints which in the conformal gauge have the form [EQUATION]', '0911.5228-1-20-11': 'We see that the Virasoro constraints are identically satisfied by the (anti)instanton solution.', '0911.5228-1-21-0': 'We are now in a position to proceed with the study of the fermionic zero modes carried by the string instanton.', '0911.5228-1-22-0': '# Fermionic equations of motion and the fermionic zero modes of the string instanton on [MATH] In a general supergravity background the equation of motion for the fermions following from the Green-Schwarz action [REF] (with the choice of superspace constraints given in Appendix A.4) is [EQUATION]', '0911.5228-1-22-1': 'Taking into account that on the mass shell the auxiliary metric [MATH] and the induced metric [MATH] are proportional to each other [EQUATION] the fermionic equations of motion take the following form which reflects the kappa-symmetry of the theory [EQUATION] where [MATH] is the matrix which appears in the kappa-symmetry projector [REF] and [MATH] is the dilatino superfield.', '0911.5228-1-23-0': 'For completeness, let us also present the equations of motion of the string bosonic modes [EQUATION] where [MATH] are torsion components and [MATH] are components of the NS-NS superfield strength with the vector indexes [MATH] and with the indices [MATH] and [MATH] standing for both the vector and the spinor indices (see Appendix A.4).', '0911.5228-1-24-0': 'At the linearized level in the [MATH] superspace the equation of motion for the fermions [REF] reduces to [EQUATION] where [MATH] is inverse of [MATH] and [EQUATION] where remember that [MATH].', '0911.5228-1-25-0': 'Note that one can alternatively derive eq. [REF] by varying the quadratic action [REF] or its Wick rotated counterpart [REF].', '0911.5228-1-26-0': '## Restriction to the instanton solution', '0911.5228-1-27-0': 'As we have discussed in Section [REF], the instanton solution is supported on the [MATH] two-dimensional subspace whose tangent space is characterized e.g. by the first two values of the [MATH] tangent space index [MATH].', '0911.5228-1-27-1': 'Restricting to this solution we have [EQUATION]', '0911.5228-1-27-2': 'It will be convenient to choose the [MATH] gamma matrices as follows [EQUATION] where [MATH] are the (re-labeled) Pauli matrices so that [MATH], and [MATH] are [MATH] Dirac gamma matrices corresponding to the four-dimensional subspace of [MATH] orthogonal to the instanton [MATH] and [MATH].', '0911.5228-1-28-0': 'The Wick rotated kappa-symmetry projection matrix [REF] then reduces to [EQUATION] and the fermionic part of the Euclidean action [REF] becomes [EQUATION]', '0911.5228-1-28-1': 'Note that in our case the fermionic terms of this two-dimensional theory differ from those of the conventional [MATH] supersymmetric [MATH] sigma-model (see [CITATION] for a review and references).', '0911.5228-1-29-0': 'In view of the form of the quadratic action [REF] and of the fermionic equation [REF] it is natural to impose on the fermionic fields the conventional kappa-symmetry gauge-fixing condition [EQUATION] which means that the fermions split into two sectors according to their chiralities in [MATH] and in the four-dimensional subspace of [MATH] orthogonal to the instanton [MATH] [EQUATION]', '0911.5228-1-29-1': 'Using the form of the [MATH] gamma-matrices [REF] we find that [EQUATION] where [EQUATION]', '0911.5228-1-29-2': 'So, the supersymmetry projection matrices [MATH] and [MATH] become [EQUATION]', '0911.5228-1-29-3': 'Their action on the two sets of the chiral fermions is [EQUATION]', '0911.5228-1-29-4': 'Note that from eqs. [REF] and [REF] it follows that all the eight [MATH] are fermions which correspond to unbroken supersymmetries of the [MATH] superbackground, while in the [MATH] sector four fermions ([MATH]) correspond to unbroken supersymmetries and other four ([MATH]) to the broken ones.', '0911.5228-1-29-5': "Note also that since for the instanton configuration the kappa-symmetry projector [REF] commutes with the 'supersymmetry' projectors [REF], it is not possible to choose the kappa-symmetry gauge-fixing condition which would put to zero all the eight 'broken-supersymmetry' fermions.", '0911.5228-1-30-0': 'For the instanton configuration the fermionic equation [REF] reduces to the following ones 0 [EQUATION]', '0911.5228-1-30-1': 'The second equation can be projected with [MATH] and [MATH] and we get [EQUATION]', '0911.5228-1-30-2': 'The covariant derivative [MATH] (defined in [REF]) contains the pullback on the instanton two-sphere of the [MATH] spin connection whose explicit form is given in Appendix B [EQUATION]', '0911.5228-1-30-3': 'Computing the pullback of the [MATH] connection, substituting it into the Dirac equations and taking into account the projection properties of the spinors we get the fermionic equations in the following form [EQUATION] where [MATH] is the inverse vielbein on [MATH], [MATH] is the intrinsic covariant derivative on the sphere of radius [MATH] with curvature [MATH] and [MATH] can be interpreted as the electromagnetic potential induced by a magnetic monopole of charge [MATH] placed at the center of the sphere.', '0911.5228-1-30-4': 'This is due to the fact that [EQUATION]', '0911.5228-1-30-5': 'Note that [MATH] and [MATH] are equivalent up to a total derivative term [EQUATION].', '0911.5228-1-30-6': 'In our parametrization of [MATH] (see Appendix B) and for a given embedding of [MATH] in [MATH], [MATH] and [MATH] have the following form in terms of the angular coordinates on [MATH] [EQUATION]', '0911.5228-1-30-7': 'We are now in a position to analyze the solutions of the fermionic equations [REF]-[REF].', '0911.5228-1-30-8': 'Eq. [REF] has the form of the Dirac equation for a fermion of mass [MATH].', '0911.5228-1-30-9': 'It is the product of [MATH] with the Killing spinor equation on the sphere [EQUATION]', '0911.5228-1-30-10': 'The Killing spinor equation on [MATH] for a two-component spinor has two non-trivial solutions [CITATION].', '0911.5228-1-30-11': 'Our [MATH] spinors carry four external indices in addition to the [MATH]-spinor index .', '0911.5228-1-30-12': 'Therefore, eq. [REF] has eight solutions and in the [MATH] sector the string instanton has eight fermionic zero modes.', '0911.5228-1-30-13': 'In spherical coordinates they have the explicit form [CITATION] [EQUATION] where [MATH] is an arbitrary constant spinor satisfying the chirality conditions [MATH].', '0911.5228-1-31-0': 'Let us now proceed with the analysis of the third fermionic equation [REF].', '0911.5228-1-31-1': 'As we have already mentioned, this equation describes the electric coupling of the fermionic field [MATH] to the monopole potential on the sphere.', '0911.5228-1-31-2': 'The electric charge of [MATH] is [MATH] for [MATH], i.e. when [MATH] is a chiral/anti-chiral two-dimensional spinor, respectively.', '0911.5228-1-31-3': 'The analysis in [CITATION] then tells us that there are non-trivial solutions of the charged Dirac equation [REF] of positive chirality when [MATH] and of negative chirality when [MATH].', '0911.5228-1-31-4': 'Since we are in the opposite situation, there are no non-trivial solutions in our case and hence [MATH].', '0911.5228-1-32-0': 'If [MATH], eq. [REF] implies that [MATH] should satisfy the massless Dirac equation [EQUATION]', '0911.5228-1-32-1': 'We observe that the electric charge of [MATH] is opposite to that of [MATH], i.e. it is [MATH] depending on whether [MATH] is chiral or anti-chiral two-dimensional spinor, i.e. whether [MATH].', '0911.5228-1-32-2': 'Now we are in the situation in which the requirement of [CITATION] for the Dirac equation [REF] to have non-trivial solutions is saturated, i.e. in our case for [MATH] of positive [MATH]-chirality [MATH] and for [MATH] of negative [MATH]-chirality [MATH].', '0911.5228-1-32-3': 'By the Atiyah-Singer index theorem there is one solution for each [MATH]-chirality of [MATH].', '0911.5228-1-32-4': 'Therefore, in the [MATH] sector the string instanton has four zero modes which have a very simple form [EQUATION] where [MATH] and [MATH] are holomorphic and anti-holomorphic spinors in the projective coordinates [MATH] and [MATH] of [MATH] which are anti-chiral in the directions transverse to the instanton, i.e. [MATH], [MATH] and [MATH].', '0911.5228-1-32-5': 'For the anti-instanton the solution takes the same form but with [MATH] and [MATH].', '0911.5228-1-33-0': 'Note that when [MATH] and in view of the form of the fermionic supervielbeins [MATH] of the supercoset [MATH] (see Appendix A.7), the non-linear fermionic equation of motion [REF] as well as the linear one [REF] involve the pull-back on the string worldsheet of the [MATH] Killing spinor operator [EQUATION] which acts on the 24 fermions [MATH] associated with the supersymmetry of [MATH] (see Appendix A.7).', '0911.5228-1-33-1': 'Therefore, if [MATH] are the 24 Killing spinors on [MATH] they solve not only the linearized equations [REF] but also the complete fermionic equations [REF].', '0911.5228-1-33-2': 'In the case of the string instanton considered above, the kappa-symmetry projector reduces the number of solutions of the pulled-back Killing spinor equation by half, leaving us with the twelve physical fermionic zero modes.', '0911.5228-1-33-3': 'It should also be noted that the fermionic zero modes found above do not contribute to the bosonic equations [REF].', '0911.5228-1-33-4': 'This guarantees that the bosonic instanton solution does not have a back reaction from the fermionic modes.', '0911.5228-1-34-0': 'Let us stress once again that, as we have shown, for the instanton solution considered above the kappa-symmetry cannot eliminate all the eight fermions [MATH] associated with the supersymmetries broken in the [MATH] background.', '0911.5228-1-34-1': 'Therefore, even if among the instanton fermionic zero modes there is no [MATH]-modes, the fluctuations around the instanton solution will have four physical fermionic degrees of freedom corresponding to the broken supersymmetries.', '0911.5228-1-35-0': 'We have thus found that the string instanton wrapping the non-trivial two-cycle inside [MATH] has twelve fermionic zero modes.', '0911.5228-1-35-1': 'As we have already mentioned, the eigth string fermionic fields [MATH] and four [MATH] correspond to twelve (of the twenty four) supersymmetries of the [MATH] background which are linearly realized on the string worldsheet.', '0911.5228-1-35-2': 'The fermionic zero modes thus play the role similar to Volkov-Akulov goldstinos [CITATION] which break supersymmetry.', '0911.5228-1-35-3': 'It would be of interest to analyze possible effects of these instantons in [MATH] superstring theory.'}

{'0911.5228-2-0-0': 'We study the string instanton wrapping a non-trivial two-cycle in [MATH] of the type IIA string theory compactified on [MATH] superspace and find that it has twelve fermionic zero modes associated with 1/2 of the supersymmetry of the background.', '0911.5228-2-0-1': 'Being goldstinos, the instanton fermionic zero modes thus spontaneously break supersymmetry of the string worldsheet theory.', '0911.5228-2-1-0': '# Introduction', '0911.5228-2-2-0': 'M-theory compactified on [MATH] and a corresponding Type IIA superstring theory compactified on [MATH] are on the bulk side of the [MATH] holography whose boundary superconformal Chern-Simons-matter theory is assumed to provide an effective worldvolume description of a stack of multiple M2-branes [CITATION].', '0911.5228-2-2-1': 'This new [MATH] correspondence shares some features with the well studied [MATH] correspondence whose bulk theory is Type IIB superstring theory compactified on [MATH] and the boundary theory is the superconformal [MATH], [MATH] super Yang-Mills theory.', '0911.5228-2-3-0': 'However, the two examples of the [MATH] correspondence are quite different.', '0911.5228-2-3-1': 'First of all, the Type IIB superstring in [MATH] is maximally supersymmetric.', '0911.5228-2-3-2': 'Its 32 supersymmetries are part of the superisometry group [MATH].', '0911.5228-2-3-3': 'On the other hand the [MATH] solution of [MATH] supergravity has 32 supersymmetries only for [MATH], while for [MATH] the theory is invariant under 24 supersymmetries and so is its ten-dimensional counterpart, the Type IIA superstring theory in the [MATH] supergravity background whose superisometries form the supergroup [MATH] [CITATION].', '0911.5228-2-3-4': 'As a result, while the Green-Schwarz action for the [MATH] superstring amounts to a worldsheet sigma-model on the supercoset space [MATH] [CITATION], the target superspace of the Green-Schwarz action of the [MATH] superstring is not a supercoset space, though it possesses the [MATH] isometry [CITATION].', '0911.5228-2-3-5': 'Only when the superstring is extended in [MATH], its dynamics can be described by the [MATH] supercoset sigma-model [CITATION].', '0911.5228-2-3-6': 'It can be obtained from the complete Green-Schwarz action [CITATION] by partially gauge fixing kappa-symmetry in a way which puts to zero the eight worldsheet fermionic modes corresponding to the eight broken supersymmetries of the [MATH] superbackground.', '0911.5228-2-3-7': 'Such a gauge fixing is admissible only when the string moves in [MATH].', '0911.5228-2-3-8': 'This gauge is, however not admissible when the string moves entirely in the [MATH] part of the superbackground.', '0911.5228-2-3-9': 'One of the consequences of this peculiar situation is that though the subsector of the [MATH] superstring theory described by the supercoset sigma-model is classically integrable [CITATION], the explicit proof of the classical integrability of the complete [MATH] superstring still remains an open problem.', '0911.5228-2-3-10': 'This is because the complete [MATH] superspace is not a supercoset space, and the methods used to prove the integrability of the [MATH] superstring [CITATION] and the supercoset sector of the [MATH] superstring [CITATION] do not apply.', '0911.5228-2-4-0': 'Another interesting peculiarity of the [MATH] superstring, which the [MATH] superstring does not have, is the existence on [MATH] of string instantons.', '0911.5228-2-4-1': 'They are formed in the Wick rotated theory by the string worldsheet wrapping a topologically non-trivial two-cycle of [MATH].', '0911.5228-2-4-2': 'This two-cycle is a [MATH] corresponding to the closed Kahler two-form [MATH] on [MATH].', '0911.5228-2-4-3': 'As we shall show, also in the case of the string instantons on [MATH] the consistent gauge fixing of kappa-symmetry does not allow reducing the string action to the supercoset sigma model, i.e. to eliminate, by using kappa-symmetry, the eight fermionic modes corresponding to the broken supersymmetries.', '0911.5228-2-5-0': 'The main goal of this paper is to study the superstring instanton on [MATH] and analyze its fermionic zero modes.', '0911.5228-2-5-1': 'In the case of branes moving in a supersymmetric background, their fermionic equations have solutions which are associated with the Killing spinors of the background that guarantee its supersymmetries.', '0911.5228-2-5-2': 'The number of physical modes of the worldvolume Dirac operator associated with the Killing spinors is equal to the number of components of the Killing spinors which are not annihilated by the kappa-symmetry projector of the brane.', '0911.5228-2-5-3': 'If the background is maximally supersymmetric, one concludes that the number of dynamical zero modes on the brane are half the number of supersymmetries, since the rank of the kappa-symmetry projector is equal to half the number of the maximal supersymmetries of the background.', '0911.5228-2-5-4': 'When the background is not maximally supersymmetric, as the [MATH] one, the number of the brane fermion zero modes associated with unbroken supersymmetries depends on how many of them are not eliminated by the kappa-symmetry projector.', '0911.5228-2-5-5': 'The action of the kappa-symmetry projector and the corresponding number of fermionic zero modes depends on how the given brane configuration is embedded into target space.', '0911.5228-2-5-6': 'In the cases with less supersymmetries the worldvolume Dirac equation may, in general, also have solutions which are not associated with unbroken supersymmetries, but with the broken ones.', '0911.5228-2-5-7': 'Thus the analysis of the brane fermionic modes in less supersymmetric backgrounds should be made case by case (see e.g. [CITATION] for a more detailed discussion of this point).', '0911.5228-2-6-0': 'As we shall see, in the [MATH] case the string instanton on [MATH] has twelve zero modes all of which are associated with unbroken supersymmetries of the background and there are no zero modes of the fermions corresponding to broken supersymmetries.', '0911.5228-2-6-1': 'It is interesting that these twelve zero modes are divided into eight and four ones which have different geometrical and physical meaning.', '0911.5228-2-6-2': 'The eight massive fermionic zero modes are four copies of the two-component Killing spinor on [MATH] and the four other fermionic modes are two copies of massless chiral and anti-chiral fermion on [MATH] electrically coupled to the electromagnetic potential created on [MATH] by a monopole placed in the center of [MATH].', '0911.5228-2-6-3': 'The monopole potential arises as part of the [MATH] spin connection pulled-back on the instanton [MATH].', '0911.5228-2-6-4': 'This, at least formally and remotely, reminds us the peculiarity of the presence of different, light and heavy, physical worldsheet degrees of freedom in a Penrose limit of the [MATH] superstring [CITATION].', '0911.5228-2-7-0': 'The presence of the string instanton and its fermionic zero modes may generate non-perturbative corrections to the string effective action, which may affect its properties and if so should be taken into account in studying, e.g. the [MATH] correspondence.', '0911.5228-2-7-1': 'The instantons may, perhaps, contribute to the worldsheet S-matrix and/or to energies of a semiclassical string.', '0911.5228-2-7-2': 'To study these effects one needs to find a way of merging the instanton and Minkowski solutions, such as spinning strings or BMN geodesics.', '0911.5228-2-7-3': 'In addition, in the presence of the instanton fermionic zero modes, the worldsheet correlator, to be non-zero, should contain a number of fermion insertions.', '0911.5228-2-8-0': 'The paper is organized as follows.', '0911.5228-2-8-1': 'In Section [REF] we review the form of the string action in the [MATH] superbackground and truncate it to the second order in fermions.', '0911.5228-2-8-2': 'In Section [REF] we describe the bosonic part of the string instanton solution and in Section [REF] we study its fermionic zero modes.', '0911.5228-2-8-3': 'Section [REF] contains a summary and brief discussion of a possible relation of the string instanton to some features of the [MATH] corespondence.', '0911.5228-2-8-4': 'Appendix A contains a description of our conventions and notation and of the geometry of the [MATH] superspace.', '0911.5228-2-8-5': 'In Appendix B we give the explicit form of the [MATH] Fubini-Study metric, vielbeins and connection which were used for the analysis of the string fermion equations.', '0911.5228-2-9-0': '# [MATH] superstring action up to the quadratic order in fermions', '0911.5228-2-10-0': 'To simplify the study of the fermionic zero modes of the string instanton we reduce the complete superstring action of [CITATION] to the quadratic order in fermions (though, the solutions we find satisfy the complete non-linear equations of motion to all orders in fermions).', '0911.5228-2-10-1': 'Alternatively, one can use the quadratic type IIA superstring action derived in [CITATION] for a generic superbackground and substitute into it the values of the supergravity fields corresponding to the [MATH] background.', '0911.5228-2-10-2': 'As a consistency check we have performed the reduction of both of the actions.', '0911.5228-2-10-3': 'We shall see that they give the same result upon a redefinition of bosonic [MATH] coordinates of the target-superspace of [CITATION].', '0911.5228-2-10-4': 'This redefinition is required due to a particular parametrization used in [CITATION] to construct an explicit form of the [MATH] supergeometry.', '0911.5228-2-11-0': 'The Green-Schwarz superstring action in a generic type IIA supergravity background has the well known form [EQUATION] where [MATH] are the worldsheet coordinates, [MATH] is an intrinsic worldsheet metric, [MATH] are worldsheet pullbacks of target superspace vector supervielbeins and [MATH] is the pull-back of the NS-NS 2-form.', '0911.5228-2-12-0': 'The kappa-symmetry transformations of the worldsheet fields [MATH] which leave the superstring action ([REF]) invariant (provided the superbackground obeys the superspace supergravity constraints) are [EQUATION] where [MATH] is a 32-component spinor parameter, [MATH] is a spinor projection matrix with [EQUATION] where [MATH] is the induced metric on the worldsheet.', '0911.5228-2-12-1': 'The explicit form of the supervielbeins [MATH] and the NS-NS 2-form [MATH] which describe the geometry of the [MATH] superspace are given in Appendix A (see also [CITATION]).', '0911.5228-2-13-0': 'Up to second order in fermions the supervielbeins and the [MATH]-field have the following form 0 [EQUATION] [EQUATION] where [MATH] is the vacuum expectation value of the dilaton, [MATH] is the radius of the [MATH] sphere whose base is [MATH], [MATH] is the eleven-dimensional Planck length related to the string tension as follows [MATH] and [MATH] is the Chern-Simons level related to the units of [MATH] and [MATH] flux which support the [MATH] solution of Type IIA supergravity.', '0911.5228-2-13-1': 'The 32-component fermionic variable [MATH] is split by projectors [MATH] and [MATH] (see Appendix A.5) into the 24-component spinors [MATH]; [MATH]) which correspond to the 24 supersymmetries of the [MATH] solution and the 8-component spinors [MATH] which correspond to the broken supersymmetries.', '0911.5228-2-13-2': 'The index [MATH] is a spinor index of [MATH] (see Appendix A for more details).', '0911.5228-2-13-3': 'The covariant derivative [MATH] is defined as follows [EQUATION] where [MATH], [MATH] and [MATH] are, respectively the vielbein, connection and Dirac-matrices of [MATH] of radius [MATH].', '0911.5228-2-13-4': '[MATH] and [MATH] are, respectively, the vielbein and connection on [MATH] of radius [MATH] and [MATH] are [MATH] gamma-matrices of [MATH].', '0911.5228-2-13-5': '[MATH] is the RR one-form potential whose field strength is the Kahler form on [MATH], [MATH].', '0911.5228-2-13-6': 'See Appendix A for more details regarding the notation and conventions.', '0911.5228-2-14-0': 'Substituting the expressions for the vielbeins [REF] and the NS-NS two-form [REF] into the action [REF] and keeping only terms up to quadratic order in fermions we get the following action [EQUATION] where [MATH] is the worldsheet pullback of the conventional [MATH] covariant derivative.', '0911.5228-2-15-0': 'The first two lines of this action coincide with the action which one gets by reducing to [MATH] the quadratic Green-Schwarz action in a generic type IIA superbackground [CITATION] .', '0911.5228-2-15-1': 'The last term in the action appeared because of our choice of parametrization of the [MATH] superspace which allowed us to write its geometry in the simplest form.', '0911.5228-2-15-2': 'It is not hard to see that the last term in [REF] can be canceled (modulo higher order terms in fermions) by making the following shift of the bosonic coordinates [MATH] of [MATH] [EQUATION]', '0911.5228-2-15-3': 'After this field redefinition the two forms of the string action become equivalent.', '0911.5228-2-16-0': 'To study the string instantons we should perform a Wick rotation of the worldsheet and the target space in the action [REF] to Euclidean signature.', '0911.5228-2-16-1': 'The Wick rotation basically consists in replacing [MATH] and [MATH], respectively with [MATH] and [MATH], replacing [MATH] with [MATH] and taking into account that the fermions [MATH] become complex spinors, since there are no Majorana spinors in ten-dimensional Euclidean space.', '0911.5228-2-16-2': 'However, the complex conjugate spinors do not appear in the Wick rotated action and, hence, the number of the fermionic degrees of freedom formally remains the same as before the Wick rotation.', '0911.5228-2-16-3': 'Note also that the Euclidean [MATH] is defined as [MATH], where [MATH] is the Wick rotated [MATH].', '0911.5228-2-16-4': 'So [MATH] as in the case of Minkowski signature.', '0911.5228-2-17-0': 'Thus, after the redefinition [REF] and the Wick rotation the action takes the following form [EQUATION] and the kappa-symmetry matrix [MATH] gets replaced by [EQUATION]', '0911.5228-2-18-0': '# String instanton wrapping a two-sphere inside [MATH] We are interested in a string whose worldsheet wraps a topologically non-trivial two-cycle inside [MATH] and thus is a stringy counterpart of the instantons of two-dimensional [MATH] sigma-models.', '0911.5228-2-18-1': 'To be topologically non-trivial this two-cycle should have a non-zero pull-back on its worldsheet of the Kahler two-form [MATH] of [MATH].', '0911.5228-2-18-2': 'Such a two-cycle is a [MATH] subspace of [MATH].', '0911.5228-2-18-3': 'To identify it, it is convenient to consider the form of the Fubini-Study metric on [MATH] given in [CITATION] [EQUATION] where [MATH], [MATH] and [MATH], and [MATH] 0 [EQUATION] are three left-invariant one-forms on [MATH] obeying [MATH] etc. (see Appendix B for more details).', '0911.5228-2-18-4': 'Notice that with this choice of the [MATH] coordinates, [MATH] and [MATH] parameterize a two-sphere of radius [MATH].', '0911.5228-2-18-5': 'This two-sphere is topologically non-trivial and associated to the Kahler form on [MATH].', '0911.5228-2-18-6': 'The string instanton wraps this sphere.', '0911.5228-2-18-7': 'For instance, if it wraps the sphere once [MATH] and [MATH] can be identified with the string worldsheet coordinates, while all other [MATH] as well as [MATH] coordinates are worldsheet constants for the instanton solution.', '0911.5228-2-18-8': 'Thus the pullback on the string instanton of the metric [REF] of [MATH] (of radius [MATH]) is the metric of the sphere of radius [MATH] [EQUATION]', '0911.5228-2-18-9': 'In this coordinate system the [MATH] vielbein [MATH] and the spin connection [MATH] can be chosen in the form [EQUATION] and the [MATH] curvature is [EQUATION]', '0911.5228-2-19-0': '## Bosonic part of the instanton solution', '0911.5228-2-20-0': 'The bosonic part of the Wick rotated string action [REF] is [EQUATION] where [MATH] and [MATH] are the vielbeins on [MATH].', '0911.5228-2-20-1': 'To discuss the instanton solution of this [MATH] sigma model it is convenient to introduce complex coordinates both on the worldsheet and in target space (see [CITATION] for a review of instantons in two-dimensional sigma models).', '0911.5228-2-20-2': 'In the conformal gauge [MATH] and in the [MATH] coordinate system on the worldsheet the action takes the form [EQUATION]', '0911.5228-2-20-3': 'To introduce complex coordinates on the target sphere it is convenient to describe it as [MATH].', '0911.5228-2-20-4': 'The Fubini-Study metric on [MATH] is [EQUATION]', '0911.5228-2-20-5': 'If we choose [MATH] to be [EQUATION] eq. [REF] takes the form of the metric on [MATH] of radius [MATH] [EQUATION]', '0911.5228-2-20-6': 'In the [MATH], [MATH] coordinate system the string action takes the following form [EQUATION]', '0911.5228-2-20-7': 'It is now obvious that a local minimum is attained if [MATH] or [MATH], i.e. the embedding is given by a holomorphic function [MATH] for the instanton or by an anti-holomorphic function [MATH] for the anti-instanton.', '0911.5228-2-20-8': 'The remaining part of the action can be shown to be a topological invariant, namely, [EQUATION] where [MATH] is the topolgical charge of the instanton and [MATH] is the [MATH] radius in the string frame.', '0911.5228-2-21-0': 'What we have just reproduced is the classical instanton solution of the two-dimensional [MATH] sigma-model [CITATION] or rather its extension to [MATH] [CITATION].', '0911.5228-2-21-1': 'The only difference is that in our case the classical string solution should also satisfy the Virasoro constraints which in the conformal gauge have the form [EQUATION]', '0911.5228-2-21-2': 'We see that the Virasoro constraints are identically satisfied by the (anti)instanton solution.', '0911.5228-2-22-0': 'Let us note that though in the [MATH] background the purely bosonic components of the NS-NS 3-form field strength [MATH] are zero, the NS-NS 2-form may have non-zero expectation values proportional to the Kahler two-form on [MATH], [MATH], where [MATH] plays the role of a constant axion.', '0911.5228-2-22-1': 'For such a two-form, [MATH] is zero since [MATH] is the closed form, [MATH].', '0911.5228-2-22-2': 'In this case also the Wess-Zumino part of the (Wick rotated) string action [REF] will contribute to the instanton action, which becomes [EQUATION]', '0911.5228-2-22-3': 'A similar situation one has in the case of string instantons on Calabi-Yau spaces [CITATION].', '0911.5228-2-22-4': 'In the context of the [MATH] correspondence, the co-homologically non-trivial [MATH] field appears from the string side in the generalization of the ABJM model to include gauge groups of a different rank proposed in [CITATION] (see the Summary below for more discussion of this point).', '0911.5228-2-23-0': 'We are now in a position to proceed with the study of the fermionic zero modes carried by the string instanton.', '0911.5228-2-24-0': '# Fermionic equations of motion and the fermionic zero modes of the string instanton on [MATH] In a general supergravity background the equation of motion for the fermions following from the Green-Schwarz action [REF] (with the choice of superspace constraints given in Appendix A.4) is [EQUATION]', '0911.5228-2-24-1': 'Taking into account that on the mass shell the auxiliary metric [MATH] and the induced metric [MATH] are proportional to each other [EQUATION] the fermionic equations of motion take the following form which reflects the kappa-symmetry of the theory [EQUATION] where [MATH] is the matrix which appears in the kappa-symmetry projector [REF] and [MATH] is the dilatino superfield.', '0911.5228-2-25-0': 'For completeness, let us also present the equations of motion of the string bosonic modes [EQUATION] where [MATH] are torsion components and [MATH] are components of the NS-NS superfield strength with the vector indexes [MATH] and with the indices [MATH] and [MATH] standing for both the vector and the spinor indices (see Appendix A.4).', '0911.5228-2-26-0': 'At the linearized level in the [MATH] superspace the equation of motion for the fermions [REF] reduces to [EQUATION] where [MATH] is inverse of [EQUATION] and [EQUATION] where remember that [MATH].', '0911.5228-2-27-0': 'Note that one can alternatively derive eq. [REF] by varying the quadratic action [REF] or its Wick rotated counterpart [REF].', '0911.5228-2-28-0': '## Restriction to the instanton solution', '0911.5228-2-29-0': 'As we have discussed in Section [REF], the instanton solution is supported on the [MATH] two-dimensional subspace whose tangent space is characterized e.g. by the first two values of the [MATH] tangent space index [MATH].', '0911.5228-2-29-1': 'Restricting to this solution we have [EQUATION]', '0911.5228-2-29-2': 'It will be convenient to choose the [MATH] gamma matrices as follows [EQUATION] where [MATH] are the (re-labeled) Pauli matrices so that [MATH], and [MATH] are [MATH] Dirac gamma matrices corresponding to the four-dimensional subspace of [MATH] orthogonal to the instanton [MATH] and [MATH].', '0911.5228-2-30-0': 'The Wick rotated kappa-symmetry projection matrix [REF] then reduces to [EQUATION] and the fermionic part of the Euclidean action [REF] becomes [EQUATION] where the metric [MATH] was defined in [REF].', '0911.5228-2-30-1': 'Note that in our case the fermionic terms of this two-dimensional theory differ from those of the conventional [MATH] supersymmetric [MATH] sigma-model (see [CITATION] for a review and references).', '0911.5228-2-31-0': 'In view of the form of the quadratic action [REF] and of the fermionic equation [REF] it is natural to impose on the fermionic fields the conventional kappa-symmetry gauge-fixing condition [EQUATION] which means that the fermions split into two sectors according to their chiralities in [MATH] and in the four-dimensional subspace of [MATH] orthogonal to the instanton [MATH] [EQUATION]', '0911.5228-2-31-1': 'Using the form of the [MATH] gamma-matrices [REF] we find that [EQUATION] where [EQUATION]', '0911.5228-2-31-2': 'So, the supersymmetry projection matrices [MATH] and [MATH] become [EQUATION]', '0911.5228-2-31-3': 'Their action on the two sets of the chiral fermions is [EQUATION]', '0911.5228-2-31-4': 'Note that from eqs. [REF] and [REF] it follows that all the eight [MATH] are fermions which correspond to unbroken supersymmetries of the [MATH] superbackground, while in the [MATH] sector four fermions ([MATH]) correspond to unbroken supersymmetries and other four ([MATH]) to the broken ones.', '0911.5228-2-31-5': "Note also that since for the instanton configuration the kappa-symmetry projector [REF] commutes with the 'supersymmetry' projectors [REF], it is not possible to choose the kappa-symmetry gauge-fixing condition which would put to zero all the eight 'broken-supersymmetry' fermions.", '0911.5228-2-31-6': 'In terms of the fields [MATH], [MATH] and [MATH] the fermionic action [REF] takes the form [EQUATION] where [MATH] is the inverse vielbein on [MATH].', '0911.5228-2-32-0': 'For the instanton configuration the fermionic equation [REF] reduces to the following ones [EQUATION]', '0911.5228-2-32-1': 'From the form of the action [REF] and the equation of motion [REF] it follows that the field [MATH] can be regarded as an auxiliary one, which can be expressed in terms of a derivative of [MATH].', '0911.5228-2-32-2': 'However, for the analysis of the solutions of eqs. [REF]-[REF] it is more convenient to consider it as an independent variable satisfying the Dirac equation [REF].', '0911.5228-2-33-0': 'The covariant derivative [MATH] (defined in [REF]) contains the pullback on the instanton two-sphere of the [MATH] spin connection whose explicit form is given in Appendix B [EQUATION]', '0911.5228-2-33-1': 'Computing the pullback of the [MATH] connection, substituting it into the Dirac equations and taking into account the projection properties of the spinors we get the fermionic equations in the following form [EQUATION] where [MATH] is the intrinsic covariant derivative on the sphere of radius [MATH] with curvature [MATH] and [MATH] can be interpreted as the electromagnetic potential induced by a magnetic monopole of charge [MATH] placed at the center of the sphere.', '0911.5228-2-33-2': 'This is due to the fact that [EQUATION]', '0911.5228-2-33-3': 'Note that [MATH] and [MATH] are equivalent up to a total derivative term [EQUATION].', '0911.5228-2-33-4': 'In our parametrization of [MATH] (see Appendix B) and for a given embedding of [MATH] in [MATH], [MATH] and [MATH] have the following form in terms of the angular coordinates on [MATH] [EQUATION]', '0911.5228-2-33-5': 'We are now in a position to analyze the solutions of the fermionic equations [REF]-[REF].', '0911.5228-2-33-6': 'Eq. [REF] has the form of the Dirac equation for a fermion of mass [MATH].', '0911.5228-2-33-7': 'It is the product of [MATH] with the Killing spinor equation on the sphere [EQUATION]', '0911.5228-2-33-8': 'The Killing spinor equation on [MATH] for a two-component spinor has two non-trivial solutions [CITATION].', '0911.5228-2-33-9': 'Our [MATH] spinors carry four external indices in addition to the [MATH]-spinor index .', '0911.5228-2-33-10': 'Therefore, eq. [REF] has eight solutions and in the [MATH] sector the string instanton has eight fermionic zero modes.', '0911.5228-2-33-11': 'In spherical coordinates they have the explicit form [CITATION] [EQUATION] where [MATH] is an arbitrary constant spinor satisfying the chirality conditions [MATH].', '0911.5228-2-34-0': 'Let us now proceed with the analysis of the third fermionic equation [REF].', '0911.5228-2-34-1': 'As we have already mentioned, this equation describes the electric coupling of the fermionic field [MATH] to the monopole potential on the sphere.', '0911.5228-2-34-2': 'The electric charge of [MATH] is [MATH] for [MATH], i.e. when [MATH] is a chiral/anti-chiral two-dimensional spinor, respectively.', '0911.5228-2-34-3': 'The analysis in [CITATION] then tells us that there are non-trivial solutions of the charged Dirac equation [REF] of positive chirality when [MATH] and of negative chirality when [MATH].', '0911.5228-2-34-4': 'Since we are in the opposite situation, there are no non-trivial solutions in our case and hence [MATH].', '0911.5228-2-35-0': 'If [MATH], eq. [REF] implies that [MATH] should satisfy the massless Dirac equation [EQUATION]', '0911.5228-2-35-1': 'We observe that the electric charge of [MATH] is opposite to that of [MATH], i.e. it is [MATH] depending on whether [MATH] is chiral or anti-chiral two-dimensional spinor, i.e. whether [MATH].', '0911.5228-2-35-2': 'Now we are in the situation in which the requirement of [CITATION] for the Dirac equation [REF] to have non-trivial solutions is saturated, i.e. in our case for [MATH] of positive [MATH]-chirality [MATH] and for [MATH] of negative [MATH]-chirality [MATH].', '0911.5228-2-35-3': 'By the Atiyah-Singer index theorem there is one solution for each [MATH]-chirality of [MATH].', '0911.5228-2-35-4': 'Therefore, in the [MATH] sector the string instanton has four zero modes which have a very simple form [EQUATION] where [MATH] and [MATH] are holomorphic and anti-holomorphic spinors in the projective coordinates [MATH] and [MATH] of [MATH] which are anti-chiral in the directions transverse to the instanton, i.e. [MATH], [MATH] and [MATH].', '0911.5228-2-35-5': 'For the anti-instanton the solution takes the same form but with [MATH] and [MATH].', '0911.5228-2-36-0': 'Note that when [MATH] and in view of the form of the fermionic supervielbeins [MATH] of the supercoset [MATH] (see Appendix A.7), the non-linear fermionic equation of motion [REF] as well as the linear one [REF] involve the pull-back on the string worldsheet of the [MATH] Killing spinor operator [EQUATION] which acts on the 24 fermions [MATH] associated with the supersymmetry of [MATH] (see Appendix A.7).', '0911.5228-2-36-1': 'Therefore, if [MATH] are the 24 Killing spinors on [MATH] they solve not only the linearized equations [REF] but also the complete fermionic equations [REF].', '0911.5228-2-36-2': 'In the case of the string instanton considered above, the kappa-symmetry projector reduces the number of solutions of the pulled-back Killing spinor equation by half, leaving us with the twelve physical fermionic zero modes.', '0911.5228-2-36-3': 'It should also be noted that the fermionic zero modes found above do not contribute to the bosonic equations [REF].', '0911.5228-2-36-4': 'This guarantees that the bosonic instanton solution does not have a back reaction from the fermionic modes.', '0911.5228-2-37-0': 'Let us stress once again that, as we have shown, for the instanton solution considered above the kappa-symmetry cannot eliminate all the eight fermions [MATH] associated with the supersymmetries broken in the [MATH] background.', '0911.5228-2-37-1': 'Therefore, even if among the instanton fermionic zero modes there is no [MATH]-modes, the fluctuations around the instanton solution will have four physical fermionic degrees of freedom corresponding to the broken supersymmetries.', '0911.5228-2-38-0': '# Summary and Discussion', '0911.5228-2-39-0': 'We have thus found that the string instanton wrapping the non-trivial two-cycle inside [MATH] has twelve fermionic zero modes.', '0911.5228-2-39-1': 'As we have already mentioned, the eight string fermionic fields [MATH] which have an effective mass [MATH] and four massless [MATH] electrically coupled to the [MATH] monopole field, correspond to twelve (of the twenty four) supersymmetries of the [MATH] background which are linearly realized on the string worldsheet.', '0911.5228-2-39-2': 'The fermionic zero modes thus play a role similar to Volkov-Akulov goldstinos [CITATION] which break supersymmetry.', '0911.5228-2-39-3': 'It would be of interest to analyze possible effects of these instantons in [MATH] superstring theory and to understand their counterparts in the boundary [MATH] theory.', '0911.5228-2-40-0': 'As was mentioned briefly in Section [REF] the instanton solution can be generalized by switching on a NS-NS field [MATH] of a non-trivial co-homology on [MATH].', '0911.5228-2-40-1': 'The coupling of the string to the [MATH]-field then results in the instanton action being shifted by a constant imaginary piece.', '0911.5228-2-40-2': 'In fact, the co-homologically non-trivial [MATH]-field arises on the string side of the [MATH] correspondence when one considers the ABJ-model [CITATION] which generalizes the ABJM construction to gauge groups of different rank, i.e. [MATH] with [MATH] instead of [MATH].', '0911.5228-2-40-3': 'The integral of [MATH] on the [MATH] cycle inside [MATH] takes a fractional value [EQUATION]', '0911.5228-2-40-4': 'In eleven dimensions this corresponds to a co-homologically nontrivial three-form potential on the 3-cycle [MATH].', '0911.5228-2-40-5': 'From the point of view of M2-branes probing a [MATH] singularity this is associated to [MATH] fractional M2-branes sitting at the singularity.', '0911.5228-2-40-6': 'The fractional M2-branes can be thought of as M5-branes wrapping the corresponding vanishing 3-cycle at the orbifold point, see [CITATION].', '0911.5228-2-41-0': 'This picture suggests that the string instanton considered in this paper should correspond in M-theory to an instantonic M2-brane, i.e. an M2-brane whose worldvolume wraps the 3-cycle [MATH].', '0911.5228-2-41-1': 'It would be interesting to find out what these instantonic M2-branes correspond to in the ABJ/ABJM gauge-theory picture.'}

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[]

[['0911.5228-1-35-3', '0911.5228-2-39-3'], ['0911.5228-2-33-10', '0911.5228-3-34-12'], ['0911.5228-2-40-3', '0911.5228-3-58-4'], ['0911.5228-2-20-6', '0911.5228-3-21-6'], ['0911.5228-2-21-0', '0911.5228-3-22-0'], ['0911.5228-2-21-1', '0911.5228-3-22-3'], ['0911.5228-2-35-4', '0911.5228-3-36-4'], ['0911.5228-1-20-8', '0911.5228-2-20-8']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0911.5228

{'0911.5228-3-0-0': 'We study the string instanton wrapping a non-trivial two-cycle in [MATH] of the type IIA string theory compactified on [MATH] superspace and find that it has twelve fermionic zero modes associated with 1/2 of the supersymmetry of the background thus manifesting that this classical instanton configuration is 1/2 BPS.', '0911.5228-3-1-0': '# Introduction', '0911.5228-3-2-0': 'M-theory compactified on [MATH] and a corresponding Type IIA superstring theory compactified on [MATH] are on the bulk side of the [MATH] holography whose boundary superconformal Chern-Simons-matter theory is assumed to provide an effective worldvolume description of a stack of multiple M2-branes [CITATION].', '0911.5228-3-2-1': 'This new [MATH] correspondence shares some features with the well studied [MATH] correspondence whose bulk theory is Type IIB superstring theory compactified on [MATH] and the boundary theory is the superconformal [MATH], [MATH] super Yang-Mills theory.', '0911.5228-3-3-0': 'However, the two examples of the [MATH] correspondence are quite different.', '0911.5228-3-3-1': 'First of all, the Type IIB superstring in [MATH] is maximally supersymmetric.', '0911.5228-3-3-2': 'Its 32 supersymmetries are part of the superisometry group [MATH].', '0911.5228-3-3-3': 'On the other hand the [MATH] solution of [MATH] supergravity has 32 supersymmetries only for [MATH], while for [MATH] the theory is invariant under 24 supersymmetries and so is its ten-dimensional counterpart, the Type IIA superstring theory in the [MATH] supergravity background whose superisometries form the supergroup [MATH] [CITATION].', '0911.5228-3-3-4': 'As a result, while the Green-Schwarz action for the [MATH] superstring amounts to a worldsheet sigma-model on the supercoset space [MATH] [CITATION], the target superspace of the Green-Schwarz action of the [MATH] superstring is not a supercoset space, though it possesses the [MATH] isometry [CITATION].', '0911.5228-3-3-5': 'Only when the superstring is extended in [MATH], its dynamics can be described by the [MATH] supercoset sigma-model [CITATION].', '0911.5228-3-3-6': 'It can be obtained from the complete Green-Schwarz action [CITATION] by partially gauge fixing kappa-symmetry in a way which puts to zero the eight worldsheet fermionic modes corresponding to the eight broken supersymmetries of the [MATH] superbackground.', '0911.5228-3-3-7': 'Such a gauge fixing is admissible only when the string moves in [MATH].', '0911.5228-3-3-8': 'This gauge is, however not admissible when the string moves entirely in the [MATH] part of the superbackground.', '0911.5228-3-3-9': 'One of the consequences of this peculiar situation is that though the subsector of the [MATH] superstring theory described by the supercoset sigma-model is classically integrable [CITATION], the explicit proof of the classical integrability of the complete [MATH] superstring still remains an open problem.', '0911.5228-3-3-10': 'This is because the complete [MATH] superspace is not a supercoset space, and the methods used to prove the integrability of the [MATH] superstring [CITATION] and of the supercoset sector of the [MATH] superstring [CITATION] do not apply.', '0911.5228-3-4-0': 'Another interesting peculiarity of the [MATH] superstring, which the [MATH] superstring does not have, is the existence on [MATH] of string instantons.', '0911.5228-3-4-1': 'They are formed in the Wick rotated theory by the string worldsheet wrapping a topologically non-trivial two-cycle of [MATH].', '0911.5228-3-4-2': 'This two-cycle is a [MATH] corresponding to the closed Kahler two-form [MATH] on [MATH].', '0911.5228-3-4-3': 'As we shall show, also in the case of the string instantons on [MATH] the consistent gauge fixing of kappa-symmetry does not allow reducing the string action to the supercoset sigma model, i.e. to eliminate, by using kappa-symmetry, the eight fermionic modes corresponding to the broken supersymmetries.', '0911.5228-3-5-0': 'The main goal of this paper is to study the superstring instanton on [MATH] and analyze its fermionic zero modes.', '0911.5228-3-5-1': 'In the case of branes moving in a supersymmetric background, their fermionic equations have solutions which are associated with the Killing spinors of the background that guarantee its supersymmetries.', '0911.5228-3-5-2': 'The number of physical modes of the worldvolume Dirac operator associated with the Killing spinors is equal to the number of components of the Killing spinors which are not annihilated by the kappa-symmetry projector of the brane.', '0911.5228-3-5-3': 'If the background is maximally supersymmetric, one concludes that the number of dynamical zero modes on the brane are half the number of supersymmetries, since the rank of the kappa-symmetry projector is equal to half the number of the maximal supersymmetries of the background.', '0911.5228-3-5-4': 'When the background is not maximally supersymmetric, as the [MATH] one, the number of the brane fermion zero modes associated with unbroken supersymmetries depends on how many of them are not eliminated by the kappa-symmetry projector.', '0911.5228-3-5-5': 'The action of the kappa-symmetry projector and the corresponding number of fermionic zero modes depends on how the given brane configuration is embedded into target space.', '0911.5228-3-5-6': 'In the cases with less supersymmetries the worldvolume Dirac equation may, in general, also have solutions which are not associated with unbroken supersymmetries, but with the broken ones.', '0911.5228-3-5-7': 'Thus the analysis of the brane fermionic modes in less supersymmetric backgrounds should be made case by case (see e.g. [CITATION] for a more detailed discussion of this point).', '0911.5228-3-6-0': 'As we shall see, in the [MATH] case the string instanton on [MATH] has twelve zero modes all of which are associated with the supersymmetries of the background and there are no zero modes of the fermions corresponding to the supersymmetries broken by [MATH].', '0911.5228-3-6-1': 'So the instanton under consideration is 1/2 BPS.', '0911.5228-3-6-2': 'It is interesting that these twelve zero modes are divided into eight and four ones which have different geometrical and physical meaning.', '0911.5228-3-6-3': 'The eight massive fermionic zero modes are four copies of the two-component Killing spinor on [MATH] and the four other fermionic modes are two copies of massless chiral and anti-chiral fermion on [MATH] electrically coupled to the electromagnetic potential created on [MATH] by a monopole placed in the center of [MATH].', '0911.5228-3-6-4': 'The monopole potential arises as part of the [MATH] spin connection pulled-back on the instanton [MATH].', '0911.5228-3-6-5': 'This, at least formally and remotely, reminds us the peculiarity of the presence of different, light and heavy, physical worldsheet degrees of freedom in a Penrose limit of the [MATH] superstring [CITATION].', '0911.5228-3-7-0': 'In M-theory compactified on [MATH], the counterpart of the string instanton considered in this paper is an Euclidean M2-brane that wraps the non-trivial 3-cycle [MATH] (for [MATH]) inside [MATH].', '0911.5228-3-8-0': 'The presence of the string instanton and its fermionic zero modes may generate non-perturbative corrections to the string effective action, which may affect its properties and if so should be taken into account in studying, e.g. the [MATH] correspondence.', '0911.5228-3-8-1': 'The instantons may, perhaps, contribute to the worldsheet S-matrix and/or to energies of a semiclassical string.', '0911.5228-3-8-2': 'To study these effects one needs to find a way of merging the instanton and Minkowski solutions, such as spinning strings or BMN geodesics.', '0911.5228-3-8-3': 'In addition, in the presence of the instanton fermionic zero modes, the worldsheet correlator, to be non-zero, should contain a number of fermion insertions.', '0911.5228-3-9-0': 'The paper is organized as follows.', '0911.5228-3-9-1': 'In Section [REF] we review the form of the string action in the [MATH] superbackground and truncate it to the second order in fermions.', '0911.5228-3-9-2': 'In Section [REF] we describe the bosonic part of the string instanton solution and in Section [REF] we study its fermionic zero modes.', '0911.5228-3-9-3': 'Section [REF] contains a summary and brief discussion of a possible relation of the string instanton to some features of the [MATH] corespondence.', '0911.5228-3-9-4': 'Appendix A contains a description of our conventions and notation and of the geometry of the [MATH] superspace.', '0911.5228-3-9-5': 'In Appendix B we give the explicit form of the [MATH] Fubini-Study metric, vielbeins and connection which were used for the analysis of the string fermion equations.', '0911.5228-3-10-0': '# [MATH] superstring action up to the quadratic order in fermions', '0911.5228-3-11-0': 'To simplify the study of the fermionic zero modes of the string instanton we reduce the complete superstring action of [CITATION] to the quadratic order in fermions (though, the solutions we find satisfy the complete non-linear equations of motion to all orders in fermions).', '0911.5228-3-11-1': 'Alternatively, one can use the quadratic type IIA superstring action derived in [CITATION] for a generic superbackground and substitute into it the values of the supergravity fields corresponding to the [MATH] background.', '0911.5228-3-11-2': 'As a consistency check we have performed the reduction of both of the actions.', '0911.5228-3-11-3': 'We shall see that they give the same result upon a redefinition of bosonic [MATH] coordinates of the target-superspace of [CITATION].', '0911.5228-3-11-4': 'This redefinition is required due to a particular parametrization used in [CITATION] to construct an explicit form of the [MATH] supergeometry.', '0911.5228-3-12-0': 'The Green-Schwarz superstring action in a generic type IIA supergravity background has the well known form [EQUATION] where [MATH] are the worldsheet coordinates, [MATH] is an intrinsic worldsheet metric, [MATH] are worldsheet pullbacks of target superspace vector supervielbeins and [MATH] is the pull-back of the NS-NS 2-form.', '0911.5228-3-13-0': 'The kappa-symmetry transformations of the worldsheet fields [MATH] which leave the superstring action ([REF]) invariant (provided the superbackground obeys the superspace supergravity constraints) are [EQUATION] where [MATH] is a 32-component spinor parameter, [MATH] is a spinor projection matrix with [EQUATION] where [MATH] is the induced metric on the worldsheet.', '0911.5228-3-13-1': 'The explicit form of the supervielbeins [MATH] and the NS-NS 2-form [MATH] which describe the geometry of the [MATH] superspace are given in Appendix A (see also [CITATION]).', '0911.5228-3-14-0': 'Up to second order in fermions the supervielbeins and the [MATH]-field have the following form 0 [EQUATION] [EQUATION] where [MATH] is the vacuum expectation value of the dilaton, [MATH] is the radius of the [MATH] sphere whose base is [MATH], [MATH] is the eleven-dimensional Planck length related to the string tension as follows [MATH] and [MATH] is the Chern-Simons level related to the units of [MATH] and [MATH] Ramond-Ramond flux which support the [MATH] solution of Type IIA supergravity.', '0911.5228-3-14-1': 'The contribution of the RR fluxes manifests itself in the presence of projectors [MATH] and [MATH] in the string action (see Appendix A.5).', '0911.5228-3-14-2': 'They split the 32-component fermionic variable [MATH] into the 24-component spinors [MATH]; [MATH]) which correspond to the 24 supersymmetries of the [MATH] solution and the 8-component spinors [MATH] which correspond to the broken supersymmetries.', '0911.5228-3-14-3': 'The index [MATH] is a spinor index of [MATH] (see Appendix A for more details).', '0911.5228-3-14-4': 'The covariant derivative [MATH] is defined as follows [EQUATION] where [MATH], [MATH] and [MATH] are, respectively the vielbein, connection and Dirac-matrices of [MATH] of radius [MATH].', '0911.5228-3-14-5': '[MATH] and [MATH] are, respectively, the vielbein and connection on [MATH] of radius [MATH] and [MATH] are [MATH] gamma-matrices of [MATH].', '0911.5228-3-14-6': '[MATH] is the RR one-form potential whose field strength is the Kahler form on [MATH], [MATH].', '0911.5228-3-14-7': 'See Appendix A for more details regarding the notation and conventions.', '0911.5228-3-15-0': 'Substituting the expressions for the vielbeins [REF] and the NS-NS two-form [REF] into the action [REF] and keeping only terms up to quadratic order in fermions we get the following action [EQUATION] where [MATH] is the worldsheet pullback of the conventional [MATH] covariant derivative.', '0911.5228-3-16-0': 'The first two lines of this action coincide with the action which one gets by reducing to [MATH] the quadratic Green-Schwarz action in a generic type IIA superbackground [CITATION].', '0911.5228-3-16-1': 'The last term in the action [REF] appeared because of our choice of parametrization of the [MATH] superspace which allowed us to write its geometry in the simplest form.', '0911.5228-3-16-2': 'It is not hard to see that the last term in [REF] can be canceled (modulo higher order terms in fermions) by making the following shift of the bosonic coordinates [MATH] of [MATH] [EQUATION]', '0911.5228-3-16-3': 'After this field redefinition the two forms of the string action become equivalent.', '0911.5228-3-17-0': 'To study the string instantons we should perform a Wick rotation of the worldsheet and the target space in the action [REF] to Euclidean signature.', '0911.5228-3-17-1': 'The Wick rotation basically consists in replacing [MATH] and [MATH], respectively with [MATH] and [MATH], replacing [MATH] with [MATH] and taking into account that the fermions [MATH] become complex spinors, since there are no Majorana spinors in ten-dimensional Euclidean space.', '0911.5228-3-17-2': 'However, the complex conjugate spinors do not appear in the Wick rotated action and, hence, the number of the fermionic degrees of freedom formally remains the same as before the Wick rotation.', '0911.5228-3-17-3': 'Note also that the Euclidean [MATH] is defined as [MATH], where [MATH] is the Wick rotated [MATH].', '0911.5228-3-17-4': 'So [MATH] as in the case of Minkowski signature.', '0911.5228-3-18-0': 'Thus, after the redefinition [REF] and the Wick rotation the action takes the following form [EQUATION] and the kappa-symmetry matrix [MATH] gets replaced by [EQUATION]', '0911.5228-3-19-0': '# String instanton wrapping a two-sphere inside [MATH] We are interested in a string whose worldsheet wraps a topologically non-trivial two-cycle inside [MATH] and thus is a stringy counterpart of the instantons of two-dimensional [MATH] sigma-models.', '0911.5228-3-19-1': 'To be topologically non-trivial this two-cycle should have a non-zero pull-back on its worldsheet of the Kahler two-form [MATH] of [MATH].', '0911.5228-3-19-2': 'Such a two-cycle is a [MATH] subspace of [MATH].', '0911.5228-3-19-3': 'To identify it, it is convenient to consider the form of the Fubini-Study metric on [MATH] given in [CITATION] [EQUATION] where [MATH], [MATH] and [MATH], and [MATH] 0 [EQUATION] are three left-invariant one-forms on [MATH] obeying [MATH] etc. (see Appendix B for more details).', '0911.5228-3-19-4': 'Notice that with this choice of the [MATH] coordinates, [MATH] and [MATH] parameterize a two-sphere of radius [MATH].', '0911.5228-3-19-5': 'This two-sphere is topologically non-trivial and associated to the Kahler form on [MATH].', '0911.5228-3-19-6': 'The string instanton wraps this sphere.', '0911.5228-3-19-7': 'For instance, if it wraps the sphere once [MATH] and [MATH] can be identified with the string worldsheet coordinates, while all other [MATH] as well as [MATH] coordinates are worldsheet constants in this case.', '0911.5228-3-19-8': 'Thus the pullback on the string instanton of the metric [REF] of [MATH] (of radius [MATH]) is the metric of the sphere of radius [MATH] [EQUATION]', '0911.5228-3-19-9': 'In this coordinate system the [MATH] vielbein [MATH] and the spin connection [MATH] can be chosen in the form [EQUATION] and the [MATH] curvature is [EQUATION]', '0911.5228-3-20-0': '## Bosonic part of the instanton solution', '0911.5228-3-21-0': 'The bosonic part of the Wick rotated string action [REF] is [EQUATION] where [MATH] and [MATH] are the vielbeins on [MATH].', '0911.5228-3-21-1': 'To discuss the instanton solution of this [MATH] sigma model it is convenient to introduce complex coordinates both on the worldsheet and in target space (see [CITATION] for a review of instantons in two-dimensional sigma models).', '0911.5228-3-21-2': 'In the conformal gauge [MATH] and in the [MATH] coordinate system on the worldsheet the action takes the form [EQUATION]', '0911.5228-3-21-3': 'To introduce complex coordinates on the target sphere it is convenient to describe it as [MATH].', '0911.5228-3-21-4': 'The Fubini-Study metric on [MATH] is [EQUATION]', '0911.5228-3-21-5': 'If we choose [MATH] to be [EQUATION] eq. [REF] takes the form of the metric on [MATH] of radius [MATH] [EQUATION]', '0911.5228-3-21-6': 'In the [MATH], [MATH] coordinate system the string action takes the following form (which is similar to that of the [MATH]-sigma model) [EQUATION]', '0911.5228-3-21-7': 'It is now obvious that a local minimum is attained if [MATH] or [MATH], i.e. the embedding is given by a holomorphic function [MATH] for the instanton or by an anti-holomorphic function [MATH] for the anti-instanton.', '0911.5228-3-21-8': 'The remaining part of the action can be shown to be a topological invariant, namely, [EQUATION] where [MATH] is the topolgical charge of the instanton and [MATH] is the [MATH] radius in the string frame.', '0911.5228-3-22-0': 'What we have just reproduced is the classical instanton solution of the two-dimensional [MATH] sigma-model [CITATION] or rather its extension to [MATH] [CITATION] which in terms of the Fubini-Studi coordinates [MATH] of [MATH] (see eq. [REF] of Appendix B) has the form [EQUATION].', '0911.5228-3-22-1': 'The difference with the [MATH] models is that in our case the string action is also invariant under worldsheet reparametrization.', '0911.5228-3-22-2': 'This means that every classical string solution must satisfy the Virasoro constraints implying that the worldsheet bosonic physical fields are associated with the string oscillations transverse to the worldsheet.', '0911.5228-3-22-3': 'For the instanton solution the string excitations along [MATH] are zero and the Virasoro constraints have the following form in the conformal gauge [EQUATION]', '0911.5228-3-22-4': 'We see that the Virasoro constraints are identically satisfied by the (anti)instanton solution.', '0911.5228-3-23-0': 'Let us note that though in the [MATH] background the purely bosonic components of the NS-NS 3-form field strength [MATH] are zero, the NS-NS 2-form may have non-zero expectation values proportional to the Kahler two-form on [MATH], [MATH], where [MATH] plays the role of a constant axion.', '0911.5228-3-23-1': 'For such a two-form, [MATH] is zero since [MATH] is the closed (but not exact) form, [MATH].', '0911.5228-3-23-2': 'In this case also the Wess-Zumino part of the (Wick rotated) string action [REF] will contribute to the instanton action, which becomes [EQUATION]', '0911.5228-3-23-3': 'A similar situation one has in the case of string instantons on Calabi-Yau spaces [CITATION].', '0911.5228-3-23-4': 'In the context of the [MATH] correspondence, the co-homologically non-trivial [MATH] field appears from the string side in the generalization of the ABJM model to include gauge groups of a different rank proposed in [CITATION] (see the Summary below for more discussion of this point).', '0911.5228-3-24-0': 'Finally, we note that the bosonic string instanton has twelve zero modes.', '0911.5228-3-24-1': 'Four of them correspond to the directions along [MATH] and eight are the instanton zero modes on [MATH] [CITATION].', '0911.5228-3-24-2': 'We are now in a position to proceed with the study of the fermionic zero modes carried by the string instanton.', '0911.5228-3-24-3': 'We shall see that their number is also twelve.', '0911.5228-3-25-0': '# Fermionic equations of motion and the fermionic zero modes of the string instanton on [MATH] In a general supergravity background the equation of motion for the fermions following from the Green-Schwarz action [REF] (with the choice of superspace constraints given in Appendix A.4) is [EQUATION]', '0911.5228-3-25-1': 'Taking into account that on the mass shell the auxiliary metric [MATH] and the induced metric [MATH] are proportional to each other [EQUATION] the fermionic equations of motion take the following form which reflects the kappa-symmetry of the theory [EQUATION] where [MATH] is the matrix which appears in the kappa-symmetry projector [REF] and [MATH] is the dilatino superfield.', '0911.5228-3-26-0': 'For completeness, let us also present the equations of motion of the string bosonic modes [EQUATION] where [MATH] are torsion components and [MATH] are components of the NS-NS superfield strength with vector indices [MATH] and with the indices [MATH] and [MATH] standing for both the vector and the spinor indices (see Appendix A.4).', '0911.5228-3-27-0': 'At the linearized level in the [MATH] superspace the equation of motion for the fermions [REF] reduces to [EQUATION] where [MATH] is inverse of [EQUATION] and [EQUATION] where remember that [MATH].', '0911.5228-3-28-0': 'Note that one can alternatively derive eq. [REF] by varying the quadratic action [REF] or its Wick rotated counterpart [REF].', '0911.5228-3-29-0': '## Restriction to the instanton solution', '0911.5228-3-30-0': 'As we have discussed in Section [REF], the instanton solution is supported on the [MATH] two-dimensional subspace whose tangent space is characterized e.g. by the first two values of the [MATH] tangent space index [MATH].', '0911.5228-3-30-1': 'Restricting to this solution we have [EQUATION]', '0911.5228-3-30-2': 'It will be convenient to choose the [MATH] gamma matrices as follows [EQUATION] where [MATH] are the (re-labeled) Pauli matrices so that [MATH], and [MATH] are [MATH] Dirac gamma matrices corresponding to the four-dimensional subspace of [MATH] orthogonal to the instanton [MATH] and [MATH].', '0911.5228-3-31-0': 'The Wick rotated kappa-symmetry projection matrix [REF] then reduces to [EQUATION] and the fermionic part of the Euclidean action [REF] becomes [EQUATION] where the metric [MATH] was defined in [REF].', '0911.5228-3-31-1': 'Note that in our case the fermionic terms of this two-dimensional theory differ from those of the conventional [MATH] supersymmetric [MATH] (or in general [MATH]) sigma-model (see [CITATION] for a review and references).', '0911.5228-3-31-2': 'For comparison, the [MATH] sigma-model Lagrangian is [EQUATION] where now [MATH]) are the complex [MATH] coordinates and [MATH] and [MATH] are independent complex [MATH]-component spinor fields, [MATH] is the Kahler (Fubini-Study) metric on [MATH] (see eq. [REF] of Appendix B for the [MATH] case), [MATH] and [MATH] and [MATH] are the [MATH] Christoffel symbol and curvature, respectively.', '0911.5228-3-32-0': 'In view of the form of the quadratic action [REF] and of the fermionic equation [REF] it is natural to impose on the fermionic fields the conventional kappa-symmetry gauge-fixing condition [EQUATION] which means that the fermions split into two sectors according to their chiralities in [MATH] and in the four-dimensional subspace of [MATH] orthogonal to the instanton [MATH] [EQUATION]', '0911.5228-3-32-1': 'Using the form of the [MATH] gamma-matrices [REF] we find that [EQUATION] where [EQUATION]', '0911.5228-3-32-2': 'So, the supersymmetry projection matrices [MATH] and [MATH] become [EQUATION]', '0911.5228-3-32-3': 'Their action on the two sets of the chiral fermions is [EQUATION]', '0911.5228-3-32-4': 'Note that from eqs. [REF] and [REF] it follows that all the eight [MATH] are fermions which correspond to unbroken supersymmetries of the [MATH] superbackground, while in the [MATH] sector four fermions ([MATH]) correspond to unbroken supersymmetries and other four ([MATH]) to the broken ones.', '0911.5228-3-32-5': "Note also that since for the instanton configuration the kappa-symmetry projector [REF] commutes with the 'supersymmetry' projectors [REF], it is not possible to choose the kappa-symmetry gauge-fixing condition which would put to zero all the eight 'broken-supersymmetry' fermions.", '0911.5228-3-32-6': 'In terms of the fields [MATH], [MATH] and [MATH] the fermionic action [REF] takes the form [EQUATION] where [MATH] is the inverse vielbein on [MATH].', '0911.5228-3-33-0': 'For the instanton configuration the fermionic equation [REF] reduces to the following ones [EQUATION]', '0911.5228-3-33-1': 'From the form of the action [REF] and the equation of motion [REF] it follows that the field [MATH] can be regarded as an auxiliary one, which can be expressed in terms of a derivative of [MATH].', '0911.5228-3-33-2': 'However, for the analysis of the solutions of eqs. [REF]-[REF] it is more convenient to consider it as an independent variable satisfying the Dirac equation [REF].', '0911.5228-3-34-0': 'The covariant derivative [MATH] (defined in [REF]) contains the pullback on the instanton two-sphere of the [MATH] spin connection whose explicit form is given in Appendix B [EQUATION]', '0911.5228-3-34-1': 'Computing the pullback of the [MATH] connection, substituting it into the Dirac equations and taking into account the projection properties of the spinors we get the fermionic equations in the following form [EQUATION] where [MATH] is the intrinsic covariant derivative on the sphere of radius [MATH] with curvature [MATH] and [MATH] can be interpreted as the electromagnetic potential induced by a magnetic monopole of charge [MATH] placed at the center of the sphere.', '0911.5228-3-34-2': 'This is due to the fact that [EQUATION]', '0911.5228-3-34-3': 'Note that [MATH] and [MATH] are equivalent up to a total derivative term [EQUATION].', '0911.5228-3-34-4': 'In our parametrization of [MATH] (see Appendix B) and for a given embedding of [MATH] in [MATH], [MATH] and [MATH] have the following form in terms of the angular coordinates on [MATH] [EQUATION]', '0911.5228-3-34-5': 'We are now in a position to analyze the solutions of the fermionic equations [REF]-[REF].', '0911.5228-3-34-6': 'Eq. [REF] has the form of the Dirac equation for a fermion of mass [MATH].', '0911.5228-3-34-7': 'It is the product of [MATH] with the Killing spinor equation on the sphere [EQUATION]', '0911.5228-3-34-8': 'The Killing spinor equation on [MATH] for a two-component spinor has two non-trivial solutions [CITATION].', '0911.5228-3-34-9': 'Our [MATH] spinors carry four (independent) external indices in addition to the [MATH]-spinor index.', '0911.5228-3-34-10': 'Therefore, eq. [REF] has eight solutions which are obviously solutions of the Dirac equation [REF].', '0911.5228-3-34-11': 'These are actually the only regular eigenspinors of the Dirac operator on the sphere with the eigenvalue [MATH] [CITATION].', '0911.5228-3-34-12': 'Thus, in the [MATH] sector the string instanton has eight fermionic zero modes which are the solutions of the Killing spinor equation [REF].', '0911.5228-3-34-13': 'In spherical coordinates they have the explicit form [CITATION] [EQUATION] where [MATH] is an arbitrary constant spinor satisfying the chirality conditions [MATH].', '0911.5228-3-35-0': 'Let us now proceed with the analysis of the third fermionic equation [REF].', '0911.5228-3-35-1': 'As we have already mentioned, this equation describes the electric coupling of the fermionic field [MATH] to the monopole potential on the sphere.', '0911.5228-3-35-2': 'The electric charge of [MATH] is [MATH] for [MATH], i.e. when [MATH] is a chiral/anti-chiral two-dimensional spinor, respectively.', '0911.5228-3-35-3': 'The analysis in [CITATION] then tells us that there are non-trivial solutions of the charged Dirac equation [REF] of positive chirality when [MATH] and of negative chirality when [MATH].', '0911.5228-3-35-4': 'Since we are in the opposite situation, there are no non-trivial solutions in our case and hence [MATH].', '0911.5228-3-36-0': 'If [MATH], eq. [REF] implies that [MATH] should satisfy the massless Dirac equation [EQUATION]', '0911.5228-3-36-1': 'We observe that the electric charge of [MATH] is opposite to that of [MATH], i.e. it is [MATH] depending on whether [MATH] is chiral or anti-chiral two-dimensional spinor, i.e. whether [MATH].', '0911.5228-3-36-2': 'Now we are in the situation in which the requirement of [CITATION] for the Dirac equation [REF] to have non-trivial solutions is saturated, i.e. in our case for [MATH] of positive [MATH]-chirality [MATH] and for [MATH] of negative [MATH]-chirality [MATH].', '0911.5228-3-36-3': 'By the Atiyah-Singer index theorem there is one solution for each [MATH]-chirality of [MATH].', '0911.5228-3-36-4': 'The general solution of [REF] has actually a very simple form [EQUATION] where [MATH] and [MATH] are holomorphic and anti-holomorphic spinors in the projective coordinates [MATH] and [MATH] of [MATH] which are anti-chiral in the directions transverse to the instanton, i.e. [MATH], [MATH] and [MATH].', '0911.5228-3-36-5': 'For the anti-instanton the solution takes the same form but with anti-holomorphic [MATH] and holomorphic [MATH].', '0911.5228-3-37-0': 'In [CITATION] it has been shown that the only normalizable solutions of the Dirac equation [REF] are those with constant [MATH] and [MATH] in [REF].', '0911.5228-3-37-1': 'This allows us to conclude that in the [MATH] sector the string instanton has four zero modes characterized by eq. [REF] with constant [MATH] and [MATH].', '0911.5228-3-37-2': 'Note that for [MATH] and [MATH] the spinor [REF] is the solution of the stronger equation [EQUATION]', '0911.5228-3-37-3': 'This equation is the projection on the instantonic sphere of the [MATH] Killing spinor equation for [MATH].', '0911.5228-3-38-0': 'To summarize, when [MATH] and in view of the form of the fermionic supervielbeins [MATH] of the supercoset [MATH] (see Appendix A.7), the non-linear fermionic equation of motion [REF] as well as the linear one [REF] involve the pull-back on the string worldsheet of the [MATH] Killing spinor operator [EQUATION] which acts on the 24 fermions [MATH] associated with the supersymmetry of [MATH] (see Appendix A.7).', '0911.5228-3-38-1': 'Therefore, if [MATH] are the 24 Killing spinors on [MATH] they solve not only the linearized equations [REF] but also the complete fermionic equations [REF].', '0911.5228-3-38-2': 'In the case of the string instanton considered above, the kappa-symmetry projector reduces the number of solutions of the pulled-back Killing spinor equation by half, leaving us with the twelve physical fermionic zero modes.', '0911.5228-3-38-3': 'It should also be noted that these fermionic zero modes do not contribute to the bosonic equations [REF].', '0911.5228-3-38-4': 'This guarantees that the bosonic instanton solution does not have a back reaction from the fermionic modes.', '0911.5228-3-39-0': 'We should note that the Dirac equations [REF]-[REF] may have (non-normalizable) solutions which are not the Killing spinors (as e.g. eq. [REF] with non-constant [MATH] and [MATH]).', '0911.5228-3-39-1': 'However, these other fermionic modes would modify the string field equations at higher order in fermions.', '0911.5228-3-39-2': 'In particular, they would produce a non-trivial contribution to the bosonic field equations [REF], i.e. back-react on the form of the purely bosonic instanton and, hence, should be discarded.', '0911.5228-3-40-0': 'Let us stress once again that, as we have shown, for the instanton solution considered above the kappa-symmetry cannot eliminate all the eight fermions [MATH] associated with the supersymmetries broken in the [MATH] background.', '0911.5228-3-40-1': 'Therefore, even if among the instanton fermionic zero modes there is no [MATH]-modes, the fluctuations around the instanton solution will have four physical fermionic degrees of freedom corresponding to the target-space supersymmetries broken by the [MATH] background.', '0911.5228-3-41-0': '## Fermionic zero modes and supersymmetry', '0911.5228-3-42-0': 'Let us discuss in more detail how the fermionic zero modes are related to supersymmetry of the [MATH] superbackground and, correspondingly, of the superstring action.', '0911.5228-3-42-1': 'At the linearized level in fermions the supersymmetry part of the [MATH] transformations acts as follows [EQUATION] where [MATH] are 24 supersymmetry parameters of [MATH] satisfying the [MATH] Killing spinor equation [EQUATION] with the explicit form of [MATH] given in eq. [REF].', '0911.5228-3-42-2': 'Note that, at the leading order in fermions, the eight fermions [MATH] are not subject to the supersymmetry transformations.', '0911.5228-3-42-3': 'The action of the isometry group [MATH] on these fermions is such that it takes the form of induced [MATH] rotations with parameters depending on [MATH], [MATH] and the [MATH] parameters [EQUATION]', '0911.5228-3-42-4': 'Thus the first nontrivial term in the supersymmetry variation of [MATH] is quadratic in fermionic fields which is beyond the linear approximation we are interested in.', '0911.5228-3-43-0': 'It is not hard to see that the quadratic string action [REF] is invariant under the supersymmetry transformations [REF] (up to quadratic order in fermions).', '0911.5228-3-43-1': 'At the same time, the action [REF], which is obtained from [REF] by the shift [REF] of the [MATH] coordinates, is invariant under the supersymmetry with the transformations of the shifted bosonic coordinates being [EQUATION]', '0911.5228-3-43-2': 'Let us now briefly recall how the target-space supersymmetry gets converted into worldsheet supersymmetry upon elimination of the un-physical fermionic degrees of freedom by gauge fixing kappa-symmetry.', '0911.5228-3-43-3': 'A more detailed discussion of such a "transmutation" of supersymmetry and its partial breaking in the Green-Schwarz formulation of superstrings and superbranes the reader may find e.g. in [CITATION].', '0911.5228-3-44-0': 'If we impose on the fermionic fields [MATH] a kappa-symmetry gauge condition as e.g. the one we have used for studying the instanton solution, eq. [REF], [EQUATION] the kappa-symmetry gauge-fixing condition will not be invariant under all the twenty-four supersymmetries [REF] but only under half of them satisfying the condition [EQUATION]', '0911.5228-3-44-1': 'In eqs. [REF] and [REF] we denoted the gauge-fixing projector by [MATH] to distinguish it from the more general projector matrix [MATH] that appears in the kappa-symmetry transformations [REF]-[REF].', '0911.5228-3-45-0': 'The target-space supersymmetries with the parameter [MATH] are those which are spontaneously broken by the presence of the string.', '0911.5228-3-45-1': 'The reason is that the remaining twelve fermionic fields [MATH] get shifted by these transformations and hence behave as Volkov-Akulov goldstinos [CITATION].', '0911.5228-3-46-0': 'The supersymmetries which remain unbroken and which become worldsheet supersymmetries are identified as follows.', '0911.5228-3-46-1': 'The gauge fixing condition [REF] is not invariant under the supersymmetry transformations [REF] with the parameter [MATH].', '0911.5228-3-46-2': 'However, this can be cured by an appropriate compensating kappa-symmetry transformation that (at the leading order in fermions) satisfies the condition [EQUATION]', '0911.5228-3-46-3': 'This condition relates the components of the [MATH]-symmetry parameter appearing in the transformation of [MATH] to the supersymmetry parameter [MATH].', '0911.5228-3-46-4': 'Since kappa-symmetry is the worldsheet fermionic symmetry which can actually be identified with the conventional local worldsheet supersymmetry [CITATION], eq. [REF] thus converts the unbroken target-space supersymmetries into worldsheet supersymmetry.', '0911.5228-3-46-5': 'Note that eq. [REF] does not involve the part of the kappa-symmetry transformation acting on the [MATH]-fermions since they are singled out with the complementary projector [MATH].', '0911.5228-3-46-6': 'This part of kappa-symmetry is fixed by the gauge condition [MATH] (see eq. [REF]).', '0911.5228-3-47-0': 'As a result, (at a leading order in fermions and bosons) under the broken and unbroken supersymmetries the worldsheet fermionic fields remaining after the gauge-fixing [REF] [MATH] and the bosonic fields [MATH] transform as follows [EQUATION] where [MATH] and [MATH] indicate the directions parallel and orthogonal to the string worldsheet, respectively, the second terms in [REF] and [REF] come from the compensating kappa-symmetry transformation [REF] that at the linearized level is just [MATH], and [MATH] stand for terms which are non-linear in fields (and their derivatives).', '0911.5228-3-48-0': 'It is instructive to notice that the leading (linear) term in the supersymmetry transformations of [MATH] along the directions trasverse to the string worldsheet contains the parameter of the unbroken supersymmetries, while along the worldsheet the linear term contains the broken supersymmetry parameter.', '0911.5228-3-48-1': 'This reflects the fact that the bosonic excitations transversal to the classical string configuration and kappa-gauge fixed fermionic fields are associated with worldsheet physical fields forming supermultiplets under the unbroken supersymmetry.', '0911.5228-3-48-2': 'At the same time the supersymmetry transformations along the string worldsheet can be compensated by an appropriated worldsheet reparametrization.', '0911.5228-3-49-0': 'For the instanton solution under consideration we have [MATH] and [MATH].', '0911.5228-3-49-1': 'So the supersymmetry transformations (at the leading order) become [EQUATION] where the dots stand for higher order terms in fermions.', '0911.5228-3-50-0': 'Under the unbroken supersymmetry transformations the fermionic zero modes induce an (isometry) transformation of the string coordinates in the transverse directions which results in a shift of the bosonic parameters characterizing the string instanton.', '0911.5228-3-50-1': "This is analogous to the supersymmetry transformations of the 'collective coordinates' of the [MATH] sigma-model instanton [CITATION].", '0911.5228-3-51-0': 'From eqs. [REF] and [REF] we also see that if we start from the purely bosonic instanton solution discussed in Section [REF], we can find at least part of the instanton fermionic zero modes by looking at the variation of the fermionic fields under supersymmetry.', '0911.5228-3-51-1': 'The form of the supersymmetry transformations implies that the bosonic instanton configuration is 1/2 BPS.', '0911.5228-3-51-2': 'Namely, the string instanton solution with [MATH] is invariant under the twelve supersymmetries [MATH].', '0911.5228-3-51-3': 'Fermionic zero modes are generated by the target-space supersymmetries (with the parameter [MATH]) which are broken by the string configuration, as we have already discussed in the end of Section [REF] where we have also shown that the instanton does not have other fermionic zero modes associated with the fields [MATH].', '0911.5228-3-51-4': 'Note that the latter cannot be obtained from the purely bosonic solution by a supersymmetry transformation since the corresponding variation of [MATH] is proportional to [MATH] itself (see eq. [REF]).', '0911.5228-3-52-0': 'Let us now compare our [MATH] superstring worldsheet theory (which has [MATH] unbroken worldsheet supersymmetries) with the supersymmetry properties of the conventional [MATH] supersymmetric [MATH] sigma-model (described by the Lagrangian [REF]) and with its instanton solutions [CITATION].', '0911.5228-3-53-0': 'The supersymmetry transformations in the [MATH] sigma-model have the following form [EQUATION] where [MATH] is now a constant complex two-component spinor parameter of [MATH] supersymmetry and the dots stand for the terms non-linear in the fields.', '0911.5228-3-53-1': 'The [MATH] sigma-model is also invariant under superconformal transformations [CITATION] [EQUATION]', '0911.5228-3-53-2': 'The superconformal transformations are similar to the rigid supersymmetries [REF] but with the complex two-component spinor parameters whose chiral and anti-chiral components are, respectively, holomorphic and anti-holomorphic [MATH].', '0911.5228-3-54-0': 'The superconformal symmetry of the [MATH] sigma-model (which is broken by quantum anomalies [CITATION]) is in a sense a counterpart of the spontaneously broken part of the target-space supersymmetry of the superstring action.', '0911.5228-3-55-0': 'If one starts from the purely bosonic instanton solution of the [MATH] sigma-model [EQUATION] one can then generate solutions of the fermionic field equations and find the corresponding fermionic zero modes by considering the supersymmetry transformations [REF] and [REF] of [MATH].', '0911.5228-3-55-1': 'In this way, taking into account that for the instanton the fields [MATH] are either holomorphic or anti-holomorphic, one finds that only half of the supersymmetry transformations [REF] and of the superconformal transformations [REF] are non-trivial, those with the parameters [MATH] and [MATH] being (anti)chiral [MATH] spinors.', '0911.5228-3-55-2': 'The fermionic zero modes obtained in this way are (anti)holomorphic (anti)chiral [MATH] spinor fields.', '0911.5228-3-55-3': 'We observe that in contrast to the case of the string instanton whose fermionic zero modes are generated by the spontaneously broken supersymmetry transformations, in the case of the [MATH] sigma-model half of the fermionic zero modes are generated by the rigid supersymmetry transformations and another half by the superconformal symmetry.', '0911.5228-3-56-0': '# Summary and Discussion', '0911.5228-3-57-0': 'We have thus found that the string instanton wrapping the non-trivial two-cycle inside [MATH] has twelve fermionic zero modes.', '0911.5228-3-57-1': 'As we have already mentioned, the eight string fermionic fields [MATH] which have an effective mass [MATH] and four massless [MATH] electrically coupled to the [MATH] monopole field, correspond to twelve (of the twenty four) supersymmetries of the [MATH] background.', '0911.5228-3-57-2': 'The fermionic zero modes thus play the role similar to Volkov-Akulov goldstinos [CITATION] and manifest partial breaking of supersymmetry.', '0911.5228-3-57-3': 'Note that in [MATH] there also exists an NS5-brane instanton wrapping the entire [MATH].', '0911.5228-3-57-4': 'It would be of interest to analyze possible effects of these instantons in [MATH] superstring theory and to understand their counterparts in the boundary [MATH] theory.', '0911.5228-3-58-0': 'As was mentioned briefly in Section [REF] the instanton solution can be generalized by switching on a NS-NS field [MATH] of a non-trivial co-homology on [MATH].', '0911.5228-3-58-1': 'The coupling of the string to the [MATH]-field then results in the instanton action being shifted by a constant imaginary piece.', '0911.5228-3-58-2': 'In fact, the co-homologically non-trivial [MATH]-field arises on the string side of the [MATH] correspondence when one considers the ABJ-model [CITATION] which generalizes the ABJM construction to gauge groups of different rank, i.e. [MATH] with [MATH] instead of [MATH].', '0911.5228-3-58-3': 'The appearance of the [MATH]-field in the string action thus results in breaking the parity-invariance of the theory.', '0911.5228-3-58-4': 'In [CITATION] it has been shown that the integral of [MATH] on the [MATH] cycle inside [MATH] takes a fractional value [EQUATION]', '0911.5228-3-58-5': 'In eleven dimensions this corresponds to a co-homologically nontrivial three-form potential on the 3-cycle [MATH].', '0911.5228-3-58-6': 'From the point of view of M2-branes probing a [MATH] singularity this is associated to [MATH] fractional M2-branes sitting at the singularity.', '0911.5228-3-58-7': 'The fractional M2-branes can be thought of as M5-branes wrapping the corresponding vanishing 3-cycle at the orbifold point, see [CITATION].', '0911.5228-3-59-0': 'This picture suggests that the string instanton considered in this paper should correspond in M-theory to an instantonic M2-brane, i.e. an M2-brane whose worldvolume wraps a 3-cycle [MATH].', '0911.5228-3-59-1': 'There is, however, a subtlety here.', '0911.5228-3-59-2': 'Namely, while in the case of the [MATH] type IIA string instanton there are an infinite number ([MATH]) of topologically different configurations, the number of non-equivalent M2-brane configurations wrapping the 3-cycle in [MATH] of the [MATH] theory is [MATH], since [MATH].', '0911.5228-3-59-3': 'The reason is that the string instanton solution has been considered in the [MATH] background of the pure type IIA [MATH] supergravity, i.e. in the limit in which (from the [MATH] perspective) the radius of the [MATH] fiber of [MATH] tends to zero ([MATH]).', '0911.5228-3-59-4': 'The consideration at finite [MATH] would require taking into account Kaluza-Klein modes and D-brane effects.', '0911.5228-3-60-0': 'It would be interesting to find out what these instantonic strings, M2-branes and NS5-branes correspond to in the ABJ/ABJM gauge-theory picture.'}

1212.0833

{'1212.0833-1-0-0': 'In this paper we give the list of all [MATH]-dimensional nilmanifolds that admit an invariant contact structure.', '1212.0833-1-1-0': '# Introduction', '1212.0833-1-2-0': 'The goal of the paper is to list all 7-dimensional nilmanifolds admitting an invariant contact structure.', '1212.0833-1-2-1': 'First, we recall some definitions.', '1212.0833-1-3-0': 'For a Lie algebra [MATH] the upper central series is the increasing sequence of ideals defined as follows: [EQUATION].', '1212.0833-1-3-1': 'In other words [MATH] is the inverse image under the canonical mapping of [MATH] onto [MATH] of the center of [MATH].', '1212.0833-1-4-0': 'The Lie algebra [MATH] is called nilpotent if there is an integer [MATH] such that [MATH].', '1212.0833-1-4-1': 'The minimal such [MATH] is called the index of nilpotency, [CITATION].', '1212.0833-1-5-0': 'A connected Lie group is nilpotent if and only if its Lie algebra is nilpotent.', '1212.0833-1-6-0': 'A nilmanifold [MATH] is a compact homogeneous space of the form [MATH], where [MATH] is a simply connected nilpotent Lie group and [MATH] is a discrete cocompact subgroup in [MATH], [CITATION].', '1212.0833-1-7-0': 'For example, an [MATH]-dimensional torus [MATH] is obviously a nilmanifold.', '1212.0833-1-7-1': 'If we consider the group [MATH] of upper triangular matrices having 1s along the diagonal then the quotient [MATH] is a nilmanifold, called the Heisenberg nilmanifold, where [MATH] is the set of matrices having integral entries.', '1212.0833-1-8-0': 'Let [MATH] be a nilmanifold and let [MATH] be the Lie algebra of the Lie group [MATH].', '1212.0833-1-8-1': 'It is well-known that the exponential map [MATH] is a global diffeomorphism and the quotient map [MATH] is the universal covering map.', '1212.0833-1-8-2': 'Hence every nilmanifold is the Eilenberg-MacLane space [MATH].', '1212.0833-1-8-3': "By Malcev's theorem a discrete group [MATH] can be realized as the fundamental group of a nilmanifold if and only if it is a finitely presented nilpotent torsion free group [CITATION].", '1212.0833-1-9-0': 'Recall that a space [MATH] is called nilpotent if the fundamental group [MATH] is nilpotent and the action of [MATH] on all homotopy groups [MATH] is nilpotent.', '1212.0833-1-9-1': 'It is clear that every nilmanifold [MATH] is a nilpotent space since [MATH] is nilpotent and all higher homotopy groups [MATH] are trivial.', '1212.0833-1-9-2': 'Hence it follows from the fundamental theorem of the rational homotopy theory [CITATION] that two nilmanifolds are homotopy equivalent if and only if the corresponding Chevalley-Eilenberg complexes are isomorphic.', '1212.0833-1-10-0': '[[CITATION]] A simply connected nilpotent Lie group [MATH] admits a discrete cocompact subgroup [MATH] if and only if there exists a basis [MATH] of the Lie algebra [MATH] of [MATH] such that the structural constants [MATH] arising in brackets [EQUATION] are rational numbers for all [MATH].', '1212.0833-1-11-0': 'Let [MATH] be a Lie algebra with a basis [MATH].', '1212.0833-1-11-1': 'Denote by [MATH] the basis for [MATH] dual to [MATH].', '1212.0833-1-11-2': 'We obtain a differential [MATH] on the exterior algebra [MATH] by defining it on degree 1 elements as [EQUATION] and extending to [MATH] as a graded bilinear derivation.', '1212.0833-1-11-3': 'Then [EQUATION] where [MATH] are the structure constants of [MATH], and it follows from duality that [EQUATION].', '1212.0833-1-11-4': 'Hence on generators the differential is expressed as [EQUATION].', '1212.0833-1-11-5': 'Note that the condition [MATH] is equivalent to the Jacobi identity in the Lie algebra.', '1212.0833-1-11-6': 'We call the differential graded algebra [EQUATION] the Chevalley-Eilenberg complex of the Lie algebra [MATH].', '1212.0833-1-12-0': 'A contact structure on a manifold [MATH] of odd dimension [MATH] is a completely non-integrable [MATH]-dimensional tangent plane distribution [MATH].', '1212.0833-1-12-1': 'In the coorientable case the distribution may be defined by a differential 1-form [MATH] as [MATH].', '1212.0833-1-12-2': 'Then the non-integrability condition can be expressed by the inequality [MATH], i.e. the form [MATH] is nowhere zero.', '1212.0833-1-12-3': 'Such differential form [MATH] is called a contact differential form.', '1212.0833-1-12-4': 'A contact structure can be viewed as an equivalence class of contact differential forms, where two forms [MATH] and [MATH] are equivalent if and only if [MATH] where [MATH] is a nowhere zero smooth function on [MATH].', '1212.0833-1-12-5': 'A contact manifold is a pair [MATH] where [MATH] is a smooth manifold of dimension [MATH] and [MATH] is a contact structure on [MATH], [CITATION].', '1212.0833-1-13-0': 'We say that a Lie algebra [MATH] of dimension [MATH] admits a contact structure if there is an element [MATH] of degree [MATH] in the Chevalley-Eilenberg complex [MATH] of [MATH] such that [MATH].', '1212.0833-1-14-0': 'We say that a contact structure (resp. a contact form) on a nilmanifold [MATH] is left (right) invariant if the contact structure (resp. a contact form) is invariant with respect to the left (resp. right) [MATH]-action on [MATH].', '1212.0833-1-15-0': 'Since every invariant diferential form on a nilmanifold [MATH] is completely determined by its values on the Lie algebra [MATH] of [MATH], we conclude that a nilmanifold [MATH] admits an invariant contact structure if and only if [MATH] admits a contact structure.', '1212.0833-1-16-0': 'Note that any invariant contact structure on [MATH] is coorientable.', '1212.0833-1-17-0': 'Clearly, the kernel of invariant contact form gives us an invariant contact structure.', '1212.0833-1-17-1': 'Conversely, for any contact form [MATH] on [MATH] we can perform the averaging of the lift of [MATH] to [MATH] rescaled down by a nowhere zero function with the finite integral and get an invariant contact form [MATH].', '1212.0833-1-17-2': 'Furthermore, [MATH] and [MATH] yield the same contact structure provided [MATH] defines an invariant contact structure.', '1212.0833-1-17-3': 'In particular, a nilmanifold [MATH] admits an invariant contact structure if and only if [MATH] admits an invariant contact form.', '1212.0833-1-18-0': 'To find all 7-dimensional nilmanifolds that admit an invariant contact structure we used the classification of 7-dimensional indecomposable nilpotent Lie algebras.', '1212.0833-1-18-1': 'Many attempts have been done on this topic.', '1212.0833-1-18-2': '(We shall see below in Section [REF] that the decomposable nilpotent Lie algebras do not admit a contact structure.)', '1212.0833-1-18-3': 'There are a few lists available: Safiullina (1964, over [MATH]) [CITATION], Romdhani (1985, over [MATH] and [MATH]) [CITATION], Seeley (1988, over [MATH]) [CITATION], Ancochea and Goze (1989, over [MATH]) [CITATION].', '1212.0833-1-18-4': 'The lists above are obtained by using different invariants.', '1212.0833-1-18-5': 'Carles [CITATION] introduced a new invariant - the weight system, compared the lists of Safiullina, Romdhani and Seeley, and found mistakes and omissions in all of them.', '1212.0833-1-18-6': 'Later in 1993 Seeley incorporated all the previous results and published his list over [MATH], [CITATION].', '1212.0833-1-18-7': 'In 1998 Gong used the Skjelbred-Sund method to classify all 7-dimensional nilpotent Lie algebras over [MATH], [CITATION].', '1212.0833-1-18-8': "We will use Gong's classification with some corrections from the list of Magnin, [CITATION].", '1212.0833-1-19-0': 'To find an invariant contact structure on a nilmanifold, we first apply Remark [REF].', '1212.0833-1-19-1': "So, we must check whether the nilpotent Lie algebras from the Gong's list admit a contact structure.", '1212.0833-1-19-2': 'To achieve this goal, we use the Sullivan minimal model theory and Corollary [REF] (see sections [REF] and [REF]).', '1212.0833-1-19-3': 'Next, for a given 7-dimensional nilpotent Lie algebra having a contact structure, we must check whether the corresponding simply connected Lie group contains a discrete cocompact subgroup.', '1212.0833-1-19-4': "For this we use the Malcev's criterion (Theorem [REF])."}

{'1212.0833-2-0-0': 'In this paper we give the list of all [MATH]-dimensional nilpotent real Lie algebras that admit a contact structure.', '1212.0833-2-0-1': 'Based on this list, we describe all [MATH]-dimensional nilmanifolds that admit an invariant contact structure.', '1212.0833-2-1-0': '# Introduction', '1212.0833-2-2-0': 'The goal of the paper is to describe all 7-dimensional nilmanifolds admitting an invariant contact structure, see Theorem [REF].', '1212.0833-2-2-1': 'To achieve this goal, we classify all 7-dimensional real nilpotent Lie algebra that admit a contact structure, see Corollary [REF].', '1212.0833-2-2-2': 'First, we recall some definitions.', '1212.0833-2-3-0': 'For a Lie algebra [MATH] the upper central series is the increasing sequence of ideals defined as follows: [EQUATION].', '1212.0833-2-3-1': 'In other words [MATH] is the inverse image under the canonical mapping of [MATH] onto [MATH] of the center of [MATH].', '1212.0833-2-4-0': 'The Lie algebra [MATH] is called nilpotent if there is an integer [MATH] such that [MATH].', '1212.0833-2-4-1': 'The minimal such [MATH] is called the index of nilpotency, [CITATION].', '1212.0833-2-5-0': 'A connected Lie group is nilpotent if and only if its Lie algebra is nilpotent.', '1212.0833-2-6-0': 'A nilmanifold [MATH] is a compact homogeneous space of the form [MATH], where [MATH] is a simply connected nilpotent Lie group and [MATH] is a discrete cocompact subgroup in [MATH], [CITATION].', '1212.0833-2-7-0': 'For example, an [MATH]-dimensional torus [MATH] is obviously a nilmanifold.', '1212.0833-2-7-1': 'If we consider the group [MATH] of upper triangular matrices having 1s along the diagonal then the quotient [MATH] is a nilmanifold, called the Heisenberg nilmanifold, where [MATH] is the set of matrices having integral entries.', '1212.0833-2-8-0': 'Let [MATH] be a nilmanifold and let [MATH] be the Lie algebra of the Lie group [MATH].', '1212.0833-2-8-1': 'It is well-known that the exponential map [MATH] is a global diffeomorphism and the quotient map [MATH] is the universal covering map.', '1212.0833-2-8-2': 'Hence every nilmanifold is the Eilenberg-MacLane space [MATH].', '1212.0833-2-8-3': "By Malcev's theorem a discrete group [MATH] can be realized as the fundamental group of a nilmanifold if and only if it is a finitely presented nilpotent torsion free group [CITATION].", '1212.0833-2-9-0': 'Recall that a space [MATH] is called nilpotent if the fundamental group [MATH] is nilpotent and the action of [MATH] on all homotopy groups [MATH] is nilpotent.', '1212.0833-2-9-1': 'It is clear that every nilmanifold [MATH] is a nilpotent space since [MATH] is nilpotent and all higher homotopy groups [MATH] are trivial.', '1212.0833-2-9-2': 'Hence, it follows from the fundamental theorem of the rational homotopy theory [CITATION] that two nilmanifolds have the same rational homotopy type if and only if the corresponding Chevalley-Eilenberg complexes are isomorphic.', '1212.0833-2-10-0': '[[CITATION]] A simply connected nilpotent Lie group [MATH] admits a discrete cocompact subgroup [MATH] if and only if there exists a basis [MATH] of the Lie algebra [MATH] of [MATH] such that the structural constants [MATH] arising in brackets [EQUATION] are rational numbers for all [MATH].', '1212.0833-2-11-0': 'Let [MATH] be a Lie algebra with a basis [MATH].', '1212.0833-2-11-1': 'Denote by [MATH] the basis for [MATH] dual to [MATH].', '1212.0833-2-11-2': 'We obtain a differential [MATH] on the exterior algebra [MATH] by defining it on degree 1 elements as [EQUATION] and extending to [MATH] as a graded bilinear derivation.', '1212.0833-2-11-3': 'Then [EQUATION] where [MATH] are the structure constants of [MATH], and it follows from duality that [EQUATION].', '1212.0833-2-11-4': 'Hence on generators the differential is expressed as [EQUATION].', '1212.0833-2-11-5': 'Note that the condition [MATH] is equivalent to the Jacobi identity in the Lie algebra.', '1212.0833-2-11-6': 'We call the differential graded algebra [EQUATION] the Chevalley-Eilenberg complex of the Lie algebra [MATH].', '1212.0833-2-12-0': 'A contact structure on a manifold [MATH] of odd dimension [MATH] is a completely non-integrable [MATH]-dimensional tangent plane distribution [MATH].', '1212.0833-2-12-1': 'In the coorientable case the distribution may be defined by a differential 1-form [MATH] as [MATH].', '1212.0833-2-12-2': 'Then the non-integrability condition can be expressed by the inequality [MATH], i.e. the form [MATH] is nowhere zero.', '1212.0833-2-12-3': 'Such differential form [MATH] is called a contact differential form.', '1212.0833-2-12-4': 'A contact structure can be viewed as an equivalence class of contact differential forms, where two forms [MATH] and [MATH] are equivalent if and only if [MATH] where [MATH] is a nowhere zero smooth function on [MATH].', '1212.0833-2-12-5': 'A contact manifold is a pair [MATH] where [MATH] is a smooth manifold of dimension [MATH] and [MATH] is a contact structure on [MATH], [CITATION].', '1212.0833-2-13-0': 'We say that a Lie algebra [MATH] of dimension [MATH] admits a contact structure if there is an element [MATH] of degree [MATH] in the Chevalley-Eilenberg complex [MATH] of [MATH] such that [MATH].', '1212.0833-2-14-0': 'We say that a contact structure (resp. a contact form) on a nilmanifold [MATH] is left (right) invariant if the contact structure (resp. a contact form) is invariant with respect to the left (resp. right) [MATH]-action on [MATH].', '1212.0833-2-15-0': 'Since every invariant diferential form on a nilmanifold [MATH] is completely determined by its values on the Lie algebra [MATH] of [MATH], we conclude that a nilmanifold [MATH] admits an invariant contact structure if and only if [MATH] admits a contact structure.', '1212.0833-2-16-0': 'Note that any invariant contact structure on [MATH] is coorientable.', '1212.0833-2-17-0': 'Clearly, the kernel of invariant contact form gives us an invariant contact structure.', '1212.0833-2-17-1': 'Conversely, for any contact form [MATH] on [MATH] we can perform the averaging of the lift of [MATH] to [MATH] rescaled down by a nowhere zero function with the finite integral and get an invariant contact form [MATH].', '1212.0833-2-17-2': 'Furthermore, [MATH] and [MATH] yield the same contact structure provided [MATH] defines an invariant contact structure.', '1212.0833-2-17-3': 'In particular, a nilmanifold [MATH] admits an invariant contact structure if and only if [MATH] admits an invariant contact form.', '1212.0833-2-18-0': 'To find all 7-dimensional nilmanifolds that admit an invariant contact structure we used the classification of 7-dimensional indecomposable nilpotent Lie algebras.', '1212.0833-2-18-1': 'Many attempts have been done on this topic.', '1212.0833-2-18-2': '(We shall see below in Section [REF] that the decomposable nilpotent Lie algebras do not admit a contact structure.)', '1212.0833-2-18-3': 'There are a few lists available: Safiullina (1964, over [MATH]) [CITATION], Romdhani (1985, over [MATH] and [MATH]) [CITATION], Seeley (1988, over [MATH]) [CITATION], Ancochea and Goze (1989, over [MATH]) [CITATION].', '1212.0833-2-18-4': 'The lists above are obtained by using different invariants.', '1212.0833-2-18-5': 'Carles [CITATION] introduced a new invariant - the weight system, compared the lists of Safiullina, Romdhani and Seeley, and found mistakes and omissions in all of them.', '1212.0833-2-18-6': 'Later in 1993 Seeley incorporated all the previous results and published his list over [MATH], [CITATION].', '1212.0833-2-18-7': 'In 1998 Gong used the Skjelbred-Sund method to classify all 7-dimensional nilpotent Lie algebras over [MATH], [CITATION].', '1212.0833-2-18-8': "We will use Gong's classification with some corrections from the list of Magnin, [CITATION].", '1212.0833-2-19-0': 'To find an invariant contact structure on a nilmanifold, we first apply Remark [REF].', '1212.0833-2-19-1': "So, we must check whether the nilpotent Lie algebras from the Gong's list admit a contact structure.", '1212.0833-2-19-2': 'To achieve this goal, we use the Sullivan minimal model theory and Corollary [REF] (see sections [REF] and [REF]).', '1212.0833-2-19-3': 'Next, for a given 7-dimensional nilpotent Lie algebra having a contact structure, we must check whether the corresponding simply connected Lie group contains a discrete cocompact subgroup.', '1212.0833-2-19-4': "For this we use the Malcev's criterion (Theorem [REF])."}

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[['1212.0833-1-9-2', '1212.0833-2-9-2']]

[]

[['1212.0833-1-2-0', '1212.0833-2-2-0'], ['1212.0833-1-0-0', '1212.0833-2-0-0'], ['1212.0833-1-0-0', '1212.0833-2-0-1']]

[]

['1212.0833-1-16-0', '1212.0833-1-18-3', '1212.0833-2-16-0', '1212.0833-2-18-3']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1212.0833

gr-qc-9712079

{'gr-qc-9712079-1-0-0': 'A spherical gravitational wave antenna is distinct from other types of gravitational wave antennas in that only a single detector is necessary to determine the direction and polarization of a gravitational wave.', 'gr-qc-9712079-1-0-1': 'We show that it is possible to solve the inverse problem in the presence of noise using only linear algebra.', 'gr-qc-9712079-1-0-2': 'Applications of this solution to general metric theories of gravity and other types of excitations also are discussed.', 'gr-qc-9712079-1-1-0': '# Introduction', 'gr-qc-9712079-1-2-0': 'Several resonant-mass gravitational wave antennas are now in continuous operation with strain sensitivities of the order [MATH] [CITATION].', 'gr-qc-9712079-1-2-1': 'With further improvements to these detectors and the addition of several large laser interferometers now under construction [CITATION], the prospects for gravitational wave astronomy are quite good.', 'gr-qc-9712079-1-2-2': 'The underlying non-gravitational physics associated with these detectors is reasonably understood and further improvements can be based on solid technological guidelines.', 'gr-qc-9712079-1-3-0': 'Many believe that the next generation of resonant-mass antennas will be of spherical shape [CITATION].', 'gr-qc-9712079-1-3-1': 'Since confirmed detection of gravitational waves will require a coincidence between several detectors, the unique features of a sphere may play an essential role in a network of gravitational wave antennas.', 'gr-qc-9712079-1-3-2': 'Two important features of a sphere are its equal sensitivity to gravitational waves from all directions and polarizations and its ability to determine the directional information and tensorial character of a gravitational wave [CITATION].', 'gr-qc-9712079-1-4-0': 'To take full advantage of these capabilities, one needs to be able to interpret the data such a detector will produce.', 'gr-qc-9712079-1-4-1': 'Recently, much work has been done to understand the output of a spherical antenna equipped with resonant transducers [CITATION].', 'gr-qc-9712079-1-4-2': 'All of these proposals operate on the principle that the response of the transducers can be transformed into a quantity that has a one-to-one correspondence with the tensorial components of a gravitational wave.', 'gr-qc-9712079-1-4-3': 'With the measurement of these components, it is possible to solve the inverse problem and obtain the direction and polarization amplitudes of a gravitational wave.', 'gr-qc-9712079-1-5-0': "The solution to the inverse problem for a noiseless spherical antenna first was outlined in the mid 1970's by Wagoner and Paik [CITATION].", 'gr-qc-9712079-1-5-1': 'More recently, the solution for a network of five noiseless bar antennas or interferometers was solved by Dhurandhar and Tinto [CITATION].', 'gr-qc-9712079-1-5-2': 'This method assumed that the detectors were co-located but oriented in different directions.', 'gr-qc-9712079-1-5-3': 'This solution is quite elegant because the exact solution can be found using straightforward algebra.', 'gr-qc-9712079-1-5-4': 'Since a sphere can be thought of as five bar detectors occupying the same space, this solution can be adapted for a spherical antenna [CITATION].', 'gr-qc-9712079-1-5-5': 'In addition, Lobo outlined a procedure that can be used if the correct theory of gravity is not general relativity but unknown [CITATION].', 'gr-qc-9712079-1-5-6': 'What all these proposals have in common is that they use basic symmetry properties of the matrices describing the detectors and their response to a gravitational wave.', 'gr-qc-9712079-1-5-7': 'This makes them intuitive and easy to visualize [CITATION].', 'gr-qc-9712079-1-6-0': 'The solution to the inverse problem in the presence of noise is more complicated.', 'gr-qc-9712079-1-6-1': 'Gursel and Tinto solved the problem for three noisy interferometers [CITATION] using a maximum likelihood method.', 'gr-qc-9712079-1-6-2': 'This solution required the measurement of the time delay of the signal between widely separated detectors to triangulate the direction of the source.', 'gr-qc-9712079-1-6-3': 'Zhou and Michelson showed that the inverse problem for a spherical antenna can be solved in the presence of noise, also using a maximum likelihood method [CITATION].', 'gr-qc-9712079-1-7-0': "What is disappointing about the maximum likelihood method is that the original simplicity of Dhurandhar and Tinto's noiseless solution is lost.", 'gr-qc-9712079-1-7-1': 'In addition, an exact solution for the spherical detector was not found, making it necessary to solve the problem numerically.', 'gr-qc-9712079-1-7-2': 'This solution can be computationally expensive, especially if the signal-to-noise ratio (SNR) is low.', 'gr-qc-9712079-1-7-3': 'For the three interferometer case, the numerical solution often can lead to an incorrect estimate at low SNR [CITATION].', 'gr-qc-9712079-1-7-4': 'This appears not to be a problem for a spherical antenna; a global maximum usually exists and is aligned with the correct direction.', 'gr-qc-9712079-1-7-5': 'This leads us to believe that the problem for a spherical antenna can be solved analytically.', 'gr-qc-9712079-1-8-0': 'On experiments with the room-temperature prototype spherical antenna (TIGA) at Louisiana State University [CITATION], we used a procedure to solve the inverse problem for impulsive excitations applied to the sphere surface.', 'gr-qc-9712079-1-8-1': "This solution was similar to Dhurandhar and Tinto's original solution for gravitational waves, but we used a perturbation argument (presented below) to take into account the finite SNR of the experiment.", 'gr-qc-9712079-1-8-2': 'The case of an impulsive excitation is more simple than a gravity wave because fewer parameters are involved.', 'gr-qc-9712079-1-8-3': 'However, we show below that this method also can be used to find an analytic solution to the inverse problem for a gravitational wave.', 'gr-qc-9712079-1-9-0': 'This article begins with a review of the interaction of a gravitational wave with an elastic sphere in Sec [REF].', 'gr-qc-9712079-1-9-1': 'The solution to the inverse problem in the absence of noise is discussed in Sec. [REF] and in the presence of noise in Sec. [REF].', 'gr-qc-9712079-1-9-2': 'Section [REF] describes the applicability of these techniques to any symmetric metric theory of gravity, and Sec. [REF] discusses their implementation for impulsive excitations.', 'gr-qc-9712079-1-9-3': 'Finally, the accuracy of the solution for finite SNR is discussed in Sec. [REF].', 'gr-qc-9712079-1-10-0': '# Interaction between a gravitational wave and an elastic sphere', 'gr-qc-9712079-1-11-0': 'The interaction between a gravitational wave and an elastic sphere has been thoroughly discussed in the literature [CITATION]; here, we follow the notation of Ref. [CITATION].', 'gr-qc-9712079-1-11-1': 'This section reviews the fundamental concepts and equations that are necessary for solving the inverse problem assuming general relativity is correct.', 'gr-qc-9712079-1-12-0': 'A gravitational wave is a traveling time-dependent deviation of the metric perturbation, denoted by [MATH].', 'gr-qc-9712079-1-12-1': 'We follow a common textbook development for the metric deviation of a gravitational wave, which finds that only the spatial components [MATH] are non-zero, and further can be taken to be transverse and traceless [CITATION].', 'gr-qc-9712079-1-12-2': 'This tensor is simplified if we initially write it in the "wave-frame," denoted by primed coordinates and indices.', 'gr-qc-9712079-1-12-3': 'This is a coordinate frame with origin at the center of mass of the detector and the [MATH] axis aligned to the propagation direction of the wave.', 'gr-qc-9712079-1-12-4': 'We restrict ourselves to detectors much smaller than the gravitational wavelength so only the time dependence of [MATH] will have significant physical effects.', 'gr-qc-9712079-1-12-5': 'A general form for the spatial components of the metric deviation in the wave-frame can be written as [EQUATION] where [MATH] and [MATH] are the wave amplitudes for the two allowed states of linear polarization, and are called the plus and cross amplitudes.', 'gr-qc-9712079-1-13-0': 'The detector is more easily described in the "lab-frame," denoted by unprimed coordinates and indices, with origin at the center of mass of the detector and the [MATH] axis aligned with the local vertical.', 'gr-qc-9712079-1-13-1': 'In this frame, the primary physical effect of a passing gravitational wave is to produce a time dependent "tidal" force density [MATH] on the material with mass density [MATH] at coordinate location [MATH].', 'gr-qc-9712079-1-13-2': 'This force is related to the metric perturbation by [EQUATION]', 'gr-qc-9712079-1-13-3': 'We can rewrite this force as the gradient of a time-dependent scalar potential [EQUATION]', 'gr-qc-9712079-1-13-4': 'It is natural to look for an alternate expression that separates the coordinate dependence into radial and angular parts.', 'gr-qc-9712079-1-13-5': 'Because the tensor [MATH] is traceless, the angular expansion can be done completely with the five second order real valued spherical harmonics [MATH], where the index [MATH].', 'gr-qc-9712079-1-13-6': 'We call the resulting time dependent expansion coefficients, denoted by [MATH], the "spherical amplitudes."', 'gr-qc-9712079-1-13-7': 'They are a complete and orthogonal representation of the cartesian metric deviation tensor [MATH].', 'gr-qc-9712079-1-13-8': 'They depend only on the two wave-frame amplitudes and the direction of propagation and are defined by [EQUATION]', 'gr-qc-9712079-1-13-9': 'To transform the metric perturbation to the lab-frame we perform the appropriate rotations using the Euler angles.', 'gr-qc-9712079-1-13-10': 'Two conventions have been commonly used in the literature on spherical detectors: the x-convention and the y-convention.', 'gr-qc-9712079-1-13-11': 'These two conventions differ only by which axis the second rotation is performed about.', 'gr-qc-9712079-1-13-12': 'In this paper we choose the y-convention of the Euler angles, but the reader should be aware that this is a potential place for confusion when comparing results by authors who have made a different choice.', 'gr-qc-9712079-1-14-0': 'The three Euler angles in the y-convention are shown in Fig. [REF].', 'gr-qc-9712079-1-14-1': 'We denote the rotation about the wave [MATH] axis by [MATH], the rotation about the new [MATH] axis ([MATH]) by [MATH] and the rotation about the final lab [MATH] axis by [MATH].', 'gr-qc-9712079-1-14-2': 'The rotation matrix for the y-convention is [EQUATION]', 'gr-qc-9712079-1-14-3': 'To transform the metric perturbation to the lab-frame we arbitrarily set the rotation [MATH] about the wave [MATH] axis equal to zero; inclusion of this rotation will only "mix" the two polarizations of the wave.', 'gr-qc-9712079-1-14-4': 'The spherical amplitudes can now be written in terms of the polarization amplitudes and the source direction [EQUATION]', 'gr-qc-9712079-1-14-5': 'The mechanics of a spherical antenna can be described by ordinary elastic theory.', 'gr-qc-9712079-1-14-6': 'One finds that the eigenfunctions of an uncoupled sphere can be written in terms of the spherical harmonics [EQUATION]', 'gr-qc-9712079-1-14-7': 'The radial eigenfunctions [MATH] and [MATH] determine the motion in the radial and tangential directions respectively and depend on the radius [MATH] and the material of the sphere [CITATION].', 'gr-qc-9712079-1-15-0': 'In general relativity, only the 5 quadrupole modes of vibration will strongly couple to the force density of a gravitational wave.', 'gr-qc-9712079-1-15-1': 'For an ideal sphere they are all degenerate, having the same eigenfrequency, and are distinguished only by their angular dependence.', 'gr-qc-9712079-1-15-2': 'The effective force [MATH] that a gravitational wave will exert on a fundamental quadrupole mode [MATH] of the sphere is given by the overlap integral between the eigenfunctions of the sphere and the gravitational tidal force [EQUATION]', 'gr-qc-9712079-1-15-3': 'Each spherical component of the gravitational field determines uniquely the effective force on the corresponding mode of the sphere, and they are all identical in magnitude.', 'gr-qc-9712079-1-15-4': 'We can interpret the effective force [MATH] in each mode as the product of: the physical mass of the sphere [MATH], an effective length [MATH] (a fraction of the sphere radius), and the gravitational acceleration [MATH].', 'gr-qc-9712079-1-15-5': 'The value of the coefficient [MATH] depends on the sphere material, but is typically [MATH] [CITATION].', 'gr-qc-9712079-1-16-0': 'By monitoring the quadrupole modes of the sphere, one has a direct measurement of the effective force on the sphere, and thus the spherical amplitudes of the gravitational wave.', 'gr-qc-9712079-1-16-1': 'The standard technique for doing so on resonant detectors is to position resonant transducers on the surface of the sphere that strongly couple to the quadrupole modes.', 'gr-qc-9712079-1-16-2': 'A number of proposals have been made for the type and positions of the transducers [CITATION].', 'gr-qc-9712079-1-16-3': 'What all of these proposals have in common is that the outputs of the transducers are combined into "mode channels" [MATH] that are constructed to have a one-to-one correspondence with the quadrupole modes of the sphere and thus the spherical amplitudes of the gravitational wave [CITATION], [EQUATION]', 'gr-qc-9712079-1-16-4': 'The mode channels can be collected to form a matrix [MATH] that in the absence of noise is equal to the cartesian strain tensor [MATH] in the lab frame [EQUATION]', 'gr-qc-9712079-1-17-0': '# Solution to the inverse problem in the absence of noise', 'gr-qc-9712079-1-18-0': 'Dhurandhar and Tinto solved the inverse problem for 5 bar antennas as well as 5 interferometers [CITATION].', 'gr-qc-9712079-1-18-1': 'Others have used their method to solve the problem for a spherical antenna [CITATION].', 'gr-qc-9712079-1-18-2': 'Their technique involves constructing a matrix that describes the response of the detector to a gravitational wave; the matrix [MATH] is related to what Dhurandhar and Tinto call the "detector response."', 'gr-qc-9712079-1-18-3': 'They found that the eigenvector of [MATH] with zero eigenvalue points in the propagation direction of the wave.', 'gr-qc-9712079-1-18-4': 'In the following we will also use this concept, but will derive the equations in the context of linear algebra as this will lead us directly into the solution for the noisy antenna.', 'gr-qc-9712079-1-18-5': 'For the remainder of this discussion we will drop the notation of time dependence [MATH] for brevity.', 'gr-qc-9712079-1-19-0': 'The strain tensor in the lab frame [MATH] is a symmetric traceless matrix.', 'gr-qc-9712079-1-19-1': 'Consequently, it can be orthogonally diagonalized and has an orthonormal set of three eigenvectors.', 'gr-qc-9712079-1-19-2': 'One can construct from the three eigenvectors a transformation matrix [MATH] that diagonalizes [MATH].', 'gr-qc-9712079-1-19-3': 'The matrix [MATH] is also orthogonal, thus it can be considered a rotation matrix (it may also include a reflection).', 'gr-qc-9712079-1-19-4': 'The physical interpretation of this transformation is to rotate the lab frame such that the [MATH] axis points in the direction of the source.', 'gr-qc-9712079-1-19-5': 'The matrix [MATH] (the eigenvectors) will tell us the angles of rotation and thus the direction of the wave.', 'gr-qc-9712079-1-20-0': 'In the wave frame, [MATH] is not normally diagonal but it can be diagonalized by rotating Eq. ([REF]) about the propagation axes using the Euler angle [MATH].', 'gr-qc-9712079-1-20-1': '[MATH] may be a constant or a function of time depending upon the situation.', 'gr-qc-9712079-1-20-2': 'This rotation only changes the polarization components of the tensor and not the direction relative to the lab frame.', 'gr-qc-9712079-1-21-0': 'To calculate the rotation matrix [MATH] we need to solve the general eigenvalue equation for the strain tensor [EQUATION]', 'gr-qc-9712079-1-21-1': 'Since [MATH] and [MATH] are equal in the absence of noise we are free to substitute [MATH] in Eq. ([REF]) for [MATH].', 'gr-qc-9712079-1-21-2': 'By inspection of Eq. ([REF]) we see that in general relativity the eigenvector of [MATH] with [MATH] points in the propagation direction of the wave.', 'gr-qc-9712079-1-21-3': 'We, therefore, need only solve Eq. ([REF]) for the corresponding eigenvector [EQUATION]', 'gr-qc-9712079-1-21-4': 'The direction can be calculated from this eigenvector by recognizing that it corresponds to the last column vector of [MATH] in Eq. ([REF]).', 'gr-qc-9712079-1-21-5': 'Dividing the elements of this column we find [EQUATION]', 'gr-qc-9712079-1-21-6': 'The unusual minus sign in Eq. ([REF]) comes from the use of the y-convention of the Euler angles.', 'gr-qc-9712079-1-21-7': 'Substituting in a particular choice of matrix elements from Eq. ([REF]) we find [EQUATION]', 'gr-qc-9712079-1-21-8': 'The [MATH] in Eq. ([REF]) illustrates the unavoidable fact that a single sphere cannot distinguish between antipodal sources.', 'gr-qc-9712079-1-21-9': 'This ambiguity is a characteristic of all gravity wave detectors, but can be removed by measuring the time delay of the signal between two widely separated antennas.', 'gr-qc-9712079-1-21-10': 'Replacing the matrix elements of [MATH] with those from Eq. ([REF]) we find [EQUATION]', 'gr-qc-9712079-1-21-11': 'This solution is valid only for a noiseless antenna; it will fail otherwise because we can no longer replace [MATH] with [MATH] and their eigenvectors and eigenvalues will no longer be equal.', 'gr-qc-9712079-1-22-0': 'Once the direction is calculated we can determine the two polarization amplitudes by taking a linear combination of Eqs. ([REF]).', 'gr-qc-9712079-1-22-1': 'These equations are actually overdetermined so several solutions exist (we have 5 equations but only 4 unknowns).', 'gr-qc-9712079-1-22-2': 'In the absence of noise any particular solution to them is valid, but in anticipation of the noisy case we will take a systematic approach to the solution.', 'gr-qc-9712079-1-23-0': 'We need only the angles [MATH] and [MATH] to rotate [MATH] to [MATH], so at this point we again set [MATH].', 'gr-qc-9712079-1-23-1': 'The amplitudes are found by equating them to the corresponding matrix elements of [MATH] in Eq. ([REF]): [MATH] and [MATH].', 'gr-qc-9712079-1-23-2': 'Because [MATH] and [MATH] are symmetric, [MATH] and [MATH] will always be identical even when noise is introduced.', 'gr-qc-9712079-1-23-3': 'However, no such restriction is placed on [MATH] and [MATH].', 'gr-qc-9712079-1-23-4': 'We will use the average [MATH] to calculate [MATH] for reasons that will become clear later.', 'gr-qc-9712079-1-24-0': 'Again, we may substitute [MATH] for [MATH].', 'gr-qc-9712079-1-24-1': 'Multiplying [MATH] for [MATH] and selecting the proper elements we find [EQUATION]', 'gr-qc-9712079-1-24-2': 'These equations can also be derived by taking a linear combination of Eqs. ([REF]).', 'gr-qc-9712079-1-24-3': 'Notice the symmetry of this particular solution: the coefficients of each component [MATH] is the same as the corresponding coefficients of [MATH] or [MATH] in Eqs. ([REF]) for [MATH].', 'gr-qc-9712079-1-25-0': '# Solution to the inverse problem in the presence of noise', 'gr-qc-9712079-1-26-0': 'Noise in the mode channels [MATH] will change the eigenvalues and eigenvectors of [MATH] such that they are no longer equal to those of [MATH].', 'gr-qc-9712079-1-26-1': 'To gain some insight into this situation, let us consider the noise as a perturbation [MATH] to the matrix [MATH] [EQUATION]', 'gr-qc-9712079-1-26-2': 'The matrix [MATH] is constructed from the noise in each mode channel [MATH], thus it has the same form as Eq. ([REF]).', 'gr-qc-9712079-1-26-3': 'The matrix [MATH] is therefore still symmetric and traceless and has the eigenvalue equation [EQUATION]', 'gr-qc-9712079-1-26-4': 'The eigenvectors of [MATH] can be expanded in terms of the eigenvectors of [MATH] [EQUATION] where the matrix [MATH] is close to the identity matrix if the perturbation is small.', 'gr-qc-9712079-1-26-5': 'However, since we do not know the values of the matrix [MATH] we cannot calculate any corrections to the matrix elements [MATH], thus the best approximation to [MATH] we can find is [MATH].', 'gr-qc-9712079-1-27-0': 'Also from perturbation theory and Eqs. ([REF]) and ([REF]) we see that the eigenvalue corresponding to the estimated direction of the wave is [MATH] if the perturbation is small.', 'gr-qc-9712079-1-27-1': 'Its magnitude will increase as the SNR decreases, but it should remain smaller than the other two eigenvalues of [MATH] for SNR [MATH].', 'gr-qc-9712079-1-27-2': 'Consequently the eigenvector corresponding to the estimated direction of the wave can be selected from the three eigenvectors of [MATH] by choosing the one whose eigenvalue is "closest" to zero.', 'gr-qc-9712079-1-27-3': 'Once [MATH] is found, it can be used to estimate the direction of the source using Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-1-28-0': 'The perturbation approach gives us a conceptual feel for the solution, but a more rigorous proof seems necessary.', 'gr-qc-9712079-1-28-1': 'Let use look at the problem in terms of an abstract vector space.', 'gr-qc-9712079-1-28-2': 'The column vectors of [MATH] can be considered a basis set for a three dimensional inner product space, say [MATH].', 'gr-qc-9712079-1-28-3': 'Since [MATH] has one column containing all zeros it can only describe a subspace of [MATH], say [MATH].', 'gr-qc-9712079-1-28-4': 'In this context, our problem is to know the best approximation from [MATH] to a vector in [MATH].', 'gr-qc-9712079-1-29-0': 'The answer lies in the Best Approximation Theorem [CITATION] which states that the best approximation to a vector [MATH] in [MATH] is the projection of [MATH] onto [MATH], [MATH].', 'gr-qc-9712079-1-29-1': 'In other words [EQUATION] for every vector [MATH] in [MATH] different from [MATH].', 'gr-qc-9712079-1-29-2': 'Here [MATH] are the column vectors of [MATH] and [MATH] are the non-zero column vectors of [MATH].', 'gr-qc-9712079-1-30-0': 'A rotation of [MATH] is also a basis for [MATH], so let us use a new basis defined by [EQUATION]', 'gr-qc-9712079-1-30-1': 'If [MATH] is diagonal, two of its column vectors point in the same direction as the non-zero column vectors of the diagonal form of [MATH].', 'gr-qc-9712079-1-30-2': 'Therefore, the projection onto [MATH] is just these two components of [MATH].', 'gr-qc-9712079-1-30-3': 'This can be stated analytically using Eq. ([REF]) to say [EQUATION] is a minimum when [MATH] diagonalizes [MATH].', 'gr-qc-9712079-1-30-4': 'Again, this occurs when [MATH] is constructed from the eigenvectors of [MATH] as described above.', 'gr-qc-9712079-1-30-5': 'This last statement is equivalent to saying [MATH] is the best approximation to [MATH] as found from perturbation theory.', 'gr-qc-9712079-1-31-0': 'It is worth noting that the above solution can also be applied to the maximum likelihood solution proposed by Zhou and Michelson [CITATION].', 'gr-qc-9712079-1-31-1': 'They did not find an exact solution to the inverse problem but instead described how one can numerically find the solution by maximizing the likelihood function (a probability density function for having a set of output data for a given set of parameter values).', 'gr-qc-9712079-1-31-2': 'Their likelihood function [MATH] can be written in our matrix notation [EQUATION] where each mode channel is assumed to have the same variance [MATH] and zero mean.', 'gr-qc-9712079-1-31-3': 'We can transform transform Eq. ([REF]) into the wave frame using a rotation matrix [MATH] [EQUATION]', 'gr-qc-9712079-1-31-4': 'We see immediately from Eqs. ([REF]) and ([REF]) that the maximum value of [MATH] occurs when [MATH] diagonalizes [MATH].', 'gr-qc-9712079-1-32-0': 'The procedure for finding the polarization amplitudes is not as rigid as for finding the direction of the wave.', 'gr-qc-9712079-1-32-1': 'Once we have estimated the wave direction we are free to choose any proper linear combination of Eqs. ([REF]) to solve for the polarization amplitudes.', 'gr-qc-9712079-1-32-2': 'The best choice will depend upon the character of the noise in the detector.', 'gr-qc-9712079-1-32-3': 'If the noise is equal in all the mode channels the best choice will be the one that equally weights the five mode channels.', 'gr-qc-9712079-1-32-4': 'As shown in Ref. [CITATION], this turns out to be the solution we presented above for the noiseless case, Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-1-33-0': 'If the noise is not equal in the five mode channels, it may be advantageous to take a different linear combination of Eqs. ([REF]) that minimizes the contribution of the more noisy mode channels.', 'gr-qc-9712079-1-33-1': 'Only two mode channels are necessary to estimate the polarization amplitudes once the direction of the wave is known, so there are many possible combinations to choose from.', 'gr-qc-9712079-1-34-0': 'Summarizing the general procedure for solving the inverse problem: we construct the matrix [MATH] from the mode channels [MATH] and compute its eigenvalues [MATH] and eigenvectors [MATH].', 'gr-qc-9712079-1-34-1': 'We choose the eigenvector with eigenvalue closest to zero and calculate the direction of the wave using Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-1-34-2': 'The polarization amplitudes can computed using Eqs. ([REF]) and ([REF]) or some other linear combination of Eqs. ([REF]) that is appropriate for the character of the noise in the detector.', 'gr-qc-9712079-1-35-0': '# Application to alternative theories of gravity', 'gr-qc-9712079-1-36-0': 'Experiments in the solar system and pulsar-timing tests have ruled out many competing theories of gravity.', 'gr-qc-9712079-1-36-1': 'However, general relativity is not the only theory of gravity that passes these weak field tests [CITATION].', 'gr-qc-9712079-1-36-2': 'One measurement that can potentially rule out certain gravitational theories is the properties of gravitational waves [CITATION], such as the speed of propagation and allowable polarization states.', 'gr-qc-9712079-1-37-0': 'It was shown above how a single sphere can measure the quadrupole components of the strain tensor, but a scalar wave can excite both the monopole mode and the quadrupole modes of a sphere [CITATION].', 'gr-qc-9712079-1-37-1': 'By monitoring both types of modes, a single spherical detector can measure all the tensor components of a gravitational wave.', 'gr-qc-9712079-1-37-2': 'This makes it possible for a single spherical detector to determine all of the six polarization states predicted by the most general symmetric metric theory of gravity [CITATION].', 'gr-qc-9712079-1-38-0': 'We can rewrite Eq. ([REF]) in terms of the electric components of the Riemann tensor [CITATION] [MATH], [EQUATION] where we now include both the [MATH] quadrupole modes and the [MATH] monopole mode.', 'gr-qc-9712079-1-38-1': 'The monopole mode of an elastic sphere is actually at a higher frequency than the quadrupole modes.', 'gr-qc-9712079-1-38-2': 'If the source is not wide-band enough for detection in both of these modes, a second sphere with the monopole mode tuned to the quadrupole modes of the first will be needed to measure this component.', 'gr-qc-9712079-1-38-3': 'If the first sphere is at relatively low frequency, one might consider making the second sphere hollow to keep it of a practical size [CITATION].', 'gr-qc-9712079-1-38-4': 'An alternative to a second sphere is to monitor the [MATH] quadrupole modes and the monopole mode of a single sphere [CITATION].', 'gr-qc-9712079-1-38-5': 'These modes are not far in frequency from each other and also have relatively large cross-sections [CITATION].', 'gr-qc-9712079-1-39-0': '[MATH] is a symmetric tensor thus it has only six independent components.', 'gr-qc-9712079-1-39-1': 'It can be written in terms of the complex Newman-Penrose parameters [CITATION] which allow the identification of the spin content of the metric theory responsible for the generation of the wave [EQUATION]', 'gr-qc-9712079-1-39-2': 'By monitoring the five quadrupole modes and the monopole mode of a sphere, one can measure all the components of [MATH], and if the direction of the wave is known one can calculate the Newman-Penrose parameters [CITATION].', 'gr-qc-9712079-1-39-3': 'Knowing these parameters can set strong limits on the validity of any metric theory of gravity that does not predict the measured polarization states [CITATION].', 'gr-qc-9712079-1-40-0': 'What we are interested in here is the converse problem: suppose we know which is the correct theory of gravity and want to calculate the direction of the wave.', 'gr-qc-9712079-1-40-1': 'To examine if the same arguments for general relativity can be used for other metric theories we need to look at the symmetries of [MATH].', 'gr-qc-9712079-1-40-2': 'To do this we divide the theories into categories using the E(2) classification scheme [CITATION] shown in Tab. [REF].', 'gr-qc-9712079-1-41-0': 'The tensor [MATH] is symmetric for all of these classes, thus it is orthogonally diagonalizable.', 'gr-qc-9712079-1-41-1': 'However, we notice that classes [MATH] and [MATH] have more degrees of freedom (direction plus polarization states) than we are capable of measuring with a single spherical detector.', 'gr-qc-9712079-1-41-2': 'These two classes are often referred to as "observer-dependent" because different observers will disagree upon which polarization states are present.', 'gr-qc-9712079-1-41-3': 'As a consequence, the polarization amplitudes for a particular observer must be known before the direction of the wave can be estimated.', 'gr-qc-9712079-1-42-0': 'For the "observer-independent" classes [MATH], [MATH], [MATH], and [MATH], the situation is more straightforward.', 'gr-qc-9712079-1-42-1': '[MATH] is obviously uninteresting as it does not predict any gravitational waves (this class along with [MATH] have essentially been ruled out by previous experiments [CITATION]).', 'gr-qc-9712079-1-42-2': 'We notice that [MATH] for the observer-independent classes can be diagonalized by a rotation [MATH] about the propagation axis, therefore, we can use the same arguments presented above for general relativity to solve for the wave direction.', 'gr-qc-9712079-1-43-0': 'The most general observer-independent class is [MATH], which has [EQUATION]', 'gr-qc-9712079-1-43-1': 'Looking at the form of [MATH] we see that the same procedure for calculating the direction of the wave in general relativity holds for all the observer-independent classes: the eigenvector with eigenvalue closest to zero points at the source.', 'gr-qc-9712079-1-43-2': 'The one exception to this statement is the case where the driving forces remain in a fixed line, for example [MATH].', 'gr-qc-9712079-1-43-3': 'In this situation the direction of the wave can only be determined within the plane defined by the two eigenvectors with eigenvalues closest to zero.', 'gr-qc-9712079-1-44-0': '# Application to impulsive excitations', 'gr-qc-9712079-1-45-0': 'Impulsive excitations are often used on resonant-mass detectors to calibrate the antenna [CITATION].', 'gr-qc-9712079-1-45-1': 'The excitations are usually administered by either a short electrical burst applied to a calibrator attached to the surface or a hammer blow.', 'gr-qc-9712079-1-45-2': 'Impulsive excitations were also used to test the analysis techniques used for experiments with the prototype spherical antenna at Louisiana State University [CITATION].', 'gr-qc-9712079-1-46-0': 'A radial impulse excitation can be easily described if we choose the [MATH] axis to be along the direction of the impulse.', 'gr-qc-9712079-1-46-1': 'By examining the quadrupole eigenfunctions of the sphere in this frame we notice that out of these modes only the [MATH] mode will be excited (other sphere modes will also be excited but their response can be removed by narrow-band filtering).', 'gr-qc-9712079-1-46-2': 'All of the other quadrupole modes have a vanishing radial component of their eigenfunctions at this location which makes their "overlap" integral with the impulse vanish.', 'gr-qc-9712079-1-46-3': 'In this frame the mode channel matrix is [EQUATION]', 'gr-qc-9712079-1-46-4': 'In the lab frame [MATH] is still given by Eq. ([REF]).', 'gr-qc-9712079-1-47-0': 'Again, the direction can be found by calculating the eigenvalues and eigenvectors of the lab frame [MATH].', 'gr-qc-9712079-1-47-1': 'In the absence of noise, the eigenvector corresponding to the direction has a non-zero eigenvalue that is opposite in sign and twice as large as the two other eigenvalues.', 'gr-qc-9712079-1-47-2': 'Using the same arguments as for general relativity we conclude that in the noisy case the correct eigenvector can be selected by choosing the one with the largest eigenvalue.', 'gr-qc-9712079-1-48-0': 'As stated above, this procedure was used on the prototype TIGA experiments.', 'gr-qc-9712079-1-48-1': 'From the mode channels, we were able to calculate the location of several impulses applied to the surface of the antenna within the accuracy of the experiment [CITATION].', 'gr-qc-9712079-1-49-0': '# Accuracy of the solution', 'gr-qc-9712079-1-50-0': 'The techniques discussed thus far will give an estimate of the wave direction and polarization but noise in the system will limit the accuracy of the estimate.', 'gr-qc-9712079-1-50-1': 'In addition, the estimation of some parameters are dependent upon others which can lead to large variances even for high SNR.', 'gr-qc-9712079-1-50-2': 'To fully discuss these effects we would need to introduce a model of the detector that includes noise from a variety of sources such as Brownian motion and noise from the motion sensors.', 'gr-qc-9712079-1-50-3': 'Such a model is beyond the scope of this article, but it is worthwhile to have a feel for the magnitude of the errors with a simplified model.', 'gr-qc-9712079-1-50-4': 'For a discussion of noise sources in a spherical antenna see Refs. [CITATION].', 'gr-qc-9712079-1-51-0': 'For this discussion we assume a wide-band white noise [MATH] added to the mode channels [EQUATION]', 'gr-qc-9712079-1-51-1': 'We assume the noise in each channel is uncorrelated but has the same variance [MATH] and zero mean.', 'gr-qc-9712079-1-51-2': 'The SNR is thus given by [EQUATION]', 'gr-qc-9712079-1-51-3': 'We assume here a burst source of gravitational waves that is linearly polarized and that the arrival time of the event is known.', 'gr-qc-9712079-1-51-4': 'We can easily obtain analytic expressions for the variances of the wave parameters by calculating the covariance matrix using the Cramer-Rao inequality [CITATION].', 'gr-qc-9712079-1-51-5': 'This will give a lower bound for the variances that is relatively accurate for high SNR.', 'gr-qc-9712079-1-51-6': 'We find the non-zero variances in this limit to be [EQUATION]', 'gr-qc-9712079-1-51-7': 'The reason some of the errors go to infinity as the angle [MATH] goes to zero can be easily seen by examining the rotation matrix of Eq. ([REF]).', 'gr-qc-9712079-1-51-8': 'We see that as [MATH] goes to zero we can no longer distinguish between the first and last rotations [MATH] and [MATH].', 'gr-qc-9712079-1-51-9': 'However, these direction dependencies of the variances are deceptive because we are not normally interested in the individual parameters but in a combination of them.', 'gr-qc-9712079-1-51-10': 'For example, if we want to calculate the amplitude of the gravitational wave [MATH], we find from the above equations [EQUATION] which is just our definition of the SNR in Eq. ([REF]).', 'gr-qc-9712079-1-51-11': 'For the direction of the source, a more appropriate measure of error is the solid angle estimation error [MATH] as defined by Gursel and Tinto [CITATION] [EQUATION]', 'gr-qc-9712079-1-51-12': 'From Eqs. ([REF]) and ([REF]) we find in the limit of high SNR [EQUATION]', 'gr-qc-9712079-1-51-13': 'From these expressions we see that the error in the estimation of the direction and amplitude of the wave is actually independent of the true direction and polarization of the wave.', 'gr-qc-9712079-1-51-14': 'This is the answer we expected since a sphere is equally sensitive to waves of all directions and polarizations.', 'gr-qc-9712079-1-52-0': 'One might consider using a different coordinate system to try to avoid the directions with very poor estimates of the two polarization amplitudes.', 'gr-qc-9712079-1-52-1': 'For example, use the xyz-convention of the Euler angles where the first and the last rotations are not the same.', 'gr-qc-9712079-1-52-2': 'This will actually not solve our problem, but instead change which directions in the sky lead to the poor estimates.', 'gr-qc-9712079-1-52-3': 'In addition, if we transform back from this coordinate system to the y-convention, we just reintroduce these errors and thus have gained nothing.', 'gr-qc-9712079-1-52-4': 'The best solution is to calculate combinations of the parameters that have variances independent of the wave direction and polarization.', 'gr-qc-9712079-1-53-0': 'Unfortunately, there does not appear to be a standard description of the wave polarization that has a direction independent variance.', 'gr-qc-9712079-1-53-1': 'For example, if we calculate the uncertainty in the angle between the two amplitudes [MATH], we find for high SNR [EQUATION]', 'gr-qc-9712079-1-53-2': 'This variance goes to infinity as [MATH] goes to zero.', 'gr-qc-9712079-1-53-3': 'The variance of the polarization ellipse also has a direction dependence and the additional problem that it is a biased estimator [CITATION].', 'gr-qc-9712079-1-53-4': 'This is also the case for the equivalent of Stokes parameters used in electromagnetism.', 'gr-qc-9712079-1-54-0': 'In the case where the direction of the source is known this problem does not exist.', 'gr-qc-9712079-1-54-1': 'The variance of each polarization amplitude is independent of the wave direction.', 'gr-qc-9712079-1-54-2': 'The variance can be calculated directly from whichever linear combination of Eqs. ([REF]) was chosen to minimize the effect of noisy mode channels.', 'gr-qc-9712079-1-54-3': 'For the case where noise in each mode channel is uncorrelated, has the same variance [MATH], and zero mean we find from Eqs. ([REF]) and ([REF]) [MATH].', 'gr-qc-9712079-1-55-0': 'To determine the values of the variance for low SNR were analytic expressions are difficult to obtain we turn to a Monte Carlo simulation.', 'gr-qc-9712079-1-55-1': 'This will also enable us to include the situation when the wrong eigenvector is chosen.', 'gr-qc-9712079-1-55-2': 'For a wave of a given direction and polarization we calculate the spherical amplitudes [MATH] and add a random number [MATH] (variance [MATH] and zero mean) to obtain the mode channels.', 'gr-qc-9712079-1-55-3': 'The direction and polarization is then calculated using the above techniques.', 'gr-qc-9712079-1-55-4': 'This procedure is repeated a large number (200) of times.', 'gr-qc-9712079-1-56-0': 'The estimation error [MATH] for a range of SNR is shown in Fig. [REF].', 'gr-qc-9712079-1-56-1': 'These results agree very well with the simulations of Zhou and Michelson [CITATION].', 'gr-qc-9712079-1-56-2': 'Also plotted is the estimation error from Eq. ([REF]), which agrees with the simulation when the SNR is high.', 'gr-qc-9712079-1-56-3': 'Varying the wave direction and polarization we obtain similar results, indicating that [MATH] is independent of the wave direction and polarization even for low SNR.', 'gr-qc-9712079-1-57-0': 'There is no need to perform the simulation for the wave amplitude [MATH].', 'gr-qc-9712079-1-57-1': 'This variance was used to define the SNR so Eq. ([REF]) is valid for all SNR.', 'gr-qc-9712079-1-58-0': 'As stated above, the variances on the polarization amplitudes are dependent on the wave direction and polarization.', 'gr-qc-9712079-1-58-1': 'Shown in Fig. [REF] is the variance on the polarization angle [MATH] for a range of SNR and several values of [MATH].', 'gr-qc-9712079-1-58-2': 'As expected, the variance increases for low values of [MATH].', 'gr-qc-9712079-1-58-3': 'Also plotted is the variance from Eq. ([REF]).', 'gr-qc-9712079-1-58-4': 'In this case the bound does not agree well with the simulation for all [MATH] and SNR [MATH].', 'gr-qc-9712079-1-59-0': '# Summary', 'gr-qc-9712079-1-60-0': 'The eigenvector solution to the inverse problem is very convenient in that the problem is reduced to solving a trivial eigenvalue problem.', 'gr-qc-9712079-1-60-1': 'No assumptions about the character of the noise were made for this solution making it applicable to spherical detectors with various types of noise sources.', 'gr-qc-9712079-1-60-2': 'The solution is computationally simple, making this technique very efficient for use in an automated data analysis system.', 'gr-qc-9712079-1-60-3': 'This feature may be important if one considers using a large number of candidate gravitational wave events in a coincidence exchange between several detectors where the source direction is used as a criterion to veto excess coincidences.', 'gr-qc-9712079-1-61-0': 'The symmetry arguments used to develop the method makes it possible to discuss its use on any symmetric metric theory of gravity as well as other types of excitations such as an impulse.', 'gr-qc-9712079-1-61-1': 'For observer-independent theories of gravity it is possible to determine the source direction using this technique, however, observer-dependent theories require prior knowledge of the polarization states of the wave before any estimate of the direction can be made.', 'gr-qc-9712079-1-62-0': 'In addition to being equally sensitive to all directions and polarizations, a sphere can also estimate the direction and amplitude of a gravitational wave with direction independent uncertainty.', 'gr-qc-9712079-1-62-1': 'However, the uncertainty on the estimate of the individual polarization amplitudes will depend upon the wave direction in the particular coordinate system chosen.', 'gr-qc-9712079-1-62-2': 'We emphasize that these uncertainties are just a result of parameter dependencies in the estimation; the sensitivity of the detector is still found to be omnidirectional.', 'gr-qc-9712079-1-62-3': 'This dependency on the source direction is not unique to a sphere, a network of bars or interferometers will also suffer from this problem [CITATION].', 'gr-qc-9712079-1-62-4': 'If the direction of the source is known to high accuracy the uncertainties on the polarization amplitudes are no longer dependent on the direction of the source.', 'gr-qc-9712079-1-63-0': 'Finally, the principles of the solution were successfully tested on experiments with the LSU prototype spherical antenna.', 'gr-qc-9712079-1-63-1': 'This practical confirmation of its validity gives us the confidence that it can be implemented on a real spherical antenna searching for gravitational waves.', 'gr-qc-9712079-1-64-0': 'I thank M. Bassan, M. Bianchi, E. Coccia, J. A. Lobo, and E. Mauceli for many useful discussions on this work.'}

{'gr-qc-9712079-2-0-0': 'A spherical gravitational wave antenna is distinct from other types of gravitational wave antennas in that only a single detector is necessary to determine the direction and polarization of a gravitational wave.', 'gr-qc-9712079-2-0-1': 'Zhou and Michelson showed that the inverse problem can be solved using the maximum likelihood method if the detector outputs are independent and have normally distributed noise with the same variance.', 'gr-qc-9712079-2-0-2': 'This paper presents an analytic solution using only linear algebra that is found to produce identical results as the maximum likelihood method but with less computational burden.', 'gr-qc-9712079-2-0-3': 'Applications of this solution to gravitational waves in alternative symmetric metric theories of gravity and impulsive excitations also are discussed.', 'gr-qc-9712079-2-1-0': '# Introduction', 'gr-qc-9712079-2-2-0': 'Several resonant-mass gravitational wave antennas are now in continuous operation with strain sensitivities of the order [MATH] [CITATION].', 'gr-qc-9712079-2-2-1': 'With further improvements to these detectors and the addition of several large laser interferometers now under construction [CITATION], the prospects for gravitational wave astronomy are quite good.', 'gr-qc-9712079-2-2-2': 'The underlying non-gravitational physics associated with these detectors is reasonably understood and further improvements can be based on solid technological guidelines.', 'gr-qc-9712079-2-3-0': 'Many believe the next generation of resonant-mass antennas will be of spherical shape [CITATION].', 'gr-qc-9712079-2-3-1': 'Confirmed detection of gravitational waves will require a coincidence between several detectors, thus the unique features of a sphere may play an essential role in a network of gravitational wave antennas.', 'gr-qc-9712079-2-3-2': 'Two important features of a sphere are its equal sensitivity to gravitational waves from all directions and polarizations and its ability to determine the directional information and tensorial character of a gravitational wave [CITATION].', 'gr-qc-9712079-2-4-0': 'To take full advantage of these capabilities, one needs to be able to interpret the data such a detector will produce.', 'gr-qc-9712079-2-4-1': 'Recently, much work has been done to understand the output of a spherical antenna equipped with resonant transducers [CITATION].', 'gr-qc-9712079-2-4-2': 'All of these proposals operate on the principle that the response of the transducers can be transformed into a quantity that has a one-to-one correspondence with the tensorial components of a gravitational wave.', 'gr-qc-9712079-2-4-3': 'With the measurement of these components, it is possible to solve the inverse problem to obtain the direction and polarization amplitudes of a gravitational wave.', 'gr-qc-9712079-2-5-0': "The solution to the inverse problem for a noiseless spherical antenna first was outlined in the mid 1970's by Wagoner and Paik [CITATION].", 'gr-qc-9712079-2-5-1': 'More recently, the solution for a network of five noiseless bar antennas or interferometers was solved by Dhurandhar and Tinto [CITATION].', 'gr-qc-9712079-2-5-2': 'This method assumed that the detectors were co-located but oriented in different directions.', 'gr-qc-9712079-2-5-3': 'This solution is quite elegant because the exact solution can be found using straightforward algebra.', 'gr-qc-9712079-2-5-4': 'Since a sphere can be thought of as five bar detectors occupying the same space, this solution can be adapted for a spherical antenna [CITATION].', 'gr-qc-9712079-2-5-5': 'In addition, Lobo outlined a procedure that can be used if the correct theory of gravity is not general relativity but unknown [CITATION].', 'gr-qc-9712079-2-5-6': 'What all these proposals have in common is that they use basic symmetry properties of the matrices describing the detectors and their response to a gravitational wave.', 'gr-qc-9712079-2-5-7': 'This makes them intuitive and easy to visualize [CITATION].', 'gr-qc-9712079-2-6-0': 'The solution to the inverse problem in the presence of noise is more complicated.', 'gr-qc-9712079-2-6-1': 'Gursel and Tinto solved the problem for three noisy interferometers using a maximum likelihood method [CITATION].', 'gr-qc-9712079-2-6-2': 'This solution required the measurement of the time delay of the signal between widely separated detectors to triangulate the direction of the source.', 'gr-qc-9712079-2-6-3': 'Zhou and Michelson showed that the inverse problem for a spherical antenna can be solved in the presence of noise, also using a maximum likelihood method [CITATION].', 'gr-qc-9712079-2-7-0': "What is disappointing about the maximum likelihood method is that the original simplicity of Dhurandhar and Tinto's noiseless solution is lost.", 'gr-qc-9712079-2-7-1': 'In addition, an exact solution for the spherical detector was not found, making it necessary to solve the problem numerically.', 'gr-qc-9712079-2-7-2': 'This solution can be computationally expensive, especially if the signal-to-noise ratio (SNR) is low.', 'gr-qc-9712079-2-7-3': 'For the three interferometer case, the numerical solution often can lead to an incorrect estimate at low SNR [CITATION].', 'gr-qc-9712079-2-7-4': 'This appears not to be a problem for a spherical antenna; a global maximum usually exists and is aligned with the correct direction.', 'gr-qc-9712079-2-7-5': 'This leads us to believe that the problem for a spherical antenna can be solved analytically.', 'gr-qc-9712079-2-8-0': 'On experiments with the room-temperature prototype spherical antenna (TIGA) at Louisiana State University, we used a procedure to solve the inverse problem for impulsive excitations applied to the sphere surface [CITATION].', 'gr-qc-9712079-2-8-1': "This solution was similar to Dhurandhar and Tinto's original solution for gravitational waves, but we used a perturbation argument (presented below) to take into account the finite SNR of the experiment.", 'gr-qc-9712079-2-8-2': 'The case of an impulsive excitation is more simple than a gravity wave because fewer parameters are involved, however, we show below that this method also can be used to find an analytic solution for gravitational waves.', 'gr-qc-9712079-2-9-0': "We begin by reviewing the response of a spherical antenna to gravitational waves in general relativity and show how Dhurandhar and Tinto's original method can be applied to solve for the wave direction and polarization amplitudes.", 'gr-qc-9712079-2-9-1': 'We then generalize the arguments to any symmetric metric theory of gravity as well as to impulsive excitations.', 'gr-qc-9712079-2-9-2': 'In Sec. [REF] we show how this technique can be extended to a noisy antenna with independent and equally sensitive detector outputs.', 'gr-qc-9712079-2-9-3': 'This solution is found to be equivalent to the maximum likelihood method under the same noise requirements.', 'gr-qc-9712079-2-9-4': 'The general approach taken allow this solution to be easily adapted to other types of excitations with similar symmetry properties.', 'gr-qc-9712079-2-9-5': 'We conclude the paper with a discussion of the limitations of the solution and possible extensions of this method.', 'gr-qc-9712079-2-10-0': '# Detector response of an elastic sphere', 'gr-qc-9712079-2-11-0': 'Dhurandhar and Tinto solved the inverse problem for 5 bar antennas as well as 5 interferometers [CITATION].', 'gr-qc-9712079-2-11-1': 'Others have used their method to solve the problem for a spherical antenna [CITATION].', 'gr-qc-9712079-2-11-2': 'Their technique involves constructing a matrix, say [MATH], that describes the response of the detector to a gravitational wave.', 'gr-qc-9712079-2-11-3': 'They found that in general relativity the eigenvector of [MATH] with zero eigenvalue points in the propagation direction of the wave.', 'gr-qc-9712079-2-11-4': 'In the following we will also use this concept, but will derive the equations in the context of linear algebra as this will lead us directly into the solution for the noisy antenna.', 'gr-qc-9712079-2-11-5': 'For a more complete discussion of the response of an elastic sphere to gravitational waves the reader is referred to Refs. [CITATION].', 'gr-qc-9712079-2-12-0': '## Detector response in general relativity', 'gr-qc-9712079-2-13-0': 'A gravitational wave is a traveling time-dependent deviation of the metric perturbation, denoted by [MATH].', 'gr-qc-9712079-2-13-1': 'We follow a common textbook development for the metric deviation of a gravitational wave, which finds that only the spatial components [MATH] are non-zero, and further can be taken to be transverse and traceless [CITATION].', 'gr-qc-9712079-2-13-2': 'This tensor is simplified if we initially write it in the "wave-frame," denoted by primed coordinates and indices.', 'gr-qc-9712079-2-13-3': 'This is a coordinate frame with origin at the center of mass of the detector and the [MATH] axis aligned to the propagation direction of the wave.', 'gr-qc-9712079-2-13-4': 'We restrict ourselves to detectors much smaller than the gravitational wavelength so only the time dependence of [MATH] will have significant physical effects.', 'gr-qc-9712079-2-13-5': 'A general form for the spatial components of the metric deviation in the wave-frame can be written as [EQUATION] where [MATH] and [MATH] are the wave amplitudes for the two allowed states of linear polarization and are called the plus and cross amplitudes.', 'gr-qc-9712079-2-14-0': 'The detector is more easily described in the "lab-frame," denoted by unprimed coordinates and indices, with origin at the center of mass of the detector and the [MATH] axis aligned with the local vertical.', 'gr-qc-9712079-2-14-1': 'In this frame, the primary physical effect of a passing gravitational wave is to produce a time dependent "tidal" force density [MATH] on the material with mass density [MATH] at coordinate location [MATH].', 'gr-qc-9712079-2-14-2': 'This force is related to the metric perturbation by [EQUATION]', 'gr-qc-9712079-2-14-3': 'It is natural to look for an alternate expression that separates the coordinate dependence into radial and angular parts.', 'gr-qc-9712079-2-14-4': 'Because the tensor [MATH] is traceless, the angular expansion can be done completely with the five second order real valued spherical harmonics [MATH], where the index [MATH].', 'gr-qc-9712079-2-14-5': 'We call the resulting time dependent expansion coefficients, denoted by [MATH], the "spherical amplitudes" [CITATION].', 'gr-qc-9712079-2-14-6': 'They are a complete and orthogonal representation of the cartesian metric deviation tensor [MATH].', 'gr-qc-9712079-2-14-7': 'They depend only on the two wave-frame amplitudes and the direction of propagation.', 'gr-qc-9712079-2-15-0': 'To transform the metric perturbation to the lab-frame we perform the appropriate rotations using the y-convention of the Euler angles shown in Fig. [REF].', 'gr-qc-9712079-2-15-1': 'We denote the rotation about the wave [MATH] axis by [MATH], the rotation about the new [MATH] axis ([MATH]) by [MATH], and the rotation about the final lab [MATH] axis by [MATH].', 'gr-qc-9712079-2-15-2': 'The rotation matrix for the y-convention is [EQUATION]', 'gr-qc-9712079-2-15-3': 'At this point we arbitrarily set the rotation [MATH] about the wave [MATH] axis equal to zero; inclusion of this rotation will only "mix" the two polarizations of the wave.', 'gr-qc-9712079-2-15-4': 'The spherical amplitudes can now be written in terms of the polarization amplitudes and the source direction [EQUATION]', 'gr-qc-9712079-2-15-5': 'The mechanics of a spherical antenna can be described by ordinary elastic theory.', 'gr-qc-9712079-2-15-6': 'One finds that the eigenfunctions of an uncoupled sphere can be written in terms of the spherical harmonics [EQUATION]', 'gr-qc-9712079-2-15-7': 'The radial eigenfunctions [MATH] and [MATH] determine the motion in the radial and tangential directions respectively and depend on the radius [MATH] and the material of the sphere [CITATION].', 'gr-qc-9712079-2-16-0': 'In general relativity, only the 5 quadrupole modes of vibration will strongly couple to the force density of a gravitational wave.', 'gr-qc-9712079-2-16-1': 'For an ideal sphere they are all degenerate, having the same eigenfrequency, and are distinguished only by their angular dependence.', 'gr-qc-9712079-2-16-2': 'The effective force [MATH] that a gravitational wave will exert on a fundamental quadrupole mode [MATH] of the sphere is given by the overlap integral between the eigenfunctions of the sphere and the gravitational tidal force [EQUATION]', 'gr-qc-9712079-2-16-3': 'Each spherical component of the gravitational field determines uniquely the effective force on the corresponding mode of the sphere and they are all identical in magnitude.', 'gr-qc-9712079-2-16-4': 'We can interpret the effective force [MATH] in each mode as the product of: the physical mass of the sphere [MATH], an effective length [MATH] (a fraction of the sphere radius), and the gravitational acceleration [MATH].', 'gr-qc-9712079-2-16-5': 'The value of the coefficient [MATH] depends on the sphere material, but is typically [MATH] [CITATION].', 'gr-qc-9712079-2-17-0': 'By monitoring the quadrupole modes of the sphere, one has a direct measurement of the effective force on the sphere and thus the spherical amplitudes of the gravitational wave.', 'gr-qc-9712079-2-17-1': 'The standard technique for doing so on resonant detectors is to position resonant transducers on the surface of the sphere that strongly couple to the quadrupole modes.', 'gr-qc-9712079-2-17-2': 'A number of proposals have been made for the type and positions of the transducers [CITATION].', 'gr-qc-9712079-2-17-3': 'What all of these proposals have in common is that the outputs of the transducers are combined into "mode channels" [MATH] that are constructed to have a one-to-one correspondence with the quadrupole modes of the sphere and thus the spherical amplitudes of the gravitational wave [CITATION], [EQUATION]', 'gr-qc-9712079-2-17-4': 'The mode channels can be collected to form a "detector response" matrix [MATH] that in the absence of noise is equal to the cartesian strain tensor [MATH] in the lab frame [EQUATION]', 'gr-qc-9712079-2-17-5': 'For the remainder of this discussion we drop the notation of time dependence [MATH] for brevity.', 'gr-qc-9712079-2-18-0': 'The strain tensor in the lab frame [MATH] is a symmetric traceless matrix.', 'gr-qc-9712079-2-18-1': 'Consequently, it can be orthogonally diagonalized and has an orthonormal set of three eigenvectors.', 'gr-qc-9712079-2-18-2': 'One can construct from the eigenvectors a transformation matrix [MATH] that diagonalizes [MATH].', 'gr-qc-9712079-2-18-3': 'The matrix [MATH] is also orthogonal, thus it can be considered a rotation matrix (it may also include a reflection).', 'gr-qc-9712079-2-18-4': 'The physical interpretation of this transformation is to rotate the lab frame such that the [MATH] axis points in the direction of the source.', 'gr-qc-9712079-2-18-5': 'The matrix [MATH] (the eigenvectors) will tell us the angles of rotation and thus the direction of the wave.', 'gr-qc-9712079-2-19-0': 'In the wave frame, [MATH] is not normally diagonal but it can be diagonalized by rotating Eq. ([REF]) about the propagation axes using the Euler angle [MATH].', 'gr-qc-9712079-2-19-1': '[MATH] may be a constant or a function of time depending upon the situation.', 'gr-qc-9712079-2-19-2': 'This rotation changes the polarization components of the tensor but not the wave direction relative to the lab frame.', 'gr-qc-9712079-2-20-0': 'To calculate the rotation matrix [MATH] we need to solve the general eigenvalue equation for the strain tensor [EQUATION]', 'gr-qc-9712079-2-20-1': 'Since [MATH] and [MATH] are equal in the absence of noise we are free to substitute [MATH] in Eq. ([REF]) for [MATH].', 'gr-qc-9712079-2-20-2': 'By inspection of Eq. ([REF]) we see that in general relativity the eigenvector of [MATH] with [MATH] points in the propagation direction of the wave.', 'gr-qc-9712079-2-20-3': 'The direction can be calculated from this eigenvector by recognizing that it corresponds to the last column vector of [MATH] in Eq. ([REF]).', 'gr-qc-9712079-2-20-4': 'Dividing the elements of this column we find [EQUATION]', 'gr-qc-9712079-2-20-5': 'The unusual minus sign in Eq. ([REF]) comes from the use of the y-convention of the Euler angles.', 'gr-qc-9712079-2-20-6': 'Expanding Eq. ([REF]) for [MATH] and substituting in a particular choice of matrix elements from Eq. ([REF]) we find [EQUATION]', 'gr-qc-9712079-2-20-7': 'This solution is valid only for a noiseless antenna; it will fail otherwise because we can no longer replace [MATH] with [MATH] and their eigenvectors and eigenvalues will no longer be equal.', 'gr-qc-9712079-2-20-8': 'The [MATH] in Eq. ([REF]) illustrates the unavoidable fact that a single sphere cannot distinguish between antipodal sources.', 'gr-qc-9712079-2-20-9': 'This ambiguity is a characteristic of all gravity wave detectors, but can be removed by measuring the time delay of the signal between two widely separated antennas.', 'gr-qc-9712079-2-21-0': 'Once the direction is calculated we can determine the two polarization amplitudes by taking a linear combination of Eqs. ([REF]).', 'gr-qc-9712079-2-21-1': 'These equations are actually overdetermined so several solutions exist (we have 5 equations but only 4 unknowns).', 'gr-qc-9712079-2-21-2': 'In the absence of noise any particular solution to them is valid, but in anticipation of the noisy case we will take a systematic approach to the solution.', 'gr-qc-9712079-2-22-0': 'We need only the angles [MATH] and [MATH] to rotate [MATH] to [MATH], so at this point we again set [MATH].', 'gr-qc-9712079-2-22-1': 'The amplitudes are found by equating them to the corresponding matrix elements of [MATH] in Eq. ([REF]).', 'gr-qc-9712079-2-22-2': 'Again, we may substitute [MATH] for [MATH] and [MATH] for [MATH] so we have [MATH] and [MATH].', 'gr-qc-9712079-2-22-3': '[MATH] and [MATH] are symmetric so [MATH] and [MATH] will always be identical even when noise is introduced.', 'gr-qc-9712079-2-22-4': 'However, no such restriction is placed on [MATH] and [MATH].', 'gr-qc-9712079-2-22-5': 'We will use the average [MATH] to calculate [MATH] for reasons that will become clear later.', 'gr-qc-9712079-2-23-0': 'Multiplying [MATH] for [MATH] and selecting the proper elements we find [EQUATION]', 'gr-qc-9712079-2-23-1': 'These equations can also be derived by taking a linear combination of Eqs. ([REF]).', 'gr-qc-9712079-2-23-2': 'This is not the only valid solution in the noiseless case, but it is particularly symmetric: the coefficients of each component [MATH] is the same as the corresponding coefficients of [MATH] or [MATH] in Eqs. ([REF]) for [MATH].', 'gr-qc-9712079-2-23-3': 'The fact that [MATH] does not contain a [MATH] contribution is an artifact of using the y-convention of the Euler angles; in other conventions this term may be non-zero.', 'gr-qc-9712079-2-24-0': '## Detector response in alternative theories of gravity', 'gr-qc-9712079-2-25-0': 'Experiments in the solar system and pulsar-timing tests have ruled out many competing theories of gravity, however, general relativity is not the only theory of gravity that passes these weak field tests [CITATION].', 'gr-qc-9712079-2-25-1': 'One measurement that can potentially rule out certain gravitational theories is the properties of gravitational waves [CITATION], such as the speed of propagation and allowable polarization states.', 'gr-qc-9712079-2-25-2': 'It was shown above how a single sphere can measure the quadrupole components of the strain tensor, but a scalar wave can excite both the monopole mode and the quadrupole modes of a sphere [CITATION].', 'gr-qc-9712079-2-25-3': 'By monitoring both types of modes, a single spherical detector can measure all the tensor components of a gravitational wave.', 'gr-qc-9712079-2-25-4': 'This makes it possible for a single spherical detector to determine all of the six polarization states predicted by the most general symmetric metric theory of gravity [CITATION].', 'gr-qc-9712079-2-26-0': 'We can rewrite Eq. ([REF]) in terms of the electric components of the Riemann tensor [CITATION] [MATH], [EQUATION] where we now include both the [MATH] quadrupole modes and the [MATH] monopole mode.', 'gr-qc-9712079-2-26-1': 'The monopole mode of an elastic sphere is actually at a higher frequency than the quadrupole modes.', 'gr-qc-9712079-2-26-2': 'If the source is not wide-band enough for detection in both of these modes, a second sphere with the monopole mode tuned to the quadrupole modes of the first will be needed to measure this component.', 'gr-qc-9712079-2-26-3': 'If the first sphere is at relatively low frequency, one might consider making the second sphere hollow to keep it of a practical size [CITATION].', 'gr-qc-9712079-2-26-4': 'An alternative to a second sphere is to monitor the [MATH] quadrupole modes and the monopole mode of a single sphere.', 'gr-qc-9712079-2-26-5': 'These modes are not far in frequency from each other and also have relatively large cross-sections [CITATION].', 'gr-qc-9712079-2-27-0': 'Expanding Eq. ([REF]) into radial and angular parts we find an additional spherical amplitude [MATH] corresponding to the [MATH] spherical harmonic.', 'gr-qc-9712079-2-27-1': 'The detector response in the lab frame can now be written as [EQUATION]', 'gr-qc-9712079-2-28-0': 'To determine how to solve the inverse problem we need to examine the form of [MATH].', 'gr-qc-9712079-2-28-1': 'It is a symmetric tensor so it has only six independent components.', 'gr-qc-9712079-2-28-2': 'It can be written in terms of the complex Newman-Penrose parameters [CITATION] which allow the identification of the spin content of the metric theory responsible for the generation of the wave [EQUATION]', 'gr-qc-9712079-2-28-3': 'We can divide the theories of gravity into categories using the E(2) classification scheme shown in Tab. [REF] [CITATION].', 'gr-qc-9712079-2-28-4': 'The tensor [MATH] is symmetric for all of these classes, thus it is orthogonally diagonalizable, but classes [MATH] and [MATH] have more degrees of freedom (direction plus polarization states) than we are capable of measuring with a single spherical detector.', 'gr-qc-9712079-2-28-5': 'These two classes are often referred to as "observer-dependent" because different observers will disagree upon which polarization states are present.', 'gr-qc-9712079-2-28-6': 'As a consequence, the polarization amplitudes for a particular observer must be known before the direction of the wave can be estimated.', 'gr-qc-9712079-2-29-0': 'For the "observer-independent" classes [MATH], [MATH], [MATH], and [MATH], the situation is more straightforward.', 'gr-qc-9712079-2-29-1': '[MATH] is obviously uninteresting as it does not predict any gravitational waves (this class along with [MATH] have essentially been ruled out by previous experiments [CITATION]).', 'gr-qc-9712079-2-29-2': 'We notice that [MATH] for the observer-independent classes can be diagonalized by a rotation [MATH] about the propagation axis, therefore, we can use the same arguments presented above for general relativity to solve for the wave direction.', 'gr-qc-9712079-2-30-0': 'The most general observer-independent class is [MATH], which has [EQUATION]', 'gr-qc-9712079-2-30-1': 'Looking at the form of [MATH] we see that the same procedure for calculating the direction of the wave in general relativity holds for all the observer-independent classes: the eigenvector of [MATH] with eigenvalue equal to zero points at the source.', 'gr-qc-9712079-2-30-2': 'The one exception to this statement is the case where the driving forces remain in a fixed line, for example [MATH].', 'gr-qc-9712079-2-30-3': 'In this situation the direction of the wave can only be determined within the plane defined by the two eigenvectors with eigenvalues equal to zero.', 'gr-qc-9712079-2-31-0': '## Detector response to impulsive excitations', 'gr-qc-9712079-2-32-0': 'Impulsive excitations are often used on resonant-mass detectors to calibrate the antenna [CITATION].', 'gr-qc-9712079-2-32-1': 'The excitations are usually administered by either a short electrical burst applied to a calibrator attached to the surface or a hammer blow.', 'gr-qc-9712079-2-32-2': 'Impulsive excitations were also used to test the analysis techniques used for experiments with the prototype spherical antenna at Louisiana State University [CITATION].', 'gr-qc-9712079-2-33-0': 'A radial impulse excitation can be easily described if we choose the [MATH] axis to be along the direction of the impulse.', 'gr-qc-9712079-2-33-1': 'By examining the quadrupole eigenfunctions of the sphere in this frame we notice that out of these modes only the [MATH] mode will be excited (other sphere modes will also be excited but their response can be removed by narrow-band filtering).', 'gr-qc-9712079-2-33-2': 'All of the other quadrupole modes have a vanishing radial component of their eigenfunctions at this location which makes their "overlap" integral with the impulse vanish.', 'gr-qc-9712079-2-33-3': 'In this frame the detector response is [EQUATION]', 'gr-qc-9712079-2-33-4': 'In the lab frame [MATH] is still given by Eq. ([REF]).', 'gr-qc-9712079-2-34-0': 'Again, the direction can be found by calculating the eigenvalues and eigenvectors of the lab frame [MATH].', 'gr-qc-9712079-2-34-1': 'In the absence of noise, the eigenvector corresponding to the direction has a non-zero eigenvalue that is opposite in sign and twice as large as the two other eigenvalues.', 'gr-qc-9712079-2-35-0': '# Solution to the inverse problem in the presence of noise', 'gr-qc-9712079-2-36-0': 'We now return to the case of a gravitational wave in general relativity to solve the inverse problem in the presence of noise.', 'gr-qc-9712079-2-36-1': 'At the end of this section we present the application of this solution to the other types of excitations mentioned above.', 'gr-qc-9712079-2-36-2': 'For this discussion we assume that the mode channels [MATH] are independent and have normally distributed noise with the same variance.', 'gr-qc-9712079-2-36-3': 'This is a reasonable assumption as the truncated icosahedral arrangement of identical transducers ideally satisfies these conditions [CITATION].', 'gr-qc-9712079-2-36-4': 'In addition, several other proposals of transducer arrangements also produce independent mode channels [CITATION] (but the sensitivity of each mode channel is different under normal conditions [CITATION]).', 'gr-qc-9712079-2-37-0': '## Solution for general relativity', 'gr-qc-9712079-2-38-0': 'Noise in the mode channels [MATH] will change the eigenvalues and eigenvectors of [MATH] such that they are no longer equal to those of [MATH].', 'gr-qc-9712079-2-38-1': 'To gain some insight into this situation, let us consider the noise as a perturbation [MATH] to the matrix [MATH] [EQUATION]', 'gr-qc-9712079-2-38-2': 'The matrix [MATH] is constructed from the noise in each mode channel [MATH], thus it has the same form as Eq. ([REF]).', 'gr-qc-9712079-2-38-3': 'The matrix [MATH] is therefore still symmetric and traceless and has the eigenvalue equation [EQUATION]', 'gr-qc-9712079-2-38-4': 'The eigenvectors of [MATH] can be expanded in terms of the eigenvectors of [MATH] [EQUATION] where the matrix [MATH] is close to the identity matrix if the perturbation is small.', 'gr-qc-9712079-2-38-5': 'However, since we do not know the values of the matrix [MATH] we cannot calculate any corrections to the matrix elements [MATH], thus the best approximation to [MATH] we can find is [MATH].', 'gr-qc-9712079-2-39-0': 'Also from perturbation theory we see that the eigenvalue corresponding to the estimated direction of the wave is [MATH] if the perturbation is small.', 'gr-qc-9712079-2-39-1': 'Its magnitude will increase as the SNR decreases, but it should remain smaller than the other two eigenvalues of [MATH] for SNR [MATH].', 'gr-qc-9712079-2-39-2': 'Consequently the eigenvector corresponding to the estimated direction of the wave can be selected from the three eigenvectors of [MATH] by choosing the one whose eigenvalue is "closest" to zero.', 'gr-qc-9712079-2-39-3': 'Once [MATH] is found, it can be used to estimate the direction of the source using Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-2-40-0': 'The perturbation approach gives us a conceptual feel for the solution, but a more rigorous proof seems necessary.', 'gr-qc-9712079-2-40-1': 'The problem we wish to solve is to estimate the direction and polarization that makes the measured five mode channels [MATH] most "look like" the expected signal from a gravitational wave, [MATH] from Eqs. ([REF]).', 'gr-qc-9712079-2-40-2': 'Zhou and Michelson used a statistical argument to justify using the least square error in their maximum likelihood method to fit for the direction and polarization [CITATION].', 'gr-qc-9712079-2-40-3': 'Given the poor statistics in this estimation (only five samples) one might question their statistical approach, nevertheless the least squares error seems to be a reasonable choice to make under the conditions on the noise stated above.', 'gr-qc-9712079-2-41-0': 'The least squares error can be written as [EQUATION]', 'gr-qc-9712079-2-41-1': 'The values of [MATH] and [MATH] that minimize [MATH] can be found by simultaneously solving the equations [MATH] and [MATH].', 'gr-qc-9712079-2-41-2': 'Doing so using Eqs. ([REF]) we find [EQUATION]', 'gr-qc-9712079-2-41-3': 'Note that Eqs. ([REF]) and ([REF]) are identical to Eqs. ([REF]) and ([REF]) found for the noiseless case.', 'gr-qc-9712079-2-41-4': 'This connection will be useful below.', 'gr-qc-9712079-2-42-0': 'We might also look for the minimum of [MATH] with respect to the direction of the wave by taking partial derivatives with respect to [MATH] and [MATH].', 'gr-qc-9712079-2-42-1': 'This procedure leads to very complicated non-linear equations whose solution is not easily obtained.', 'gr-qc-9712079-2-42-2': 'For this reason we instead will look at how [MATH] varies close to our eigenvector solution.', 'gr-qc-9712079-2-42-3': 'We begin by rewriting [MATH] in terms of the detector response [EQUATION]', 'gr-qc-9712079-2-42-4': 'The inner product [MATH] can be interpreted as the distance between [MATH] and [MATH] which we know to be invariant to rotations.', 'gr-qc-9712079-2-42-5': 'If [MATH] is the matrix that diagonalizes [MATH] such that [MATH], we can write [EQUATION]', 'gr-qc-9712079-2-42-6': 'It is now clear that the least squares fit is the matrix [MATH] that minimizes the distance [MATH].', 'gr-qc-9712079-2-42-7': 'Geometrically, this minimum occurs when [MATH] is the projection of [MATH] onto [MATH].', 'gr-qc-9712079-2-42-8': 'Given that [MATH] is diagonal one might guess that this minimum occurs when [MATH] is also diagonal.', 'gr-qc-9712079-2-42-9': 'Let us proceed to prove this conjecture.', 'gr-qc-9712079-2-43-0': 'Let [MATH] be the matrix constructed from the eigenvectors of [MATH] so that [MATH] where [MATH] is a diagonal matrix.', 'gr-qc-9712079-2-43-1': 'Let us also assume [MATH] differs from [MATH] by a small rotation [MATH] such that [EQUATION] where [MATH] is a skew-symmetric matrix with zeros along the diagonal.', 'gr-qc-9712079-2-43-2': 'Substituting Eq. ([REF]) into Eq. ([REF]) and keeping terms only up to [MATH] we find [EQUATION]', 'gr-qc-9712079-2-43-3': 'Expanding Eq. ([REF]) and remembering that the trace is invariant under cyclic permutations of the matrices in a product and that [MATH] we find [EQUATION]', 'gr-qc-9712079-2-43-4': 'All the first order terms in [MATH] have vanished so we have proven that [MATH] is stationary near [MATH].', 'gr-qc-9712079-2-43-5': 'To show this point is a minimum we need to evaluate the second order terms in [MATH].', 'gr-qc-9712079-2-44-0': '[MATH] can be written in terms of a unit vector [MATH] representing the axis of rotation, so the square of this matrix is given by [EQUATION]', 'gr-qc-9712079-2-44-1': 'Recalling the procedure for deriving Eqs. ([REF]) and ([REF]) in the noiseless case and that they are identical to the least squares minimum Eqs. ([REF]) and ([REF]) we can set [EQUATION]', 'gr-qc-9712079-2-44-2': 'The matrix [MATH] is also traceless so [MATH].', 'gr-qc-9712079-2-44-3': 'Now the second order terms can be written as [EQUATION]', 'gr-qc-9712079-2-44-4': 'Using [MATH] and [MATH] we find that [MATH] is not guaranteed to be positive for all possible real values of [MATH] and [MATH].', 'gr-qc-9712079-2-44-5': 'Fortunately we may also assume we have ordered the eigenvalues of [MATH] such that [MATH] is the eigenvalue closest to zero.', 'gr-qc-9712079-2-44-6': 'Now we have an additional condition [MATH], where [MATH].', 'gr-qc-9712079-2-44-7': 'Substituting this into Eq. ([REF]) we find [EQUATION]', 'gr-qc-9712079-2-44-8': 'By inspection we see that [MATH] is always positive under the conditions stated above, therefore [MATH] is always a minimum near [MATH].', 'gr-qc-9712079-2-45-0': 'We further used a Monte Carlo type simulation to show that this point is always the global minimum of [MATH].', 'gr-qc-9712079-2-45-1': 'For a wave of a given direction and polarization we calculated the spherical amplitudes [MATH] and added a random number (variance [MATH] and zero mean) to obtain the mode channels.', 'gr-qc-9712079-2-45-2': 'The direction and polarization were estimated using the eigenvector method as well as by numerically finding the minimum of [MATH] from Eq. ([REF]).', 'gr-qc-9712079-2-45-3': 'We found the two methods gave identical results, even for high values of [MATH], confirming that [MATH] is a global minimum of [MATH].', 'gr-qc-9712079-2-45-4': 'Therefore, the diagonal form of [MATH] is the best approximation to [MATH] and can be used to estimate the direction and polarization of the wave.', 'gr-qc-9712079-2-46-0': 'Both the maximum likelihood method and the eigenvector solution minimize the mean square error under the conditions on the noise stated above, therefore, produce the same answer for the estimated values.', 'gr-qc-9712079-2-46-1': 'However, the eigenvector solution is more straightforward and computationally simple.', 'gr-qc-9712079-2-46-2': 'We construct the matrix [MATH] from the mode channels [MATH] and compute its eigenvalues [MATH] and eigenvectors [MATH].', 'gr-qc-9712079-2-46-3': 'We choose the eigenvector with eigenvalue closest to zero and estimate the direction of the wave using Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-2-46-4': 'The polarization amplitudes can be estimated using Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-2-47-0': '## Extensions of the eigenvector solution', 'gr-qc-9712079-2-48-0': 'The detector response matrix [MATH] for other metric theories of gravity as well as for impulse excitations satisfy the symmetry arguments used in the discussion for general relativity.', 'gr-qc-9712079-2-48-1': 'This means we can easily adapt the noiseless solutions to the case where noise is present in the same fashion.', 'gr-qc-9712079-2-49-0': 'For observer-independent gravitational theories the method for estimating the direction of the wave is identical to that of general relativity: the eigenvector of the detector response matrix with eigenvalue closest to zero can be used to estimate the direction of the source.', 'gr-qc-9712079-2-49-1': 'Observer-dependent theories require prior knowledge of the polarization states of the wave before any estimate of the direction can be made.', 'gr-qc-9712079-2-49-2': 'Once these are known it should be straightforward to adapt the eigenvector technique to estimate the direction.', 'gr-qc-9712079-2-49-3': 'In the case of an impulsive excitation, the eigenvector corresponding to the direction has an eigenvalue that is opposite in sign and greater in magnitude than the two other eigenvalues.', 'gr-qc-9712079-2-49-4': 'Converting the eigenvectors to a direction again comes from Eqs. ([REF]) and ([REF]).', 'gr-qc-9712079-2-50-0': '# Discussion', 'gr-qc-9712079-2-51-0': 'The eigenvector solution is very convenient in that the inverse problem is reduced to solving a trivial eigenvalue problem.', 'gr-qc-9712079-2-51-1': 'The solution is computationally simple, making this technique very efficient for use in an automated data analysis system.', 'gr-qc-9712079-2-51-2': 'This feature may be important if one considers using a large number of candidate gravitational wave events in a coincidence exchange between several detectors where the source direction is used as a criterion to veto excess coincidences.', 'gr-qc-9712079-2-52-0': 'The main restrictions on the eigenvector solution is that the mode channels must be independent and the noise normally distributed with equal variance.', 'gr-qc-9712079-2-52-1': 'These restrictions can ideally be satisfied for a number of transducer arrangements [CITATION].', 'gr-qc-9712079-2-52-2': 'We found that the eigenvector solution corresponds exactly to the maximum likelihood method under these conditions.', 'gr-qc-9712079-2-52-3': 'It may be possible to apply this solution when the noise is not gaussian or is different for each mode channel, however further research is necessary to verify this extension.', 'gr-qc-9712079-2-53-0': 'Through a number of numerical simulations as well as examination of the work of others [CITATION] we found that the errors due to the noise on a direction estimation are independent of the source location and wave amplitude for a given SNR.', 'gr-qc-9712079-2-53-1': 'However, the estimation of the polarization amplitudes using Eqs. ([REF]) and ([REF]) lead to direction dependent uncertainties.', 'gr-qc-9712079-2-53-2': 'For example, Fig. [REF] shows the variance on the polarization angle [MATH] for a range of SNR and several values of [MATH] found from a Monte Carlo type simulation.', 'gr-qc-9712079-2-53-3': 'Notice that the variance increases for low values of [MATH].', 'gr-qc-9712079-2-53-4': 'This realization is disturbing given that a spherical antenna is equally sensitive to waves from all directions and polarizations.', 'gr-qc-9712079-2-54-0': 'One might consider using a different coordinate system to try to avoid the directions with very poor estimates of the two polarization amplitudes.', 'gr-qc-9712079-2-54-1': 'For example, use the xyz-convention of the Euler angles where the first and the last rotations are not the same.', 'gr-qc-9712079-2-54-2': 'This actually will not solve our problem, but instead change the directions in the sky which lead to the poor estimates.', 'gr-qc-9712079-2-54-3': 'If we transform back from this coordinate system to the y-convention we just reintroduce the errors and thus have gained nothing.', 'gr-qc-9712079-2-55-0': 'This dependency on the source direction is not unique to a sphere, a network of bars or interferometers will also suffer from this problem [CITATION].', 'gr-qc-9712079-2-55-1': 'This leads us to believe that we are excluding a piece of information from our procedures.', 'gr-qc-9712079-2-55-2': 'The solution may lie in using the information from the two other eigenvectors of the detector response.', 'gr-qc-9712079-2-55-3': 'In the above derivations these eigenvectors were simply discarded, but they also contain information about the gravitational wave that may eliminate these direction dependent errors.', 'gr-qc-9712079-2-55-4': 'This approach will be the topic of a future paper [CITATION].', 'gr-qc-9712079-2-56-0': 'While there are a few limitations to the eigenvector solution, its simplicity makes it easily extendable to other types of excitations.', 'gr-qc-9712079-2-56-1': 'As discussed above, impulsive excitations can be located using this technique.', 'gr-qc-9712079-2-56-2': 'As a practical example we recall that this solution was successfully tested on experiments with the LSU prototype spherical antenna [CITATION].', 'gr-qc-9712079-2-56-3': 'This practical confirmation of its validity gives us the confidence that it can be implemented on a real spherical antenna searching for gravitational waves.', 'gr-qc-9712079-2-57-0': 'I thank M. Bassan, M. Bianchi, E. Coccia, and E. Mauceli for many useful discussions on this work.', 'gr-qc-9712079-2-57-1': 'In particular, I thank J. A. Lobo for his assistance and advice.', 'gr-qc-9712079-2-57-2': 'I also thank the referee for suggesting part of the argument presented in Sec. [REF].'}

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['gr-qc-9712079-1-42-1', 'gr-qc-9712079-2-29-1'], ['gr-qc-9712079-1-42-2', 'gr-qc-9712079-2-29-2'], ['gr-qc-9712079-1-43-0', 'gr-qc-9712079-2-30-0'], ['gr-qc-9712079-1-43-2', 'gr-qc-9712079-2-30-2'], ['gr-qc-9712079-1-16-1', 'gr-qc-9712079-2-17-1'], ['gr-qc-9712079-1-16-2', 'gr-qc-9712079-2-17-2'], ['gr-qc-9712079-1-16-3', 'gr-qc-9712079-2-17-3'], ['gr-qc-9712079-1-46-0', 'gr-qc-9712079-2-33-0'], ['gr-qc-9712079-1-46-1', 'gr-qc-9712079-2-33-1'], ['gr-qc-9712079-1-46-2', 'gr-qc-9712079-2-33-2'], ['gr-qc-9712079-1-46-4', 'gr-qc-9712079-2-33-4'], ['gr-qc-9712079-1-24-1', 'gr-qc-9712079-2-23-0'], ['gr-qc-9712079-1-24-2', 'gr-qc-9712079-2-23-1'], ['gr-qc-9712079-1-62-3', 'gr-qc-9712079-2-55-0'], ['gr-qc-9712079-1-18-0', 'gr-qc-9712079-2-11-0'], ['gr-qc-9712079-1-18-1', 'gr-qc-9712079-2-11-1'], ['gr-qc-9712079-1-18-4', 'gr-qc-9712079-2-11-4'], ['gr-qc-9712079-1-6-0', 'gr-qc-9712079-2-6-0'], ['gr-qc-9712079-1-6-2', 'gr-qc-9712079-2-6-2'], ['gr-qc-9712079-1-6-3', 'gr-qc-9712079-2-6-3'], ['gr-qc-9712079-1-20-0', 'gr-qc-9712079-2-19-0'], ['gr-qc-9712079-1-20-1', 'gr-qc-9712079-2-19-1'], ['gr-qc-9712079-1-63-1', 'gr-qc-9712079-2-56-3'], ['gr-qc-9712079-1-19-0', 'gr-qc-9712079-2-18-0'], ['gr-qc-9712079-1-19-1', 'gr-qc-9712079-2-18-1'], ['gr-qc-9712079-1-19-3', 'gr-qc-9712079-2-18-3'], ['gr-qc-9712079-1-19-4', 'gr-qc-9712079-2-18-4'], ['gr-qc-9712079-1-19-5', 'gr-qc-9712079-2-18-5'], ['gr-qc-9712079-1-52-0', 'gr-qc-9712079-2-54-0'], ['gr-qc-9712079-1-52-1', 'gr-qc-9712079-2-54-1'], ['gr-qc-9712079-1-60-2', 'gr-qc-9712079-2-51-1'], ['gr-qc-9712079-1-60-3', 'gr-qc-9712079-2-51-2'], ['gr-qc-9712079-1-12-0', 'gr-qc-9712079-2-13-0'], ['gr-qc-9712079-1-12-1', 'gr-qc-9712079-2-13-1'], ['gr-qc-9712079-1-12-2', 'gr-qc-9712079-2-13-2'], ['gr-qc-9712079-1-12-3', 'gr-qc-9712079-2-13-3'], ['gr-qc-9712079-1-12-4', 'gr-qc-9712079-2-13-4'], ['gr-qc-9712079-1-27-1', 'gr-qc-9712079-2-39-1'], ['gr-qc-9712079-1-27-2', 'gr-qc-9712079-2-39-2'], ['gr-qc-9712079-1-27-3', 'gr-qc-9712079-2-39-3'], ['gr-qc-9712079-1-15-0', 'gr-qc-9712079-2-16-0'], ['gr-qc-9712079-1-15-1', 'gr-qc-9712079-2-16-1'], ['gr-qc-9712079-1-15-2', 'gr-qc-9712079-2-16-2'], ['gr-qc-9712079-1-15-4', 'gr-qc-9712079-2-16-4'], ['gr-qc-9712079-1-15-5', 'gr-qc-9712079-2-16-5'], ['gr-qc-9712079-1-3-2', 'gr-qc-9712079-2-3-2'], ['gr-qc-9712079-1-23-0', 'gr-qc-9712079-2-22-0'], ['gr-qc-9712079-1-23-3', 'gr-qc-9712079-2-22-4'], ['gr-qc-9712079-1-23-4', 'gr-qc-9712079-2-22-5'], ['gr-qc-9712079-1-45-0', 'gr-qc-9712079-2-32-0'], ['gr-qc-9712079-1-45-1', 'gr-qc-9712079-2-32-1'], ['gr-qc-9712079-1-45-2', 'gr-qc-9712079-2-32-2'], ['gr-qc-9712079-1-4-0', 'gr-qc-9712079-2-4-0'], ['gr-qc-9712079-1-4-1', 'gr-qc-9712079-2-4-1'], ['gr-qc-9712079-1-4-2', 'gr-qc-9712079-2-4-2'], ['gr-qc-9712079-1-7-0', 'gr-qc-9712079-2-7-0'], ['gr-qc-9712079-1-7-1', 'gr-qc-9712079-2-7-1'], ['gr-qc-9712079-1-7-2', 'gr-qc-9712079-2-7-2'], ['gr-qc-9712079-1-7-3', 'gr-qc-9712079-2-7-3'], ['gr-qc-9712079-1-7-4', 'gr-qc-9712079-2-7-4'], 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[]

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[]

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/gr-qc/9712079

hep-ph-9801399

{'hep-ph-9801399-1-0-0': 'Ill-defined pinch singularities characteristic of out of equilibrium thermal field theories are analyzed.', 'hep-ph-9801399-1-0-1': 'We identify two mechanisms which eliminate pinching: the first is based on the vanishing of phase space at the singular point (Threshold effect).', 'hep-ph-9801399-1-0-2': 'It is effective in QED with massive electron and massless photon.', 'hep-ph-9801399-1-0-3': 'In massless QCD this mechanism fails, but the pinches cancel due to the second mechanism, i.e. due to the spinor/tensor structure of the single self-energy insertion contribution to propagator.', 'hep-ph-9801399-1-0-4': 'The constraints imposed on distribution functions are very reasonable.', 'hep-ph-9801399-1-1-0': '# Introduction', 'hep-ph-9801399-1-2-0': 'Out of equilibrium thermal field theories have recently attracted much interest.', 'hep-ph-9801399-1-2-1': 'From the experimental point of view, very interesting are the various aspects of heavy-ion collisions and the related hot QCD plasma.', 'hep-ph-9801399-1-2-2': 'The supposedly gluon dominated stage is of special interest.', 'hep-ph-9801399-1-3-0': 'Contrary to the equilibrium case[MATH] where pinch, collinear and infrared problems have been successfully controlled[MATH], out of equilibrium theory[MATH] is suffering from them to these days.', 'hep-ph-9801399-1-3-1': 'However progress has been made in this field, too.', 'hep-ph-9801399-1-4-0': 'Weldon[MATH] observes that the out of equilibrium pinch singularity does not cancel; hence it spoils analyticity and causality.', 'hep-ph-9801399-1-4-1': 'The problem gets worse with more than one self-energy insertions.', 'hep-ph-9801399-1-5-0': 'Bedaque argues that in out of equilibrium theory the time extension should be finite.', 'hep-ph-9801399-1-5-1': 'Thus the time integration limits from [MATH] to [MATH], which are responsible for the appearance of pinches, have to be abandoned as unphysical[MATH].', 'hep-ph-9801399-1-6-0': 'Le Bellac and Mabilat[MATH] show that pinching singularities give contribution of order [MATH], where [MATH] is the deviation from equilibrium.', 'hep-ph-9801399-1-6-1': 'They also find that collinear singularities cancel in scalar theory, and in QCD using physical gauges, but not in the case of covariant gauges.', 'hep-ph-9801399-1-6-2': 'Niegawa[MATH] finds that the pinch-like term contains divergent part which cancels collinear singularities in covariant gauge.', 'hep-ph-9801399-1-7-0': 'Altherr and Seibert find that in the massive [MATH] theory pinch singularity does not occure due to the kinematical constraint[MATH].', 'hep-ph-9801399-1-8-0': 'Altherr suggests a regularization method in which the propagator is modified by the width [MATH] which is an arbitrary function of momentum, to be calculated in a self-consistent way.', 'hep-ph-9801399-1-8-1': 'In the [MATH] theory for small deviations from equilibrium, [MATH] is found to be just the usual equilibrium damping rate[MATH].', 'hep-ph-9801399-1-9-0': 'This recipe has been justified in the resummed Schwinger-Dyson series in various problems with pinching[MATH].', 'hep-ph-9801399-1-10-0': 'Baier, Dirks and Redlich[MATH] calculate the [MATH] self-energy contribution to the pion propagator, regulating pinch contributions by the damping rate.', 'hep-ph-9801399-1-10-1': 'In a subsequent papers with Schiff[MATH] they calculate the quark propagator within the HTL approximation[MATH]; in the resummed Schwinger-Dyson series the pinch is naturally regulated by [MATH].', 'hep-ph-9801399-1-11-0': 'Carrington, Defu and Thoma[MATH] find that in the HTL approximation to the resummed photon propagator no pinch singularities appear.', 'hep-ph-9801399-1-12-0': 'Niegawa[MATH] introduces a notion of renormalized particle-number density.', 'hep-ph-9801399-1-12-1': 'He finds, that in the appropriately redefined calculation scheme the amplitudes and reaction rates are free from pinch singularities.', 'hep-ph-9801399-1-13-0': 'In our previous paper[MATH] we discussed extensively two interesting aspects of the problem: pinch singularities as they appear in the resummed Schwinger-Dyson series and, separately, pinches which appear in the approximating the propagator by bare propagator plus the contribution from single self-energy insertion.', 'hep-ph-9801399-1-14-0': 'By pinching singularity we understand the contour which passes between two infinitely close poles [EQUATION] where [MATH].', 'hep-ph-9801399-1-14-1': 'It is controlled by some parameter, e.g. [MATH].', 'hep-ph-9801399-1-14-2': 'For finite [MATH], the expression is regular.', 'hep-ph-9801399-1-14-3': 'However, when [MATH] tends to zero, the integration path is "pinched" between the two poles, and the expression is ill-defined.', 'hep-ph-9801399-1-14-4': 'Integration gives the [MATH] contribution plus regular terms.By performing a simple (but forbidden) decomposition of [MATH] into [MATH], one gets the related ill-defined [MATH] expression.', 'hep-ph-9801399-1-15-0': 'Similar to ([REF]) is the expression which corresponds to the resummed Schwinger-Dyson series.', 'hep-ph-9801399-1-15-1': '[EQUATION] where [MATH], and [MATH] (which appears in the next expression) are proportional to [MATH], and [MATH] where [MATH], [MATH] and [MATH] are the components of the self-energy matrix to be defined in the next chapter.', 'hep-ph-9801399-1-16-0': 'In this expression pinching is absent[MATH] if [MATH] at the value of [MATH] which satisfies [MATH].', 'hep-ph-9801399-1-17-0': 'In some limiting cases where [MATH] it is important to observe, that as [MATH] the behaviour of [MATH] and the behaviour of [MATH] are related, resulting again with the elimination of pinching[MATH].', 'hep-ph-9801399-1-18-0': 'The expression, which corresponds to the single self-energy insertion approximation to the propagator, is similar to the above: [EQUATION]', 'hep-ph-9801399-1-18-1': 'One can rewrite the integral as [EQUATION]', 'hep-ph-9801399-1-18-2': 'If it happens that [EQUATION] then the integral ([REF]) decomposes into two pieces which, although possibly divergent, do not suffer from pinching.', 'hep-ph-9801399-1-19-0': 'Ther are two cases when the function [MATH] is even identically zero in the vicinity of [MATH] point: in thermal equilibrium it is so due to the detailed balance relations, in massive [MATH] theory out of equilibrium it is so due to the mass shell condition[MATH].', 'hep-ph-9801399-1-19-1': 'The last mechanism works also in out of equilibrium QED if the small photon mass [MATH] is introduced.', 'hep-ph-9801399-1-19-2': 'But such elimination of pinching can be missleading: the domain of [MATH], where [MATH], shrinks to a point as [MATH].', 'hep-ph-9801399-1-19-3': 'But we shall show the elimination of pinching also in the [MATH] case.', 'hep-ph-9801399-1-20-0': 'In this paper we identify two mechanisms leading to the relation ([REF]).', 'hep-ph-9801399-1-20-1': 'They are based on observation that in the pinch-like contribution loop particles have to be on mass shell.', 'hep-ph-9801399-1-21-0': 'First mechanism is effective in out of equilibrium QED: in the pinch-like contribution to electron propagator phase space vanishes linearly as [MATH] .', 'hep-ph-9801399-1-21-1': 'In the pinch-like contribution to the photon propagator the domain of integration is shifted to infinity as [MATH].', 'hep-ph-9801399-1-21-2': 'For distributions disappearing fast enough at large energies the contribution again vanishes linearly in the [MATH] limit.', 'hep-ph-9801399-1-21-3': 'This mechanism is valid also in QCD in the cases when massive quarks appear.', 'hep-ph-9801399-1-22-0': 'In the case of out of equilibrium massless QCD the phase space does not vanish but there is an alternative mechanism: the spinor/tensor structure in all the cases leads to the relation ([REF]).', 'hep-ph-9801399-1-23-0': 'Also in the case of out of equilibrium massless QCD introduction of small gluon mass does not help.', 'hep-ph-9801399-1-23-1': 'In this case processes like [MATH] are kinematically allowed,spinor/tensor structure is modified, and [MATH] does not vanish in the [MATH] limit.', 'hep-ph-9801399-1-24-0': 'In a few cases none of the mentioned mechanisms works and one has indeed to sum the Schwinger-Dyson series.', 'hep-ph-9801399-1-24-1': 'It is the case of the [MATH] loop in the [MATH] self-energy .', 'hep-ph-9801399-1-24-2': 'Even in the limit of zero pion mass [MATH] vanishes only as [MATH] and the relation ([REF]) is not fullfilled.', 'hep-ph-9801399-1-24-3': 'Similar is the problem of electroweak interactions involving decays of [MATH] and [MATH] bosons, Higgs particle decay, etc.', 'hep-ph-9801399-1-24-4': 'Second important case is massless [MATH] theory.', 'hep-ph-9801399-1-24-5': 'Here unlike in the massless QCD there is no spin factor to provide [MATH] factor necesarry to obtain ([REF]).', 'hep-ph-9801399-1-25-0': 'The densities are restricted only mildly: they should be cut off at high energies at least as [MATH] to obtain a finite total particle density; for non-zero [MATH], they should be finite; for [MATH] near zero, they should not diverge faster than [MATH], electron (positron) distribution should have finite derivative.', 'hep-ph-9801399-1-25-1': 'Further restrictions may come from Slavnov-Taylor identities[MATH], but they are not crucial for our analysis.', 'hep-ph-9801399-1-26-0': 'The paper is organized as follows.', 'hep-ph-9801399-1-27-0': 'In section 2 we analyze the Schwinger-Dyson equation in the Keldysh representation, solve it formally and identify pinch-like expressions.', 'hep-ph-9801399-1-27-1': 'In the case of one-loop self-energy insertions, we find that the Keldysh component ([MATH]) of the self-energy is responsible for pinches.', 'hep-ph-9801399-1-27-2': 'Further we find that the non-zero Keldysh component requires loop particles to be on shell.', 'hep-ph-9801399-1-28-0': 'In section 3 we analyze functions such as [MATH], [MATH] and [MATH] and investigate their threshold properties.', 'hep-ph-9801399-1-29-0': 'In section 4 we show that the electron and photon propagators, calculated in the single self-energy insertion approximation, are free from pinching.', 'hep-ph-9801399-1-30-0': 'In section 5 we analyze pinch-like expressions in the [MATH], [MATH] and ghost-ghost contribution to gluon propagator, quark propagator and ghost propagator in the single self-energy insertion approximation.', 'hep-ph-9801399-1-30-1': 'We find that, in all the cases, the spinor/tensor factor [MATH] contains factor [MATH] which is enough to eliminate pinching.', 'hep-ph-9801399-1-31-0': 'In section 6 we briefly recollect the main results of the paper.', 'hep-ph-9801399-1-32-0': '# Propagators and the Schwinger-Dyson equation', 'hep-ph-9801399-1-33-0': 'We start[MATH] by defining out of equilibrium thermal propagators for bosons, in the case that one can ignore the variations of slow variables in Wigner functions[MATH]: [EQUATION]', 'hep-ph-9801399-1-33-1': 'For particles with additional degrees of freedom relations ([REF])-([REF]) are provided with extra factors [MATH] for spin 1/2, [MATH] for vector particle, etc., and similarly for internal degrees of freedom.', 'hep-ph-9801399-1-33-2': 'To keep the discussion as general as possible, we show this factors explicitly only when necessary.', 'hep-ph-9801399-1-33-3': 'The propagator defined by relations ([REF])-([REF]) satisfies the important condition [EQUATION]', 'hep-ph-9801399-1-33-4': 'In the case of equilibrium, we have [EQUATION]', 'hep-ph-9801399-1-33-5': 'To obtain the corresponding relations for fermions we only need to make the substitution [EQUATION]', 'hep-ph-9801399-1-33-6': 'In the case of equilibrium, for fermions we have [EQUATION]', 'hep-ph-9801399-1-33-7': 'Out of equilibrium, [MATH] and [MATH] will be some given functions of [MATH].', 'hep-ph-9801399-1-34-0': 'To transform to the Keldysh form, one defines the matrix Q[MATH] as [EQUATION]', 'hep-ph-9801399-1-34-1': 'Now [EQUATION]', 'hep-ph-9801399-1-34-2': 'We need [MATH] expressed through [MATH] and [MATH]: [EQUATION]', 'hep-ph-9801399-1-34-3': 'Again for fermions, [MATH] is equal to [EQUATION]', 'hep-ph-9801399-1-34-4': 'The proper self-energy [EQUATION] satisfies the condition [EQUATION]', 'hep-ph-9801399-1-34-5': "It is also transformed to the Keldysh form (in Niemi's paper there is a misprint using [MATH] instead of [MATH]) [EQUATION]", 'hep-ph-9801399-1-34-6': 'We also find [EQUATION]', 'hep-ph-9801399-1-34-7': 'The calculation of the [MATH] matrix gives (propagators [MATH] and [MATH] in the self-energy matrix and in the Schwinger-Dyson equation are also given by ([REF]) to ([REF]), with the spin indices suppressed to keep the discussion as general as possible) [EQUATION]', 'hep-ph-9801399-1-34-8': 'Simple exercise, with the help of ([REF]), will convince us that only on-shell loop-particle momenta contribute to [MATH].', 'hep-ph-9801399-1-34-9': 'The Schwinger-Dyson equation [EQUATION] can be written in the Keldysh form [EQUATION]', 'hep-ph-9801399-1-34-10': 'By expanding ([REF]), we deduce the contribution from the single self-energy insertion to be of the form [EQUATION] which is evidently well-defined, and the Keldysh component suspected for pinches; [EQUATION]', 'hep-ph-9801399-1-34-11': 'It is easy to obtain a solution[MATH] for [MATH] and [MATH] using the form ([REF]).', 'hep-ph-9801399-1-34-12': 'One observes that the equations for [MATH] and [MATH] are simple and the solution is straightforward: [EQUATION]', 'hep-ph-9801399-1-34-13': 'To calculate [MATH], we can use the solution ([REF]) for [MATH] and [MATH]: [EQUATION]', 'hep-ph-9801399-1-34-14': 'Now we eliminate [MATH] with the help of ([REF]): [EQUATION]', 'hep-ph-9801399-1-34-15': 'The first term in ([REF]) is not always zero, but it is a well-defined expression!', 'hep-ph-9801399-1-34-16': 'The second term in ([REF]) is potentially ill-defined (or pinch-like).', 'hep-ph-9801399-1-34-17': 'The pinch-like contribution appears only in this equation; thus it is the key for the whole problem of pinch singularities.', 'hep-ph-9801399-1-34-18': 'In the one-loop approximation, it requires loop particles to be on mass shell.', 'hep-ph-9801399-1-34-19': 'This will be enough to remove ill-defined expressions in all studied cases.', 'hep-ph-9801399-1-35-0': 'We start with ([REF]).', 'hep-ph-9801399-1-35-1': 'After substituting ([REF]) into ([REF]), we obtain the regular term plus pinch-like contribution: [EQUATION]', 'hep-ph-9801399-1-35-2': 'For equilibrium densities, we have [MATH] , and expression ([REF]) vanishes identically.', 'hep-ph-9801399-1-35-3': 'This is also true in the case of fermions.', 'hep-ph-9801399-1-36-0': 'Expression ([REF]) is the only one suspected for pinch singularities at the single self-energy insertion level.', 'hep-ph-9801399-1-36-1': 'The function [MATH] ([REF]) belongs to the type of functions characterized by the fact that both loop particles have to be on mass shell.', 'hep-ph-9801399-1-36-2': 'It is analyzed in detail in section 3 and 4 (for threshold effect) and in section 5 (for spin effect).', 'hep-ph-9801399-1-36-3': 'With the help of this analyzis we show in this paper that the relation ([REF]) transforms into [EQUATION] where [MATH] is [MATH] multiplied by spinor/tensor factors included in the definition of [MATH].', 'hep-ph-9801399-1-36-4': 'Important for cancellation of pinches is the finitness of the limit [EQUATION]', 'hep-ph-9801399-1-36-5': 'The index [MATH] indicates that the limiting value [MATH] is approached either from below or from above, and this two values are generally different.', 'hep-ph-9801399-1-36-6': 'As a consequence the right hand side of expression ([REF]) behaves locally as (the connection between [MATH] and [MATH] is obvious) [EQUATION] and the term proportional to [MATH] is capable of producing logarithmyc singularity.', 'hep-ph-9801399-1-37-0': 'Furthermore we were unable to eliminate pinches related to the double, triple, ... self-energy insertion contribution to propagator.', 'hep-ph-9801399-1-37-1': 'But their sum [MATH] is free from pinching, under the assumption that also the resummed Schwinger-Dyson series is free from pinching.', 'hep-ph-9801399-1-38-0': '# Threshold factor', 'hep-ph-9801399-1-39-0': 'In this section we analyze the phase space of the loop integral with both loop particles on mass shell.', 'hep-ph-9801399-1-39-1': 'Special care is devoted to the behaviour of this integral near thresholds.', 'hep-ph-9801399-1-39-2': 'In this analysis the densities are constrained only mildly: they are supposed to be finite and smooth, with possible exception at zero energy.', 'hep-ph-9801399-1-39-3': 'We also assume that the total density of particles is finite.', 'hep-ph-9801399-1-39-4': 'The expressions are written for all particles beeing bosons,and spins are not specified, change to fermion is elementary.', 'hep-ph-9801399-1-40-0': 'To obtain the integrals over the products of [MATH] and [MATH], we start with a useful relation: [EQUATION]', 'hep-ph-9801399-1-40-1': 'Similar relations could be obtained for higher powers of [MATH] and [MATH].', 'hep-ph-9801399-1-40-2': 'For example, for the nth power of [MATH] real part of the integral will be obtained by substituting [MATH] for of [MATH].', 'hep-ph-9801399-1-40-3': 'Now we easily calculate the [MATH]: [EQUATION] [MATH] is the factor dependent on spin and internal degrees of freedom.', 'hep-ph-9801399-1-41-0': 'As we assume that the zero-temperature renormalization has already been performed, the zero-temperature part is in fact eliminated by counter terms and only the thermal part remains: [EQUATION]', 'hep-ph-9801399-1-41-1': 'Now, starting from ([REF]) to ([REF]), we calculate [MATH] and [MATH].', 'hep-ph-9801399-1-41-2': '[EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-9801399-1-41-3': 'It is useful to define [MATH] as [EQUATION]', 'hep-ph-9801399-1-41-4': "After integrating over [MATH]'s one obtains expressions of the general form [EQUATION] where [MATH], [EQUATION]", 'hep-ph-9801399-1-41-5': 'Let us start with the [MATH] case.', 'hep-ph-9801399-1-41-6': 'Solution of [MATH] gives the integration limits: [EQUATION] or [EQUATION]', 'hep-ph-9801399-1-41-7': 'Assume now that [MATH].', 'hep-ph-9801399-1-41-8': 'In that case, at threshold, the limits shrink to the value [EQUATION]', 'hep-ph-9801399-1-41-9': 'In expression ([REF])we expand the integrand for small [MATH].', 'hep-ph-9801399-1-41-10': 'Then, after integration, the result takes the form [EQUATION] with the coefficient [MATH] given by [EQUATION]', 'hep-ph-9801399-1-41-11': 'Thus we can isolate the threshold effect [EQUATION]', 'hep-ph-9801399-1-41-12': 'Relation ([REF]) is the key to further discussion of the threshold effect.', 'hep-ph-9801399-1-41-13': 'One should observe that relation ([REF]) also holds in the case when the integrand cannot be expanded in powers of [MATH], but then it holds only locally near the threshold.', 'hep-ph-9801399-1-42-0': 'We obtain this also for higher dimension (D=6 for example).', 'hep-ph-9801399-1-43-0': 'Relation ([REF]) puts some limits on the behaviour of density functions: they should not tend to infinity at any value of [MATH]; near [MATH], owing to the presence of the factor [MATH], they should not rise faster than [MATH].', 'hep-ph-9801399-1-44-0': 'Owing to ([REF]) and ([REF]) the function [MATH] has the following properties important for cancellation of pinches.', 'hep-ph-9801399-1-45-0': 'It vanishes between the thresholds, i.e. the domain [MATH] is forbidden ([MATH]).', 'hep-ph-9801399-1-46-0': 'It is (in principle) different from zero, in the allowed domain [MATH] and [MATH].', 'hep-ph-9801399-1-46-1': 'An exception to this rule are occasional zeros owing to the specific form of densities.', 'hep-ph-9801399-1-47-0': 'The behaviour at the boundaries (i.e. in the allowed region near threshold) depends on the masses [MATH] and [MATH] and there are a few possibilities.', 'hep-ph-9801399-1-48-0': 'If both masses are non-zero and different ([MATH]) then there are two thresholds and [MATH] behaves as [MATH] in the allowed region near the threshold [MATH].', 'hep-ph-9801399-1-49-0': 'If one of the masses is zero ([MATH] or [MATH]), then ([REF]) gives that the thresholds are identical (i.e. forbidden domain shrinks to zero) and one obtains the [MATH] behaviour near [MATH].', 'hep-ph-9801399-1-50-0': 'If the masses are equal but different from zero ([MATH]), then there are two thresholds with different behaviour.', 'hep-ph-9801399-1-50-1': 'The function [MATH] behaves as [MATH] in the allowed region near the threshold [MATH] .', 'hep-ph-9801399-1-51-0': 'However at the other threshold, namely at [MATH], the physical region is determined by [MATH] and the above discussion does not apply.', 'hep-ph-9801399-1-51-1': 'In fact, the integration limits ([REF]) or ([REF]) are valid, but the region between [MATH] and [MATH] is now excluded from integration.', 'hep-ph-9801399-1-51-2': 'One has to integrate over the domain [MATH] and [MATH].', 'hep-ph-9801399-1-51-3': 'This leads to the limitation in the high energy behaviour of the density functions.', 'hep-ph-9801399-1-52-0': 'If both masses vanish ([MATH]), the thresholds coincide, there is no forbidden region and no threshold behaviour.', 'hep-ph-9801399-1-52-1': 'The behaviour depends on the spin of the particles involved.', 'hep-ph-9801399-1-52-2': 'For scalars, the leading term in expansion of I does not vanish.', 'hep-ph-9801399-1-53-0': 'The case of vanishing masses ([MATH]) for particles with spin exhibits a peculiar behaviour.', 'hep-ph-9801399-1-53-1': 'In all studied examples (see the section 5 for details), [MATH] behaves as [MATH] as [MATH].', 'hep-ph-9801399-1-54-0': '# Pinch Singularities in QED', 'hep-ph-9801399-1-55-0': '## Pinch Singularities in Electron Propagator', 'hep-ph-9801399-1-56-0': 'In this subsection we apply the results of previous section to cancel the pinching singularity appearing in a single self-energy insertion approximation to electron propagator.', 'hep-ph-9801399-1-56-1': 'To do so we have to substitute [MATH], [MATH], [MATH], [MATH] and [MATH], where [MATH] and [MATH] ar given nonequilibrium ditributions of electrons and photons into the relations ([REF]), ([REF]),([REF]) and ([REF]).', 'hep-ph-9801399-1-56-2': 'The thresholds are now identical [EQUATION] and the integration limits are [EQUATION] or [EQUATION]', 'hep-ph-9801399-1-56-3': 'Then, with the help of ([REF]), we define [EQUATION]', 'hep-ph-9801399-1-56-4': 'The trace factor [MATH] is calculated with loop particles on mass shell [EQUATION]', 'hep-ph-9801399-1-56-5': 'In the calculation of term proportional to [MATH] we have to use the trick [EQUATION]', 'hep-ph-9801399-1-56-6': 'For [MATH] we can decompose vector [MATH] as [EQUATION] where, in the heat bath frame with [MATH] axis oriented along vector [MATH] [EQUATION]', 'hep-ph-9801399-1-56-7': 'The transverse component of [MATH], [MATH] vanishes after the integration over [MATH].', 'hep-ph-9801399-1-57-0': 'Finally we obtain [EQUATION]', 'hep-ph-9801399-1-57-1': 'Now we can study the limit [EQUATION]', 'hep-ph-9801399-1-57-2': 'It is easy to find that [MATH] is finite provided that [MATH] and [MATH].', 'hep-ph-9801399-1-57-3': 'The last condition is easy to investigate by limiting procedure: [EQUATION]', 'hep-ph-9801399-1-57-4': 'One should observe here that the integration limits imply that the limit [MATH] is taken from below for [MATH], and from above for [MATH].', 'hep-ph-9801399-1-57-5': 'The two limits lead to different values of [MATH].', 'hep-ph-9801399-1-57-6': 'Only the first term in ([REF]) can create problems.', 'hep-ph-9801399-1-57-7': 'We rewrite it as [MATH].', 'hep-ph-9801399-1-57-8': 'As the relation ([REF]) should be valid at any [MATH] we can integrate over [MATH] to find that the photon distribution should not grow faster than [MATH] as [MATH] appraches zero, while the derivative of the electron distribution [MATH] should be finite at any [MATH] [EQUATION]', 'hep-ph-9801399-1-57-9': 'Under the very reasonable conditions ([REF]) and ([REF]) electron propagator is free from pinches.', 'hep-ph-9801399-1-58-0': 'It is interesting to observe the discontinuity of [MATH] at the point [MATH].', 'hep-ph-9801399-1-58-1': 'This feature will be repeated in massless QCD.', 'hep-ph-9801399-1-59-0': 'It is worth observing that [MATH] is gauge independent, at least within the class of covariant gauges.', 'hep-ph-9801399-1-60-0': '## Pinch Singularities in Photon Propagator', 'hep-ph-9801399-1-61-0': 'To consider the pinching singularity appearing in a single self-energy insertion approximation to photon propagator we have to substitute [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-9801399-1-61-1': 'There are two thresholds but for massless photon relevant is only [MATH] and the domain where [MATH].', 'hep-ph-9801399-1-61-2': 'The integration limits are given by the same expression ([REF]), but now we have to integrate over the the domain [MATH] and [MATH].', 'hep-ph-9801399-1-61-3': 'As [MATH] we find [MATH] and [MATH].', 'hep-ph-9801399-1-61-4': 'The integration domain is stil infinite but it is shifted toward [MATH] where one expects that particle disstribution vanish.', 'hep-ph-9801399-1-61-5': '[EQUATION]', 'hep-ph-9801399-1-61-6': 'To calculate [MATH] for [MATH] loop we parametrize loop momentum [MATH] by introduction of an intermediary variable [MATH] perpendicular to [MATH].', 'hep-ph-9801399-1-61-7': '[MATH] is the mass of loop particles.', 'hep-ph-9801399-1-61-8': '[EQUATION]', 'hep-ph-9801399-1-61-9': 'At the end of the calculation we eliminate [MATH] in favour of [MATH].', 'hep-ph-9801399-1-61-10': 'After all possible singular denominators are cancelled, one can set [MATH].', 'hep-ph-9801399-1-61-11': '[EQUATION]', 'hep-ph-9801399-1-61-12': 'Using the relation ([REF]) we obtain [EQUATION]', 'hep-ph-9801399-1-61-13': 'In the integration over [MATH] the terms proportional to [MATH] dominate and [MATH] if [EQUATION] [MATH] is given by [EQUATION]', 'hep-ph-9801399-1-61-14': 'Asuming that the distributions obey inverse power low at large energies [MATH], [MATH] and [MATH] we find that the terms linear in densities dominate.', 'hep-ph-9801399-1-61-15': 'Thus one finds for [MATH] [EQUATION]', 'hep-ph-9801399-1-61-16': 'It follows that ([REF]) is finite (in fact it vanishes) if [MATH].', 'hep-ph-9801399-1-61-17': 'This is exactly the condition [EQUATION]', 'hep-ph-9801399-1-61-18': 'Thus the pinching singularity is cancelled in photon propagator under the condition that electron and positron distributions are such that total number of particles should be finite.', 'hep-ph-9801399-1-62-0': 'Also in the photon propagator, the quantity [MATH] does not depend on gauge parameter.', 'hep-ph-9801399-1-63-0': 'Expression ([REF]) is not valid for [MATH].', 'hep-ph-9801399-1-64-0': '# Pinch Singularities in Massless QCD', 'hep-ph-9801399-1-65-0': 'In this section we consider the case of massless QCD.', 'hep-ph-9801399-1-65-1': 'Pinching singularities, related to massive quarks, are eliminated by the methods of previous section.', 'hep-ph-9801399-1-66-0': 'In self-energy insertions related to gluon, quark and ghost propagator, masses in the loop as well as the masses of propagated particles are zero.', 'hep-ph-9801399-1-66-1': 'Thus the methods of previous section do not produce expected result.', 'hep-ph-9801399-1-66-2': 'Attention is turned to the spin degrees of freedom, i.e. to the function [MATH] of the integrand in ([REF]) to ([REF]).', 'hep-ph-9801399-1-66-3': 'In the calculation of [MATH] it is anticipated that loop particles have to be on mass shell.', 'hep-ph-9801399-1-66-4': 'In that case [MATH] provides an extra [MATH] factor in all the cases considered, in which not all particles are scalars.', 'hep-ph-9801399-1-66-5': 'This [MATH] factor suffices for the elimination of pinching singularities.', 'hep-ph-9801399-1-67-0': 'The integration limits are now [EQUATION]', 'hep-ph-9801399-1-67-1': 'The difference [MATH] is finite and there is no threshold effect.', 'hep-ph-9801399-1-68-0': 'It is worth observing that for [MATH] we have to integrate between [MATH] and [MATH], while for [MATH] the integration domain is [MATH] and [MATH].', 'hep-ph-9801399-1-68-1': 'This leads to two limits [MATH] in all cases of massless QCD.', 'hep-ph-9801399-1-69-0': 'By inspection of the final results ([REF]),([REF]) and ([REF]) we find that the case [MATH] requires integrability of the function [MATH] leading to the ([REF]) condition on quark, gluon and ghost distribution functions.', 'hep-ph-9801399-1-70-0': 'By using ([REF]), we again introduce the intermediary variable [MATH] perpendicular to [MATH]; now we have to set [MATH].', 'hep-ph-9801399-1-71-0': '## Self-Energy Insertions Contributing to Gluon Propagator', 'hep-ph-9801399-1-72-0': 'The [MATH] related to gluon propagator is the sum [EQUATION] where the terms in the sum are defined as [EQUATION]', 'hep-ph-9801399-1-72-1': 'Pinching singularities, related to massive quarks, are eliminated by the methods of previous section.', 'hep-ph-9801399-1-72-2': 'Tensor [MATH] related to the massless quark-antiquark contribution to the gluon self-energy is [EQUATION]', 'hep-ph-9801399-1-72-3': 'As [MATH] contains only [MATH] and [MATH] projectors, relation ([REF]) guarrantees that the result does not depend on gauge parameter.', 'hep-ph-9801399-1-73-0': 'Relation ([REF]) contains only terms proportional to [MATH], and the [MATH] is finite.', 'hep-ph-9801399-1-74-0': 'For the ghost-ghost contribution to the gluon self-energy tensor [MATH] is given by [EQUATION]', 'hep-ph-9801399-1-74-1': 'The antisymmetric part vanishes after integration, so we have left it out from the final result in ([REF]).', 'hep-ph-9801399-1-75-0': 'Tensor [MATH] for the gluon-gluon contribution to the gluon self-energy is [EQUATION]', 'hep-ph-9801399-1-75-1': 'Expressions ([REF]), ([REF]) and ([REF]) have been obtained by substitution of ([REF]), ([REF]) and ([REF]), and, finally by eliminating the variable [MATH] in favour of [MATH].', 'hep-ph-9801399-1-75-2': 'Tensor [MATH] is linear in [MATH], thus it vanishes after integration over [MATH].', 'hep-ph-9801399-1-76-0': 'We note here that, in the Feynman gauge ([MATH]), the operator [MATH] is absent from the gluon self-energy!', 'hep-ph-9801399-1-76-1': 'Consequently, the relation originating from Slavnov-Taylor identities (proved in [CITATION] for equilibrium densities) [MATH] is fulfilled at [MATH] only if [MATH].', 'hep-ph-9801399-1-76-2': 'Thus the contributions to [MATH] from the ghost-ghost and gluon-gluon self-energies mutually cancel, imposing restrictions on the densities related to unphysical degrees of freedom.', 'hep-ph-9801399-1-76-3': 'As it does not interfere with the cancellation of pinches, the problem of unphysical degrees of freedom will be discussed elsewhere .', 'hep-ph-9801399-1-77-0': 'Finally in all three cases we need ([REF]).', 'hep-ph-9801399-1-78-0': 'The above expressions ([REF]), ([REF]) and ([REF]) for the ghost-ghost, quark-antiquark and gluon-gluon contributions to the gluon self-energy contain only terms proportional to [MATH].', 'hep-ph-9801399-1-78-1': 'Thus the function [MATH] approaches finite value [MATH].', 'hep-ph-9801399-1-78-2': 'Thus we have shown that under the condition ([REF]) single self-energy contribution to gluon propagator is free from pinching.', 'hep-ph-9801399-1-79-0': '## Quark-Gluon Self-Energy Contribution to Quark Propagator', 'hep-ph-9801399-1-80-0': 'The [MATH] spinor for quark-gluon contribution to massless quark propagator is defined as [EQUATION]', 'hep-ph-9801399-1-80-1': 'In the self-energy of a massless quark coupled to a gluon the spin factor [MATH] is given by [EQUATION]', 'hep-ph-9801399-1-80-2': 'Relevant to our further discussion is the product (as [MATH] term will be integrated out we omit it) [EQUATION] which contains the damping factor [MATH].', 'hep-ph-9801399-1-81-0': 'By inserting ([REF]) into ([REF]) we obtain ([REF]) free from pinches.', 'hep-ph-9801399-1-82-0': 'To calculate [MATH] we need the limit [EQUATION]', 'hep-ph-9801399-1-82-1': 'From ([REF]) we conclude that [MATH] does not depend on gauge parameter.', 'hep-ph-9801399-1-82-2': 'This supports the conclusion of Niegawa[MATH] that the coefficient of the collinear singularity contribution from pinch-like term does not depend on gauge.', 'hep-ph-9801399-1-83-0': 'Ommiting the details, we observe that, also in the Coulomb gauge, the pinching is absent from quark propagator.', 'hep-ph-9801399-1-84-0': '## Ghost-Gluon Self-Energy Contribution to Ghost Propagator', 'hep-ph-9801399-1-85-0': 'The [MATH] factor is defined as [EQUATION]', 'hep-ph-9801399-1-85-1': 'The [MATH] factor for ghost-gluon contribution to the ghost self-energy is [EQUATION]', 'hep-ph-9801399-1-85-2': 'The factor [MATH] ensures the absence of pinch singularity and a well-defined perturbative result.', 'hep-ph-9801399-1-86-0': '## Scalar -Photon Self-Energy Contribution to Scalar Propagator', 'hep-ph-9801399-1-87-0': 'Although not part of massless QCD the massless scalar boson interacting with photon is treated by the same methods.', 'hep-ph-9801399-1-88-0': 'The [MATH] factor is defined as [EQUATION]', 'hep-ph-9801399-1-88-1': 'The [MATH] factor for massless scalar-photon contribution to the scalar self-energy [EQUATION] clearly exhibits the [MATH] damping factor!', 'hep-ph-9801399-1-89-0': '# Conclusion', 'hep-ph-9801399-1-90-0': 'Studying the out of equilibrium Schwinger-Dyson equation, we have found that ill-defined pinch-like expressions appear exclusively in the Keldysh component ([MATH]) of the resummed propagator ([REF]), or the single self-energy insertion approximation to it ([REF]).', 'hep-ph-9801399-1-90-1': 'This component does not vanish only in the expressions with the Keldysh component ([REF]) ([MATH]; or [MATH] for single self-energy approximation) of the self-energy matrix.', 'hep-ph-9801399-1-90-2': 'This then requires that loop particles be on mass shell.', 'hep-ph-9801399-1-90-3': 'This is the crucial point to eliminate pinch singularities.', 'hep-ph-9801399-1-91-0': 'We have identified two basic mechanisms for the elimination of pinching: threshold and spin effect.', 'hep-ph-9801399-1-92-0': 'For massive electron and massless photon (or quark and gluon) it is the threshold effect in the phase space integration which produces critical [MATH] or [MATH] dumping.', 'hep-ph-9801399-1-93-0': 'In the case of massless quark, ghost and gluon this mechanism fails, but spinor/tensor structure of the self-energy provides extra [MATH] dumping factor.', 'hep-ph-9801399-1-94-0': 'We have found that in QED the pinching singularities appearing in the single self-energy insertion approximation to the electron and the photon propagator are absent under very reasonable conditions: the distribution function should be finite, exceptionally photon distribution is allowed to diverge as [MATH] as [MATH]; the derivative of electron distribution should be finite; total density of electrons and photons should be finite.', 'hep-ph-9801399-1-95-0': 'For QCD identical conditions are imposed on the distribution of massive quarks and the distribution of gluon; distributions of massless quarks and ghosts (observe here that in the covariant gauge ghost distribution is not required to be identically zero) should be integrable functions; they are limited by the finitness of the total density.', 'hep-ph-9801399-1-96-0': 'In the preceeding sections we have, thus, shown that all pinch-like expressions appearing in QED and QCD (with massless and massive quarks!)', 'hep-ph-9801399-1-96-1': 'at the single self-energy insertion level do transform into well-defined expressions.', 'hep-ph-9801399-1-96-2': 'Many other theories behave in that way.', 'hep-ph-9801399-1-96-3': 'But there are important exceptions: all the theories in which the lowest order processes are kinematically allowed do not acquire well defined expressions at this level.', 'hep-ph-9801399-1-96-4': 'These are: electroweak interactions, processes involving Higgs and two light particles, [MATH] meson and two [MATH] mesons, [MATH], [MATH] and other heavy particles decaying into the pair of light particles.', 'hep-ph-9801399-1-96-5': 'etc.', 'hep-ph-9801399-1-96-6': 'Second important exception is massless [MATH] theory.', 'hep-ph-9801399-1-96-7': 'Here unlike in the massless QCD there are no spin factors to provide ([REF]).', 'hep-ph-9801399-1-96-8': 'In these cases, one has to resort to the resummed Schwinger-Dyson series.', 'hep-ph-9801399-1-97-0': 'The main result of the present paper is the cancellation of pinching singularities at the single self-energy insertion level in QED- and QCD-like theories, This then together with the reported[MATH] cancellation of collinear singularities allows the extraction of the useful physical information that the imaginary parts of the two-loop diagrams contain.', 'hep-ph-9801399-1-97-1': 'This is not the case for three-loop diagrams for some of them contain double self-energy insertions.', 'hep-ph-9801399-1-97-2': 'In that case one has, again, to resort to the sofistication of resummed propagators.'}

{'hep-ph-9801399-2-0-0': 'We analyze ill-defined pinch singularities characteristic of out of equilibrium thermal field theories.', 'hep-ph-9801399-2-0-1': 'We identify two mechanisms that eliminate pinching even at the single self-energy insertion approximation to the propagator: the first is based on the vanishing of phase space at the singular point (threshold effect).', 'hep-ph-9801399-2-0-2': 'It is effective in QED with a massive electron and a massless photon.', 'hep-ph-9801399-2-0-3': 'In massless QCD, this mechanism fails, but the pinches cancel owing to the second mechanism, i.e., owing to the spinor/tensor structure of the single self-energy insertion contribution to the propagator.', 'hep-ph-9801399-2-0-4': 'The constraints imposed on distribution functions are very reasonable.', 'hep-ph-9801399-2-1-0': '# Introduction', 'hep-ph-9801399-2-2-0': 'Out of equilibrium thermal field theories have recently attracted much interest.', 'hep-ph-9801399-2-2-1': 'From the experimental point of view, various aspects of heavy-ion collisions and the related hot QCD plasma are of considerable interest, in particular the supposedly gluon-dominated stage.', 'hep-ph-9801399-2-3-0': 'Contrary to the equilibrium case[MATH] where pinch, collinear, and infrared problems have been successfully controlled[MATH], out of equilibrium theory[MATH] has suffered from them to these days.', 'hep-ph-9801399-2-3-1': 'However, progress has been made in this field, too.', 'hep-ph-9801399-2-4-0': 'Weldon[MATH] has observed that the out of equilibrium pinch singularity does not cancel; hence it spoils analyticity and causality.', 'hep-ph-9801399-2-4-1': 'The problem gets worse with more than one self-energy insertions.', 'hep-ph-9801399-2-5-0': 'Bedaque has argued that in out of equilibrium theory the time extension should be finite.', 'hep-ph-9801399-2-5-1': 'Thus, the time integration limits from [MATH] to [MATH], which are responsible for the appearance of pinches, have to be abandoned as unphysical[MATH].', 'hep-ph-9801399-2-6-0': 'Le Bellac and Mabilat[MATH] have found that collinear singularities cancel in scalar theory, and in QCD using physical gauges, but not in the case of covariant gauges.', 'hep-ph-9801399-2-6-1': 'Niegawa[MATH] has found that the pinch-like term contains a divergent part that cancels collinear singularities in the covariant gauge.', 'hep-ph-9801399-2-7-0': 'In their discussion of the pinch-like term Le Bellac and Mabilat[MATH] try to avoid the problems with pinching singularity by substituting the bare retarded photon (gluon) propagator with the resummed Schwinger-Dyson series calculated in the HTL[MATH] approximation: [EQUATION] with the spectral function given by [MATH].', 'hep-ph-9801399-2-7-1': 'In the above expressions [MATH] is the thermal photon (gluon) mass given by [MATH] for the gluon; note that they assume a small deviation from equilibrium), and [MATH] .', 'hep-ph-9801399-2-7-2': 'In equilibrium, at high temperatures and low momenta, this substitution is necessary in order to obtain the results correct to the leading order in [MATH].', 'hep-ph-9801399-2-7-3': 'One expects that similar methods work also for a narrow class of particle distributions corresponding to "high temperatures" out of equilibrium.', 'hep-ph-9801399-2-7-4': 'However, their expression for pinch-like contribution differs from the one following from our general expression for the resummed Schwinger-Dyson series ([REF]).', 'hep-ph-9801399-2-7-5': 'At medium and large photon momenta the elimination of pinching by the use of the HTL approximated propagator is no more justified.', 'hep-ph-9801399-2-8-0': 'Altherr and Seibert have found that in massive [MATH] theory pinch singularity does not occur owing to the kinematical constraint[MATH].', 'hep-ph-9801399-2-8-1': 'This result is restricted to the case of one-loop self-energies.', 'hep-ph-9801399-2-9-0': 'Altherr has suggested a regularization method in which the propagator is modified by the width [MATH] which is an arbitrary function of momentum to be calculated in a self-consistent way.', 'hep-ph-9801399-2-9-1': 'In [MATH] theory, for small deviations from equilibrium, [MATH] was found to be just the usual equilibrium damping rate[MATH].', 'hep-ph-9801399-2-10-0': 'This recipe has been justified in the resummed Schwinger-Dyson series in various problems with pinching[MATH].', 'hep-ph-9801399-2-11-0': 'Baier, Dirks, and Redlich[MATH] have calculated the [MATH] self-energy contribution to the pion propagator, regulating pinch contributions by the damping rate.', 'hep-ph-9801399-2-11-1': 'In subsequent papers with Schiff[MATH] they have calculated the quark propagator within the HTL approximation[MATH]; in the resummed Schwinger-Dyson series, the pinch is naturally regulated by [MATH].', 'hep-ph-9801399-2-12-0': 'Carrington, Defu, and Thoma[MATH] have found that no pinch singularities appear in the HTL approximation to the resummed photon propagator .', 'hep-ph-9801399-2-13-0': 'Niegawa[MATH] has introduced the notion of renormalized particle-number density.', 'hep-ph-9801399-2-13-1': 'He has found that, in the appropriately redefined calculation scheme, the amplitudes and reaction rates are free from pinch singularities.', 'hep-ph-9801399-2-14-0': 'By pinching singularity we understand the contour passing between two infinitely close poles: [EQUATION] where [MATH].', 'hep-ph-9801399-2-14-1': 'It is controlled by some parameter, e.g., [MATH].', 'hep-ph-9801399-2-14-2': 'For finite [MATH], the expression is regular.', 'hep-ph-9801399-2-14-3': 'However, when [MATH] tends to zero, the integration path is "pinched" between the two poles, and the expression is ill-defined.', 'hep-ph-9801399-2-14-4': 'Integration gives an [MATH] contribution plus regular terms.', 'hep-ph-9801399-2-14-5': 'By performing a simple decomposition of [MATH] into [MATH], one obtains the related ill-defined [MATH] expression.', 'hep-ph-9801399-2-15-0': 'The following expression, which is similar to ([REF]), corresponds to the resummed Schwinger-Dyson series: [EQUATION] where [MATH] and [MATH] (which appears in ([REF])) are, respectively, proportional to [MATH] and [MATH], where [MATH], [MATH], and [MATH] are the components of the self-energy matrix to be defined in Sec. III.', 'hep-ph-9801399-2-16-0': 'In expression ([REF]), pinching is absent[MATH] if [MATH] at a value of [MATH] satisfying [MATH].', 'hep-ph-9801399-2-17-0': 'The expression corresponding to the single self-energy insertion approximation to the propagator is similar to ([REF]): [EQUATION]', 'hep-ph-9801399-2-17-1': 'One can rewrite the integral as [EQUATION]', 'hep-ph-9801399-2-17-2': 'If it happens that [EQUATION] then the integral ([REF]) decomposes into two pieces that, although possibly divergent, do not suffer from pinching.', 'hep-ph-9801399-2-18-0': 'There are two cases in which the function [MATH] is even identically zero in the vicinity of the [MATH] point: in thermal equilibrium, because of detailed balance relations; in massive [MATH] theory out of equilibrium, owing to the mass shell condition[MATH].', 'hep-ph-9801399-2-18-1': 'The latter mechanism also works in out of equilibrium QED if a small photon mass [MATH] is introduced.', 'hep-ph-9801399-2-18-2': 'However, this elimination of pinching can be misleading: the domain of [MATH], where [MATH], shrinks to a point as [MATH].', 'hep-ph-9801399-2-18-3': 'We shall show that the elimination of pinching also occurs in the [MATH] case.', 'hep-ph-9801399-2-19-0': 'In this paper we identify two mechanisms leading to relation ([REF]).', 'hep-ph-9801399-2-19-1': 'They are based on the observation that in the pinch-like contribution loop particles have to be on mass shell.', 'hep-ph-9801399-2-20-0': 'The first mechanism is effective in out of equilibrium QED: in the pinch-like contribution to the electron propagator, phase space vanishes linearly as [MATH] .', 'hep-ph-9801399-2-20-1': 'In the pinch-like contribution to the photon propagator, the domain of integration is shifted to infinity as [MATH].', 'hep-ph-9801399-2-20-2': 'For distributions disappearing rapidly enough at large energies, the contribution again vanishes linearly in the [MATH] limit.', 'hep-ph-9801399-2-20-3': 'This mechanism is also valid in QCD in the cases with massive quarks.', 'hep-ph-9801399-2-21-0': 'In out of equilibrium massless QCD, phase space does not vanish, but there is an alternative mechanism: the spinor/tensor structure in all cases leads to relation ([REF]).', 'hep-ph-9801399-2-22-0': 'Also, in out of equilibrium massless QCD, introduction of a small gluon mass does not help.', 'hep-ph-9801399-2-22-1': 'In this case, processes like [MATH] are kinematically allowed, the spinor/tensor structure is modified, and [MATH] does not vanish in the [MATH] limit.', 'hep-ph-9801399-2-23-0': 'In a few cases, none of the mentioned mechanisms works and one has to sum the Schwinger-Dyson series.', 'hep-ph-9801399-2-23-1': 'This is the case of the [MATH] loop in the [MATH] self-energy .', 'hep-ph-9801399-2-23-2': 'Even in the limit of zero pion mass, [MATH] vanishes only as [MATH] and relation ([REF]) is not fulfilled.', 'hep-ph-9801399-2-23-3': 'A similar problem appears in electroweak interactions involving decays of [MATH] and [MATH] bosons, decay of Higgs particles, etc.', 'hep-ph-9801399-2-23-4': 'Another important case is massless [MATH] theory.', 'hep-ph-9801399-2-23-5': 'In contrast to massless QCD, massless [MATH] theory contains no spin factor to provide a [MATH] factor necessary to obtain ([REF]).', 'hep-ph-9801399-2-24-0': 'The densities are restricted only mildly: they should be cut off at high energies, at least as [MATH], in order to obtain a finite total particle density; for nonzero [MATH], they should be finite; for [MATH] near zero, they should not diverge more rapidly than [MATH], the electron (positron) distribution should have a finite derivative.', 'hep-ph-9801399-2-24-1': 'Further restrictions may come from Slavnov-Taylor identities[MATH], but they are not crucial for our analysis.', 'hep-ph-9801399-2-25-0': 'When necessary we assume that the zero-temperature renormalization has already been performed.', 'hep-ph-9801399-2-25-1': 'The "finite temperature"[MATH] renormalization out of equilibrium[MATH] may give rise to new problems formally similar to those treated here.', 'hep-ph-9801399-2-25-2': 'Their treatment is beyond the scope of this paper.', 'hep-ph-9801399-2-26-0': 'The paper is organized as follows.', 'hep-ph-9801399-2-27-0': 'In Sec. II we analyze the Schwinger-Dyson equation in the Keldysh representation, solve it formally, and identify pinch-like expressions.', 'hep-ph-9801399-2-27-1': 'For one-loop self-energy insertions, we find that the Keldysh component ([MATH]) of the self-energy is responsible for pinches.', 'hep-ph-9801399-2-27-2': 'We further find that the nonzero Keldysh component requires loop particles to be on shell.', 'hep-ph-9801399-2-28-0': 'In Sec. III we analyze functions such as [MATH], [MATH], and [MATH], and investigate their threshold properties.', 'hep-ph-9801399-2-29-0': 'In Sec.IV we show that the electron and photon propagators, calculated in the single self-energy insertion approximation, are free from pinching.', 'hep-ph-9801399-2-30-0': 'In Sec. V we analyze pinch-like expressions in the [MATH], [MATH], and ghost-ghost contributions to the gluon propagator, the quark propagator and the ghost propagator in the single self-energy insertion approximation.', 'hep-ph-9801399-2-30-1': 'We find that, in all the cases, the spinor/tensor factor [MATH] contains a factor [MATH] that is sufficient to eliminate pinching.', 'hep-ph-9801399-2-31-0': 'In Sec. VI we briefly recollect the main results of the paper.', 'hep-ph-9801399-2-32-0': '# Propagators and the Schwinger-Dyson equation', 'hep-ph-9801399-2-33-0': 'We start[MATH] by defining out of equilibrium thermal propagators for bosons, in the case when we can ignore the variations of slow variables in Wigner functions[MATH]: [EQUATION]', 'hep-ph-9801399-2-33-1': 'For particles with additional degrees of freedom, relations ([REF])-([REF]) are provided with extra factors [MATH] for spin 1/2, [MATH] for vector particle, etc., and similarly for internal degrees of freedom.', 'hep-ph-9801399-2-33-2': 'To keep the discussion as general as possible, we show these factors explicitly only when necessary.', 'hep-ph-9801399-2-33-3': 'The propagator defined by relations ([REF])-([REF]) satisfies the important condition [EQUATION]', 'hep-ph-9801399-2-33-4': 'In the case of equilibrium, we have [EQUATION]', 'hep-ph-9801399-2-33-5': 'To obtain the corresponding relations for fermions, we only need to make the substitution [EQUATION]', 'hep-ph-9801399-2-33-6': 'In the case of equilibrium, for fermions we have [EQUATION]', 'hep-ph-9801399-2-33-7': 'Out of equilibrium, [MATH] and [MATH] will be some given functions of [MATH].', 'hep-ph-9801399-2-34-0': 'To transform into the Keldysh form, one defines the matrix Q as [EQUATION]', 'hep-ph-9801399-2-34-1': 'Now [EQUATION]', 'hep-ph-9801399-2-34-2': 'We need [MATH] expressed through [MATH] and [MATH]: [EQUATION]', 'hep-ph-9801399-2-34-3': 'Again for fermions, [MATH] is equal to [EQUATION]', 'hep-ph-9801399-2-34-4': 'The proper self-energy [EQUATION] satisfies the condition [EQUATION]', 'hep-ph-9801399-2-34-5': "It is also transformed into the Keldysh form (in Niemi's paper there is a misprint using [MATH] instead of [MATH]): [EQUATION]", 'hep-ph-9801399-2-34-6': 'We also find [EQUATION]', 'hep-ph-9801399-2-34-7': 'The "cutting rules" (refs. [CITATION], see also ref. [CITATION] for application of the rules out of equilibrium) will convince us that only on-shell loop-particle momenta contribute to [MATH] and [MATH].', 'hep-ph-9801399-2-35-0': 'The calculation of the [MATH] matrix gives (propagators [MATH] and [MATH] in the self-energy matrix and in the Schwinger-Dyson equation are also given by ([REF]) to ([REF]), with the spin indices suppressed to keep the discussion as general as possible): [EQUATION]', 'hep-ph-9801399-2-35-1': 'The Schwinger-Dyson equation [EQUATION] can be written in the Keldysh form as [EQUATION]', 'hep-ph-9801399-2-35-2': 'By expanding ([REF]), we deduce the contribution from the single self-energy insertion to be of the form [EQUATION] which is evidently well defined, and the Keldysh component suspected for pinching: [EQUATION]', 'hep-ph-9801399-2-35-3': 'It is easy to obtain a solution[MATH] for [MATH] and [MATH] using the form ([REF]).', 'hep-ph-9801399-2-35-4': 'One observes that the equations for [MATH] and [MATH] are simple and the solution is straightforward: [EQUATION]', 'hep-ph-9801399-2-35-5': 'To calculate [MATH], we can use the solution ([REF]) for [MATH] and [MATH]: [EQUATION]', 'hep-ph-9801399-2-35-6': 'Now we eliminate [MATH] with the help of ([REF]): [EQUATION]', 'hep-ph-9801399-2-35-7': 'The first term in ([REF]) is not always zero, but it does not contain pinching singularities!', 'hep-ph-9801399-2-35-8': 'The second term in ([REF]) is potentially ill-defined (or pinch-like).', 'hep-ph-9801399-2-35-9': 'The pinch-like contribution appears only in this equation; thus it is the key to the whole problem of pinch singularities.', 'hep-ph-9801399-2-35-10': 'In the one-loop approximation, it requires loop particles to be on mass shell.', 'hep-ph-9801399-2-35-11': 'This will be sufficient to remove ill-defined expressions in all studied cases.', 'hep-ph-9801399-2-36-0': 'Equation ([REF]) differs from the one used in Ref. [CITATION].', 'hep-ph-9801399-2-36-1': 'Indeed in Ref. [CITATION] Im[MATH] is used instead of [MATH].', 'hep-ph-9801399-2-37-0': 'We start with ([REF]).', 'hep-ph-9801399-2-37-1': 'After substituting ([REF]) into ([REF]), we obtain the regular term plus the pinch-like contribution: [EQUATION]', 'hep-ph-9801399-2-37-2': 'For equilibrium densities, we have [MATH] , and expression ([REF]) vanishes identically.', 'hep-ph-9801399-2-37-3': 'This is also true for fermions.', 'hep-ph-9801399-2-38-0': 'Expression ([REF]) is the only one suspected of pinch singularities at the single self-energy insertion level.', 'hep-ph-9801399-2-38-1': 'The function [MATH] in ([REF]) belongs to the type of functions characterized by the fact that both loop particles have to be on mass shell.', 'hep-ph-9801399-2-38-2': 'It is analyzed in detail in Secs. III and IV (for threshold effect) and in Sec. V (for spin effect).', 'hep-ph-9801399-2-38-3': 'With the help of this analysis we show that relation ([REF]) transforms into [EQUATION] where [MATH] is [MATH] multiplied by spinor/tensor factors included in the definition of [MATH].', 'hep-ph-9801399-2-38-4': 'The finiteness of the limit [EQUATION] is important for cancellation of pinches.', 'hep-ph-9801399-2-38-5': 'The index [MATH] indicates that the limiting value [MATH] is approached from either below or above, and these two values are generally different.', 'hep-ph-9801399-2-38-6': 'To isolate the potentially divergent terms, we express the function [MATH] in terms of functions that are symmetric ([MATH]) and antisymmetric ([MATH]) around the value [MATH]: [EQUATION]', 'hep-ph-9801399-2-38-7': 'These functions are given by [EQUATION]', 'hep-ph-9801399-2-38-8': 'Locally (around the value [MATH]), these functions are related to the limits [MATH] by [EQUATION]', 'hep-ph-9801399-2-38-9': 'As a consequence, the right-hand side of expression ([REF]) behaves locally as [EQUATION] and the term proportional to [MATH] is capable of producing logarithmic singularity.', 'hep-ph-9801399-2-39-0': 'Furthermore, we were unable to eliminate pinches related to the double, triple, etc., self-energy insertion contributions to the propagator.', 'hep-ph-9801399-2-40-0': '# Threshold factor', 'hep-ph-9801399-2-41-0': 'In this section we analyze the phase space of the loop integral with both loop particles on mass shell.', 'hep-ph-9801399-2-41-1': 'Special care is devoted to the behavior of this integral near thresholds.', 'hep-ph-9801399-2-41-2': 'In this analysis the densities are constrained only mildly: they are supposed to be finite and smooth, with a possible exception at zero energy.', 'hep-ph-9801399-2-41-3': 'We also assume that the total density of particles is finite.', 'hep-ph-9801399-2-41-4': 'The expressions are written for all particles being bosons, and spins are not specified; change to fermions is elementary.', 'hep-ph-9801399-2-42-0': 'To obtain the integrals over the products of [MATH] and [MATH], we start with a useful relation: [EQUATION] where [MATH], as a function of [MATH], is some polynomial of order 0, 1, 2, or 3.', 'hep-ph-9801399-2-43-0': 'Relation ([REF]) is obtained as an average of the results obtained by closing the integration path through the upper and through the lower semi-plane.', 'hep-ph-9801399-2-44-0': 'In the defining relations ([REF]) to ([REF]) the particle distribution [MATH] (and similarly for [MATH]) appears only in the expression where it is multiplied by [MATH].', 'hep-ph-9801399-2-44-1': 'Thus there is freedom to replace [MATH] by its on mass shell value [MATH], where [MATH].', 'hep-ph-9801399-2-44-2': 'The physical results should not be altered by this replacement.', 'hep-ph-9801399-2-44-3': 'Then we write [MATH], defined in ([REF]), as [MATH], and similarly for [MATH].', 'hep-ph-9801399-2-44-4': 'This substitution matters when one wants to perform integrals in ([REF]) to ([REF]) with the help of ([REF]).', 'hep-ph-9801399-2-44-5': 'With this replacement, [MATH] in ([REF]) does not depend on densities and ([REF]) can be proved as stated above, without need to discuss the analytic properties of [MATH] and [MATH].', 'hep-ph-9801399-2-44-6': 'Without our replacement, one would be immediately stack with the question how these quantities (i.e., [MATH] and [MATH]), supposed to be known, within some uncertainty, along the real axis, behave in the complex plane very far from the real axis!', 'hep-ph-9801399-2-45-0': 'Similar relations could be obtained for higher powers of [MATH] and [MATH].', 'hep-ph-9801399-2-45-1': 'For example, for the nth power of [MATH], the real part of the integral will be obtained by substituting [MATH] instead of [MATH].', 'hep-ph-9801399-2-45-2': 'Now we easily calculate [MATH] as [EQUATION] [MATH] is the factor dependent on spin and internal degrees of freedom.', 'hep-ph-9801399-2-45-3': 'The thermal part of [MATH] is given by: [EQUATION]', 'hep-ph-9801399-2-45-4': 'At equilibrium equation ([REF]) (after necessary boson[MATH]fermion conversion) agrees with the known results[MATH].', 'hep-ph-9801399-2-46-0': 'Now, starting from ([REF]) to ([REF]), we calculate [MATH] and [MATH].', 'hep-ph-9801399-2-46-1': '[EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-9801399-2-46-2': 'At equilibrium equations ([REF]) to ([REF]) agree (after setting F=-1 for scalar case) with the corresponding equations obtained for boson-boson intermediate state[MATH].', 'hep-ph-9801399-2-47-0': 'It is useful to define [MATH] as [EQUATION]', 'hep-ph-9801399-2-47-1': "After integrating over [MATH]'s, one obtains expressions of the general form [EQUATION] where [MATH], [EQUATION] [MATH] is the angle between vector [MATH] and [MATH] axes.", 'hep-ph-9801399-2-48-0': 'Let us start with the [MATH] case.', 'hep-ph-9801399-2-48-1': 'Solution of [MATH] gives the integration limits [EQUATION] or [EQUATION]', 'hep-ph-9801399-2-48-2': 'Assume now that [MATH].', 'hep-ph-9801399-2-48-3': 'In this case, at the threshold, the limits shrink to the value [EQUATION]', 'hep-ph-9801399-2-48-4': 'Near the threshold, it is convenient to replace the integration variable by [MATH].', 'hep-ph-9801399-2-48-5': 'Now, for [MATH] such small, that the integration limits [MATH] are both of the same sign as [MATH], we have [MATH].', 'hep-ph-9801399-2-49-0': 'We define the coefficient [MATH] by [EQUATION]', 'hep-ph-9801399-2-49-1': 'Now the expression ([REF]) can be approximated by [EQUATION]', 'hep-ph-9801399-2-49-2': 'Relation ([REF]) is the key to further discussion of the threshold effect.', 'hep-ph-9801399-2-50-0': 'We obtain this also for higher dimension (D=6, for example).', 'hep-ph-9801399-2-51-0': 'Relation ([REF]) put some limits on the behavior of density functions: they should not tend to infinity at any value of [MATH]; near [MATH], owing to the presence of the factor [MATH], they should not rise more rapidly than [MATH].', 'hep-ph-9801399-2-52-0': 'Owing to ([REF]) and ([REF]), the function [MATH] has the following properties important for cancellation of pinches.', 'hep-ph-9801399-2-53-0': 'It vanishes between the thresholds, i.e., the domain [MATH] is forbidden ([MATH]).', 'hep-ph-9801399-2-53-1': 'If it happens that the bare mass [MATH] belongs to this domain, the single self-energy insertion will be free of pinching.', 'hep-ph-9801399-2-53-2': 'In this case, multiple (double, triple, etc.) self-energy insertions will also be free of pinching.', 'hep-ph-9801399-2-53-3': 'Massive [MATH] theory[MATH] is a good example of this case.', 'hep-ph-9801399-2-54-0': 'It is (in principle) different from zero in the allowed domain [MATH] and [MATH].', 'hep-ph-9801399-2-54-1': 'In this case, one cannot get rid of pinching.', 'hep-ph-9801399-2-54-2': 'This situation appears in the [MATH] interaction[MATH].', 'hep-ph-9801399-2-54-3': 'An exception to this rule are occasional zeros owing to the specific form of densities.', 'hep-ph-9801399-2-55-0': 'The behavior at the boundaries (i.e., in the allowed region near the threshold) depends on the masses [MATH] and [MATH] and there are a few possibilities.', 'hep-ph-9801399-2-56-0': 'If both masses are nonzero and different ([MATH]), then there are two thresholds and [MATH] behaves as [MATH] in the allowed region near the threshold [MATH].', 'hep-ph-9801399-2-56-1': 'For [MATH], the power [MATH] is not large enough to suppress pinching.', 'hep-ph-9801399-2-57-0': 'If one of the masses is zero ([MATH] or [MATH]), then ([REF]) gives that the thresholds are identical (i.e., the forbidden domain shrinks to zero) and one obtains the [MATH] behavior near [MATH].', 'hep-ph-9801399-2-57-1': 'This case (for [MATH]) is promising.', 'hep-ph-9801399-2-57-2': 'The elimination of pinching in the electron propagator, considered in Sec.IV, is one of important examples.', 'hep-ph-9801399-2-58-0': 'If the masses are equal but different from zero ([MATH]), then there are two thresholds with different behavior.', 'hep-ph-9801399-2-58-1': 'The function [MATH] behaves as [MATH] in the allowed region near the threshold [MATH],and this behavior cannot eliminate pinching in the supposed case [MATH] .', 'hep-ph-9801399-2-59-0': 'However, at the other threshold, namely at [MATH], the physical region is determined by [MATH] and the above discussion does not apply.', 'hep-ph-9801399-2-59-1': 'In fact, the integration limits ([REF]) or ([REF]) are valid, but the region between [MATH] and [MATH] is now excluded from integration.', 'hep-ph-9801399-2-59-2': 'One has to integrate over the domain [MATH].', 'hep-ph-9801399-2-59-3': 'This leads to the limitation in the high-energy behavior of the density functions.', 'hep-ph-9801399-2-59-4': 'An important example of such behavior, elimination of pinching in the photon propagator ([MATH]), is discussed in Sec.IV.', 'hep-ph-9801399-2-60-0': 'If both masses vanish ([MATH]), the thresholds coincide, there is no forbidden region and no threshold behavior.', 'hep-ph-9801399-2-60-1': 'The behavior depends on the spin of the particles involved.', 'hep-ph-9801399-2-60-2': 'For scalars, the leading term in the expansion of [MATH] does not vanish.', 'hep-ph-9801399-2-60-3': 'Pinching is not eliminated.', 'hep-ph-9801399-2-61-0': 'The case of vanishing masses ([MATH]) for particles with spin exhibits a peculiar behavior.', 'hep-ph-9801399-2-61-1': 'In all studied examples (see Sec.V for details), [MATH] behaves as [MATH] as [MATH], which promises the elimination of pinching.', 'hep-ph-9801399-2-62-0': '# Pinch Singularities in QED', 'hep-ph-9801399-2-63-0': '## Pinch Singularities in the Electron Propagator', 'hep-ph-9801399-2-64-0': 'In this subsection we apply the results of preceding section to cancel the pinching singularity appearing in a single self-energy insertion approximation to the electron propagator.', 'hep-ph-9801399-2-64-1': 'To do so, we have to substitute [MATH], [MATH], [MATH], [MATH], and [MATH], where [MATH] and [MATH] are given non-equilibrium distributions of electrons and photons in relations ([REF]), ([REF]), ([REF]), and ([REF]).', 'hep-ph-9801399-2-64-2': 'The thresholds are now identical [EQUATION] and the integration limits are [EQUATION] or [EQUATION]', 'hep-ph-9801399-2-64-3': 'At threshold the limits shrink to the value [EQUATION]', 'hep-ph-9801399-2-64-4': 'Then, with the help of ([REF]), we define [EQUATION]', 'hep-ph-9801399-2-64-5': 'The trace factor [MATH] is calculated with loop particles on mass shell: [EQUATION]', 'hep-ph-9801399-2-64-6': 'In calculating the term proportional to [MATH], we have to use the trick [EQUATION]', 'hep-ph-9801399-2-64-7': 'For [MATH], we can decompose the vector [MATH] as [EQUATION] where, in the heat-bath frame with the [MATH] axis oriented along the vector [MATH], we have [EQUATION]', 'hep-ph-9801399-2-64-8': 'The transverse component of [MATH], [MATH] vanishes after integration over [MATH].', 'hep-ph-9801399-2-65-0': 'Finally, we obtain [EQUATION]', 'hep-ph-9801399-2-65-1': 'Now we can study the limit [EQUATION]', 'hep-ph-9801399-2-65-2': 'It is easy to find that [MATH] is finite provided that [MATH] and [MATH].', 'hep-ph-9801399-2-65-3': 'The last condition is easy to investigate using the limiting procedure: [EQUATION]', 'hep-ph-9801399-2-65-4': 'One should observe here that the integration limits imply that the limit [MATH] is taken from below for [MATH], and from above for [MATH].', 'hep-ph-9801399-2-65-5': 'The two limits lead to different values of [MATH].', 'hep-ph-9801399-2-65-6': 'Only the first term in ([REF]) can give rise to problems.', 'hep-ph-9801399-2-65-7': 'We rewrite it as [MATH].', 'hep-ph-9801399-2-65-8': 'As relation ([REF]) should be valid at any [MATH], we can integrate over [MATH] to find that the photon distribution should not grow more rapidly than [MATH] as [MATH] approaches zero, while the derivative of the electron distribution [MATH] should be finite at any [MATH]: [EQUATION]', 'hep-ph-9801399-2-65-9': 'Under the very reasonable conditions ([REF]) and ([REF]) the electron propagator is free from pinches.', 'hep-ph-9801399-2-66-0': 'It is interesting to observe the discontinuity of [MATH] at the point [MATH].', 'hep-ph-9801399-2-66-1': 'This feature will be repeated in massless QCD.', 'hep-ph-9801399-2-67-0': 'It is worth observing that [MATH] is gauge independent, at least within the class of covariant gauges.', 'hep-ph-9801399-2-68-0': '## Pinch Singularities in the Photon Propagator', 'hep-ph-9801399-2-69-0': 'To consider the pinching singularity appearing in a single self-energy insertion approximation to the photon propagator, we have to make the substitutions [MATH], [MATH], [MATH], and [MATH].', 'hep-ph-9801399-2-69-1': 'There are two thresholds, but only [MATH] and the domain where [MATH] are relevant to a massless photon.', 'hep-ph-9801399-2-69-2': 'The integration limits are given by the same expression ([REF]), but now we have to integrate over the domain [MATH].', 'hep-ph-9801399-2-69-3': 'As [MATH], we find [MATH] and [MATH].', 'hep-ph-9801399-2-69-4': 'The integration domain is still infinite but is shifted toward [MATH] where one expects that the particle distribution vanishes: [EQUATION]', 'hep-ph-9801399-2-69-5': 'To calculate [MATH] for the [MATH] loop, we parametrize the loop momentum [MATH] by introducing an intermediary variable [MATH] perpendicular to [MATH].', 'hep-ph-9801399-2-69-6': '[MATH] is the mass of loop particles: [EQUATION]', 'hep-ph-9801399-2-69-7': 'At the end of the calculation we eliminate [MATH] in favor of [MATH].', 'hep-ph-9801399-2-69-8': 'After all possible singular denominators are canceled, one can set [MATH].', 'hep-ph-9801399-2-69-9': '[EQUATION]', 'hep-ph-9801399-2-69-10': 'Using relation ([REF]) we obtain [EQUATION]', 'hep-ph-9801399-2-69-11': 'In the integration over [MATH] the terms proportional to [MATH] dominate and [MATH] if [EQUATION]', 'hep-ph-9801399-2-69-12': 'Here [MATH] is given by [EQUATION]', 'hep-ph-9801399-2-69-13': 'Assuming that the distributions obey the inverse-power law at large energies [MATH] and [MATH], we find that the terms linear in densities dominate.', 'hep-ph-9801399-2-69-14': 'Thus, for [MATH], one finds [EQUATION]', 'hep-ph-9801399-2-69-15': 'It follows that ([REF]) is finite (in fact, it vanishes) if [MATH].', 'hep-ph-9801399-2-69-16': 'Similar analysis for electron propagator at [MATH] (thus outside of our analysis of pinch singularities) leads to [MATH].', 'hep-ph-9801399-2-69-17': 'This is exactly the condition [EQUATION]', 'hep-ph-9801399-2-69-18': 'Thus the pinching singularity is canceled in the photon propagator under the condition that the electron and positron distributions should be such that the total number of particles is finite.', 'hep-ph-9801399-2-70-0': 'Also, in the photon propagator, the quantity [MATH] does not depend on the gauge parameter.', 'hep-ph-9801399-2-71-0': 'Expression ([REF]) is not valid for [MATH].', 'hep-ph-9801399-2-72-0': '# Pinch Singularities in Massless QCD', 'hep-ph-9801399-2-73-0': 'In this section we consider the case of massless QCD.', 'hep-ph-9801399-2-73-1': 'Pinching singularities, related to massive quarks, are eliminated by the methods used in the preceding section.', 'hep-ph-9801399-2-74-0': 'In self-energy insertions related to gluon, quark, and ghost propagators, the masses in the loop as well as the masses of the propagated particles are zero.', 'hep-ph-9801399-2-74-1': 'Thus, the methods of the preceding section do not produce the expected result.', 'hep-ph-9801399-2-74-2': 'Attention is turned to the spin degrees of freedom, i.e., to the function [MATH] of the integrand in ([REF]) to ([REF]).', 'hep-ph-9801399-2-74-3': 'In the calculation of [MATH] it has been anticipated that the loop particles have to be on mass shell.', 'hep-ph-9801399-2-74-4': 'In this case, [MATH] provides an extra [MATH] factor in all the cases considered, in which not all particles are scalars.', 'hep-ph-9801399-2-74-5': 'This [MATH] factor suffices for the elimination of pinching singularities.', 'hep-ph-9801399-2-75-0': 'The integration limits are now [EQUATION]', 'hep-ph-9801399-2-75-1': 'The difference [MATH] is finite and there is no threshold effect.', 'hep-ph-9801399-2-76-0': 'It is worth observing that for [MATH], we have to integrate between [MATH] and [MATH], whereas for [MATH], the integration domain is [MATH].', 'hep-ph-9801399-2-76-1': 'This leads to two limits, [MATH], in all cases of massless QCD.', 'hep-ph-9801399-2-77-0': 'By inspection of the final results ([REF]),([REF]), and ([REF]), we find that the case [MATH] requires integrability of the function [MATH] leading to the condition ([REF]) on the quark, gluon, and ghost distribution functions.', 'hep-ph-9801399-2-78-0': 'By using ([REF]), we again introduce the intermediary variable [MATH] perpendicular to [MATH]; now we have to set [MATH].', 'hep-ph-9801399-2-79-0': '## Self-Energy Insertions Contributing to the Gluon Propagator', 'hep-ph-9801399-2-80-0': 'The function [MATH] related to the gluon propagator is the sum [EQUATION] where the terms in the sum are defined as [EQUATION]', 'hep-ph-9801399-2-80-1': 'Pinching singularities, related to massive quarks, are eliminated by the methods used in the preceding section.', 'hep-ph-9801399-2-80-2': 'The tensor [MATH] related to the massless quark-antiquark contribution to the gluon self-energy is [EQUATION]', 'hep-ph-9801399-2-80-3': 'As [MATH] contains only [MATH] and [MATH] projectors, relation ([REF]) guarantees that the result does not depend on the gauge parameter.', 'hep-ph-9801399-2-81-0': 'Relation ([REF]) contains only terms proportional to [MATH], and [MATH] is finite.', 'hep-ph-9801399-2-82-0': 'For the ghost-ghost contribution to the gluon self-energy, the tensor [MATH] is given by [EQUATION]', 'hep-ph-9801399-2-82-1': 'The antisymmetric part vanishes after integration, so we have left it out from the final result in ([REF]).', 'hep-ph-9801399-2-83-0': 'The tensor [MATH] for the gluon-gluon contribution to the gluon self-energy is [EQUATION]', 'hep-ph-9801399-2-83-1': 'Expressions ([REF]), ([REF]), and ([REF]) have been obtained by substitution of ([REF]), ([REF]), and ([REF]) and, finally, by eliminating the variable [MATH] in favor of [MATH].', 'hep-ph-9801399-2-83-2': 'The tensor [MATH] is linear in [MATH], thus it vanishes after integration over [MATH].', 'hep-ph-9801399-2-84-0': 'We note here that, in the Feynman gauge ([MATH]), the operator [MATH] is absent from the gluon self-energy!', 'hep-ph-9801399-2-84-1': 'Consequently, the relation originating from Slavnov-Taylor identities (proved in [CITATION] for equilibrium densities), [MATH], is fulfilled at [MATH] only if [MATH].', 'hep-ph-9801399-2-84-2': 'Thus the contributions to [MATH] from the ghost-ghost and gluon-gluon self-energies mutually cancel, imposing restrictions on the densities related to unphysical degrees of freedom.', 'hep-ph-9801399-2-84-3': 'As it does not interfere with the cancellation of pinches, the problem of unphysical degrees of freedom will be discussed elsewhere.', 'hep-ph-9801399-2-85-0': 'Finally, we need ([REF]) in all three cases.', 'hep-ph-9801399-2-86-0': 'Expressions ([REF]), ([REF]), and ([REF]) for the ghost-ghost, quark-antiquark, and gluon-gluon contributions to the gluon self-energy contain only terms proportional to [MATH].', 'hep-ph-9801399-2-86-1': 'The function [MATH] approaches the finite value [MATH].', 'hep-ph-9801399-2-87-0': 'Thus we have shown that the single self-energy contribution to the gluon propagator is free from pinching under the condition ([REF]) .', 'hep-ph-9801399-2-88-0': '## Quark-Gluon Self-Energy Contribution to the Quark Propagator', 'hep-ph-9801399-2-89-0': 'The [MATH] spinor for the quark-gluon contribution to the massless quark propagator is defined as [EQUATION]', 'hep-ph-9801399-2-89-1': 'In the self-energy of a massless quark coupled to a gluon the spin factor [MATH] is given by [EQUATION]', 'hep-ph-9801399-2-89-2': 'For our further discussion, we need the product [EQUATION] which contains the damping factor [MATH].', 'hep-ph-9801399-2-89-3': 'The term [MATH] will be integrated out.', 'hep-ph-9801399-2-90-0': 'By inserting ([REF]) into ([REF]), we obtain ([REF]) free from pinches.', 'hep-ph-9801399-2-91-0': 'To calculate [MATH], we need the limit [EQUATION]', 'hep-ph-9801399-2-91-1': 'From ([REF]) we conclude that [MATH] does not depend on the gauge parameter.', 'hep-ph-9801399-2-92-0': 'Omitting details, we observe that pinching is absent from the quark propagator, also in the Coulomb gauge.', 'hep-ph-9801399-2-93-0': '## Ghost-Gluon Self-Energy Contribution to the Ghost Propagator', 'hep-ph-9801399-2-94-0': 'The [MATH] factor is defined as [EQUATION]', 'hep-ph-9801399-2-94-1': 'The [MATH] factor for the ghost-gluon contribution to the ghost self-energy is [EQUATION]', 'hep-ph-9801399-2-94-2': 'The factor [MATH] ensures the absence of pinch singularity and a well-defined perturbative result.', 'hep-ph-9801399-2-95-0': '## Scalar-Photon Self-Energy Contribution to the Scalar Propagator', 'hep-ph-9801399-2-96-0': 'Although the massless scalar boson interacting with a photon is not part of massless QCD , it is treated using the same methods.', 'hep-ph-9801399-2-97-0': 'The [MATH] factor is defined as [EQUATION]', 'hep-ph-9801399-2-97-1': 'The [MATH] factor for the massless scalar-photon contribution to the scalar self-energy, [EQUATION] clearly exhibits the [MATH] damping factor!', 'hep-ph-9801399-2-98-0': '# Conclusion', 'hep-ph-9801399-2-99-0': 'Studying the out of equilibrium Schwinger-Dyson equation, we have found that ill-defined pinch-like expressions appear exclusively in the Keldysh component ([MATH]) of the resummed propagator ([REF]), or in the single self-energy insertion approximation to it ([REF]).', 'hep-ph-9801399-2-99-1': 'This component does not vanish only in the expressions with the Keldysh component ([REF]) ([MATH] or [MATH] for the single self-energy approximation) of the self-energy matrix.', 'hep-ph-9801399-2-99-2': 'This then requires that loop particles be on mass shell.', 'hep-ph-9801399-2-99-3': 'This is the crucial point to eliminate pinch singularities.', 'hep-ph-9801399-2-100-0': 'We have identified two basic mechanisms for the elimination of pinching: the threshold and the spin effects.', 'hep-ph-9801399-2-101-0': 'For a massive electron and a massless photon (or quark and gluon) it is the threshold effect in the phase space integration that produces, respectively, the critical [MATH] or [MATH] damping factors.', 'hep-ph-9801399-2-102-0': 'In the case of a massless quark, ghost, and gluon, this mechanism fails, but the spinor/tensor structure of the self-energy provides an extra [MATH] damping factor.', 'hep-ph-9801399-2-103-0': 'We have found that, in QED, the pinching singularities appearing in the single self-energy insertion approximation to the electron and the photon propagators are absent under very reasonable conditions: the distribution function should be finite, exceptionally the photon distribution is allowed to diverge as [MATH] as [MATH]; the derivative of the electron distribution should be finite; the total density of electrons should be finite.', 'hep-ph-9801399-2-104-0': 'For QCD, identical conditions are imposed on the distribution of massive quarks and the distribution of gluons; the distributions of massless quarks and ghosts (observe here that in the covariant gauge, the ghost distribution is not required to be identically zero) should be integrable functions; they are limited by the finiteness of the total density.', 'hep-ph-9801399-2-105-0': 'In the preceding sections we have shown that all pinch-like expressions appearing in QED and QCD (with massless and massive quarks!)', 'hep-ph-9801399-2-105-1': 'at the single self-energy insertion level do transform into well-defined expressions.', 'hep-ph-9801399-2-105-2': 'Many other theories behave in such a way.', 'hep-ph-9801399-2-105-3': 'However, there are important exceptions: all theories in which lowest-order processes are kinematically allowed do not acquire well-defined expressions at this level.', 'hep-ph-9801399-2-105-4': 'These are electroweak interactions, processes involving Higgs and two light particles, a [MATH] meson and two [MATH] mesons, [MATH], [MATH], and other heavy particles decaying into a pair of light particles, etc.', 'hep-ph-9801399-2-105-5': 'The second important exception is massless [MATH] theory.', 'hep-ph-9801399-2-105-6': 'This theory, in contrast to massless QCD, contains no spin factors to provide ([REF]).', 'hep-ph-9801399-2-105-7': 'In these cases, one has to resort to the resummed Schwinger-Dyson series.', 'hep-ph-9801399-2-106-0': 'The main result of the present paper is the cancellation of pinching singularities at the single self-energy insertion level in QED- and QCD-like theories.', 'hep-ph-9801399-2-106-1': 'This, together with the reported[MATH] cancellation of collinear singularities, allows the extraction of useful physical information contained in the imaginary parts of the two-loop diagrams.', 'hep-ph-9801399-2-106-2': 'This is not the case with three-loop diagrams, because some of them contain double self-energy insertions.', 'hep-ph-9801399-2-106-3': 'In this case, one again has to resort to the sophistication of resummed propagators.'}

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['hep-ph-9801399-1-74-1', 'hep-ph-9801399-2-82-1'], ['hep-ph-9801399-1-14-2', 'hep-ph-9801399-2-14-2'], ['hep-ph-9801399-1-14-3', 'hep-ph-9801399-2-14-3'], ['hep-ph-9801399-1-51-1', 'hep-ph-9801399-2-59-1'], ['hep-ph-9801399-1-61-17', 'hep-ph-9801399-2-69-17'], ['hep-ph-9801399-1-46-1', 'hep-ph-9801399-2-54-3'], ['hep-ph-9801399-1-9-0', 'hep-ph-9801399-2-10-0'], ['hep-ph-9801399-1-85-0', 'hep-ph-9801399-2-94-0'], ['hep-ph-9801399-1-85-2', 'hep-ph-9801399-2-94-2'], ['hep-ph-9801399-1-76-0', 'hep-ph-9801399-2-84-0'], ['hep-ph-9801399-1-76-2', 'hep-ph-9801399-2-84-2'], ['hep-ph-9801399-1-58-0', 'hep-ph-9801399-2-66-0'], ['hep-ph-9801399-1-58-1', 'hep-ph-9801399-2-66-1'], ['hep-ph-9801399-1-90-2', 'hep-ph-9801399-2-99-2'], ['hep-ph-9801399-1-90-3', 'hep-ph-9801399-2-99-3'], ['hep-ph-9801399-1-18-1', 'hep-ph-9801399-2-17-1'], ['hep-ph-9801399-1-88-0', 'hep-ph-9801399-2-97-0'], ['hep-ph-9801399-1-56-2', 'hep-ph-9801399-2-64-2'], ['hep-ph-9801399-1-56-3', 'hep-ph-9801399-2-64-4'], ['hep-ph-9801399-1-33-3', 'hep-ph-9801399-2-33-3'], ['hep-ph-9801399-1-33-4', 'hep-ph-9801399-2-33-4'], ['hep-ph-9801399-1-33-6', 'hep-ph-9801399-2-33-6'], ['hep-ph-9801399-1-33-7', 'hep-ph-9801399-2-33-7'], ['hep-ph-9801399-1-4-1', 'hep-ph-9801399-2-4-1'], ['hep-ph-9801399-1-96-1', 'hep-ph-9801399-2-105-1'], ['hep-ph-9801399-1-96-8', 'hep-ph-9801399-2-105-7'], ['hep-ph-9801399-1-41-1', 'hep-ph-9801399-2-46-0'], ['hep-ph-9801399-1-41-3', 'hep-ph-9801399-2-47-0'], ['hep-ph-9801399-1-41-5', 'hep-ph-9801399-2-48-0'], ['hep-ph-9801399-1-41-7', 'hep-ph-9801399-2-48-2'], ['hep-ph-9801399-1-34-2', 'hep-ph-9801399-2-34-2'], ['hep-ph-9801399-1-34-3', 'hep-ph-9801399-2-34-3'], ['hep-ph-9801399-1-34-4', 'hep-ph-9801399-2-34-4'], ['hep-ph-9801399-1-34-6', 'hep-ph-9801399-2-34-6'], ['hep-ph-9801399-1-34-11', 'hep-ph-9801399-2-35-3'], ['hep-ph-9801399-1-34-12', 'hep-ph-9801399-2-35-4'], ['hep-ph-9801399-1-34-13', 'hep-ph-9801399-2-35-5'], ['hep-ph-9801399-1-34-14', 'hep-ph-9801399-2-35-6'], ['hep-ph-9801399-1-34-16', 'hep-ph-9801399-2-35-8'], ['hep-ph-9801399-1-34-18', 'hep-ph-9801399-2-35-10'], ['hep-ph-9801399-1-40-1', 'hep-ph-9801399-2-45-0'], ['hep-ph-9801399-1-69-0', 'hep-ph-9801399-2-77-0'], ['hep-ph-9801399-1-44-0', 'hep-ph-9801399-2-52-0'], ['hep-ph-9801399-1-97-1', 'hep-ph-9801399-2-106-2'], ['hep-ph-9801399-1-48-0', 'hep-ph-9801399-2-56-0'], ['hep-ph-9801399-1-93-0', 'hep-ph-9801399-2-102-0'], ['hep-ph-9801399-1-0-2', 'hep-ph-9801399-2-0-2'], ['hep-ph-9801399-1-0-3', 'hep-ph-9801399-2-0-3'], ['hep-ph-9801399-1-39-1', 'hep-ph-9801399-2-41-1'], ['hep-ph-9801399-1-39-2', 'hep-ph-9801399-2-41-2'], ['hep-ph-9801399-1-37-0', 'hep-ph-9801399-2-39-0'], ['hep-ph-9801399-1-23-1', 'hep-ph-9801399-2-22-1'], ['hep-ph-9801399-1-11-0', 'hep-ph-9801399-2-12-0'], ['hep-ph-9801399-1-66-0', 'hep-ph-9801399-2-74-0'], ['hep-ph-9801399-1-66-2', 'hep-ph-9801399-2-74-2'], ['hep-ph-9801399-1-66-3', 'hep-ph-9801399-2-74-3'], ['hep-ph-9801399-1-66-4', 'hep-ph-9801399-2-74-4'], ['hep-ph-9801399-1-82-0', 'hep-ph-9801399-2-91-0'], ['hep-ph-9801399-1-82-1', 'hep-ph-9801399-2-91-1'], ['hep-ph-9801399-1-47-0', 'hep-ph-9801399-2-55-0'], ['hep-ph-9801399-1-49-0', 'hep-ph-9801399-2-57-0'], ['hep-ph-9801399-1-57-3', 'hep-ph-9801399-2-65-3'], ['hep-ph-9801399-1-57-8', 'hep-ph-9801399-2-65-8'], ['hep-ph-9801399-1-57-9', 'hep-ph-9801399-2-65-9'], ['hep-ph-9801399-1-81-0', 'hep-ph-9801399-2-90-0'], ['hep-ph-9801399-1-68-0', 'hep-ph-9801399-2-76-0'], ['hep-ph-9801399-1-68-1', 'hep-ph-9801399-2-76-1'], ['hep-ph-9801399-1-87-0', 'hep-ph-9801399-2-96-0'], ['hep-ph-9801399-1-21-0', 'hep-ph-9801399-2-20-0'], ['hep-ph-9801399-1-21-1', 'hep-ph-9801399-2-20-1'], ['hep-ph-9801399-1-21-2', 'hep-ph-9801399-2-20-2'], ['hep-ph-9801399-1-52-0', 'hep-ph-9801399-2-60-0'], ['hep-ph-9801399-1-52-1', 'hep-ph-9801399-2-60-1'], ['hep-ph-9801399-1-52-2', 'hep-ph-9801399-2-60-2'], ['hep-ph-9801399-1-43-0', 'hep-ph-9801399-2-51-0'], ['hep-ph-9801399-1-65-1', 'hep-ph-9801399-2-73-1'], ['hep-ph-9801399-1-62-0', 'hep-ph-9801399-2-70-0'], ['hep-ph-9801399-1-22-0', 'hep-ph-9801399-2-21-0'], ['hep-ph-9801399-1-29-0', 'hep-ph-9801399-2-29-0'], ['hep-ph-9801399-1-35-1', 'hep-ph-9801399-2-37-1'], ['hep-ph-9801399-1-25-0', 'hep-ph-9801399-2-24-0'], ['hep-ph-9801399-1-36-0', 'hep-ph-9801399-2-38-0'], ['hep-ph-9801399-1-36-1', 'hep-ph-9801399-2-38-1'], ['hep-ph-9801399-1-36-3', 'hep-ph-9801399-2-38-3'], ['hep-ph-9801399-1-36-5', 'hep-ph-9801399-2-38-5'], ['hep-ph-9801399-1-36-6', 'hep-ph-9801399-2-38-9'], ['hep-ph-9801399-1-75-0', 'hep-ph-9801399-2-83-0'], ['hep-ph-9801399-1-75-1', 'hep-ph-9801399-2-83-1'], ['hep-ph-9801399-1-75-2', 'hep-ph-9801399-2-83-2'], ['hep-ph-9801399-1-80-0', 'hep-ph-9801399-2-89-0'], ['hep-ph-9801399-1-74-0', 'hep-ph-9801399-2-82-0'], ['hep-ph-9801399-1-72-0', 'hep-ph-9801399-2-80-0'], ['hep-ph-9801399-1-72-1', 'hep-ph-9801399-2-80-1'], ['hep-ph-9801399-1-72-2', 'hep-ph-9801399-2-80-2'], ['hep-ph-9801399-1-72-3', 'hep-ph-9801399-2-80-3'], ['hep-ph-9801399-1-14-0', 'hep-ph-9801399-2-14-0'], ['hep-ph-9801399-1-14-1', 'hep-ph-9801399-2-14-1'], ['hep-ph-9801399-1-91-0', 'hep-ph-9801399-2-100-0'], ['hep-ph-9801399-1-19-1', 'hep-ph-9801399-2-18-1'], ['hep-ph-9801399-1-19-3', 'hep-ph-9801399-2-18-3'], ['hep-ph-9801399-1-5-0', 'hep-ph-9801399-2-5-0'], ['hep-ph-9801399-1-5-1', 'hep-ph-9801399-2-5-1'], ['hep-ph-9801399-1-51-0', 'hep-ph-9801399-2-59-0'], ['hep-ph-9801399-1-51-2', 'hep-ph-9801399-2-59-2'], ['hep-ph-9801399-1-51-3', 'hep-ph-9801399-2-59-3'], ['hep-ph-9801399-1-20-0', 'hep-ph-9801399-2-19-0'], ['hep-ph-9801399-1-20-1', 'hep-ph-9801399-2-19-1'], ['hep-ph-9801399-1-61-0', 'hep-ph-9801399-2-69-0'], ['hep-ph-9801399-1-61-2', 'hep-ph-9801399-2-69-2'], ['hep-ph-9801399-1-61-6', 'hep-ph-9801399-2-69-5'], ['hep-ph-9801399-1-61-9', 'hep-ph-9801399-2-69-7'], ['hep-ph-9801399-1-61-10', 'hep-ph-9801399-2-69-8'], ['hep-ph-9801399-1-61-12', 'hep-ph-9801399-2-69-10'], ['hep-ph-9801399-1-61-13', 'hep-ph-9801399-2-69-11'], ['hep-ph-9801399-1-61-14', 'hep-ph-9801399-2-69-13'], ['hep-ph-9801399-1-61-16', 'hep-ph-9801399-2-69-15'], ['hep-ph-9801399-1-61-18', 'hep-ph-9801399-2-69-18'], ['hep-ph-9801399-1-46-0', 'hep-ph-9801399-2-54-0'], ['hep-ph-9801399-1-53-0', 'hep-ph-9801399-2-61-0'], ['hep-ph-9801399-1-6-1', 'hep-ph-9801399-2-6-0'], ['hep-ph-9801399-1-6-2', 'hep-ph-9801399-2-6-1'], ['hep-ph-9801399-1-10-0', 'hep-ph-9801399-2-11-0'], ['hep-ph-9801399-1-10-1', 'hep-ph-9801399-2-11-1'], ['hep-ph-9801399-1-50-0', 'hep-ph-9801399-2-58-0'], ['hep-ph-9801399-1-12-0', 'hep-ph-9801399-2-13-0'], ['hep-ph-9801399-1-45-0', 'hep-ph-9801399-2-53-0'], ['hep-ph-9801399-1-85-1', 'hep-ph-9801399-2-94-1'], ['hep-ph-9801399-1-28-0', 'hep-ph-9801399-2-28-0'], ['hep-ph-9801399-1-76-1', 'hep-ph-9801399-2-84-1'], ['hep-ph-9801399-1-76-3', 'hep-ph-9801399-2-84-3'], ['hep-ph-9801399-1-8-0', 'hep-ph-9801399-2-9-0'], ['hep-ph-9801399-1-8-1', 'hep-ph-9801399-2-9-1'], ['hep-ph-9801399-1-90-0', 'hep-ph-9801399-2-99-0'], 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['hep-ph-9801399-1-96-3', 'hep-ph-9801399-2-105-3'], ['hep-ph-9801399-1-96-4', 'hep-ph-9801399-2-105-4'], ['hep-ph-9801399-1-96-6', 'hep-ph-9801399-2-105-5'], ['hep-ph-9801399-1-73-0', 'hep-ph-9801399-2-81-0'], ['hep-ph-9801399-1-94-0', 'hep-ph-9801399-2-103-0'], ['hep-ph-9801399-1-41-6', 'hep-ph-9801399-2-48-1'], ['hep-ph-9801399-1-41-8', 'hep-ph-9801399-2-48-3'], ['hep-ph-9801399-1-34-0', 'hep-ph-9801399-2-34-0'], ['hep-ph-9801399-1-34-5', 'hep-ph-9801399-2-34-5'], ['hep-ph-9801399-1-34-7', 'hep-ph-9801399-2-35-0'], ['hep-ph-9801399-1-34-9', 'hep-ph-9801399-2-35-1'], ['hep-ph-9801399-1-34-10', 'hep-ph-9801399-2-35-2'], ['hep-ph-9801399-1-34-17', 'hep-ph-9801399-2-35-9'], ['hep-ph-9801399-1-34-19', 'hep-ph-9801399-2-35-11'], ['hep-ph-9801399-1-40-2', 'hep-ph-9801399-2-45-1'], ['hep-ph-9801399-1-40-3', 'hep-ph-9801399-2-45-2'], ['hep-ph-9801399-1-78-0', 'hep-ph-9801399-2-86-0'], ['hep-ph-9801399-1-78-1', 'hep-ph-9801399-2-86-1'], ['hep-ph-9801399-1-78-2', 'hep-ph-9801399-2-87-0'], ['hep-ph-9801399-1-24-0', 'hep-ph-9801399-2-23-0'], ['hep-ph-9801399-1-24-1', 'hep-ph-9801399-2-23-1'], ['hep-ph-9801399-1-24-2', 'hep-ph-9801399-2-23-2'], ['hep-ph-9801399-1-24-4', 'hep-ph-9801399-2-23-4']]

[]

[['hep-ph-9801399-1-97-0', 'hep-ph-9801399-2-106-0'], ['hep-ph-9801399-1-97-0', 'hep-ph-9801399-2-106-1'], ['hep-ph-9801399-1-97-2', 'hep-ph-9801399-2-106-3'], ['hep-ph-9801399-1-0-0', 'hep-ph-9801399-2-0-0'], ['hep-ph-9801399-1-0-1', 'hep-ph-9801399-2-0-1'], ['hep-ph-9801399-1-39-4', 'hep-ph-9801399-2-41-4'], ['hep-ph-9801399-1-23-0', 'hep-ph-9801399-2-22-0'], ['hep-ph-9801399-1-66-1', 'hep-ph-9801399-2-74-1'], ['hep-ph-9801399-1-57-0', 'hep-ph-9801399-2-65-0'], ['hep-ph-9801399-1-57-6', 'hep-ph-9801399-2-65-6'], ['hep-ph-9801399-1-21-3', 'hep-ph-9801399-2-20-3'], ['hep-ph-9801399-1-35-3', 'hep-ph-9801399-2-37-3'], ['hep-ph-9801399-1-2-1', 'hep-ph-9801399-2-2-1'], ['hep-ph-9801399-1-2-2', 'hep-ph-9801399-2-2-1'], ['hep-ph-9801399-1-36-2', 'hep-ph-9801399-2-38-2'], ['hep-ph-9801399-1-36-4', 'hep-ph-9801399-2-38-4'], ['hep-ph-9801399-1-80-2', 'hep-ph-9801399-2-89-2'], ['hep-ph-9801399-1-14-4', 'hep-ph-9801399-2-14-4'], ['hep-ph-9801399-1-19-0', 'hep-ph-9801399-2-18-0'], ['hep-ph-9801399-1-19-2', 'hep-ph-9801399-2-18-2'], ['hep-ph-9801399-1-61-1', 'hep-ph-9801399-2-69-1'], ['hep-ph-9801399-1-61-4', 'hep-ph-9801399-2-69-4'], ['hep-ph-9801399-1-61-7', 'hep-ph-9801399-2-69-6'], ['hep-ph-9801399-1-61-15', 'hep-ph-9801399-2-69-14'], ['hep-ph-9801399-1-92-0', 'hep-ph-9801399-2-101-0'], ['hep-ph-9801399-1-53-1', 'hep-ph-9801399-2-61-1'], ['hep-ph-9801399-1-50-1', 'hep-ph-9801399-2-58-1'], ['hep-ph-9801399-1-12-1', 'hep-ph-9801399-2-13-1'], ['hep-ph-9801399-1-18-0', 'hep-ph-9801399-2-17-0'], ['hep-ph-9801399-1-56-1', 'hep-ph-9801399-2-64-1'], ['hep-ph-9801399-1-56-5', 'hep-ph-9801399-2-64-6'], ['hep-ph-9801399-1-83-0', 'hep-ph-9801399-2-92-0'], ['hep-ph-9801399-1-31-0', 'hep-ph-9801399-2-31-0'], ['hep-ph-9801399-1-96-2', 'hep-ph-9801399-2-105-2'], ['hep-ph-9801399-1-96-7', 'hep-ph-9801399-2-105-6'], ['hep-ph-9801399-1-15-1', 'hep-ph-9801399-2-15-0'], ['hep-ph-9801399-1-41-4', 'hep-ph-9801399-2-47-1'], ['hep-ph-9801399-1-34-15', 'hep-ph-9801399-2-35-7'], ['hep-ph-9801399-1-40-0', 'hep-ph-9801399-2-42-0'], ['hep-ph-9801399-1-24-3', 'hep-ph-9801399-2-23-3'], ['hep-ph-9801399-1-24-5', 'hep-ph-9801399-2-23-5']]

[]

['hep-ph-9801399-1-26-0', 'hep-ph-9801399-1-34-1', 'hep-ph-9801399-1-41-2', 'hep-ph-9801399-1-42-0', 'hep-ph-9801399-1-61-5', 'hep-ph-9801399-1-61-8', 'hep-ph-9801399-1-61-11', 'hep-ph-9801399-1-63-0', 'hep-ph-9801399-1-77-0', 'hep-ph-9801399-1-96-5', 'hep-ph-9801399-2-16-0', 'hep-ph-9801399-2-26-0', 'hep-ph-9801399-2-34-1', 'hep-ph-9801399-2-46-1', 'hep-ph-9801399-2-50-0', 'hep-ph-9801399-2-69-3', 'hep-ph-9801399-2-69-9', 'hep-ph-9801399-2-71-0', 'hep-ph-9801399-2-85-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/9801399

1311.5156

{'1311.5156-1-0-0': 'The amplitude of double Higgs boson production by the gluon fusion, [MATH], is known to be small due to cancellation between the graphs with the boson trilinear coupling and those with the coupling to the top quark.', '1311.5156-1-0-1': 'For this reason a study of this process was suggested as a sensitive probe of the Higgs sector nonlinearity.', '1311.5156-1-0-2': 'We calculate in a closed analytical form this amplitude at the threshold of the two bosons, where the cancellation is the strongest, and discuss the origin of the small value of the amplitude.', '1311.5156-1-0-3': 'We also note that the cancellation in the double boson production is in fact a part of a more general phenomenon of suppression of similar threshold amplitudes for multiple boson production, which, although not directly relevant to the actual top quark and the Higgs boson, can be useful in other studies.', '1311.5156-1-1-0': 'With the observation [CITATION] of what is most likely the long anticipated Higgs boson of the Standard Model, a further study of the Higgs sector becomes a matter of practical feasibility.', '1311.5156-1-1-1': 'In particular the nonlinear terms in this sector, describing the interaction between the bosons, are most fundamentally related to the underlying framework of the Standard Model.', '1311.5156-1-1-2': 'Thus a test of the self interaction in the Higgs sector would certainly justify overcoming the experimental difficulties that such study inevitably entails.', '1311.5156-1-1-3': 'The specific process in which the Higgs trilinear coupling can be measured at a hadron collider is the double boson production by gluon fusion [CITATION]: [MATH].', '1311.5156-1-1-4': 'At the lowest loop level this process is contributed by two types of graphs shown in Fig. 1, the box diagram and the triangle diagram with the trilinear coupling between the bosons.', '1311.5156-1-1-5': 'It has been noticed some time ago [CITATION] that with the standard couplings the contributions of these two types of graphs exactly cancel in the limit where the mass [MATH] of the top quark is much larger than any kinematical invariant in the process, which also implies that [MATH] with [MATH] being the mass of the [MATH] boson.', '1311.5156-1-1-6': 'With the actual masses, [MATH]GeV and [MATH]GeV the cancellation is not complete, but still the cross section calculated [CITATION] with the gluon distribution functions at the LHC energies is greatly suppressed in comparison with what would be given by only one type of graphs in Fig. 1.', '1311.5156-1-1-7': 'This suppression of the double Higgs boson production by gluon fusion implies an enhanced relative importance of higher loop corrections [CITATION] and of any nonstandard couplings [CITATION], thus providing an advantageous opportunity for studying the latter effects.', '1311.5156-1-2-0': 'Furthermore, a study of the process [MATH] may include not only the measurement of the total cross section, but also of the distribution in the invariant [MATH] for this process, in particular near the threshold at [MATH], where the effect of the cancellation of the standard contributions is the strongest.', '1311.5156-1-2-1': 'It thus appears interesting to analyze in more detail the threshold limit of the amplitude for the the double boson production.', '1311.5156-1-2-2': 'It should be noted that a full calculation of both the triangle and the box graphs is available [CITATION] at an arbitrary kinematics in terms of the Passarino-Veltman integrals [CITATION], however any actual calculations are so far done by numerical routines, which also somewhat obscures the reasons for the suppression of the process.', '1311.5156-1-2-3': "In this paper we calculate the threshold amplitude in a closed analytical form for arbitrary ratio [MATH] and argue that the 'residual' cancellation between the box and triangle graphs at the actual values of [MATH] and [MATH] results from a combination of the exact cancellation in the limit [MATH], the analytical properties of the amplitude, and the zero of the major absorptive part of the amplitude at [MATH] (in addition to the trivial zeros at [MATH] and [MATH]), which can be traced to the property of 'nullification' [CITATION] i.e. of exact vanishing of the on-shell sum of the tree level threshold amplitudes for [MATH] at the special mass ratio [MATH] with integer [MATH] and [MATH].", '1311.5156-1-2-4': 'The cancellation in the one loop amplitude makes the process sensitive to higher loop corrections.', '1311.5156-1-2-5': 'In particular the top quark loop correction to the boson trilinear coupling [CITATION] produces a singular in the limit [MATH] contribution to the amplitude, whose numerical value almost equals that of the one loop term.', '1311.5156-1-2-6': 'In the concluding part we also illustrate that the cancellation between different graphs for the double Higgs boson production is in fact a part of more general phenomenon of a similar cancellation in the threshold amplitudes for multiple boson production.', '1311.5156-1-2-7': 'Although phenomenologically this behavior is not very significant for the actual top quark and the Higgs boson, it can prove to be relevant in other studies.', '1311.5156-1-3-0': 'The amplitude for the process [MATH] at the threshold is described by one form factor [MATH] and can be written in terms of the momenta [MATH] and the polarization (and color) amplitudes [MATH] of the gluons as [EQUATION] where [MATH] is the QCD coupling constant.', '1311.5156-1-3-1': 'In the limit [MATH] the field [MATH] can be replaced by a constant and the form factor [MATH] can be found [CITATION] by considering the top quark loop for the vacuum polarization with the mass [MATH] rescaled in the constant background: [MATH], where [MATH]GeV is the Higgs field vacuum expectation value.', '1311.5156-1-3-2': 'Proceeding in this way one finds [EQUATION]', '1311.5156-1-3-3': "Clearly, the quadratic in [MATH] term in the latter expansion corresponds to the contribution of the box graph, while the linear in [MATH] term describes the contribution of the triangle diagram with the subsequent 'self proliferation' of the bosons.", '1311.5156-1-3-4': 'With the Standard Model couplings one readily finds for the two bosons produced at the threshold [EQUATION] and verifies the exact cancellation in Eq. ([REF]) between the box and the triangle.', '1311.5156-1-4-0': 'For finite masses [MATH] and [MATH] the form factor can be written in terms of a dimensionless function [MATH] of the ratio [MATH] as [MATH].', '1311.5156-1-4-1': 'The contribution [MATH] of the triangle graph to the function [MATH] can be readily found by a simple adaptation of the analytical expression [CITATION] for the amplitude for the coupling of the Higgs boson to two photons (or gluons): [EQUATION] where the branch of the function [MATH] is defined in such a way that on the upper side of the cut at positive real [MATH] it reads as [EQUATION]', '1311.5156-1-4-2': 'The contribution [MATH] of the box type graphs to the function [MATH] can be found using its analytical and asymptotic properties.', '1311.5156-1-4-3': 'Indeed, the function [MATH] vanishes in the limit corresponding to zero top quark mass, [MATH], and also at [MATH] due to the discussed above low energy theorem.', '1311.5156-1-4-4': 'This function is real at real [MATH] in the interval [MATH] and has a right cut at positive [MATH] starting from [MATH] and a left cut at [MATH].', '1311.5156-1-4-5': 'This implies that the function [MATH] can be fully restored from its imaginary part on the cuts.', '1311.5156-1-4-6': "The imaginary part can be found from unitarity (or, equivalently, using the Cutkosky's cutting rules).", '1311.5156-1-4-7': 'The corresponding cuts for both the triangle and the box graphs are shown in Fig. 2.', '1311.5156-1-4-8': 'The triangle graph has only one cut which results together with the cuts of the box diagram of the type in Fig. 2b in the discontinuity of the function [MATH] starting at [MATH].', '1311.5156-1-4-9': 'The cuts of the type shown in Fig. 2c contribute to the discontinuity at positive [MATH] starting at [MATH], while the cuts of the type in Fig.2d give rise to the discontinuity in [MATH] at negative [MATH] such that [MATH].', '1311.5156-1-4-10': 'It should be noted that for the purpose of calculation of the box graphs the parameter [MATH] refers to the momentum transferred by a scalar source in the vertex in the graph, [MATH].', '1311.5156-1-4-11': "Thus setting this momentum space-like [MATH], corresponding to the 'physical region' for the cut of the type of Fig. 2d does not lead to any inconsistency.", '1311.5156-1-4-12': 'It is for this reason that the analytical continuation to negative [MATH] should be considered as to a negative [MATH] while preserving [MATH] positive, since the parameter [MATH] enters the diagrams as dynamical in the propagator of the quark.', '1311.5156-1-5-0': 'In what follows we denote the imaginary part of the function [MATH] resulting from the cuts of each type in Fig. 2 as respectively Im[MATH], Im[MATH], Im[MATH] and Im[MATH].', '1311.5156-1-5-1': "It is a simple exercise to verify (even before the integration over the phase space of [MATH]) that the expressions arising from the cuts in Fig. 2a and 2b are related: [EQUATION] which implies, given Eq.([REF]), that these two expressions combine in the total '[MATH]-channel' absorptive part Im[MATH] of [MATH], corresponding to the process [MATH] with on-shell quarks, having the form [EQUATION] where [MATH] stands for the step function.", '1311.5156-1-5-2': "One can notice that the expression in Eq.([REF]) has a nontrivial zero at [MATH] in a complete agreement with the 'nullification' property for the amplitudes describing the production of scalars at threshold by on-shell fermions [CITATION].", '1311.5156-1-6-0': 'For the imaginary part generated by the cuts of the type in Fig. 2c and Fig. 2d we find after a straightforward calculation [EQUATION] and [EQUATION]', '1311.5156-1-6-1': 'A strong similarity between the expressions in Eq. ([REF]) and ([REF]) is apparent, but the reason for it is not.', '1311.5156-1-7-0': 'The full absorptive part of the function [MATH] is given by the sum of the expressions ([REF]), ([REF]) and ([REF]).', '1311.5156-1-7-1': 'The discontinuity at the cuts and the condition that [MATH] goes to zero at [MATH] and that it is also vanishing at [MATH] is sufficient to restore the full expression for [MATH] (e.g. by using the dispersion relation).', '1311.5156-1-7-2': 'The result can be written in a closed analytical form as [EQUATION]', '1311.5156-1-7-3': 'Numerically, the actual masses of the top quark and the Higgs boson correspond to the value [MATH], where the expression ([REF]) gives [MATH].', '1311.5156-1-7-4': 'This value is more than ten times smaller and of the opposite sign compared with the contribution of the triangle graph alone: [MATH].', '1311.5156-1-7-5': 'Clearly, such significant cancellation implies that besides the vanishing of [MATH] at [MATH], the coefficients of the Taylor expansion for [MATH] are quite small: [EQUATION]', '1311.5156-1-7-6': "It can be noted that had one ignored the right and left 'far' cuts for [MATH] starting at [MATH] and restored this function from the absorptive part ([REF]) alone (but still using the condition of [MATH] vanishing at zero and infinity), the result would be the function [EQUATION] which gives a reasonably close approximation for [MATH] at [MATH], so that, in this sense, the cut starting at [MATH] gives the major contribution to the full result in this domain of [MATH].", '1311.5156-1-8-0': 'Using the above formulas one can also estimate, in the limit of small [MATH], the behavior of the amplitude slightly above the threshold, namely at the value of [MATH] for the two bosons such that [MATH], but still [MATH].', '1311.5156-1-8-1': 'Indeed, the box graph depends on the parameter [MATH] and in this limit can be taken at its threshold value.', '1311.5156-1-8-2': 'The only dependence on [MATH] then arises from the Higgs boson propagator in the diagram with triangle.', '1311.5156-1-8-3': 'The contribution of the triangle graph to the amplitude [MATH] is still described by one form factor [MATH] as in Eq. ([REF]) which is given by [EQUATION]', '1311.5156-1-8-4': 'The first (constant) term cancels against the box graph contribution, and one can see that the form factor deviates toward negative values above the threshold.', '1311.5156-1-8-5': 'This is the same negative sign as the threshold amplitude ([REF]) at small, but nonzero, values of [MATH].', '1311.5156-1-8-6': 'Therefore it can be concluded that the discussed cancellation between the box and triangle contributions is the strongest at the threshold.', '1311.5156-1-9-0': 'The small value of the leading Standard Model term for the amplitude of [MATH] makes it very sensitive, besides possible nonstandard effects, to higher order corrections.', '1311.5156-1-9-1': 'At small [MATH] the most important correction at the threshold arises from the modification of the Higgs trilinear coupling by a top quark loop as shown in Fig. 3.', '1311.5156-1-9-2': '(This is also the only next to leading order correction enhanced by the number of colors [MATH].)', '1311.5156-1-9-3': 'Indeed, in terms of the function [MATH] this correction, [MATH], is proportional to [MATH] and is thus singular in [MATH] at [MATH].', '1311.5156-1-9-4': 'The coefficient of the singularity is recently calculated [CITATION] by using the effective Higgs potential generated by the top quark loop, and the result reads as [EQUATION] which numerical value is only slightly smaller than the leading order result in Eq. ([REF]).', '1311.5156-1-10-0': 'Before concluding our discussion we would like to mention, as a theoretical side remark, that the cancellation at [MATH] between the one loop graphs at the threshold for the process [MATH] is not limited to double Higgs boson production, but also takes place at the thresholds for the processes of [MATH]-boson production, [MATH] with even [MATH].', '1311.5156-1-10-1': 'It should be mentioned that this behavior is relevant only when [MATH] and thus it appears to be not relevant for the actual Higgs boson and the top quark, but can still be useful in other studies.', '1311.5156-1-10-2': 'In order to establish this behavior one can use the technique of generating functions for calculating the amplitudes at multiboson thresholds [CITATION], which automatically takes into account all the tree-type sub-graphs generated by the self interaction in the Higgs sector.', '1311.5156-1-10-3': 'Within this approach one calculates the vacuum polarization top quark loop [MATH] in the classical background Higgs field depending on the Euclidean time [MATH] as [EQUATION] where [MATH], so that the field [MATH] is the familiar solution to the classical equations of motion: [MATH].', '1311.5156-1-10-4': 'Each threshold amplitude [MATH] is then found as the [MATH]-th derivative with respect to [MATH] of [MATH] at [MATH].', '1311.5156-1-10-5': 'In other words the latter is the generating function for all the amplitudes [MATH] as [EQUATION]', '1311.5156-1-10-6': "In the limit, when the ratio [MATH] is very small, one can consider the variation of the background field on the scale [MATH] as adiabatic and use the 'free' expression for the quark loop with a varying mass [MATH].", '1311.5156-1-10-7': 'The generating amplitude [MATH], as well as all the amplitudes [MATH], have the same one form factor structure as in Eq.([REF]), [EQUATION] and one can write for the generating form factor the expression [EQUATION] and, upon the Taylor expansion, the formula for the threshold form factors [MATH]: [EQUATION]', '1311.5156-1-10-8': 'Clearly, these form factors are vanishing at even [MATH] as a simple consequence of [MATH] in Eq.([REF]) being an odd function of [MATH].', '1311.5156-1-10-9': 'It is interesting to note that for odd [MATH], where the result in Eq.([REF]) is nonzero, there is still a certain cancellation between the (poligon) graphs taking place.', '1311.5156-1-10-10': 'Indeed, the contribution to [MATH] of the triangle graph alone can be evaluated similarly to Eq.([REF]): [EQUATION] where the production amplitude [MATH] can be found in Ref. [CITATION].', '1311.5156-1-10-11': 'The triangle contribution is thus larger than the full result ([REF]) by the factor [MATH]: [MATH].', '1311.5156-1-11-0': 'Lacking a full calculation of the amplitude [MATH] beyond the adiabatic in [MATH] approximation, the form factors [MATH] are not known at arbitrary [MATH].', '1311.5156-1-11-1': 'We thus can note here only a limited result regarding a generalization of Eq. ([REF]) to [MATH].', '1311.5156-1-11-2': 'Namely the imaginary part of [MATH] on the unitary cut at [MATH], associated with the process [MATH] with on-shell quarks, is uniquely determined by the zeros [CITATION] of the [MATH] amplitude as a function of [MATH] and the matrix element [MATH] and is given by [EQUATION] although it is not clear at present whether this cut dominates the behavior of the form factor [MATH] for a general [MATH] as it does for [MATH].', '1311.5156-1-12-0': 'In summary.', '1311.5156-1-12-1': 'We have derived the closed analytical expression in Eq. ([REF]) for the amplitude of [MATH] at the threshold of the two Higgs bosons, where the cancellation between the tringle and the box graphs of Fig. 1 is the strongest.', '1311.5156-1-12-2': "The reasons for this cancellation are traced to the vanishing of the amplitude as a function of the mass ratio [MATH] at both [MATH] and [MATH] and, to an extent, to the 'extra' zero of the imaginary part of the amplitude on the 'major' cut, related to the property of 'nullification' of the on-shell threshold amplitudes [MATH].", '1311.5156-1-12-3': 'The strong cancellation between the one loop contributions leads to that the main, in the limit [MATH], two loop correction is numerically comparable to the one loop result for the actual masses of the Higgs boson and the top quark.', '1311.5156-1-12-4': 'We have also illustrated that the cancellation in the double boson production amplitude is in fact a part of a more general phenomenon of suppression of multi boson production by two gluons at the corresponding thresholds.', '1311.5156-1-13-0': 'This work is supported, in part, by the DOE grant DE-FG02-94ER40823.'}

{'1311.5156-2-0-0': 'The amplitude of double Higgs boson production by the gluon fusion, [MATH], is known to be small due to cancellation between the graphs with the boson trilinear coupling and those with the coupling to the top quark.', '1311.5156-2-0-1': 'For this reason a study of this process was suggested as a sensitive probe of the Higgs sector nonlinearity.', '1311.5156-2-0-2': 'We calculate in a closed analytical form this amplitude at the threshold of the two bosons, where the cancellation is the strongest, and discuss the origin of the small value of the amplitude.', '1311.5156-2-0-3': 'We also note that the cancellation in the double boson production is in fact a part of a more general phenomenon of suppression of similar threshold amplitudes for multiple boson production, which, although not directly relevant to the actual top quark and the Higgs boson, can be useful in other studies.', '1311.5156-2-1-0': 'With the observation [CITATION] of what is most likely the long anticipated Higgs boson of the Standard Model, a further study of the Higgs sector becomes a matter of practical feasibility.', '1311.5156-2-1-1': 'In particular the nonlinear terms in this sector, describing the interaction between the bosons, are most fundamentally related to the underlying framework of the Standard Model.', '1311.5156-2-1-2': 'Thus a test of the self interaction in the Higgs sector would certainly justify overcoming the experimental difficulties that such study inevitably entails.', '1311.5156-2-1-3': 'The specific process in which the Higgs trilinear coupling can be measured at a hadron collider is the double boson production by gluon fusion [CITATION]: [MATH].', '1311.5156-2-1-4': 'At the lowest loop level this process is contributed by two types of graphs shown in Fig. 1, the box diagram and the triangle diagram with the trilinear coupling between the bosons.', '1311.5156-2-1-5': 'It has been noticed some time ago [CITATION] that with the standard couplings the contributions of these two types of graphs exactly cancel in the limit where the mass [MATH] of the top quark is much larger than any kinematical invariant in the process, which also implies that [MATH] with [MATH] being the mass of the [MATH] boson.', '1311.5156-2-1-6': 'With the actual masses, [MATH]GeV and [MATH]GeV the cancellation is not complete, but still the cross section calculated [CITATION] with the gluon distribution functions at the LHC energies is greatly suppressed in comparison with what would be given by only one type of graphs in Fig. 1.', '1311.5156-2-1-7': 'This suppression of the double Higgs boson production by gluon fusion implies an enhanced relative importance of higher loop corrections [CITATION] and of any nonstandard couplings [CITATION], thus providing an advantageous opportunity for studying the latter effects.', '1311.5156-2-2-0': 'Furthermore, a study of the process [MATH] may include not only the measurement of the total cross section, but also of the distribution in the invariant [MATH] for this process, in particular near the threshold at [MATH], where the effect of the cancellation of the standard contributions is the strongest.', '1311.5156-2-2-1': 'It thus appears interesting to analyze in more detail the threshold limit of the amplitude for the the double boson production.', '1311.5156-2-2-2': 'It should be noted that a full calculation of both the triangle and the box graphs is available [CITATION] at an arbitrary kinematics in terms of the Passarino-Veltman integrals [CITATION], however any actual calculations are so far done by numerical routines, which also somewhat obscures the reasons for the suppression of the process.', '1311.5156-2-2-3': "In this paper we calculate the threshold amplitude in a closed analytical form for arbitrary ratio [MATH] and argue that the 'residual' cancellation between the box and triangle graphs at the actual values of [MATH] and [MATH] results from a combination of the exact cancellation in the limit [MATH], the analytical properties of the amplitude, and the zero of the major absorptive part of the amplitude at [MATH] (in addition to the trivial zeros at [MATH] and [MATH]), which can be traced to the property of 'nullification' [CITATION] i.e. of exact vanishing of the on-shell sum of the tree level threshold amplitudes for [MATH] at the special mass ratio [MATH] with integer [MATH] and [MATH].", '1311.5156-2-2-4': 'The cancellation in the one loop amplitude makes the process sensitive to higher loop corrections.', '1311.5156-2-2-5': 'In particular the top quark loop correction to the boson trilinear coupling [CITATION] produces a singular in the limit [MATH] contribution to the amplitude, whose numerical value almost equals that of the one loop term.', '1311.5156-2-2-6': 'In the concluding part we also illustrate that the cancellation between different graphs for the double Higgs boson production is in fact a part of more general phenomenon of a similar cancellation in the threshold amplitudes for multiple boson production.', '1311.5156-2-2-7': 'Although phenomenologically this behavior is not very significant for the actual top quark and the Higgs boson, it can prove to be relevant in other studies.', '1311.5156-2-3-0': 'The amplitude for the process [MATH] at the threshold is described by one form factor [MATH] and can be written in terms of the momenta [MATH] and the polarization (and color) amplitudes [MATH] of the gluons as [EQUATION] where [MATH] is the QCD coupling constant.', '1311.5156-2-3-1': 'In the limit [MATH] the field [MATH] can be replaced by a constant and the form factor [MATH] can be found [CITATION] by considering the top quark loop for the vacuum polarization with the mass [MATH] rescaled in the constant background: [MATH], where [MATH]GeV is the Higgs field vacuum expectation value.', '1311.5156-2-3-2': 'Proceeding in this way one finds [EQUATION]', '1311.5156-2-3-3': "Clearly, the quadratic in [MATH] term in the latter expansion corresponds to the contribution of the box graph, while the linear in [MATH] term describes the contribution of the triangle diagram with the subsequent 'self proliferation' of the bosons.", '1311.5156-2-3-4': 'With the Standard Model couplings one readily finds for the two bosons produced at the threshold [EQUATION] and verifies the exact cancellation in Eq. ([REF]) between the box and the triangle.', '1311.5156-2-4-0': 'For finite masses [MATH] and [MATH] the form factor can be written in terms of a dimensionless function [MATH] of the ratio [MATH] as [MATH].', '1311.5156-2-4-1': 'The contribution [MATH] of the triangle graph to the function [MATH] can be readily found by a simple adaptation of the analytical expression [CITATION] for the amplitude for the coupling of the Higgs boson to two photons (or gluons): [EQUATION] where the branch of the function [MATH] is defined in such a way that on the upper side of the cut at positive real [MATH] it reads as [EQUATION]', '1311.5156-2-4-2': 'The contribution [MATH] of the box type graphs to the function [MATH] can be found using its analytical and asymptotic properties.', '1311.5156-2-4-3': 'Indeed, the function [MATH] vanishes in the limit corresponding to zero top quark mass, [MATH], and also at [MATH] due to the discussed above low energy theorem.', '1311.5156-2-4-4': 'This function is real at real [MATH] in the interval [MATH] and has a right cut at positive [MATH] starting from [MATH] and a left cut at [MATH].', '1311.5156-2-4-5': 'This implies that the function [MATH] can be fully restored from its imaginary part on the cuts.', '1311.5156-2-4-6': "The imaginary part can be found from unitarity (or, equivalently, using the Cutkosky's cutting rules).", '1311.5156-2-4-7': 'The corresponding cuts for both the triangle and the box graphs are shown in Fig. 2.', '1311.5156-2-4-8': 'The triangle graph has only one cut which results together with the cuts of the box diagram of the type in Fig. 2b in the discontinuity of the function [MATH] starting at [MATH].', '1311.5156-2-4-9': 'The cuts of the type shown in Fig. 2c contribute to the discontinuity at positive [MATH] starting at [MATH], while the cuts of the type in Fig.2d give rise to the discontinuity in [MATH] at negative [MATH] such that [MATH].', '1311.5156-2-4-10': 'It should be noted that for the purpose of calculation of the box graphs the parameter [MATH] refers to the momentum transferred by a scalar source in the vertex in the graph, [MATH].', '1311.5156-2-4-11': "Thus setting this momentum space-like [MATH], corresponding to the 'physical region' for the cut of the type of Fig. 2d does not lead to any inconsistency.", '1311.5156-2-4-12': 'It is for this reason that the analytical continuation to negative [MATH] should be considered as to a negative [MATH] while preserving [MATH] positive, since the parameter [MATH] enters the diagrams as dynamical in the propagator of the quark.', '1311.5156-2-5-0': 'In what follows we denote the imaginary part of the function [MATH] resulting from the cuts of each type in Fig. 2 as respectively Im[MATH], Im[MATH], Im[MATH] and Im[MATH].', '1311.5156-2-5-1': "It is a simple exercise to verify (even before the integration over the phase space of [MATH]) that the expressions arising from the cuts in Fig. 2a and 2b are related: [EQUATION] which implies, given Eq.([REF]), that these two expressions combine in the total '[MATH]-channel' absorptive part Im[MATH] of [MATH], corresponding to the process [MATH] with on-shell quarks, having the form [EQUATION] where [MATH] stands for the step function.", '1311.5156-2-5-2': "One can notice that the expression in Eq.([REF]) has a nontrivial zero at [MATH] in a complete agreement with the 'nullification' property for the amplitudes describing the production of scalars at threshold by on-shell fermions [CITATION].", '1311.5156-2-6-0': 'For the imaginary part generated by the cuts of the type in Fig. 2c and Fig. 2d we find after a straightforward calculation [EQUATION] and [EQUATION]', '1311.5156-2-6-1': 'A strong similarity between the expressions in Eq. ([REF]) and ([REF]) is apparent, but the reason for it is not.', '1311.5156-2-7-0': 'The full absorptive part of the function [MATH] is given by the sum of the expressions ([REF]), ([REF]) and ([REF]).', '1311.5156-2-7-1': 'The discontinuity at the cuts and the condition that [MATH] goes to zero at [MATH] and that it is also vanishing at [MATH] is sufficient to restore the full expression for [MATH] (e.g. by using the dispersion relation).', '1311.5156-2-7-2': 'The result can be written in a closed analytical form as [EQUATION]', '1311.5156-2-7-3': 'Numerically, the actual masses of the top quark and the Higgs boson correspond to the value [MATH], where the expression ([REF]) gives [MATH].', '1311.5156-2-7-4': 'This value is more than ten times smaller and of the opposite sign compared with the contribution of the triangle graph alone: [MATH].', '1311.5156-2-7-5': 'Clearly, such significant cancellation implies that besides the vanishing of [MATH] at [MATH], the coefficients of the Taylor expansion for [MATH] are quite small: [EQUATION]', '1311.5156-2-7-6': "It can be noted that had one ignored the right and left 'far' cuts for [MATH] starting at [MATH] and restored this function from the absorptive part ([REF]) alone (but still using the condition of [MATH] vanishing at zero and infinity), the result would be the function [EQUATION] which gives a reasonably close approximation for [MATH] at [MATH], so that, in this sense, the cut starting at [MATH] gives the major contribution to the full result in this domain of [MATH].", '1311.5156-2-8-0': 'Using the above formulas one can also estimate, in the limit of small [MATH], the behavior of the amplitude slightly above the threshold, namely at the value of [MATH] for the two bosons such that [MATH], but still [MATH].', '1311.5156-2-8-1': 'Indeed, the box graph depends on the parameter [MATH] and in this limit can be taken at its threshold value.', '1311.5156-2-8-2': 'The only dependence on [MATH] then arises from the Higgs boson propagator in the diagram with triangle.', '1311.5156-2-8-3': 'The contribution of the triangle graph to the amplitude [MATH] is still described by one form factor [MATH] as in Eq. ([REF]) which is given by [EQUATION]', '1311.5156-2-8-4': 'The first (constant) term cancels against the box graph contribution, and one can see that the form factor deviates toward negative values above the threshold.', '1311.5156-2-8-5': 'This is the same negative sign as the threshold amplitude ([REF]) at small, but nonzero, values of [MATH].', '1311.5156-2-8-6': 'Therefore it can be concluded that the discussed cancellation between the box and triangle contributions is the strongest at the threshold.', '1311.5156-2-9-0': 'The small value of the leading Standard Model term for the amplitude of [MATH] makes it very sensitive, besides possible nonstandard effects, to higher order corrections.', '1311.5156-2-9-1': 'At small [MATH] the most important correction at the threshold arises from the modification of the Higgs trilinear coupling by a top quark loop as shown in Fig. 3.', '1311.5156-2-9-2': '(This is also the only next to leading order correction enhanced by the number of colors [MATH].)', '1311.5156-2-9-3': 'Indeed, in terms of the function [MATH] this correction, [MATH], is proportional to [MATH] and is thus singular in [MATH] at [MATH].', '1311.5156-2-9-4': 'The coefficient of the singularity is recently calculated [CITATION] by using the effective Higgs potential generated by the top quark loop, and the result reads as [EQUATION] which numerical value is only slightly smaller than the leading order result in Eq. ([REF]).', '1311.5156-2-10-0': 'Before concluding our discussion we would like to mention, as a theoretical side remark, that the cancellation at [MATH] between the one loop graphs at the threshold for the process [MATH] is not limited to double Higgs boson production, but also takes place at the thresholds for the processes of [MATH]-boson production, [MATH] with even [MATH].', '1311.5156-2-10-1': 'It should be mentioned that this behavior is relevant only when [MATH] and thus it appears to be not relevant for the actual Higgs boson and the top quark, but can still be useful in other studies.', '1311.5156-2-10-2': 'In order to establish this behavior one can use the technique of generating functions for calculating the amplitudes at multiboson thresholds [CITATION], which automatically takes into account all the tree-type sub-graphs generated by the self interaction in the Higgs sector.', '1311.5156-2-10-3': 'Within this approach one calculates the vacuum polarization top quark loop [MATH] in the classical background Higgs field depending on the Euclidean time [MATH] as [EQUATION] where [MATH], so that the field [MATH] is the familiar solution to the classical equations of motion: [MATH].', '1311.5156-2-10-4': 'Each threshold amplitude [MATH] is then found as the [MATH]-th derivative with respect to [MATH] of [MATH] at [MATH].', '1311.5156-2-10-5': 'In other words the latter is the generating function for all the amplitudes [MATH] as [EQUATION]', '1311.5156-2-10-6': "In the limit, when the ratio [MATH] is very small, one can consider the variation of the background field on the scale [MATH] as adiabatic and use the 'free' expression for the quark loop with a varying mass [MATH].", '1311.5156-2-10-7': 'The generating amplitude [MATH], as well as all the amplitudes [MATH], have the same one form factor structure as in Eq.([REF]), [EQUATION] and one can write for the generating form factor the expression [EQUATION] and, upon the Taylor expansion, the formula for the threshold form factors [MATH]: [EQUATION]', '1311.5156-2-10-8': 'Clearly, these form factors are vanishing at even [MATH] as a simple consequence of [MATH] in Eq.([REF]) being an odd function of [MATH].', '1311.5156-2-10-9': 'It is interesting to note that for odd [MATH], where the result in Eq.([REF]) is nonzero, there is still a certain cancellation between the (poligon) graphs taking place.', '1311.5156-2-10-10': 'Indeed, the contribution to [MATH] of the triangle graph alone can be evaluated similarly to Eq.([REF]): [EQUATION] where the production amplitude [MATH] can be found in Ref. [CITATION].', '1311.5156-2-10-11': 'The triangle contribution is thus larger than the full result ([REF]) by the factor [MATH]: [MATH].', '1311.5156-2-11-0': 'Lacking a full calculation of the amplitude [MATH] beyond the adiabatic in [MATH] approximation, the form factors [MATH] are not known at arbitrary [MATH].', '1311.5156-2-11-1': 'We thus can note here only a limited result regarding a generalization of Eq. ([REF]) to [MATH].', '1311.5156-2-11-2': 'Namely the imaginary part of [MATH] on the unitary cut at [MATH], associated with the process [MATH] with on-shell quarks, is uniquely determined by the zeros [CITATION] of the [MATH] amplitude as a function of [MATH] and the matrix element [MATH] and is given by [EQUATION] although it is not clear at present whether this cut dominates the behavior of the form factor [MATH] for a general [MATH] as it does for [MATH].', '1311.5156-2-12-0': 'In summary.', '1311.5156-2-12-1': 'We have derived the closed analytical expression in Eq. ([REF]) for the amplitude of [MATH] at the threshold of the two Higgs bosons, where the cancellation between the tringle and the box graphs of Fig. 1 is the strongest.', '1311.5156-2-12-2': "The reasons for this cancellation are traced to the vanishing of the amplitude as a function of the mass ratio [MATH] at both [MATH] and [MATH] and, to an extent, to the 'extra' zero of the imaginary part of the amplitude on the 'major' cut, related to the property of 'nullification' of the on-shell threshold amplitudes [MATH].", '1311.5156-2-12-3': 'The strong cancellation between the one loop contributions leads to that the main, in the limit [MATH], two loop correction is numerically comparable to the one loop result for the actual masses of the Higgs boson and the top quark.', '1311.5156-2-12-4': 'We have also illustrated that the cancellation in the double boson production amplitude is in fact a part of a more general phenomenon of suppression of multi boson production by two gluons at the corresponding thresholds.', '1311.5156-2-13-0': 'This work is supported, in part, by the DOE grant DE-FG02-94ER40823.'}

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[]

[]

[]

[]

['1311.5156-1-12-0', '1311.5156-2-12-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1311.5156

cond-mat-0005497

{'cond-mat-0005497-1-0-0': 'Spin wave excitations were measured in the ferromagnetic phase of Nd[MATH]Sr[MATH]MnO[MATH] by neutron scattering.', 'cond-mat-0005497-1-0-1': 'This compound is located in proximity to the A-type antiferromagnetic state, and it shows a clear anisotropy and anomalous softening of the spin wave excitations.', 'cond-mat-0005497-1-0-2': 'The softening in the ferromagnetic phase is induced by the orbital ordering.', 'cond-mat-0005497-1-1-0': '.5 Keywords: Magnetoresistance - transition metals; Spin waves; Neutron scattering', 'cond-mat-0005497-1-2-0': 'height 0.4pt 1.5]', 'cond-mat-0005497-1-3-0': 'Recent experimental and theoretical studies revealed that the ordering of the Mn [MATH] orbitals plays a crucial role to determine physical properties in the perovskite manganites in addition to the well-known double exchange interactions.', 'cond-mat-0005497-1-3-1': 'Especially, the strong influence of the orbital ordering on the magnetism and the transport properties in the antiferromagnetic (AFM) state are widely recognized.', 'cond-mat-0005497-1-3-2': 'For example, the metallic A-type AFM state, in which ferromagnetic (FM) planes stack antiferromagnetically, was suggested to be attributed to the ordering of [MATH] orbitals of Mn ions in the FM planes.', 'cond-mat-0005497-1-3-3': 'The anisotropic interactions of the [MATH] orbitals introduce a noticeable two dimensional character in the spin fluctuations[CITATION].', 'cond-mat-0005497-1-4-0': 'On the other hand, the influence of the orbital ordering in the FM or paramagnetic (PM) state is not clear yet.', 'cond-mat-0005497-1-4-1': 'Accordingly, a neutron scattering studies on the FM and PM phase of Pr[MATH]Sr[MATH]MnO[MATH] and Nd[MATH]Sr[MATH]MnO[MATH], which are located in proximity to the A-type AFM state, were performed in order to examine the influence of the orbital ordering on the spin fluctuations.', 'cond-mat-0005497-1-4-2': 'The samples are single crystals, and the measurements were performed on the [MATH] scattering plane (in the [MATH] setting).', 'cond-mat-0005497-1-4-3': 'Because the space of this paper is limited, we will only present the results for Nd[MATH]Sr[MATH]MnO[MATH].', 'cond-mat-0005497-1-4-4': 'Similar results were obtained for Pr[MATH]Sr[MATH]MnO[MATH], which will be described elsewhere.', 'cond-mat-0005497-1-5-0': 'Nd[MATH]Sr[MATH]MnO[MATH] shows the FM transitions at [MATH] K. With decreasing temperature, the magnetic structure is switched to the CE-type AFM structure at [MATH] K as a first order transition [CITATION].', 'cond-mat-0005497-1-5-1': 'In addition, we found that the canted A-type AFM structure appears as a second order transition in the FM phase below [MATH] K. Its AFM propagation vector is (0,1,0) in reciprocal lattice unit.', 'cond-mat-0005497-1-5-2': 'In this paper, we will denote the direction within or between the FM planes as intraplane or interplane direction, respectively.', 'cond-mat-0005497-1-6-0': 'Figure [REF] (a) shows the spin wave dispersion curves which were measured from the zone center to the zone boundary in the FM phase.', 'cond-mat-0005497-1-6-1': 'Two interesting features were observed.', 'cond-mat-0005497-1-6-2': 'First, the spin wave excitations show strongly anisotropic behavior, which is similar to the A-type AFM phase[CITATION].', 'cond-mat-0005497-1-6-3': 'Secondly, the dispersion curves exhibit severe softening near the zone boundary.', 'cond-mat-0005497-1-6-4': 'Especially, the softening of the dispersion curve along the interplane direction is so strong that the dispersion resembles that of the A-type AFM state.', 'cond-mat-0005497-1-6-5': 'We think these results suggest the existence of the influence of the [MATH]-type orbital ordering in the FM phase.', 'cond-mat-0005497-1-6-6': 'In the following, we shall discuss relations between softening of the interplane spin waves and the orbital ordering in detail.', 'cond-mat-0005497-1-6-7': 'The discussion on the anisotropy of the spin waves has been reported elsewhere [CITATION].', 'cond-mat-0005497-1-7-0': 'In order to parametrize the softening phenomenogically, we have fitted the observed dispersions to the Heisenberg model for ferromagnets, although the localized spin model is too naive.', 'cond-mat-0005497-1-7-1': 'It is necessary to take the second nearest neighbor exchanges along the interplane direction into account to reproduce the softening of the interplane dispersion.', 'cond-mat-0005497-1-7-2': 'Therefore, we fitted the data by the following equation: [EQUATION] where [MATH] and [MATH] are the exchange integrals for the nearest neighbors and those for the second nearest neighbors along the interplane direction.', 'cond-mat-0005497-1-7-3': 'The results are shown in Fig. [REF] (b).', 'cond-mat-0005497-1-7-4': 'Alternatively, we have fitted the data to the Heisenberg model for antiferromagnet, because the interplane dispersions resemble the one for the A-type antiferromagnet.', 'cond-mat-0005497-1-7-5': 'Here, we adopted only the exchanges between the first nearest neighbors.', 'cond-mat-0005497-1-7-6': 'Then, the equation for the dispersion becomes [EQUATION] where [MATH].', 'cond-mat-0005497-1-7-7': 'The results of the fitting by Eq. ([REF]) is indicated in Fig. [REF] (c).', 'cond-mat-0005497-1-8-0': 'Although Eq. ([REF]) can reproduce the softening, it fails to describe the data for the small [MATH] region, and [MATH] becomes very small compared to [MATH]: [MATH] meV and [MATH] meV for 220 K, while [MATH] meV and [MATH] meV for 175 K.', 'cond-mat-0005497-1-8-1': 'In contrast, the observed data are rather well described by the AFM dispersion (Eq. [REF]) except for high-[MATH] region at 220 K.', 'cond-mat-0005497-1-8-2': 'In this case, however, [MATH] meV for 220 K and [MATH] meV for 175 K) in spite of the fact that the long range FM ordering is formed.', 'cond-mat-0005497-1-9-0': 'The exchange parameters obtained by these procedures are clearly anomalous, and it should be attributed to the inadequacy of the localized Heisenberg spin model.', 'cond-mat-0005497-1-9-1': 'It is clear that theory which takes into account the relevant microscopic mechanism has to be developed.', 'cond-mat-0005497-1-9-2': 'We think the fluctuations of the exchange energy induced by the [MATH]-type orbital correlation may be responsible for the softening of the interplane dispersion.', 'cond-mat-0005497-1-9-3': 'This scenario may also explain the temperature dependence of the interplane dispersion.', 'cond-mat-0005497-1-9-4': 'As shown in Fig. [REF], the interplane dispersion has a gap at the zone boundary above [MATH], which reflects that the orbital ordering is not fully stabilized yet.', 'cond-mat-0005497-1-9-5': 'With decreasing temperature, the orbital ordering becomes stable.', 'cond-mat-0005497-1-9-6': 'As a result, the energy gap decreases and eventually vanishes in the canted A-type AFM phase, resulting in a complete A-type AFM dispersion as shown in Fig. [REF] (c).', 'cond-mat-0005497-1-9-7': 'The transition to the A-type AFM states may be regarded as a transition by the softening of magnons, on the analogy of the structural transition by the softening of phonons.', 'cond-mat-0005497-1-9-8': 'We would like to note that the similar softening of the FM spin wave dispersion and its temperature dependence was reproduced by the recent theoretical calculations which adopted the orbital correlations [CITATION].', 'cond-mat-0005497-1-10-0': 'In summary, we found an anomalous softening of the magnon dispersions in the FM phase of Nd[MATH]Sr[MATH]MnO[MATH].', 'cond-mat-0005497-1-10-1': 'The softening is attributed to the influence of the orbital ordering in the FM phase, and the FM to A-type AFM transition is induced by the softening of the magnon.', 'cond-mat-0005497-1-11-0': 'This work was supported by a Grant-In-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan and by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.'}

{'cond-mat-0005497-2-0-0': 'Spin wave excitations were measured in the ferromagnetic phase of Nd[MATH]Sr[MATH]MnO[MATH] by neutron scattering.', 'cond-mat-0005497-2-0-1': 'This compound is located in proximity to the A-type antiferromagnetic state, and it shows a clear anisotropy and anomalous softening of the spin wave excitations.', 'cond-mat-0005497-2-0-2': 'The softening in the ferromagnetic phase is induced by the orbital ordering.', 'cond-mat-0005497-2-1-0': '.5 Keywords: Magnetoresistance - transition metals; Spin waves; Neutron scattering', 'cond-mat-0005497-2-2-0': 'height 0.4pt 1.5]', 'cond-mat-0005497-2-3-0': 'Recent experimental and theoretical studies revealed that the ordering of the Mn [MATH] orbitals plays a crucial role to determine physical properties in the perovskite manganites in addition to the well-known double exchange interactions.', 'cond-mat-0005497-2-3-1': 'Especially, the strong influence of the orbital ordering on the magnetism and the transport properties in the antiferromagnetic (AFM) state are widely recognized.', 'cond-mat-0005497-2-3-2': 'For example, the metallic A-type AFM state, in which ferromagnetic (FM) planes stack antiferromagnetically, was suggested to be attributed to the ordering of [MATH] orbitals of Mn ions in the FM planes.', 'cond-mat-0005497-2-3-3': 'The anisotropic interactions of the [MATH] orbitals introduce a noticeable two dimensional character in the spin fluctuations[CITATION].', 'cond-mat-0005497-2-4-0': 'On the other hand, the influence of the orbital ordering in the FM or paramagnetic (PM) state is not clear yet.', 'cond-mat-0005497-2-4-1': 'Accordingly, a neutron scattering studies on the FM and PM phase of Pr[MATH]Sr[MATH]MnO[MATH] and Nd[MATH]Sr[MATH]MnO[MATH], which are located in proximity to the A-type AFM state, were performed in order to examine the influence of the orbital ordering on the spin fluctuations.', 'cond-mat-0005497-2-4-2': 'The samples are single crystals, and the measurements were performed on the [MATH] scattering plane (in the [MATH] setting).', 'cond-mat-0005497-2-4-3': 'Because the space of this paper is limited, we will only present the results for Nd[MATH]Sr[MATH]MnO[MATH].', 'cond-mat-0005497-2-4-4': 'Similar results were obtained for Pr[MATH]Sr[MATH]MnO[MATH], which will be described elsewhere.', 'cond-mat-0005497-2-5-0': 'Nd[MATH]Sr[MATH]MnO[MATH] shows the FM transitions at [MATH] K. With decreasing temperature, the magnetic structure is switched to the CE-type AFM structure at [MATH] K as a first order transition [CITATION].', 'cond-mat-0005497-2-5-1': 'In addition, we found that the canted A-type AFM structure appears as a second order transition in the FM phase below [MATH] K. Its AFM propagation vector is (0,1,0) in reciprocal lattice unit.', 'cond-mat-0005497-2-5-2': 'In this paper, we will denote the direction within or between the FM planes as intraplane or interplane direction, respectively.', 'cond-mat-0005497-2-6-0': 'Figure [REF] (a) shows the spin wave dispersion curves which were measured from the zone center to the zone boundary in the FM phase.', 'cond-mat-0005497-2-6-1': 'Two interesting features were observed.', 'cond-mat-0005497-2-6-2': 'First, the spin wave excitations show strongly anisotropic behavior, which is similar to the A-type AFM phase[CITATION].', 'cond-mat-0005497-2-6-3': 'Secondly, the dispersion curves exhibit severe softening near the zone boundary.', 'cond-mat-0005497-2-6-4': 'Especially, the softening of the dispersion curve along the interplane direction is so strong that the dispersion resembles that of the A-type AFM state.', 'cond-mat-0005497-2-6-5': 'We think these results suggest the existence of the influence of the [MATH]-type orbital ordering in the FM phase.', 'cond-mat-0005497-2-6-6': 'In the following, we shall discuss relations between softening of the interplane spin waves and the orbital ordering in detail.', 'cond-mat-0005497-2-6-7': 'The discussion on the anisotropy of the spin waves has been reported elsewhere [CITATION].', 'cond-mat-0005497-2-7-0': 'In order to parametrize the softening phenomenogically, we have fitted the observed dispersions to the Heisenberg model for ferromagnets, although the localized spin model is too naive.', 'cond-mat-0005497-2-7-1': 'It is necessary to take the second nearest neighbor exchanges along the interplane direction into account to reproduce the softening of the interplane dispersion.', 'cond-mat-0005497-2-7-2': 'Therefore, we fitted the data by the following equation: [EQUATION] where [MATH] and [MATH] are the exchange integrals for the nearest neighbors and those for the second nearest neighbors along the interplane direction.', 'cond-mat-0005497-2-7-3': 'The results are shown in Fig. [REF] (b).', 'cond-mat-0005497-2-7-4': 'Alternatively, we have fitted the data to the Heisenberg model for antiferromagnet, because the interplane dispersions resemble the one for the A-type antiferromagnet.', 'cond-mat-0005497-2-7-5': 'Here, we adopted only the exchanges between the first nearest neighbors.', 'cond-mat-0005497-2-7-6': 'Then, the equation for the dispersion becomes [EQUATION] where [MATH].', 'cond-mat-0005497-2-7-7': 'The results of the fitting by Eq. ([REF]) is indicated in Fig. [REF] (c).', 'cond-mat-0005497-2-8-0': 'Although Eq. ([REF]) can reproduce the softening, it fails to describe the data for the small [MATH] region, and [MATH] becomes very small compared to [MATH]: [MATH] meV and [MATH] meV for 220 K, while [MATH] meV and [MATH] meV for 175 K.', 'cond-mat-0005497-2-8-1': 'In contrast, the observed data are rather well described by the AFM dispersion (Eq. [REF]) except for high-[MATH] region at 220 K.', 'cond-mat-0005497-2-8-2': 'In this case, however, [MATH] meV for 220 K and [MATH] meV for 175 K) in spite of the fact that the long range FM ordering is formed.', 'cond-mat-0005497-2-9-0': 'The exchange parameters obtained by these procedures are clearly anomalous, and it should be attributed to the inadequacy of the localized Heisenberg spin model.', 'cond-mat-0005497-2-9-1': 'It is clear that theory which takes into account the relevant microscopic mechanism has to be developed.', 'cond-mat-0005497-2-9-2': 'We think the fluctuations of the exchange energy induced by the [MATH]-type orbital correlation may be responsible for the softening of the interplane dispersion.', 'cond-mat-0005497-2-9-3': 'This scenario may also explain the temperature dependence of the interplane dispersion.', 'cond-mat-0005497-2-9-4': 'As shown in Fig. [REF], the interplane dispersion has a gap at the zone boundary above [MATH], which reflects that the orbital ordering is not fully stabilized yet.', 'cond-mat-0005497-2-9-5': 'With decreasing temperature, the orbital ordering becomes stable.', 'cond-mat-0005497-2-9-6': 'As a result, the energy gap decreases and eventually vanishes in the canted A-type AFM phase, resulting in a complete A-type AFM dispersion as shown in Fig. [REF] (c).', 'cond-mat-0005497-2-9-7': 'The transition to the A-type AFM states may be regarded as a transition by the softening of magnons, on the analogy of the structural transition by the softening of phonons.', 'cond-mat-0005497-2-9-8': 'We would like to note that the similar softening of the FM spin wave dispersion and its temperature dependence was reproduced by the recent theoretical calculations which adopted the orbital correlations [CITATION].', 'cond-mat-0005497-2-10-0': 'In summary, we found an anomalous softening of the magnon dispersions in the FM phase of Nd[MATH]Sr[MATH]MnO[MATH].', 'cond-mat-0005497-2-10-1': 'The softening is attributed to the influence of the orbital ordering in the FM phase, and the FM to A-type AFM transition is induced by the softening of the magnon.', 'cond-mat-0005497-2-11-0': 'This work was supported by a Grant-In-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan and by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.'}

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['cond-mat-0005497-1-1-0', 'cond-mat-0005497-1-2-0', 'cond-mat-0005497-2-1-0', 'cond-mat-0005497-2-2-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/cond-mat/0005497

0905.0858

{'0905.0858-1-0-0': 'We perform cosmological N-body simulations of the Dvali-Gabadadze-Porrati braneworld model, by solving the full non-linear equations of motion for the scalar degree of freedom in this model, the brane bending mode.', '0905.0858-1-0-1': 'While coupling universally to matter, the brane-bending mode has self-interactions that become important as soon as the density field becomes non-linear.', '0905.0858-1-0-2': 'These self-interactions lead to a suppression of the field in high-density environments, and restore gravity to General Relativity.', '0905.0858-1-0-3': 'The code uses a multi-grid relaxation scheme to solve the non-linear field equation in the quasi-static approximation.', '0905.0858-1-0-4': 'We perform simulations of a flat self-accelerating DGP model without cosmological constant.', '0905.0858-1-0-5': 'However, we expect our main results to apply qualitatively to a range of braneworld cosmologies.', '0905.0858-1-0-6': 'The results of the DGP simulations are compared with standard gravity simulations assuming the same expansion history, and with DGP simulations using the linearized equation for the brane bending mode.', '0905.0858-1-0-7': 'This allows us to isolate the effects of the non-linear self-couplings of the field which are noticeable already on quasi-linear scales.', '0905.0858-1-0-8': 'We present results on the matter power spectrum and the halo mass function, and discuss the behavior of the brane bending mode within cosmological structure formation.', '0905.0858-1-0-9': 'We find that, independently of CMB constraints, the self-accelerating DGP model is strongly constrained by current weak lensing and cluster abundance measurements.', '0905.0858-1-1-0': '# Introduction', '0905.0858-1-2-0': 'The observed acceleration of the universe [CITATION] poses a fascinating challenge to physicists and cosmologists: it points towards new physics at an unusually low energy scale ([MATH]eV), or large length scale ([MATH]Mpc).', '0905.0858-1-2-1': 'For this reason, no natural explanation appears to exist up to now.', '0905.0858-1-2-2': 'Many attempts have been made to go beyond the minimal explanation, a cosmological constant or vacuum energy.', '0905.0858-1-2-3': 'They can be broadly classified into two categories: acceleration is due to an additional, smooth stress-energy component with negative pressure (dark energy [CITATION]); or it is caused by gravity itself, through modifications to General Relativity (GR) on cosmological scales.', '0905.0858-1-2-4': 'While there are strong constraints on deviations from GR in the Solar System [CITATION], gravity is remarkably weakly constrained on cosmological length scales.', '0905.0858-1-2-5': 'This provides an independent motivation for studying modified gravity models in the context of cosmology.', '0905.0858-1-3-0': 'Smooth dark energy models have enough freedom to reproduce essentially any background expansion history of the universe.', '0905.0858-1-3-1': 'In order to distinguish modified gravity from the smooth dark energy scenario, it is thus necessary to study the growth of cosmological structure.', '0905.0858-1-3-2': 'A wealth of observables can be used for this purpose [CITATION], e.g. weak lensing, galaxy-CMB cross-correlation, galaxy cluster abundances, and many more.', '0905.0858-1-3-3': 'However, almost all of these measurements are affected by non-linearities in the matter density field, or have most of their information on non-linear scales.', '0905.0858-1-3-4': 'All viable modified gravity models include a non-linear mechanism to restore General Relativity in high-density environments, which is necessary in order to satisfy Solar System constraints.', '0905.0858-1-3-5': 'This mechanism has to be taken into account in order to make accurate predictions for observables on non-linear scales in modified gravity.', '0905.0858-1-3-6': 'For one representative of [MATH] modified gravity, [CITATION] showed that the effects of this non-linear chameleon mechanism are significant for models that satisfy Solar System constraints.', '0905.0858-1-4-0': 'One of the most popular modified gravity scenarios is the Dvali-Gabadadze-Porrati (DGP) model [CITATION].', '0905.0858-1-4-1': 'In this model, matter and radiation are confined to a 4-dimensional brane in a 5-dimensional bulk spacetime.', '0905.0858-1-4-2': 'While gravity propagates in five dimensions on the largest scales, it becomes 4-dimensional below a certain crossover scale [MATH].', '0905.0858-1-4-3': 'On scales smaller than [MATH], and when gravity is weak, DGP gravity can be described as an effective four-dimensional scalar-tensor theory [CITATION].', '0905.0858-1-4-4': 'The scalar field, referred to as brane bending mode, is massless but has non-linear derivative interactions which suppress the field in high-density environments, restoring GR locally and allowing DGP gravity to pass Solar System constraints.', '0905.0858-1-5-0': 'Applied to homogeneous and isotropic cosmology, the DGP model allows for two solutions [CITATION].', '0905.0858-1-5-1': 'In one branch of the theory, the self-accelerating branch, the effective weakening of gravity on large scales leads to an accelerated late-time expansion of the brane, without any cosmological constant.', '0905.0858-1-5-2': 'The scalar brane-bending mode mediates a repulsive force, leading to a smaller effective gravitational constant on large scales.', '0905.0858-1-5-3': 'In the other solution, the normal branch, there is no accleration, and the brane-bending mode mediates an attractive force.', '0905.0858-1-5-4': 'In linear perturbation theory around its de Sitter limit, the brane bending mode in the self-accelerating DGP model has been shown to have the wrong sign in the kinetic term ("ghost") [CITATION], suggesting an instability, although the situation in the non-linear case is not clear [CITATION].', '0905.0858-1-5-5': 'The normal branch is free of the ghost.', '0905.0858-1-6-0': 'Linear perturbation theory around a cosmological background in DGP has been derived in, e.g. [CITATION].', '0905.0858-1-6-1': 'In this approach, it is possible to predict the expansion history, CMB anisotropies, and the ISW effect in DGP [CITATION].', '0905.0858-1-6-2': 'Using this information, [CITATION] have shown that the self-accelerating DGP model without cosmological constant, which has only one free parameter, [MATH], is disfavored at the [MATH] level by current CMB and Supernova data.', '0905.0858-1-6-3': 'This is due to the earlier onset of acceleration in DGP, and the additional suppression of growth through the brane-bending mode.', '0905.0858-1-7-0': 'While the self-accelerating branch of DGP is thus challenged by theory and observations, much work is being done on extending the DGP model to higher co-dimensions, in the context of degravitation [CITATION].', '0905.0858-1-7-1': 'These higher-dimensional models are expected to bring the expansion history close to that of [MATH]CDM, while exhibiting a similar effective scalar-tensor regime as in the original DGP model.', '0905.0858-1-8-0': 'So far, studies of the DGP model in the context of cosmology have dealt with the linearized theory, valid on large scales.', '0905.0858-1-8-1': 'However, as soon as perturbations in the matter density become of order unity, the non-linear interactions of the brane bending mode become important (e.g., [CITATION]).', '0905.0858-1-8-2': 'Hence, in order to make predictions for observables in the non-linear regime, the full non-linear equations of motion must be solved in conjunction with the evolution of the matter perturbations.', '0905.0858-1-8-3': 'In this paper, we present the results of an N-body simulation of the self-accelerating DGP model, which self-consistently solves the non-linear equation for the brane-bending mode and its effect on the motion of particles.', '0905.0858-1-8-4': 'We solve the equation of motion in the quasi-static approximation, neglecting time derivatives with respect to spatial derivatives, as is usually done in N-body simulations (Section [REF] presents a consistency test of this approximation).', '0905.0858-1-9-0': 'We present results on the matter power spectrum, the halo mass function (abundance of dark matter halos), and the behavior of the brane bending mode in the cosmological context.', '0905.0858-1-9-1': 'These results can be used to strengthen constraints on the self-accelerating DGP model significantly, e.g., through observations of weak lensing and the abundance of galaxy clusters.', '0905.0858-1-9-2': 'More generally, the simulations presented here can serve as starting point for building a model of non-linear structure formation in braneworld scenarios, and for benchmarking perturbative approaches [CITATION].', '0905.0858-1-10-0': 'In Section [REF], we describe the DGP model, and present the relevant equations and analytical test cases.', '0905.0858-1-10-1': 'Section [REF] presents the code, which is benchmarked in Section [REF].', '0905.0858-1-10-2': 'The cosmological simulations are described in Section [REF].', '0905.0858-1-10-3': 'Results and their impact on cosmological constraints on the DGP model are presented in Section [REF] and Section [REF].', '0905.0858-1-10-4': 'We conclude in Section [REF].', '0905.0858-1-11-0': '# DGP model', '0905.0858-1-12-0': '## Background evolution', '0905.0858-1-13-0': 'The Dvali-Gabadadze-Porrati model [CITATION] consists of a spatially three-dimensional brane in a 4+1 dimensional Minkowski bulk.', '0905.0858-1-13-1': 'Matter and all interactions except gravity are confined to the brane.', '0905.0858-1-13-2': 'The gravitational action consists of the five-dimensional Einstein-Hilbert action plus a term which leads to the 4D gravity limit on small scales: [EQUATION]', '0905.0858-1-13-3': 'Here, [MATH], [MATH] stand for the bulk coordinates and metric, while [MATH], [MATH] are the induced coordinates and metric on the brane, and [MATH], [MATH] denote the corresponding Ricci scalars.', '0905.0858-1-13-4': 'In the following, we will drop the [MATH] notation where no confusion can arise.', '0905.0858-1-13-5': 'The boundary term is added to the action in order to ensure that variation with respect to [MATH] leads to the correct five-dimensional Einstein equations (e.g., [CITATION]).', '0905.0858-1-14-0': 'The two gravitational constants, or Planck masses [MATH], [MATH] appearing in Eq. ([REF]) can be related via a length scale, the crossover scale: [EQUATION]', '0905.0858-1-14-1': 'On scales above [MATH], gravity becomes five-dimensional, with forces falling off as [MATH].', '0905.0858-1-14-2': 'Below [MATH], gravity is four-dimensional, but not Einsteinian gravity, a point to which we return below.', '0905.0858-1-15-0': 'Since all matter is thought as confined to the brane, the five-dimensional metric has to obey non-trivial junction conditions over the brane [CITATION].', '0905.0858-1-15-1': 'Assuming an empty Minkowski bulk, a spatially flat brane, and a homogeneous and isotropic matter distribution on the brane, the junction conditions lead to the following analogue of the Friedmann equation for the scale factor of the induced metric on the brane: [EQUATION]', '0905.0858-1-15-2': 'The junction conditions leave two possible branches of the theory, determined by the sign on the left hand side of Eq. ([REF]).', '0905.0858-1-15-3': "In the following, we focus on the branch with the '[MATH]' sign, which asymptotes to a late-time de Sitter-Universe, [MATH], and is correspondingly called the accelerating branch.", '0905.0858-1-15-4': 'Then, in the matter-dominated epoch, neglecting radiation, and again assuming no cosmological constant or curvature, Eq. ([REF]) can be rewritten as: [EQUATION] and [MATH] is the average matter density today.', '0905.0858-1-15-5': 'This expansion history is clearly different from [MATH]CDM, and corresponds to an effective dark energy with [MATH] in the matter-dominated era at high redshifts.', '0905.0858-1-16-0': '## Cosmological perturbations and brane bending mode', '0905.0858-1-17-0': 'The propagation of light and particles on the DGP brane is completely determined by the perturbed 4D Friedmann-Robertson-Walker metric: [EQUATION]', '0905.0858-1-17-1': 'However, in order to determine the evolution of the metric potentials on the brane, it is necessary to solve the full 5D Einstein equations [CITATION].', '0905.0858-1-17-2': 'An additional scalar degree of freedom associated with local displacements of the brane appears, the so-called brane-bending mode [MATH] which couples to matter.', '0905.0858-1-17-3': 'In our convention, [MATH] is dimensionless, instead of being scaled to the Planck mass [MATH].', '0905.0858-1-18-0': 'In the decoupling limit of DGP [CITATION], when setting gravitational interactions to 0, the self-interactions of the [MATH] field remain constant.', '0905.0858-1-18-1': 'Hence, while perturbations of the metric higher than linear order can be neglected for cosmological studies, it is crucial to consider the self-interactions in the brane-bending mode.', '0905.0858-1-19-0': 'In the quasi-static regime, for scales [MATH], time derivatives can be neglected with respect to spatial derivatives, and the equation for the brane-bending mode reads (e.g., [CITATION]): [EQUATION] where the derivatives are with respect to comoving coordinates [MATH], [MATH] is the matter density perturbation, and the function [MATH] (for the accelerating branch) is given by: [EQUATION]', '0905.0858-1-19-1': 'Note that [MATH] is always negative in the self-accelerating branch.', '0905.0858-1-19-2': 'In Section [REF], we show results of a consistency test of the quasi-static assumption used in Eq. ([REF]).', '0905.0858-1-20-0': 'The brane-bending mode influences the dynamics of particles through the dynamical potential [MATH], which, assuming the same boundary conditions for [MATH] and [MATH], is given by: [EQUATION] where the Newtonian potential [MATH] satisfies the usual Poisson equation: [EQUATION]', '0905.0858-1-20-1': 'In contrast, the propagation of photons, determined by the lensing potential [MATH], is not directly affected by [MATH].', '0905.0858-1-21-0': '## Weak brane regime', '0905.0858-1-22-0': 'If the gradient of the [MATH] field is small, i.e. for small gravitational accelerations, Eq. ([REF]) can be linearized, yielding a standard Poisson equation: [EQUATION]', '0905.0858-1-22-1': 'We will refer to a cosmology with this linearized equation as the linearized DGP model.', '0905.0858-1-22-2': 'In this regime, also called weak-brane phase, [MATH] becomes proportional to the Newtonian potential [MATH], corresponding to a constant rescaling of the gravitational constant through Eq. ([REF]): [EQUATION]', '0905.0858-1-22-3': 'Substituting the linear solution into Eq. ([REF]), we obtain a rough estimate for the overdensity [MATH] at which the non-linear interactions become important: [EQUATION]', '0905.0858-1-22-4': 'For a self-accelerating DGP model that leads to cosmic acceleration today, [MATH], [MATH], and the prefactor in Eq. ([REF]) is of order unity today.', '0905.0858-1-22-5': 'Hence, the self-coupling of the brane bending mode [MATH] becomes important as soon as the matter density field becomes non-linear, [MATH].', '0905.0858-1-23-0': '## Vainshtein effect', '0905.0858-1-24-0': 'In general, Eq. ([REF]) is difficult to solve in full generality due to the non-linearity in the derivative terms.', '0905.0858-1-24-1': 'However, an instructive test case, a spherically symmetric matter distribution, can be solved analytically.', '0905.0858-1-24-2': 'In the spherically symmetric case, Eq. ([REF]) becomes [CITATION]: [EQUATION]', '0905.0858-1-24-3': 'For simplicity, we assume a spherical mass [MATH] of radius [MATH] with uniform density, and set [MATH].', '0905.0858-1-24-4': 'Then, we can integrate Eq. ([REF]) once and obtain the gravitational acceleration in DGP: [EQUATION] where [MATH] is the Newtonian acceleration of the spherical mass, and: [EQUATION]', '0905.0858-1-24-5': 'Here, [MATH] denotes a characteristic scale of the solution, the Vainshtein radius: [EQUATION] and [MATH] is the gravitational radius of the mass.', '0905.0858-1-24-6': 'For very large distances, [MATH], [MATH] approaches the constant [MATH], which exactly matches the linear solution, Eq. ([REF]).', '0905.0858-1-24-7': 'Note also that by substituting [MATH] in Eq. ([REF]), we see that the non-linearity criterion is directly proportional to [MATH].', '0905.0858-1-25-0': 'In the opposite limit, [MATH], [MATH] becomes suppressed with respect to the linear solution, and [MATH] approaches the small constant [MATH] inside the mass (assuming [MATH]).', '0905.0858-1-25-1': 'This Vainshtein effect [CITATION] amounts to restoring GR in deep potential wells, while far away from the mass, gravity is in the scalar-tensor regime.', '0905.0858-1-26-0': 'Fig. [REF] shows the analytical solution for the [MATH] field and the deviation from the Newtonian acceleration, as well as the code results which agree well with the analytical solution in the regime of validity (see Section [REF]).', '0905.0858-1-27-0': '## Plane wave solution', '0905.0858-1-28-0': 'Another exact solution to Eq. ([REF]) exists: for a simple plane wave density perturbation, [MATH].', '0905.0858-1-28-1': 'In this case, the two non-linear terms exactly cancel, and we are left with the linear solution, Eq. ([REF]).', '0905.0858-1-28-2': 'In Appendix [REF] this is used to test our code for perturbations of different wavelengths.', '0905.0858-1-28-3': 'Also, the plane wave and spherically symmetric solutions can be considered as two limiting cases for understanding the [MATH] field behavior in the cosmological context.', '0905.0858-1-28-4': 'We will return to this point in Section [REF].', '0905.0858-1-29-0': '# Code implementation', '0905.0858-1-30-0': 'We use an extension of the code first described in [CITATION] for our simulations of DGP gravity.', '0905.0858-1-30-1': 'The code is a standard particle-mesh N-body code [CITATION] assuming collisionless dark matter only, with a fixed grid of size [MATH] and periodic boundary conditions, and uses second-order accurate leapfrog integration for the particle propagation.', '0905.0858-1-30-2': 'The major addition is a relaxation solver for the non-linear equation of the brane bending mode [MATH], Eq. ([REF]).', '0905.0858-1-30-3': 'The simulation proceeds by performing steps [MATH] in the scale factor [MATH].', '0905.0858-1-31-0': 'Except for the different expansion history, the equations for particle propagation are identical to those in standard CDM simulations, and are expressed in terms of comoving coordinates [MATH], [MATH]: [EQUATION]', '0905.0858-1-31-1': 'Here, [MATH] and [MATH] are the position and momentum of particle [MATH] in code units (see Appendix [REF]), and [MATH] is taken from Eq. ([REF]).', '0905.0858-1-31-2': 'The main modification enters in the determination of [MATH].', '0905.0858-1-31-3': 'One time step of the N-body simulation proceeds as follows:', '0905.0858-1-32-0': 'Thus, in each time step we have to solve for the brane-bending mode [MATH] which contributes to the dynamical potential [MATH] via Eq. ([REF]).', '0905.0858-1-32-1': 'In other words, given the density field, we have to solve the non-linear elliptical differential equation Eq. ([REF]).', '0905.0858-1-32-2': 'We use a Gauss-Seidel relaxation scheme together with the Newton method.', '0905.0858-1-32-3': 'A crucial tool is the multigrid [CITATION], a hierarchy of coarser grids which are essential to speeding up the convergence [CITATION]: as the relaxation scheme, which operates locally, gets rid of small-scale errors very efficiently, the coarser grids quickly reduce the long-wavelength error modes which are hard to eliminate on the fine grid alone.', '0905.0858-1-32-4': 'For details of the implementation, see Appendix [REF].', '0905.0858-1-33-0': 'While the [MATH] field is in general well-behaved, the convergence properties become worse for strongly inhomogeneous density fields, because the non-linearities in Eq. ([REF]) are in the derivatives of the [MATH] field.', '0905.0858-1-33-1': 'In order to reach the desired convergence in our cosmological simulations, it is necessary to smooth the density field entering the r.h.s. of Eq. ([REF]) with a Gaussian kernel [MATH], with [MATH] set to the size of a grid cell.', '0905.0858-1-33-2': 'This smoothes over the noise in the density field due to the discreteness of particles.', '0905.0858-1-34-0': '# Code tests', '0905.0858-1-35-0': 'In this section, we present different tests our code was put through to benchmark its performance and limitations.', '0905.0858-1-35-1': 'We focus on tests applying to the modified gravity sector.', '0905.0858-1-35-2': 'For the results of standard N-body code tests for this code, see [CITATION].', '0905.0858-1-36-0': '## Spherical mass test', '0905.0858-1-37-0': 'In order to study how well the code reproduces the exact result for a spherical mass in DGP, we start out with a grid with [MATH] grid cells and [MATH] particles.', '0905.0858-1-37-1': 'We arbitrarily assume a box size of [MATH] and set [MATH].', '0905.0858-1-37-2': 'All particles are moved into a spherical mass of radius [MATH] and uniform density, which then corresponds to a mass of [MATH] (assuming [MATH]).', '0905.0858-1-37-3': 'Given the density field assigned from the particle positions, we then use the relaxation solver to solve for [MATH], and measure the field values and radial acceleration throughout the box.', '0905.0858-1-38-0': 'For the purposes of this test, we can vary the two parameters [MATH] and [MATH] in Eq. ([REF]) independently.', '0905.0858-1-38-1': '[MATH] determines the strength of the additional force mediated by [MATH] [e.g., Eq. ([REF])], which is attractive for positive [MATH] and repulsive for [MATH].', '0905.0858-1-38-2': 'For given [MATH], [MATH] controls the Vainshtein radius, i.e. the scale where the self-interactions of the [MATH] field become important.', '0905.0858-1-39-0': 'Fig. [REF] shows the field solution (left panel) and relative deviation of the acceleration from the Newtonian value (right panel) as a function of distance from the center, [MATH], for [MATH], [MATH] grid cells, corresponding to [MATH], and [MATH].', '0905.0858-1-39-1': 'The agreement with the analytical solution (thick lines) is generally very good, except at large radii, where the periodic boundary conditions become important, and around [MATH], where artifacts of interpolating a sphere onto a cubic grid become visible.', '0905.0858-1-39-2': 'At distances of order the size of the grid cells, the force resolution becomes worse leading to a growing scatter in the measured acceleration.', '0905.0858-1-39-3': 'Note that the analytical solution necessarily assumes an isolated mass, since the superposition principle cannot be used due to the non-linear [MATH] field equation.', '0905.0858-1-39-4': 'For this reason, we added an arbitrary zero-point to the [MATH] field in order to match the analytical solution in the left panel of Fig. [REF].', '0905.0858-1-39-5': 'Of course, such a zero-point does not affect any observable quantity, such as the acceleration shown in the right panel.', '0905.0858-1-40-0': 'We tested the field solution for several different values of [MATH], [MATH], [MATH], and [MATH].', '0905.0858-1-40-1': 'Given the caveats pointed out, we found very good convergence to the analytical solution.', '0905.0858-1-40-2': 'In particular, increasing [MATH] reduces the interpolation artifacts at [MATH], while decreasing [MATH] reduces the impact of the periodic boundary conditions.', '0905.0858-1-40-3': 'Solutions for different values of [MATH] are shown in both panels of Fig. [REF].', '0905.0858-1-40-4': 'While the solid lines show the exact solution to the full equation, Eq. ([REF]), the dotted lines show the linearized result, Eq. ([REF]).', '0905.0858-1-40-5': 'Clearly, for the parameters chosen, the effect of the non-linearity is significant for [MATH] of a few [MATH] or less.', '0905.0858-1-41-0': '## Convergence and resolution tests', '0905.0858-1-42-0': 'In most test cases we studied, the density field is sufficiently smooth that the relaxation converges to the desired low tolerance.', '0905.0858-1-42-1': 'However, for density fields with considerable small-scale inhomogeneities, such as the cosmological density field at late times, we find that this level is not achievable.', '0905.0858-1-42-2': 'This is because the non-linearity of the [MATH] equation is important as soon as the overdensity [MATH] becomes of order unity, which in the N-body simulation corresponds to [MATH] particle per grid cell.', '0905.0858-1-42-3': 'Equivalently, the Vainshtein radius for a single particle in our N-body simulation is of order the grid scale (see Fig. [REF], bottom left).', '0905.0858-1-42-4': 'Thus, the [MATH] field reacts much more strongly to noise due to the discreteness of particles than the Newtonian potential, leading to increased residual errors of the approximate solution.', '0905.0858-1-43-0': 'If we write Eq. ([REF]) in code units (see Appendix [REF]) as [MATH], then for an approximate solution [MATH], [MATH] is the dimensionless residual, where [MATH] is the size of a grid cell.', '0905.0858-1-43-1': 'In the following, we use the RMS of the dimensionless residual, [MATH] as a benchmark for the convergence of the field solution.', '0905.0858-1-43-2': 'We performed tests with sine wave density fields of various wavelengths in order to determine what residual [MATH] is acceptable in our simulations (see Appendix [REF]).', '0905.0858-1-43-3': 'In case of a pure sine wave, the non-linearity in Eq. ([REF]) vanishes, and the exact non-linear solution is identical to the linearized solution.', '0905.0858-1-43-4': 'We found that residuals of [MATH] are safe, as for residuals at that level, the errors in the solution are negligible compared to the unavoidable truncation errors from taking numerical derivatives on the grid.', '0905.0858-1-44-0': 'In order to reduce the noise in the density field in the cosmological simulations and improve the solution to the acceptable level of [MATH], we increase the number of particles from [MATH] to [MATH], and smooth the density field entering the r.h.s. of Eq. ([REF]) as described in Section [REF].', '0905.0858-1-44-1': 'With these steps, the solution converges with a residual [MATH] for all box sizes (left panel of Fig. [REF]).', '0905.0858-1-44-2': 'Note that the dimensionless RMS residuals never exceed a fraction of [MATH] of the RMS of the [MATH] field solution.', '0905.0858-1-45-0': 'Since the force resolution in our simulations is in any case limited to scales above [MATH], a smoothing on the grid scale is not expected to degrade the resolution of the simulations significantly.', '0905.0858-1-45-1': 'We checked this by running linearized DGP simulations given by Eq. ([REF]) (Section [REF]) using the same smoothing of the r.h.s. of Eq. ([REF]), and comparing the resulting power spectrum to that of linearized DGP simulations without smoothing, for the same initial conditions.', '0905.0858-1-45-2': 'The result is shown in the right panel of Fig. [REF].', '0905.0858-1-45-3': 'As expected, the effect of smoothing increases towards larger [MATH].', '0905.0858-1-45-4': 'The smoothing effect on the matter power spectrum is positive, since the smoothing removes power in the [MATH] field on the smallest scales, and [MATH] mediates a repulsive force.', '0905.0858-1-45-5': 'This leads to an enhancement in the matter power spectrum.', '0905.0858-1-46-0': 'The smoothing effect remains below 4% for [MATH], which we adopt as the maximum wave number considered for each box, where [MATH] is the Nyquist frequency of the grid.', '0905.0858-1-46-1': 'From our studies with different smoothing radii [MATH], we found that the smoothing effects on full DGP simulations show a similar [MATH]-dependence as those of the linearized DGP simulations, but are smaller by a factor of [MATH].', '0905.0858-1-46-2': 'This is understandable since the [MATH] field is suppressed in dense regions in the full simulations, reducing the effect of the smoothing.', '0905.0858-1-46-3': 'Taking into account this factor, we correct the power spectra measured in the full DGP simulations for the smoothing effects using the curves shown in Fig. [REF] (right panel).', '0905.0858-1-46-4': 'Note that in any case these effects are at the level of few percent or less.', '0905.0858-1-47-0': 'In order to assess the effect of the finite grid resolution, we also performed simulations with [MATH] and [MATH], and [MATH].', '0905.0858-1-47-1': 'Fig. [REF] (right panel) also shows the deviation of the power spectrum in these low-resolution simulations from the [MATH] simulation with the same initial conditions.', '0905.0858-1-47-2': 'The vertical lines indicate [MATH] for each case, i.e. the maximum [MATH] we consider for each simulation box.', '0905.0858-1-47-3': 'Below [MATH], the deviations are less than 10% for either [MATH] or [MATH].', '0905.0858-1-47-4': 'Note also that the resolution effects are independent of the type of simulation (DGP, linearized DGP, or GR).', '0905.0858-1-47-5': 'Hence, they cancel when measuring the deviation of [MATH] between different simulation types, and we expect this deviation to be accurate to the percent level for the range in [MATH] we consider.', '0905.0858-1-48-0': '## Quasi-static approximation', '0905.0858-1-49-0': 'In solving for the brane bending mode [MATH], we have assumed that all time-derivative terms in the full equation of motion of [MATH] (in terms of physical coordinates): [EQUATION] can be neglected with respect to the spatial derivatives in Eq. ([REF]).', '0905.0858-1-49-1': 'Up to second order, there are three time derivative terms which have been neglected: [EQUATION] [MATH] and [MATH] appear as [MATH] in the equations of motion.', '0905.0858-1-49-2': 'While we cannot rigorously prove that the quasi-static approximation holds, we can perform a consistency test by measuring the terms Eqs. ([REF])-([REF]) in the simulations, and checking whether they are indeed small.', '0905.0858-1-49-3': 'The time derivatives of [MATH] are calculated in each grid cell using the two neighboring time steps.', '0905.0858-1-49-4': 'We then take the RMS of each term over the grid, [MATH], and compare with the RMS of the spatial Laplacian, [MATH].', '0905.0858-1-49-5': 'Fig. [REF] presents [MATH] and [MATH] relative to the spatial derivatives for our largest and smallest box sizes, and shows that they are well under control.', '0905.0858-1-49-6': 'The relative magnitude of all time derivatives are of order a few [MATH] or less.', '0905.0858-1-49-7': 'This is in keeping with the expectation that [MATH], and [MATH] in our simulations.', '0905.0858-1-49-8': 'We conclude that the quasi-static approach is a self-consistent approximation.', '0905.0858-1-50-0': '# Cosmological simulations', '0905.0858-1-51-0': "We performed a suite of cosmological simulations for three different types of gravity: unmodified GR, corresponding to a smooth dark energy model with the same expansion history as DGP, referred to as QCDM; linearized DGP gravity, using the linearized [MATH] equation of motion [Eq. ([REF])], corresponding to a time-dependent rescaling of Newton's constant; and full DGP, solving for the full non-linear [MATH] solution [Eq. ([REF])].", '0905.0858-1-51-1': 'Comparing linearized DGP with full DGP allows us to study the Vainshtein effect in a cosmological setting.', '0905.0858-1-52-0': 'The cosmological parameters are those of the best-fit flat self-accelerating DGP model to WMAP 5yr data [CITATION] and are summarized in Tab. [REF].', '0905.0858-1-52-1': 'We generated the initial conditions at [MATH]) from a modified transfer function output of CAMB [CITATION] for a flat [MATH]CDM model with [MATH].', '0905.0858-1-52-2': 'The [MATH]CDM transfer function was corrected for small early-time modified gravity effects in the DGP model using the PPF approach, as detailed in Appendix [REF].', '0905.0858-1-53-0': 'The simulations were run with [MATH] grid cells on a side, and [MATH] particles, i.e. one particle per grid cell, to reduce the shot noise in the density field (see Section [REF]).', '0905.0858-1-53-1': 'We performed six simulations each for four different box sizes, from [MATH] up to [MATH] (Tab. [REF]).', '0905.0858-1-53-2': 'On an 8-core machine, the QCDM and linearized DGP runs require [MATH]10h of computing time, while the full DGP simulations require about 300h.', '0905.0858-1-54-0': 'Fig. [REF] (left panel) shows the combined power spectrum from all box sizes for the QCDM simulations, including bootstrap errors.', '0905.0858-1-54-1': 'Also shown is the non-linear power spectrum for QCDM, calculated from the linear power spectrum using the halofit procedure [CITATION].', '0905.0858-1-54-2': 'The power spectrum measured in each box is used up to [MATH] (see Tab. [REF]), and different boxes are combined weighting by volume.', '0905.0858-1-54-3': 'The lower panel of Fig. [REF] (left) shows the power spectrum relative to the halofit prediction, measured separately for each box size.', '0905.0858-1-54-4': 'The power spectra measured in different boxes clearly agree within the errors, and the deviations from the halofit prediction are within the accuracy ([MATH]%) expected from this fitting procedure (especially given the significant departures from [MATH]CDM of the simulated expansion history).', '0905.0858-1-55-0': 'Fig. [REF] also shows the non-linear power spectrum for a flat [MATH]CDM cosmology fixed to the same [MATH], [MATH], and primordial normalization, corresponding to a linear normalization of [MATH] today.', '0905.0858-1-55-1': 'Note that due to the different expansion history and the earlier onset of acceleration in DGP, linear growth is suppressed by [MATH]% in QCDM relative to [MATH]CDM.', '0905.0858-1-55-2': 'In the following, we will always compare the DGP results with QCDM, so that the expansion history effects are taken out, and all deviations are strictly due to modifications of gravity.', '0905.0858-1-56-0': 'We have also measured the mass function of dark matter halos in our simulations.', '0905.0858-1-56-1': 'The spherical overdensity halo-finding and combination of different simulation boxes was done as described in [CITATION].', '0905.0858-1-56-2': 'As our grid resolution is the same as in the simulations analyzed in [CITATION], while we have a factor of 8 more particles, we conservatively increase the minimum required number of particles from 800 to 6400.', '0905.0858-1-56-3': 'The corresponding mass thresholds are given in Tab. [REF].', '0905.0858-1-56-4': 'We define the halo mass [MATH] as the mass enclosed within a sphere of radius [MATH], so that the average density within the sphere is 200 times the average matter density in the universe, [MATH].', '0905.0858-1-57-0': 'Fig. [REF] (right panel) shows the mass function, [MATH], measured in QCDM simulations, in comparison with the prediction calculated from the linear QCDM power spectrum using the fitting formula from Tinker et al.[CITATION].', '0905.0858-1-57-1': 'The lower panel shows the relative deviations from the prediction, separately for each simulation box.', '0905.0858-1-57-2': 'Within the mass range accessible with our simulations, [MATH], the agreement with the prediction and among different boxes is good.', '0905.0858-1-57-3': 'We also show the predicted mass function for a [MATH]CDM model with the same initial power spectrum in Fig. [REF].', '0905.0858-1-57-4': 'Apparently, the number of halos above [MATH] is significantly reduced due to the suppressed growth in QCDM.', '0905.0858-1-58-0': '# Results', '0905.0858-1-59-0': 'In order to get a visual impression of some of the physics in DGP N-body simulations, we show slices through one simulation box at [MATH]) in Fig. [REF].', '0905.0858-1-59-1': 'The slices are 64 cells thick, and for each pixel we take the maximum absolute values of each quantity over the thickness of the slice, for better visibility.', '0905.0858-1-59-2': 'The density field (top left; in logarithmic scale) is difficult to distinguish visually from the QCDM result for the same run, as the effects on the power spectrum are at the [MATH]% level (Section [REF]).', '0905.0858-1-59-3': 'The top right panel in Fig. [REF] shows the dynamical potential [MATH], exhibiting the potential wells of massive collapsed structures.', '0905.0858-1-59-4': 'The brane-bending mode [MATH] is shown in the lower left panel.', '0905.0858-1-59-5': 'On linear scales, [MATH] is proportional to the potential [MATH] (Section [REF]), but evidently does not follow the potential within deeper potential wells, making it appear smoother.', '0905.0858-1-59-6': 'To make this more clear, we show the difference of the [MATH] solution from the linear value [Eq. ([REF])], [MATH], in the lower right panel of Fig. [REF]: [MATH] is suppressed ([MATH]) in overdense regions, showing the Vainshtein effect at work in a cosmological setting.', '0905.0858-1-59-7': 'Note that quite low-mass structures which are not conspicuous in the potential [MATH] already lead to a suppression of [MATH].', '0905.0858-1-59-8': 'See Section [REF] for a discussion of the Vainshtein effect in dark matter halos.', '0905.0858-1-60-0': '## Power spectrum', '0905.0858-1-61-0': 'Fig. [REF] (left panel) shows the relative deviation of the linearized and full DGP power spectra from the QCDM result at redshift 0.', '0905.0858-1-61-1': 'The error bars are obtained from the 6 separate runs using a bootstrap procedure.', '0905.0858-1-61-2': 'By comparing simulation runs with the same initial conditions, and then averaging the deviations, most of the cosmic variance cancels out and we are able to obtain significantly less scatter.', '0905.0858-1-61-3': 'Also shown is the predicted deviation in the linear power spectrum.', '0905.0858-1-61-4': 'Note first that in the self-accelerated DGP branch, the scalar field mediates a repulsive force, leading to a suppression of the growth of structure.', '0905.0858-1-61-5': 'Due to the scale-invariant modification of gravity in linearized quasi-static DGP [Eq. ([REF])], the predicted linear deviation is also scale-independent.', '0905.0858-1-61-6': 'On linear scales, [MATH], both linearized and full DGP simulations agree well with the linear prediction.', '0905.0858-1-62-0': 'Apparently, the full DGP result departs significantly from linearized DGP on quasi-linear to non-linear scales.', '0905.0858-1-62-1': 'The Vainshtein effect begins to operate wherever overdensities become of order unity, and suppresses the deviation from GR.', '0905.0858-1-62-2': 'Note that small effects of the non-linear [MATH] equation can already be seen on quite large scales, [MATH] corresponding to [MATH], not far from the acoustic features in the matter power spectrum.', '0905.0858-1-62-3': 'While we do not expect dramatic effects on cosmological parameter constraints from BAO, a modeling of these non-linear effects will be necessary for precision constraints in the context of DGP and similar braneworld models.', '0905.0858-1-63-0': 'We also show the deviation of [MATH] from [MATH], where [MATH] is calculated from the corresponding linear power spectrum using the standard halofit prescription.', '0905.0858-1-63-1': 'halofit describes the linearized DGP power spectrum reasonably well up to [MATH], in agreement with the findings of [CITATION], while it follows neither the linearized nor full DGP at higher [MATH].', '0905.0858-1-63-2': 'We have not explored whether phenomenological modifications of halofit allow for an improvement of the fit.', '0905.0858-1-64-0': 'The dashed points at high [MATH] extend the range in frequency up to [MATH] of our smallest box, [MATH].', '0905.0858-1-64-1': 'While resolution effects are still expected to cancel out to first order in this representation, these points only serve to indicate that the trends seen for full and linear DGP at lower [MATH] continue towards smaller scales.', '0905.0858-1-65-0': 'The right panel of Fig. [REF] shows the evolution of the modified gravity effects on the power spectrum as function of redshift.', '0905.0858-1-65-1': 'On large scales [MATH], the deviation is almost scale-free and evolves as predicted by linear theory.', '0905.0858-1-65-2': 'At earlier times, the density field is non-linear only on smaller scales.', '0905.0858-1-65-3': 'Hence, the Vainshtein effect becomes visible in the power spectrum only at higher [MATH].', '0905.0858-1-65-4': 'However, it does affect the power spectrum deviation significantly already at [MATH].', '0905.0858-1-66-0': '## Halo mass function', '0905.0858-1-67-0': 'The abundance of massive dark matter halos is a sensitive probe of the growth of structure in the Universe [CITATION].', '0905.0858-1-67-1': 'In particular, the number of the most massive halos which host galaxy clusters depends exponentially on the amplitude of the matter power spectrum.', '0905.0858-1-67-2': 'Fig. [REF] shows the effect of the modification of gravity in DGP on the halo mass function, relative to the QCDM effective dark energy cosmology.', '0905.0858-1-67-3': 'We investigated the effect of the smoothing used in the full DGP simulations by comparing linearized DGP simulations with and without smoothing, as done for the power spectrum (Section [REF]).', '0905.0858-1-67-4': 'Above our rather conservative mass threshold for each box, we only found a small effect of the smoothing on halo masses in the linearized DGP simulations.', '0905.0858-1-67-5': 'Since the [MATH] field is in any case suppressed within halos due to the Vainshtein mechanism, we expect the smoothing effects to be even smaller for the full DGP simulations.', '0905.0858-1-68-0': 'As expected, the full DGP simulations are somewhat closer to QCDM than the linearized DGP case in Fig. [REF].', '0905.0858-1-68-1': 'The suppression in the mass function is significant for [MATH], reaching more than 30% for massive cluster-size halos.', '0905.0858-1-68-2': 'The suppression is smaller for lower-mass halos.', '0905.0858-1-68-3': 'This is presumably because these halos formed earlier in cosmic history, when the modified gravity effects of DGP were significantly smaller.', '0905.0858-1-69-0': 'Note that when compared to [MATH]CDM, this suppression comes in addition to the larger suppression of the mass function due to the expansion history of DGP (see dashed line in Fig. [REF]).', '0905.0858-1-69-1': 'However, we expect the effect of the DGP modification of gravity to be fairly independent of the expansion history.', '0905.0858-1-69-2': 'In particular, we expect a similar effect on the halo mass function in generalized braneworld models which exhibit an expansion history close to [MATH]CDM [CITATION].', '0905.0858-1-69-3': 'Hence, the mass function is expected to be a sensitive probe for braneworld modified gravity models, as is the case for [MATH] gravity [CITATION].', '0905.0858-1-69-4': 'Note that for the normal branch of braneworld gravity, the sign of the effect in Fig. [REF] will be reversed, leading to an enhancement of the number of massive halos.', '0905.0858-1-70-0': '## Brane bending mode and Vainshtein effect', '0905.0858-1-71-0': 'Fig. [REF] (left panel) shows the average profiles of the brane-bending mode [MATH], the dynamical potential [MATH], and the lensing potential [MATH] of dark matter halos in the full DGP simulations.', '0905.0858-1-71-1': 'The brane-bending mode has been scaled by [MATH], so that it agrees with [MATH] for the linearized solution.', '0905.0858-1-71-2': 'We only use our highest resolution simulation box for this measurement ([MATH]).', '0905.0858-1-71-3': 'The inner radius of the profiles is given by one grid cell.', '0905.0858-1-71-4': 'The profiles were measured by spherically averaging many lines of sight for each halo with [MATH], and then stacked, scaling each profile to the radius [MATH] of the respective halo.', '0905.0858-1-71-5': 'This was done in order to reduce the significant scatter in the profiles.', '0905.0858-1-72-0': 'While the lensing potential [MATH] obeys the standard Poisson equation in DGP, the dynamical potential [MATH] receives a contribution from the brane bending mode [Eq. ([REF])].', '0905.0858-1-72-1': 'Apparently, the [MATH] field flattens out towards the halo center, which is in qualitative agreement with the non-linear field solution for a spherical mass (see, e.g. Fig. [REF]).', '0905.0858-1-72-2': 'Note that [MATH] turns away from the linear solution when the overdensity is of order a few, in agreement with the estimate in Section [REF].', '0905.0858-1-73-0': 'The quantity which is actually observable however is the gradient of the field, shown in Fig. [REF] (right panel), which gives essentially the deviation of the acceleration from the Newtonian acceleration.', '0905.0858-1-73-1': 'For each halo, we measure the radial gradient relative to the halo center.', '0905.0858-1-73-2': 'We again scaled [MATH] by [MATH] to match the linearized solution to [MATH].', '0905.0858-1-73-3': 'From the halo exterior towards the interior, the gradient of [MATH] grows initially, turns around at [MATH], and shrinks for even smaller radii.', '0905.0858-1-73-4': 'As is clearly shown by comparing the dynamical potential [MATH] with its Newtonian value, [MATH], any observable deviations from GR are indeed suppressed within massive halos.', '0905.0858-1-73-5': 'We find that for halos between [MATH] and [MATH], the deviation in the acceleration is around [MATH] in the inner parts, while it is suppressed down to [MATH] for halos around [MATH].', '0905.0858-1-73-6': 'This trend with mass is expected, given that [MATH] for the spherically symmetric solution.', '0905.0858-1-74-0': 'In order to model the behavior of the brane bending mode in the cosmological context, we can make use of the two limiting cases presented in Section [REF].', '0905.0858-1-74-1': 'One is the spherically symmetric solution (Section [REF]), which can be used in modeling a spherical collapse of dark matter halos in DGP [CITATION].', '0905.0858-1-74-2': 'Moreover, Khoury and Wyman [CITATION] have based the approximate field solution in their N-body simulation on this limiting case.', '0905.0858-1-74-3': 'More realistically however, cosmic structure forms by collapsing into non-radially symmetric structures such as filaments.', '0905.0858-1-74-4': 'The plane wave density perturbation can serve as another, albeit rather extreme, test case.', '0905.0858-1-74-5': 'For such a perturbation, the non-linear self-coupling of [MATH] vanishes (Section [REF]).', '0905.0858-1-74-6': 'Hence, we expect the Vainshtein effect to be typically weakened in a non-radially symmetric setting, compared to the spherically symmetric case.', '0905.0858-1-75-0': 'In their ansatz, Khoury and Wyman assumed that the spherically symmetric solution Eqs. ([REF])-([REF]) holds wherever the field becomes non-linear.', '0905.0858-1-75-1': 'Since in this solution, the gravitational constant is rescaled locally by [MATH], the Poisson equation for [MATH] can be written in this ansatz as: [EQUATION] where [MATH] is now a function of the local overdensity.', '0905.0858-1-75-2': 'Substituting [MATH] in Eq. ([REF]) [or Eq. ([REF])] we see again that [MATH], with an order unity coefficient.', '0905.0858-1-75-3': 'Khoury and Wyman [CITATION] chose: [EQUATION]', '0905.0858-1-75-4': 'Hence, in the DGP model we simulated, [MATH].', '0905.0858-1-75-5': 'Given the density field in our DGP simulations, we can solve Eq. ([REF]) by Fourier transform as done in [CITATION], subtract the Newtonian potential, and compare the brane bending mode [MATH] from this ansatz with our numerical solution of the full [MATH] equation.', '0905.0858-1-76-0': 'Fig. [REF] (right panel) shows the result for stacked halo profiles of the radial gradient of [MATH].', '0905.0858-1-76-1': 'While [MATH] shows a qualitatively similar behavior to the full solution, the suppression due to the Vainshtein effect appears to set in at significantly larger radii in [MATH] than in the full solution.', '0905.0858-1-76-2': 'This is qualitatively in agreement with our expectation that non-radially symmetric configurations experience a weaker non-linear suppression.', '0905.0858-1-76-3': 'For [MATH], the non-linearity is even mildly acting at large distances from the halo, so that we do not see the average profile approach the linear solution in Fig. [REF].', '0905.0858-1-76-4': 'This is because there is significant scatter in the profiles for different halos, and also among different lines of sight for a single halo.', '0905.0858-1-76-5': 'The non-linearity tends to suppress [MATH] in higher density environments and vice-versa for low density regions, leading to a suppression of the average [MATH] profile.', '0905.0858-1-76-6': 'Note also that there is some freedom in adjusting the order-unity coefficient for [MATH] [Eq. ([REF])] in this ansatz.', '0905.0858-1-76-7': 'If we reduce [MATH] by a factor of [MATH], [MATH] follows the full solution more closely at large radii.', '0905.0858-1-76-8': 'We plan on investigating the possibility of calibrating the (computationally much less expensive) ansatz of Eqs. ([REF])-([REF]) with the full simulations.', '0905.0858-1-77-0': 'Although we did not compare our results with a full simulation based on the spherically symmetric approach of [CITATION], these results seem to indicate that this approximation overestimates the non-linear suppression of the brane-bending mode, which might affect observables such as the power spectrum or halo mass function.', '0905.0858-1-77-1': 'We point out however that the crossover scale [MATH] in the braneworld-inspired model considered in [CITATION] is an order of magnitude smaller than our [MATH], and the non-linearity in Eqs. ([REF])-([REF]) only becomes effective at much higher overdensities [MATH], so that the solution is possibly less sensitive to this approximation.', '0905.0858-1-78-0': '# Constraints on the self-accelerating DGP model', '0905.0858-1-79-0': 'We now briefly discuss the impact of our simulation results on cosmological constraints on the self-accelerating DGP model without [MATH].', '0905.0858-1-79-1': 'First, regarding, baryon acoustic oscillations, our results for the DGP power spectrum on quasi-linear scales show that non-linear effects are not significantly enhanced at the BAO scale.', '0905.0858-1-79-2': 'We estimate that any modified gravity corrections to the BAO scale are below the percent level, and hence well within the uncertainties of current BAO measurements [CITATION].', '0905.0858-1-79-3': 'Including baryon acoustic oscillations will increase the power of the combined CMB and Supernova constraints on the self-accelerating DGP model [CITATION] with non-zero curvature to [MATH] [CITATION].', '0905.0858-1-80-0': 'Second, weak lensing measurements constrain the amplitude of the matter power spectrum today, which in this model is significantly smaller due to the suppression of growth by the earlier onset of acceleration and the repulsive force mediated by the brane-bending mode.', '0905.0858-1-80-1': 'The linear power spectrum normalization at [MATH] in the models we simulated is [MATH] and [MATH], while for a [MATH]CDM model with the same primordial normalization, it is [MATH].', '0905.0858-1-80-2': 'We found that the non-linear matter power spectrum in the full DGP simulations is always below that of QCDM (up to [MATH], Fig. [REF]).', '0905.0858-1-80-3': 'Hence, the suppression of the non-linear power spectrum in the self-accelerating DGP model corresponds to a reduction in the inferred linear normalization [MATH] today of at least 0.1, with respect to a [MATH]CDM model with the same initial conditions.', '0905.0858-1-80-4': 'Such a deviation is disfavored by about 1.5 standard deviations by current weak lensing measurements [CITATION].', '0905.0858-1-81-0': 'The abundance of massive dark matter halos is probed by cluster surveys.', '0905.0858-1-81-1': 'We showed in Section [REF] that the abundance of halos above [MATH] is suppressed by around 20% relative to QCDM, which itself only predicts half as many halos in that mass range as a [MATH]CDM model with the same primordial power.', '0905.0858-1-81-2': 'The latter again corresponds to a shift in [MATH] by 0.1, which is constrained to more than [MATH] by X-ray cluster measurements [CITATION], when taking into account the systematic errors.', '0905.0858-1-82-0': '# Conclusions', '0905.0858-1-83-0': 'The N-body simulations of DGP gravity we presented here show how the self-interactions of the brane bending mode [MATH], which are responsible for restoring General Relativity in dense environments, influence the formation of structure in the universe.', '0905.0858-1-83-1': 'In the self-accelerating DGP model we simulated, this scalar field mediates a repulsive force, effectively weakening gravity.', '0905.0858-1-83-2': 'We indeed find that the field solution turns away from the linearized solution whenever the matter overdensity [MATH] becomes of order a few, and that the repulsive force is suppressed by more than an order of magnitude compared to its linearized value in the center of massive halos.', '0905.0858-1-83-3': 'We also compared our full solution to that obtained in the approximate ansatz of [CITATION].', '0905.0858-1-83-4': 'While their ansatz agrees well with our results in the densest regions, it seems to overestimate the non-linear suppression in less dense regions, such as the outer regions and environments of halos.', '0905.0858-1-83-5': 'This is in line with our finding that the non-linear suppression of [MATH] is weaker for non-spherically symmetric configurations.', '0905.0858-1-84-0': 'The non-linear matter power spectrum in the DGP simulations shows that the suppression due to the repulsive [MATH] field is amplified on non-linear scales for the linearized DGP simulations.', '0905.0858-1-84-1': 'In the full DGP simulations, this suppression is smaller, and eventually turns around on Mpc scales.', '0905.0858-1-84-2': 'The deviations between linerized and full DGP power spectra are noticeable already on quasi-linear scales, [MATH].', '0905.0858-1-84-3': 'At the BAO scale, modified gravity effects on the power spectrum are at the percent-level.', '0905.0858-1-85-0': 'We found that the abundance of massive dark matter halos is significantly suppressed in the DGP simulations, compared to a standard gravity simulation with the same expansion history.', '0905.0858-1-85-1': 'Again, the non-linear suppression of the [MATH] field alleviates this suppression, and hence has to be taken into account when using observations to constrain this type of modified gravity.', '0905.0858-1-85-2': 'The effect on the halo mass function is in fact large enough to make this an interesting observational probe of more general braneworld scenarios.', '0905.0858-1-86-0': 'Independently of the CMB constraints [CITATION], our results on the non-linear structure formation strongly constrain the self-accelerating DGP model (without [MATH]).', '0905.0858-1-86-1': 'In the future, we plan to extend our simulations to the normal branch of DGP, and more general braneworld models [CITATION], and attempt a modeling of the modified gravity effects, in the context of the halo model for example.', '0905.0858-1-86-2': 'This model can then serve as a framework for cosmological constraints on braneworld gravity which take into account the non-linear mechanisms inherent to this model consistently.', '0905.0858-1-86-3': 'Simulations of both [MATH] gravity and DGP have shown that most interesting phenomena appear on quasi-linear scales, and in the abundance and environments of clusters.', '0905.0858-1-86-4': 'Fortunately, these scales are well accessible to observations, e.g., through weak lensing, the Lyman-[MATH] forest, and cluster surveys, which can then be used as precision probes of gravity on cosmological scales.', '0905.0858-1-87-0': 'I am indebted to Scott Dodelson, Wayne Hu, and Andrey Kravtsov for invaluable input and guidance.', '0905.0858-1-87-1': 'I would like to thank Hiro Oyaizu and Marcos Lima for our previous collaborative work on the code and analysis, and Angela Olinto, Mike Gladders, Cora Dvorkin, Sam Leitner, Michael Mortonson, and Amol Upadhye for discussions.', '0905.0858-1-87-2': 'The support of the Fermilab computing staff is gratefully acknowledged.', '0905.0858-1-88-0': 'The simulations used in this work have been performed on the Joint Fermilab - KICP Supercomputing Cluster, supported by grants from Fermilab, Kavli Institute for Cosmological Physics, and the University of Chicago.', '0905.0858-1-88-1': 'This work was supported by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142.', '0905.0858-1-89-0': '# Code implementation and discretization', '0905.0858-1-90-0': 'This appendix describes details of the N-body code implementation, focusing on the relaxation solver for Eq. ([REF]) as the main non-standard part of the code.', '0905.0858-1-90-1': 'The code is written in C++, uses OpenMP for parallelization of most time-critical operations, and employs FFTW [CITATION] for the Fast Fourier Transforms.', '0905.0858-1-91-0': '## Code units', '0905.0858-1-92-0': 'The comoving code units follow the convention used in [CITATION], where [EQUATION] and [EQUATION]', '0905.0858-1-92-1': 'In the above definitions, [MATH] is the comoving simulation box size in [MATH], [MATH] is the number of grid cells in each direction, [MATH] is the Hubble parameter today, [MATH] is the critical density today, and [MATH] is the fraction of non-relativistic matter today relative to the critical density.', '0905.0858-1-93-0': 'Transformed to code units, the equation for [MATH] becomes: [EQUATION]', '0905.0858-1-93-1': 'Here, [MATH] acts with respect to code coordinates, [MATH], [MATH], and [MATH] is the matter overdensity on the grid, determined from the particle positions.', '0905.0858-1-93-2': 'Note that we have not yet scaled [MATH] to [MATH].', '0905.0858-1-93-3': 'After solving Eq. ([REF]), [MATH] is added to the standard Newtonian potential, which is solved for using a Fast Fourier Transform, to obtain the dynamical potential in DGP: [EQUATION]', '0905.0858-1-94-0': '## Discretization of [MATH] equation', '0905.0858-1-95-0': 'We employ standard second-order symmetric differences for the discretization of the second derivatives in Eq. ([REF]): [EQUATION] and correspondingly for derivatives with respect to [MATH], [MATH].', '0905.0858-1-95-1': 'Here, [MATH] stand for the grid indices, and the step size [MATH] for the base grid and [MATH] for the grid of refinement level [MATH].', '0905.0858-1-95-2': 'The finite differences are evaluated before squaring in calculating the non-linear terms in Eq. ([REF]).', '0905.0858-1-96-0': 'We have tried different discretizations, such as going to higher order in the finite differences, and solving for the deviation of [MATH] from the solution of the linearized equation [Eq. ([REF])], instead of solving for [MATH] itself.', '0905.0858-1-96-1': 'The performance of those discretizations is comparable to or worse than the simple discretization Eqs. ([REF])-([REF]).', '0905.0858-1-96-2': 'This is understandable, since going to higher order amounts to making the derivative operations less local in order to be sensitive to error modes of longer wavelength.', '0905.0858-1-96-3': 'However, in our multigrid relaxation scheme, this is already done efficiently by the coarser grids, so going to higher order in a single relaxation does not improve performance.', '0905.0858-1-96-4': 'Hence, we stay with the straightforward discretization, Eqs. ([REF])-([REF]).', '0905.0858-1-97-0': '## Relaxation algorithm', '0905.0858-1-98-0': 'In general, a relaxation scheme operates by iteratively obtaining a better approximation to the solution [MATH] given a previous guess, [MATH].', '0905.0858-1-98-1': 'On the grid, the solution is updated successively for each cell [MATH]: [EQUATION] where [MATH] is the discretized field equation solved for [MATH].', '0905.0858-1-98-2': 'In case of a linear field equation, [MATH] can be solved for straightforwardly.', '0905.0858-1-98-3': "However, in our case the field equation is non-linear in [MATH], and we instead solve for [MATH] iteratively via Newton's method.", '0905.0858-1-98-4': 'Writing the field equation Eq. ([REF]) as [MATH], we need to solve: [EQUATION] for [MATH], where we have suppressed the dependences of [MATH] on neighboring grid cells.', '0905.0858-1-98-5': "By expanding [MATH] in a Taylor series around the current approximation, one step of Newton's method works by updating [MATH] with: [EQUATION]", '0905.0858-1-98-6': 'Our relaxation step is thus given by Eq. ([REF]), where [MATH] is determined from Eq. ([REF]) and the discretization Eqs. ([REF])-([REF]).', '0905.0858-1-98-7': 'In practice, the order in which we loop over grid cells in performing the relaxation Eq. ([REF]) is important, since each relaxation step depends on the neighboring grid cells.', '0905.0858-1-98-8': 'In particular, we need to take care when parallelizing the relaxation.', '0905.0858-1-98-9': 'We use a generalized red-black scheme (see, e.g., [CITATION]), by successively running over cells with [MATH] modulo 4 = [MATH], [MATH].', '0905.0858-1-98-10': 'This was done in order to break dependences due to neighboring cells within each set, in particular due to the mixed derivatives Eq. ([REF]), so that each set can be efficiently parallelized.', '0905.0858-1-98-11': 'We experimented with different ordering schemes and found that this choice performed best.', '0905.0858-1-99-0': 'Now, assume we have an approximate solution [MATH] to the field equation Eq. ([REF]) on the fine grid [MATH], which differs from the true solution [MATH] by the error [MATH], [MATH].', '0905.0858-1-99-1': 'Further, the residual is defined as [MATH].', '0905.0858-1-99-2': 'Since [MATH] by assumption, we obtain the following residual equation: [EQUATION]', '0905.0858-1-99-3': 'We expect that the relaxation on the fine grid has removed most small-scale error modes, so [MATH] mainly consists of longer wavelength modes.', '0905.0858-1-99-4': 'Thus, we will solve for [MATH] on the coarser grid [MATH], and then correct the approximate solution [MATH] for this error.', '0905.0858-1-99-5': 'On the coarse grid [MATH], Eq. ([REF]) reads: [EQUATION]', '0905.0858-1-99-6': 'Here, the superscripts denote which grid a given quantity is defined on, and [MATH] is the restriction operator mapping fields from the fine grid to the coarse grid.', '0905.0858-1-99-7': 'Note that Eq. ([REF]) is of the same form as the [MATH] equation on the coarse grid, [MATH], except with a different right-hand side.', '0905.0858-1-99-8': 'Hence, Eq. ([REF]) is solved for [MATH] using the same algorithm as the original equation, but at one grid level higher.', '0905.0858-1-99-9': 'Once Eq. ([REF]) is solved, we can correct [MATH] for the error: [EQUATION] where [MATH] is the interpolation operator mapping the error from the coarse grid to the fine grid.', '0905.0858-1-100-0': 'In summary, the multigrid relaxation proceeds as follows:', '0905.0858-1-101-0': 'Correct [MATH] using [MATH] [Eq. ([REF])].', '0905.0858-1-102-0': 'Correct [MATH] using [MATH] [Eq. ([REF])].', '0905.0858-1-103-0': 'Thus, the relaxation proceeds as a nested loop going down to the coarsest grid, and then correcting the solutions on the successively finer grids.', '0905.0858-1-103-1': 'One such iteration through all grid levels is called a V-cycle.', '0905.0858-1-103-2': 'The coarsest grid we use has 4 cells on a side, which corresponds to a refinement of [MATH] for a [MATH] base grid.', '0905.0858-1-104-0': 'For the interpolation operator [MATH], we use standard bilinear interpolation (consistent with the cloud-in-cell scheme used for assigning densities and measuring accelerations on the grid).', '0905.0858-1-104-1': 'For the restriction [MATH], we use full-weighting, the transpose of the bilinear interpolation operator.', '0905.0858-1-104-2': 'See the appendix in [CITATION] for an explicit definition.', '0905.0858-1-105-0': 'In order to ensure convergence at all times, we adopt a large value of [MATH], at the expense of computing time.', '0905.0858-1-105-1': 'Usually, one V-cycle reduces the residual [MATH] by 1-2 orders of magnitude.', '0905.0858-1-105-2': 'We stop the relaxation when either [MATH] is reached, or the convergence stalls at a certain level of [MATH].', '0905.0858-1-105-3': 'Either of this usually happens within 4 V-cycles.', '0905.0858-1-105-4': 'The value of [MATH] for each time step is logged, allowing for a monitoring of the convergence status in the simulations.', '0905.0858-1-106-0': '# Sine wave test', '0905.0858-1-107-0': 'In this section, we use the fact that the full [MATH] solution is identical to the linearized solution for a plane wave perturbation (Section [REF]) to test our code for perturbations on different length scales.', '0905.0858-1-107-1': 'We consider a simple sine wave density perturbation, with [MATH]-vector chosen to lie along the [MATH]-axis: [EQUATION]', '0905.0858-1-107-2': 'In this case, the solution to the non-linear [MATH] equation is identical to the linear solution: [EQUATION]', '0905.0858-1-107-3': 'Hence, the exact acceleration is given by: [EQUATION]', '0905.0858-1-107-4': 'For a given level of residuals [MATH], we now demand that the errors due to the approximate solution of the [MATH] equation of motion are small compared to the unavoidable truncation errors which are encurred by taking the gradient of the potential on the grid.', '0905.0858-1-108-0': 'In the following, we set [MATH], [MATH], and the number of grid cells is set to [MATH].', '0905.0858-1-108-1': 'The truncation errors are measured by assigning the analytical solution for the potential to each grid point, and comparing the exact acceleration to the one obtained by derivation and interpolation from the grid.', '0905.0858-1-108-2': 'The truncation errors for two sine waves of different wavelength are shown as deviations in the acceleration in Fig. [REF] (red points).', '0905.0858-1-108-3': 'We scale the deviations to the RMS acceleration of the sine wave, [MATH], in order to avoid far outliers obtained near the zeros of the sine wave when scaling to the exact solution at each point.', '0905.0858-1-108-4': 'For shorter wavelength modes, the truncation error increases and reaches order unity for waves on the Nyquist scale, as expected.', '0905.0858-1-109-0': 'In order to estimate the additional error in the acceleration made due to insufficient convergence of the solution, we compare the acceleration measured from the non-linear field solution coming out of the relaxation solver, to the corresponding solution of the linearized [MATH] equation Eq. ([REF]).', '0905.0858-1-109-1': 'We vary the parameters [MATH] and [MATH] which control the strength of the non-linearity in Eq. ([REF]).', '0905.0858-1-109-2': 'In Fig. [REF], we show the measured deviations in the acceleration from the linear solution for different values of [MATH] and [MATH], and in particular for the largest values that reached convergence.', '0905.0858-1-109-3': 'The final residuals of the [MATH] solution in each case are also shown.', '0905.0858-1-110-0': 'Apparently, the errors in the acceleration are completely negligible for residuals [MATH], consistent with results we found for other test cases such as the spherical mass.', '0905.0858-1-110-1': 'For sufficiently strong non-linearity and short wavelengths, residuals of order [MATH] are reached which lead to measurable deviations in the acceleration.', '0905.0858-1-110-2': 'However, for the residual values up to [MATH] that we explored, the error due to the incomplete convergence is very small compared to the truncation error in all cases.', '0905.0858-1-110-3': 'Hence, we conclude that for residuals smaller than the conservative bound of [MATH], the [MATH] solution we obtain is sufficiently accurate in order not to bias the particle dynamics.', '0905.0858-1-111-0': '# Correcting Initial Conditions for DGP', '0905.0858-1-112-0': 'Due to the additional term [MATH] in the Friedmann equation, the self-accelerating DGP model starts to deviate from a [MATH]CDM expansion history at relatively high redshifts.', '0905.0858-1-112-1': 'In addition, there are modifications to the growth of horizon-scale modes.', '0905.0858-1-112-2': 'For this reason, even at the initial redshift of our simulations, [MATH], there are small departures in the matter transfer function in DGP as compared to a [MATH]CDM model with the same early-Universe parameters [MATH].', '0905.0858-1-113-0': 'In order to take these differences into account, we calculate the dark matter transfer function [MATH] at [MATH] for DGP and QCDM, using the PPF approach described in [CITATION], and for the corresponding flat [MATH]CDM model.', '0905.0858-1-113-1': 'We do not include the effects of radiation on the transfer function, which is not necessary since we are only interested in the relative deviation of the DGP [MATH] from [MATH]CDM.', '0905.0858-1-113-2': 'Fig. [REF] shows the relative deviation in [MATH] from [MATH]CDM for DGP and QCDM.', '0905.0858-1-113-3': 'Both QCDM and DGP transfer functions are slightly suppressed on all scales due to the effects of the earlier onset of acceleration.', '0905.0858-1-113-4': 'On super-horizon scales, the transfer function is further suppressed in DGP caused by the transition to 5D gravity on very large scales.', '0905.0858-1-113-5': 'In contrast, [MATH] is less suppressed in QCDM because of the fluctuations in the effective dark energy.', '0905.0858-1-114-0': 'We correct the transfer function obtained from CAMB for these small differences at [MATH] by multiplying [MATH] with [MATH], where [MATH] is a simple [MATH] function fit to the DGP curve shown in Fig. [REF].', '0905.0858-1-114-1': 'Hence, these initial conditions are not quite correct for the QCDM simulations.', '0905.0858-1-114-2': 'However, the differences are very small on the scales probed by our simulations ([MATH]).'}

{'0905.0858-2-0-0': 'We perform cosmological N-body simulations of the Dvali-Gabadadze-Porrati braneworld model, by solving the full non-linear equations of motion for the scalar degree of freedom in this model, the brane bending mode.', '0905.0858-2-0-1': 'While coupling universally to matter, the brane-bending mode has self-interactions that become important as soon as the density field becomes non-linear.', '0905.0858-2-0-2': 'These self-interactions lead to a suppression of the field in high-density environments, and restore gravity to General Relativity.', '0905.0858-2-0-3': 'The code uses a multi-grid relaxation scheme to solve the non-linear field equation in the quasi-static approximation.', '0905.0858-2-0-4': 'We perform simulations of a flat self-accelerating DGP model without cosmological constant.', '0905.0858-2-0-5': 'However, the type of non-linear interactions of the brane-bending mode, which are the focus of this study, are generic to a wide class of braneworld cosmologies.', '0905.0858-2-0-6': 'The results of the DGP simulations are compared with standard gravity simulations assuming the same expansion history, and with DGP simulations using the linearized equation for the brane bending mode.', '0905.0858-2-0-7': 'This allows us to isolate the effects of the non-linear self-couplings of the field which are noticeable already on quasi-linear scales.', '0905.0858-2-0-8': 'We present results on the matter power spectrum and the halo mass function, and discuss the behavior of the brane bending mode within cosmological structure formation.', '0905.0858-2-0-9': 'We find that, independently of CMB constraints, the self-accelerating DGP model is strongly constrained by current weak lensing and cluster abundance measurements.', '0905.0858-2-1-0': '# Introduction', '0905.0858-2-2-0': 'The observed acceleration of the universe [CITATION] poses a fascinating challenge to physicists and cosmologists: it points towards new physics at an unusually low energy scale ([MATH]eV), or large length scale ([MATH]Mpc).', '0905.0858-2-2-1': 'For this reason, no natural explanation appears to exist up to now.', '0905.0858-2-2-2': 'Many attempts have been made to go beyond the minimal explanation, a cosmological constant or vacuum energy.', '0905.0858-2-2-3': 'They can be broadly classified into two categories: acceleration is due to an additional, smooth stress-energy component with negative pressure (dark energy [CITATION]); or it is caused by gravity itself, through modifications to General Relativity (GR) on cosmological scales.', '0905.0858-2-2-4': 'While there are strong constraints on deviations from GR in the Solar System [CITATION], gravity is remarkably weakly constrained on cosmological length scales.', '0905.0858-2-2-5': 'This provides an independent motivation for studying modified gravity models in the context of cosmology.', '0905.0858-2-3-0': 'Smooth dark energy models have enough freedom to reproduce essentially any background expansion history of the universe.', '0905.0858-2-3-1': 'In order to distinguish modified gravity from the smooth dark energy scenario, it is thus necessary to study the growth of cosmological structure.', '0905.0858-2-3-2': 'A wealth of observables can be used for this purpose [CITATION], e.g. weak lensing, galaxy-CMB cross-correlation, galaxy cluster abundances, and many more.', '0905.0858-2-3-3': 'However, almost all of these measurements are affected by non-linearities in the matter density field, or have most of their information on non-linear scales.', '0905.0858-2-3-4': 'All viable modified gravity models include a non-linear mechanism to restore General Relativity in high-density environments, which is necessary in order to satisfy Solar System constraints.', '0905.0858-2-3-5': 'This mechanism has to be taken into account in order to make accurate predictions for observables on non-linear scales in modified gravity.', '0905.0858-2-3-6': 'For one representative of [MATH] modified gravity, [CITATION] showed that the effects of this non-linear chameleon mechanism are significant for models that satisfy Solar System constraints.', '0905.0858-2-4-0': 'One of the most popular modified gravity scenarios is the Dvali-Gabadadze-Porrati (DGP) model [CITATION].', '0905.0858-2-4-1': 'In this model, matter and radiation are confined to a 4-dimensional brane in a 5-dimensional bulk spacetime.', '0905.0858-2-4-2': 'While gravity propagates in five dimensions on the largest scales, it becomes 4-dimensional below a certain crossover scale [MATH].', '0905.0858-2-4-3': 'On scales smaller than [MATH], and when gravity is weak, DGP gravity can be described as an effective four-dimensional scalar-tensor theory [CITATION].', '0905.0858-2-4-4': 'The scalar field, referred to as brane bending mode, is massless but has non-linear derivative interactions which suppress the field in high-density environments, restoring GR locally and allowing DGP gravity to pass Solar System constraints.', '0905.0858-2-5-0': 'Applied to homogeneous and isotropic cosmology, the DGP model allows for two solutions [CITATION].', '0905.0858-2-5-1': 'In one branch of the theory, the self-accelerating branch, the effective weakening of gravity on large scales leads to an accelerated late-time expansion of the brane, without any cosmological constant.', '0905.0858-2-5-2': 'The scalar brane-bending mode mediates a repulsive force, leading to a smaller effective gravitational constant on large scales.', '0905.0858-2-5-3': 'In the other solution, the normal branch, there is no accleration, and the brane-bending mode mediates an attractive force.', '0905.0858-2-5-4': 'In linear perturbation theory around its de Sitter limit, the brane bending mode in the self-accelerating DGP model has been shown to have the wrong sign in the kinetic term ("ghost") [CITATION], suggesting an instability, although the situation in the non-linear case is not clear [CITATION].', '0905.0858-2-5-5': 'The normal branch is free of the ghost.', '0905.0858-2-6-0': 'Linear perturbation theory around a cosmological background in DGP has been derived in, e.g. [CITATION].', '0905.0858-2-6-1': 'In this approach, it is possible to predict the expansion history, CMB anisotropies, and the ISW effect in DGP [CITATION].', '0905.0858-2-6-2': 'Using this information, [CITATION] have shown that the self-accelerating DGP model without cosmological constant, which has only one free parameter, [MATH], is disfavored at the [MATH] level by current CMB and Supernova data.', '0905.0858-2-6-3': 'This is due to the earlier onset of acceleration in DGP, and the additional suppression of growth through the brane-bending mode.', '0905.0858-2-7-0': 'While the self-accelerating branch of DGP is thus challenged by theory and observations, much work is being done on extending the DGP model to higher co-dimensions, in the context of degravitation [CITATION].', '0905.0858-2-7-1': 'These higher-dimensional models are expected to bring the expansion history close to that of [MATH]CDM, while exhibiting a similar effective scalar-tensor regime as in the original DGP model (see also [CITATION]).', '0905.0858-2-7-2': 'The form of the non-linear interactions of the brane-bending mode are also generic to these generalized models (see also [CITATION]).', '0905.0858-2-7-3': 'Hence, it will be straightforward to perform simulations of these models once the expansion history and evolution of linear perturbations is worked out.', '0905.0858-2-7-4': 'In addition, the standard DGP model is able to satisfy the cosmological constraints if a cosmological constant (brane tension) is included [CITATION].', '0905.0858-2-8-0': 'So far, studies of the DGP model in the context of cosmology have mostly dealt with the linearized theory, valid on large scales.', '0905.0858-2-8-1': 'However, as soon as perturbations in the matter density become of order unity, the non-linear interactions of the brane bending mode become important (e.g., [CITATION]).', '0905.0858-2-8-2': 'Hence, in order to make predictions for observables in the non-linear regime, the full non-linear equations of motion must be solved in conjunction with the evolution of the matter perturbations.', '0905.0858-2-8-3': 'Recently, perturbative approaches have been developed to extend the predictions into the quasi-linear regime [CITATION].', '0905.0858-2-8-4': 'Also, Khoury and Wyman [CITATION] have used a spherically-symmetric approximation of the brane-bending mode interactions in their N-body simulation of braneworld models.', '0905.0858-2-8-5': 'In this paper, we present the results of an N-body simulation of the self-accelerating DGP model, which self-consistently solves the full non-linear equation for the brane-bending mode and its effect on the motion of particles (we compare our results with the approximation used in [CITATION] in Section [REF]).', '0905.0858-2-8-6': 'We solve the equation of motion in the quasi-static approximation, neglecting time derivatives with respect to spatial derivatives, as is usually done in N-body simulations (Section [REF] presents a consistency test of this approximation).', '0905.0858-2-9-0': 'We present results on the matter power spectrum, the halo mass function (abundance of dark matter halos), and the behavior of the brane bending mode in the cosmological context.', '0905.0858-2-9-1': 'These results can be used to strengthen constraints on the self-accelerating DGP model significantly, e.g., through observations of weak lensing and the abundance of galaxy clusters.', '0905.0858-2-9-2': 'More generally, the simulations presented here can serve as starting point for building a model of non-linear structure formation in braneworld scenarios, and for benchmarking perturbative approaches [CITATION].', '0905.0858-2-10-0': 'In Section [REF], we describe the DGP model, and present the relevant equations and analytical test cases.', '0905.0858-2-10-1': 'Section [REF] presents the code, which is benchmarked in Section [REF].', '0905.0858-2-10-2': 'The cosmological simulations are described in Section [REF].', '0905.0858-2-10-3': 'Results and their impact on cosmological constraints on the DGP model are presented in Section [REF] and Section [REF].', '0905.0858-2-10-4': 'We conclude in Section [REF].', '0905.0858-2-11-0': '# DGP model', '0905.0858-2-12-0': '## Background evolution', '0905.0858-2-13-0': 'The Dvali-Gabadadze-Porrati model [CITATION] consists of a spatially three-dimensional brane in a 4+1 dimensional Minkowski bulk.', '0905.0858-2-13-1': 'Matter and all interactions except gravity are confined to the brane.', '0905.0858-2-13-2': 'The gravitational action consists of the five-dimensional Einstein-Hilbert action plus a term which leads to the 4D gravity limit on small scales: [EQUATION]', '0905.0858-2-13-3': 'Here, [MATH], [MATH] stand for the bulk coordinates and metric, while [MATH], [MATH] are the induced coordinates and metric on the brane, and [MATH], [MATH] denote the corresponding Ricci scalars.', '0905.0858-2-13-4': 'In the following, we will drop the [MATH] notation where no confusion can arise.', '0905.0858-2-13-5': 'The boundary term is added to the action in order to ensure that variation with respect to [MATH] leads to the correct five-dimensional Einstein equations (e.g., [CITATION]).', '0905.0858-2-14-0': 'The two gravitational constants, or Planck masses [MATH], [MATH] appearing in Eq. ([REF]) can be related via a length scale, the crossover scale: [EQUATION]', '0905.0858-2-14-1': 'On scales above [MATH], gravity becomes five-dimensional, with forces falling off as [MATH].', '0905.0858-2-14-2': 'Below [MATH], gravity is four-dimensional, but not Einsteinian gravity, a point to which we return below.', '0905.0858-2-15-0': 'Since all matter is thought as confined to the brane, the five-dimensional metric has to obey non-trivial junction conditions over the brane [CITATION].', '0905.0858-2-15-1': 'Assuming an empty Minkowski bulk, a spatially flat brane, and a homogeneous and isotropic matter distribution on the brane, the junction conditions lead to the following analogue of the Friedmann equation for the scale factor of the induced metric on the brane: [EQUATION]', '0905.0858-2-15-2': 'The junction conditions leave two possible branches of the theory, determined by the sign on the left hand side of Eq. ([REF]).', '0905.0858-2-15-3': "In the following, we focus on the branch with the '[MATH]' sign, which asymptotes to a late-time de Sitter-Universe, [MATH], and is correspondingly called the accelerating branch.", '0905.0858-2-15-4': 'Then, in the matter-dominated epoch, neglecting radiation, and again assuming no cosmological constant or curvature, Eq. ([REF]) can be rewritten as: [EQUATION] and [MATH] is the average matter density today.', '0905.0858-2-15-5': 'This expansion history is clearly different from [MATH]CDM, and corresponds to an effective dark energy with [MATH] in the matter-dominated era at high redshifts.', '0905.0858-2-16-0': '## Cosmological perturbations and brane bending mode', '0905.0858-2-17-0': 'The propagation of light and particles on the DGP brane is completely determined by the perturbed 4D Friedmann-Robertson-Walker metric: [EQUATION]', '0905.0858-2-17-1': 'However, in order to determine the evolution of the metric potentials on the brane, it is necessary to solve the full 5D Einstein equations [CITATION].', '0905.0858-2-17-2': 'An additional scalar degree of freedom associated with local displacements of the brane appears, the so-called brane-bending mode [MATH] which couples to matter.', '0905.0858-2-17-3': 'In our convention, [MATH] is dimensionless, instead of being scaled to the Planck mass [MATH].', '0905.0858-2-18-0': 'In the decoupling limit of DGP [CITATION], when setting gravitational interactions to 0, the self-interactions of the [MATH] field remain constant.', '0905.0858-2-18-1': 'Hence, while perturbations of the metric higher than linear order can be neglected for cosmological studies, it is crucial to consider the self-interactions in the brane-bending mode.', '0905.0858-2-19-0': 'In the quasi-static regime, for scales [MATH], time derivatives can be neglected with respect to spatial derivatives, and the equation for the brane-bending mode reads (e.g., [CITATION]): [EQUATION] where the derivatives are with respect to comoving coordinates [MATH], [MATH] is the matter density perturbation, and the function [MATH] (for the accelerating branch) is given by: [EQUATION]', '0905.0858-2-19-1': 'Note that [MATH] is always negative in the self-accelerating branch.', '0905.0858-2-19-2': 'In Section [REF], we show results of a consistency test of the quasi-static assumption used in Eq. ([REF]).', '0905.0858-2-20-0': 'The brane-bending mode influences the dynamics of particles through the dynamical potential [MATH], which, assuming the same boundary conditions for [MATH] and [MATH], is given by: [EQUATION] where the Newtonian potential [MATH] satisfies the usual Poisson equation: [EQUATION]', '0905.0858-2-20-1': 'In contrast, the propagation of photons, determined by the lensing potential [MATH], is not directly affected by [MATH].', '0905.0858-2-21-0': '## Weak brane regime', '0905.0858-2-22-0': 'If the gradient of the [MATH] field is small, i.e. for small gravitational accelerations, Eq. ([REF]) can be linearized, yielding a standard Poisson equation: [EQUATION]', '0905.0858-2-22-1': 'We will refer to a cosmology with this linearized equation as the linearized DGP model.', '0905.0858-2-22-2': 'In this regime, also called weak-brane phase, [MATH] becomes proportional to the Newtonian potential [MATH], corresponding to a constant rescaling of the gravitational constant through Eq. ([REF]): [EQUATION]', '0905.0858-2-22-3': 'Substituting the linear solution into Eq. ([REF]), we obtain a rough estimate for the overdensity [MATH] at which the non-linear interactions become important: [EQUATION]', '0905.0858-2-22-4': 'For a self-accelerating DGP model that leads to cosmic acceleration today, [MATH], [MATH], and the prefactor in Eq. ([REF]) is of order unity today.', '0905.0858-2-22-5': 'Hence, the self-coupling of the brane bending mode [MATH] becomes important as soon as the matter density field becomes non-linear, [MATH].', '0905.0858-2-23-0': '## Vainshtein effect', '0905.0858-2-24-0': 'In general, Eq. ([REF]) is difficult to solve in full generality due to the non-linearity in the derivative terms.', '0905.0858-2-24-1': 'However, an instructive test case, a spherically symmetric matter distribution, can be solved analytically.', '0905.0858-2-24-2': 'In the spherically symmetric case, Eq. ([REF]) becomes [CITATION]: [EQUATION]', '0905.0858-2-24-3': 'For simplicity, we assume a spherical mass [MATH] of radius [MATH] with uniform density, and set [MATH].', '0905.0858-2-24-4': 'Then, we can integrate Eq. ([REF]) once and obtain the gravitational acceleration in DGP: [EQUATION] where [MATH] is the Newtonian acceleration of the spherical mass, and: [EQUATION]', '0905.0858-2-24-5': 'Here, [MATH] denotes a characteristic scale of the solution, the Vainshtein radius: [EQUATION] and [MATH] is the gravitational radius of the mass.', '0905.0858-2-24-6': 'For very large distances, [MATH], [MATH] approaches the constant [MATH], which exactly matches the linear solution, Eq. ([REF]).', '0905.0858-2-24-7': 'Note also that by substituting [MATH] in Eq. ([REF]), we see that the non-linearity criterion is directly proportional to [MATH].', '0905.0858-2-25-0': 'In the opposite limit, [MATH], [MATH] becomes suppressed with respect to the linear solution, and [MATH] approaches the small constant [MATH] inside the mass (assuming [MATH]).', '0905.0858-2-25-1': 'This Vainshtein effect [CITATION] amounts to restoring GR in deep potential wells, while far away from the mass, gravity is in the scalar-tensor regime.', '0905.0858-2-26-0': 'Fig. [REF] shows the analytical solution for the [MATH] field and the deviation from the Newtonian acceleration, as well as the code results which agree well with the analytical solution in the regime of validity (see Section [REF]).', '0905.0858-2-27-0': '## Plane wave solution', '0905.0858-2-28-0': 'Another exact solution to Eq. ([REF]) exists: for a simple plane wave density perturbation, [MATH].', '0905.0858-2-28-1': 'In this case, the two non-linear terms exactly cancel, and we are left with the linear solution, Eq. ([REF]).', '0905.0858-2-28-2': 'In Appendix [REF] this is used to test our code for perturbations of different wavelengths.', '0905.0858-2-28-3': 'Also, the plane wave and spherically symmetric solutions can be considered as two limiting cases for understanding the [MATH] field behavior in the cosmological context.', '0905.0858-2-28-4': 'We will return to this point in Section [REF].', '0905.0858-2-29-0': '# Code implementation', '0905.0858-2-30-0': 'We use an extension of the code first described in [CITATION] for our simulations of DGP gravity.', '0905.0858-2-30-1': 'The code is a standard particle-mesh N-body code [CITATION] assuming collisionless dark matter only, with a fixed grid of size [MATH] and periodic boundary conditions, and uses second-order accurate leapfrog integration for the particle propagation.', '0905.0858-2-30-2': 'The major addition is a relaxation solver for the non-linear equation of the brane bending mode [MATH], Eq. ([REF]).', '0905.0858-2-30-3': 'The simulation proceeds by performing steps [MATH] in the scale factor [MATH].', '0905.0858-2-31-0': 'Except for the different expansion history, the equations for particle propagation are identical to those in standard CDM simulations, and are expressed in terms of comoving coordinates [MATH], [MATH]: [EQUATION]', '0905.0858-2-31-1': 'Here, [MATH] and [MATH] are the position and momentum of particle [MATH] in code units (see Appendix [REF]), and [MATH] is taken from Eq. ([REF]).', '0905.0858-2-31-2': 'The main modification enters in the determination of [MATH].', '0905.0858-2-31-3': 'One time step of the N-body simulation proceeds as follows:', '0905.0858-2-32-0': 'Thus, in each time step we have to solve for the brane-bending mode [MATH] which contributes to the dynamical potential [MATH] via Eq. ([REF]).', '0905.0858-2-32-1': 'In other words, given the density field, we have to solve the non-linear elliptical differential equation Eq. ([REF]).', '0905.0858-2-32-2': 'We use a Gauss-Seidel relaxation scheme together with the Newton method.', '0905.0858-2-32-3': 'A crucial tool is the multigrid [CITATION], a hierarchy of coarser grids which are essential to speeding up the convergence [CITATION]: as the relaxation scheme, which operates locally, gets rid of small-scale errors very efficiently, the coarser grids quickly reduce the long-wavelength error modes which are hard to eliminate on the fine grid alone.', '0905.0858-2-32-4': 'For details of the implementation, see Appendix [REF].', '0905.0858-2-33-0': 'While the [MATH] field is in general well-behaved, the convergence properties become worse for strongly inhomogeneous density fields, because the non-linearities in Eq. ([REF]) are in the derivatives of the [MATH] field.', '0905.0858-2-33-1': 'In order to reach the desired convergence in our cosmological simulations, it is necessary to smooth the density field entering the r.h.s. of Eq. ([REF]) with a Gaussian kernel [MATH], with [MATH] set to the size of a grid cell.', '0905.0858-2-33-2': 'This smoothes over the noise in the density field due to the discreteness of particles.', '0905.0858-2-34-0': '# Code tests', '0905.0858-2-35-0': 'In this section, we present different tests our code was put through to benchmark its performance and limitations.', '0905.0858-2-35-1': 'We focus on tests applying to the modified gravity sector.', '0905.0858-2-35-2': 'For the results of standard N-body code tests for this code, see [CITATION].', '0905.0858-2-36-0': '## Spherical mass test', '0905.0858-2-37-0': 'In order to study how well the code reproduces the exact result for a spherical mass in DGP, we start out with a grid with [MATH] grid cells and [MATH] particles.', '0905.0858-2-37-1': 'We arbitrarily assume a box size of [MATH] and set [MATH].', '0905.0858-2-37-2': 'All particles are moved into a spherical mass of radius [MATH] and uniform density, which then corresponds to a mass of [MATH] (assuming [MATH]).', '0905.0858-2-37-3': 'Given the density field assigned from the particle positions, we then use the relaxation solver to solve for [MATH], and measure the field values and radial acceleration throughout the box.', '0905.0858-2-38-0': 'For the purposes of this test, we can vary the two parameters [MATH] and [MATH] in Eq. ([REF]) independently.', '0905.0858-2-38-1': '[MATH] determines the strength of the additional force mediated by [MATH] [e.g., Eq. ([REF])], which is attractive for positive [MATH] and repulsive for [MATH].', '0905.0858-2-38-2': 'For given [MATH], [MATH] controls the Vainshtein radius, i.e. the scale where the self-interactions of the [MATH] field become important.', '0905.0858-2-39-0': 'Fig. [REF] shows the field solution (left panel) and relative deviation of the acceleration from the Newtonian value (right panel) as a function of distance from the center, [MATH], for [MATH], [MATH] grid cells, corresponding to [MATH], and [MATH].', '0905.0858-2-39-1': 'The agreement with the analytical solution (thick lines) is generally very good, except at large radii, where the periodic boundary conditions become important, and around [MATH], where artifacts of interpolating a sphere onto a cubic grid become visible.', '0905.0858-2-39-2': 'At distances of order the size of the grid cells, the force resolution becomes worse leading to a growing scatter in the measured acceleration.', '0905.0858-2-39-3': 'Note that the analytical solution necessarily assumes an isolated mass, since the superposition principle cannot be used due to the non-linear [MATH] field equation.', '0905.0858-2-39-4': 'For this reason, we added an arbitrary zero-point to the [MATH] field in order to match the analytical solution in the left panel of Fig. [REF].', '0905.0858-2-39-5': 'Of course, such a zero-point does not affect any observable quantity, such as the acceleration shown in the right panel.', '0905.0858-2-40-0': 'We tested the field solution for several different values of [MATH], [MATH], [MATH], and [MATH].', '0905.0858-2-40-1': 'Given the caveats pointed out, we found very good convergence to the analytical solution.', '0905.0858-2-40-2': 'In particular, increasing [MATH] reduces the interpolation artifacts at [MATH], while decreasing [MATH] reduces the impact of the periodic boundary conditions.', '0905.0858-2-40-3': 'Solutions for different values of [MATH] are shown in both panels of Fig. [REF].', '0905.0858-2-40-4': 'While the solid lines show the exact solution to the full equation, Eq. ([REF]), the dotted lines show the linearized result, Eq. ([REF]).', '0905.0858-2-40-5': 'Clearly, for the parameters chosen, the effect of the non-linearity is significant for [MATH] of a few [MATH] or less.', '0905.0858-2-41-0': '## Convergence and resolution tests', '0905.0858-2-42-0': 'In most test cases we studied, the density field is sufficiently smooth that the relaxation converges to the desired low tolerance.', '0905.0858-2-42-1': 'However, for density fields with considerable small-scale inhomogeneities, such as the cosmological density field at late times, we find that this level is not achievable.', '0905.0858-2-42-2': 'This is because the non-linearity of the [MATH] equation is important as soon as the overdensity [MATH] becomes of order unity, which in the N-body simulation corresponds to [MATH] particle per grid cell.', '0905.0858-2-42-3': 'Equivalently, the Vainshtein radius for a single particle in our N-body simulation is of order the grid scale (see Fig. [REF], bottom left).', '0905.0858-2-42-4': 'Thus, the [MATH] field reacts much more strongly to noise due to the discreteness of particles than the Newtonian potential, leading to increased residual errors of the approximate solution.', '0905.0858-2-43-0': 'If we write Eq. ([REF]) in code units (see Appendix [REF]) as [MATH], then for an approximate solution [MATH], [MATH] is the dimensionless residual, where [MATH] is the size of a grid cell.', '0905.0858-2-43-1': 'In the following, we use the RMS of the dimensionless residual, [MATH] as a benchmark for the convergence of the field solution.', '0905.0858-2-43-2': 'We performed tests with sine wave density fields of various wavelengths in order to determine what residual [MATH] is acceptable in our simulations (see Appendix [REF]).', '0905.0858-2-43-3': 'In case of a pure sine wave, the non-linearity in Eq. ([REF]) vanishes, and the exact non-linear solution is identical to the linearized solution.', '0905.0858-2-43-4': 'We found that residuals of [MATH] are safe, as for residuals at that level, the errors in the solution are negligible compared to the unavoidable truncation errors from taking numerical derivatives on the grid.', '0905.0858-2-44-0': 'In order to reduce the noise in the density field in the cosmological simulations and improve the solution to the acceptable level of [MATH], we increase the number of particles from [MATH] to [MATH], and smooth the density field entering the r.h.s. of Eq. ([REF]) as described in Section [REF].', '0905.0858-2-44-1': 'With these steps, the solution converges with a residual [MATH] for all box sizes (left panel of Fig. [REF]).', '0905.0858-2-44-2': 'Note that the dimensionless RMS residuals never exceed a fraction of [MATH] of the RMS of the [MATH] field solution.', '0905.0858-2-45-0': 'Since the force resolution in our simulations is in any case limited to scales above [MATH], a smoothing on the grid scale is not expected to degrade the resolution of the simulations significantly.', '0905.0858-2-45-1': 'We checked this by running linearized DGP simulations given by Eq. ([REF]) (Section [REF]) using the same smoothing of the r.h.s. of Eq. ([REF]), and comparing the resulting power spectrum to that of linearized DGP simulations without smoothing, for the same initial conditions.', '0905.0858-2-45-2': 'The result is shown in the right panel of Fig. [REF].', '0905.0858-2-45-3': 'As expected, the effect of smoothing increases towards larger [MATH].', '0905.0858-2-45-4': 'The smoothing effect on the matter power spectrum is positive, since the smoothing removes power in the [MATH] field on the smallest scales, and [MATH] mediates a repulsive force.', '0905.0858-2-45-5': 'This leads to an enhancement in the matter power spectrum.', '0905.0858-2-46-0': 'The smoothing effect remains below 4% for [MATH], which we adopt as the maximum wave number considered for each box, where [MATH] is the Nyquist frequency of the grid.', '0905.0858-2-46-1': 'From our studies with different smoothing radii [MATH], we found that the smoothing effects on full DGP simulations show a similar [MATH]-dependence as those of the linearized DGP simulations, but are smaller by a factor of [MATH].', '0905.0858-2-46-2': 'This is understandable since the [MATH] field is suppressed in dense regions in the full simulations, reducing the effect of the smoothing.', '0905.0858-2-46-3': 'Taking into account this factor, we correct the power spectra measured in the full DGP simulations for the smoothing effects using the curves shown in Fig. [REF] (right panel).', '0905.0858-2-46-4': 'Note that in any case these effects are at the level of few percent or less.', '0905.0858-2-47-0': 'In order to assess the effect of the finite grid resolution, we also performed simulations with [MATH] and [MATH], and [MATH].', '0905.0858-2-47-1': 'Fig. [REF] (right panel) also shows the deviation of the power spectrum in these low-resolution simulations from the [MATH] simulation with the same initial conditions.', '0905.0858-2-47-2': 'The vertical lines indicate [MATH] for each case, i.e. the maximum [MATH] we consider for each simulation box.', '0905.0858-2-47-3': 'Below [MATH], the deviations are less than 10% for either [MATH] or [MATH].', '0905.0858-2-47-4': 'Note also that the resolution effects are independent of the type of simulation (DGP, linearized DGP, or GR).', '0905.0858-2-47-5': 'Hence, they cancel when measuring the deviation of [MATH] between different simulation types, and we expect this deviation to be accurate to the percent level for the range in [MATH] we consider.', '0905.0858-2-48-0': '## Quasi-static approximation', '0905.0858-2-49-0': 'In solving for the brane bending mode [MATH], we have assumed that all time-derivative terms in the full equation of motion of [MATH] (in terms of physical coordinates): [EQUATION] can be neglected with respect to the spatial derivatives in Eq. ([REF]).', '0905.0858-2-49-1': 'Up to second order, there are three time derivative terms which have been neglected: [EQUATION] [MATH] and [MATH] appear as [MATH] in the equations of motion.', '0905.0858-2-49-2': 'While we cannot rigorously prove that the quasi-static approximation holds, we can perform a consistency test by measuring the terms Eqs. ([REF])-([REF]) in the simulations, and checking whether they are indeed small.', '0905.0858-2-49-3': 'The time derivatives of [MATH] are calculated in each grid cell using the two neighboring time steps.', '0905.0858-2-49-4': 'We then take the RMS of each term over the grid, [MATH], and compare with the RMS of the spatial Laplacian, [MATH].', '0905.0858-2-49-5': 'Fig. [REF] presents [MATH] and [MATH] relative to the spatial derivatives for our largest and smallest box sizes, and shows that they are well under control.', '0905.0858-2-49-6': 'The relative magnitude of all time derivatives are of order a few [MATH] or less.', '0905.0858-2-49-7': 'This is in keeping with the expectation that [MATH], and [MATH] in our simulations.', '0905.0858-2-49-8': 'We conclude that the quasi-static approach is a self-consistent approximation.', '0905.0858-2-50-0': '# Cosmological simulations', '0905.0858-2-51-0': "We performed a suite of cosmological simulations for three different types of gravity: unmodified GR, corresponding to a smooth dark energy model with the same expansion history as DGP, referred to as QCDM; linearized DGP gravity, using the linearized [MATH] equation of motion [Eq. ([REF])], corresponding to a time-dependent rescaling of Newton's constant; and full DGP, solving for the full non-linear [MATH] solution [Eq. ([REF])].", '0905.0858-2-51-1': 'Comparing linearized DGP with full DGP allows us to study the Vainshtein effect in a cosmological setting.', '0905.0858-2-52-0': 'The cosmological parameters are those of the best-fit flat self-accelerating DGP model to WMAP 5yr data [CITATION] and are summarized in Tab. [REF].', '0905.0858-2-52-1': 'We generated the initial conditions at [MATH]) from a modified transfer function output of CAMB [CITATION] for a flat [MATH]CDM model with [MATH].', '0905.0858-2-52-2': 'The [MATH]CDM transfer function was corrected for small early-time modified gravity effects in the DGP model using the PPF approach, as detailed in Appendix [REF].', '0905.0858-2-53-0': 'The simulations were run with [MATH] grid cells on a side, and [MATH] particles, i.e. one particle per grid cell, to reduce the shot noise in the density field (see Section [REF]).', '0905.0858-2-53-1': 'We performed six simulations each for four different box sizes, from [MATH] up to [MATH] (Tab. [REF]).', '0905.0858-2-53-2': 'On an 8-core machine, the QCDM and linearized DGP runs require [MATH]10h of computing time, while the full DGP simulations require about 300h.', '0905.0858-2-54-0': 'Fig. [REF] (left panel) shows the combined power spectrum from all box sizes for the QCDM simulations, including bootstrap errors.', '0905.0858-2-54-1': 'Also shown is the non-linear power spectrum for QCDM, calculated from the linear power spectrum using the halofit procedure [CITATION].', '0905.0858-2-54-2': 'The power spectrum measured in each box is used up to [MATH] (see Tab. [REF]), and different boxes are combined weighting by volume.', '0905.0858-2-54-3': 'The lower panel of Fig. [REF] (left) shows the power spectrum relative to the halofit prediction, measured separately for each box size.', '0905.0858-2-54-4': 'The power spectra measured in different boxes clearly agree within the errors, and the deviations from the halofit prediction are within the accuracy ([MATH]%) expected from this fitting procedure (especially given the significant departures from [MATH]CDM of the simulated expansion history).', '0905.0858-2-55-0': 'Fig. [REF] also shows the non-linear power spectrum for a flat [MATH]CDM cosmology fixed to the same [MATH], [MATH], and primordial normalization, corresponding to a linear normalization of [MATH] today.', '0905.0858-2-55-1': 'Note that due to the different expansion history and the earlier onset of acceleration in DGP, the linear growth factor is suppressed by [MATH] in QCDM relative to [MATH]CDM.', '0905.0858-2-55-2': 'The repulsive brane-bending mode in DGP suppresses growth further, leading to a suppression in the linear regime of [MATH] relative to [MATH]CDM.', '0905.0858-2-55-3': 'In the following, we will always compare the DGP results with QCDM, so that the expansion history effects are taken out, and all deviations are strictly due to modifications of gravity.', '0905.0858-2-56-0': 'We have also measured the mass function of dark matter halos in our simulations.', '0905.0858-2-56-1': 'The spherical overdensity halo-finding and combination of different simulation boxes was done as described in [CITATION].', '0905.0858-2-56-2': 'As our grid resolution is the same as in the simulations analyzed in [CITATION], while we have a factor of 8 more particles, we conservatively increase the minimum required number of particles from 800 to 6400.', '0905.0858-2-56-3': 'The corresponding mass thresholds are given in Tab. [REF].', '0905.0858-2-56-4': 'We define the halo mass [MATH] as the mass enclosed within a sphere of radius [MATH], so that the average density within the sphere is 200 times the average matter density in the universe, [MATH].', '0905.0858-2-57-0': 'Fig. [REF] (right panel) shows the mass function, [MATH], measured in QCDM simulations, in comparison with the prediction calculated from the linear QCDM power spectrum using the fitting formula from Tinker et al.[CITATION].', '0905.0858-2-57-1': 'The lower panel shows the relative deviations from the prediction, separately for each simulation box.', '0905.0858-2-57-2': 'Within the mass range accessible with our simulations, [MATH], the agreement with the prediction and among different boxes is good.', '0905.0858-2-57-3': 'We also show the predicted mass function for a [MATH]CDM model with the same initial power spectrum in Fig. [REF].', '0905.0858-2-57-4': 'Apparently, the number of halos above [MATH] is significantly reduced due to the suppressed growth in QCDM.', '0905.0858-2-58-0': '# Results', '0905.0858-2-59-0': 'In order to get a visual impression of some of the physics in DGP N-body simulations, we show slices through one simulation box at [MATH]) in Fig. [REF].', '0905.0858-2-59-1': 'The slices are 64 cells thick, and for each pixel we take the maximum absolute values of each quantity over the thickness of the slice, for better visibility.', '0905.0858-2-59-2': 'The density field (top left; in logarithmic scale) is difficult to distinguish visually from the QCDM result for the same run, as the effects on the power spectrum are at the [MATH]% level (Section [REF]).', '0905.0858-2-59-3': 'The top right panel in Fig. [REF] shows the dynamical potential [MATH], exhibiting the potential wells of massive collapsed structures.', '0905.0858-2-59-4': 'The brane-bending mode [MATH] is shown in the lower left panel.', '0905.0858-2-59-5': 'On linear scales, [MATH] is proportional to the potential [MATH] (Section [REF]), but evidently does not follow the potential within deeper potential wells, making it appear smoother.', '0905.0858-2-59-6': 'To make this more clear, we show the difference of the [MATH] solution from the linear value [Eq. ([REF])], [MATH], in the lower right panel of Fig. [REF]: [MATH] is suppressed ([MATH]) in overdense regions, showing the Vainshtein effect at work in a cosmological setting.', '0905.0858-2-59-7': 'Note that quite low-mass structures which are not conspicuous in the potential [MATH] already lead to a suppression of [MATH].', '0905.0858-2-59-8': 'See Section [REF] for a discussion of the Vainshtein effect in dark matter halos.', '0905.0858-2-60-0': '## Power spectrum', '0905.0858-2-61-0': 'Fig. [REF] (left panel) shows the relative deviation of the linearized and full DGP power spectra from the QCDM result at redshift 0.', '0905.0858-2-61-1': 'The error bars are obtained from the 6 separate runs using a bootstrap procedure.', '0905.0858-2-61-2': 'By comparing simulation runs with the same initial conditions, and then averaging the deviations, most of the cosmic variance cancels out and we are able to obtain significantly less scatter.', '0905.0858-2-61-3': 'Also shown is the predicted deviation in the linear power spectrum.', '0905.0858-2-61-4': 'Note first that in the self-accelerated DGP branch, the scalar field mediates a repulsive force, leading to a suppression of the growth of structure.', '0905.0858-2-61-5': 'Due to the scale-invariant modification of gravity in linearized quasi-static DGP [Eq. ([REF])], the predicted linear deviation is also scale-independent.', '0905.0858-2-61-6': 'On linear scales, [MATH], both linearized and full DGP simulations agree well with the linear prediction.', '0905.0858-2-62-0': 'Apparently, the full DGP result departs significantly from linearized DGP on quasi-linear to non-linear scales.', '0905.0858-2-62-1': 'The Vainshtein effect begins to operate wherever overdensities become of order unity, and suppresses the deviation from GR.', '0905.0858-2-62-2': 'Note that small effects of the non-linear [MATH] equation can already be seen on quite large scales, [MATH] corresponding to [MATH], not far from the acoustic features in the matter power spectrum.', '0905.0858-2-62-3': 'While we do not expect dramatic effects on cosmological parameter constraints from BAO, a modeling of these non-linear effects will be necessary for precision constraints in the context of DGP and similar braneworld models.', '0905.0858-2-63-0': 'We also show the deviation of [MATH] from [MATH], where [MATH] is calculated from the corresponding linear power spectrum using the standard halofit prescription.', '0905.0858-2-63-1': 'halofit describes the linearized DGP power spectrum reasonably well up to [MATH], in agreement with the findings of [CITATION], while it follows neither the linearized nor full DGP at higher [MATH].', '0905.0858-2-63-2': 'We have not explored whether phenomenological modifications of halofit allow for an improvement of the fit.', '0905.0858-2-64-0': 'The dashed points at high [MATH] extend the range in frequency up to [MATH] of our smallest box, [MATH].', '0905.0858-2-64-1': 'While resolution effects are still expected to cancel out to first order in this representation, these points only serve to indicate that the trends seen for full and linear DGP at lower [MATH] continue towards smaller scales.', '0905.0858-2-65-0': 'The right panel of Fig. [REF] shows the evolution of the modified gravity effects on the power spectrum as function of redshift.', '0905.0858-2-65-1': 'On large scales [MATH], the deviation is almost scale-free and evolves as predicted by linear theory.', '0905.0858-2-65-2': 'At earlier times, the density field is non-linear only on smaller scales.', '0905.0858-2-65-3': 'Hence, the Vainshtein effect becomes visible in the power spectrum only at higher [MATH].', '0905.0858-2-65-4': 'However, it does affect the power spectrum deviation significantly already at [MATH].', '0905.0858-2-66-0': '## Halo mass function', '0905.0858-2-67-0': 'The abundance of massive dark matter halos is a sensitive probe of the growth of structure in the Universe [CITATION].', '0905.0858-2-67-1': 'In particular, the number of the most massive halos which host galaxy clusters depends exponentially on the amplitude of the matter power spectrum.', '0905.0858-2-67-2': 'Fig. [REF] shows the effect of the modification of gravity in DGP on the halo mass function, relative to the QCDM effective dark energy cosmology.', '0905.0858-2-67-3': 'We investigated the effect of the smoothing used in the full DGP simulations by comparing linearized DGP simulations with and without smoothing, as done for the power spectrum (Section [REF]).', '0905.0858-2-67-4': 'Above our rather conservative mass threshold for each box, we only found a small effect of the smoothing on halo masses in the linearized DGP simulations.', '0905.0858-2-67-5': 'Since the [MATH] field is in any case suppressed within halos due to the Vainshtein mechanism, we expect the smoothing effects to be even smaller for the full DGP simulations.', '0905.0858-2-68-0': 'As expected, the full DGP simulations are somewhat closer to QCDM than the linearized DGP case in Fig. [REF].', '0905.0858-2-68-1': 'The suppression in the mass function is significant for [MATH], reaching more than 30% for massive cluster-size halos.', '0905.0858-2-68-2': 'The suppression is smaller for lower-mass halos.', '0905.0858-2-68-3': 'This is presumably because these halos formed earlier in cosmic history, when the modified gravity effects of DGP were significantly smaller.', '0905.0858-2-69-0': 'Note that when compared to [MATH]CDM, this suppression comes in addition to the larger suppression of the mass function due to the expansion history of DGP (see dashed line in Fig. [REF]).', '0905.0858-2-69-1': 'However, we expect the effect of the DGP modification of gravity to be fairly independent of the expansion history.', '0905.0858-2-69-2': 'In particular, we expect a similar effect on the halo mass function in generalized braneworld models which exhibit an expansion history close to [MATH]CDM [CITATION].', '0905.0858-2-69-3': 'Hence, the mass function is expected to be a sensitive probe for braneworld modified gravity models, as is the case for [MATH] gravity [CITATION].', '0905.0858-2-69-4': 'Note that for the normal branch of braneworld gravity, the sign of the effect in Fig. [REF] will be reversed, leading to an enhancement of the number of massive halos.', '0905.0858-2-70-0': '## Brane bending mode and Vainshtein effect', '0905.0858-2-71-0': 'Fig. [REF] (left panel) shows the average profiles of the brane-bending mode [MATH], the dynamical potential [MATH], and the lensing potential [MATH] of dark matter halos in the full DGP simulations.', '0905.0858-2-71-1': 'The brane-bending mode has been scaled by [MATH], so that it agrees with [MATH] for the linearized solution.', '0905.0858-2-71-2': 'We only use our highest resolution simulation box for this measurement ([MATH]).', '0905.0858-2-71-3': 'The inner radius of the profiles is given by one grid cell.', '0905.0858-2-71-4': 'The profiles were measured by spherically averaging many lines of sight for each halo with [MATH], and then stacked, scaling each profile to the radius [MATH] of the respective halo.', '0905.0858-2-71-5': 'This was done in order to reduce the significant scatter in the profiles.', '0905.0858-2-72-0': 'While the lensing potential [MATH] obeys the standard Poisson equation in DGP, the dynamical potential [MATH] receives a contribution from the brane bending mode [Eq. ([REF])].', '0905.0858-2-72-1': 'Apparently, the [MATH] field flattens out towards the halo center, which is in qualitative agreement with the non-linear field solution for a spherical mass (see, e.g. Fig. [REF]).', '0905.0858-2-72-2': 'Note that [MATH] turns away from the linear solution when the overdensity is of order a few, in agreement with the estimate in Section [REF].', '0905.0858-2-73-0': 'The quantity which is actually observable however is the gradient of the field, shown in Fig. [REF] (right panel), which gives essentially the deviation of the acceleration from the Newtonian acceleration.', '0905.0858-2-73-1': 'For each halo, we measure the radial gradient relative to the halo center.', '0905.0858-2-73-2': 'We again scaled [MATH] by [MATH] to match the linearized solution to [MATH].', '0905.0858-2-73-3': 'From the halo exterior towards the interior, the gradient of [MATH] grows initially, turns around at [MATH], and shrinks for even smaller radii.', '0905.0858-2-73-4': 'As is clearly shown by comparing the dynamical potential [MATH] with its Newtonian value, [MATH], any observable deviations from GR are indeed suppressed within massive halos.', '0905.0858-2-73-5': 'We find that for halos between [MATH] and [MATH], the deviation in the acceleration is around [MATH] in the inner parts, while it is suppressed down to [MATH] for halos around [MATH].', '0905.0858-2-73-6': 'This trend with mass is expected, given that [MATH] for the spherically symmetric solution.', '0905.0858-2-74-0': 'In order to model the behavior of the brane bending mode in the cosmological context, we can make use of the two limiting cases presented in Section [REF].', '0905.0858-2-74-1': 'One is the spherically symmetric solution (Section [REF]), which can be used in modeling a spherical collapse of dark matter halos in DGP [CITATION].', '0905.0858-2-74-2': 'Moreover, Khoury and Wyman [CITATION] have based the approximate field solution in their N-body simulation on this limiting case.', '0905.0858-2-74-3': 'More realistically however, cosmic structure forms by collapsing into non-radially symmetric structures such as filaments.', '0905.0858-2-74-4': 'The plane wave density perturbation can serve as another, albeit rather extreme, test case.', '0905.0858-2-74-5': 'For such a perturbation, the non-linear self-coupling of [MATH] vanishes (Section [REF]).', '0905.0858-2-74-6': 'Hence, we expect the Vainshtein effect to be typically weakened in a non-radially symmetric setting, compared to the spherically symmetric case.', '0905.0858-2-75-0': 'In their ansatz, Khoury and Wyman assumed that the spherically symmetric solution Eqs. ([REF])-([REF]) holds wherever the field becomes non-linear.', '0905.0858-2-75-1': 'Since in this solution, the gravitational constant is rescaled locally by [MATH], the Poisson equation for [MATH] can be written in this ansatz as: [EQUATION] where [MATH] is now a function of the local overdensity.', '0905.0858-2-75-2': 'Substituting [MATH] in Eq. ([REF]) [or Eq. ([REF])] we see again that [MATH], with an order unity coefficient.', '0905.0858-2-75-3': 'Khoury and Wyman [CITATION] chose: [EQUATION]', '0905.0858-2-75-4': 'Hence, in the DGP model we simulated, [MATH].', '0905.0858-2-75-5': 'Given the density field in our DGP simulations, we can solve Eq. ([REF]) by Fourier transform as done in [CITATION], subtract the Newtonian potential, and compare the brane bending mode [MATH] from this ansatz with our numerical solution of the full [MATH] equation.', '0905.0858-2-76-0': 'Fig. [REF] (right panel) shows the result for stacked halo profiles of the radial gradient of [MATH].', '0905.0858-2-76-1': 'While [MATH] shows a qualitatively similar behavior to the full solution, the suppression due to the Vainshtein effect appears to set in at significantly larger radii in [MATH] than in the full solution.', '0905.0858-2-76-2': 'This is qualitatively in agreement with our expectation that non-radially symmetric configurations experience a weaker non-linear suppression.', '0905.0858-2-76-3': 'The fact that [MATH] does not approach the linear solution at large radii is a result of the approximation Eq. ([REF]), as was derived in the appendix of [CITATION].', '0905.0858-2-76-4': 'The precise tensorial structure of Eq. ([REF]) restores [MATH] to the linear solution at large distances [MATH].', '0905.0858-2-76-5': 'It would be worth investigating whether this simplified (and computationally much less expensive) approach can be extended to recover the linear [MATH] solution at large scales, an essential feature of the full brane-bending mode interactions.', '0905.0858-2-77-0': 'Although we did not compare our results with a full simulation based on the spherically symmetric approach of [CITATION], these results seem to indicate that this approximation overestimates the non-linear suppression of the brane-bending mode, which might affect observables such as the power spectrum or halo mass function.', '0905.0858-2-77-1': 'We point out however that the crossover scale [MATH] in the braneworld-inspired model considered in [CITATION] is an order of magnitude smaller than our [MATH], and the non-linearity in Eqs. ([REF])-([REF]) only becomes effective at much higher overdensities [MATH], so that for the models studied in [CITATION], the solution is possibly less sensitive to this approximation.', '0905.0858-2-78-0': '# Constraints on the self-accelerating DGP model', '0905.0858-2-79-0': 'We now briefly discuss the impact of our simulation results on cosmological constraints on the self-accelerating DGP model without [MATH].', '0905.0858-2-79-1': 'First, regarding, baryon acoustic oscillations, our results for the DGP power spectrum on quasi-linear scales show that non-linear effects are not significantly enhanced at the BAO scale.', '0905.0858-2-79-2': 'We estimate that any modified gravity corrections to the BAO scale are below the percent level, and hence well within the uncertainties of current BAO measurements [CITATION].', '0905.0858-2-79-3': 'Including baryon acoustic oscillations will increase the power of the combined CMB and Supernova constraints on the self-accelerating DGP model [CITATION] with non-zero curvature to [MATH] [CITATION].', '0905.0858-2-80-0': 'Second, weak lensing measurements constrain the amplitude of the matter power spectrum today, which in this model is significantly smaller due to the suppression of growth by the earlier onset of acceleration and the repulsive force mediated by the brane-bending mode.', '0905.0858-2-80-1': 'The linear power spectrum normalization at [MATH] in the models we simulated is [MATH] and [MATH], while for a [MATH]CDM model with the same primordial normalization, it is [MATH].', '0905.0858-2-80-2': 'We found that the non-linear matter power spectrum in the full DGP simulations is always below that of QCDM (up to [MATH], Fig. [REF]).', '0905.0858-2-80-3': 'Hence, the suppression of the non-linear power spectrum in the self-accelerating DGP model corresponds to a reduction in the inferred linear normalization [MATH] today of at least 0.1, with respect to a [MATH]CDM model with the same initial conditions.', '0905.0858-2-80-4': 'Such a deviation is disfavored by about 1.5 standard deviations by current weak lensing measurements [CITATION].', '0905.0858-2-81-0': 'The abundance of massive dark matter halos is probed by cluster surveys.', '0905.0858-2-81-1': 'We showed in Section [REF] that the abundance of halos above [MATH] is suppressed by around 20% relative to QCDM, which itself only predicts half as many halos in that mass range as a [MATH]CDM model with the same primordial power.', '0905.0858-2-81-2': 'The latter again corresponds to a shift in [MATH] by 0.1, which is constrained to more than [MATH] by X-ray cluster measurements [CITATION], when taking into account the systematic errors.', '0905.0858-2-82-0': '# Conclusions', '0905.0858-2-83-0': 'The N-body simulations of DGP gravity we presented here show how the self-interactions of the brane bending mode [MATH], which are responsible for restoring General Relativity in dense environments, influence the formation of structure in the universe.', '0905.0858-2-83-1': 'In the self-accelerating DGP model we simulated, this scalar field mediates a repulsive force, effectively weakening gravity.', '0905.0858-2-83-2': 'We indeed find that the field solution turns away from the linearized solution whenever the matter overdensity [MATH] becomes of order a few, and that the repulsive force is suppressed by more than an order of magnitude compared to its linearized value in the center of massive halos.', '0905.0858-2-83-3': 'We also compared our full solution to that obtained in the approximate ansatz of [CITATION].', '0905.0858-2-83-4': 'While their ansatz agrees well with our results in the densest regions, it seems to overestimate the non-linear suppression in less dense regions, such as the outer regions and environments of halos.', '0905.0858-2-83-5': 'This is in line with our finding that the non-linear suppression of [MATH] is weaker for non-spherically symmetric configurations.', '0905.0858-2-84-0': 'The non-linear matter power spectrum in the DGP simulations shows that the suppression due to the repulsive [MATH] field is amplified on non-linear scales for the linearized DGP simulations.', '0905.0858-2-84-1': 'In the full DGP simulations, this suppression is smaller, and eventually turns around on Mpc scales.', '0905.0858-2-84-2': 'The deviations between linerized and full DGP power spectra are noticeable already on quasi-linear scales, [MATH].', '0905.0858-2-84-3': 'At the BAO scale, modified gravity effects on the power spectrum are at the percent-level.', '0905.0858-2-85-0': 'We found that the abundance of massive dark matter halos is significantly suppressed in the DGP simulations, compared to a standard gravity simulation with the same expansion history.', '0905.0858-2-85-1': 'Again, the non-linear suppression of the [MATH] field alleviates this suppression, and hence has to be taken into account when using observations to constrain this type of modified gravity.', '0905.0858-2-85-2': 'The effect on the halo mass function is in fact large enough to make this an interesting observational probe of more general braneworld scenarios.', '0905.0858-2-86-0': 'Independently of the CMB constraints [CITATION], our results on the non-linear structure formation strongly constrain the self-accelerating DGP model (without [MATH]).', '0905.0858-2-86-1': 'In the future, we plan to extend our simulations to the normal branch of DGP, and more general braneworld(-inspired) models [CITATION].', '0905.0858-2-86-2': 'The key part of the simulations, solving for the non-linear interactions of the brane-bending mode, is generic to all of these models, and hence the code will be readily generalized to these cases.', '0905.0858-2-86-3': 'We also plan to attempt a modeling of the modified gravity effects, in the context of the halo model for example.', '0905.0858-2-86-4': 'This model can then serve as a framework for cosmological constraints on braneworld gravity which take into account the non-linear mechanisms inherent to this model consistently.', '0905.0858-2-86-5': 'Simulations of both [MATH] gravity and DGP have shown that most interesting phenomena appear on scales of 1 to tens of Mpc, and in the abundance and environments of clusters.', '0905.0858-2-86-6': 'Fortunately, these scales are well accessible to observations, e.g., through weak lensing, the Lyman-[MATH] forest, and cluster surveys, which can then be used as precision probes of gravity on cosmological scales.', '0905.0858-2-87-0': 'I am indebted to Scott Dodelson, Wayne Hu, and Andrey Kravtsov for invaluable input and guidance.', '0905.0858-2-87-1': 'I would like to thank Hiro Oyaizu and Marcos Lima for our previous collaborative work on the code and analysis, and Angela Olinto, Mike Gladders, Cora Dvorkin, Sam Leitner, Michael Mortonson, and Amol Upadhye for discussions.', '0905.0858-2-87-2': 'The support of the Fermilab computing staff is gratefully acknowledged.', '0905.0858-2-88-0': 'The simulations used in this work have been performed on the Joint Fermilab - KICP Supercomputing Cluster, supported by grants from Fermilab, Kavli Institute for Cosmological Physics, and the University of Chicago.', '0905.0858-2-88-1': 'This work was supported by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142.', '0905.0858-2-89-0': '# Code implementation and discretization', '0905.0858-2-90-0': 'This appendix describes details of the N-body code implementation, focusing on the relaxation solver for Eq. ([REF]) as the main non-standard part of the code.', '0905.0858-2-90-1': 'The code is written in C++, uses OpenMP for parallelization of most time-critical operations, and employs FFTW [CITATION] for the Fast Fourier Transforms.', '0905.0858-2-91-0': '## Code units', '0905.0858-2-92-0': 'The comoving code units follow the convention used in [CITATION], where [EQUATION] and [EQUATION]', '0905.0858-2-92-1': 'In the above definitions, [MATH] is the comoving simulation box size in [MATH], [MATH] is the number of grid cells in each direction, [MATH] is the Hubble parameter today, [MATH] is the critical density today, and [MATH] is the fraction of non-relativistic matter today relative to the critical density.', '0905.0858-2-93-0': 'Transformed to code units, the equation for [MATH] becomes: [EQUATION]', '0905.0858-2-93-1': 'Here, [MATH] acts with respect to code coordinates, [MATH], [MATH], and [MATH] is the matter overdensity on the grid, determined from the particle positions.', '0905.0858-2-93-2': 'Note that we have not yet scaled [MATH] to [MATH].', '0905.0858-2-93-3': 'After solving Eq. ([REF]), [MATH] is added to the standard Newtonian potential, which is solved for using a Fast Fourier Transform, to obtain the dynamical potential in DGP: [EQUATION]', '0905.0858-2-94-0': '## Discretization of [MATH] equation', '0905.0858-2-95-0': 'We employ standard second-order symmetric differences for the discretization of the second derivatives in Eq. ([REF]): [EQUATION] and correspondingly for derivatives with respect to [MATH], [MATH].', '0905.0858-2-95-1': 'Here, [MATH] stand for the grid indices, and the step size [MATH] for the base grid and [MATH] for the grid of refinement level [MATH].', '0905.0858-2-95-2': 'The finite differences are evaluated before squaring in calculating the non-linear terms in Eq. ([REF]).', '0905.0858-2-96-0': 'We have tried different discretizations, such as going to higher order in the finite differences, and solving for the deviation of [MATH] from the solution of the linearized equation [Eq. ([REF])], instead of solving for [MATH] itself.', '0905.0858-2-96-1': 'The performance of those discretizations is comparable to or worse than the simple discretization Eqs. ([REF])-([REF]).', '0905.0858-2-96-2': 'This is understandable, since going to higher order amounts to making the derivative operations less local in order to be sensitive to error modes of longer wavelength.', '0905.0858-2-96-3': 'However, in our multigrid relaxation scheme, this is already done efficiently by the coarser grids, so going to higher order in a single relaxation does not improve performance.', '0905.0858-2-96-4': 'Hence, we stay with the straightforward discretization, Eqs. ([REF])-([REF]).', '0905.0858-2-97-0': '## Relaxation algorithm', '0905.0858-2-98-0': 'In general, a relaxation scheme operates by iteratively obtaining a better approximation to the solution [MATH] given a previous guess, [MATH].', '0905.0858-2-98-1': 'On the grid, the solution is updated successively for each cell [MATH]: [EQUATION] where [MATH] is the discretized field equation solved for [MATH].', '0905.0858-2-98-2': 'In case of a linear field equation, [MATH] can be solved for [MATH] straightforwardly.', '0905.0858-2-98-3': "However, in our case the field equation is non-linear in [MATH], and we instead solve for [MATH] iteratively via Newton's method.", '0905.0858-2-98-4': 'Writing the field equation Eq. ([REF]) as [MATH], we need to solve: [EQUATION] for [MATH], where we have suppressed the dependences of [MATH] on neighboring grid cells.', '0905.0858-2-98-5': "By expanding [MATH] in a Taylor series around the current approximation, one step of Newton's method works by updating [MATH] with: [EQUATION]", '0905.0858-2-98-6': 'Our relaxation step is thus given by Eq. ([REF]), where [MATH] is determined from Eq. ([REF]) and the discretization Eqs. ([REF])-([REF]).', '0905.0858-2-98-7': 'In practice, the order in which we loop over grid cells in performing the relaxation Eq. ([REF]) is important, since each relaxation step depends on the neighboring grid cells.', '0905.0858-2-98-8': 'In particular, we need to take care when parallelizing the relaxation.', '0905.0858-2-98-9': 'We use a generalized red-black scheme (see, e.g., [CITATION]), by successively running over cells with [MATH] modulo 4 = [MATH], [MATH].', '0905.0858-2-98-10': 'This was done in order to break dependences due to neighboring cells within each set, in particular due to the mixed derivatives Eq. ([REF]), so that each set can be efficiently parallelized.', '0905.0858-2-98-11': 'We experimented with different ordering schemes and found that this choice performed best.', '0905.0858-2-99-0': 'Now, assume we have an approximate solution [MATH] to the field equation Eq. ([REF]) on the fine grid [MATH], which differs from the true solution [MATH] by the error [MATH], [MATH].', '0905.0858-2-99-1': 'Further, the residual is defined as [MATH].', '0905.0858-2-99-2': 'Since [MATH] by assumption, we obtain the following residual equation: [EQUATION]', '0905.0858-2-99-3': 'We expect that the relaxation on the fine grid has removed most small-scale error modes, so [MATH] mainly consists of longer wavelength modes.', '0905.0858-2-99-4': 'Thus, we will solve for [MATH] on the coarser grid [MATH], and then correct the approximate solution [MATH] for this error.', '0905.0858-2-99-5': 'On the coarse grid [MATH], Eq. ([REF]) reads: [EQUATION]', '0905.0858-2-99-6': 'Here, the superscripts denote which grid a given quantity is defined on, and [MATH] is the restriction operator mapping fields from the fine grid to the coarse grid.', '0905.0858-2-99-7': 'Note that Eq. ([REF]) is of the same form as the [MATH] equation on the coarse grid, [MATH], except with a different right-hand side.', '0905.0858-2-99-8': 'Hence, Eq. ([REF]) is solved for [MATH] using the same algorithm as the original equation, but at one grid level higher.', '0905.0858-2-99-9': 'Once Eq. ([REF]) is solved, we can correct [MATH] for the error: [EQUATION] where [MATH] is the interpolation operator mapping the error from the coarse grid to the fine grid.', '0905.0858-2-100-0': 'In summary, the multigrid relaxation proceeds as follows:', '0905.0858-2-101-0': 'Correct [MATH] using [MATH] [Eq. ([REF])].', '0905.0858-2-102-0': 'Correct [MATH] using [MATH] [Eq. ([REF])].', '0905.0858-2-103-0': 'Thus, the relaxation proceeds as a nested loop going down to the coarsest grid, and then correcting the solutions on the successively finer grids.', '0905.0858-2-103-1': 'One such iteration through all grid levels is called a V-cycle.', '0905.0858-2-103-2': 'The coarsest grid we use has 4 cells on a side, which corresponds to a refinement of [MATH] for a [MATH] base grid.', '0905.0858-2-104-0': 'For the interpolation operator [MATH], we use standard bilinear interpolation (consistent with the cloud-in-cell scheme used for assigning densities and measuring accelerations on the grid).', '0905.0858-2-104-1': 'For the restriction [MATH], we use full-weighting, the transpose of the bilinear interpolation operator.', '0905.0858-2-104-2': 'See the appendix in [CITATION] for an explicit definition.', '0905.0858-2-105-0': 'In order to ensure convergence at all times, we adopt a large value of [MATH], at the expense of computing time.', '0905.0858-2-105-1': 'Usually, one V-cycle reduces the residual [MATH] by 1-2 orders of magnitude.', '0905.0858-2-105-2': 'We stop the relaxation when either [MATH] is reached, or the convergence stalls at a certain level of [MATH].', '0905.0858-2-105-3': 'Either of this usually happens within 4 V-cycles.', '0905.0858-2-105-4': 'The value of [MATH] for each time step is logged, allowing for a monitoring of the convergence status in the simulations.', '0905.0858-2-106-0': '# Sine wave test', '0905.0858-2-107-0': 'In this section, we use the fact that the full [MATH] solution is identical to the linearized solution for a plane wave perturbation (Section [REF]) to test our code for perturbations on different length scales.', '0905.0858-2-107-1': 'We consider a simple sine wave density perturbation, with [MATH]-vector chosen to lie along the [MATH]-axis: [EQUATION]', '0905.0858-2-107-2': 'In this case, the solution to the non-linear [MATH] equation is identical to the linear solution: [EQUATION]', '0905.0858-2-107-3': 'Hence, the exact acceleration is given by: [EQUATION]', '0905.0858-2-107-4': 'For a given level of residuals [MATH], we now demand that the errors due to the approximate solution of the [MATH] equation of motion are small compared to the unavoidable truncation errors which are encurred by taking the gradient of the potential on the grid.', '0905.0858-2-108-0': 'In the following, we set [MATH], [MATH], and the number of grid cells is set to [MATH].', '0905.0858-2-108-1': 'The truncation errors are measured by assigning the analytical solution for the potential to each grid point, and comparing the exact acceleration to the one obtained by derivation and interpolation from the grid.', '0905.0858-2-108-2': 'The truncation errors for two sine waves of different wavelength are shown as deviations in the acceleration in Fig. [REF] (red points).', '0905.0858-2-108-3': 'We scale the deviations to the RMS acceleration of the sine wave, [MATH], in order to avoid far outliers obtained near the zeros of the sine wave when scaling to the exact solution at each point.', '0905.0858-2-108-4': 'For shorter wavelength modes, the truncation error increases and reaches order unity for waves on the Nyquist scale, as expected.', '0905.0858-2-109-0': 'In order to estimate the additional error in the acceleration made due to insufficient convergence of the solution, we compare the acceleration measured from the non-linear field solution coming out of the relaxation solver, to the corresponding solution of the linearized [MATH] equation Eq. ([REF]).', '0905.0858-2-109-1': 'We vary the parameters [MATH] and [MATH] which control the strength of the non-linearity in Eq. ([REF]).', '0905.0858-2-109-2': 'In Fig. [REF], we show the measured deviations in the acceleration from the linear solution for different values of [MATH] and [MATH], and in particular for the largest values that reached convergence.', '0905.0858-2-109-3': 'The final residuals of the [MATH] solution in each case are also shown.', '0905.0858-2-110-0': 'Apparently, the errors in the acceleration are completely negligible for residuals [MATH], consistent with results we found for other test cases such as the spherical mass.', '0905.0858-2-110-1': 'For sufficiently strong non-linearity and short wavelengths, residuals of order [MATH] are reached which lead to measurable deviations in the acceleration.', '0905.0858-2-110-2': 'However, for the residual values up to [MATH] that we explored, the error due to the incomplete convergence is very small compared to the truncation error in all cases.', '0905.0858-2-110-3': 'Hence, we conclude that for residuals smaller than the conservative bound of [MATH], the [MATH] solution we obtain is sufficiently accurate in order not to bias the particle dynamics.', '0905.0858-2-111-0': '# Correcting Initial Conditions for DGP', '0905.0858-2-112-0': 'Due to the additional term [MATH] in the Friedmann equation, the self-accelerating DGP model starts to deviate from a [MATH]CDM expansion history at relatively high redshifts.', '0905.0858-2-112-1': 'In addition, there are modifications to the growth of horizon-scale modes.', '0905.0858-2-112-2': 'For this reason, even at the initial redshift of our simulations, [MATH], there are small departures in the matter transfer function in DGP as compared to a [MATH]CDM model with the same early-Universe parameters [MATH].', '0905.0858-2-113-0': 'In order to take these differences into account, we calculate the dark matter transfer function [MATH] at [MATH] for DGP and QCDM, using the PPF approach described in [CITATION], and for the corresponding flat [MATH]CDM model.', '0905.0858-2-113-1': 'We do not include the effects of radiation on the transfer function, which is not necessary since we are only interested in the relative deviation of the DGP [MATH] from [MATH]CDM.', '0905.0858-2-113-2': 'Fig. [REF] shows the relative deviation in [MATH] from [MATH]CDM for DGP and QCDM.', '0905.0858-2-113-3': 'Both QCDM and DGP transfer functions are slightly suppressed on all scales due to the effects of the earlier onset of acceleration.', '0905.0858-2-113-4': 'On super-horizon scales, the transfer function is further suppressed in DGP caused by the transition to 5D gravity on very large scales.', '0905.0858-2-113-5': 'In contrast, [MATH] is less suppressed in QCDM because of the fluctuations in the effective dark energy.', '0905.0858-2-114-0': 'We correct the transfer function obtained from CAMB for these small differences at [MATH] by multiplying [MATH] with [MATH], where [MATH] is a simple [MATH] function fit to the DGP curve shown in Fig. [REF].', '0905.0858-2-114-1': 'Hence, these initial conditions are not quite correct for the QCDM simulations.', '0905.0858-2-114-2': 'However, the differences are very small on the scales probed by our simulations ([MATH]).'}

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'0905.0858-2-22-3'], ['0905.0858-1-22-4', '0905.0858-2-22-4'], ['0905.0858-1-22-5', '0905.0858-2-22-5'], ['0905.0858-1-25-0', '0905.0858-2-25-0'], ['0905.0858-1-25-1', '0905.0858-2-25-1'], ['0905.0858-1-107-0', '0905.0858-2-107-0'], ['0905.0858-1-107-1', '0905.0858-2-107-1'], ['0905.0858-1-107-2', '0905.0858-2-107-2'], ['0905.0858-1-107-3', '0905.0858-2-107-3'], ['0905.0858-1-107-4', '0905.0858-2-107-4']]

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[]

[['0905.0858-1-86-1', '0905.0858-2-86-1'], ['0905.0858-1-86-1', '0905.0858-2-86-3']]

[]

['0905.0858-1-31-3', '0905.0858-1-100-0', '0905.0858-1-101-0', '0905.0858-1-102-0', '0905.0858-2-31-3', '0905.0858-2-100-0', '0905.0858-2-101-0', '0905.0858-2-102-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0905.0858

1706.02388

{'1706.02388-1-0-0': 'We consider the four-point correlator of the stress-energy tensor multiplet in [MATH] SYM.', '1706.02388-1-0-1': "In the planar limit and at large 't Hooft coupling such correlator is given by the corresponding holographic correlation function in IIB supergravity on [MATH].", '1706.02388-1-0-2': "We consider subleading corrections in the number of colours, i.e. order [MATH], at large 't Hooft coupling.", '1706.02388-1-0-3': 'This corresponds to loop corrections to the supergravity result.', '1706.02388-1-0-4': 'Consistency conditions, most notably crossing symmetry, constrain the form of such corrections and lead to a complete determination of the spectrum of leading twist intermediate operators.', '1706.02388-1-1-0': 'Introduction.', '1706.02388-1-1-1': 'The prototypical example of [MATH] correspondence relates [MATH] Super Yang-Mills (SYM) in four dimensions to type IIB string theory on [MATH] [CITATION].', '1706.02388-1-1-2': 'Even after 20 years of its original formulation, and in spite of tremendous progress in many directions, non-protected quantities have only been explored in certain corners, or edges, of the parameter space.', '1706.02388-1-1-3': "One particularly interesting corner corresponds to planar SYM with large t'Hooft coupling [MATH], which is dual to classical supergravity on the bulk.", '1706.02388-1-1-4': 'In this regime single trace chiral primary operators (CPO) of weight [MATH] map to supergravity fields with mass [MATH], and their correlation functions can in principle be computed by tree-level Witten diagrams on [MATH]: [EQUATION]', '1706.02388-1-1-5': 'Three point correlators of arbitrary CPOs, as well as four point correlator of the stress tensor multiplet, were computed long ago [CITATION].', '1706.02388-1-1-6': 'Recently, an elegant algorithm based on symmetries and consistency conditions to determine the four-point correlator of arbitrary CPOs was proposed in [CITATION], see also [CITATION].', '1706.02388-1-2-0': "In this note we consider subleading corrections in [MATH] to correlators in the large t' Hooft coupling regime.", '1706.02388-1-2-1': 'This corresponds to quantum corrections on the gravity side.', '1706.02388-1-2-2': 'Although some progress has been made for specific contributions, see [CITATION], loop diagrams in [MATH] are a largely unexplored subject, mostly due to technical difficulties that prohibit direct computations.', '1706.02388-1-2-3': 'A systematic study of loops in [MATH] based on symmetries was started in [CITATION] for sectors of CFTs.', '1706.02388-1-2-4': 'The main tool of [CITATION] were methods of the analytic bootstrap, started in [CITATION], and in particular its algebraic formulation [CITATION].', '1706.02388-1-2-5': 'In this letter we would like to report the first complete results to order [MATH] for a full fledge CFT, namely [MATH] SYM.', '1706.02388-1-2-6': 'We will focus on the four point correlator of the lowest component of the stress tensor multiplet [MATH].', '1706.02388-1-2-7': "In the planar limit and at large 't Hooft coupling the intermediate operators consist of double trace operators of spin [MATH] and dimension [EQUATION] where [MATH].", '1706.02388-1-2-8': 'The leading order correction [MATH] is given by the supergravity result.', '1706.02388-1-2-9': 'In this letter we study the consequences of superconformal symmetry, consistency of the OPE and crossing symmetry for the subleading corrections.', '1706.02388-1-2-10': 'The analysis of crossing symmetry to order [MATH] is highly complicated by mixing among double trace operators.', '1706.02388-1-2-11': 'Namely, there is more than one intermediate operator for a given twist and spin.', '1706.02388-1-2-12': 'From the bulk point of view this corresponds to take into account all Kaluza Klein (KK)-modes.', '1706.02388-1-2-13': 'In order to solve this mixing problem we study general correlators [MATH] in the supergravity approximation.', '1706.02388-1-2-14': 'This allows to disentangle the contribution of each double trace operator/KK-mode and apply the methods of [CITATION] for the present case.', '1706.02388-1-2-15': 'This leads to an expression for [MATH] valid to all orders in inverse powers of the spin for each KK-mode.', '1706.02388-1-2-16': 'These expansions can be resummed exactly.', '1706.02388-1-3-0': 'In addition crossing symmetry allows the addition of solutions with finite support in the spin.', '1706.02388-1-3-1': 'From the bulk perspective these ambiguities correspond to unknown coefficients in front of the counterterms to be added in the presence of divergences.', '1706.02388-1-3-2': 'For the present case we expect such extra solutions to be absent for spin two and higher.', '1706.02388-1-3-3': 'With this, for instance, for the leading twist operator of spin two we obtain [EQUATION]', '1706.02388-1-3-4': 'Similar results can be obtained for any spin.', '1706.02388-1-3-5': 'In principle our algorithm fixes also the OPE coefficients.', '1706.02388-1-4-0': 'Stress-tensor correlator in [MATH] SYM.', '1706.02388-1-4-1': 'In [MATH] SYM the stress tensor sits in a half-BPS multiplet.', '1706.02388-1-4-2': 'The lowest component of this multiplet is a scalar operator [MATH] of dimension two transforming in the [MATH] of the [MATH]symmetry group [MATH].', '1706.02388-1-4-3': 'Its four-point correlator takes the form [EQUATION] where the sum runs over representations in the tensor product [MATH] and we have introduced the standard cross-ratios [EQUATION]', '1706.02388-1-4-4': 'Superconformal symmetry allows writing all contributions [MATH] in terms of a single non-protected function [MATH] satisfying the following crossing relation.', '1706.02388-1-4-5': '[EQUATION] where [MATH] is the central charge.', '1706.02388-1-4-6': 'See [CITATION] for a detailed discussion.', '1706.02388-1-4-7': 'This function can be decomposed into the contribution from operators in (semi-)short multiplets and operators in long multiplets [EQUATION] where [MATH] is independent of the coupling constant and can be found in [CITATION], and [MATH] admits a decomposition in superconformal blocks [EQUATION] where the sum runs over superconformal primary operators in long multiplets, in the singlet representation of [MATH], with twist (dimension minus the spin) [MATH] and even spin [MATH].', '1706.02388-1-4-8': '[MATH] denotes the square of the OPE coefficients.', '1706.02388-1-4-9': 'It is convenient to write the conformal blocks in terms of cross-ratios [MATH] such that [MATH].', '1706.02388-1-4-10': 'In terms of these [EQUATION] where [MATH] is the standard hypergeometric function.', '1706.02388-1-4-11': 'In the strict limit of infinite central charge [MATH] reduces to the generalised free fields result [MATH], which agrees with the large [MATH] result in the Born approximation (free theory).', '1706.02388-1-4-12': 'The intermediate operators correspond to towers of double trace operators of twist [MATH] and OPE coefficients [EQUATION]', '1706.02388-1-4-13': 'The four-point correlator admits an expansion around large [MATH] or equivalently large central charge [MATH]: [EQUATION]', '1706.02388-1-4-14': 'Accordingly the intermediate operators acquire corrections [EQUATION]', '1706.02388-1-4-15': "In this note we will focus in the limit of large 't Hooft coupling [MATH].", '1706.02388-1-4-16': 'In this regime there is no new operators appearing in the OPE at this order and [MATH] can be computed from the classical supergravity result in [CITATION].', '1706.02388-1-4-17': 'This leads to the following correction for the spectrum and OPE coefficients [CITATION] [EQUATION] where [MATH].', '1706.02388-1-4-18': 'It is important to note that for a given [MATH] and [MATH] there is more than one superconformal primary in the singlet of [MATH], except for [MATH].', '1706.02388-1-4-19': 'The above corrections should then be interpreted as (weighted-)averages.', '1706.02388-1-4-20': 'This will be very important below.', '1706.02388-1-5-0': 'From leading to subleading corrections.', '1706.02388-1-5-1': 'Our aim is to compute [MATH] and [MATH] from crossing symmetry.', '1706.02388-1-5-2': 'Since [MATH] receives contributions only up to order [MATH], the crossing equation for [MATH] is simply [EQUATION]', '1706.02388-1-5-3': 'We will follow the same strategy as in [CITATION]: start by determining the piece proportional to [MATH] in [MATH] from the CFT data at order [MATH].', '1706.02388-1-5-4': 'By crossing symmetry this will lead to a precise divergence proportional to [MATH].', '1706.02388-1-5-5': 'Matching this divergence then fixes [MATH] and [MATH] to all orders in [MATH].', '1706.02388-1-5-6': 'Plugging ([REF]) into the conformal block decomposition and expanding up to order [MATH], we find [EQUATION] where we have introduced [MATH].', '1706.02388-1-5-7': 'In particular the piece proportional to [MATH] is [EQUATION]', '1706.02388-1-5-8': 'A serious obstacle to compute this is the mixing among double trace operators [MATH].', '1706.02388-1-5-9': 'They have the same twist and spin at zeroth order and transform under the same representation of [MATH].', '1706.02388-1-5-10': 'Hence, the sum in ([REF]) should contain an extra index [MATH] - which we leave implicit- to account for degenerate operators at tree-level.', '1706.02388-1-5-11': 'For the same reason, quantities above should be interpreted as averages over these families, weighted by their respective OPE coefficient at zeroth order.', '1706.02388-1-5-12': 'Therefore, the weighted average [MATH] does not follow from the leading order result, except for [MATH], for which there is a unique state.', '1706.02388-1-5-13': 'In order to tackle this mixing problem we consider the complete families of four-point correlators [MATH] in the supergravity approximation.', '1706.02388-1-5-14': 'This is done in the appendix and it leads to the following remarkable structure for the average in question [EQUATION] where [MATH].', '1706.02388-1-5-15': 'The coefficients [MATH] have been computed explicitly up to twist 10.', '1706.02388-1-5-16': 'We will see however that they can be fixed, for any value of the twist, by resorting to crossing symmetry.', '1706.02388-1-5-17': 'In order to understand this, consider the following sequence of twist conformal blocks (TCB) [EQUATION]', '1706.02388-1-5-18': 'We then propose the following expansion: [EQUATION]', '1706.02388-1-5-19': 'This allows to write the piece proportional to [MATH] in [MATH] in terms of TCB and the coefficients [MATH].', '1706.02388-1-5-20': 'Furthermore, crossing plus consistency with the CPW expansion, e.g. absence of [MATH], fixes the range of [MATH] to be [MATH].', '1706.02388-1-5-21': 'From the explicit expression for TCB found in the appendix, we can extract the contribution proportional to [MATH].', '1706.02388-1-5-22': '[EQUATION] where the functions [MATH] are defined in the appendix.', '1706.02388-1-5-23': 'This contribution should be crossing symmetric by itself.', '1706.02388-1-5-24': 'This imposes a set of linear constraints on the coefficients [MATH].', '1706.02388-1-5-25': 'Now we note the following remarkable fact: the expansion ([REF]) is consistent with this set of constraints and furthermore, the constraints fix uniquely the coefficients [MATH] for all twists!', '1706.02388-1-5-26': 'Up to twist 10 these coefficients agree precisely with the ones found by explicit computations.', '1706.02388-1-6-0': 'Having found the averages [MATH] we can now turn into the sums [EQUATION]', '1706.02388-1-6-1': 'These sums can be decomposed into individual contributions, corresponding to the insertion of single or double poles in [MATH] into the sums defining the TCB.', '1706.02388-1-6-2': 'Denoting by [MATH] the quadratic Casimir with eigenfunction [MATH] and eigenvalue [MATH] we obtain [EQUATION] which gives a differential equation for the components of the sums ([REF]).', '1706.02388-1-6-3': 'This can be easily solved case by case.', '1706.02388-1-6-4': 'The sums have the following structure [EQUATION] for instance, for the first case we obtain [EQUATION]', '1706.02388-1-7-0': 'Spectrum at order [MATH].', '1706.02388-1-7-1': 'We will now consider the crossing equation ([REF]).', '1706.02388-1-7-2': 'Our strategy will be to expand it around [MATH] and focus in terms proportional to different powers of [MATH] and [MATH].', '1706.02388-1-7-3': 'Note that the [MATH] dependence in ([REF]) will only arise when the derivative hits [MATH].', '1706.02388-1-7-4': 'On the other hand the behaviour around [MATH] is more subtle and one needs to perform the sum over the spin.', '1706.02388-1-7-5': 'The piece proportional to [MATH] has already been discussed in the previous section regarding the problem of mixing.', '1706.02388-1-7-6': 'The relation proportional to [MATH] leads to [EQUATION]', '1706.02388-1-7-7': 'In this note we will restrict ourselves to corrections to the spectrum of leading twist operators [MATH].', '1706.02388-1-7-8': 'This amounts to consider the small [MATH] limit of the relation above.', '1706.02388-1-7-9': 'On the l.h.s. only terms with [MATH] will survive.', '1706.02388-1-7-10': 'On the other hand, note that all terms on the r.h.s will contribute to this limit.', '1706.02388-1-7-11': 'The sum over derivatives of conformal blocks with the extra insertion [MATH] can be computed with some effort.', '1706.02388-1-7-12': 'With this result together with the derivative relation for [MATH] the above relation in the small [MATH] limit can be expressed as follows [EQUATION] where [MATH] is the small [MATH] limit of [MATH] and we have introduced [MATH].', '1706.02388-1-7-13': 'All the terms except the first one in the above relation are exactly computable.', '1706.02388-1-7-14': 'Crossing symmetry then implies that [MATH] should be such that its insertion produces a [MATH] divergence times a fully fixed expansion in powers of [MATH].', '1706.02388-1-7-15': 'This problem can be solved by proposing the following expansion [EQUATION]', '1706.02388-1-7-16': 'Hence the first term in ([REF]) can be written in terms of TCB [MATH] at [MATH].', '1706.02388-1-7-17': 'As before, crossing symmetry plus consistency with the CPW expansion fixes the range [MATH].', '1706.02388-1-7-18': 'From the procedure outlined in the appendix one can compute the contribution proportional to [MATH] for [MATH] for [MATH].', '1706.02388-1-7-19': 'This allows to determine all coefficients [MATH], and hence [MATH] to all orders in [MATH].', '1706.02388-1-7-20': 'The result can be organised as to make manifest the contribution from each KK-mode.', '1706.02388-1-7-21': 'We start by representing [MATH] as follows [EQUATION] where [MATH] and [MATH] for [MATH].', '1706.02388-1-7-22': 'Each term inside the sum represents the contribution from the [MATH]th KK mode, or more precisely the intermediate double trace operators [MATH].', '1706.02388-1-7-23': 'We can then compute the contribution to [MATH] from each KK-mode.', '1706.02388-1-7-24': 'From the bulk point of view, this has the interpretation of an expansion into KK-modes running along the loop.', '1706.02388-1-7-25': 'Following the steps outlined above, we can compute [MATH] to all orders in [MATH].', '1706.02388-1-7-26': 'Remarkably, the resulting series can be resummed exactly.', '1706.02388-1-7-27': 'For the massless KK-modes one obtains [EQUATION]', '1706.02388-1-7-28': 'Taking into account only the massless KK-mode should be equivalent to doing the bulk computation in 5d super-gravity.', '1706.02388-1-7-29': 'Note that in this case the answer is convergent and finite for all values of the spin.', '1706.02388-1-7-30': 'This is consistent with the fact that 5d supergravity is free of divergences at one loop.', '1706.02388-1-7-31': 'For [MATH] the results have the following structure [EQUATION] where [MATH] and [MATH] are polynomials such that this contribution starts at order [MATH] at large [MATH].', '1706.02388-1-7-32': 'An important comment is in order.', '1706.02388-1-7-33': 'Even though the contribution of each KK-mode leads to an asymptotic series in [MATH], the sum of all of them leads to a convergent series.', '1706.02388-1-7-34': 'This is in tune with the analysis of [CITATION].', '1706.02388-1-7-35': 'Let us now consider the contribution of a generic KK-mode for finite/small values of the spins.', '1706.02388-1-7-36': 'The general structure is [EQUATION] for some polynomials [MATH].', '1706.02388-1-7-37': 'At large [MATH] we find the following behaviour [EQUATION] which implies the sum over KK-modes is convergent for all values of the spin.', '1706.02388-1-7-38': 'In particular for spin zero and two we obtain [EQUATION] which leads to the result quoted in the introduction.', '1706.02388-1-8-0': 'Discussion.', '1706.02388-1-8-1': "In this note we have reported the first complete results for the CFT data of unprotected operators in [MATH] SYM to order [MATH] and at large 't Hooft coupling.", '1706.02388-1-8-2': 'A more detailed exposition will appear in [CITATION].', '1706.02388-1-8-3': 'There are several open questions that would be nice to understand.', '1706.02388-1-9-0': 'It would be interesting to compute explicitly [MATH] for [MATH].', '1706.02388-1-9-1': 'Once this is found it would be interesting to study its large [MATH] behaviour and compare it to the expectations from the bulk perspective.', '1706.02388-1-9-2': 'It would be important to understand if solutions with finite support in the spin are present.', '1706.02388-1-9-3': 'Preliminary results show that crossing symmetry does not require any non-analytical corrections at finite spin.', '1706.02388-1-9-4': 'On the other hand crossing symmetry allows the addition of any of the truncated solutions constructed in [CITATION].', '1706.02388-1-9-5': 'From the bulk perspective these solutions correspond to counterterms that need to be added to render the computation finite.', '1706.02388-1-9-6': 'The 10d supergravity computation contains a quadratic divergence proportional to [MATH], see [CITATION] eq. 4.2.', '1706.02388-1-9-7': 'This will lead to a contribution which becomes large for large [MATH] but has support only for spin zero.', '1706.02388-1-9-8': 'We expect that other extra solutions are not present.', '1706.02388-1-9-9': 'Note that this ambiguity is already present at leading order in [MATH].', '1706.02388-1-9-10': 'In this case, all truncated solutions are forbidden by requiring consistency with the flat space limit, see e.g.[CITATION].', '1706.02388-1-9-11': 'Presumably consistency with the flat space amplitude to order [MATH] will also forbid most extra solutions.', '1706.02388-1-9-12': 'Relatedly, there are several results in the literature that bound the behaviour of [MATH] for large [MATH], see e.g. [CITATION], it would be interesting to extend these results to the order we are considering.', '1706.02388-1-10-0': 'Leading order corrections in [MATH] are in principle possible to consider.', '1706.02388-1-10-1': 'At leading order they correspond to the addition of the first truncated solution with a known coefficient [CITATION].', '1706.02388-1-10-2': "At order [MATH] one would have to 'square' the supergravity contribution plus this contribution.", '1706.02388-1-10-3': 'Since the latter is truncated in the spin, the extra sums involved are very simple.', '1706.02388-1-10-4': 'This computation is expected to lead to divergences, since the first truncated solution grows much faster, with [MATH], than supergravity.', '1706.02388-1-10-5': 'One could also consider the exchange of a finite number of single trace operators, combining the results of [CITATION] with the methods of this note.', '1706.02388-1-11-0': 'It would be interesting to write the full result for the four-point correlator in space time.', '1706.02388-1-11-1': 'From the results of this note and [CITATION] some pieces can already be written explicitly.', '1706.02388-1-11-2': 'It would be fascinating if these pieces, together with crossing symmetry, fix the full answer.', '1706.02388-1-11-3': 'As advocated in [CITATION], Mellin space could be the right language for such an endeavour.', '1706.02388-1-11-4': 'This would be the first step to extend the results of [CITATION] to include loop corrections.', '1706.02388-1-12-0': 'The expansion in [MATH] for non-protected quantities in the context of [MATH] duality is a largely unexplored subject.', '1706.02388-1-12-1': 'Our result opens a window to study this problem systematically and quantitatively.', '1706.02388-1-13-0': 'We are grateful to G. Arutyunov, Z. Komargodski, J. Maldacena, A. Zhiboedov and specially Ofer Aharony for useful discussions.', '1706.02388-1-13-1': 'We would like to thank the ICTP-SAIFR in Sao Paulo for their hospitality during part of this work.', '1706.02388-1-13-2': 'The work of L.F.A was supported by ERC STG grant 306260.', '1706.02388-1-13-3': 'L.F.A. is a Wolfson Royal Society Research Merit Award holder.', '1706.02388-1-13-4': 'The work of A.B. is partially supported by Templeton Award 52476 of A. Strominger and by Simons Investigator Award from the Simons Foundation of X. Yin.', '1706.02388-1-14-0': '# Appendices', '1706.02388-1-15-0': 'The Mixing problem.', '1706.02388-1-15-1': 'A technical obstacle in inferring the [MATH] piece of the correlator at order [MATH] is mixing.', '1706.02388-1-15-2': 'For a given twist [MATH] all double trace operators [MATH] mix, and the eigenfunctions of the Hamiltonian are certain combinations of those [EQUATION] where the dependence on the spin is implicit.', '1706.02388-1-15-3': 'We will consider this problem at leading order in [MATH].', '1706.02388-1-15-4': 'We can choose the double trace operators [MATH] to be canonically normalised, such that the coefficients [MATH] form an orthonormal matrix in this basis.', '1706.02388-1-15-5': 'In order to solve the mixing problem we consider general correlators [MATH].', '1706.02388-1-15-6': 'At zeroth order the above operators appear in this correlator with OPE coefficient [EQUATION] where [MATH] could also depend on the spin and the twist.', '1706.02388-1-15-7': 'Note that this sum is proportional to [MATH].', '1706.02388-1-15-8': 'At order [MATH] these operators acquire an anomalous dimension [MATH].', '1706.02388-1-15-9': 'From the explicit conformal block decomposition for the correlator [MATH] in the supergravity approximation we can read off [EQUATION] the averages [MATH] for a given twist can be conveniently packed in a mixing matrix [MATH].', '1706.02388-1-15-10': 'For instance, for [MATH] we have to analyse the correlators with [MATH], which can be found in [CITATION].', '1706.02388-1-15-11': 'This leads to the following mixing matrix [EQUATION] where [MATH] for [MATH].', '1706.02388-1-15-12': 'We have analysed this problem for several values of [MATH], using the explicit supergravity results in [CITATION].', '1706.02388-1-15-13': 'The averages [MATH] at order [MATH] are given by the elements of [MATH].', '1706.02388-1-15-14': 'We are interested in [MATH].', '1706.02388-1-15-15': 'For instance, for the case [MATH] we obtain [EQUATION]', '1706.02388-1-15-16': 'In all cases we found the remarkable pattern ([REF]) quoted in the body of the note.', '1706.02388-1-15-17': 'As seen there, this structure together with crossing symmetry is enough to fix all coefficients [MATH].', '1706.02388-1-15-18': 'These values also agree with the ones found by explicit computations.', '1706.02388-1-16-0': 'Twist conformal blocks.', '1706.02388-1-16-1': 'The zeroth order correlator can be expressed in terms of twist conformal blocks [MATH], defined as the contribution from operators with twist [MATH].', '1706.02388-1-16-2': '[EQUATION]', '1706.02388-1-16-3': 'The explicit form of (super-)conformal blocks leads to the following structure [EQUATION] by plugging this structure into ([REF]) and expanding around [MATH] the functions [MATH] can be found [EQUATION] where [MATH].', '1706.02388-1-16-4': 'Next we define the sequence of functions [MATH] corresponding to extra insertions of [MATH] [EQUATION] [MATH] is the eigenvalue of a specific quadratic Casimir operator.', '1706.02388-1-16-5': 'More precisely, [MATH] admits the same factorisation as in ([REF]) with [MATH], where the functions [MATH] satisfy the following recursive relation [EQUATION] and [MATH].', '1706.02388-1-16-6': 'With this recursion relation together with the expression for [MATH] we can find [MATH] exactly for the first few values of [MATH] and also as various expansions.', '1706.02388-1-16-7': 'The divergent behaviour as [MATH] for [MATH] will be important for us.', '1706.02388-1-16-8': 'From the explicit answer we see [EQUATION]', '1706.02388-1-16-9': 'It can be then seen that [EQUATION] with [MATH] as [MATH] and [EQUATION]', '1706.02388-1-16-10': 'The functions [MATH] can be build recursively to any desired order.'}

{'1706.02388-2-0-0': 'We consider the four-point correlator of the stress-energy tensor multiplet in [MATH] SYM.', '1706.02388-2-0-1': "In the planar limit and at large 't Hooft coupling such correlator is given by the corresponding holographic correlation function in IIB supergravity on [MATH].", '1706.02388-2-0-2': "We consider subleading corrections in the number of colours, i.e. order [MATH], at large 't Hooft coupling.", '1706.02388-2-0-3': 'This corresponds to loop corrections to the supergravity result.', '1706.02388-2-0-4': 'Consistency conditions, most notably crossing symmetry, constrain the form of such corrections and lead to a complete determination of the spectrum of leading twist intermediate operators.', '1706.02388-2-1-0': 'Introduction.', '1706.02388-2-1-1': 'The prototypical example of [MATH] correspondence relates [MATH] Super Yang-Mills (SYM) in four dimensions to type IIB string theory on [MATH] [CITATION].', '1706.02388-2-1-2': 'Even after 20 years of its original formulation, and in spite of tremendous progress in many directions, non-protected quantities have only been explored in certain corners, or edges, of the parameter space.', '1706.02388-2-1-3': "One particularly interesting corner corresponds to planar SYM with large t'Hooft coupling [MATH], which is dual to classical supergravity on the bulk.", '1706.02388-2-1-4': 'In this regime single trace chiral primary operators (CPO) of weight [MATH], [MATH], map to supergravity fields with mass [MATH], and their correlation functions can in principle be computed by tree-level Witten diagrams on [MATH]: [EQUATION]', '1706.02388-2-1-5': 'Three point correlators of arbitrary CPOs, as well as four point correlator of the stress tensor multiplet, were computed long ago [CITATION].', '1706.02388-2-1-6': 'Recently, an elegant algorithm based on symmetries and consistency conditions to determine the four-point correlator of arbitrary CPOs was proposed in [CITATION], see also [CITATION].', '1706.02388-2-2-0': "In this note we consider subleading corrections in [MATH] to correlators in the large t' Hooft coupling regime.", '1706.02388-2-2-1': 'This corresponds to quantum corrections on the gravity side.', '1706.02388-2-2-2': 'Although some progress has been made for specific contributions, see [CITATION], loop diagrams in [MATH] are a largely unexplored subject, mostly due to technical difficulties that prohibit direct computations.', '1706.02388-2-2-3': 'The analytic bootstrap was initiated in [CITATION] and developed into a powerful algebraic machinery in [CITATION].', '1706.02388-2-2-4': 'This algebraic formulation allowed a systematic study of loops in [MATH] based on symmetries, started in [CITATION] for sectors of CFTs.', '1706.02388-2-2-5': 'In this letter we would like to report the first complete results to order [MATH] for a full fledge CFT, namely [MATH] SYM.', '1706.02388-2-2-6': 'We will focus on the four point correlator of the lowest component of the stress tensor multiplet [MATH].', '1706.02388-2-2-7': "In the planar limit and at large 't Hooft coupling the intermediate operators consist of double trace operators of spin [MATH] and dimension [EQUATION] where [MATH].", '1706.02388-2-2-8': 'The leading order correction [MATH] is given by the supergravity result.', '1706.02388-2-2-9': 'In this letter we study the consequences of superconformal symmetry, consistency of the OPE and crossing symmetry for the subleading corrections.', '1706.02388-2-2-10': 'The analysis of crossing symmetry to order [MATH] is highly complicated by mixing among double trace operators.', '1706.02388-2-2-11': 'Namely, there is more than one intermediate operator for a given twist and spin.', '1706.02388-2-2-12': 'From the bulk point of view this corresponds to take into account all Kaluza Klein (KK)-modes.', '1706.02388-2-2-13': 'In order to solve this mixing problem we study general correlators [MATH] in the supergravity approximation.', '1706.02388-2-2-14': 'This allows to disentangle the contribution of each double trace operator/KK-mode and apply the methods of [CITATION] for the present case.', '1706.02388-2-2-15': 'This leads to an expression for [MATH] valid to all orders in inverse powers of the spin for each KK-mode.', '1706.02388-2-2-16': 'These expansions can be resummed exactly.', '1706.02388-2-3-0': 'In addition crossing symmetry allows the addition of solutions with finite support in the spin.', '1706.02388-2-3-1': 'From the bulk perspective these ambiguities correspond to unknown coefficients in front of possible counterterms.', '1706.02388-2-3-2': 'For the present case we expect such extra solutions to be absent for spin two and higher.', '1706.02388-2-3-3': 'With this, for instance, for the leading twist operators of spin two and four we obtain [EQUATION]', '1706.02388-2-3-4': 'Similar results can be obtained for any spin.', '1706.02388-2-3-5': 'In principle our algorithm fixes also the OPE coefficients.', '1706.02388-2-4-0': 'Note added: Shortly after our paper appeared in arXiv, an independent computation was presented [CITATION].', '1706.02388-2-4-1': 'Our results appear to be in full agreement.', '1706.02388-2-5-0': 'Stress-tensor correlator in [MATH] SYM.', '1706.02388-2-5-1': 'In [MATH] SYM the stress tensor sits in a half-BPS multiplet.', '1706.02388-2-5-2': 'The lowest component of this multiplet is a scalar operator [MATH] of dimension two transforming in the [MATH] of the [MATH]symmetry group [MATH].', '1706.02388-2-5-3': 'Its four-point correlator takes the form [EQUATION] where the sum runs over representations in the tensor product [MATH] and we have introduced the standard cross-ratios [EQUATION]', '1706.02388-2-5-4': 'Superconformal symmetry allows writing all contributions [MATH] in terms of a single non-protected function [MATH] satisfying the following crossing relation.', '1706.02388-2-5-5': '[EQUATION] where [MATH] is the central charge.', '1706.02388-2-5-6': 'See [CITATION] for a detailed discussion.', '1706.02388-2-5-7': 'This function can be decomposed into the contribution from operators in (semi-)short multiplets and operators in long multiplets [EQUATION] where [MATH] is independent of the coupling constant and can be found in [CITATION], and [MATH] admits a decomposition in superconformal blocks [EQUATION] where the sum runs over superconformal primary operators in long multiplets, in the singlet representation of [MATH], with twist (dimension minus the spin) [MATH] and even spin [MATH].', '1706.02388-2-5-8': '[MATH] denotes the square of the OPE coefficients.', '1706.02388-2-5-9': 'It is convenient to write the conformal blocks in terms of cross-ratios [MATH] such that [MATH].', '1706.02388-2-5-10': 'In terms of these [EQUATION] where [MATH] is the standard hypergeometric function.', '1706.02388-2-5-11': 'In the strict limit of infinite central charge [MATH] reduces to the generalised free fields result [MATH], which agrees with the large [MATH] result in the Born approximation (free theory).', '1706.02388-2-5-12': 'The intermediate operators correspond to towers of double trace operators of twist [MATH] and OPE coefficients [EQUATION]', '1706.02388-2-5-13': 'The four-point correlator admits an expansion around large [MATH] or equivalently large central charge [MATH]: [EQUATION]', '1706.02388-2-5-14': 'Accordingly the intermediate operators acquire corrections [EQUATION]', '1706.02388-2-5-15': "In this note we will focus in the limit of large 't Hooft coupling [MATH].", '1706.02388-2-5-16': 'In this regime there is no new operators appearing in the OPE at this order and [MATH] can be computed from the classical supergravity result in [CITATION].', '1706.02388-2-5-17': 'This leads to the following correction for the spectrum and OPE coefficients [CITATION] [EQUATION] where [MATH].', '1706.02388-2-5-18': 'It is important to note that for a given [MATH] and [MATH] there is more than one superconformal primary in the singlet of [MATH], except for [MATH].', '1706.02388-2-5-19': 'The above corrections should then be interpreted as (weighted-)averages.', '1706.02388-2-5-20': 'This will be very important below.', '1706.02388-2-6-0': 'From leading to subleading corrections.', '1706.02388-2-6-1': 'Our aim is to compute [MATH] and [MATH] from crossing symmetry.', '1706.02388-2-6-2': 'Since [MATH] receives contributions only up to order [MATH], the crossing equation for [MATH] is simply [EQUATION]', '1706.02388-2-6-3': 'We will follow the same strategy as in [CITATION]: start by determining the piece proportional to [MATH] in [MATH] from the CFT data at order [MATH].', '1706.02388-2-6-4': 'By crossing symmetry this will lead to a precise divergence proportional to [MATH].', '1706.02388-2-6-5': 'Matching this divergence then fixes [MATH] and [MATH] to all orders in [MATH].', '1706.02388-2-6-6': 'Plugging ([REF]) into the conformal block decomposition and expanding up to order [MATH], we find [EQUATION] where we have introduced [MATH].', '1706.02388-2-6-7': 'In particular the piece proportional to [MATH] is [EQUATION]', '1706.02388-2-6-8': 'A serious obstacle to compute this is the mixing among double trace operators [MATH].', '1706.02388-2-6-9': 'They have the same twist and spin at zeroth order and transform under the same representation of [MATH].', '1706.02388-2-6-10': 'Hence, the sum in ([REF]) should contain an extra index [MATH] - which we leave implicit- to account for degenerate operators at tree-level.', '1706.02388-2-6-11': 'For the same reason, quantities above should be interpreted as averages over these families, weighted by their respective OPE coefficient at zeroth order.', '1706.02388-2-6-12': 'Therefore, the weighted average [MATH] does not follow from the leading order result, except for [MATH], for which there is a unique state.', '1706.02388-2-6-13': 'In order to tackle this mixing problem we consider the complete families of four-point correlators [MATH] in the supergravity approximation.', '1706.02388-2-6-14': 'This is done in the appendix and it leads to the following remarkable structure for the average in question [EQUATION] where [MATH].', '1706.02388-2-6-15': 'The coefficients [MATH] have been computed explicitly up to twist 10.', '1706.02388-2-6-16': 'We will see however that they can be fixed, for any value of the twist, by resorting to crossing symmetry.', '1706.02388-2-6-17': 'In order to understand this, consider the following sequence of twist conformal blocks (TCB) [EQUATION]', '1706.02388-2-6-18': 'We then propose the following expansion: [EQUATION]', '1706.02388-2-6-19': 'This allows to write the piece proportional to [MATH] in [MATH] in terms of TCB and the coefficients [MATH].', '1706.02388-2-6-20': 'Furthermore, crossing plus consistency with the CPW expansion, e.g. absence of [MATH], fixes the range of [MATH] to be [MATH].', '1706.02388-2-6-21': 'From the explicit expression for TCB found in the appendix, we can extract the contribution proportional to [MATH].', '1706.02388-2-6-22': '[EQUATION] where the functions [MATH] are defined in the appendix.', '1706.02388-2-6-23': 'This contribution should be crossing symmetric by itself.', '1706.02388-2-6-24': 'This imposes a set of linear constraints on the coefficients [MATH].', '1706.02388-2-6-25': 'Now we note the following remarkable fact: the expansion ([REF]) is consistent with this set of constraints and furthermore, the constraints fix uniquely the coefficients [MATH] for all twists!', '1706.02388-2-6-26': 'Up to twist 10 these coefficients agree precisely with the ones found by explicit computations.', '1706.02388-2-7-0': 'Having found the averages [MATH] we can now turn into the sums [EQUATION]', '1706.02388-2-7-1': 'These sums can be decomposed into individual contributions, corresponding to the insertion of single or double poles in [MATH] into the sums defining the TCB.', '1706.02388-2-7-2': 'Denoting by [MATH] the quadratic Casimir with eigenfunction [MATH] and eigenvalue [MATH] we obtain [EQUATION] which gives a differential equation for the components of the sums ([REF]).', '1706.02388-2-7-3': 'This can be easily solved case by case.', '1706.02388-2-7-4': 'The sums have the following structure [EQUATION] for instance, for the first case we obtain [EQUATION]', '1706.02388-2-8-0': 'Spectrum at order [MATH].', '1706.02388-2-8-1': 'We will now consider the crossing equation ([REF]).', '1706.02388-2-8-2': 'Our strategy will be to expand it around [MATH] and focus in terms proportional to different powers of [MATH] and [MATH].', '1706.02388-2-8-3': 'Note that the [MATH] dependence in ([REF]) will only arise when the derivative hits [MATH].', '1706.02388-2-8-4': 'On the other hand the behaviour around [MATH] is more subtle and one needs to perform the sum over the spin.', '1706.02388-2-8-5': 'The piece proportional to [MATH] has already been discussed in the previous section regarding the problem of mixing.', '1706.02388-2-8-6': 'The relation proportional to [MATH] leads to [EQUATION]', '1706.02388-2-8-7': 'In this note we will restrict ourselves to corrections to the spectrum of leading twist operators [MATH].', '1706.02388-2-8-8': 'This amounts to consider the small [MATH] limit of the relation above.', '1706.02388-2-8-9': 'On the l.h.s. only terms with [MATH] will survive.', '1706.02388-2-8-10': 'On the other hand, note that all terms on the r.h.s will contribute to this limit.', '1706.02388-2-8-11': 'The sum over derivatives of conformal blocks with the extra insertion [MATH] can be computed with some effort.', '1706.02388-2-8-12': 'With this result together with the derivative relation for [MATH] the above relation in the small [MATH] limit can be expressed as follows [EQUATION] where [MATH] is the small [MATH] limit of [MATH] and we have introduced [MATH].', '1706.02388-2-8-13': 'All the terms except the first one in the above relation are exactly computable.', '1706.02388-2-8-14': 'Crossing symmetry then implies that [MATH] should be such that its insertion produces a [MATH] divergence times a fully fixed expansion in powers of [MATH].', '1706.02388-2-8-15': 'This problem can be solved by proposing the following expansion [EQUATION]', '1706.02388-2-8-16': 'Hence the first term in ([REF]) can be written in terms of TCB [MATH] at [MATH].', '1706.02388-2-8-17': 'As before, crossing symmetry plus consistency with the CPW expansion fixes the range [MATH].', '1706.02388-2-8-18': 'From the procedure outlined in the appendix one can compute the contribution proportional to [MATH] for [MATH] for [MATH].', '1706.02388-2-8-19': 'This allows to determine all coefficients [MATH], and hence [MATH] to all orders in [MATH].', '1706.02388-2-8-20': 'The result can be organised as to make manifest the contribution from each KK-mode.', '1706.02388-2-8-21': 'We start by representing [MATH] as follows [EQUATION] where [MATH].', '1706.02388-2-8-22': 'Each term inside the sum represents the contribution from the [MATH]th KK mode, or more precisely the intermediate double trace operators [MATH].', '1706.02388-2-8-23': 'We can then compute the contribution to [MATH] from each KK-mode.', '1706.02388-2-8-24': 'From the bulk point of view, this has the interpretation of an expansion into KK-modes running along the loop.', '1706.02388-2-8-25': 'Following the steps outlined above, we can compute [MATH] to all orders in [MATH].', '1706.02388-2-8-26': 'Remarkably, the resulting series can be resummed exactly.', '1706.02388-2-8-27': 'For the massless KK-modes one obtains [EQUATION]', '1706.02388-2-8-28': 'Taking into account only the massless KK-mode should be equivalent to doing the bulk computation in 5d super-gravity.', '1706.02388-2-8-29': 'Note that in this case the answer is convergent and finite for all values of the spin.', '1706.02388-2-8-30': 'This is consistent with the fact that 5d supergravity is free of divergences at one loop.', '1706.02388-2-8-31': 'For [MATH] the results have the following structure [EQUATION] where [MATH] and [MATH] are polynomials such that this contribution starts at order [MATH] at large [MATH].', '1706.02388-2-8-32': 'An important comment is in order.', '1706.02388-2-8-33': 'Even though the contribution of each KK-mode leads to an asymptotic series in [MATH], the sum of all of them leads to a convergent series.', '1706.02388-2-8-34': 'This is in tune with the analysis of [CITATION].', '1706.02388-2-8-35': 'Let us now consider the contribution of a generic KK-mode for finite/small values of the spins.', '1706.02388-2-8-36': 'The general structure is [EQUATION] for some polynomials [MATH].', '1706.02388-2-8-37': 'At large [MATH] we find the following behaviour [EQUATION] since [MATH], this implies the sum over [MATH] is actually divergent for spin zero!', '1706.02388-2-8-38': 'Note that this agrees with the presence of a quadratic divergence in the 10d supergravity computation.', '1706.02388-2-8-39': 'For spin two and higher we get a convergent sum.', '1706.02388-2-8-40': 'For instance, for the first cases we obtain [EQUATION] which leads to the results quoted in the introduction.', '1706.02388-2-8-41': 'Similar results are obtained for arbitrary spin [CITATION].', '1706.02388-2-9-0': 'Discussion.', '1706.02388-2-9-1': "In this note we have reported the first complete results for the CFT data of unprotected operators in [MATH] SYM to order [MATH] and at large 't Hooft coupling.", '1706.02388-2-9-2': 'A more detailed exposition will appear in [CITATION].', '1706.02388-2-9-3': 'There are several open questions that would be nice to adress.', '1706.02388-2-10-0': 'It would be interesting to compute explicitly [MATH] for [MATH].', '1706.02388-2-10-1': 'Once this is found it would be interesting to study its large [MATH] behaviour and compare it to the expectations from the bulk perspective.', '1706.02388-2-10-2': 'It would be important to understand if solutions with finite support in the spin are present.', '1706.02388-2-10-3': 'Preliminary results show that crossing symmetry does not require any non-analytical corrections at finite spin.', '1706.02388-2-10-4': 'On the other hand crossing symmetry allows the addition of any of the truncated solutions constructed in [CITATION].', '1706.02388-2-10-5': 'From the bulk perspective these solutions correspond to counterterms, e.g. the ones that need to be added to render the computation finite.', '1706.02388-2-10-6': 'The 10d supergravity computation contains a quadratic divergence proportional to [MATH], see [CITATION] eq. 4.2.', '1706.02388-2-10-7': 'This will lead to a contribution which becomes large for large [MATH] but has support only for spin zero.', '1706.02388-2-10-8': 'Indeed, this divergence appears to be visible in our computation, when summing over KK-modes for spin zero.', '1706.02388-2-10-9': 'We expect that other extra solutions are not present.', '1706.02388-2-10-10': 'Note that this ambiguity is already present at leading order in [MATH].', '1706.02388-2-10-11': 'In this case, all truncated solutions are forbidden by requiring consistency with the flat space limit, see e.g.[CITATION].', '1706.02388-2-10-12': 'Presumably consistency with the flat space amplitude to order [MATH] will also forbid most extra solutions.', '1706.02388-2-10-13': 'Relatedly, there are several results in the literature that bound the behaviour of [MATH] for large [MATH], see e.g. [CITATION], it would be interesting to extend these results to the order we are considering.', '1706.02388-2-11-0': 'Leading order corrections in [MATH] are in principle possible to consider.', '1706.02388-2-11-1': 'At leading order they correspond to the addition of the first truncated solution with a known coefficient [CITATION].', '1706.02388-2-11-2': "At order [MATH] one would have to 'square' the supergravity contribution plus this contribution.", '1706.02388-2-11-3': 'Since the latter is truncated in the spin, the extra sums involved are very simple.', '1706.02388-2-11-4': 'This computation is also expected to lead to divergences, since the first truncated solution grows much faster, with [MATH], than supergravity.', '1706.02388-2-11-5': 'One could also consider the exchange of a finite number of single trace operators, combining the results of [CITATION] with the methods of this note.', '1706.02388-2-12-0': 'It would be interesting to study the full four-point correlator in space time.', '1706.02388-2-12-1': 'In this note we have computed explicitly the piece proportional to [MATH], which should encode the full physical information about the correlator [CITATION].', '1706.02388-2-12-2': 'For instance, we have seen that from this piece, through crossing, the CFT data follows to all orders in [MATH], and from this the four point correlator can be reconstructed, up to pieces which contribute only for finite values of the spin given in [CITATION].', '1706.02388-2-12-3': 'It would be interesting to study this problem in Mellin space.', '1706.02388-2-12-4': 'This would be the first step to extend the results of [CITATION] to include loop corrections.', '1706.02388-2-13-0': 'The expansion in [MATH] for non-protected quantities in the context of [MATH] duality is a largely unexplored subject.', '1706.02388-2-13-1': 'Our result opens a window to study this problem systematically and quantitatively.', '1706.02388-2-14-0': 'We are grateful to G. Arutyunov, Z. Komargodski, J. Maldacena, A. Zhiboedov and specially Ofer Aharony for useful discussions.', '1706.02388-2-14-1': 'We would like to thank the ICTP-SAIFR in Sao Paulo for their hospitality during part of this work.', '1706.02388-2-14-2': 'The work of L.F.A was supported by ERC STG grant 306260.', '1706.02388-2-14-3': 'L.F.A. is a Wolfson Royal Society Research Merit Award holder.', '1706.02388-2-14-4': 'The work of A.B. is partially supported by Templeton Award 52476 of A. Strominger and by Simons Investigator Award from the Simons Foundation of X. Yin.', '1706.02388-2-15-0': '# Appendices', '1706.02388-2-16-0': 'The Mixing problem.', '1706.02388-2-16-1': 'A technical obstacle in inferring the [MATH] piece of the correlator at order [MATH] is mixing.', '1706.02388-2-16-2': 'For a given twist [MATH] all double trace operators [MATH] mix, and the eigenfunctions of the Hamiltonian are certain combinations of those [EQUATION] where the dependence on the spin is implicit.', '1706.02388-2-16-3': 'We will consider this problem at leading order in [MATH].', '1706.02388-2-16-4': 'We can choose the double trace operators [MATH] to be canonically normalised, such that the coefficients [MATH] form an orthonormal matrix in this basis.', '1706.02388-2-16-5': 'In order to solve the mixing problem we consider general correlators [MATH].', '1706.02388-2-16-6': 'At zeroth order the above operators appear in this correlator with OPE coefficient [EQUATION] where [MATH] could also depend on the spin and the twist.', '1706.02388-2-16-7': 'Note that this sum is proportional to [MATH].', '1706.02388-2-16-8': 'At order [MATH] these operators acquire an anomalous dimension [MATH].', '1706.02388-2-16-9': 'From the explicit conformal block decomposition for the correlator [MATH] in the supergravity approximation we can read off [EQUATION] the averages [MATH] for a given twist can be conveniently packed in a mixing matrix [MATH].', '1706.02388-2-16-10': 'For instance, for [MATH] we have to analyse the correlators with [MATH], which can be found in [CITATION].', '1706.02388-2-16-11': 'This leads to the following mixing matrix [EQUATION] where [MATH] for [MATH].', '1706.02388-2-16-12': 'We have analysed this problem for several values of [MATH], using the explicit supergravity results in [CITATION].', '1706.02388-2-16-13': 'The averages [MATH] at order [MATH] are given by the elements of [MATH].', '1706.02388-2-16-14': 'We are interested in [MATH].', '1706.02388-2-16-15': 'For instance, for the case [MATH] we obtain [EQUATION]', '1706.02388-2-16-16': 'In all cases we found the remarkable pattern ([REF]) quoted in the body of the note.', '1706.02388-2-16-17': 'As seen there, this structure together with crossing symmetry is enough to fix all coefficients [MATH].', '1706.02388-2-16-18': 'These values also agree with the ones found by explicit computations.', '1706.02388-2-17-0': 'Twist conformal blocks.', '1706.02388-2-17-1': 'The zeroth order correlator can be expressed in terms of twist conformal blocks [MATH], defined as the contribution from operators with twist [MATH].', '1706.02388-2-17-2': '[EQUATION]', '1706.02388-2-17-3': 'The explicit form of (super-)conformal blocks leads to the following structure [EQUATION] by plugging this structure into ([REF]) and expanding around [MATH] the functions [MATH] can be found [EQUATION] where [MATH].', '1706.02388-2-17-4': 'Next we define the sequence of functions [MATH] corresponding to extra insertions of [MATH] [EQUATION] [MATH] is the eigenvalue of a specific quadratic Casimir operator.', '1706.02388-2-17-5': 'More precisely, [MATH] admits the same factorisation as in ([REF]) with [MATH], where the functions [MATH] satisfy the following recursive relation [EQUATION] and [MATH].', '1706.02388-2-17-6': 'With this recursion relation together with the expression for [MATH] we can find [MATH] exactly for the first few values of [MATH] and also as various expansions.', '1706.02388-2-17-7': 'The divergent behaviour as [MATH] for [MATH] will be important for us.', '1706.02388-2-17-8': 'From the explicit answer we see [EQUATION]', '1706.02388-2-17-9': 'It can be then seen that [EQUATION] with [MATH] as [MATH] and [EQUATION]', '1706.02388-2-17-10': 'The functions [MATH] can be build recursively to any desired order.'}

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['1706.02388-1-7-24', '1706.02388-2-8-24'], ['1706.02388-1-7-25', '1706.02388-2-8-25'], ['1706.02388-1-7-26', '1706.02388-2-8-26'], ['1706.02388-1-7-27', '1706.02388-2-8-27'], ['1706.02388-1-7-28', '1706.02388-2-8-28'], ['1706.02388-1-7-29', '1706.02388-2-8-29'], ['1706.02388-1-7-30', '1706.02388-2-8-30'], ['1706.02388-1-7-31', '1706.02388-2-8-31'], ['1706.02388-1-7-32', '1706.02388-2-8-32'], ['1706.02388-1-7-33', '1706.02388-2-8-33'], ['1706.02388-1-7-34', '1706.02388-2-8-34'], ['1706.02388-1-7-35', '1706.02388-2-8-35'], ['1706.02388-1-7-36', '1706.02388-2-8-36'], ['1706.02388-1-1-1', '1706.02388-2-1-1'], ['1706.02388-1-1-2', '1706.02388-2-1-2'], ['1706.02388-1-1-3', '1706.02388-2-1-3'], ['1706.02388-1-1-5', '1706.02388-2-1-5'], ['1706.02388-1-1-6', '1706.02388-2-1-6'], ['1706.02388-1-6-0', '1706.02388-2-7-0'], ['1706.02388-1-6-1', '1706.02388-2-7-1'], ['1706.02388-1-6-2', '1706.02388-2-7-2'], ['1706.02388-1-6-3', '1706.02388-2-7-3'], ['1706.02388-1-6-4', '1706.02388-2-7-4'], ['1706.02388-1-8-1', '1706.02388-2-9-1'], ['1706.02388-1-8-2', '1706.02388-2-9-2'], ['1706.02388-1-2-0', '1706.02388-2-2-0'], ['1706.02388-1-2-1', '1706.02388-2-2-1'], ['1706.02388-1-2-2', '1706.02388-2-2-2'], ['1706.02388-1-2-5', '1706.02388-2-2-5'], ['1706.02388-1-2-6', '1706.02388-2-2-6'], ['1706.02388-1-2-7', '1706.02388-2-2-7'], ['1706.02388-1-2-8', '1706.02388-2-2-8'], ['1706.02388-1-2-9', '1706.02388-2-2-9'], ['1706.02388-1-2-10', '1706.02388-2-2-10'], ['1706.02388-1-2-11', '1706.02388-2-2-11'], ['1706.02388-1-2-12', '1706.02388-2-2-12'], ['1706.02388-1-2-13', '1706.02388-2-2-13'], ['1706.02388-1-2-14', '1706.02388-2-2-14'], ['1706.02388-1-2-15', '1706.02388-2-2-15'], ['1706.02388-1-2-16', '1706.02388-2-2-16'], ['1706.02388-1-15-1', '1706.02388-2-16-1'], ['1706.02388-1-15-2', '1706.02388-2-16-2'], ['1706.02388-1-15-3', '1706.02388-2-16-3'], ['1706.02388-1-15-4', '1706.02388-2-16-4'], ['1706.02388-1-15-5', '1706.02388-2-16-5'], ['1706.02388-1-15-6', '1706.02388-2-16-6'], ['1706.02388-1-15-7', '1706.02388-2-16-7'], ['1706.02388-1-15-8', '1706.02388-2-16-8'], ['1706.02388-1-15-9', '1706.02388-2-16-9'], ['1706.02388-1-15-10', '1706.02388-2-16-10'], ['1706.02388-1-15-11', '1706.02388-2-16-11'], ['1706.02388-1-15-12', '1706.02388-2-16-12'], ['1706.02388-1-15-13', '1706.02388-2-16-13'], ['1706.02388-1-15-14', '1706.02388-2-16-14'], ['1706.02388-1-15-15', '1706.02388-2-16-15'], ['1706.02388-1-15-16', '1706.02388-2-16-16'], ['1706.02388-1-15-17', '1706.02388-2-16-17'], ['1706.02388-1-15-18', '1706.02388-2-16-18'], ['1706.02388-1-16-1', '1706.02388-2-17-1'], ['1706.02388-1-16-3', '1706.02388-2-17-3'], ['1706.02388-1-16-4', '1706.02388-2-17-4'], ['1706.02388-1-16-5', '1706.02388-2-17-5'], ['1706.02388-1-16-6', '1706.02388-2-17-6'], ['1706.02388-1-16-7', '1706.02388-2-17-7'], ['1706.02388-1-16-8', '1706.02388-2-17-8'], ['1706.02388-1-16-9', '1706.02388-2-17-9'], ['1706.02388-1-16-10', '1706.02388-2-17-10'], ['1706.02388-1-4-0', '1706.02388-2-5-0'], ['1706.02388-1-4-1', '1706.02388-2-5-1'], ['1706.02388-1-4-2', '1706.02388-2-5-2'], ['1706.02388-1-4-3', '1706.02388-2-5-3'], ['1706.02388-1-4-4', '1706.02388-2-5-4'], ['1706.02388-1-4-5', '1706.02388-2-5-5'], ['1706.02388-1-4-6', '1706.02388-2-5-6'], ['1706.02388-1-4-7', '1706.02388-2-5-7'], ['1706.02388-1-4-8', '1706.02388-2-5-8'], ['1706.02388-1-4-9', '1706.02388-2-5-9'], ['1706.02388-1-4-10', '1706.02388-2-5-10'], ['1706.02388-1-4-11', '1706.02388-2-5-11'], ['1706.02388-1-4-12', '1706.02388-2-5-12'], ['1706.02388-1-4-13', '1706.02388-2-5-13'], ['1706.02388-1-4-14', '1706.02388-2-5-14'], ['1706.02388-1-4-15', '1706.02388-2-5-15'], ['1706.02388-1-4-16', '1706.02388-2-5-16'], ['1706.02388-1-4-17', '1706.02388-2-5-17'], ['1706.02388-1-4-18', '1706.02388-2-5-18'], ['1706.02388-1-4-19', '1706.02388-2-5-19'], ['1706.02388-1-4-20', '1706.02388-2-5-20'], ['1706.02388-1-10-0', '1706.02388-2-11-0'], ['1706.02388-1-10-1', '1706.02388-2-11-1'], ['1706.02388-1-10-2', '1706.02388-2-11-2'], ['1706.02388-1-10-3', '1706.02388-2-11-3'], ['1706.02388-1-10-5', '1706.02388-2-11-5'], ['1706.02388-1-5-0', '1706.02388-2-6-0'], ['1706.02388-1-5-1', '1706.02388-2-6-1'], ['1706.02388-1-5-2', '1706.02388-2-6-2'], ['1706.02388-1-5-3', '1706.02388-2-6-3'], ['1706.02388-1-5-4', '1706.02388-2-6-4'], ['1706.02388-1-5-5', '1706.02388-2-6-5'], ['1706.02388-1-5-6', '1706.02388-2-6-6'], ['1706.02388-1-5-7', '1706.02388-2-6-7'], ['1706.02388-1-5-8', '1706.02388-2-6-8'], ['1706.02388-1-5-9', '1706.02388-2-6-9'], ['1706.02388-1-5-10', '1706.02388-2-6-10'], ['1706.02388-1-5-11', '1706.02388-2-6-11'], ['1706.02388-1-5-12', '1706.02388-2-6-12'], ['1706.02388-1-5-13', '1706.02388-2-6-13'], ['1706.02388-1-5-14', '1706.02388-2-6-14'], ['1706.02388-1-5-15', '1706.02388-2-6-15'], ['1706.02388-1-5-16', '1706.02388-2-6-16'], ['1706.02388-1-5-17', '1706.02388-2-6-17'], ['1706.02388-1-5-18', '1706.02388-2-6-18'], ['1706.02388-1-5-19', '1706.02388-2-6-19'], ['1706.02388-1-5-20', '1706.02388-2-6-20'], ['1706.02388-1-5-21', '1706.02388-2-6-21'], ['1706.02388-1-5-22', '1706.02388-2-6-22'], ['1706.02388-1-5-23', '1706.02388-2-6-23'], ['1706.02388-1-5-24', '1706.02388-2-6-24'], ['1706.02388-1-5-25', '1706.02388-2-6-25'], ['1706.02388-1-5-26', '1706.02388-2-6-26'], ['1706.02388-1-9-0', '1706.02388-2-10-0'], ['1706.02388-1-9-1', '1706.02388-2-10-1'], ['1706.02388-1-9-2', '1706.02388-2-10-2'], ['1706.02388-1-9-3', '1706.02388-2-10-3'], ['1706.02388-1-9-4', '1706.02388-2-10-4'], ['1706.02388-1-9-6', '1706.02388-2-10-6'], ['1706.02388-1-9-7', '1706.02388-2-10-7'], ['1706.02388-1-9-8', '1706.02388-2-10-9'], ['1706.02388-1-9-9', '1706.02388-2-10-10'], ['1706.02388-1-9-10', '1706.02388-2-10-11'], ['1706.02388-1-9-11', '1706.02388-2-10-12'], ['1706.02388-1-9-12', '1706.02388-2-10-13'], ['1706.02388-1-13-0', '1706.02388-2-14-0'], ['1706.02388-1-13-1', '1706.02388-2-14-1'], ['1706.02388-1-13-2', '1706.02388-2-14-2'], ['1706.02388-1-13-3', '1706.02388-2-14-3'], ['1706.02388-1-13-4', '1706.02388-2-14-4'], ['1706.02388-1-11-4', '1706.02388-2-12-4'], ['1706.02388-1-3-0', '1706.02388-2-3-0'], ['1706.02388-1-3-2', '1706.02388-2-3-2'], ['1706.02388-1-3-4', '1706.02388-2-3-4'], ['1706.02388-1-3-5', '1706.02388-2-3-5'], ['1706.02388-1-12-0', '1706.02388-2-13-0'], ['1706.02388-1-12-1', '1706.02388-2-13-1'], ['1706.02388-1-0-0', '1706.02388-2-0-0'], ['1706.02388-1-0-1', '1706.02388-2-0-1'], ['1706.02388-1-0-2', '1706.02388-2-0-2'], ['1706.02388-1-0-3', '1706.02388-2-0-3'], ['1706.02388-1-0-4', '1706.02388-2-0-4'], ['1706.02388-1-7-21', '1706.02388-2-8-21'], ['1706.02388-1-1-4', '1706.02388-2-1-4'], ['1706.02388-1-8-3', '1706.02388-2-9-3'], ['1706.02388-1-10-4', '1706.02388-2-11-4'], ['1706.02388-1-9-5', '1706.02388-2-10-5'], ['1706.02388-1-11-0', '1706.02388-2-12-0'], ['1706.02388-1-3-3', '1706.02388-2-3-3'], ['1706.02388-1-7-37', '1706.02388-2-8-37'], ['1706.02388-1-7-38', '1706.02388-2-8-40'], ['1706.02388-1-2-3', '1706.02388-2-2-4'], ['1706.02388-1-2-4', '1706.02388-2-2-3'], ['1706.02388-1-3-1', '1706.02388-2-3-1']]

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['1706.02388-1-7-24', '1706.02388-2-8-24'], ['1706.02388-1-7-25', '1706.02388-2-8-25'], ['1706.02388-1-7-26', '1706.02388-2-8-26'], ['1706.02388-1-7-27', '1706.02388-2-8-27'], ['1706.02388-1-7-28', '1706.02388-2-8-28'], ['1706.02388-1-7-29', '1706.02388-2-8-29'], ['1706.02388-1-7-30', '1706.02388-2-8-30'], ['1706.02388-1-7-31', '1706.02388-2-8-31'], ['1706.02388-1-7-32', '1706.02388-2-8-32'], ['1706.02388-1-7-33', '1706.02388-2-8-33'], ['1706.02388-1-7-34', '1706.02388-2-8-34'], ['1706.02388-1-7-35', '1706.02388-2-8-35'], ['1706.02388-1-7-36', '1706.02388-2-8-36'], ['1706.02388-1-1-1', '1706.02388-2-1-1'], ['1706.02388-1-1-2', '1706.02388-2-1-2'], ['1706.02388-1-1-3', '1706.02388-2-1-3'], ['1706.02388-1-1-5', '1706.02388-2-1-5'], ['1706.02388-1-1-6', '1706.02388-2-1-6'], ['1706.02388-1-6-0', '1706.02388-2-7-0'], ['1706.02388-1-6-1', '1706.02388-2-7-1'], ['1706.02388-1-6-2', '1706.02388-2-7-2'], ['1706.02388-1-6-3', '1706.02388-2-7-3'], ['1706.02388-1-6-4', '1706.02388-2-7-4'], ['1706.02388-1-8-1', '1706.02388-2-9-1'], ['1706.02388-1-8-2', '1706.02388-2-9-2'], ['1706.02388-1-2-0', '1706.02388-2-2-0'], ['1706.02388-1-2-1', '1706.02388-2-2-1'], ['1706.02388-1-2-2', '1706.02388-2-2-2'], ['1706.02388-1-2-5', '1706.02388-2-2-5'], ['1706.02388-1-2-6', '1706.02388-2-2-6'], ['1706.02388-1-2-7', '1706.02388-2-2-7'], ['1706.02388-1-2-8', '1706.02388-2-2-8'], ['1706.02388-1-2-9', '1706.02388-2-2-9'], ['1706.02388-1-2-10', '1706.02388-2-2-10'], ['1706.02388-1-2-11', '1706.02388-2-2-11'], ['1706.02388-1-2-12', '1706.02388-2-2-12'], ['1706.02388-1-2-13', '1706.02388-2-2-13'], ['1706.02388-1-2-14', '1706.02388-2-2-14'], ['1706.02388-1-2-15', '1706.02388-2-2-15'], ['1706.02388-1-2-16', '1706.02388-2-2-16'], ['1706.02388-1-15-1', '1706.02388-2-16-1'], ['1706.02388-1-15-2', '1706.02388-2-16-2'], ['1706.02388-1-15-3', '1706.02388-2-16-3'], ['1706.02388-1-15-4', '1706.02388-2-16-4'], ['1706.02388-1-15-5', '1706.02388-2-16-5'], ['1706.02388-1-15-6', '1706.02388-2-16-6'], ['1706.02388-1-15-7', '1706.02388-2-16-7'], ['1706.02388-1-15-8', '1706.02388-2-16-8'], ['1706.02388-1-15-9', '1706.02388-2-16-9'], ['1706.02388-1-15-10', '1706.02388-2-16-10'], ['1706.02388-1-15-11', '1706.02388-2-16-11'], ['1706.02388-1-15-12', '1706.02388-2-16-12'], ['1706.02388-1-15-13', '1706.02388-2-16-13'], ['1706.02388-1-15-14', '1706.02388-2-16-14'], ['1706.02388-1-15-15', '1706.02388-2-16-15'], ['1706.02388-1-15-16', '1706.02388-2-16-16'], ['1706.02388-1-15-17', '1706.02388-2-16-17'], ['1706.02388-1-15-18', '1706.02388-2-16-18'], ['1706.02388-1-16-1', '1706.02388-2-17-1'], ['1706.02388-1-16-3', '1706.02388-2-17-3'], ['1706.02388-1-16-4', '1706.02388-2-17-4'], ['1706.02388-1-16-5', '1706.02388-2-17-5'], ['1706.02388-1-16-6', '1706.02388-2-17-6'], ['1706.02388-1-16-7', '1706.02388-2-17-7'], ['1706.02388-1-16-8', '1706.02388-2-17-8'], ['1706.02388-1-16-9', '1706.02388-2-17-9'], ['1706.02388-1-16-10', '1706.02388-2-17-10'], ['1706.02388-1-4-0', '1706.02388-2-5-0'], ['1706.02388-1-4-1', '1706.02388-2-5-1'], ['1706.02388-1-4-2', '1706.02388-2-5-2'], ['1706.02388-1-4-3', '1706.02388-2-5-3'], ['1706.02388-1-4-4', '1706.02388-2-5-4'], ['1706.02388-1-4-5', '1706.02388-2-5-5'], ['1706.02388-1-4-6', '1706.02388-2-5-6'], ['1706.02388-1-4-7', '1706.02388-2-5-7'], ['1706.02388-1-4-8', '1706.02388-2-5-8'], ['1706.02388-1-4-9', '1706.02388-2-5-9'], ['1706.02388-1-4-10', '1706.02388-2-5-10'], ['1706.02388-1-4-11', '1706.02388-2-5-11'], ['1706.02388-1-4-12', '1706.02388-2-5-12'], ['1706.02388-1-4-13', '1706.02388-2-5-13'], ['1706.02388-1-4-14', '1706.02388-2-5-14'], ['1706.02388-1-4-15', '1706.02388-2-5-15'], ['1706.02388-1-4-16', '1706.02388-2-5-16'], ['1706.02388-1-4-17', '1706.02388-2-5-17'], ['1706.02388-1-4-18', '1706.02388-2-5-18'], ['1706.02388-1-4-19', '1706.02388-2-5-19'], ['1706.02388-1-4-20', '1706.02388-2-5-20'], ['1706.02388-1-10-0', '1706.02388-2-11-0'], ['1706.02388-1-10-1', '1706.02388-2-11-1'], ['1706.02388-1-10-2', '1706.02388-2-11-2'], ['1706.02388-1-10-3', '1706.02388-2-11-3'], ['1706.02388-1-10-5', '1706.02388-2-11-5'], ['1706.02388-1-5-0', '1706.02388-2-6-0'], ['1706.02388-1-5-1', '1706.02388-2-6-1'], ['1706.02388-1-5-2', '1706.02388-2-6-2'], ['1706.02388-1-5-3', '1706.02388-2-6-3'], ['1706.02388-1-5-4', '1706.02388-2-6-4'], ['1706.02388-1-5-5', '1706.02388-2-6-5'], ['1706.02388-1-5-6', '1706.02388-2-6-6'], ['1706.02388-1-5-7', '1706.02388-2-6-7'], ['1706.02388-1-5-8', '1706.02388-2-6-8'], ['1706.02388-1-5-9', '1706.02388-2-6-9'], ['1706.02388-1-5-10', '1706.02388-2-6-10'], ['1706.02388-1-5-11', '1706.02388-2-6-11'], ['1706.02388-1-5-12', '1706.02388-2-6-12'], ['1706.02388-1-5-13', '1706.02388-2-6-13'], ['1706.02388-1-5-14', '1706.02388-2-6-14'], ['1706.02388-1-5-15', '1706.02388-2-6-15'], ['1706.02388-1-5-16', '1706.02388-2-6-16'], ['1706.02388-1-5-17', '1706.02388-2-6-17'], ['1706.02388-1-5-18', '1706.02388-2-6-18'], ['1706.02388-1-5-19', '1706.02388-2-6-19'], ['1706.02388-1-5-20', '1706.02388-2-6-20'], ['1706.02388-1-5-21', '1706.02388-2-6-21'], ['1706.02388-1-5-22', '1706.02388-2-6-22'], ['1706.02388-1-5-23', '1706.02388-2-6-23'], ['1706.02388-1-5-24', '1706.02388-2-6-24'], ['1706.02388-1-5-25', '1706.02388-2-6-25'], ['1706.02388-1-5-26', '1706.02388-2-6-26'], ['1706.02388-1-9-0', '1706.02388-2-10-0'], ['1706.02388-1-9-1', '1706.02388-2-10-1'], ['1706.02388-1-9-2', '1706.02388-2-10-2'], ['1706.02388-1-9-3', '1706.02388-2-10-3'], ['1706.02388-1-9-4', '1706.02388-2-10-4'], ['1706.02388-1-9-6', '1706.02388-2-10-6'], ['1706.02388-1-9-7', '1706.02388-2-10-7'], ['1706.02388-1-9-8', '1706.02388-2-10-9'], ['1706.02388-1-9-9', '1706.02388-2-10-10'], ['1706.02388-1-9-10', '1706.02388-2-10-11'], ['1706.02388-1-9-11', '1706.02388-2-10-12'], ['1706.02388-1-9-12', '1706.02388-2-10-13'], ['1706.02388-1-13-0', '1706.02388-2-14-0'], ['1706.02388-1-13-1', '1706.02388-2-14-1'], ['1706.02388-1-13-2', '1706.02388-2-14-2'], ['1706.02388-1-13-3', '1706.02388-2-14-3'], ['1706.02388-1-13-4', '1706.02388-2-14-4'], ['1706.02388-1-11-4', '1706.02388-2-12-4'], ['1706.02388-1-3-0', '1706.02388-2-3-0'], ['1706.02388-1-3-2', '1706.02388-2-3-2'], ['1706.02388-1-3-4', '1706.02388-2-3-4'], ['1706.02388-1-3-5', '1706.02388-2-3-5'], ['1706.02388-1-12-0', '1706.02388-2-13-0'], ['1706.02388-1-12-1', '1706.02388-2-13-1'], ['1706.02388-1-0-0', '1706.02388-2-0-0'], ['1706.02388-1-0-1', '1706.02388-2-0-1'], ['1706.02388-1-0-2', '1706.02388-2-0-2'], ['1706.02388-1-0-3', '1706.02388-2-0-3'], ['1706.02388-1-0-4', '1706.02388-2-0-4']]

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[]

[['1706.02388-1-7-37', '1706.02388-2-8-37'], ['1706.02388-1-7-38', '1706.02388-2-8-40'], ['1706.02388-1-2-3', '1706.02388-2-2-4'], ['1706.02388-1-2-4', '1706.02388-2-2-3'], ['1706.02388-1-3-1', '1706.02388-2-3-1']]

[]

['1706.02388-1-1-0', '1706.02388-1-8-0', '1706.02388-1-15-0', '1706.02388-1-16-0', '1706.02388-1-16-2', '1706.02388-2-1-0', '1706.02388-2-9-0', '1706.02388-2-16-0', '1706.02388-2-17-0', '1706.02388-2-17-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1706.02388

1707.00341

{'1707.00341-1-0-0': 'Criticality represents a specific point in the parameter space of a higher-derivative gravity theory, where the linearized field equations become degenerate.', '1707.00341-1-0-1': 'In 4D Critical Gravity, the Lagrangian contains a Weyl-squared term, which does not modify the asymptotic form of the curvature.', '1707.00341-1-0-2': 'The Weyl[MATH] coupling is chosen such that it eliminates the massive scalar mode and it renders the massive spin-2 mode massless.', '1707.00341-1-0-3': 'In doing so, the theory turns consistent around the critical point.', '1707.00341-1-0-4': 'Here, we employ the Noether-Wald method to derive the conserved quantities for the action of Critical Gravity.', '1707.00341-1-0-5': 'It is manifest from this energy definition that, at the critical point, the mass is identically zero for Einstein spacetimes, what is a defining property of the theory.', '1707.00341-1-0-6': 'As the entropy is obtained from the Noether-Wald charges at the horizon, it is evident that it also vanishes for any Einstein black hole.', '1707.00341-1-1-0': '# Introduction', '1707.00341-1-2-0': 'General Relativity (GR) is a successful theory of gravity at a classical level but it lacks of consistency in a quantum regime because it is not renormalizable.', '1707.00341-1-2-1': 'On the other hand, in the low energy limit of String Theory, which should be finite to all orders, there appear contributions that are quadratic in the curvature.', '1707.00341-1-2-2': 'As a consequence, higher curvature extensions of Einstein gravity are expected to give rise to a gravity theory with a better ultraviolet behavior.', '1707.00341-1-2-3': 'Early work on the subject has suggested that this class of theories should be renormalizable .', '1707.00341-1-3-0': 'Lower-dimensional examples have been extensively studied in recent literature.', '1707.00341-1-3-1': 'They are regarded as insightful toy models which capture essential features of [MATH] gravity.', '1707.00341-1-3-2': 'One of them is New Massive Gravity (NMG) , a parity-even three-dimensional theory which describes two propagating massive spin-2 modes, in contrast to [MATH] Einstein gravity which is topological.', '1707.00341-1-3-3': 'Picking up the conventional sign of the Einstein-Hilbert action, the energy of the massive excitations is negative (ghost modes), while the mass of the Banados-Teitelboim-Zanelli (BTZ) black hole is positive.', '1707.00341-1-3-4': 'Clearly, this inconsistency persists even if one reverses the sign of the kinetic term.', '1707.00341-1-3-5': 'A physically reasonable theory arises at a specific point of parametric space, where the massive spin-2 field turns massless .', '1707.00341-1-3-6': 'At this particular point, both the energy of the graviton and the mass of the BTZ black hole vanish identically .', '1707.00341-1-3-7': 'Furthermore, both central charges turn into zero, what leads to a vanishing entropy .', '1707.00341-1-3-8': 'Another feature of the theory is the presence of new modes with logarithmic behavior at the critical point .', '1707.00341-1-3-9': 'These modes are eliminated when standard Brown-Henneaux boundary conditions are considered.', '1707.00341-1-3-10': 'Relaxing the asymptotic conditions to include log terms switches on new holographic sources at the boundary .', '1707.00341-1-4-0': 'Another theory in three dimensions sharing similar features with NMG is Topologically Massive Gravity (TMG) .', '1707.00341-1-4-1': 'The corresponding critical point defines the concept of Chiral Gravity.', '1707.00341-1-4-2': 'However, in this case, the central charges are different from each other due to a parity-violating term in the action.', '1707.00341-1-4-3': 'As a consequence, neither mass nor entropy vanish for BTZ black holes at the chiral point.', '1707.00341-1-5-0': 'The generalization of the concept of criticality, present in these models, to four dimensions is given by theories which include quadratic terms in the curvature with particular couplings on top of the Einstein-Hilbert action.', '1707.00341-1-5-1': 'The most general form of a gravity action with quadratic-curvature corrections in [MATH] is given by [EQUATION] where [MATH] and [MATH] are arbitrary couplings, and [MATH] is the cosmological constant in terms of the AdS radius [MATH].', '1707.00341-1-5-2': 'The Riemann-squared term is not present, as it can be always traded off by the Gauss-Bonnet (GB) invariant plus the curvature-squared terms present in the action ([REF]).', '1707.00341-1-5-3': 'The GB term does not affect the field equations in the bulk but it does modify the boundary dynamics.', '1707.00341-1-6-0': 'This class of theories leads to equations of motion (EOM) with up to four derivatives in the metric.', '1707.00341-1-6-1': 'Generically, they describe modes that represent a massless spin-2 graviton, a massive spin-2 field and a massive scalar.', '1707.00341-1-6-2': 'For a quadratic-curvature gravity theory with arbitrary coupling constants, perturbations around a given background would give rise to ghosts.', '1707.00341-1-6-3': 'The problem with the sign of the energy of these modes can be circumvented by a sign flip of the constant in front of Einstein kinetic term.', '1707.00341-1-6-4': 'On the other hand, Einstein black holes are solutions to the theory defined by Eq. ([REF]).', '1707.00341-1-6-5': 'Therefore, the change in the sign mentioned above would lead to a negative mass for Schwarzschild-AdS black hole.', '1707.00341-1-6-6': 'Needless to say, this picture is clearly unphysical as the energy of the perturbations around a background and the mass of a black hole carry opposite signs.', '1707.00341-1-6-7': 'In view of this general obstruction to obtain a four-dimensional gravity theory which is free of the inconsistencies discussed above, it was quite surprising when the authors of Ref. pointed out the fact that, for the particular couplings [MATH] and [MATH], the massive scalar is eliminated and the massive spin-2 mode turns massless.', '1707.00341-1-6-8': 'This choice renders the theory physically sensible around the critical point.', '1707.00341-1-6-9': 'This fact is confirmed by using the Ostrogradsky method for Lagrangians with derivatives of higher order: the energy for the massive mode vanishes for the critical value of the couplings.', '1707.00341-1-7-0': 'From the point of view of the energy of the black holes of the theory, one can use the Abbott-Deser-Tekin (ADT) formula to evaluate the mass of Schwarzschild-AdS solution, what results in [EQUATION] where [MATH] is the mass parameter in the solution.', '1707.00341-1-7-1': 'The general formula Eq. ([REF]), makes evident that, for the critical condition mentioned above, the mass for Schwarzschild-AdS black hole vanishes.', '1707.00341-1-7-2': 'In the present work, as an alternative to Deser-Tekin procedure, we employ Noether-Wald method to compute the charges in Critical Gravity.', '1707.00341-1-7-3': 'This full (non-linearized) expression derived in this way has a remarkable property: the energy of any Einstein space is identically zero, as long anticipated in Ref.', '1707.00341-1-8-0': '# Deser-Tekin energy in [MATH] quadratic-curvature gravity', '1707.00341-1-9-0': 'As mentioned in the previous section, in Refs., the authors provide a generic definition of energy for an arbitrary curvature-squared gravity theory.', '1707.00341-1-9-1': 'That definition of the energy is obtained as an extension of the Abbott-Deser method .', '1707.00341-1-10-0': 'In order to obtain the ADT mass for a general asymptotically AdS (AAdS) solution, we need to write down the metric of the spacetime in the form of [MATH], where [MATH] is the metric of the background and [MATH] is the perturbation tensor.', '1707.00341-1-10-1': 'Such construction leaves the first-order variation of field equations as [EQUATION] where [MATH] and [MATH] are the linearized expression of Einstein tensor and Ricci scalar, respectively.', '1707.00341-1-10-2': 'The tensor [MATH] is the contribution of fourth order in the derivatives to the field equations.', '1707.00341-1-10-3': 'The equation ([REF]) has to be equal to an effective energy-momentum tensor [MATH], which is covariantly conserved.', '1707.00341-1-10-4': 'One can write a conserved current, for a set of Killing fields [MATH] that represents the isometries of the background [EQUATION]', '1707.00341-1-10-5': 'In order to evaluate the mass of a gravitational object, the Killing vector needs to be timelike, at least, at infinity.', '1707.00341-1-10-6': 'Whenever there is a current which is conserved, one is able to write down [MATH] as the divergence of a 2-form prepotential, i.e., [EQUATION]', '1707.00341-1-10-7': 'One can consider a spacetime foliated by a normal (radial) direction [MATH] [EQUATION] where [MATH] is the induced metric on [MATH], and its radial evolution is defined by the unit vector [MATH].', '1707.00341-1-11-0': 'In this coordinate frame, the conserved charge can be expressed as an integral on the co-dimension two surface [MATH] [EQUATION]', '1707.00341-1-11-1': 'Here, [MATH] is a surface normal vector that defines the integration for a fixed time and radius.', '1707.00341-1-11-2': 'For the case of curvature-squared gravity in four dimensions, the conserved quantity adopts the form [EQUATION]', '1707.00341-1-12-0': '# Critical Gravity', '1707.00341-1-13-0': 'In Ref. , the energy of the graviton modes in quadratic-curvature gravity was studied.', '1707.00341-1-13-1': 'These excitations come from the linearized EOM ([REF]).', '1707.00341-1-13-2': 'The choice [MATH] leads to a traceless perturbation ([MATH]) which eliminates the massive scalar mode.', '1707.00341-1-13-3': 'Consequently, the equation for the propagating mode takes the form [EQUATION]', '1707.00341-1-13-4': 'The first factor of the equation describes the propagation of a massless graviton in an AdS background while the second one represents a massive spin-2 field.', '1707.00341-1-13-5': 'It is clear that the latter becomes massless by imposing the critical value [MATH].', '1707.00341-1-13-6': 'This particular coupling produces the fourth order equation [EQUATION] which reflects the appearance of both massless and logarithmic modes .', '1707.00341-1-13-7': 'In order to obtain the energy of the excitations, the authors in Ref. followed a Hamiltonian approach.', '1707.00341-1-13-8': 'For an unrestricted value of[MATH], the action up to quadratic order in [MATH] is [EQUATION]', '1707.00341-1-13-9': 'Using the Ostrogradsky method for higher-derivative Lagrangians, one obtains the following conjugate momenta [EQUATION]', '1707.00341-1-13-10': 'Due to the fact that the Lagrangian is time independent, the Hamiltonian can be written as its time average, that is [EQUATION]', '1707.00341-1-13-11': 'Evaluating for the case of massless and massive propagating modes, one obtains the following expressions for the corresponding on-shell energies [EQUATION] where the subscripts [MATH] and [MATH] stand for massless graviton and massive spin-2 field, respectively.', '1707.00341-1-14-0': 'In a gravity theory with quadratic terms in the curvature, where the couplings are related as [MATH], there is only a specific value of [MATH] that kills the negative energy states.', '1707.00341-1-14-1': 'More specifically, from Eqs. ([REF],[REF]) it is shown that for [MATH], the energy of both the massless and the massive modes is zero.', '1707.00341-1-14-2': 'Hence, all the ghosts disappear leading to a consistent theory of gravity.', '1707.00341-1-15-0': 'On the other hand, the generic expression for the energy of the black holes in this gravity theory is given by Eq.([REF]).', '1707.00341-1-15-1': 'For any static black hole, the only nonvanishing contribution comes from the first term on the right hand side of Eq.([REF]).', '1707.00341-1-15-2': 'In particular, for a Schwarzschild-AdS black hole, the ADT charge leads to the result in Eq.([REF]).', '1707.00341-1-15-3': 'Is is easy to notice that, for the particular value of the couplings which define Critical Gravity ([MATH], [MATH]), the mass of the black hole vanish.', '1707.00341-1-16-0': 'In what follows, we provide an alternative formula of conserved charges in Critical Gravity, which makes manifest the fact that the energy for Einstein black holes is identically zero.', '1707.00341-1-17-0': '# Noether-Wald charges in Critical Gravity', '1707.00341-1-18-0': 'A general prescription to define conserved charges in an arbitrary theory of gravity was given in Refs..', '1707.00341-1-18-1': 'For the purpose of the discussion below, we will restrict ourselves to the case where Lagrangian density is a functional only of the metric and the curvature, [MATH].', '1707.00341-1-18-2': 'For a given set of Killing vectors [MATH], the Noether current is written down as [EQUATION]', '1707.00341-1-18-3': 'For simplicity, we assume that the surface term [MATH] is separable into a part that contains variations of the Christoffel symbol and another part that contains variations of the metric.', '1707.00341-1-18-4': 'As we are interested in diffeomorphic charges for gravity, all the variations are replaced by a Lie derivative along the vector [MATH].', '1707.00341-1-19-0': 'Using the Killing equation, [MATH], one can notice that first term in Eq. ([REF]) vanishes.', '1707.00341-1-19-1': 'The same relation, this time for the Lie derivative of the Christoffel connection, would produce a combination of double covariant derivatives and curvatures.', '1707.00341-1-19-2': 'This casts the current, for a generic gravity theory, in the form [EQUATION]', '1707.00341-1-19-3': 'Here, the tensor [MATH] is the functional derivative of [MATH] with respect to the spacetime Riemann tensor [MATH], that is, [EQUATION]', '1707.00341-1-19-4': 'It can be shown, by means of the general form of the field equations for these class of gravity theories, that the last two terms on the right hand side of ([REF]) form the EOM contracted with the Killing field.', '1707.00341-1-19-5': 'Thus, on-shell, the first term on the right side of ([REF]) is the only nonvanishing part.', '1707.00341-1-20-0': 'As the tensor [MATH] satisfies Bianchi identity, the conserved current turns into a total derivative [EQUATION]', '1707.00341-1-20-1': 'As the Noether current [MATH] can be written as [MATH], the conserved charge is expressed as an integral on the co-dimension two surface [MATH] [EQUATION] as mentioned previously in Section [REF].', '1707.00341-1-20-2': 'Finally the conserved charge is written as [EQUATION]', '1707.00341-1-20-3': 'An alternative form for the action of Critical Gravity considers the difference between Weyl[MATH] and the GB term [MATH], as the GB invariant term does not alter the bulk dynamics [EQUATION]', '1707.00341-1-20-4': 'We can split the action in two parts: the first one is the MacDowell-Mansouri action, [MATH], which is given by the Einstein-Hilbert plus GB terms, the latter with a fixed coupling .', '1707.00341-1-20-5': 'In Einstein gravity, this corresponds to a built-in renormalized AdS action .', '1707.00341-1-20-6': 'The second part is minus the action of Conformal Gravity [MATH].', '1707.00341-1-20-7': 'Using the Noether-Wald formula for the current ([REF]) for the first part, the functional derivative with respect to the Riemann tensor of the Lagrangian in [MATH] produces [EQUATION] whereas, for the Conformal Gravity part [MATH], we get [EQUATION]', '1707.00341-1-20-8': 'Using the Noether-Wald formula ([REF]), the total charge for the theory [EQUATION]', '1707.00341-1-20-9': 'By definition, the Weyl tensor is [EQUATION]', '1707.00341-1-20-10': 'For Einstein spaces, [MATH], the Weyl tensor adopts the particular form [EQUATION] where the right hand side, is referred to as AdS curvature', '1707.00341-1-21-0': 'Using the above fact, the conserved quantity in Critical Gravity is identically zero for Einstein spaces.', '1707.00341-1-22-0': '# Electric part of the Weyl tensor and Einstein modes in Conformal Gravity', '1707.00341-1-23-0': 'CG in four dimensions is invariant under local Weyl rescalings of the metric ([MATH]).', '1707.00341-1-23-1': 'Solutions to CG are Bach-flat geometries, which include Einstein spacetimes.', '1707.00341-1-24-0': 'From a holographic viewpoint, asymptotically AdS space in CG are endowed with new sources at the conformal boundary.', '1707.00341-1-24-1': 'Indeed, we can set any AAdS spacetime in Fefferman-Graham (FG) form of the metric [EQUATION] where the metric [MATH] is expanded as a power series around the boundary [MATH], i.e., [EQUATION]', '1707.00341-1-24-2': 'Here, the ellipsis denotes higher-order terms which do not enter into the holographic description of [MATH] AAdS spaces.', '1707.00341-1-25-0': 'The presence of the term [MATH] reflects the fact the space contains a non-Einstein part.', '1707.00341-1-25-1': 'By demanding the vanishing of the linear term on [MATH], one recovers the Einstein branch, with only even powers of [MATH] in the expansion.', '1707.00341-1-25-2': 'This is achieved by imposing a Neumann boundary condition on the metric, [MATH] .', '1707.00341-1-26-0': 'On the other hand, the Noether-Wald charge for Conformal Gravity is proportional to the Weyl tensor, as shown by Eq. ([REF]).', '1707.00341-1-26-1': 'However, it is not obvious whether, for Einstein spaces, the holographic modes of CG at the boundary are contained in the electric part of the Weyl tensor [EQUATION] as it is the case in Einstein gravity.', '1707.00341-1-27-0': 'As Einstein spaces are solutions of the EOM of CG in the bulk, we restrict the discussion to the surface term in the variation of [MATH], that is, [EQUATION] where [MATH] is the Einstein part of the Weyl tensor ([REF]).', '1707.00341-1-28-0': 'The second term in the above relation can be eliminated using the Bianchi identity of second kind.', '1707.00341-1-28-1': 'A projection of all indices to the boundary can be performed by taking the explicit form of the normal vector [MATH] in Gaussian coordinates.', '1707.00341-1-28-2': 'Then, the surface term takes the form [EQUATION]', '1707.00341-1-28-3': 'In Gauss normal frame ([REF]), the relevant components of the Christoffel symbol are [EQUATION] where [MATH] is the extrinsic curvature at [MATH].', '1707.00341-1-28-4': 'Equipped with this result, the variation of the action is written as [EQUATION] after some algebraic manipulation and index relabelling.', '1707.00341-1-29-0': 'The rest of the proof relies on a power-counting argument in the radial coordinate [MATH].', '1707.00341-1-29-1': 'In order to do so, it is required to expand the tensorial quantities which appear at the surface term.', '1707.00341-1-30-0': 'First, we consider the FG expansion for Einstein spacetimes, where [MATH] and [MATH] with the metric at the conformal boundary given by [EQUATION]', '1707.00341-1-30-1': 'From this form of the metric, the following expressions are straightforwardly derived [EQUATION] where [MATH] is the Schouten tensor defined for the boundary metric [MATH], i.e., [EQUATION]', '1707.00341-1-30-2': 'In a similar fashion, one can compute the fall-off of the different components of the spacetime Weyl tensor.', '1707.00341-1-30-3': 'Here, we just write down the ones which are of relevance for this holographic discussion [EQUATION] where [MATH] correspond to the boundary Weyl tensor and the indices of [MATH] are raised and lowered with the metric [MATH].', '1707.00341-1-30-4': 'Replacing all the above quantities in Eq. ([REF]), we realize that the first term and third terms in the integrand are of order [MATH].', '1707.00341-1-30-5': 'That implies that these terms do not contribute in the limit [MATH].', '1707.00341-1-30-6': 'In turn, the only nonvanishing contribution comes from the second term in Eq. ([REF]) as [EQUATION] expressed in terms of the holographic Einstein modes.', '1707.00341-1-31-0': 'One can take a few steps back in the expansion of the boundary quantities and appropriately covariantize the last result, in order to express it in terms of the subtrace of the spacetime Weyl tensor [EQUATION]', '1707.00341-1-31-1': 'Due to the fact that the Weyl tensor is traceless ([MATH]), its subtrace can be traded off by the electric part of the Weyl tensor [EQUATION]', '1707.00341-1-32-0': 'As a consequence, the variation of the Conformal Gravity action is [EQUATION]', '1707.00341-1-33-0': 'for the Einstein modes of the theory.', '1707.00341-1-33-1': 'At the same time, this means that the definition of conserved quantities for that sector of CG can be mapped to the notion of Conformal Mass in [MATH].', '1707.00341-1-34-0': '# Conclusions', '1707.00341-1-35-0': 'In the present work, we have shown that, in Critical Gravity, the energy of any Einstein solution vanishes identically.', '1707.00341-1-35-1': 'This proof does not make use of any particular Einstein black hole, nor relies on charge formulas obtained from the linearization of the field equations.', '1707.00341-1-35-2': 'In this respect, charge expression ([REF]) provides the explicit realization of a claim originally stated in Ref. The holographic derivation in Section [REF] confirms the fact that the boundary stress tensor for the total action ([REF]) is zero, in a similar way as in Ref. When one goes beyond Einstein spaces, the expression ([REF]) is able to capture the effects due to the presence of higher-derivative terms in the curvature.', '1707.00341-1-35-3': 'Indeed, as it was shown in Ref., only the non-Einstein modes of the Weyl tensor survive in the surface term form the variation of the Critical Gravity action.', '1707.00341-1-35-4': 'As a matter of fact, the boundary contributions are expressible in terms of the Bach tensor, what enormously simplify the computation of holographic correlation functions at the critical point .', '1707.00341-1-35-5': 'Noether-Wald charges provides the black hole entropy in a given gravity theory, when evaluated at the horizon [MATH], [EQUATION]', '1707.00341-1-35-6': 'As the condition in the Weyl tensor ([REF]) holds throughout the spacetime for Einstein solutions, it is evident from the above formula that the entropy vanishes in Critical Gravity.', '1707.00341-1-35-7': 'The addition of topological invariants to the four-dimensional AdS gravity action has led to energy definitions which are finite , but also has provided insight on the problem of holography for asymptotically AdS spaces in Einstein gravity .', '1707.00341-1-35-8': 'The result presented here indicates that the Gauss-Bonnet term also plays a role in the holographic description of gravity beyond Einstein theory.'}

{'1707.00341-2-0-0': 'Criticality represents a specific point in the parameter space of a higher-derivative gravity theory, where the linearized field equations become degenerate.', '1707.00341-2-0-1': 'In 4D Critical Gravity, the Lagrangian contains a Weyl-squared term, which does not modify the asymptotic form of the curvature.', '1707.00341-2-0-2': 'The Weyl[MATH] coupling is chosen such that it eliminates the massive scalar mode and it renders the massive spin-2 mode massless.', '1707.00341-2-0-3': 'In doing so, the theory turns consistent around the critical point.Here, we employ the Noether-Wald method to derive the conserved quantities for the action of Critical Gravity.', '1707.00341-2-0-4': 'It is manifest from this energy definition that, at the critical point, the mass is identically zero for Einstein spacetimes, what is a defining property of the theory.', '1707.00341-2-0-5': 'As the entropy is obtained from the Noether-Wald charges at the horizon, it is evident that it also vanishes for any Einstein black hole.', '1707.00341-2-1-0': '# Introduction', '1707.00341-2-2-0': 'General Relativity (GR) is a successful theory of gravity at a classical level but it lacks of consistency in a quantum regime because it is not renormalizable.', '1707.00341-2-2-1': 'On the other hand, in the low energy limit of String Theory, which should be finite to all orders, there appear contributions that are quadratic in the curvature.', '1707.00341-2-2-2': 'As a consequence, higher curvature extensions of Einstein gravity are expected to give rise to a gravity theory with a better ultraviolet behavior.', '1707.00341-2-2-3': 'Early work on the subject has suggested that this class of theories should be renormalizable .', '1707.00341-2-3-0': 'Lower-dimensional examples have been extensively studied in recent literature.', '1707.00341-2-3-1': 'They are regarded as insightful toy models which capture essential features of [MATH] gravity.', '1707.00341-2-3-2': 'One of them is New Massive Gravity (NMG) , a parity-even three-dimensional theory which describes two propagating massive spin-2 modes, in contrast to [MATH] Einstein gravity which is topological.', '1707.00341-2-3-3': 'Picking up the conventional sign of the Einstein-Hilbert action, the energy of the massive excitations is negative (ghost modes), while the mass of the Banados-Teitelboim-Zanelli (BTZ) black hole is positive.', '1707.00341-2-3-4': 'Clearly, this inconsistency persists even if one reverses the sign of the kinetic term.', '1707.00341-2-3-5': 'A physically reasonable theory arises at a specific point of parametric space, where the massive spin-2 field turns massless .', '1707.00341-2-3-6': 'At this particular point, both the energy of the graviton and the mass of the BTZ black hole vanish identically .', '1707.00341-2-3-7': 'Furthermore, both central charges turn into zero, what leads to a vanishing entropy .', '1707.00341-2-3-8': 'Another feature of the theory is the presence of new modes with logarithmic behavior at the critical point .', '1707.00341-2-3-9': 'These modes are eliminated when standard Brown-Henneaux boundary conditions are considered.', '1707.00341-2-3-10': 'Relaxing the asymptotic conditions to include log terms switches on new holographic sources at the boundary .', '1707.00341-2-4-0': 'Another theory in three dimensions sharing similar features with NMG is Topologically Massive Gravity(TMG) .', '1707.00341-2-4-1': 'The corresponding critical point defines the concept of Chiral Gravity.', '1707.00341-2-4-2': 'However, in this case, the central charges are different from each other due to a parity-violating term in the action.', '1707.00341-2-4-3': 'As a consequence, neither mass nor entropy vanish for BTZ black holes at the chiral point.', '1707.00341-2-5-0': 'The generalization of the concept of criticality, present in these models, to four dimensions is given by theories which include quadratic terms in the curvature with particular couplings on top of the Einstein-Hilbert action.', '1707.00341-2-5-1': 'The most general form of a gravity action with quadratic-curvature corrections in [MATH] is given by [EQUATION] where [MATH] and [MATH] are arbitrary couplings, and [MATH] is the cosmological constant in terms of the AdS radius [MATH].', '1707.00341-2-5-2': 'The Riemann-squared term is not present, as it can be always traded off by the Gauss-Bonnet (GB) invariant plus the curvature-squared terms present in the action ([REF]).', '1707.00341-2-5-3': 'The GB term does not affect the field equations in the bulk but it does modify the boundary dynamics.', '1707.00341-2-6-0': 'This class of theories leads to equations of motion (EOM) with up to four derivatives in the metric.', '1707.00341-2-6-1': 'Generically, they describe modes that represent a massless spin-2 graviton, a massive spin-2 field and a massive scalar.', '1707.00341-2-6-2': 'For a quadratic-curvature gravity theory with arbitrary coupling constants, perturbations around a given background would give rise to ghosts.', '1707.00341-2-6-3': 'The problem with the sign of the energy of these modes can be circumvented by a sign flip of the constant in front of Einstein kinetic term.', '1707.00341-2-6-4': 'On the other hand, Einstein black holes are solutions to the theory defined by Eq. ([REF]).', '1707.00341-2-6-5': 'Therefore, the change in the sign mentioned above would lead to a negative mass for Schwarzschild-AdS black hole.', '1707.00341-2-6-6': 'Needless to say, this picture is clearly unphysical as the energy of the perturbations around a background and the mass of a black hole carry opposite signs.', '1707.00341-2-7-0': 'In view of this general obstruction to obtain a four-dimensional gravity theory which is free of the inconsistencies discussed above, it was quite surprising when the authors of Ref. pointed out the fact that, for the particular couplings [MATH] and [MATH], the massive scalar is eliminated and the massive spin-2 mode turns massless.', '1707.00341-2-7-1': 'This choice renders the theory physically sensible around the critical point.', '1707.00341-2-7-2': 'This fact is confirmed by using the Ostrogradsky method for Lagrangians with derivatives of higher order: the energy for the massive mode vanishes for the critical value of the couplings.', '1707.00341-2-7-3': 'From the point of view of the energy of the black holes of the theory, one can use the Abbott-Deser-Tekin (ADT) formula to evaluate the mass of Schwarzschild-AdS solution, what results in [EQUATION] where [MATH] is the mass parameter in the solution.', '1707.00341-2-7-4': 'The general formula Eq. ([REF]), makes evident that, for the critical condition mentioned above, the mass for Schwarzschild-AdS black hole vanishes.', '1707.00341-2-8-0': 'In the present work, as an alternative to Deser-Tekin procedure, we employ Noether-Wald method to compute the charges in Critical Gravity.', '1707.00341-2-8-1': 'This full (non-linearized) expression derived in this way has a remarkable property: the energy of any Einstein space is identically zero, as long anticipated in Ref.', '1707.00341-2-9-0': '# Deser-Tekin energy in [MATH] quadratic-curvature gravity', '1707.00341-2-10-0': 'As mentioned in the previous section, in Refs., the authors provide a generic definition of energy for an arbitrary curvature-squared gravity theory.', '1707.00341-2-10-1': 'That definition of the energy is obtained as an extension of the Abbott-Deser method .', '1707.00341-2-11-0': 'In order to obtain the ADT mass for a general asymptotically AdS (AAdS) solution, we need to write down the metric of the spacetime in the form of [MATH], where [MATH] is the metric of the background and [MATH] is the perturbation tensor.', '1707.00341-2-11-1': 'Such construction leaves the first-order variation of field equations as [EQUATION] where [MATH] and [MATH] are the linearized expression of Einstein tensor and Ricci scalar, respectively.', '1707.00341-2-11-2': 'The tensor [MATH] is the contribution of fourth order in the derivatives to the field equations.', '1707.00341-2-11-3': 'The equation ([REF]) has to be equal to an effective energy-momentum tensor [MATH], which is covariantly conserved.', '1707.00341-2-11-4': 'One can write a conserved current, for a set of Killing fields [MATH] that represents the isometries of the background [EQUATION]', '1707.00341-2-11-5': 'In order to evaluate the mass of a gravitational object, the Killing vector needs to be timelike, at least, at infinity.', '1707.00341-2-11-6': 'Whenever there is a current which is conserved, one is able to write down [MATH] as the divergence of a 2-form prepotential, i.e., [EQUATION]', '1707.00341-2-11-7': 'One can consider a spacetime foliated by a normal (radial) direction [MATH] [EQUATION] where [MATH] is the induced metric on [MATH], and its radial evolution is defined by the unit vector [MATH].', '1707.00341-2-12-0': 'In this coordinate frame, the conserved charge can be expressed as an integral on the co-dimension two surface [MATH] [EQUATION]', '1707.00341-2-12-1': 'Here, [MATH] is a surface normal vector that defines the integration for a fixed time and radius.', '1707.00341-2-12-2': 'For the case of curvature-squared gravity in four dimensions, the conserved quantity adopts the form [EQUATION]', '1707.00341-2-13-0': '# Critical Gravity', '1707.00341-2-14-0': 'In Ref. , the energy of the graviton modes in quadratic-curvature gravity was studied.', '1707.00341-2-14-1': 'These excitations come from the linearized EOM ([REF]).', '1707.00341-2-14-2': 'The choice [MATH] leads to a traceless perturbation ([MATH]) which eliminates the massive scalar mode.', '1707.00341-2-14-3': 'Consequently, the equation for the propagating mode takes the form [EQUATION]', '1707.00341-2-14-4': 'The first factor of the equation describes the propagation of a massless graviton in an AdS background while the second one represents a massive spin-2 field.', '1707.00341-2-15-0': 'It is clear that the latter becomes massless by imposing the critical value [MATH].', '1707.00341-2-15-1': 'This particular coupling produces the fourth order equation [EQUATION] which reflects the appearance of both massless and logarithmic modes .', '1707.00341-2-16-0': 'In order to obtain the energy of the excitations, the authors in Ref. followed a Hamiltonian approach.', '1707.00341-2-16-1': 'For an unrestricted value of[MATH], the action up to quadratic order in [MATH] is [EQUATION]', '1707.00341-2-16-2': 'Using the Ostrogradsky method for higher-derivative Lagrangians, one obtains the following conjugate momenta [EQUATION]', '1707.00341-2-16-3': 'Due to the fact that the Lagrangian is time independent, the Hamiltonian can be written as its time average, that is [EQUATION]', '1707.00341-2-16-4': 'Evaluating for the case of massless and massive propagating modes, one obtains the following expressions for the corresponding on-shell energies [EQUATION] where the subscripts [MATH] and [MATH] stand for massless graviton and massive spin-2 field, respectively.', '1707.00341-2-17-0': 'In a gravity theory with quadratic terms in the curvature, where the couplings are related as [MATH], there is only a specific value of [MATH] that kills the negative energy states.', '1707.00341-2-17-1': 'More specifically, from Eqs. ([REF],[REF]) it is shown that for [MATH], the energy of both the massless and the massive modes is zero.', '1707.00341-2-17-2': 'Hence, all the ghosts disappear leading to a consistent theory of gravity.', '1707.00341-2-18-0': 'Therefore, the action of Critical Gravity reads [EQUATION]', '1707.00341-2-18-1': 'On the other hand, the generic expression for the energy of the black holes in this gravity theory is given by Eq.([REF]).', '1707.00341-2-18-2': 'For any static black hole, the only nonvanishing contribution comes from the first term on the right hand side of Eq.([REF]).', '1707.00341-2-18-3': 'In particular, for a Schwarzschild-AdS black hole, the ADT charge leads to the result in Eq.([REF]).', '1707.00341-2-18-4': 'Is is easy to notice that, for the particular value of the couplings which define Critical Gravity ([MATH], [MATH]), the mass of the black hole vanish.', '1707.00341-2-19-0': 'In what follows, we provide an alternative formula of conserved charges in Critical Gravity, which makes manifest the fact that the energy for Einstein black holes is identically zero.', '1707.00341-2-20-0': '# Noether-Wald charges in Critical Gravity', '1707.00341-2-21-0': 'A general prescription to define conserved charges in an arbitrary theory of gravity was given in Refs..', '1707.00341-2-21-1': 'For the purpose of the discussion below, we will restrict ourselves to the case where Lagrangian density is a functional only of the metric and the curvature, [MATH].', '1707.00341-2-21-2': 'For a given set of Killing vectors [MATH], the Noether current is written down as [EQUATION]', '1707.00341-2-21-3': 'For simplicity, we assume that the surface term [MATH] is separable into a part that contains variations of the Christoffel symbol and another part that contains variations of the metric.', '1707.00341-2-21-4': 'As we are interested in diffeomorphic charges for gravity, all the variations are replaced by a Lie derivative along the vector [MATH].', '1707.00341-2-22-0': 'Using the Killing equation, [MATH], one can notice that first term in Eq. ([REF]) vanishes.', '1707.00341-2-22-1': 'The same relation, this time for the Lie derivative of the Christoffel connection, would produce a combination of double covariant derivatives and curvatures.', '1707.00341-2-22-2': 'This casts the current, for a generic gravity theory, in the form [EQUATION]', '1707.00341-2-22-3': 'Here, the tensor [MATH] is the functional derivative of [MATH] with respect to the spacetime Riemann tensor [MATH], that is, [EQUATION]', '1707.00341-2-22-4': 'It can be shown, by means of the general form of the field equations for these class of gravity theories, that the last two terms on the right hand side of ([REF]) form the EOM contracted with the Killing field.', '1707.00341-2-23-0': 'Thus, on-shell, the first term on the right side of ([REF]) is the only nonvanishing part.', '1707.00341-2-24-0': 'As the tensor [MATH] satisfies Bianchi identity, the conserved current turns into a total derivative [EQUATION]', '1707.00341-2-24-1': 'As the Noether current [MATH] can be written as [MATH], the conserved charge is expressed as an integral on the co-dimension two surface [MATH] [EQUATION] as mentioned previously in Section [REF].', '1707.00341-2-24-2': 'Finally the conserved charge is written as [EQUATION]', '1707.00341-2-24-3': 'An alternative form for the action of Critical Gravity considers the difference between Weyl[MATH] and the GB term [MATH], as the GB invariant term does not alter the bulk dynamics [EQUATION]', '1707.00341-2-24-4': 'We can split the action in two parts: the first one is the MacDowell-Mansouri action, [MATH], which is given by the Einstein-Hilbert plus GB terms, the latter with a fixed coupling .', '1707.00341-2-24-5': 'In Einstein gravity, this corresponds to a built-in renormalized AdS action .', '1707.00341-2-24-6': 'The second part is minus the action of Conformal Gravity [MATH].', '1707.00341-2-25-0': 'Using the Noether-Wald formula for the current ([REF]) for the first part, the functional derivative with respect to the Riemann tensor of the Lagrangian in [MATH] produces [EQUATION] whereas, for the Conformal Gravity part [MATH], we get [EQUATION]', '1707.00341-2-25-1': 'Using the Noether-Wald formula ([REF]), the total charge for the theory [EQUATION]', '1707.00341-2-25-2': 'By definition, the Weyl tensor is [EQUATION]', '1707.00341-2-25-3': 'For Einstein spaces, [MATH], the Weyl tensor adopts the particular form [EQUATION] where the right hand side, is referred to as AdS curvature Using the above fact, the conserved quantity in Critical Gravity is identically zero for Einstein spaces.', '1707.00341-2-26-0': '# Electric part of the Weyl tensor and Einstein modes in Conformal Gravity', '1707.00341-2-27-0': 'CG in four dimensions is invariant under local Weyl rescalings of the metric ([MATH]).', '1707.00341-2-27-1': 'Solutions to CG are Bach-flat geometries, which include Einstein spacetimes.', '1707.00341-2-28-0': 'From a holographic viewpoint, asymptotically AdS space in CG are endowed with new sources at the conformal boundary.', '1707.00341-2-28-1': 'Indeed, we can set any AAdS spacetime in Fefferman-Graham (FG) form of the metric [EQUATION] where the metric [MATH] is expanded as a power series around the boundary [MATH], i.e., [EQUATION]', '1707.00341-2-28-2': 'Here, the ellipsis denotes higher-order terms which do not enter into the holographic description of [MATH] AAdS spaces.', '1707.00341-2-29-0': 'The presence of the term [MATH] reflects the fact the space contains a non-Einstein part.', '1707.00341-2-29-1': 'By demanding the vanishing of the linear term on [MATH], one recovers the Einstein branch, with only even powers of [MATH] in the expansion.', '1707.00341-2-29-2': 'This is achieved by imposing a Neumann boundary condition on the metric, [MATH] .', '1707.00341-2-30-0': 'On the other hand, the Noether-Wald charge for Conformal Gravity is proportional to the Weyl tensor, as shown by Eq. ([REF]).', '1707.00341-2-30-1': 'However, it is not obvious whether, for Einstein spaces, the holographic modes of CG at the boundary are contained in the electric part of the Weyl tensor [EQUATION] as it is the case in Einstein gravity.', '1707.00341-2-31-0': 'As Einstein spaces are solutions of the EOM of CG in the bulk, we restrict the discussion to the surface term in the variation of [MATH], that is, [EQUATION] where [MATH] is the Einstein part of the Weyl tensor ([REF]).', '1707.00341-2-32-0': 'The second term in the above relation can be eliminated using the Bianchi identity of second kind.', '1707.00341-2-32-1': 'A projection of all indices to the boundary can be performed by taking the explicit form of the normal vector [MATH] in Gaussian coordinates.', '1707.00341-2-32-2': 'Then, the surface term takes the form [EQUATION]', '1707.00341-2-32-3': 'In Gauss normal frame ([REF]), the relevant components of the Christoffel symbol are [EQUATION] where [MATH] is the extrinsic curvature at [MATH].', '1707.00341-2-32-4': 'Equipped with this result, the variation of the action is written as [EQUATION] after some algebraic manipulation and index relabeling.', '1707.00341-2-33-0': 'The rest of the proof relies on a power-counting argument in the radial coordinate [MATH].', '1707.00341-2-33-1': 'In order to do so, it is required to expand the tensorial quantities which appear at the surface term.', '1707.00341-2-34-0': 'First, we consider the FG expansion for Einstein spacetimes, where [MATH] and [MATH] with the metric at the conformal boundary given by [EQUATION]', '1707.00341-2-34-1': 'From this form of the metric, the following expressions are straightforwardly derived [EQUATION] where [MATH] is the Schouten tensor defined for the boundary metric [MATH], i.e., [EQUATION]', '1707.00341-2-34-2': 'In a similar fashion, one can compute the fall-off of the different components of the spacetime Weyl tensor.', '1707.00341-2-34-3': 'Here, we just write down the ones which are of relevance for this holographic discussion [EQUATION] where [MATH] correspond to the boundary Weyl tensor and the indices of [MATH] are raised and lowered with the metric [MATH].', '1707.00341-2-35-0': 'Replacing all the above quantities in Eq. ([REF]), we realize that the first term and third terms in the integrand are of order [MATH].', '1707.00341-2-35-1': 'That implies that these terms do not contribute in the limit [MATH].', '1707.00341-2-35-2': 'In turn, the only nonvanishing contribution comes from the second term in Eq. ([REF]) as [EQUATION] expressed in terms of the holographic Einstein modes.', '1707.00341-2-36-0': 'One can take a few steps back in the expansion of the boundary quantities and appropriately covariantize the last result, in order to express it in terms of the subtrace of the spacetime Weyl tensor [EQUATION]', '1707.00341-2-36-1': 'Due to the fact that the Weyl tensor is traceless ([MATH]), its subtrace can be traded off by the electric part of the Weyl tensor [EQUATION]', '1707.00341-2-36-2': 'As a consequence, the variation of the Conformal Gravity action is [EQUATION] for the Einstein modes of the theory.', '1707.00341-2-36-3': 'At the same time, this means that the definition of conserved quantities for that sector of CG can be mapped to the notion of Conformal Mass in [MATH] .', '1707.00341-2-37-0': '# Conclusions', '1707.00341-2-38-0': 'In the present work, we have shown that, in Critical Gravity, the energy of any Einstein solution vanishes identically.', '1707.00341-2-38-1': 'This proof does not make use of any particular Einstein black hole, nor relies on charge formulas obtained from the linearization of the field equations.', '1707.00341-2-38-2': 'In this respect, charge expression ([REF]) provides the explicit realization of a claim originally stated in Ref. The holographic derivation in Section [REF] confirms the fact that the boundary stress tensor for the total action ([REF]) is zero, in a similar way as in Ref. When one goes beyond Einstein spaces, the expression ([REF]) is able to capture the effects due to the presence of higher-derivative terms in the curvature.', '1707.00341-2-38-3': 'Indeed, as it was shown in Ref., only the non-Einstein modes of the Weyl tensor survive in the surface term form the variation of the Critical Gravity action.', '1707.00341-2-38-4': 'As a matter of fact, the boundary contributions are expressible in terms of the Bach tensor, what enormously simplify the computation of holographic correlation functions at the critical point .', '1707.00341-2-38-5': 'Noether-Wald charges provides the black hole entropy in a given gravity theory, when evaluated at the horizon [MATH], [EQUATION]', '1707.00341-2-38-6': 'As the condition in the Weyl tensor ([REF]) holds throughout the spacetime for Einstein solutions, it is evident from the above formula that the entropy vanishes in Critical Gravity.', '1707.00341-2-38-7': 'The addition of topological invariants to the four-dimensional AdS gravity action has led to energy definitions which are finite , but also has provided insight on the problem of holography for asymptotically AdS spaces in Einstein gravity .', '1707.00341-2-38-8': 'The result presented here indicates that the Gauss-Bonnet term also plays a role in the holographic description of gravity beyond Einstein theory.'}

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['1707.00341-1-35-7', '1707.00341-2-38-7'], ['1707.00341-1-35-8', '1707.00341-2-38-8'], ['1707.00341-1-27-0', '1707.00341-2-31-0'], ['1707.00341-1-18-0', '1707.00341-2-21-0'], ['1707.00341-1-18-1', '1707.00341-2-21-1'], ['1707.00341-1-18-2', '1707.00341-2-21-2'], ['1707.00341-1-18-3', '1707.00341-2-21-3'], ['1707.00341-1-18-4', '1707.00341-2-21-4'], ['1707.00341-1-11-0', '1707.00341-2-12-0'], ['1707.00341-1-11-1', '1707.00341-2-12-1'], ['1707.00341-1-11-2', '1707.00341-2-12-2'], ['1707.00341-1-9-0', '1707.00341-2-10-0'], ['1707.00341-1-9-1', '1707.00341-2-10-1'], ['1707.00341-1-25-0', '1707.00341-2-29-0'], ['1707.00341-1-25-1', '1707.00341-2-29-1'], ['1707.00341-1-25-2', '1707.00341-2-29-2'], ['1707.00341-1-14-0', '1707.00341-2-17-0'], ['1707.00341-1-14-1', '1707.00341-2-17-1'], ['1707.00341-1-14-2', '1707.00341-2-17-2'], ['1707.00341-1-29-0', '1707.00341-2-33-0'], ['1707.00341-1-29-1', '1707.00341-2-33-1'], ['1707.00341-1-0-0', '1707.00341-2-0-0'], ['1707.00341-1-0-1', '1707.00341-2-0-1'], ['1707.00341-1-0-2', '1707.00341-2-0-2'], ['1707.00341-1-0-5', '1707.00341-2-0-4'], ['1707.00341-1-0-6', '1707.00341-2-0-5'], ['1707.00341-1-23-0', '1707.00341-2-27-0'], ['1707.00341-1-23-1', '1707.00341-2-27-1'], ['1707.00341-1-3-0', '1707.00341-2-3-0'], ['1707.00341-1-3-1', '1707.00341-2-3-1'], ['1707.00341-1-3-2', '1707.00341-2-3-2'], ['1707.00341-1-3-3', '1707.00341-2-3-3'], ['1707.00341-1-3-4', '1707.00341-2-3-4'], ['1707.00341-1-3-5', '1707.00341-2-3-5'], ['1707.00341-1-3-6', '1707.00341-2-3-6'], ['1707.00341-1-3-7', '1707.00341-2-3-7'], ['1707.00341-1-3-8', '1707.00341-2-3-8'], ['1707.00341-1-3-9', '1707.00341-2-3-9'], ['1707.00341-1-3-10', '1707.00341-2-3-10'], ['1707.00341-1-10-0', '1707.00341-2-11-0'], ['1707.00341-1-10-1', '1707.00341-2-11-1'], ['1707.00341-1-10-2', '1707.00341-2-11-2'], ['1707.00341-1-10-3', '1707.00341-2-11-3'], ['1707.00341-1-10-4', '1707.00341-2-11-4'], ['1707.00341-1-10-5', '1707.00341-2-11-5'], ['1707.00341-1-10-6', '1707.00341-2-11-6'], ['1707.00341-1-10-7', '1707.00341-2-11-7'], ['1707.00341-1-28-0', '1707.00341-2-32-0'], ['1707.00341-1-28-1', '1707.00341-2-32-1'], ['1707.00341-1-28-2', '1707.00341-2-32-2'], ['1707.00341-1-28-3', '1707.00341-2-32-3'], ['1707.00341-1-16-0', '1707.00341-2-19-0'], ['1707.00341-1-2-0', '1707.00341-2-2-0'], ['1707.00341-1-2-1', '1707.00341-2-2-1'], ['1707.00341-1-2-2', '1707.00341-2-2-2'], ['1707.00341-1-2-3', '1707.00341-2-2-3'], ['1707.00341-2-8-0', '1707.00341-3-8-0'], ['1707.00341-2-8-1', '1707.00341-3-8-1'], ['1707.00341-2-25-0', '1707.00341-3-25-0'], ['1707.00341-2-25-1', '1707.00341-3-25-1'], ['1707.00341-2-25-2', '1707.00341-3-25-2'], ['1707.00341-2-25-3', '1707.00341-3-25-3'], ['1707.00341-2-28-0', '1707.00341-3-28-0'], ['1707.00341-2-28-1', '1707.00341-3-28-1'], ['1707.00341-2-28-2', '1707.00341-3-28-2'], ['1707.00341-2-21-0', '1707.00341-3-21-0'], ['1707.00341-2-21-1', '1707.00341-3-21-1'], ['1707.00341-2-21-2', '1707.00341-3-21-2'], ['1707.00341-2-21-3', 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['1707.00341-1-13-6', '1707.00341-2-15-1'], ['1707.00341-1-13-7', '1707.00341-2-16-0'], ['1707.00341-1-13-8', '1707.00341-2-16-1'], ['1707.00341-1-13-9', '1707.00341-2-16-2'], ['1707.00341-1-13-10', '1707.00341-2-16-3'], ['1707.00341-1-13-11', '1707.00341-2-16-4'], ['1707.00341-1-30-0', '1707.00341-2-34-0'], ['1707.00341-1-30-1', '1707.00341-2-34-1'], ['1707.00341-1-30-2', '1707.00341-2-34-2'], ['1707.00341-1-30-3', '1707.00341-2-34-3'], ['1707.00341-1-30-4', '1707.00341-2-35-0'], ['1707.00341-1-30-5', '1707.00341-2-35-1'], ['1707.00341-1-30-6', '1707.00341-2-35-2'], ['1707.00341-1-31-0', '1707.00341-2-36-0'], ['1707.00341-1-31-1', '1707.00341-2-36-1'], ['1707.00341-1-20-7', '1707.00341-2-25-0'], ['1707.00341-1-20-8', '1707.00341-2-25-1'], ['1707.00341-1-20-9', '1707.00341-2-25-2'], ['1707.00341-1-20-0', '1707.00341-2-24-0'], ['1707.00341-1-20-1', '1707.00341-2-24-1'], ['1707.00341-1-20-2', '1707.00341-2-24-2'], ['1707.00341-1-20-3', '1707.00341-2-24-3'], ['1707.00341-1-20-4', '1707.00341-2-24-4'], ['1707.00341-1-20-5', '1707.00341-2-24-5'], ['1707.00341-1-20-6', '1707.00341-2-24-6'], ['1707.00341-1-6-7', '1707.00341-2-7-0'], ['1707.00341-1-6-8', '1707.00341-2-7-1'], ['1707.00341-1-6-9', '1707.00341-2-7-2'], ['1707.00341-1-7-0', '1707.00341-2-7-3'], ['1707.00341-1-7-1', '1707.00341-2-7-4'], ['1707.00341-1-6-0', '1707.00341-2-6-0'], ['1707.00341-1-6-1', '1707.00341-2-6-1'], ['1707.00341-1-6-2', '1707.00341-2-6-2'], ['1707.00341-1-6-3', '1707.00341-2-6-3'], ['1707.00341-1-6-4', '1707.00341-2-6-4'], ['1707.00341-1-6-5', '1707.00341-2-6-5'], ['1707.00341-1-6-6', '1707.00341-2-6-6'], ['1707.00341-1-7-2', '1707.00341-2-8-0'], ['1707.00341-1-7-3', '1707.00341-2-8-1']]

[['1707.00341-1-4-0', '1707.00341-2-4-0'], ['1707.00341-1-28-4', '1707.00341-2-32-4'], ['1707.00341-1-32-0', '1707.00341-2-36-2'], ['1707.00341-1-33-1', '1707.00341-2-36-3']]

[]

[['1707.00341-1-0-3', '1707.00341-2-0-3'], ['1707.00341-1-0-4', '1707.00341-2-0-3'], ['1707.00341-1-20-10', '1707.00341-2-25-3'], ['1707.00341-1-21-0', '1707.00341-2-25-3']]

[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1707.00341

{'1707.00341-3-0-0': 'Criticality represents a specific point in the parameter space of a higher-derivative gravity theory, where the linearized field equations become degenerate.', '1707.00341-3-0-1': 'In 4D Critical Gravity, the Lagrangian contains a Weyl-squared term, which does not modify the asymptotic form of the curvature.', '1707.00341-3-0-2': 'The Weyl[MATH] coupling is chosen such that it eliminates the massive scalar mode and it renders the massive spin-2 mode massless.', '1707.00341-3-0-3': 'In doing so, the theory turns consistent around the critical point.Here, we employ the Noether-Wald method to derive the conserved quantities for the action of Critical Gravity.', '1707.00341-3-0-4': 'It is manifest from this energy definition that, at the critical point, the mass is identically zero for Einstein spacetimes, what is a defining property of the theory.', '1707.00341-3-0-5': 'As the entropy is obtained from the Noether-Wald charges at the horizon, it is evident that it also vanishes for any Einstein black hole.', '1707.00341-3-1-0': '# Introduction', '1707.00341-3-2-0': 'General Relativity (GR) is a successful theory of gravity at a classical level but it lacks of consistency in a quantum regime because it is not renormalizable.', '1707.00341-3-2-1': 'On the other hand, in the low energy limit of String Theory, which should be finite to all orders, there appear contributions that are quadratic in the curvature.', '1707.00341-3-2-2': 'As a consequence, higher curvature extensions of Einstein gravity are expected to give rise to a gravity theory with a better ultraviolet behavior.', '1707.00341-3-2-3': 'Early work on the subject has suggested that this class of theories should be renormalizable .', '1707.00341-3-3-0': 'Lower-dimensional examples have been extensively studied in recent literature.', '1707.00341-3-3-1': 'They are regarded as insightful toy models which capture essential features of [MATH] gravity.', '1707.00341-3-3-2': 'One of them is New Massive Gravity (NMG) , a parity-even three-dimensional theory which describes two propagating massive spin-2 modes, in contrast to [MATH] Einstein gravity which is topological.', '1707.00341-3-3-3': 'Picking up the conventional sign of the Einstein-Hilbert action, the energy of the massive excitations is negative (ghost modes), while the mass of the Banados-Teitelboim-Zanelli (BTZ) black hole is positive.', '1707.00341-3-3-4': 'Clearly, this inconsistency persists even if one reverses the sign of the kinetic term.', '1707.00341-3-3-5': 'A physically reasonable theory arises at a specific point of parametric space, where the massive spin-2 field turns massless .', '1707.00341-3-3-6': 'At this particular point, both the energy of the graviton and the mass of the BTZ black hole vanish identically .', '1707.00341-3-3-7': 'Furthermore, both central charges turn into zero, what leads to a vanishing entropy .', '1707.00341-3-3-8': 'Another feature of the theory is the presence of new modes with logarithmic behavior at the critical point .', '1707.00341-3-3-9': 'These modes are eliminated when standard Brown-Henneaux boundary conditions are considered.', '1707.00341-3-3-10': 'Relaxing the asymptotic conditions to include log terms switches on new holographic sources at the boundary .', '1707.00341-3-4-0': 'Another theory in three dimensions sharing similar features with NMG is Topologically Massive Gravity(TMG) .', '1707.00341-3-4-1': 'The corresponding critical point defines the concept of Chiral Gravity.', '1707.00341-3-4-2': 'However, in this case, the central charges are different from each other due to a parity-violating term in the action.', '1707.00341-3-4-3': 'As a consequence, neither mass nor entropy vanish for BTZ black holes at the chiral point.', '1707.00341-3-5-0': 'The generalization of the concept of criticality, present in these models, to four dimensions is given by theories which include quadratic terms in the curvature with particular couplings on top of the Einstein-Hilbert action.', '1707.00341-3-5-1': 'The most general form of a gravity action with quadratic-curvature corrections in [MATH] is given by [EQUATION] where [MATH] and [MATH] are arbitrary couplings, and [MATH] is the cosmological constant in terms of the AdS radius [MATH].', '1707.00341-3-5-2': 'The Riemann-squared term is not present, as it can be always traded off by the Gauss-Bonnet (GB) invariant plus the curvature-squared terms present in the action ([REF]).', '1707.00341-3-5-3': 'The GB term does not affect the field equations in the bulk but it does modify the boundary dynamics.', '1707.00341-3-6-0': 'This class of theories leads to equations of motion (EOM) with up to four derivatives in the metric.', '1707.00341-3-6-1': 'Generically, they describe modes that represent a massless spin-2 graviton, a massive spin-2 field and a massive scalar.', '1707.00341-3-6-2': 'For a quadratic-curvature gravity theory with arbitrary coupling constants, perturbations around a given background would give rise to ghosts.', '1707.00341-3-6-3': 'The problem with the sign of the energy of these modes can be circumvented by a sign flip of the constant in front of Einstein kinetic term.', '1707.00341-3-6-4': 'On the other hand, Einstein black holes are solutions to the theory defined by Eq. ([REF]).', '1707.00341-3-6-5': 'Therefore, the change in the sign mentioned above would lead to a negative mass for Schwarzschild-AdS black hole.', '1707.00341-3-6-6': 'Needless to say, this picture is clearly unphysical as the energy of the perturbations around a background and the mass of a black hole carry opposite signs.', '1707.00341-3-7-0': 'In view of this general obstruction to obtain a four-dimensional gravity theory which is free of the inconsistencies discussed above, it was quite surprising when the authors of Ref. pointed out the fact that, for the particular couplings [MATH] and [MATH], the massive scalar is eliminated and the massive spin-2 mode turns massless.', '1707.00341-3-7-1': 'This choice renders the theory physically sensible around the critical point.', '1707.00341-3-7-2': 'This fact is confirmed by using the Ostrogradsky method for Lagrangians with derivatives of higher order: the energy for the massive mode vanishes for the critical value of the couplings.', '1707.00341-3-7-3': 'From the point of view of the energy of the black holes of the theory, one can use the Abbott-Deser-Tekin (ADT) formula to evaluate the mass of Schwarzschild-AdS solution, what results in [EQUATION] where [MATH] is the mass parameter in the solution.', '1707.00341-3-7-4': 'The general formula Eq. ([REF]), makes evident that, for the critical condition mentioned above, the mass for Schwarzschild-AdS black hole vanishes.', '1707.00341-3-8-0': 'In the present work, as an alternative to Deser-Tekin procedure, we employ Noether-Wald method to compute the charges in Critical Gravity.', '1707.00341-3-8-1': 'This full (non-linearized) expression derived in this way has a remarkable property: the energy of any Einstein space is identically zero, as long anticipated in Ref.', '1707.00341-3-9-0': '# Deser-Tekin energy in [MATH] quadratic-curvature gravity', '1707.00341-3-10-0': 'As mentioned in the previous section, in Refs., the authors provide a generic definition of energy for an arbitrary curvature-squared gravity theory.', '1707.00341-3-10-1': 'That definition of the energy is obtained as an extension of the Abbott-Deser method .', '1707.00341-3-11-0': 'In order to obtain the ADT mass for a general asymptotically AdS (AAdS) solution, we need to write down the metric of the spacetime in the form of [MATH], where [MATH] is the metric of the background and [MATH] is the perturbation tensor.', '1707.00341-3-11-1': 'Such construction leaves the first-order variation of field equations as [EQUATION] where [MATH] and [MATH] are the linearized expression of Einstein tensor and Ricci scalar, respectively.', '1707.00341-3-11-2': 'The tensor [MATH] is the contribution of fourth order in the derivatives to the field equations.', '1707.00341-3-11-3': 'The equation ([REF]) has to be equal to an effective energy-momentum tensor [MATH], which is covariantly conserved.', '1707.00341-3-11-4': 'One can write a conserved current, for a set of Killing fields [MATH] that represents the isometries of the background [EQUATION]', '1707.00341-3-11-5': 'In order to evaluate the mass of a gravitational object, the Killing vector needs to be timelike, at least, at infinity.', '1707.00341-3-11-6': 'Whenever there is a current which is conserved, one is able to write down [MATH] as the divergence of a 2-form prepotential, i.e., [EQUATION]', '1707.00341-3-11-7': 'One can consider a spacetime foliated by a normal (radial) direction [MATH] [EQUATION] where [MATH] is the induced metric on [MATH], and its radial evolution is defined by the unit vector [MATH].', '1707.00341-3-12-0': 'In this coordinate frame, the conserved charge can be expressed as an integral on the co-dimension two surface [MATH] [EQUATION]', '1707.00341-3-12-1': 'Here, [MATH] is a surface normal vector that defines the integration for a fixed time and radius.', '1707.00341-3-12-2': 'For the case of curvature-squared gravity in four dimensions, the conserved quantity adopts the form [EQUATION]', '1707.00341-3-13-0': '# Critical Gravity', '1707.00341-3-14-0': 'In Ref. , the energy of the graviton modes in quadratic-curvature gravity was studied.', '1707.00341-3-14-1': 'These excitations come from the linearized EOM ([REF]).', '1707.00341-3-14-2': 'The choice [MATH] leads to a traceless perturbation ([MATH]) which eliminates the massive scalar mode.', '1707.00341-3-14-3': 'Consequently, the equation for the propagating mode takes the form [EQUATION]', '1707.00341-3-14-4': 'The first factor of the equation describes the propagation of a massless graviton in an AdS background while the second one represents a massive spin-2 field.', '1707.00341-3-15-0': 'It is clear that the latter becomes massless by imposing the critical value [MATH].', '1707.00341-3-15-1': 'This particular coupling produces the fourth order equation [EQUATION] which reflects the appearance of both massless and logarithmic modes .', '1707.00341-3-16-0': 'In order to obtain the energy of the excitations, the authors in Ref. followed a Hamiltonian approach.', '1707.00341-3-16-1': 'For an unrestricted value of[MATH], the action up to quadratic order in [MATH] is [EQUATION]', '1707.00341-3-16-2': 'Using the Ostrogradsky method for higher-derivative Lagrangians, one obtains the following conjugate momenta [EQUATION]', '1707.00341-3-16-3': 'Due to the fact that the Lagrangian is time independent, the Hamiltonian can be written as its time average, that is [EQUATION]', '1707.00341-3-16-4': 'Evaluating for the case of massless and massive propagating modes, one obtains the following expressions for the corresponding on-shell energies [EQUATION] where the subscripts [MATH] and [MATH] stand for massless graviton and massive spin-2 field, respectively.', '1707.00341-3-17-0': 'In a gravity theory with quadratic terms in the curvature, where the couplings are related as [MATH], there is only a specific value of [MATH] that kills the negative energy states.', '1707.00341-3-17-1': 'More specifically, from Eqs. ([REF],[REF]) it is shown that for [MATH], the energy of both the massless and the massive modes is zero.', '1707.00341-3-17-2': 'Hence, all the ghosts disappear leading to a consistent theory of gravity.', '1707.00341-3-18-0': 'Therefore, the action of Critical Gravity reads [EQUATION]', '1707.00341-3-18-1': 'On the other hand, the generic expression for the energy of the black holes in this gravity theory is given by Eq.([REF]).', '1707.00341-3-18-2': 'For any static black hole, the only nonvanishing contribution comes from the first term on the right hand side of Eq.([REF]).', '1707.00341-3-18-3': 'In particular, for a Schwarzschild-AdS black hole, the ADT charge leads to the result in Eq.([REF]).', '1707.00341-3-18-4': 'Is is easy to notice that, for the particular value of the couplings which define Critical Gravity ([MATH], [MATH]), the mass of the black hole vanish.', '1707.00341-3-19-0': 'In what follows, we provide an alternative formula of conserved charges in Critical Gravity, which makes manifest the fact that the energy for Einstein black holes is identically zero.', '1707.00341-3-20-0': '# Noether-Wald charges in Critical Gravity', '1707.00341-3-21-0': 'A general prescription to define conserved charges in an arbitrary theory of gravity was given in Refs..', '1707.00341-3-21-1': 'For the purpose of the discussion below, we will restrict ourselves to the case where Lagrangian density is a functional only of the metric and the curvature, [MATH].', '1707.00341-3-21-2': 'For a given set of Killing vectors [MATH], the Noether current is written down as [EQUATION]', '1707.00341-3-21-3': 'For simplicity, we assume that the surface term [MATH] is separable into a part that contains variations of the Christoffel symbol and another part that contains variations of the metric.', '1707.00341-3-21-4': 'As we are interested in diffeomorphic charges for gravity, all the variations are replaced by a Lie derivative along the vector [MATH].', '1707.00341-3-22-0': 'Using the Killing equation, [MATH], one can notice that first term in Eq. ([REF]) vanishes.', '1707.00341-3-22-1': 'The same relation, this time for the Lie derivative of the Christoffel connection, would produce a combination of double covariant derivatives and curvatures.', '1707.00341-3-22-2': 'This casts the current, for a generic gravity theory, in the form [EQUATION]', '1707.00341-3-22-3': 'Here, the tensor [MATH] is the functional derivative of [MATH] with respect to the spacetime Riemann tensor [MATH], that is, [EQUATION]', '1707.00341-3-22-4': 'It can be shown, by means of the general form of the field equations for these class of gravity theories, that the last two terms on the right hand side of ([REF]) form the EOM contracted with the Killing field.', '1707.00341-3-23-0': 'Thus, on-shell, the first term on the right side of ([REF]) is the only nonvanishing part.', '1707.00341-3-24-0': 'As the tensor [MATH] satisfies Bianchi identity, the conserved current turns into a total derivative [EQUATION]', '1707.00341-3-24-1': 'As the Noether current [MATH] can be written as [MATH], the conserved charge is expressed as an integral on the co-dimension two surface [MATH] [EQUATION] as mentioned previously in Section [REF].', '1707.00341-3-24-2': 'Finally the conserved charge is written as [EQUATION]', '1707.00341-3-24-3': 'An alternative form for the action of Critical Gravity considers the difference between Weyl[MATH] and the GB term [MATH], as the GB invariant term does not alter the bulk dynamics [EQUATION]', '1707.00341-3-24-4': 'We can split the action in two parts: the first one is the MacDowell-Mansouri action, [MATH], which is given by the Einstein-Hilbert plus GB terms, the latter with a fixed coupling .', '1707.00341-3-24-5': 'In Einstein gravity, this corresponds to a built-in renormalized AdS action .', '1707.00341-3-24-6': 'The second part is minus the action of Conformal Gravity [MATH].', '1707.00341-3-25-0': 'Using the Noether-Wald formula for the current ([REF]) for the first part, the functional derivative with respect to the Riemann tensor of the Lagrangian in [MATH] produces [EQUATION] whereas, for the Conformal Gravity part [MATH], we get [EQUATION]', '1707.00341-3-25-1': 'Using the Noether-Wald formula ([REF]), the total charge for the theory [EQUATION]', '1707.00341-3-25-2': 'By definition, the Weyl tensor is [EQUATION]', '1707.00341-3-25-3': 'For Einstein spaces, [MATH], the Weyl tensor adopts the particular form [EQUATION] where the right hand side, is referred to as AdS curvature Using the above fact, the conserved quantity in Critical Gravity is identically zero for Einstein spaces.', '1707.00341-3-26-0': '# Electric part of the Weyl tensor and Einstein modes in Conformal Gravity', '1707.00341-3-27-0': 'CG in four dimensions is invariant under local Weyl rescalings of the metric ([MATH]).', '1707.00341-3-27-1': 'Solutions to CG are Bach-flat geometries, which include Einstein spacetimes.', '1707.00341-3-28-0': 'From a holographic viewpoint, asymptotically AdS space in CG are endowed with new sources at the conformal boundary.', '1707.00341-3-28-1': 'Indeed, we can set any AAdS spacetime in Fefferman-Graham (FG) form of the metric [EQUATION] where the metric [MATH] is expanded as a power series around the boundary [MATH], i.e., [EQUATION]', '1707.00341-3-28-2': 'Here, the ellipsis denotes higher-order terms which do not enter into the holographic description of [MATH] AAdS spaces.', '1707.00341-3-29-0': 'The presence of the term [MATH] reflects the fact the space contains a non-Einstein part.', '1707.00341-3-29-1': 'By demanding the vanishing of the linear term on [MATH], one recovers the Einstein branch, with only even powers of [MATH] in the expansion.', '1707.00341-3-29-2': 'This is achieved by imposing a Neumann boundary condition on the metric, [MATH] .', '1707.00341-3-30-0': 'On the other hand, the Noether-Wald charge for Conformal Gravity is proportional to the Weyl tensor, as shown by Eq. ([REF]).', '1707.00341-3-30-1': 'However, it is not obvious whether, for Einstein spaces, the holographic modes of CG at the boundary are contained in the electric part of the Weyl tensor [EQUATION] as it is the case in Einstein gravity.', '1707.00341-3-31-0': 'As Einstein spaces are solutions of the EOM of CG in the bulk, we restrict the discussion to the surface term in the variation of [MATH], that is, [EQUATION] where [MATH] is the Einstein part of the Weyl tensor ([REF]).', '1707.00341-3-32-0': 'The second term in the above relation can be eliminated using the Bianchi identity of second kind.', '1707.00341-3-32-1': 'A projection of all indices to the boundary can be performed by taking the explicit form of the normal vector [MATH] in Gaussian coordinates.', '1707.00341-3-32-2': 'Then, the surface term takes the form [EQUATION]', '1707.00341-3-32-3': 'In Gauss normal frame ([REF]), the relevant components of the Christoffel symbol are [EQUATION] where [MATH] is the extrinsic curvature at [MATH].', '1707.00341-3-32-4': 'Equipped with this result, the variation of the action is written as [EQUATION] after some algebraic manipulation and index relabeling.', '1707.00341-3-33-0': 'The rest of the proof relies on a power-counting argument in the radial coordinate [MATH].', '1707.00341-3-33-1': 'In order to do so, it is required to expand the tensorial quantities which appear at the surface term.', '1707.00341-3-34-0': 'First, we consider the FG expansion for Einstein spacetimes, where [MATH] and [MATH] with the metric at the conformal boundary given by [EQUATION]', '1707.00341-3-34-1': 'From this form of the metric, the following expressions are straightforwardly derived [EQUATION] where [MATH] is the Schouten tensor defined for the boundary metric [MATH], i.e., [EQUATION]', '1707.00341-3-34-2': 'In a similar fashion, one can compute the fall-off of the different components of the spacetime Weyl tensor.', '1707.00341-3-34-3': 'Here, we just write down the ones which are of relevance for this holographic discussion [EQUATION] where [MATH] correspond to the boundary Weyl tensor and the indices of [MATH] are raised and lowered with the metric [MATH].', '1707.00341-3-35-0': 'Replacing all the above quantities in Eq. ([REF]), we realize that the first term and third terms in the integrand are of order [MATH].', '1707.00341-3-35-1': 'That implies that these terms do not contribute in the limit [MATH].', '1707.00341-3-35-2': 'In turn, the only nonvanishing contribution comes from the second term in Eq. ([REF]) as [EQUATION] expressed in terms of the holographic Einstein modes.', '1707.00341-3-36-0': 'One can take a few steps back in the expansion of the boundary quantities and appropriately covariantize the last result, in order to express it in terms of the subtrace of the spacetime Weyl tensor [EQUATION]', '1707.00341-3-36-1': 'Due to the fact that the Weyl tensor is traceless ([MATH]), its subtrace can be traded off by the electric part of the Weyl tensor [EQUATION]', '1707.00341-3-36-2': 'As a consequence, the variation of the Conformal Gravity action is [EQUATION] for the Einstein modes of the theory.', '1707.00341-3-36-3': 'At the same time, this means that the definition of conserved quantities for that sector of CG can be mapped to the notion of Conformal Mass in [MATH] .', '1707.00341-3-37-0': '# Conclusions', '1707.00341-3-38-0': 'In the present work, we have shown that, in Critical Gravity, the energy of any Einstein solution vanishes identically.', '1707.00341-3-38-1': 'This proof does not make use of any particular Einstein black hole, nor relies on charge formulas obtained from the linearization of the field equations.', '1707.00341-3-38-2': 'In this respect, charge expression ([REF]) provides the explicit realization of a claim originally stated in Ref. The holographic derivation in Section [REF] confirms the fact that the boundary stress tensor for the total action ([REF]) is zero, in a similar way as in Ref. When one goes beyond Einstein spaces, the expression ([REF]) is able to capture the effects due to the presence of higher-derivative terms in the curvature.', '1707.00341-3-38-3': 'Indeed, as it was shown in Ref., only the non-Einstein modes of the Weyl tensor survive in the surface term form the variation of the Critical Gravity action.', '1707.00341-3-38-4': 'As a matter of fact, the boundary contributions are expressible in terms of the Bach tensor, what enormously simplify the computation of holographic correlation functions at the critical point .', '1707.00341-3-38-5': 'Noether-Wald charges provides the black hole entropy in a given gravity theory, when evaluated at the horizon [MATH], [EQUATION]', '1707.00341-3-38-6': 'As the condition in the Weyl tensor ([REF]) holds throughout the spacetime for Einstein solutions, it is evident from the above formula that the entropy vanishes in Critical Gravity.', '1707.00341-3-38-7': 'The addition of topological invariants to the four-dimensional AdS gravity action has led to energy definitions which are finite , but also has provided insight on the problem of holography for asymptotically AdS spaces in Einstein gravity .', '1707.00341-3-38-8': 'The result presented here indicates that the Gauss-Bonnet term also plays a role in the holographic description of gravity beyond Einstein theory.'}

astro-ph-9906371

{'astro-ph-9906371-1-0-0': 'The bright Seyfert 1 galaxy MCG[MATH]6-30-15 shows large variability on a variety of time scales.', 'astro-ph-9906371-1-0-1': 'We study the long time scale variability via Rossi X-ray Timing Explorer-All Sky Monitor (RXTE-ASM) observations, which reveal [MATH] variability on month to year long time scales.', 'astro-ph-9906371-1-0-2': 'We study the short time scale variablity using a set of simultaneous archival observations that were obtained from RXTE and the Advanced Satellite for Cosmology and Astrophysics (ASCA).', 'astro-ph-9906371-1-0-3': 'These RXTE observations span nearly [MATH]sec and indicate that the X-ray Fourier Power Spectral Density has a root mean square (rms) variability of 20%, is flat from approximately [MATH]-[MATH]Hz, and then steepens into a power law [MATH] with [MATH].', 'astro-ph-9906371-1-0-4': 'A further steepening to [MATH] occurs between [MATH]-[MATH]Hz.', 'astro-ph-9906371-1-0-5': 'The shape and rms amplitude are comparable to what has been observed in NGC 5548 and Cyg X-1, albeit with break frequencies that differ by a factor of [MATH] and [MATH], respectively.', 'astro-ph-9906371-1-0-6': 'If we take the break frequencies as indicative of the mass of the central black hole, then this mass may be as low as [MATH].', 'astro-ph-9906371-1-0-7': 'A [MATH]ks lead of the 0.5-1keV ASCA band compared to the 8-15keV RXTE band was also found.', 'astro-ph-9906371-1-0-8': 'Scaling from similar soft X-ray leads observed in NGC 5548 and Cyg X-1, this also is consistent with a relatively low central black hole mass.', 'astro-ph-9906371-1-1-0': '# Introduction', 'astro-ph-9906371-1-2-0': 'The type 1 Seyfert galaxy MCG[MATH]6-30-15 has in recent years been the subject of intense study owing to the discovery by the Advanced Satellite for Cosmology and Astrophysics (ASCA) of a resolved, broad iron K[MATH] fluorescent line in its hard X-ray spectrum (##1 ##2tanaka:95b).', 'astro-ph-9906371-1-2-1': 'The shape of the line is consistent with a gravitationally and Doppler shifted emission line which originates from near the inner edge of an accretion disk around a black hole.', 'astro-ph-9906371-1-2-2': 'In the X-rays, MCG[MATH]6-30-15 is also one of the brighter and more variable type 1 Seyferts.', 'astro-ph-9906371-1-2-3': 'It is therefore hoped that by examining the correlated variability between the ionizing X-ray continuum and various components of the Fe K[MATH] line, one can obtain a size scale for the system and thence a mass for the black hole (##1 ##2reynolds:99a).', 'astro-ph-9906371-1-2-4': 'Such "reverberation mapping", however, requires that the various components of the line be spectrally resolved on time scales shorter than the intrinsic response time scale of the line-emitting material.', 'astro-ph-9906371-1-3-0': 'At present, ASCA and BeppoSAX are the only X-ray telescopes in operation which have the spectral resolution to measure fluxes in different line components separately.', 'astro-ph-9906371-1-3-1': 'Unfortunately, the required integration times ([MATH]ks) to obtain this resolution are longer than the light travel time across the inner edge of an accretion disk around a [MATH] Schwarzschild black hole.', 'astro-ph-9906371-1-3-2': 'Furthermore, the fits to the iron line appear to require a Kerr geometry (although see ##1 ##2reynolds:97b) in order to explain the broad red wing of the line and the lack of significant emission blueward of 6.4keV, thus making the relevant time scales even shorter.', 'astro-ph-9906371-1-3-3': 'Nonetheless, over longer time scales, significant variability has been measured by ASCA in the shape of the Fe K[MATH] line from MCG[MATH]6-30-15 (##1 ##2iwasawa:96a), and a partially resolved line shape has been obtained for a brief flare during a recent observing campaign for a relatively short integration time of 5ks (##1 ##2iwasawa:99a).', 'astro-ph-9906371-1-4-0': 'These time-resolved spectral investigations suggest that the broad and narrow components of the iron line are correlated differently with the flux state depending on the time scale investigated.', 'astro-ph-9906371-1-4-1': 'For integrations [MATH]s, the narrow component varies with the continuum flux whilst the broad component appears to be anti-correlated.', 'astro-ph-9906371-1-4-2': 'In contrast, ##1 (##2)iwasawa:96a claim that on shorter time scales the broad component responds immediately to flux changes whilst the narrow component remains constant.', 'astro-ph-9906371-1-4-3': 'They suggest that multiple X-ray flares occur on the disk surface near the inner edge and that the flares are localized so as to illuminate only a relatively small region of the disk.', 'astro-ph-9906371-1-4-4': 'The line fluxes from these small regions make contributions to narrow ranges in line redshift and thus produce very complex temporal behavior.', 'astro-ph-9906371-1-4-5': 'In support of this interpretation, ##1 (##2)iwasawa:99a consider the iron line associated with a very short, bright flare from MCG[MATH]6-30-15 observed during 1997 August by ASCA (see Fig. [REF]).', 'astro-ph-9906371-1-4-6': 'The line was very redshifted and had little or no emission blueward of 6keV.', 'astro-ph-9906371-1-4-7': 'Its shape was consistent with having originated entirely from a small region at [MATH] on the approaching side of the disk.', 'astro-ph-9906371-1-4-8': 'The flare lasted about 4ks, which is also approximately the orbital period at this radius for a [MATH] black hole.', 'astro-ph-9906371-1-4-9': 'In order for the line not to be significantly more smeared by the orbital motion, ##1 (##2)iwasawa:99a estimate a black hole mass of [MATH].', 'astro-ph-9906371-1-5-0': 'The major complication that this model faces is the large degree of variability which has been seen in the X-ray flux of MCG[MATH]6-30-15 (Fig. [REF]).', 'astro-ph-9906371-1-5-1': 'In particular, during the performance verification phase of ASCA, ##1 (##2)reynolds:95a noted that the 0.5-10keV flux increased by a factor 1.5 over 100s, i.e., the dynamical time scale at the inner edge of disk surrounding a maximal Kerr, [MATH] black hole.', 'astro-ph-9906371-1-5-2': 'Typically one expects the bulk of the X-ray variability to occur on dynamical time scales or longer.', 'astro-ph-9906371-1-5-3': 'It is clearly important to determine whether this variability represented a rare, rapid event or if such rapid time scales are truly characteristic of the behavior of MCG[MATH]6-30-15.', 'astro-ph-9906371-1-6-0': 'One might expect that characteristic time scales should scale with the mass of the central object (see, for example, the discussion of ##1 ##2edelson:99a); therefore, in this work we try to gauge the size of the system and the mass of the black hole in MCG[MATH]6-30-15 by studying its characteristic X-ray variability properties in comparison to other black hole systems such as Cyg X-1 and NGC 5548.', 'astro-ph-9906371-1-6-1': 'For the high-frequency variability analysis, we use the 1997 August simultaneous ASCA/Rossi X-ray Timing Explorer (RXTE) observation discussed by ##1 (##2)iwasawa:99a (see also ##1 ##2lee:98a).', 'astro-ph-9906371-1-6-2': 'In the analysis that follows, we use ASCA screening criteria as outlined by ##1 (##2)brandt:96a, except that we use the more stringent criteria of 7 GeV/c for the rigidity and an elevation angle of [MATH].', 'astro-ph-9906371-1-6-3': 'Data from both SIS detectors are combined into a single lightcurve.', 'astro-ph-9906371-1-6-4': 'For the RXTE data, we use screening criteria and analysis techniques appropriate for faint sources, as we have previously discussed in ##1 (##2)chiang:99a.', 'astro-ph-9906371-1-6-5': 'Specifically, we only analyze top layer data from proportional counter units 0, 1, and 2.', 'astro-ph-9906371-1-7-0': '# Power Spectra', 'astro-ph-9906371-1-8-0': 'First we study the long-term behavior of MCG[MATH]6-30-15 by public data (##1 ##2lochner:97a) from the All Sky Monitor (ASM) on RXTE (see ##1 ##2levine:96a).', 'astro-ph-9906371-1-8-1': 'We use the total (1.3-12.2keV) ASM rate, and furthermore we bin the data into 45 evenly spaced 28day time bins .', 'astro-ph-9906371-1-8-2': 'The data for the ASM observations are presented in Fig. [REF]b, with the date of the pointed ASCA/RXTE observation indicated by an arrow.', 'astro-ph-9906371-1-9-0': 'The count rate light curve shows significant variability with fluctuations on a variety of time scales.', 'astro-ph-9906371-1-9-1': 'The total root mean square variability is 53% of the mean, which is to be compared to the value of 32% (calculated from the ASM lightcurve error bars) expected from noise fluctuations alone.', 'astro-ph-9906371-1-9-2': 'The difference between these two values is over 3-[MATH] significant, and the total excess variability present on 1month to 3year time scales is [MATH].', 'astro-ph-9906371-1-10-0': 'We investigate the frequency-dependence of this variability by calculating the power spectral density (PSD) via a discrete fast Fourier transform (FFT) of the ASM lightcurve.', 'astro-ph-9906371-1-10-1': 'This PSD (and all PSD to follow) is normalized using the one-sided normalization of ##1 (##2)belloni:90a, where integrating over positive frequencies yields the total mean square variability relative to the squared mean.', 'astro-ph-9906371-1-10-2': 'We further logarithmically bin the PSD over frequencies [MATH].', 'astro-ph-9906371-1-10-3': 'The results are presented in Fig. [REF]a, where we also show a Monte Carlo estimate of the mean PSD noise level.', 'astro-ph-9906371-1-10-4': 'Although the total variability is above the noise level, no clear trend for the frequency dependence emerges.', 'astro-ph-9906371-1-10-5': 'The PSD is roughly consistent with being flat, however, and it is unlikely that any steep power law trend extends down to the lowest observed frequencies.', 'astro-ph-9906371-1-11-0': 'To investigate the high-frequency end of the power spectrum, we use the RXTE 8-15keV and the ASCA 0.5-1keV lightcurves.', 'astro-ph-9906371-1-11-1': 'Both lightcurves contain a number of data gaps, especially on the orbital time scale of [MATH]ks due to blockage by the earth and passage through the South Atlantic Anomaly.', 'astro-ph-9906371-1-11-2': 'We therefore use the techniques of ##1 (##2)lomb:76a and ##1 (##2)scargle:82a to calculate the PSD.', 'astro-ph-9906371-1-11-3': "Note that even using these techniques, significant variability power appears on 5ks time scales due to data 'clumping' on the orbital time scales.", 'astro-ph-9906371-1-12-0': 'The results, binned over [MATH], are also presented in Fig. [REF]a.', 'astro-ph-9906371-1-12-1': 'The ASCA lightcurve shows 30% rms variability, whereas the RXTE lightcurve shows 20% rms variability.', 'astro-ph-9906371-1-12-2': 'Both PSD have comparable shapes, specifically, flat below [MATH]Hz and [MATH] above.', 'astro-ph-9906371-1-12-3': 'Any further breaks in the power spectra are difficult to assess due to power on the orbital time scales.', 'astro-ph-9906371-1-13-0': 'At frequencies [MATH]Hz there are enough contiguous data segments to be able to calculate the PSD using standard FFT methods.', 'astro-ph-9906371-1-13-1': 'In Fig. [REF]b we show the calculated, background and noise-subtracted PSD for the 1.8-3.6keV and 8-15keV RXTE bands.', 'astro-ph-9906371-1-13-2': 'Below [MATH]Hz, where signal to noise is greatest, both PSD are consistent with being [MATH] and having 6 rms variability.', 'astro-ph-9906371-1-13-3': 'From the generated background lightcurves, we estimate that background fluctuations contribute at most 1.5 and 2 rms variability, respectively, to these PSD at [MATH]Hz.', 'astro-ph-9906371-1-13-4': 'As the PSD of the background lightcurves tend to be slightly steeper than [MATH], there may be some trend for background fluctuations to steepen the observed high-frequency PSD.', 'astro-ph-9906371-1-14-0': '# Time Delays', 'astro-ph-9906371-1-15-0': 'To assess the significance of any time delays between the low energy ASCA lightcurve and the high energy RXTE lightcurve, we use the Z-transformed Discrete Cross-Correlation Function (ZDCF) of ##1 (##2)alexander:97a, which is based upon the DCF method of ##1 (##2)edelson:88a.', 'astro-ph-9906371-1-15-1': 'Auto-correlation functions can also be computed by this method (see below).', 'astro-ph-9906371-1-16-0': 'This method has been previously applied to a simultaneous Extreme Ultraviolet Explorer (EUVE)/ASCA/RXTE observation of NGC 5548 (##1 ##2chiang:99a), where it was noted that the low energy ASCA band leads the high energy RXTE band by 5ks.', 'astro-ph-9906371-1-16-1': 'The application of the ZDCF to the MCG[MATH]6-30-15 data is presented in Fig. [REF], where we have used lightcurves binned with 512s resolution.', 'astro-ph-9906371-1-16-2': 'A positive delay indicates that the latter lightcurve lags the former.', 'astro-ph-9906371-1-16-3': 'Fitting a parabola to the results, we find that the ASCA band leads the RXTE band by [MATH]ks (90% confidence levels, as determined by Monte Carlo simulations; see ##1 ##2chiang:99a).', 'astro-ph-9906371-1-16-4': 'Note that PSD derived from autocorrelations calculated via the ZDCF yield identical results to those presented in Fig. [REF]a.', 'astro-ph-9906371-1-17-0': 'At high Fourier frequency, we have used the 1.8-3.6keV and 8-15keV RXTE lightcurves discussed above to search for frequency-dependent time lags using standard FFT techniques (see, for example, ##1 ##2nowak:99a).', 'astro-ph-9906371-1-17-1': '(There were an insufficient number of uninterupted, strictly simultaneous ASCA/RXTE lightcurves for these purposes.)', 'astro-ph-9906371-1-17-2': 'No significant time delays were found, and the 1-[MATH] upper limits were 50-100s in the [MATH]-[MATH]Hz range.', 'astro-ph-9906371-1-18-0': '# Discussion', 'astro-ph-9906371-1-19-0': 'The MCG[MATH]6-30-15 PSD breaks to being [MATH] at [MATH]Hz, and then breaks to being approximately [MATH] between [MATH]-[MATH]Hz.', 'astro-ph-9906371-1-19-1': 'This is to be compared to the Cyg X-1 PSD which has a comparable rms amplitude and shape, and has a set of PSD breaks at frequencies between 0.03-0.3Hz and 1-10Hz (##1 ##2belloni:90a,nowak:99a).', 'astro-ph-9906371-1-19-2': 'The black hole mass in Cyg X-1 is estimated to be [MATH] (##1 ##2herrero:95a); therefore, if these break frequencies scale with mass, then the central black hole mass of MCG[MATH]6-30-15 could be as low as [MATH].', 'astro-ph-9906371-1-19-3': 'NGC 5548, which is believed to have a central black hole mass of [MATH] (##1 ##2done:95a,chiang:96a), shows a similar PSD with break frequencies at [MATH]Hz and between [MATH]-[MATH]Hz (##1 ##2chiang:99a).', 'astro-ph-9906371-1-19-4': 'Again, if the break frequencies scale with mass, then the central black hole mass for MCG[MATH]6-30-15 could be several orders of magnitude lower than that for NGC 5548.', 'astro-ph-9906371-1-20-0': 'To date, the most carefully studied X-ray PSD for any AGN is that for NGC 3516 (##1 ##2edelson:99a).', 'astro-ph-9906371-1-20-1': 'There the PSD was seen to break from nearly flat at [MATH]<[MATH][MATH]Hz, and then gradually steepen into an [MATH] power law up to frequencies as high as [MATH]Hz.', 'astro-ph-9906371-1-20-2': 'NGC 3516 is seen to be intermediate between NGC 5548 and MCG[MATH]6-30-15.', 'astro-ph-9906371-1-20-3': 'Based upon these measurements (and upon other factors, such as the source luminosity), ##1 (##2)edelson:99a argue for a black hole mass in the range of [MATH]-[MATH], i.e., intermediate to the masses of NGC 5548 and MCG[MATH]6-30-15 discussed above.', 'astro-ph-9906371-1-21-0': 'The observed RXTE/ASCA lag for MCG[MATH]6-30-15 is also intermediate between the X-ray lags seen in Cyg X-1 (##1 ##2miyamoto:88a,nowak:99a) and NGC 5548 (##1 ##2chiang:99a).', 'astro-ph-9906371-1-21-1': 'The time lag observed in NGC 5548, effectively measured on the [MATH] portion of its PSD, is 5ks.', 'astro-ph-9906371-1-21-2': 'Time lags on the flat portion of the NGC 5548 PSD could be considerably longer (##1 ##2chiang:99a).', 'astro-ph-9906371-1-21-3': 'On the flat/[MATH] portion of the Cyg X-1 PSD, X-ray time delays are [MATH]s, whilst on the [MATH] portion of the PSD the X-ray time lags are [MATH]-[MATH]s.', 'astro-ph-9906371-1-21-4': 'The MCG[MATH]6-30-15 time lags cover a similar dynamic range from 1.6ks (overall lag) to [MATH]s (high frequency lag).', 'astro-ph-9906371-1-21-5': 'The observed MCG[MATH]6-30-15 time delays therefore scale to the observed NGC 5548 and Cyg X-1 time delays in an analogous manner as their respective PSD scale to one another.', 'astro-ph-9906371-1-22-0': 'If the characteristic variability and lag times are indicative of mass, then a mass as low as [MATH] may be required for the central black hole of MCG[MATH]6-30-15.', 'astro-ph-9906371-1-22-1': 'Assuming a bolometric luminosity of [MATH] (##1 ##2reynolds:97a), this would imply that MCG[MATH]6-30-15 is emitting at [MATH] of its Eddington rate, which is large but still plausible.', 'astro-ph-9906371-1-22-2': 'A relatively low central black hole mass would make the large amplitude, rapid variability reported by ##1 (##2)reynolds:95a much easier to understand, whereas a mass as large as the [MATH] discussed by ##1 (##2)iwasawa:99a seems very unlikely.', 'astro-ph-9906371-1-22-3': 'Alternative models to localized flares appearing solely on one side of the disk should therefore be considered (e.g., ##1 ##2reynolds:97b).', 'astro-ph-9906371-1-23-0': 'Given a low mass, the [MATH]ks ASCA lead discussed above is then most likely associated with characteristic viscous time scales in the disk, rather than light crossing time scales.', 'astro-ph-9906371-1-23-1': 'Even models that describe X-ray time delays with flares, e.g., ##1 (##2)poutanen:99a, require correlations among trains of flares on viscous time scales in order to produce the longest observed lags.', 'astro-ph-9906371-1-23-2': 'The [MATH]<[MATH][MATH]s lags seen at high frequency likely provide an upper limit to the Compton diffusion time scale (see the discussion of ##1 ##2nowak:99a).', 'astro-ph-9906371-1-23-3': 'These time scales are problematic for any future hopes of performing "reverberation mapping" of the iron K[MATH] line in this system with Constellation-X as it will require [MATH]ks integration times to study the line profile of MCG[MATH]6-30-15 (##1 ##2young:99a).', 'astro-ph-9906371-1-24-0': 'There are several caveats that need to be mentioned.', 'astro-ph-9906371-1-24-1': 'First, we do not know the scaling of the break frequencies with fractional Eddington luminosity in either galactic black hole candidates or AGN.', 'astro-ph-9906371-1-24-2': 'This is an especially important consideration as the mass estimates discussed above imply a large range of Eddington luminosity ratios.', 'astro-ph-9906371-1-24-3': 'Second, the [MATH] ASM lightcurve variability has no simple analogy in Cyg X-1, where ultra-low frequency noise is typically associated with dipping activity due to obscuration by the accretion stream (##1 ##2angelini:94a).', 'astro-ph-9906371-1-24-4': 'Considering all the evidence for rapid variability and extremely short time lags discussed above, however, a low mass for MCG[MATH]6-30-15 seems to us very compelling.', 'astro-ph-9906371-1-24-5': 'With the advent of the X-ray Multiple Mirror (XMM) mission, which is has large effective area and is capable of extremely long, uninterupted observations, these analyses will become more detailed for NGC 5548 and MCG[MATH]6-30-15, and will allow one to develop a statistical sample of numerous other AGN.', 'astro-ph-9906371-1-25-0': 'We thank Ron Remillard for generating the ASM lightcurve of MCG[MATH]6-30-15.', 'astro-ph-9906371-1-25-1': 'We would like to thank Omer Blaes, Katja Pottschmidt, Norm Murray, and Jorn Wilms for useful conversations.', 'astro-ph-9906371-1-25-2': 'This work has been financed by NASA Grants NAG5-4731, NAG5-7723, and NAG5-6337.', 'astro-ph-9906371-1-25-3': 'This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center.'}

{'astro-ph-9906371-2-0-0': 'The bright Seyfert 1 galaxy MCG[MATH]6-30-15 shows large variability on a variety of time scales.', 'astro-ph-9906371-2-0-1': 'We study the [MATH]<[MATH][MATH]day time scale variability using a set of simultaneous archival observations that were obtained from RXTE and the Advanced Satellite for Cosmology and Astrophysics (ASCA).', 'astro-ph-9906371-2-0-2': 'The RXTE observations span nearly [MATH]sec and indicate that the X-ray Fourier Power Spectral Density has an rms variability of 16%, is flat from approximately [MATH]-[MATH]Hz, and then steepens into a power law [MATH] with [MATH]>[MATH][MATH].', 'astro-ph-9906371-2-0-3': 'A further steepening to [MATH] occurs between [MATH]-[MATH]Hz.', 'astro-ph-9906371-2-0-4': 'The shape and rms amplitude are comparable to what has been observed in NGC 5548 and Cyg X-1, albeit with break frequencies that differ by a factor of [MATH] and [MATH], respectively.', 'astro-ph-9906371-2-0-5': 'If the break frequencies are indicative of the central black hole mass, then this mass may be as low as [MATH].', 'astro-ph-9906371-2-0-6': 'An upper limit of [MATH]ks for the relative lag between the 0.5-2keV ASCA band compared to the 8-15keV RXTE band was also found.', 'astro-ph-9906371-2-0-7': 'Again by analogy with NGC 5548 and Cyg X-1, this limit is consistent with a relatively low central black hole mass.', 'astro-ph-9906371-2-1-0': '# Introduction', 'astro-ph-9906371-2-2-0': 'The type 1 Seyfert galaxy MCG[MATH]6-30-15 has in recent years been the subject of intense study owing to the discovery by the Advanced Satellite for Cosmology and Astrophysics (ASCA) of a resolved, broad iron K[MATH] fluorescent line in its hard X-ray spectrum (##1 ##2tanaka:95b).', 'astro-ph-9906371-2-2-1': 'The shape of the line is consistent with a gravitationally and Doppler shifted emission line which originates from near the inner edge of an accretion disk around a black hole.', 'astro-ph-9906371-2-2-2': 'In the X-rays, MCG[MATH]6-30-15 is also one of the brighter and more variable type 1 Seyferts.', 'astro-ph-9906371-2-2-3': 'It is therefore hoped that by examining the correlated variability between the ionizing X-ray continuum and various components of the Fe K[MATH] line, one can obtain a size scale for the system and thence a mass for the black hole (##1 ##2reynolds:99a).', 'astro-ph-9906371-2-2-4': 'This requires, however, that the various components of the line be spectrally resolved on time scales shorter than the intrinsic response time scale of the line-emitting material.', 'astro-ph-9906371-2-3-0': 'At present, ASCA, BeppoSAX, and Chandra are the only X-ray telescopes with the spectral resolution to measure fluxes in different line components separately.', 'astro-ph-9906371-2-3-1': 'Unfortunately, the integration times ([MATH]ks) required to obtain this resolution are longer than the light travel time across the inner edge of an accretion disk around a [MATH] Schwarzschild black hole.', 'astro-ph-9906371-2-3-2': 'Furthermore, the fits to the iron line appear to require a Kerr geometry (although see ##1 ##2reynolds:97b) in order to explain the broad red wing of the line and the lack of significant emission blueward of 6.4keV, thus making the relevant time scales even shorter.', 'astro-ph-9906371-2-3-3': 'Nonetheless, over longer time scales, ASCA has measured significant variability in the shape of the MCG[MATH]6-30-15 Fe K[MATH] line (##1 ##2iwasawa:96a).', 'astro-ph-9906371-2-4-0': 'Time-resolved spectral investigations (##1 ##2iwasawa:96a,iwasawa:99a) suggest that the broad and narrow components of the iron line are correlated differently with the flux state depending on the time scale investigated.', 'astro-ph-9906371-2-4-1': 'For integrations [MATH]s, the narrow component varies with the continuum flux whilst the broad component appears to be anti-correlated.', 'astro-ph-9906371-2-4-2': 'In contrast, on shorter time scales the broad component responds immediately to flux changes whilst the narrow component remains constant.', 'astro-ph-9906371-2-4-3': '##1 (##2)iwasawa:99a suggest that multiple, localized X-ray flares occur on the disk surface near the inner edge and illuminate only a relatively small region of the disk.', 'astro-ph-9906371-2-4-4': 'These small regions make contributions to narrow ranges in line redshift and thus produce very complex temporal behavior.', 'astro-ph-9906371-2-5-0': 'In support of this interpretation, ##1 (##2)iwasawa:99a consider the iron line associated with a very short, bright flare from MCG[MATH]6-30-15 observed during 1997 August by ASCA (see Fig. [REF]).', 'astro-ph-9906371-2-5-1': 'The line was very redshifted and had little or no emission blueward of 6keV.', 'astro-ph-9906371-2-5-2': 'Its shape was consistent with having originated entirely from a small region at [MATH] on the approaching side of the disk.', 'astro-ph-9906371-2-5-3': 'The flare lasted about 4ks, which is also approximately the orbital period at this radius for a [MATH] black hole.', 'astro-ph-9906371-2-5-4': 'In order for the line not to be significantly more smeared by the orbital motion, ##1 (##2)iwasawa:99a estimate a black hole mass of [MATH].', 'astro-ph-9906371-2-5-5': 'Alternatively, they suggest that a much lower mass is possible if all the line emission arises from [MATH].', 'astro-ph-9906371-2-6-0': 'A major complication that a high-mass model faces is the large amplitude and rapid X-ray variability of MCG[MATH]6-30-15 (Fig. [REF]).', 'astro-ph-9906371-2-6-1': 'In particular, during the performance verification phase of ASCA, ##1 (##2)reynolds:95a noted that the 0.5-10keV flux increased by a factor 1.5 over 100s, i.e., the dynamical time scale at the inner disk edge surrounding a maximal Kerr, [MATH] black hole.', 'astro-ph-9906371-2-6-2': 'Typically one expects the bulk of the X-ray variability to occur on dynamical time scales or longer.', 'astro-ph-9906371-2-6-3': 'It is clearly important to determine whether this variability represented a rare, rapid event or if such time scales are truly characteristic of the behavior of MCG[MATH]6-30-15.', 'astro-ph-9906371-2-7-0': 'One might expect that characteristic time scales should scale with the mass of the central object (see, e.g., the discussions of ##1 ##2mchardy:88a and ##1 ##2edelson:99a); therefore, in this work we try to gauge the size of the system and the mass of the black hole in MCG[MATH]6-30-15 by studying its characteristic X-ray variability properties in comparison to other black hole systems such as Cyg X-1 and NGC 5548.', 'astro-ph-9906371-2-7-1': 'For the high-frequency variability analysis, we use the 1997 August simultaneous ASCA/Rossi X-ray Timing Explorer (RXTE) observation discussed by ##1 (##2)iwasawa:99a (see also ##1 ##2lee:98a).', 'astro-ph-9906371-2-7-2': 'In our analysis we screen the ASCA data as outlined by ##1 (##2)brandt:96a, except that we use the more stringent criteria of 7 GeV/c for the rigidity and an elevation angle of [MATH].', 'astro-ph-9906371-2-7-3': 'Data from both SIS detectors are combined into a single lightcurve.', 'astro-ph-9906371-2-7-4': 'For the RXTE data, we use screening criteria and analysis techniques appropriate for faint sources, as we have previously discussed in ##1 (##2)chiang:99a.', 'astro-ph-9906371-2-7-5': 'Specifically, we only analyze top layer data from proportional counter units 0, 1, and 2.', 'astro-ph-9906371-2-8-0': '# Power Spectra', 'astro-ph-9906371-2-9-0': 'MCG[MATH]6-30-15 is detected by the All Sky Monitor (ASM; ##1 ##2levine:96a) on RXTE with a mean count rate of 0.44cps in the 1.3-12.2keV band.', 'astro-ph-9906371-2-9-1': 'Currently, there is a 0.1 cps uncertainty in the zero level offset, as well as comparable magnitude systematic time-dependent variations due to, for example, the solar angular position relative to the source (Remillard 1999, priv.', 'astro-ph-9906371-2-9-2': 'comm.)', 'astro-ph-9906371-2-9-3': 'Assessing the very low frequency ([MATH]<[MATH][MATH]Hz) X-ray variability of MCG[MATH]6-30-15 is therefore problematic.', 'astro-ph-9906371-2-9-4': 'Efforts are currently underway, however, to revise the ASM data reduction process in order to minimize such systematic effects (Remillard 1999, priv.', 'astro-ph-9906371-2-9-5': 'comm.)', 'astro-ph-9906371-2-9-6': '; therefore, an ultra-low frequency variability study of MCG[MATH]6-30-15 in principle will be feasible in the near future.', 'astro-ph-9906371-2-10-0': 'We are able, however, to investigate the high-frequency power spectral density (PSD) of MCG[MATH]6-30-15 by using the RXTE 8-15keV and the ASCA 0.5-2keV lightcurves binned on 4096sec time scales.', 'astro-ph-9906371-2-10-1': 'Both lightcurves contain data gaps, especially on the orbital time scale of [MATH]ks due to blockage by the Earth and passage through the South Atlantic Anomaly.', 'astro-ph-9906371-2-10-2': 'We therefore use the techniques of ##1 (##2)lomb:76a and ##1 (##2)scargle:82a to calculate the PSD, and we only consider frequencies [MATH]<[MATH][MATH]Hz (evenly sampled in intervals of the inverse of the observation duration).', 'astro-ph-9906371-2-10-3': 'Higher time resolution lightcurves showed excess power on the [MATH]ks orbital time scale.', 'astro-ph-9906371-2-10-4': 'The results, binned over the greater of four contiguous frequency bins or logarithmically over [MATH], are presented in Fig. [REF].', 'astro-ph-9906371-2-10-5': 'Here we use a one-sided normalization where integrating over positive frequencies yields the total mean square variability relative to the squared mean for the particular lightcurve being analyzed.', 'astro-ph-9906371-2-10-6': 'The ASCA lightcurve shows 28% rms variability, whereas the RXTE lightcurve shows 16% rms variability.', 'astro-ph-9906371-2-10-7': 'Both PSD have comparable shapes, i.e., flat from [MATH]-[MATH]Hz and slightly steeper than [MATH] at higher frequencies.', 'astro-ph-9906371-2-11-0': 'We have fit a broken power law to the data, assuming error bars equal to the average PSD value divided by the square root of the number of frequency bins averaged over.', 'astro-ph-9906371-2-11-1': 'Although such error estimates are only valid for averages made from independent frequency bins (which is not strictly true for the Lomb-Scargle periodogram; ##1 ##2scargle:82a), this should provide a rough estimate, especially as the lightcurves are nearly evenly sampled.', 'astro-ph-9906371-2-11-2': 'The best fit PSD slopes are [MATH] for the PCA and [MATH] for ASCA (errors are [MATH]).', 'astro-ph-9906371-2-11-3': 'The break frequencies are found to be [MATH]Hz for the PCA, and [MATH]Hz for ASCA.', 'astro-ph-9906371-2-11-4': 'These break frequencies are consistent with the value found by ##1 (##2)mchardy:98a.', 'astro-ph-9906371-2-12-0': 'At frequencies [MATH]Hz there are enough contiguous data segments to be able to calculate the PSDs using standard FFT methods (##1 ##2nowak:99a, and extensive references therein).', 'astro-ph-9906371-2-12-1': 'In Fig. [REF] we show the calculated, background and noise-subtracted, PSD for the 1.8-3.6keV and 8-15keV RXTE bands.', 'astro-ph-9906371-2-12-2': 'Below [MATH]Hz, where signal to noise is greatest, both PSDs are consistent with being [MATH] and having 6 rms variability.', 'astro-ph-9906371-2-12-3': 'From the generated background lightcurves, we estimate that background fluctuations contribute at most 1.5 and 2 rms variability, respectively, to these PSDs at [MATH]Hz.', 'astro-ph-9906371-2-12-4': 'As the PSDs of the background lightcurves tend to be slightly steeper than [MATH], there may be some trend for background fluctuations to steepen the observed high-frequency PSDs, and, in fact, ##1 (##2)yaqoob:97a find a slightly flatter ([MATH]) high-frequency PSD for MCG[MATH]6-30-15.', 'astro-ph-9906371-2-13-0': '# Time Delays', 'astro-ph-9906371-2-14-0': 'In order to examine time delays between the low energy ASCA lightcurve and the high energy RXTE lightcurve, we use the Z-transformed Discrete Cross-Correlation Function (ZDCF) of ##1 (##2)alexander:97a, which is based upon the DCF method of ##1 (##2)edelson:88a.', 'astro-ph-9906371-2-14-1': 'Auto-correlation functions can also be computed by this procedure, and we note that PSD derived from autocorrelations calculated via the ZDCF yield identical results to those presented in Fig. [REF].', 'astro-ph-9906371-2-14-2': 'We use the Monte Carlo methods described by ##1 (##2)peterson:98a to assess the significance of any uncertainties due to flux measurement errors as well as uneven or incomplete sampling.', 'astro-ph-9906371-2-15-0': 'We have previously applied these methods to a simultaneous Extreme Ultraviolet Explorer (EUVE)/ASCA/RXTE observation of NGC 5548 (##1 ##2chiang:99a), where we found evidence for the low energy ASCA band leading the high energy RXTE band by 5ks.', 'astro-ph-9906371-2-15-1': 'The results for the MCG[MATH]6-30-15 data are shown in Fig. [REF], for which we have used lightcurves binned at 512s resolution.', 'astro-ph-9906371-2-15-2': 'A positive delay indicates that the RXTE light curve lags the ASCA light curve.', 'astro-ph-9906371-2-15-3': 'Due to the ambiguities associated with interpreting cross-correlation results, particularly for such small delays, we considered three different measures of the "lag".', 'astro-ph-9906371-2-15-4': 'For each Monte Carlo trial, we estimate the characteristic lag by (1) fitting a parabola to the ZDCF values to find the location of the peak, (2) computing the centroid of the ZDCF over positive values bracketing the maximum value, and (3) using the location of the actual maximum value of the ZDCF.', 'astro-ph-9906371-2-15-5': 'In all three cases, our simulations yield evidence for a positive lag at various degrees of significance: [MATH]ks, [MATH]ks, [MATH] (90% C.L.).', 'astro-ph-9906371-2-15-6': 'Although the fitted peak estimate is consistent with a positive lag, both the centroid and ZDCF-maximum estimates are formally consistent with zero giving an upper limit of [MATH]<[MATH][MATH]ks.', 'astro-ph-9906371-2-16-0': 'At high Fourier frequency, we have used the 1.8-3.6keV and 8-15keV RXTE lightcurves discussed above to search for frequency-dependent time lags using standard FFT techniques.', 'astro-ph-9906371-2-16-1': '(A complete discussion of such methods, including calculation of error bars, is presented in ##1 ##2nowak:99a and references therein).', 'astro-ph-9906371-2-16-2': 'No significant time delays were found, and the 1-[MATH] upper limits were 50-100s in the [MATH]-[MATH]Hz range.', 'astro-ph-9906371-2-16-3': 'Note that there were an insufficient number of uninterrupted, strictly simultaneous ASCA/RXTE lightcurves to allow calculation of their relative time delays via direct FFT methods.', 'astro-ph-9906371-2-17-0': '# Discussion', 'astro-ph-9906371-2-18-0': 'The MCG[MATH]6-30-15 PSD breaks to being [MATH] at [MATH]Hz, and then breaks to being approximately [MATH] between [MATH]-[MATH]Hz.', 'astro-ph-9906371-2-18-1': 'This is to be compared to the Cyg X-1 PSD which has a comparable rms amplitude and shape, and has a set of PSD breaks at frequencies between 0.03-0.3Hz and 1-10Hz (##1 ##2nowak:99a, and references therein).', 'astro-ph-9906371-2-18-2': 'The black hole mass in Cyg X-1 is estimated to be [MATH] (##1 ##2herrero:95a); therefore, if these break frequencies scale with mass, then the central black hole mass of MCG[MATH]6-30-15 could be as low as [MATH].', 'astro-ph-9906371-2-18-3': 'NGC 5548, which is believed to have a central black hole mass of [MATH] (##1 ##2done:96a,chiang:96a,peterson:99a), shows a similar PSD with break frequencies at [MATH]Hz and between [MATH]-[MATH]Hz (##1 ##2chiang:99a).', 'astro-ph-9906371-2-18-4': 'Again, if the break frequencies scale with mass, then the central black hole mass for MCG[MATH]6-30-15 could be several orders of magnitude lower than that for NGC 5548.', 'astro-ph-9906371-2-19-0': 'To date, the most carefully studied X-ray PSD for any AGN is that for NGC 3516 (##1 ##2edelson:99a).', 'astro-ph-9906371-2-19-1': '(See also ##1 ##2mchardy:98a who present preliminary results for a number of AGN, including MCG[MATH]6-30-15.)', 'astro-ph-9906371-2-19-2': 'For NGC 3516 the PSD was seen to break from nearly flat at [MATH]<[MATH][MATH]Hz, and then gradually steepen into an [MATH] power law up to frequencies as high as [MATH]Hz.', 'astro-ph-9906371-2-19-3': 'NGC 3516 is seen to be intermediate between NGC 5548 and MCG[MATH]6-30-15.', 'astro-ph-9906371-2-19-4': 'Based upon these measurements (and upon other factors, such as the source luminosity), ##1 (##2)edelson:99a argue for a black hole mass in the range of [MATH]-[MATH], i.e., intermediate to the masses of NGC 5548 and MCG[MATH]6-30-15 discussed above.', 'astro-ph-9906371-2-20-0': 'The RXTE/ASCA lag upper limit for MCG[MATH]6-30-15 is also intermediate between the observed X-ray lags for Cyg X-1 (##1 ##2miyamoto:88a,nowak:99a) and NGC 5548 (##1 ##2chiang:99a).', 'astro-ph-9906371-2-20-1': 'The time lag observed in NGC 5548, effectively measured on the [MATH] portion of its PSD, is 5ks.', 'astro-ph-9906371-2-20-2': 'Time lags on the flat portion of the NGC 5548 PSD could be considerably longer (##1 ##2chiang:99a).', 'astro-ph-9906371-2-20-3': 'Near the PSD break from flat to [MATH], the X-ray time lags in Cyg X-1 are [MATH]s, whilst on the [MATH] portion of the PSD the X-ray time lags are [MATH]-[MATH]s.', 'astro-ph-9906371-2-20-4': 'The MCG[MATH]6-30-15 time lags may cover a similar dynamic range from [MATH]ks (overall lag) to [MATH]s (high frequency lag).', 'astro-ph-9906371-2-21-0': 'If the characteristic variability and lag times are indicative of mass, then a mass as low as [MATH] may be required for the central black hole of MCG[MATH]6-30-15.', 'astro-ph-9906371-2-21-1': 'Assuming a bolometric luminosity of [MATH] (##1 ##2reynolds:97a), this would imply that MCG[MATH]6-30-15 is emitting at [MATH] of its Eddington rate, which is large but still plausible.', 'astro-ph-9906371-2-21-2': 'A relatively low central black hole mass would make the large amplitude, rapid variability reported by ##1 (##2)reynolds:95a much easier to understand, whereas a mass as large as the [MATH], the upper end discussed by ##1 (##2)iwasawa:99a, seems very unlikely.', 'astro-ph-9906371-2-21-3': 'The [MATH]<[MATH][MATH]s lags seen at high frequency then likely provide an upper limit to the Compton diffusion time scale (see the discussion of ##1 ##2nowak:99a).', 'astro-ph-9906371-2-21-4': 'These time scales are problematic for future hopes of simultaneously temporally and spectrally resolving the iron K[MATH] line in this system with Constellation-X as it will require [MATH]ks integration times to study the line profile of MCG[MATH]6-30-15 (##1 ##2young:99a).', 'astro-ph-9906371-2-22-0': 'A 30% Eddington luminosity indicates that it is worthwhile to search for ultra-low frequency ([MATH]<[MATH][MATH]Hz) variability in excess of [MATH] rms, that is, above an extrapolation of the flat part of the ASCA and RXTE-PCA PSDs.', 'astro-ph-9906371-2-22-1': "'Very high state', i.e., [MATH]>[MATH][MATH], galactic black hole candidates can show PSDs that are flat below [MATH]Hz, are approximately proportional to [MATH] PSD between [MATH]-[MATH]Hz, are flat again between [MATH]-[MATH]Hz, and then break into an [MATH] PSD at higher frequencies.", 'astro-ph-9906371-2-22-2': "(Specifically, see Fig. 4b of ##1 ##2miyamoto:91a, which shows a 'very high state' PSD of GX339[MATH]4.)", 'astro-ph-9906371-2-22-3': "The low frequency portion of the 'very high state' PSD has no simple analogy in the (usually observed) low/hard state of Cyg X-1, where ultra-low frequency noise is typically associated with dipping activity due to obscuration by the accretion stream (##1 ##2angelini:94a).", 'astro-ph-9906371-2-22-4': "Previous models, for example, have associated 'very high state' low-frequency variability with fluctuations of a viscously unstable [MATH]-disk (##1 ##2nowak:94a, and references therein).", 'astro-ph-9906371-2-23-0': 'This highlights the major caveat that needs to be mentioned; we do not know the average PSD shape nor the scaling of the break frequencies as a function of fractional Eddington luminosity in either galactic black hole candidates or AGN.', 'astro-ph-9906371-2-23-1': 'This is an especially important consideration as the mass estimates discussed above imply a large range of Eddington luminosity ratios.', 'astro-ph-9906371-2-23-2': 'Considering all the evidence for rapid variability and extremely short time lags in MCG[MATH]6-30-15 discussed above, however, a low mass for MCG[MATH]6-30-15 seems to us very compelling.', 'astro-ph-9906371-2-23-3': 'With the advent of the X-ray Multiple Mirror (XMM) mission, which has large effective area and is capable of extremely long, uninterrupted observations, these analyses will become more detailed for NGC 5548 and MCG[MATH]6-30-15, and will allow one to develop a statistical sample of numerous other AGN.', 'astro-ph-9906371-2-24-0': 'We thank R. Remillard for generating an ASM lightcurve of MCG[MATH]6-30-15, and O. Blaes, K. Pottschmidt, N. Murray, and J. Wilms for useful conversations.', 'astro-ph-9906371-2-24-1': 'This work has been financed by NASA Grants NAG5-4731, NAG5-7723, and NAG5-6337.', 'astro-ph-9906371-2-24-2': 'This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center.'}

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[]

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[]

['astro-ph-9906371-1-25-2', 'astro-ph-9906371-2-9-2', 'astro-ph-9906371-2-9-5', 'astro-ph-9906371-2-24-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/9906371

1011.2501

{'1011.2501-1-0-0': '[MATH] Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA', '1011.2501-1-1-0': 'The search for extra-solar planets has led to the surprising discovery of many Jupiter-like planets in very close proximity to their host star[CITATION], the so-called "hot Jupiters."', '1011.2501-1-1-1': 'Even more surprisingly, many of these hot Jupiters have orbits that are eccentric or highly inclined with respect to the equator of the star, and some (about [MATH]) appear to be in retrograde orbits[CITATION].', '1011.2501-1-1-2': 'How they get so close to the star in such orbits remains an open question.', '1011.2501-1-1-3': 'Slow migration though a protoplanetary disk[CITATION][MATH][CITATION] would produce orbits with low eccentricities and inclinations.', '1011.2501-1-1-4': 'Some models[CITATION][MATH][CITATION] invoke a companion star in the system, which perturbs the inner orbit and can produce increases in eccentricity and inclination but not retrograde orbits.', '1011.2501-1-1-5': 'Here we show that the presence of an additional, moderately inclined and eccentric massive planet in the system can naturally explain close, inclined, eccentric, and even retrograde orbits.', '1011.2501-1-1-6': 'We provide a complete and accurate treatment of the secular dynamics including both the key octupole-order effects and tidal friction.', '1011.2501-1-1-7': 'The flow of angular momentum from the inner orbit to the orbit of the perturber can lead to both high eccentricities and inclinations, and even flip the inner orbit.', '1011.2501-1-1-8': 'Previous treatments of the secular dynamics focusing on stellar-mass perturbers would not allow for such an outcome.', '1011.2501-1-1-9': 'In our treatment the component of the inner orbit\'s angular momentum perpendicular to the stellar equatorial plane can change sign; a brief excursion to very high eccentricity during the chaotic evolution of the inner orbit can then lead to rapid "tidal capture," forming a retrograde hot Jupiter.', '1011.2501-1-2-0': 'Despite many attempts[CITATION][MATH][CITATION], there is no model that can account for all the properties of the known hot Jupiter (HJ) systems.', '1011.2501-1-2-1': 'One model suggests that HJs formed far away from the star and slowly spiraled in, losing angular momentum and orbital energy to the protoplanetary disk[CITATION][MATH][CITATION].', '1011.2501-1-2-2': 'This "migration" process should produce planets with low orbital inclinations and eccentricities.', '1011.2501-1-2-3': 'However, many HJs are observed to be on orbits with high eccentricities, and misaligned with respect to the rotation axis of the star (as measured through the Rossiter-McLaughlin effect[CITATION]) and some of these ([MATH] out of [MATH]) appear to be in retrograde orbits.', '1011.2501-1-2-4': 'Secular perturbations from a distant binary star companion can produce increases in the eccentricity and inclination of a planetary orbit[CITATION], but they cannot produce a retrograde orbit.', '1011.2501-1-2-5': "During the evolution to high eccentricity, tidal dissipation near pericenter can force the planet's orbit to decay, potentially forming a HJ[CITATION][MATH][CITATION].", '1011.2501-1-2-6': 'Another mechanism to produce a tilted orbit is via planet-planet scattering[CITATION], possibly combined with other perturbers and tidal friction[CITATION].', '1011.2501-1-3-0': 'In our new treatment we allow for the magnitude and orientation of both orbital angular momenta to change (see Figure 1).', '1011.2501-1-3-1': 'The additional body (either an outer planet or a brown-dwarf companion) gravitationally perturbs the inner planet on time scales long compared to the orbital period (i.e., we consider the secular evolution of the system).', '1011.2501-1-3-2': 'We define the orientation of the inner orbit so that a prograde (retrograde) orbit has [MATH]), where [MATH] is the inclination of the inner orbit with respect to the total angular momentum, assumed parallel to the stellar rotation axis.', '1011.2501-1-3-3': 'We assume a hierarchical configuration, with the outer perturber in a much wider orbit than the inner one.', '1011.2501-1-3-4': 'In the secular approximation the orbits may change shape and orientation but the semi-major axes (SMA) are strictly conserved in the absence of tidal dissipation[CITATION][MATH][CITATION].', '1011.2501-1-3-5': 'In particular, the Kozai-Lidov mechanism[CITATION][MATH][CITATION][MATH][CITATION] produces large-amplitude oscillations of the eccentricity and inclination when the initial relative inclination between the inner and outer orbits is sufficiently large ( [MATH]).', '1011.2501-1-4-0': 'We have derived the secular evolution equations to octupole order using Hamiltonian perturbation theory[CITATION][MATH][CITATION][MATH][CITATION].', '1011.2501-1-4-1': 'In contrast to previous derivations of "Kozai-type" evolution, our treatment allows for changes in the [MATH]-components of the orbital angular momenta (i.e., the components along the total angular momentum) [MATH] and [MATH] (see supplementary material).', '1011.2501-1-4-2': 'The octupole-order equations allow us to calculate more accurately the evolution of systems with more closely coupled orbits and with planetary-mass perturbers.', '1011.2501-1-4-3': 'The octupole-level terms can give rise to fluctuations in the eccentricity maxima to arbitrarily high values[CITATION][MATH][CITATION], in contrast to the regular evolution in the quadrupole potential[CITATION][MATH][CITATION][MATH][CITATION], where the amplitude of eccentricity oscillations is constant.', '1011.2501-1-4-4': 'Many previous studies of secular perturbations in hierarchical triples considered a stellar-mass perturber, for which [MATH] is very nearly constant[CITATION][MATH][CITATION][MATH][CITATION].', '1011.2501-1-4-5': 'Moreover, the assumption that [MATH] is constant has been built into previous derivations[CITATION][MATH][CITATION].', '1011.2501-1-4-6': 'However, this assumption is only valid as long as [MATH], which is not the case in comparable-mass systems (with two planets).', '1011.2501-1-4-7': 'Unfortunately, an immediate consequence of this assumption is that a prograde orbit can never be turned into a retrograde orbit.', '1011.2501-1-4-8': 'Figure [REF] shows the evolution of a representative system (here without tidal effects for simplicity): the inner planet oscillates between prograde and retrograde orbits as angular momentum flows back and forth between the two orbits.', '1011.2501-1-5-0': 'Previous calculations of planet migration through "Kozai cycles with tidal friction" (KCTF)[CITATION][MATH][CITATION][MATH][CITATION][MATH][CITATION] produced a slow, gradual spiral-in of the inner planet.', '1011.2501-1-5-1': 'Instead, our more accurate treatment shows that the eccentricity can occasionally reach a much higher value than in the regular "Kozai cycles" calculated to quadrupole order.', '1011.2501-1-5-2': 'Thus, the pericenter distance will occasionally shrink on a short time scale (compared to the Kozai period), and the planet can then suddenly be tidally captured by the star.', '1011.2501-1-5-3': 'We propose to call this "Kozai capture".', '1011.2501-1-5-4': 'Kozai capture provides a new way to form HJs.', '1011.2501-1-5-5': 'If the capture happens after the inner orbit has flipped the HJ will appear in a retrograde orbit.', '1011.2501-1-5-6': 'This is illustrated in Figure 2.', '1011.2501-1-5-7': 'During the evolution of the system the inner orbit shrinks in steps (Fig. 2c) whenever the dissipation becomes significant, i.e., near unusually high eccentricity maxima.', '1011.2501-1-5-8': 'The inner orbit can then eventually become tidally circularized.', '1011.2501-1-5-9': 'This happens near the end of the evolution, on a very short time scale (see Fig. 2, right panels).', '1011.2501-1-5-10': 'In this final step, the inner orbit completely and quickly decouples from the outer perturber, and the orbital angular momenta then become constant.', '1011.2501-1-5-11': 'Therefore, the final SMA for the HJ is [MATH], where [MATH] is the pericenter distance at the beginning of the capture phase[CITATION].', '1011.2501-1-6-0': 'The same type of evolution shown in Figure 2 is seen with a broad range of initial conditions.', '1011.2501-1-6-1': "Our mechanism requires that the outer perturber's orbit start with high inclination ([MATH]).", '1011.2501-1-6-2': 'The particular configuration in Figure 2 has a very wide outer orbit similar to those of directly imaged planets such as Fomalhaut b[CITATION] and HR 8799b[CITATION].', '1011.2501-1-6-3': 'In this case the inner Jupiter could have formed in its original location in accordance with the standard core accretion model[CITATION].', '1011.2501-1-6-4': 'An alternative path to such a configuration involves strong planet-planet scattering in a closely packed initial system of several giant planets[CITATION].', '1011.2501-1-6-5': 'Independent of any particular planet formation mechanism, we predict that systems with misaligned HJs should also contain a much more distant massive planet on an inclined orbit.'}

{'1011.2501-2-0-0': 'About [MATH] of "hot Jupiters" are actually orbiting counter to the spin axis of the star[CITATION].', '1011.2501-2-0-1': 'Perturbations from a binary star companion[CITATION] can produce high inclinations, but cannot explain orbits that are retrograde with respect to the total angular momentum of the system.', '1011.2501-2-0-2': 'Such orbits can be produced through secular perturbations in hierarchical triple-star systems[CITATION].', '1011.2501-2-0-3': 'Here we report a similar application to planetary bodies, including both the key octupole-order effects and tidal friction, and find that it can produce hot Jupiters in orbits that are retrograde with respect to the total angular momentum.', '1011.2501-2-0-4': 'With stellar mass perturbers such an outcome is not possible[CITATION].', '1011.2501-2-0-5': "With planetary perturbers the inner orbit's angular momentum component parallel to the total angular momentum need not be constant[CITATION].", '1011.2501-2-0-6': 'In fact, as we show here, it can even change sign, leading to a retrograde orbit.', '1011.2501-2-0-7': 'A brief excursion to very high eccentricity during the chaotic evolution of the inner orbit can then lead to rapid capture, forming a retrograde hot Jupiter.', '1011.2501-2-1-0': 'Despite many attempts[CITATION], there is no model that can account for all the properties of the known hot Jupiter (HJ) systems.', '1011.2501-2-1-1': 'One model suggests that HJs formed far away from the star and slowly spiraled in, losing angular momentum and orbital energy to the protoplanetary disk[CITATION].', '1011.2501-2-1-2': 'This "migration" process should produce planets with low orbital inclinations and eccentricities.', '1011.2501-2-1-3': 'However, many HJs are observed to be on orbits with high eccentricities, and misaligned with the spin axis of the star (as measured through the Rossiter-McLaughlin effect[CITATION]) and some of these ([MATH] out of [MATH]) even appear to be orbiting counter to the spin of the star.', '1011.2501-2-1-4': 'In a second model, secular perturbations from a distant binary star companion can produce increases in the eccentricity and inclination of a planetary orbit[CITATION].', '1011.2501-2-1-5': "During the evolution to high eccentricity, tidal dissipation near pericenter can force the planet's orbit to decay, potentially forming a misaligned HJ[CITATION].", '1011.2501-2-1-6': 'Recently, secular chaos involving several planets has also been proposed as a way to form HJs on eccentric and misaligned orbits[CITATION].', '1011.2501-2-1-7': 'A different class of models to produce a tilted orbit is via planet-planet scattering[CITATION], possibly combined with other perturbers and tidal friction[CITATION].', '1011.2501-2-1-8': 'In such models the initial configuration is a densly-packed system of planets and the final tilted orbit is a result of dynamical scattering among the planets, in contrast to the secular interactions we study here.', '1011.2501-2-2-0': 'In our general treatment of secular interactions between two orbiting bodies we allow for the magnitude and orientation of both orbital angular momenta to change (see Figure 1).', '1011.2501-2-2-1': 'The outer body (here either a planet or a brown-dwarf) gravitationally perturbs the inner planet on time scales long compared to the orbital period (i.e., we consider the secular evolution of the system).', '1011.2501-2-2-2': 'We define the orientation of the inner orbit with respect to the invariable plane of the system (perpendicular to the total angular momentum): a prograde (retrograde) orbit has [MATH]), where [MATH] is the inclination of the inner orbit with respect to the total angular momentum vector.', '1011.2501-2-2-3': 'Note that the word "retrograde" is also used in the literature to indicate orbital motion counter to the stellar spin.', '1011.2501-2-2-4': 'The directly observed parameter is actually the projected angle between the spin axis of the star and the orbital angular momentum of a HJ.', '1011.2501-2-2-5': 'Our proposed mechanism can produce HJs that are "retrograde" both with respect to the stellar spin and with respect to the total angular momentum.', '1011.2501-2-2-6': 'By contrast, a stellar companion can only succeed in the former.', '1011.2501-2-2-7': 'See the online Supplementary Information for more details; henceforth we will use the term "retrograde" only to indicate an orbit with [MATH] as define above.', '1011.2501-2-3-0': 'We assume a hierarchical configuration, with the outer perturber on a much wider orbit than the inner one.', '1011.2501-2-3-1': 'In the secular approximation the orbits may change shape and orientation but the semi-major axes are strictly conserved in the absence of tidal dissipation[CITATION].', '1011.2501-2-3-2': 'In particular, the Kozai-Lidov mechanism[CITATION] produces large-amplitude oscillations of the eccentricity and inclination when the initial relative inclination between the inner and outer orbits is sufficiently large ( [MATH]).', '1011.2501-2-4-0': 'We have derived the secular evolution equations to octupole order using Hamiltonian perturbation theory[CITATION].', '1011.2501-2-4-1': 'In contrast to previous derivations of "Kozai-type" evolution, our treatment allows for changes in the [MATH]-components of the orbital angular momenta (i.e., the components along the total angular momentum) [MATH] and [MATH] (see Supplementary Information).', '1011.2501-2-4-2': 'The octupole-order equations allow us to calculate the evolution of systems with more closely coupled orbits and with planetary-mass perturbers.', '1011.2501-2-4-3': 'The octupole-level terms can give rise to fluctuations in the eccentricity maxima to arbitrarily high values[CITATION], in contrast to the regular evolution in the quadrupole potential[CITATION], where the amplitude of eccentricity oscillations is constant.', '1011.2501-2-5-0': 'Many previous studies of secular perturbations in hierarchical triples considered a stellar-mass perturber, for which [MATH] is very nearly constant[CITATION].', '1011.2501-2-5-1': 'Moreover, the assumption that [MATH] is constant has been built into previous derivations[CITATION].', '1011.2501-2-5-2': 'However, this assumption is only valid as long as [MATH], which is not the case in comparable-mass systems (e.g., with two planets).', '1011.2501-2-5-3': 'Unfortunately, an immediate consequence of this assumption is that an orbit that is prograde relative to the total angular momentum always remains prograde.', '1011.2501-2-5-4': 'Figure [REF] shows the evolution of a representative system (here without tidal effects for simplicity): the inner planet oscillates between prograde and retrograde orbits (with respect to the total angular momentum) as angular momentum flows back and forth between the two orbits.', '1011.2501-2-6-0': 'Previous calculations of planet migration through "Kozai cycles with tidal friction"[CITATION] produced a slow, gradual spiral-in of the inner planet.', '1011.2501-2-6-1': 'Instead, our treatment shows that the eccentricity can occasionally reach a much higher value than in the regular "Kozai cycles" calculated to quadrupole order.', '1011.2501-2-6-2': 'Thus, the pericenter distance will occasionally shrink on a short time scale (compared to the Kozai period), and the planet can then suddenly be tidally captured by the star.', '1011.2501-2-6-3': 'We propose to call this "Kozai capture."', '1011.2501-2-7-0': 'Kozai capture provides a new way to form HJs.', '1011.2501-2-7-1': 'If the capture happens after the inner orbit has flipped the HJ will appear in a retrograde orbit.', '1011.2501-2-7-2': 'This is illustrated in Figure 2.', '1011.2501-2-7-3': 'During the evolution of the system the inner orbit shrinks in steps (Fig. 2c) whenever the dissipation becomes significant, i.e., near unusually high eccentricity maxima.', '1011.2501-2-7-4': 'The inner orbit can then eventually become tidally circularized.', '1011.2501-2-7-5': 'This happens near the end of the evolution, on a very short time scale (see Fig. 2, right panels).', '1011.2501-2-7-6': 'In this final step, the inner orbit completely and quickly decouples from the outer perturber, and the orbital angular momenta then become constant.', '1011.2501-2-7-7': 'Therefore, the final semi-major axis for the HJ is [MATH], where [MATH] is the pericenter distance at the beginning of the capture phase[CITATION].', '1011.2501-2-8-0': 'The same type of evolution shown in Figure 2 is seen with a broad range of initial conditions.', '1011.2501-2-8-1': 'There are two main routes to forming a HJ through the dynamical evolution of the systems we consider here.', '1011.2501-2-8-2': 'In the first, tidal friction slowly damps the growing eccentricity of the inner planet, resulting in circularized, prograde HJs.', '1011.2501-2-8-3': 'In the second, a sudden high-eccentricity spike in the orbital evolution of the inner planet is accompanied by a flip of its orbit.', '1011.2501-2-8-4': 'The planet is then quickly circularized into a retrograde short-period orbit.', '1011.2501-2-8-5': 'We can estimate the relative frequencies of these two types of outcomes using Monte Carlo simulations.', '1011.2501-2-8-6': 'Given the vast parameter space for initial conditions, a complete study of the statistics is beyond the scope of this Letter (but see Naoz et al., in preparation).', '1011.2501-2-8-7': 'However, we can provide a representative example: consider systems where the inner planet was formed in situ at [MATH]AU with zero obliquity (orbit in the stellar equatorial plane) and with some small eccentricity [MATH], while the outer planet has [MATH]AU.', '1011.2501-2-8-8': 'The masses are [MATH] and [MATH].', '1011.2501-2-8-9': 'We draw the eccentricity of the outer orbit from a uniform distribution and the mutual inclination from a distribution uniform in [MATH] between [MATH] and [MATH] (i.e., isotropic among prograde orbits).', '1011.2501-2-8-10': 'For this case we find that, among all HJs that are formed, about [MATH] are in truly retrograde motion (i.e., with respect to the total angular momentum) and about [MATH] are orbiting counter to the stellar spin axis.', '1011.2501-2-8-11': 'Note that the latter fraction is significantly larger than what previous studies have obtained with stellar-mass perturbers (at most [MATH] [CITATION]).', '1011.2501-2-8-12': 'The high observed incidence of planets orbiting counter to the stellar spin axis[CITATION] may suggest that planet-planet secular interactions are an important part of their dynamical history.', '1011.2501-2-9-0': 'Our mechanism requires that two coupled orbits start with a relatively high mutual inclination ([MATH]).', '1011.2501-2-9-1': 'The particular configuration in Figure 2 has a very wide outer orbit similar to those of directly imaged planets such as Fomalhaut b[CITATION] and HR 8799b[CITATION].', '1011.2501-2-9-2': 'In this case the inner Jupiter could have formed in its original location in accordance with the standard core accretion model[CITATION] on a nearly circular orbit.', '1011.2501-2-9-3': 'An alternative path to such a configuration involves strong planet-planet scattering in a closely packed initial system of several giant planets[CITATION].', '1011.2501-2-9-4': 'Independent of any particular planet formation mechanism, we predict that systems with misaligned HJs should also contain a much more distant massive planet or brown dwarf on an inclined orbit.', '1011.2501-2-10-0': 'We would like to thank Dan Fabrycky and Hagai Perets for discussions.', '1011.2501-2-10-1': 'SN acknowledges support from a Gruber Foundation Fellowship and from the National Post Doctoral Award Program for Advancing Women in Science (Weizmann Institute of Science).', '1011.2501-2-10-2': 'Simulations for this project were performed on the HPC cluster fugu funded by an NSF MRI award.', '1011.2501-2-10-3': '[Contributions] SN preformed a numerical calculations with some help from JT.', '1011.2501-2-10-4': 'All authors developed the mathematical model, discussed the physical interpretation of the results and jointly wrote the manuscript.', '1011.2501-2-10-5': '[Competing Interests] The authors declare that they have no competing financial interests.', '1011.2501-2-10-6': '[Correspondence] Correspondence and requests for materials should be addressed to S.N (email: [email protected]).', '1011.2501-2-11-0': 'Supplementary Information Octupole-order Evolution Equations and Angular Momentum Conservation', '1011.2501-2-12-0': 'Our derivation corrects an error in previous Hamiltonian derivations of the secular evolution equations.', '1011.2501-2-13-0': 'We consider a hierarchical triple system consisting of an inner binary ([MATH] and [MATH]) and a third body ([MATH]) in a wider exterior orbit.', '1011.2501-2-13-1': "We describe the system using canonical variables, known as Delaunay's elements, which provide a particularly convenient dynamical description of our three-body system.", '1011.2501-2-13-2': 'The coordinates are chosen to be the mean anomalies, [MATH] and [MATH], the longitudes of ascending nodes, [MATH] and [MATH], and the arguments of periastron, [MATH] and [MATH], where subscripts [MATH] denote the inner and outer orbits, respectively.', '1011.2501-2-13-3': 'Their conjugate momenta are: [EQUATION] where [MATH] is the gravitational constant, and [EQUATION] where [MATH] and [MATH] are the absolute values of the angular momentum vectors ([MATH] and [MATH]), and [MATH] and [MATH] are the z-components of these vectors.', '1011.2501-2-14-0': 'We choose to work in a coordinate system where the total initial angular momentum of the system lies along the [MATH] axis.', '1011.2501-2-14-1': 'The transformation to this coordinate system is known as the elimination of the nodes[MATH]; the [MATH]-[MATH] plane in this coordinate system is known as the invariable plane.', '1011.2501-2-14-2': 'Figure [REF] shows the resulting configuration of the orbits.', '1011.2501-2-14-3': 'We obtain simple relations between [MATH], [MATH], [MATH] and [MATH], using [MATH]: [EQUATION] where the relation for [MATH] comes from setting [MATH] (and similarly for [MATH]).', '1011.2501-2-14-4': 'Because total angular momentum is conserved by the evolution of the system, we must have [MATH], implying that [EQUATION]', '1011.2501-2-14-5': 'The Hamiltonian for the three-body system can be transformed into the form [EQUATION] where [MATH] and [MATH] represent the Keplerian interaction between bodies 1 and 2 and the central body, and [MATH] represents the interaction between body 1 and body 2.', '1011.2501-2-14-6': 'The Kepler Hamiltonians depend only on the momenta [MATH] and [MATH], while the interaction Hamiltonian, [MATH], depends on all the coordinates and momenta.', '1011.2501-2-14-7': 'Due to the rotational symmetry of the problem, [MATH] depends on [MATH] and [MATH] only through the combination [MATH].', '1011.2501-2-14-8': 'Because we are interested in secular effects, we average the Hamiltonian over the coordinates (angles) [MATH] and [MATH], obtaining the secular Hamiltonian [EQUATION] where [EQUATION]', '1011.2501-2-14-9': "For simplicity we first focus on the quadrupole approximation, where the error is more easily shown; it is then straightforward to see its effects at all orders in the hierarchical triple system's secular dynamics expansion.", '1011.2501-2-14-10': 'The quadrupole Hamiltonian results from expanding [MATH] to second order footnote in [MATH]: [EQUATION]', '1011.2501-2-14-11': 'The resulting quadrupole-order Hamiltonian, [MATH], depends only on the coordinates [MATH], [MATH], and [MATH], with the latter two appearing only in the combination [MATH]: [EQUATION]', '1011.2501-2-14-12': 'Previous calculations[MATH] eliminated [MATH] and [MATH] from the Hamiltonian using eq. ([REF]), obtaining a quadrupole Hamiltonian that depends only on [MATH].', '1011.2501-2-14-13': 'But, this is incorrect!', '1011.2501-2-14-14': 'Such a Hamiltonian would imply that all quantities in eq. ([REF]) are constant except [MATH], i.e. that eq. ([REF]) is incorrect.', '1011.2501-2-14-15': 'Thus the previously used formalism did not conserve angular momentum.', '1011.2501-2-14-16': 'The initial Hamiltonian is spherically symmetric, and therefore does conserve angular momentum; the correct quadrupole Hamiltonian does as well.', '1011.2501-2-14-17': 'Because the correct quadrupole Hamiltonian depends on [MATH] and [MATH] through the combination [MATH], we have [EQUATION] or [EQUATION]', '1011.2501-2-14-18': 'The mathematical error affects all orders in secular perturbations.', '1011.2501-2-14-19': 'The independence of the secular quadrupole Hamiltonian on [MATH] was the source[MATH] of the famous relation [MATH].', '1011.2501-2-14-20': 'In the correct derivation, this relation does not always hold.', '1011.2501-2-14-21': 'However, in a certain limit, it does.', '1011.2501-2-14-22': 'From eq. ([REF]), we see that [EQUATION]', '1011.2501-2-14-23': 'When [MATH], we have [EQUATION]', '1011.2501-2-14-24': 'At the quadrupole level [MATH] is independent of [MATH], so [MATH], implying [EQUATION] when [MATH].', '1011.2501-2-14-25': 'This is precisely the limit considered in previous works[MATH], so their conclusion that [MATH] is correct (though not for the reason they claim), but the limit where [MATH] is not sufficient for our work.', '1011.2501-2-15-0': 'In some later studies, the assumption that [MATH] was built into the calculations of secular evolution for various astrophysical systems[MATH], even when the condition [MATH] was not satisfied.', '1011.2501-2-15-1': 'Moreover many previous studies simply set [MATH], which is repeating the same error.', '1011.2501-2-15-2': 'In fact, given the mutual inclination [MATH], the inner and outer inclinations [MATH] and [MATH] are set by the conservation of total angular momentum: [EQUATION]', '1011.2501-2-15-3': 'Tidal Friction', '1011.2501-2-16-0': 'We adopt the tidal evolution equations of Ref. 16, which are based on the equilibrium tide model of Ref. 32.', '1011.2501-2-16-1': 'The complete equations can be found in Ref. 2, eqs A1-A5.', '1011.2501-2-16-2': 'Following their approach (see their eq. A10) we set the tidal quality factors [MATH] [see also Ref. 33].', '1011.2501-2-16-3': 'This means that the viscous times of the star and planet remain constant; the representative values we adopt here are [MATH] yr for the star and [MATH] yr for the planet, which correspond to [MATH] and [MATH], respectively, for a 1-day period.', '1011.2501-2-17-0': 'Comparison to Observations', '1011.2501-2-18-0': "The observable parameter from the Rossiter-McLaughlin effect is the projected angle between the star's spin and the orbital angular momentum (the projected obliquity)[MATH].", '1011.2501-2-18-1': 'Here instead we focus on the true angle between the orbital angular momentum of the inner planet and the total angular momentum.', '1011.2501-2-18-2': 'Projection effects can cause these two quantities to differ in magnitude, or even sign.', '1011.2501-2-19-0': 'Moreover, several mechanisms have been proposed in the literature that could, under certain assumptions, directly affect the spin axis of the star.', '1011.2501-2-19-1': 'These mechanisms can re-align the stellar spin axis through tidal interactions with either a slowly spinning star[MATH] or with the outer convective layer of a sufficiently cold star[MATH].', '1011.2501-2-19-2': 'Additionally, a magnetic interaction between the star and the protoplanetary disk could also lead to misalignment between the stellar spin and the disk[MATH].', '1011.2501-2-20-0': 'These effects can potentially complicate the interpretation of any specific observation.', '1011.2501-2-20-1': 'Nevertheless, if hot Jupiters are produced by the simple mechanism described here, many of their orbits should indeed be observed with large projected obliquities.'}

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[]

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['1011.2501-2-11-0', '1011.2501-2-15-3', '1011.2501-2-17-0', '1011.2501-3-11-0', '1011.2501-3-15-3', '1011.2501-3-17-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1011.2501

{'1011.2501-3-0-0': 'About 25 per cent of ‘hot Jupiters’ (extrasolar Jovian-mass planets with close-in orbits) are actually orbiting counter to the spin direction of the star[CITATION].', '1011.2501-3-0-1': 'Perturbations from a distant binary star companion[CITATION] can produce high inclinations, but cannot explain orbits that are retrograde with respect to the total angular momentum of the system.', '1011.2501-3-0-2': 'Such orbits in a stellar context can be produced through secular (that is, long term) perturbations in hierarchical triple-star systems.', '1011.2501-3-0-3': 'Here we report a similar application to planetary bodies, including both the key octupole-order effects and tidal friction, and find that it can produce hot Jupiters in orbits that are retrograde with respect to the total angular momentum.', '1011.2501-3-0-4': 'With distant stellar mass perturbers such an outcome is not possible[CITATION].', '1011.2501-3-0-5': "With planetary perturbers the inner orbit's angular momentum component parallel to the total angular momentum need not be constant[CITATION].", '1011.2501-3-0-6': 'In fact, as we show here, it can even change sign, leading to a retrograde orbit.', '1011.2501-3-0-7': 'A brief excursion to very high eccentricity during the chaotic evolution of the inner orbit can then lead to rapid capture, forming a retrograde hot Jupiter.', '1011.2501-3-1-0': 'Despite many attempts[CITATION], there is no model that can account for all the properties of the known hot Jupiter (HJ) systems.', '1011.2501-3-1-1': 'One model suggests that HJs formed far away from the star and slowly spiraled in, losing angular momentum and orbital energy to the protoplanetary disk[CITATION].', '1011.2501-3-1-2': 'This "migration" process should produce planets with low orbital inclinations and eccentricities.', '1011.2501-3-1-3': 'However, many HJs are observed to be on orbits with high eccentricities, and misaligned with the spin direction of the star (as measured through the Rossiter-McLaughlin effect[CITATION]) and some of these ([MATH] out of [MATH]) even appear to be orbiting counter to the spin of the star.', '1011.2501-3-1-4': 'In a second model, secular perturbations from a distant binary star companion can produce increases in the eccentricity and inclination of a planetary orbit[CITATION].', '1011.2501-3-1-5': "During the evolution to high eccentricity, tidal dissipation near pericenter can force the planet's orbit to decay, potentially forming a misaligned HJ[CITATION].", '1011.2501-3-1-6': 'Recently, secular chaos involving several planets has also been proposed as a way to form HJs on eccentric and misaligned orbits[CITATION].', '1011.2501-3-1-7': 'A different class of models to produce a tilted orbit is via planet-planet scattering[CITATION], possibly combined with other perturbers and tidal friction[CITATION].', '1011.2501-3-1-8': 'In such models the initial configuration is a densly-packed system of planets and the final tilted orbit is a result of dynamical scattering among the planets, in contrast to the secular interactions we study here.', '1011.2501-3-2-0': 'In our general treatment of secular interactions between two orbiting bodies we allow for the magnitude and orientation of both orbital angular momenta to change (see Figure 1).', '1011.2501-3-2-1': 'The outer body (here either a planet or a brown-dwarf) gravitationally perturbs the inner planet on time scales long compared to the orbital period (i.e., we consider the secular evolution of the system).', '1011.2501-3-2-2': 'We define the orientation of the inner orbit with respect to the invariable plane of the system (perpendicular to the total angular momentum): a prograde (retrograde) orbit has [MATH]), where [MATH] is the inclination of the inner orbit with respect to the total angular momentum vector.', '1011.2501-3-2-3': 'Note that the word "retrograde" is also used in the literature to indicate orbital motion counter to the stellar spin.', '1011.2501-3-2-4': 'The directly observed parameter is actually the projected angle between the spin axis of the star and the orbital angular momentum of a HJ.', '1011.2501-3-2-5': 'Our proposed mechanism can produce HJs that are "retrograde" both with respect to the stellar spin and with respect to the total angular momentum.', '1011.2501-3-2-6': 'By contrast, a distant stellar companion can only succeed in the former.', '1011.2501-3-2-7': 'See the online Supplementary Information for more details; henceforth we will use the term "retrograde" only to indicate an orbit with [MATH] as define above.', '1011.2501-3-3-0': 'We assume a hierarchical configuration, with the outer perturber on a much wider orbit than the inner one.', '1011.2501-3-3-1': 'In the secular approximation the orbits may change shape and orientation but the semi-major axes are strictly conserved in the absence of tidal dissipation[CITATION].', '1011.2501-3-3-2': 'In particular, the Kozai-Lidov mechanism[CITATION] produces large-amplitude oscillations of the eccentricity and inclination when the initial relative inclination between the inner and outer orbits is sufficiently large ( [MATH]).', '1011.2501-3-4-0': 'We have derived the secular evolution equations to octupole order using Hamiltonian perturbation theory[CITATION].', '1011.2501-3-4-1': 'In contrast to previous derivations of "Kozai-type" evolution, our treatment allows for changes in the [MATH]-components of the orbital angular momenta (i.e., the components along the total angular momentum) [MATH] and [MATH] (see Supplementary Information).', '1011.2501-3-4-2': 'The octupole-order equations allow us to calculate the evolution of systems with more closely coupled orbits and with planetary-mass perturbers.', '1011.2501-3-4-3': 'The octupole-level terms can give rise to fluctuations in the eccentricity maxima to arbitrarily high values[CITATION], in contrast to the regular evolution in the quadrupole potential[CITATION], where the amplitude of eccentricity oscillations is constant.', '1011.2501-3-5-0': 'Many previous studies of secular perturbations in hierarchical triples considered a stellar-mass perturber, for which [MATH] is very nearly constant[CITATION].', '1011.2501-3-5-1': 'Moreover, the assumption that [MATH] is constant has been built into previous derivations[CITATION].', '1011.2501-3-5-2': 'However, this assumption is only valid as long as [MATH], which is not the case in comparable-mass systems (e.g., with two planets).', '1011.2501-3-5-3': 'Unfortunately, an immediate consequence of this assumption is that an orbit that is prograde relative to the total angular momentum always remains prograde.', '1011.2501-3-5-4': 'Figure [REF] shows the evolution of a representative system (here without tidal effects for simplicity): the inner planet oscillates between prograde and retrograde orbits (with respect to the total angular momentum) as angular momentum flows back and forth between the two orbits.', '1011.2501-3-6-0': 'Previous calculations of planet migration through "Kozai cycles with tidal friction"[CITATION] produced a slow, gradual spiral-in of the inner planet.', '1011.2501-3-6-1': 'Instead, our treatment shows that the eccentricity can occasionally reach a much higher value than in the regular "Kozai cycles" calculated to quadrupole order.', '1011.2501-3-6-2': 'Thus, the pericenter distance will occasionally shrink on a short time scale (compared to the Kozai period), and the planet can then suddenly be tidally captured by the star.', '1011.2501-3-6-3': 'We propose to call this "Kozai capture."', '1011.2501-3-7-0': 'Kozai capture provides a new way to form HJs.', '1011.2501-3-7-1': 'If the capture happens after the inner orbit has flipped the HJ will appear in a retrograde orbit.', '1011.2501-3-7-2': 'This is illustrated in Figure 2.', '1011.2501-3-7-3': 'During the evolution of the system the inner orbit shrinks in steps (Fig. 2c) whenever the dissipation becomes significant, i.e., near unusually high eccentricity maxima.', '1011.2501-3-7-4': 'The inner orbit can then eventually become tidally circularized.', '1011.2501-3-7-5': 'This happens near the end of the evolution, on a very short time scale (see Fig. 2, right panels).', '1011.2501-3-7-6': 'In this final step, the inner orbit completely and quickly decouples from the outer perturber, and the orbital angular momenta then become constant.', '1011.2501-3-7-7': 'Therefore, the final semi-major axis for the HJ is [MATH], where [MATH] is the pericenter distance at the beginning of the capture phase[CITATION].', '1011.2501-3-8-0': 'The same type of evolution shown in Figure 2 is seen with a broad range of initial conditions.', '1011.2501-3-8-1': 'There are two main routes to forming a HJ through the dynamical evolution of the systems we consider here.', '1011.2501-3-8-2': 'In the first, tidal friction slowly damps the growing eccentricity of the inner planet, resulting in circularized, prograde HJs.', '1011.2501-3-8-3': 'In the second, a sudden high-eccentricity spike in the orbital evolution of the inner planet is accompanied by a flip of its orbit.', '1011.2501-3-8-4': 'The planet is then quickly circularized into a retrograde short-period orbit.', '1011.2501-3-8-5': 'We can estimate the relative frequencies of these two types of outcomes using Monte Carlo simulations.', '1011.2501-3-8-6': 'Given the vast parameter space for initial conditions, a complete study of the statistics is beyond the scope of this Letter (but see Naoz et al., in preparation).', '1011.2501-3-8-7': 'However, we can provide a representative example: consider systems where the inner planet was formed in situ at [MATH]AU with zero obliquity (orbit in the stellar equatorial plane) and with some small eccentricity [MATH], while the outer planet has [MATH]AU.', '1011.2501-3-8-8': 'The masses are [MATH] and [MATH].', '1011.2501-3-8-9': 'We draw the eccentricity of the outer orbit from a uniform distribution and the mutual inclination from a distribution uniform in [MATH] between [MATH] and [MATH] (i.e., isotropic among prograde orbits).', '1011.2501-3-8-10': 'For this case we find that, among all HJs that are formed, about [MATH] are in truly retrograde motion (i.e., with respect to the total angular momentum) and about [MATH] are orbiting counter to the stellar spin direction.', '1011.2501-3-8-11': 'Note that the latter fraction is significantly larger than what previous studies have obtained with stellar-mass perturbers (at most [MATH] [CITATION]).', '1011.2501-3-8-12': 'The high observed incidence of planets orbiting counter to the stellar spin direction[CITATION] may suggest that planet-planet secular interactions are an important part of their dynamical history.', '1011.2501-3-9-0': 'Our mechanism requires that two coupled orbits start with a relatively high mutual inclination ([MATH]).', '1011.2501-3-9-1': 'The particular configuration in Figure 2 has a very wide outer orbit similar to those of directly imaged planets such as Fomalhaut b[CITATION] and HR 8799b[CITATION].', '1011.2501-3-9-2': 'In this case the inner Jupiter could have formed in its original location in accordance with the standard core accretion model[CITATION] on a nearly circular orbit.', '1011.2501-3-9-3': 'An alternative path to such a configuration involves strong planet-planet scattering in a closely packed initial system of several giant planets[CITATION].', '1011.2501-3-9-4': 'Independent of any particular planet formation mechanism, we predict that systems with misaligned HJs should also contain a much more distant massive planet or brown dwarf on an inclined orbit.', '1011.2501-3-10-0': 'We would like to thank Dan Fabrycky and Hagai Perets for discussions.', '1011.2501-3-10-1': 'SN acknowledges support from a Gruber Foundation Fellowship and from the National Post Doctoral Award Program for Advancing Women in Science (Weizmann Institute of Science).', '1011.2501-3-10-2': 'Simulations for this project were performed on the HPC cluster fugu funded by an NSF MRI award.', '1011.2501-3-10-3': '[Contributions] SN preformed a numerical calculations with some help from JT.', '1011.2501-3-10-4': 'All authors developed the mathematical model, discussed the physical interpretation of the results and jointly wrote the manuscript.', '1011.2501-3-10-5': '[Competing Interests] The authors declare that they have no competing financial interests.', '1011.2501-3-10-6': '[Correspondence] Correspondence and requests for materials should be addressed to S.N (email: [email protected]).', '1011.2501-3-11-0': 'Supplementary Information Octupole-order Evolution Equations and Angular Momentum Conservation', '1011.2501-3-12-0': 'Our derivation corrects an error in previous Hamiltonian derivations of the secular evolution equations.', '1011.2501-3-13-0': 'We consider a hierarchical triple system consisting of an inner binary ([MATH] and [MATH]) and a third body ([MATH]) in a wider exterior orbit.', '1011.2501-3-13-1': "We describe the system using canonical variables, known as Delaunay's elements, which provide a particularly convenient dynamical description of our three-body system.", '1011.2501-3-13-2': 'The coordinates are chosen to be the mean anomalies, [MATH] and [MATH], the longitudes of ascending nodes, [MATH] and [MATH], and the arguments of periastron, [MATH] and [MATH], where subscripts [MATH] denote the inner and outer orbits, respectively.', '1011.2501-3-13-3': 'Their conjugate momenta are: [EQUATION] where [MATH] is the gravitational constant, and [EQUATION] where [MATH] and [MATH] are the absolute values of the angular momentum vectors ([MATH] and [MATH]), and [MATH] and [MATH] are the z-components of these vectors.', '1011.2501-3-14-0': 'We choose to work in a coordinate system where the total initial angular momentum of the system lies along the [MATH] axis.', '1011.2501-3-14-1': 'The transformation to this coordinate system is known as the elimination of the nodes[MATH]; the [MATH]-[MATH] plane in this coordinate system is known as the invariable plane.', '1011.2501-3-14-2': 'Figure [REF] shows the resulting configuration of the orbits.', '1011.2501-3-14-3': 'We obtain simple relations between [MATH], [MATH], [MATH] and [MATH], using [MATH]: [EQUATION] where the relation for [MATH] comes from setting [MATH] (and similarly for [MATH]).', '1011.2501-3-14-4': 'Because total angular momentum is conserved by the evolution of the system, we must have [MATH], implying that [EQUATION]', '1011.2501-3-14-5': 'The Hamiltonian for the three-body system can be transformed into the form [EQUATION] where [MATH] and [MATH] represent the Keplerian interaction between bodies 1 and 2 and the central body, and [MATH] represents the interaction between body 1 and body 2.', '1011.2501-3-14-6': 'The Kepler Hamiltonians depend only on the momenta [MATH] and [MATH], while the interaction Hamiltonian, [MATH], depends on all the coordinates and momenta.', '1011.2501-3-14-7': 'Due to the rotational symmetry of the problem, [MATH] depends on [MATH] and [MATH] only through the combination [MATH].', '1011.2501-3-14-8': 'Because we are interested in secular effects, we average the Hamiltonian over the coordinates (angles) [MATH] and [MATH], obtaining the secular Hamiltonian [EQUATION] where [EQUATION]', '1011.2501-3-14-9': "For simplicity we first focus on the quadrupole approximation, where the error is more easily shown; it is then straightforward to see its effects at all orders in the hierarchical triple system's secular dynamics expansion.", '1011.2501-3-14-10': 'The quadrupole Hamiltonian results from expanding [MATH] to second order footnote in [MATH]: [EQUATION]', '1011.2501-3-14-11': 'The resulting quadrupole-order Hamiltonian, [MATH], depends only on the coordinates [MATH], [MATH], and [MATH], with the latter two appearing only in the combination [MATH]: [EQUATION]', '1011.2501-3-14-12': 'Previous calculations[MATH] eliminated [MATH] and [MATH] from the Hamiltonian using eq. ([REF]), obtaining a quadrupole Hamiltonian that depends only on [MATH].', '1011.2501-3-14-13': 'But, this is incorrect!', '1011.2501-3-14-14': 'Such a Hamiltonian would imply that all quantities in eq. ([REF]) are constant except [MATH], i.e. that eq. ([REF]) is incorrect.', '1011.2501-3-14-15': 'Thus the previously used formalism did not conserve angular momentum.', '1011.2501-3-14-16': 'The initial Hamiltonian is spherically symmetric, and therefore does conserve angular momentum; the correct quadrupole Hamiltonian does as well.', '1011.2501-3-14-17': 'Because the correct quadrupole Hamiltonian depends on [MATH] and [MATH] through the combination [MATH], we have [EQUATION] or [EQUATION]', '1011.2501-3-14-18': 'The mathematical error affects all orders in secular perturbations.', '1011.2501-3-14-19': 'The independence of the secular quadrupole Hamiltonian on [MATH] was the source[MATH] of the famous relation [MATH].', '1011.2501-3-14-20': 'In the correct derivation, this relation does not always hold.', '1011.2501-3-14-21': 'However, in a certain limit, it does.', '1011.2501-3-14-22': 'From eq. ([REF]), we see that [EQUATION]', '1011.2501-3-14-23': 'When [MATH], we have [EQUATION]', '1011.2501-3-14-24': 'At the quadrupole level [MATH] is independent of [MATH], so [MATH], implying [EQUATION] when [MATH].', '1011.2501-3-14-25': 'This is precisely the limit considered in previous works[MATH], so their conclusion that [MATH] is correct (though not for the reason they claim), but the limit where [MATH] is not sufficient for our work.', '1011.2501-3-15-0': 'In some later studies, the assumption that [MATH] was built into the calculations of secular evolution for various astrophysical systems[MATH], even when the condition [MATH] was not satisfied.', '1011.2501-3-15-1': 'Moreover many previous studies simply set [MATH], which is repeating the same error.', '1011.2501-3-15-2': 'In fact, given the mutual inclination [MATH], the inner and outer inclinations [MATH] and [MATH] are set by the conservation of total angular momentum: [EQUATION]', '1011.2501-3-15-3': 'Tidal Friction', '1011.2501-3-16-0': 'We adopt the tidal evolution equations of Ref. 16, which are based on the equilibrium tide model of Ref. 32.', '1011.2501-3-16-1': 'The complete equations can be found in Ref. 2, eqs A1-A5.', '1011.2501-3-16-2': 'Following their approach (see their eq. A10) we set the tidal quality factors [MATH] [see also Ref. 33].', '1011.2501-3-16-3': 'This means that the viscous times of the star and planet remain constant; the representative values we adopt here are [MATH] yr for the star and [MATH] yr for the planet, which correspond to [MATH] and [MATH], respectively, for a 1-day period.', '1011.2501-3-17-0': 'Comparison to Observations', '1011.2501-3-18-0': "The observable parameter from the Rossiter-McLaughlin effect is the projected angle between the star's spin and the orbital angular momentum (the projected obliquity)[MATH].", '1011.2501-3-18-1': 'Here instead we focus on the true angle between the orbital angular momentum of the inner planet and the total angular momentum.', '1011.2501-3-18-2': 'Projection effects can cause these two quantities to differ in magnitude, or even sign.', '1011.2501-3-19-0': 'Moreover, several mechanisms have been proposed in the literature that could, under certain assumptions, directly affect the spin axis of the star.', '1011.2501-3-19-1': 'These mechanisms can re-align the stellar spin axis through tidal interactions with either a slowly spinning star[MATH] or with the outer convective layer of a sufficiently cold star[MATH].', '1011.2501-3-19-2': 'Additionally, a magnetic interaction between the star and the protoplanetary disk could also lead to misalignment between the stellar spin and the disk[MATH].', '1011.2501-3-20-0': 'These effects can potentially complicate the interpretation of any specific observation.', '1011.2501-3-20-1': 'Nevertheless, if hot Jupiters are produced by the simple mechanism described here, many of their orbits should indeed be observed with large projected obliquities.'}

1408.1779

{'1408.1779-1-0-0': 'Accretion of fields by black holes is a subject of great interest in physics.', '1408.1779-1-0-1': 'It is known that accretion plays a fundamental role in active galactic nuclei and in the evolution of black holes.', '1408.1779-1-0-2': 'Accretion of fundamental fields is often related to the study of absorption cross section.', '1408.1779-1-0-3': 'Basically all black holes which absorption of fields has been studied so far present singularities.', '1408.1779-1-0-4': 'However, even within general relativity, it is possible to construct regular black holes: objects with event horizons but without singularities.', '1408.1779-1-0-5': 'Many physically motivated regular black hole solutions have been proposed in the past years, demanding the understanding of their absorption properties.', '1408.1779-1-0-6': 'We study the absorption of planar massless scalar waves by Bardeen regular black holes.', '1408.1779-1-0-7': 'We compare the absorption cross section of Bardeen and Reissner-Nordstrom black holes, showing that the former always have a bigger absorption cross section for fixed values of the field frequency and of the normalized black hole charge.', '1408.1779-1-0-8': 'We also show that it is possible for a Bardeen black hole to have the same high-frequency limit of a Reissner-Nordstrom black hole.', '1408.1779-1-0-9': 'Our results suggest that, in mid-to-high-frequency regimes, regular black holes can have compatible properties with black holes with singularities, as far as absorption is concerned.', '1408.1779-1-1-0': '# Introduction', '1408.1779-1-2-0': 'One of the most intriguing predictions of the general relativity (GR) is the existence of black holes (BHs).', '1408.1779-1-2-1': 'BHs became a paradigm in physics, and are believed to populate the galaxies [CITATION].', '1408.1779-1-2-2': 'Within standard GR, black holes are simple objects, described only by their mass, angular momentum and charge [CITATION].', '1408.1779-1-2-3': 'However, standard black holes suffer from one of the main problems of GR: the presence of singularities.', '1408.1779-1-2-4': 'Our physical knowledge breaks down at singularities.', '1408.1779-1-2-5': 'Although generally hidden by a horizon, and protected by the Penrose conjecture [CITATION] (see also [CITATION] for a review), singularities are expected to exist within GR, according to the singularity theorems developed by Hawking and collaborators [CITATION].', '1408.1779-1-3-0': 'Singularities are expected to be better understood with an improved theory of gravity (whether an extension or a modification of GR) [CITATION].', '1408.1779-1-3-1': 'Notwithstanding, within GR it is possible to obtain BH solutions without singularities.', '1408.1779-1-3-2': 'Bardeen presented a BH solution without singularities that satisfies the weak energy condition in GR [CITATION].', '1408.1779-1-3-3': "Although Bardeen's solution has its theoretical motivation in the studies of BH spacetimes with no singularities, a stronger physical motivation for it was missing until it was shown that the Bardeen's BH is a solution of GR with a nonlinear magnetic monopole, i.e., a solution of the Einstein's equations coupled to a nonlinear electrodynamics [CITATION].", '1408.1779-1-3-4': 'Apart from this, further works with other physically motivated regular BHs can be found in the literature (see, e.g., [CITATION]).', '1408.1779-1-4-0': 'One way to test the physics of BHs is analyzing test fields around them.', '1408.1779-1-4-1': 'In this context, there are the quasinormal modes: natural oscillation frequencies of the fields with physically motivated boundary conditions [CITATION].', '1408.1779-1-4-2': 'An extensive survey of quasinormal modes of test charged scalar fields around different types of regular BHs was presented in Ref. [CITATION].', '1408.1779-1-4-3': 'Quasinormal modes of the Dirac field were investigated in Ref. [CITATION] and of the massive scalar fields in Hayward regular BH in Ref. [CITATION].', '1408.1779-1-4-4': 'Quasinormal modes have an interesting relation with scattering processes in BH spacetimes.', '1408.1779-1-4-5': 'This relation can be seen, for instance, in the scattering of Gaussian packets by BHs [CITATION].', '1408.1779-1-5-0': 'Another important aspect of BHs is how they absorb matter and fields around them, i.e., their accretion rate.', '1408.1779-1-5-1': 'Accretion has an important role in the phenomenology of active galactic nuclei, and can be considered as an important agent to the mass growth of their BH hosts (see, e.g., Refs. [CITATION] and the references therein).', '1408.1779-1-5-2': 'Along more than 45 years, absorption of scalar fields has been studied extensively in many BH scenarios (see, e.g., Refs. [CITATION] and the references therein).', '1408.1779-1-5-3': 'The initial field configuration is usually taken to be plane waves at infinity and the problem is often directed to compute the absorption cross section of the field.', '1408.1779-1-5-4': 'Also, in the classical (high-frequency) limit, absorption cross sections are directly related with the shadows of BHs [CITATION].', '1408.1779-1-5-5': 'Moreover, the case of planar waves absorption have many features in common with the case of accretion of a fluid moving with constant velocity towards a BH (see, e.g., Ref. [CITATION]), which turns out to be important in the phenomenology of extreme mass-ratio inspirals [CITATION].', '1408.1779-1-6-0': 'In this paper we address the problem of how regular BHs absorb fields, focusing in the analysis of the absorption cross section of planar massless scalar waves by a Bardeen regular BH.', '1408.1779-1-6-1': 'Generically, the line element of spherically symmetric BH spacetimes can be written in the standard spherical coordinate system as: [EQUATION] where the function [MATH] depends on the particular BH under consideration.', '1408.1779-1-6-2': 'As we shall see, the Bardeen BH has a structure very similar to that of a standard electrically charged BH within GR, i.e., of a Reissner-Nordstrom (RN) BH.', '1408.1779-1-6-3': 'Because of that, we shall compare our results with the RN BH ones [CITATION].', '1408.1779-1-7-0': 'The remainder of this paper is organized as follows.', '1408.1779-1-7-1': 'In Sec. [REF] we review some aspects of the Bardeen regular BHs.', '1408.1779-1-7-2': 'In Sec. [REF] we revisit the main aspects of the absorption cross section of planar massless scalar waves in spherically symmetric BH spacetimes.', '1408.1779-1-7-3': 'We also present the results in the low- and high-frequency regimes for the massless scalar absorption cross section of Bardeen BHs.', '1408.1779-1-7-4': 'In Sec. [REF] we exhibit a selection of our numerical results.', '1408.1779-1-7-5': 'We compare our results for the Bardeen regular BH with the results for the RN BH.', '1408.1779-1-7-6': 'Also, we discuss the possibility of having a Bardeen BH with a similar absorption cross section of a RN BH.', '1408.1779-1-7-7': 'Throughout the paper we use natural units, for which [MATH].', '1408.1779-1-8-0': '# Bardeen regular black holes', '1408.1779-1-9-0': 'As mentioned in the Introduction, the Bardeen BH was one of the first regular BH solutions presented in the literature [CITATION].', '1408.1779-1-9-1': 'Later, it received the physical interpretation of a BH with a nonlinear magnetic monopole [CITATION].', '1408.1779-1-9-2': 'Nonlinear electrodynamics theories within GR are generically described by the action [EQUATION] where [MATH] is the Ricci scalar, [MATH] is the Lagrangian of the electromagnetic field, [MATH], with [MATH] being the standard electromagnetic field strength, and [MATH] is the determinant of the metric [MATH].', '1408.1779-1-9-3': 'For the theory that generates the Bardeen regular BH, we have that [EQUATION] where [MATH], [MATH] is the magnetic charge and [MATH] is the mass of the configuration [CITATION].', '1408.1779-1-9-4': 'The line element of the Bardeen BH is given by Eq. [REF], with [EQUATION]', '1408.1779-1-9-5': 'The Bardeen solution has a structure similar to the RN spacetime, presenting two horizons up to some value of the BH charge.', '1408.1779-1-9-6': 'For [MATH], the two horizons coincide and we have the so-called extremal BH.', '1408.1779-1-9-7': 'In this paper, we shall consider [MATH].', '1408.1779-1-10-0': 'For later comparison, it is instructive to mention explicitly the RN solution.', '1408.1779-1-10-1': 'The line element of the RN spacetime is given by Eq. [REF], with [EQUATION] where, in this case, [MATH] is the electric charge of the BH.', '1408.1779-1-10-2': 'The extreme case of the RN BH is given by [MATH].', '1408.1779-1-10-3': 'Note that we are using the same symbol ([MATH]) for both magnetic (Bardeen BH) and electric (RN BH) charge.', '1408.1779-1-10-4': 'In order to better compare both spacetimes, we shall present our results in terms of the normalized charge [MATH].', '1408.1779-1-11-0': '# Absorption cross section', '1408.1779-1-12-0': '## Partial-waves approach', '1408.1779-1-13-0': 'A massless scalar field [MATH] is described by the Klein-Gordon equation, namely [EQUATION]', '1408.1779-1-13-1': 'Here we are considering a minimally coupled scalar field.', '1408.1779-1-14-0': 'A monochromatic wave with frequency [MATH] in a spherically symmetric background can be written as [EQUATION] where [MATH] are the standard scalar spherical harmonics.', '1408.1779-1-14-1': 'Substituting the expansion [REF] in Eq. [REF], and using the properties of the spherical harmonics, we get the following radial equation for [MATH]: [EQUATION] in which [MATH] is the tortoise coordinate, defined through [MATH], and [EQUATION] is the scalar field potential.', '1408.1779-1-14-2': 'Plots of [MATH] for Bardeen and Schwarzschild BHs are shown in Fig. [REF].', '1408.1779-1-14-3': 'The scalar field potential [MATH] is localized, in the sense that it goes to zero at the event horizon and at infinity [CITATION].', '1408.1779-1-14-4': 'We are interested in a solution that represents a wave coming from the past null infinity.', '1408.1779-1-14-5': 'Such a solution can be written using the so-called [MATH] modes, i.e. [EQUATION] with [EQUATION] where the coefficients [MATH] and [MATH] are obtained by requiring the functions [MATH] and [MATH] to be solutions of Eq. [REF] far from the BH and close to the event horizon, respectively.', '1408.1779-1-14-6': '[MATH] and [MATH] are the reflection and transmission coefficients, respectively, and are related through [EQUATION]', '1408.1779-1-14-7': 'Using the solution [REF], the absorption cross section of planar massless scalar waves can be written as [EQUATION] with [MATH] being the partial absorption cross sections, given by [EQUATION]', '1408.1779-1-15-0': '## Low- and high-frequency limits', '1408.1779-1-16-0': 'In the low-frequency regime, it has been proven that the absorption cross section of massless scalar fields by static BHs [CITATION], as well as stationary BHs [CITATION], tends to the area of the BH horizon.', '1408.1779-1-16-1': 'Our numerical results agree remarkably well with this low-frequency limit.', '1408.1779-1-16-2': 'In Fig. [REF] we plot the area of the event horizon for the Bardeen and RN BHs, as function of the normalized charge.', '1408.1779-1-16-3': 'We can see that the event horizon area of a Bardeen BH is bigger than the corresponding one of the RN BH with the same normalized charge.', '1408.1779-1-17-0': 'In the high-frequency limit, a massless scalar wave can be described by the propagation of a null vector, which follows a null geodesic.', '1408.1779-1-17-1': 'Therefore, in this limit we can consider the classical capture cross section of null geodesics to describe the absorption cross section of massless fields.', '1408.1779-1-18-0': 'Geodesics around Bardeen BHs were also studied in [CITATION].', '1408.1779-1-18-1': 'Here we consider null geodesics in spherically symmetric BHs.', '1408.1779-1-18-2': 'Their motion is described by the Lagrangian [MATH], that satisfies [EQUATION] in which we consider, without loss of generality, the motion in the plane [MATH].', '1408.1779-1-18-3': 'The overdot indicates derivative with respect to the affine parameter of the curve.', '1408.1779-1-18-4': 'Considering the conserved quantities, namely the energy [MATH] and angular momentum [MATH] (see, e.g., Ref. [CITATION]), the equation of motion becomes: [EQUATION] which can be regarded as an energy balance equation with the effective potential [EQUATION]', '1408.1779-1-18-5': 'The high-frequency limit of the absorption cross section, also called geometric cross section, [MATH], is then found by computing the classical capture radius of light rays in the spacetime under investigation.', '1408.1779-1-18-6': 'For spherically symmetric spacetimes, the null geodesic radius [MATH] is obtained through [MATH], with the prime denoting derivative with respect to [MATH].', '1408.1779-1-18-7': 'The critical impact parameter is given by [MATH], with [MATH] being characteristic of the null circular geodesic.', '1408.1779-1-18-8': 'Therefore, we have [EQUATION] and [EQUATION]', '1408.1779-1-18-9': 'With Eq. [REF] one finds the value of [MATH], and by substituting this value in Eq. [REF] one finds the capture (or geometric) cross section [MATH].', '1408.1779-1-19-0': 'In Fig. [REF] we compare the capture cross section of the Bardeen BH with the RN BH case, as a function of the normalized charge.', '1408.1779-1-19-1': 'In general, a Bardeen BH have a bigger capture cross section, compared with the RN BH with the same value of [MATH].', '1408.1779-1-20-0': 'An improvement of the high-frequency approximation to compute the absorption cross section for spherically symmetric BHs was proposed in Ref. [CITATION].', '1408.1779-1-20-1': 'It was shown that the oscillatory part of the absorption cross section in the eikonal limit can be written as [EQUATION] where [MATH], with [MATH] being the Lyapunov exponent of the null geodesic [CITATION], and [MATH] being the angular velocity of the null geodesic.', '1408.1779-1-20-2': 'Therefore, we can write the high-frequency absorption cross section as [EQUATION]', '1408.1779-1-20-3': 'Equation [REF] is usually refered in the literature as the sinc approximation.', '1408.1779-1-20-4': 'In Fig. [REF] we compare the results obtained through Eq. [REF] with the full numerical computation of the absorption cross section, given by Eq. [REF].', '1408.1779-1-20-5': 'It is interesting to note that, although Eq. [REF] is obtained within the assumption of high frequencies, it is still a very good approximation for intermediate frequency values.', '1408.1779-1-21-0': '# Results', '1408.1779-1-22-0': 'We have computed numerically the absorption cross sections of planar massless scalar waves impinging on Bardeen BHs.', '1408.1779-1-22-1': 'In this section we show a selection of our results.', '1408.1779-1-23-0': 'In Fig. [REF] we present the partial absorption cross section [given by Eq. [REF]] for [MATH] and [MATH] and for different values of [MATH].', '1408.1779-1-23-1': 'We see that for [MATH] the limit [MATH] results in [MATH], in agreement with the result mentioned in Sec. [REF].', '1408.1779-1-24-0': 'In Fig. [REF] we present the total absorption cross section [given by Eq. [REF]] in the Bardeen BH case, for [MATH] and [MATH], as well as in the Schwarzschild BH case.', '1408.1779-1-24-1': 'The horizontal dotted lines are the high-frequency limits in each case.', '1408.1779-1-24-2': 'We see that the increasing of the monopole charge implies in a decreasing of the absorption cross section, in agreement with the increasing of the scattering potential (cf. Fig. [REF]), as well as with the decreasing of the horizon area (cf. Fig. [REF]).', '1408.1779-1-24-3': 'The sum of the partial absorption cross sections generates the oscillatory profile shown in the plots of Fig. [REF].', '1408.1779-1-25-0': 'In Fig. [REF] we compare the absorption cross section of Bardeen and RN BHs, for the same values of [MATH].', '1408.1779-1-25-1': 'As already mentioned in Sec. [REF] (cf. Fig. [REF]), the high-frequency limit of the absorption cross section of the Bardeen BH is bigger than the correspondent RN BH case with the same value of [MATH].', '1408.1779-1-25-2': 'We verified that this behavior (bigger absorption for the Bardeen BH) also applies to the total absorption cross section as a whole, for any fixed value of the frequency [MATH], for the same normalized charge [MATH].', '1408.1779-1-25-3': 'This is in accordance with the fact that the scalar field potential for the RN BH is always bigger than the corresponding one for the Bardeen BH, as it is shown in Fig. [REF], where we plot the case in which [MATH].', '1408.1779-1-25-4': 'Larger values of [MATH] present a similar behavior.', '1408.1779-1-26-0': 'Although for the same values of [MATH] the Bardeen BH has a bigger absorption cross section than the corresponding RN BH, for different values of the normalized charge [MATH] they can have the same capture cross section, i.e. the same high-frequency limit of the absorption cross section.', '1408.1779-1-26-1': 'In Fig. [REF] we plot the values of the normalized charge for which the capture (or high-frequency absorption) cross section is the same for Bardeen and RN BHs.', '1408.1779-1-26-2': 'We can see from Fig. [REF] that the RN BH must have a lower value of the normalized charge in order to have the same capture cross section of a Bardeen BH.', '1408.1779-1-27-0': 'The equality between the high-frequency values of the absorption cross section of RN and Bardeen BHs with different normalized charges raises the following question: Can a Bardeen BH produce the same absorption spectrum of a RN BH?', '1408.1779-1-27-1': 'To answer this, we have computed the absorption cross section for configurations which have the same high-frequency limits.', '1408.1779-1-27-2': 'Some results are shown in Fig. [REF], where we plot the configurations for which [MATH] are chosen to be [MATH] and [MATH].', '1408.1779-1-27-3': 'We can see that the low-frequency absorption cross section is different, although not only the high-frequency limits are the same, but also the oscillation profile are similar.', '1408.1779-1-28-0': 'The similarity of the oscillation profile of Bardeen and RN BHs with the same [MATH] can be understood as follows.', '1408.1779-1-28-1': 'From Eq. [REF], we see that the oscillation pattern depends on [MATH].', '1408.1779-1-28-2': 'Since we use configurations with the same capture cross section, the angular velocity ([MATH]) of the null geodesics are also the same, once that [MATH].', '1408.1779-1-28-3': 'Therefore, the frequency of oscillation of the two configurations will be similar.', '1408.1779-1-29-0': 'The above scenario suggests that a regular black hole can, in principle, mimic a black hole with singularities, as far as mid-to-high-frequency absorption cross section is concerned.', '1408.1779-1-29-1': 'However, we should note that, as Fig. [REF] shows, there is no complete correspondence between Bardeen and RN BHs absorption spectra with the same capture cross section, as it can be seen in Fig. [REF].', '1408.1779-1-29-2': 'Moreover, as it can be verified in Fig. [REF], for a Bardeen BH with [MATH], the corresponding RN BH with the same value of the capture cross section has a normalized charge [MATH].', '1408.1779-1-29-3': 'Therefore, for a RN BH with a charge [MATH] there is no correspondent Bardeen BH with the same capture cross section.', '1408.1779-1-30-0': '# Final remarks', '1408.1779-1-31-0': 'In this paper we presented a study of the absorption properties of regular black holes: objects which have event horizons but not singularities.', '1408.1779-1-31-1': 'For that purpose, we analyzed the case of an asymptotic planar massless scalar wave impinging upon a Bardeen regular black hole [CITATION].', '1408.1779-1-32-0': 'We computed numerically the massless scalar absorption cross section of Bardeen regular black holes showing that the generic oscillation behavior of spherical black holes with singularities, like the Schwarzschild and Reissner-Nordstrom ones, is also present in the case of Bardeen regular black holes.', '1408.1779-1-32-1': 'The increasing of the monopole charge, starting from the Schwarzschild black hole case (for which [MATH]), implies in a decreasing of the absorption cross section.', '1408.1779-1-32-2': 'Our numerical results are in full agreement with the low- and high-frequency limits of the absorption cross section, which can be obtained analytically.', '1408.1779-1-33-0': 'We compared the massless absorption cross section of a Bardeen black hole with the one of a Reissner-Nordstrom black hole with the same value of the normalized charge [MATH].', '1408.1779-1-33-1': 'We obtained that the behavior of the absorption cross section is qualitatively similar in both cases, but the Bardeen case always presents a bigger absorption cross section than the Reissner-Nordstrom case, for any fixed values of [MATH].', '1408.1779-1-34-0': 'Based on the behavior of null geodesics, we have shown that the capture cross section of a Bardeen black hole is always bigger than the corresponding one of a Reissner-Nordstrom black holes with the same value of [MATH].', '1408.1779-1-34-1': 'We have also shown that a Bardeen black hole can have the same capture cross section of a Reissner-Nordstrom black hole with a different value of [MATH].', '1408.1779-1-35-0': 'We computed numerically the massless scalar absorption cross section for arbitrary frequencies by Bardeen and Reissner-Nordstrom black holes with the same high-frequency limit.', '1408.1779-1-35-1': 'We concluded that, more than having the same capture cross section, the oscillation of the absorption cross section is similar for both cases.', '1408.1779-1-35-2': 'This comes from the fact that the oscillation depends on the angular velocity of the null circular geodesic, which is the same for the two cases.', '1408.1779-1-35-3': 'Our results suggest that some regular black holes could be mimicked by black holes with singularities, as far as mid-to-high-frequency absorption properties are concerned.', '1408.1779-1-35-4': 'The differences between the two cases manifest mainly in the low-frequency regime.', '1408.1779-1-36-0': 'The authors would like to thank Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES), and Fundacao Amazonia Paraense de Amparo a Pesquisa (FAPESPA) for partial financial support.'}

{'1408.1779-2-0-0': 'Accretion of fields by black holes is a subject of great interest in physics.', '1408.1779-2-0-1': 'It is known that accretion plays a fundamental role in active galactic nuclei and in the evolution of black holes.', '1408.1779-2-0-2': 'Accretion of fundamental fields is often related to the study of absorption cross section.', '1408.1779-2-0-3': 'Basically all black holes for which absorption of fields has been studied so far present singularities.', '1408.1779-2-0-4': 'However, even within general relativity, it is possible to construct regular black holes: objects with event horizons but without singularities.', '1408.1779-2-0-5': 'Many physically motivated regular black hole solutions have been proposed in the past years, demanding the understanding of their absorption properties.', '1408.1779-2-0-6': 'We study the absorption of planar massless scalar waves by Bardeen regular black holes.', '1408.1779-2-0-7': 'We compare the absorption cross section of Bardeen and Reissner-Nordstrom black holes, showing that the former always have a bigger absorption cross section for fixed values of the field frequency and of the normalized black hole charge.', '1408.1779-2-0-8': 'We also show that it is possible for a Bardeen black hole to have the same high-frequency absorption cross section of a Reissner-Nordstrom black hole.', '1408.1779-2-0-9': 'Our results suggest that, in mid-to-high-frequency regimes, regular black holes can have compatible properties with black holes with singularities, as far as absorption is concerned.', '1408.1779-2-1-0': '# Introduction', '1408.1779-2-2-0': 'One of the most intriguing predictions of general relativity (GR) is the existence of black holes (BHs).', '1408.1779-2-2-1': 'BHs became a paradigm in physics, and are believed to populate the galaxies [CITATION].', '1408.1779-2-2-2': 'Within standard GR, black holes are simple objects, described only by their mass, angular momentum and charge [CITATION].', '1408.1779-2-2-3': 'However, standard black holes suffer from one of the main problems of GR: the presence of singularities.', '1408.1779-2-2-4': 'Our physical knowledge breaks down at singularities.', '1408.1779-2-2-5': 'Although generally hidden by a horizon, and protected by the Penrose conjecture [CITATION] (see also [CITATION] for a review), singularities are expected to exist within GR, according to the singularity theorems developed by Hawking and collaborators [CITATION].', '1408.1779-2-3-0': 'Singularities are expected to be better understood with an improved theory of gravity (whether an extension or a modification of GR) [CITATION].', '1408.1779-2-3-1': 'Notwithstanding, within GR it is possible to obtain BH solutions without singularities.', '1408.1779-2-3-2': 'Bardeen presented a BH solution without singularities that satisfies the weak energy condition in GR [CITATION].', '1408.1779-2-3-3': "Although Bardeen's solution has its theoretical motivation in the studies of BH spacetimes with no singularities, a stronger physical motivation for it was missing until it was shown that the Bardeen BH is a solution of GR with a nonlinear magnetic monopole, i.e., a solution of the Einstein's equations coupled to a nonlinear electrodynamics [CITATION].", '1408.1779-2-3-4': 'Apart from this, further works with other physically motivated regular BHs can be found in the literature (see, e.g., Refs. [CITATION]).', '1408.1779-2-4-0': 'One way to test the physics of BHs is analyzing test fields around them.', '1408.1779-2-4-1': 'In this context, there are the quasinormal modes: natural oscillation frequencies of the fields with physically motivated boundary conditions [CITATION].', '1408.1779-2-4-2': 'An extensive survey of quasinormal modes of test charged scalar fields around different types of regular BHs was presented in Ref. [CITATION].', '1408.1779-2-4-3': 'Quasinormal modes of the Dirac field were investigated in Ref. [CITATION] and of the massive scalar fields in Hayward regular BHs in Ref. [CITATION].', '1408.1779-2-4-4': 'Quasinormal modes have an interesting relation with scattering processes in BH spacetimes.', '1408.1779-2-4-5': 'This relation can be seen, for instance, in the scattering of Gaussian packets by BHs [CITATION].', '1408.1779-2-5-0': 'Another important aspect of BHs is how they absorb matter and fields around them, i.e., their accretion rate.', '1408.1779-2-5-1': 'Accretion has an important role in the phenomenology of active galactic nuclei, and can be considered as an important agent to the mass growth of their BH hosts (see, e.g., Refs. [CITATION] and the references therein).', '1408.1779-2-5-2': 'Along more than 45 years, absorption of scalar fields has been studied extensively in many BH scenarios (see, e.g., Refs. [CITATION] and the references therein).', '1408.1779-2-5-3': 'The initial field configuration is usually taken to be plane waves at infinity and the problem is often directed to compute the absorption cross section of the field.', '1408.1779-2-5-4': 'Also, in the classical (high-frequency) limit, absorption cross sections are directly related with the shadows of BHs [CITATION].', '1408.1779-2-5-5': 'Moreover, the case of planar waves absorption has many features in common with the case of accretion of a fluid moving with constant velocity toward a BH (see, e.g., Ref. [CITATION]), which turns out to be important in the phenomenology of extreme mass-ratio inspirals [CITATION].', '1408.1779-2-6-0': 'In this paper we address the problem of how regular BHs absorb fields, focusing in the analysis of the absorption cross section of planar massless scalar waves by a Bardeen regular BH.', '1408.1779-2-6-1': 'Generically, the line element of spherically symmetric BH spacetimes can be written in the standard spherical coordinate system as [EQUATION] where the function [MATH] depends on the particular BH under consideration.', '1408.1779-2-6-2': 'As we shall see, the Bardeen BH has a structure very similar to that of a standard electrically charged BH within GR, i.e., of a Reissner-Nordstrom (RN) BH.', '1408.1779-2-6-3': 'Because of that, we shall compare our results with the RN BH ones [CITATION].', '1408.1779-2-7-0': 'The remainder of this paper is organized as follows.', '1408.1779-2-7-1': 'In Sec. [REF] we review some aspects of the Bardeen regular BHs.', '1408.1779-2-7-2': 'In Sec. [REF] we revisit the main aspects of the absorption cross section of planar massless scalar waves in spherically symmetric BH spacetimes.', '1408.1779-2-7-3': 'We also present the results in the low- and high-frequency regimes for the massless scalar absorption cross section of Bardeen BHs.', '1408.1779-2-7-4': 'In Sec. [REF] we exhibit a selection of our numerical results.', '1408.1779-2-7-5': 'We compare our results for the Bardeen regular BH with the results for the RN BH.', '1408.1779-2-7-6': 'Also, we discuss the possibility of having a Bardeen BH with a similar absorption cross section of a RN BH.', '1408.1779-2-7-7': 'We present our final remarks in Sec. [REF].', '1408.1779-2-7-8': 'Throughout the paper we use natural units, for which [MATH].', '1408.1779-2-8-0': '# Bardeen regular black holes', '1408.1779-2-9-0': 'As mentioned in the Introduction, the Bardeen BH was one of the first regular BH solutions presented in the literature [CITATION].', '1408.1779-2-9-1': 'Later, it received the physical interpretation of a BH with a nonlinear magnetic monopole [CITATION].', '1408.1779-2-9-2': 'Nonlinear electrodynamics theories within GR are generically described by the action [EQUATION] where [MATH] is the Ricci scalar; [MATH] is the Lagrangian of the electromagnetic field; [MATH]; with [MATH] being the standard electromagnetic field strength; and [MATH] is the determinant of the metric [MATH].', '1408.1779-2-9-3': 'For the theory that generates the Bardeen regular BH, we have that [EQUATION] where [MATH], [MATH] is the magnetic charge and [MATH] is the mass of the configuration [CITATION].', '1408.1779-2-9-4': 'The line element of the Bardeen BH is given by Eq. [REF], with [EQUATION]', '1408.1779-2-9-5': 'The Bardeen solution has a structure similar to the RN spacetime, presenting two horizons up to some value of the BH charge.', '1408.1779-2-9-6': 'For [MATH], the two horizons coincide and we have the so-called extremal BH.', '1408.1779-2-9-7': 'In this paper, we shall consider [MATH].', '1408.1779-2-10-0': 'For later comparison, it is instructive to mention explicitly the RN solution.', '1408.1779-2-10-1': 'The line element of the RN spacetime is given by Eq. [REF], with [EQUATION] where, in this case, [MATH] is the electric charge of the BH.', '1408.1779-2-10-2': 'The extreme case of the RN BH is given by [MATH].', '1408.1779-2-10-3': 'Note that we are using the same symbol ([MATH]) for both magnetic (Bardeen BH) and electric (RN BH) charge.', '1408.1779-2-10-4': 'To better compare both spacetimes, we shall present our results in terms of the normalized charge [MATH].', '1408.1779-2-11-0': '# Absorption cross section', '1408.1779-2-12-0': '## Partial-waves approach', '1408.1779-2-13-0': 'A massless scalar field [MATH] is described by the Klein-Gordon equation, namely, [EQUATION]', '1408.1779-2-13-1': 'Here we are considering a minimally coupled scalar field.', '1408.1779-2-14-0': 'A monochromatic wave with frequency [MATH] in a spherically symmetric background can be written as [EQUATION] where [MATH] are the standard scalar spherical harmonics.', '1408.1779-2-14-1': 'Substituting the expansion [REF] in Eq. [REF], and using the properties of the spherical harmonics, we get the following radial equation for [MATH]: [EQUATION] in which [MATH] is the tortoise coordinate, defined through [MATH], and [EQUATION] is the scalar field potential.', '1408.1779-2-14-2': 'Plots of [MATH] for Bardeen and Schwarzschild BHs are shown in Fig. [REF].', '1408.1779-2-14-3': 'The scalar field potential [MATH] is localized, in the sense that it goes to zero at the event horizon and at infinity [CITATION].', '1408.1779-2-14-4': 'We are interested in a solution that represents a wave coming from the past null infinity.', '1408.1779-2-14-5': 'Such a solution can be written using the so-called [MATH] modes, i.e. [EQUATION] with [EQUATION] where the coefficients [MATH] and [MATH] are obtained by requiring the functions [MATH] and [MATH] to be solutions of Eq. [REF] far from the BH and close to the event horizon, respectively.', '1408.1779-2-14-6': '[MATH] and [MATH] are the reflection and transmission coefficients, respectively, and are related through [EQUATION]', '1408.1779-2-14-7': 'Using the solution [REF], the absorption cross section of planar massless scalar waves can be written as [EQUATION] with [MATH] being the partial absorption cross sections, given by [EQUATION]', '1408.1779-2-15-0': '## Low- and high-frequency limits', '1408.1779-2-16-0': 'In the low-frequency regime, it has been proven that the absorption cross section of massless scalar fields by static BHs [CITATION], as well as stationary BHs [CITATION], tends to the area of the BH horizon.', '1408.1779-2-16-1': 'Our numerical results agree remarkably well with this low-frequency limit.', '1408.1779-2-16-2': 'In Fig. [REF] we plot the area of the event horizon for the Bardeen and RN BHs, as a function of the normalized charge.', '1408.1779-2-16-3': 'We can see that the event horizon area of a Bardeen BH is bigger than the corresponding one of the RN BH with the same normalized charge.', '1408.1779-2-17-0': 'In the high-frequency limit, a massless scalar wave can be described by the propagation of a null vector, which follows a null geodesic.', '1408.1779-2-17-1': 'Therefore, in this limit we can consider the classical capture cross section of null geodesics to describe the absorption cross section of massless fields.', '1408.1779-2-18-0': 'Geodesics around Bardeen BHs were also studied in Ref. [CITATION].', '1408.1779-2-18-1': 'Here we consider null geodesics in spherically symmetric BHs.', '1408.1779-2-18-2': 'Their motion is described by the Lagrangian [MATH], that satisfies [EQUATION] in which we consider, without loss of generality, the motion in the plane [MATH].', '1408.1779-2-18-3': 'The overdot indicates derivative with respect to the affine parameter of the curve.', '1408.1779-2-18-4': 'Considering the conserved quantities, namely the energy [MATH] and angular momentum [MATH] (see, e.g., Ref. [CITATION]), the equation of motion becomes [EQUATION] which can be regarded as an energy balance equation with the effective potential [EQUATION]', '1408.1779-2-18-5': 'The high-frequency limit of the absorption cross section, also called geometric cross section, [MATH], is then found by computing the classical capture radius of light rays in the spacetime under investigation.', '1408.1779-2-18-6': 'For spherically symmetric spacetimes, the null geodesic radius [MATH] is obtained through [MATH], with the prime denoting derivative with respect to [MATH].', '1408.1779-2-18-7': 'The critical impact parameter is given by [MATH], with [MATH] being characteristic of the null circular geodesic.', '1408.1779-2-18-8': 'Therefore, we have [EQUATION] and [EQUATION]', '1408.1779-2-18-9': 'With Eq. [REF] one finds the value of [MATH], and by substituting this value in Eq. [REF] one finds the capture (or geometric) cross section [MATH].', '1408.1779-2-19-0': 'In Fig. [REF] we compare the capture cross section of the Bardeen BH with the RN BH case, as a function of the normalized charge.', '1408.1779-2-19-1': 'In general, a Bardeen BH has a bigger capture cross section, compared with the RN BH with the same value of [MATH].', '1408.1779-2-20-0': 'An improvement of the high-frequency approximation to compute the absorption cross section for spherically symmetric BHs was proposed in Ref. [CITATION].', '1408.1779-2-20-1': 'It was shown that the oscillatory part of the absorption cross section in the eikonal limit can be written as [EQUATION] where [MATH], with [MATH] being the Lyapunov exponent of the null geodesic [CITATION], and [MATH] being the angular velocity of the null geodesic.', '1408.1779-2-20-2': 'Therefore, we can write the high-frequency absorption cross section as [EQUATION]', '1408.1779-2-20-3': 'Equation [REF] is usually referred to in the literature as the sinc approximation.', '1408.1779-2-20-4': 'In Fig. [REF] we compare the results obtained through Eq. [REF] with the full numerical computation of the absorption cross section, given by Eq. [REF].', '1408.1779-2-20-5': 'It is interesting to note that, although Eq. [REF] is obtained within the assumption of high frequencies, it is still a very good approximation for intermediate frequency values.', '1408.1779-2-21-0': '# Results', '1408.1779-2-22-0': 'We have computed numerically the absorption cross section of planar massless scalar waves impinging on Bardeen BHs.', '1408.1779-2-22-1': 'In this section we show a selection of our results.', '1408.1779-2-23-0': 'In Fig. [REF] we present the partial absorption cross sections [given by Eq. [REF]] for [MATH] and [MATH] and for different values of [MATH].', '1408.1779-2-23-1': 'We see that for [MATH] the limit [MATH] results in [MATH], in agreement with the result mentioned in Sec. [REF].', '1408.1779-2-24-0': 'In Fig. [REF] we present the total absorption cross section [given by Eq. [REF]] in the Bardeen BH case, for [MATH] and [MATH], as well as in the Schwarzschild BH case.', '1408.1779-2-24-1': 'The horizontal lines are the high-frequency limits in each case.', '1408.1779-2-24-2': 'We see that the increasing of the monopole charge implies in a decreasing of the absorption cross section, in agreement with the increasing of the scattering potential (cf. Fig. [REF]), as well as with the decreasing of the horizon area (cf. Fig. [REF]).', '1408.1779-2-24-3': 'The sum of the partial absorption cross sections generates the oscillatory profile shown in the plots of Fig. [REF].', '1408.1779-2-25-0': 'In Fig. [REF] we compare the absorption cross section of Bardeen and RN BHs, for the same values of [MATH].', '1408.1779-2-25-1': 'As already mentioned in Sec. [REF] (cf. Fig. [REF]), the high-frequency limit of the absorption cross section of the Bardeen BH is bigger than the correspondent RN BH case with the same value of [MATH].', '1408.1779-2-25-2': 'We verified that this behavior (bigger absorption for the Bardeen BH) also applies to the total absorption cross section as a whole, for any fixed value of the frequency [MATH], for the same normalized charge [MATH].', '1408.1779-2-25-3': 'This is in accordance with the fact that the scalar field potential for the RN BH is always bigger than the corresponding one for the Bardeen BH, as it is shown in Fig. [REF], where we plot the case in which [MATH].', '1408.1779-2-25-4': 'Larger values of [MATH] present a similar behavior.', '1408.1779-2-26-0': 'Although for the same values of [MATH] the Bardeen BH has a bigger absorption cross section than the corresponding RN BH, for different values of the normalized charge [MATH] they can have the same capture cross section, i.e. the same high-frequency limit of the absorption cross section.', '1408.1779-2-26-1': 'In Fig. [REF] we plot the values of the normalized charge for which the capture (or high-frequency absorption) cross section is the same for Bardeen and RN BHs.', '1408.1779-2-26-2': 'We can see from Fig. [REF] that the RN BH must have a lower value of the normalized charge in order to have the same capture cross section of a Bardeen BH.', '1408.1779-2-27-0': 'The equality between the high-frequency values of the absorption cross section of RN and Bardeen BHs with different normalized charges raises the following question: Can a Bardeen BH produce the same absorption spectrum of a RN BH?', '1408.1779-2-27-1': 'To answer this, we have computed the absorption cross section for configurations which have the same high-frequency limits.', '1408.1779-2-27-2': 'Some results are shown in Fig. [REF], where we plot the configurations for which [MATH] are chosen to be [MATH] and [MATH].', '1408.1779-2-27-3': 'We can see that the low-frequency absorption cross section is different, although not only the high-frequency limits are the same, but also the oscillation profiles are similar.', '1408.1779-2-28-0': 'The similarity of the oscillation profile of Bardeen and RN BHs with the same [MATH] can be understood as follows.', '1408.1779-2-28-1': 'From Eq. [REF], we see that the oscillation pattern depends on [MATH].', '1408.1779-2-28-2': 'Since we use configurations with the same capture cross section, the angular velocity ([MATH]) of the null geodesics are also the same, once [MATH].', '1408.1779-2-28-3': 'Therefore, the frequency of oscillation of the two configurations will be similar.', '1408.1779-2-29-0': 'The above scenario suggests that a regular black hole can, in principle, mimic a black hole with singularities, as far as mid-to-high-frequency absorption cross section is concerned.', '1408.1779-2-29-1': 'However, we should note that, as Fig. [REF] shows, there is no complete correspondence between Bardeen and RN BHs absorption spectra with the same capture cross section, as it can be seen in Fig. [REF].', '1408.1779-2-29-2': 'Moreover, as it can be verified in Fig. [REF], for a Bardeen BH with [MATH], the corresponding RN BH with the same value of the capture cross section has a normalized charge [MATH].', '1408.1779-2-29-3': 'Therefore, for a RN BH with a charge [MATH] there is no correspondent Bardeen BH with the same capture cross section.', '1408.1779-2-30-0': '# Final remarks', '1408.1779-2-31-0': 'In this paper we presented a study of the absorption properties of regular black holes: objects which have event horizons but not singularities.', '1408.1779-2-31-1': 'For that purpose, we analyzed the case of an asymptotic planar massless scalar wave impinging upon a Bardeen regular black hole [CITATION].', '1408.1779-2-32-0': 'We computed numerically the massless scalar absorption cross section of Bardeen regular black holes showing that the generic oscillation behavior of spherical black holes with singularities, like the Schwarzschild and Reissner-Nordstrom ones, is also present in the case of Bardeen regular black holes.', '1408.1779-2-32-1': 'The increasing of the monopole charge, starting from the Schwarzschild black hole case (for which [MATH]), implies a decreasing of the absorption cross section.', '1408.1779-2-32-2': 'Our numerical results are in full agreement with the low- and high-frequency limits of the absorption cross section, which can be obtained analytically.', '1408.1779-2-33-0': 'We compared the massless absorption cross section of a Bardeen black hole with the one of a Reissner-Nordstrom black hole with the same value of the normalized charge [MATH].', '1408.1779-2-33-1': 'We obtained that the behavior of the absorption cross section is qualitatively similar in both cases, but the Bardeen case always presents a bigger absorption cross section than the Reissner-Nordstrom case, for any fixed values of [MATH].', '1408.1779-2-34-0': 'Based on the behavior of null geodesics, we have shown that the capture cross section of a Bardeen black hole is always bigger than the corresponding one of a Reissner-Nordstrom black hole with the same value of [MATH].', '1408.1779-2-34-1': 'We have also shown that a Bardeen black hole can have the same capture cross section of a Reissner-Nordstrom black hole with a different value of [MATH].', '1408.1779-2-35-0': 'We computed numerically the massless scalar absorption cross section for arbitrary frequencies by Bardeen and Reissner-Nordstrom black holes with the same high-frequency limit.', '1408.1779-2-35-1': 'We concluded that, more than having the same capture cross section, the oscillation of the absorption cross section is similar for both cases.', '1408.1779-2-35-2': 'This comes from the fact that the oscillation depends on the angular velocity of the null circular geodesic, which is the same for the two cases.', '1408.1779-2-35-3': 'Our results suggest that some regular black holes could be mimicked by black holes with singularities, as far as mid-to-high-frequency absorption properties are concerned.', '1408.1779-2-35-4': 'The differences between the two cases manifest mainly in the low-frequency regime.', '1408.1779-2-36-0': 'The authors would like to thank Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES), and Fundacao Amazonia Paraense de Amparo a Pesquisa (FAPESPA) for partial financial support.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1408.1779

1801.04888

{'1801.04888-1-0-0': 'Visible light communications (VLC) is an emerging technology with a promise of viable solution to spectrum crunch problem in conventional radio frequency bands.', '1801.04888-1-0-1': 'In this work, we consider a VLC system where mobile users are randomly changing their horizontal location and vertical orientation.', '1801.04888-1-0-2': 'The non-orthogonal multiple access (NOMA) strategy with full channel state information (CSI) feedback is adopted to serve these users with improved spectral efficiency.', '1801.04888-1-0-3': 'To reduce computational burden and overhead due to tracking and feeding back the full CSI, we consider various limited feedback schemes where users are ordered based on their distance and vertical angle information instead of full CSI.', '1801.04888-1-0-4': 'Comprehensive numerical results verify the superiority of NOMA as compared to orthogonal multiple access while compensating the loss in user rates due to the random receiver orientation.', '1801.04888-1-0-5': 'In addition, vertical angle information based limited feedback schemes are observed to achieve satisfactory performance as compared to full CSI feedback, while conventional distance feedback scheme shows poor performance in this realistic VLC scenario.', '1801.04888-1-1-0': 'Visible light communications (VLC), non-orthogonal multiple access (NOMA), random receiver orientation, user pairing, limited feedback.', '1801.04888-1-2-0': '# Introduction', '1801.04888-1-3-0': 'Visible light communications (VLC) is a promising technology for wireless 5G networks and beyond by leveraging the broad license-free optical spectrum at wavelengths of [MATH]-[MATH] nm [CITATION].', '1801.04888-1-3-1': 'Together with developments on light emitting diode (LED) as the primary illumination source, VLC networks appear as a viable solution for simultaneous illumination and communication at low power consumption and with high durability [CITATION].', '1801.04888-1-3-2': 'As recent research efforts reveal the power of VLC transmission being capable of achieving a speed of more than multiple Gigabits per second [CITATION], this emerging technology enables ever increasing data-rate demanding mobile applications for next generation wireless networks.', '1801.04888-1-4-0': 'Towards improving the performance of multiuser VLC networks even more, a recent strategy of non-orthogonal multiple access (NOMA) appears as a powerful technology suggesting to serve multiple users at the same time and frequency slot, hence in a non-orthogonal fashion [CITATION].', '1801.04888-1-4-1': 'The NOMA strategy has been recently considered for VLC networks with a limited attention.', '1801.04888-1-4-2': 'In [CITATION], NOMA is considered in a VLC scenario and the performance is compared with orthogonal frequency division multiple acces (OFDMA) scheme.', '1801.04888-1-4-3': 'The performance analysis of NOMA is conducted for VLC networks in [CITATION] with various considerations involving lighting quality and power allocation.', '1801.04888-1-4-4': 'For a VLC NOMA system, a multiple-input multiple-output (MIMO) setting is explored in [CITATION], bit-error-rate (BER) analysis is performed in [CITATION], user sum rate maximization is conducted in [CITATION], a location based user grouping scheme is offered in [CITATION], and a phase pre-distorted symbol detection method is proposed in [CITATION] with a better error performance as compared to SIC based schemes.', '1801.04888-1-5-0': 'VLC networks involving NOMA transmission have two main drawbacks which are the overhead of full CSI feedback and availability of line-of-sight (LOS) links.', '1801.04888-1-5-1': 'Since the NOMA transmitter needs to order users according to their channel qualities, the channel gains should be estimated at user side and fed back to the transmitter, which causes computational burden while tracking unknown channel gains, and link overheads during feeding back this information.', '1801.04888-1-5-2': 'In addition, VLC transmission highly relies on LOS links, which may not be readily available all the time, especially when the receiving direction towards LED is outside the field-of-view (FOV) of the receiver.', '1801.04888-1-5-3': 'The problem of LOS link unavailability naturally arises in VLC networks with mobile users because of random user orientations around receiving direction, which is rigorously handled in [CITATION].', '1801.04888-1-6-0': 'In this paper we consider a multiuser VLC network, where mobile users with varying locations are served by NOMA transmission with various full and limited feedback schemes, and the receiver direction is changing randomly in the vertical plane.', '1801.04888-1-6-1': 'To the best of our knowledge, this realistic VLC scenario has not been studied in the literature before.', '1801.04888-1-6-2': 'We investigate the performance of NOMA with ideal full CSI feedback as well as various limited feedback schemes involving distance and vertical angle information.', '1801.04888-1-6-3': 'We show through comprehensive numerical results that NOMA is superior to conventional orthogonal multiple access (OMA) while compensating the loss in user rates due to the random orientation of VLC receivers.', '1801.04888-1-6-4': 'Furthermore, we show that the vertical angle based limited feedback schemes can achieve a satisfactory level of user rates as compared to full CSI feedback.', '1801.04888-1-6-5': 'Because of the unique feature of this realistic VLC network (with random receiver orientation), distance only feedback scheme falls short of angle based alternatives although distance feedback is a reasonable candidate for radio frequency (RF) networks [CITATION].', '1801.04888-1-7-0': 'The rest of the paper is organized as follows.', '1801.04888-1-7-1': 'Section [REF] introduces the system model.', '1801.04888-1-7-2': 'Section [REF] considers the NOMA transmission with various feedback schemes.', '1801.04888-1-7-3': 'Section [REF] presents the numerical results and Section [REF] concludes the paper with some final remarks.', '1801.04888-1-8-0': 'Notations: [MATH] denotes the continuous uniform distribution over the interval [MATH].', '1801.04888-1-8-1': '[MATH] is Kronecker delta taking [MATH] if [MATH], and [MATH] otherwise.', '1801.04888-1-9-0': '# System Model', '1801.04888-1-10-0': 'We consider an indoor VLC downlink transmission scenario involving a single transmitting LED and [MATH] users each of which is equipped with a receiving photodetector.', '1801.04888-1-10-1': 'The interaction between the LED and [MATH]th user over a LoS link is depicted in Fig. [REF].', '1801.04888-1-10-2': 'The corresponding direct current (DC) channel gain is given as [EQUATION] where [MATH] is the vertical distance between the LED and the plane including all the users, [MATH] is the horizontal distance of the [MATH]th user to the LED, [MATH] and [MATH] are the angle of irradiance and incidence of the [MATH]th user, respectively, [MATH] is the Lambertian order with [MATH] being the half-power beamwidth of the LED, [MATH] and [MATH] are the detection area and field of view (FOV) of the photodetectors.', '1801.04888-1-10-3': 'The notation [MATH] represents a rectangular function given as [EQUATION] and, hence, the channel gain of the [MATH]th user is nonzero only if [MATH] is smaller than [MATH], or equivalently LED is inside the receiver FOV.', '1801.04888-1-11-0': 'We assume that the users are non-stationary within both the horizontal and vertical planes such that they are continuously changing their locations and orientations.', '1801.04888-1-11-1': 'In particular, the horizontal distance [MATH] of [MATH]th user (representing the location in the horizontal plane) is assumed to follow a uniform distribution with [MATH] m.', '1801.04888-1-11-2': 'In addition, the vertical orientation of [MATH]th user is also varying around a mean vertical angle [MATH], which is picked up from a uniform distribution with [MATH], within a maximum deviation angle of [MATH].', '1801.04888-1-11-3': "As a result, the [MATH]th user's orientation or, equivalently, the vertical angle [MATH] takes a value from a uniform distribution with [MATH].", '1801.04888-1-12-0': 'Furthermore, we assume that the distance [MATH], mean vertical angle [MATH], and vertical angle [MATH] of [MATH]th user do not change during a single transmission period over which the respective user rates are evaluated.', '1801.04888-1-12-1': 'In subsequent transmission periods, all these variables take new values from their respective distributions, where we assume that [MATH] and [MATH] are varying much slowly as compared to [MATH], and, hence, have relatively large coherence time.', '1801.04888-1-12-2': 'As a result, each user is changing its location and mean vertical direction slowly, although relatively small variations in actual vertical angle happen much quickly.', '1801.04888-1-12-3': 'Hence, [MATH] and [MATH] are good candidates for limited feedback schemes since they can be tracked with less computational burden.', '1801.04888-1-12-4': 'In the meanwhile, any limited feedback scheme involving [MATH] or [MATH] only will degrade the user rates since they do not capture the status of the receiver direction being inside or outside the FOV, which we call FOV status and represented by [MATH] in [REF].', '1801.04888-1-12-5': 'In the next section, we will consider this compromise between limited feedback (with lower computational burden and overhead) and better user rates during the NOMA transmission.', '1801.04888-1-13-0': '# Non-Orthogonal Multiple Access', '1801.04888-1-14-0': 'In this section, we investigate the details of the NOMA transmission for the VLC downlink setting presented in Section [REF] with a special attention to full CSI and some low-rate limited feedback schemes.', '1801.04888-1-15-0': '## NOMA Transmission and User Rates', '1801.04888-1-16-0': 'In NOMA transmission, multiple users of sufficiently different channel qualities are served simultaneously in the same frequency band, which results in a non-orthogonal transmission strategy.', '1801.04888-1-16-1': 'The respective messages of users paired for NOMA transmission, which are therefore referred to as NOMA users, are weighted by suitable power allocation coefficients each of which is inversely proportional to the channel quality of respective NOMA user, as sketched in Fig. [REF].', '1801.04888-1-16-2': 'The weighted messages are then combined together along with the superposition coding principle, and sent to all users.', '1801.04888-1-16-3': 'Each NOMA user decodes its own message after decoding, if any, messages of relatively weaker users allocated with more power, while treating the messages of stronger users as noise.', '1801.04888-1-16-4': 'In the meanwhile, the decoded messages of weaker users are cancelled from the received signal through successive interference cancellation (SIC) approach.', '1801.04888-1-17-0': 'Without any loss of generality, we assume that [MATH]th user has the [MATH]th largest nonzero channel gain among [MATH] users involved in NOMA transmission with [MATH].', '1801.04888-1-17-1': 'The SINR at [MATH]th user while decoding the message of a weaker [MATH]th user with [MATH] is given as [EQUATION] where [MATH] is the power allocation coefficient of [MATH]th user such that [MATH], and [MATH] is the transmit signal-to-noise ratio (SNR).', '1801.04888-1-17-2': 'Note that [REF] implicitly assumes that the message of any [MATH]th user with [MATH] and, hence, having a weaker channel gain than that of [MATH]th user has already been decoded successfully and subtracted from the received signal as per SIC strategy.', '1801.04888-1-17-3': 'Furthermore, SINR of [MATH]th user while decoding its own message is defined as [EQUATION]', '1801.04888-1-17-4': 'At any NOMA user, the overall decoding mechanism is assumed to be in outage if instantaneous user rates associated with either of [REF] or [REF] do not meet the respective target rates of NOMA users based on their preset quality-of-service (QoS) requirements.', '1801.04888-1-17-5': 'In particular, assuming [MATH] and [MATH] are instantaneous rates associated with [REF] and [REF], respectively, outage probability of [MATH]th user is defined as [EQUATION] where [MATH] is the QoS based target rate of [MATH]th user.', '1801.04888-1-17-6': 'Defining [MATH], outage probability in [REF] is given in terms of SINR as follows [EQUATION] and, the respective sum rate is accordingly defined as [EQUATION]', '1801.04888-1-17-7': 'For the OMA transmission, we assume that all resources are allocated to a single user while it is served equally during [MATH] of the total transmission period, hence results in the following sum rate expression [EQUATION] where [MATH].', '1801.04888-1-18-0': '## Optimal and Low-Rate Feedback Schemes', '1801.04888-1-19-0': 'In NOMA transmission, users to be served are chosen among a total of [MATH] users based on their relative channel qualities.', '1801.04888-1-19-1': 'It is therefore vital for NOMA transmitter to order users according to their channel qualities based on the information fed back from users.', '1801.04888-1-19-2': 'The optimal strategy from this perspective is to order users according to their full CSI as follows [EQUATION] where we label the user having the [MATH]th stronger channel gain as [MATH]th user, without any loss of generality.', '1801.04888-1-20-0': 'Because the channel gains need to be tracked continuously to enable full CSI feedback, some limited feedback alternatives on channel qualities are considered to relieve computational complexity at user side.', '1801.04888-1-20-1': 'To this end, we investigate three different limited feedback schemes (instead of full CSI feedback), which involve distance [MATH], vertical angle [MATH], and mean vertical angle [MATH] information separately, and result in the following ordered user sets [EQUATION] respectively.', '1801.04888-1-20-2': 'The respective partial channel gain expressions in these limited feedback based orders are obtained using [REF] and available feedback information as follows [EQUATION] where [REF] is obtained by substituting the geometrical equivalent of [MATH] according to the setting in Fig. [REF].', '1801.04888-1-21-0': 'Since distance based feedback scheme does not involve angle information, respective partial channel gain in [REF] does not perform any check regarding FOV status (whether the receiving direction is inside FOV or not).', '1801.04888-1-21-1': 'As we present in Section [REF], the lack of angle information and, hence, FOV status check in distance feedback scheme results in degraded user rates since receiver direction is changing randomly in the vertical plane.', '1801.04888-1-21-2': 'It is therefore more convenient to consider angle based feedback schemes when the receiver orientation is random, which improves achievable rates by incorporating FOV status into user ordering, as shown in [REF] and [REF] via the function [MATH].', '1801.04888-1-22-0': 'The coherence time of the receiver orientation captured by the vertical angle [MATH] can be anticipated to be much less than that for the horizontal distance [MATH] (representing geometrical position), and, hence, it should be estimated more frequently.', '1801.04888-1-22-1': 'Although this discussion might bring up some doubt regarding the feasibility of vertical angle based feedback, the mean vertical angle [MATH] can be reasonably assumed to be changing much slowly, and, hence, [MATH] appears as a powerful alternative for limited feedback schemes since it relieves the computational burden by requesting update for its value less frequently.', '1801.04888-1-23-0': '# Numerical Results', '1801.04888-1-24-0': 'We assume a total of [MATH] users, which are randomly deployed over a plane where the horizontal distance [MATH] follows a uniform distribution with [MATH] m.', '1801.04888-1-24-1': 'The vertical angle [MATH] and mean vertical angle follow uniform distribution with [MATH] and [MATH], respectively, where [MATH], [MATH], and [MATH] unless otherwise stated.', '1801.04888-1-24-2': 'We assume that the LED is vertically off the horizontal plane including all the users by [MATH] m with a half power beamwidth of [MATH], the receiver area of the photodetector is [MATH], and FOV of the receiver is [MATH].', '1801.04888-1-25-0': 'We choose two users at a time ([MATH]) for NOMA transmission based on channel qualities as described in Section [REF].', '1801.04888-1-25-1': 'In particular, we label [MATH]th and [MATH]th users as the weaker and the stronger ones, respectively, where [MATH] and [MATH].', '1801.04888-1-25-2': 'We assume that respective power coefficients are [MATH] and [MATH], while the target data rates are [MATH] bit/s/Hz and [MATH] bit/s/Hz.', '1801.04888-1-26-0': 'The sum rates for NOMA transmission with random receiver orientation are depicted in Fig. [REF] for full CSI feedback as well as the limited feedback of distance, vertical angle, and mean vertical angle only cases.', '1801.04888-1-26-1': 'For comparison purposes, sum rates for NOMA transmission with fixed receiver orientation, where all the receivers are looking up directly towards ceiling, and OMA transmission with random receiver orientation are also provided assuming full CSI feedback.', '1801.04888-1-26-2': 'We observe that the random receiver orientation with even a small vertical deviation of [MATH] causes [MATH] dB more transmit power to achieve the maximum sum rate of [MATH] bit/s/Hz, when the NOMA strategy with full CSI feedback is adopted.', '1801.04888-1-26-3': 'The respective sum rate of OMA is observed to achieve the same maximum sum rate with [MATH] dB more transmit power as compared to NOMA, which highlights NOMA as a powerful candidate while dealing with adverse effects of random receiver orientation.', '1801.04888-1-27-0': 'When we consider computationally more efficient feedback alternatives for NOMA, vertical angle information performs the best among the others where it attains the maximum achievable sum rate at a transmit power of [MATH] dB off NOMA with the full CSI feedback, and outperforms OMA with full CSI feedback.', '1801.04888-1-27-1': 'When we relax the angle information by feeding back the mean vertical angle only, which relieves the burden of tracking the angle information for mobile users, the respective sum rate is now away from the exact vertical angle case by [MATH] dB and saturates at [MATH] bit/s/Hz.', '1801.04888-1-27-2': 'Hence, the degradation in sum rate with respect to maximum achievable level is only [MATH] when mean of the vertical angle values are employed.', '1801.04888-1-27-3': 'The distance feedback is observed to perform the worst of all, and the resulting sum rate saturates at a very small value of [MATH] bit/s/Hz only ([MATH] of the maximum achievable sum rate), which can be considered as the cost of not involving vertical angle information and, hence, FOV status check, and is far below the mean vertical angle feedback case.', '1801.04888-1-28-0': 'As a remark, sum rates for two variants of limited angle feedback are higher than that for the full CSI feedback at low transmit power, which seems interesting.', '1801.04888-1-28-1': 'In this low power regime, the stronger ([MATH]th) user is always in outage as shown in Fig. [REF], and the benefit of NOMA should, therefore, not be expected for this single user transmission case.', '1801.04888-1-28-2': 'Note that weaker ([MATH]th) user can always be served better in this regime as [MATH] gets larger (channel quality improves), and this is what the angle feedback variants unintentionally do whenever the respective user ordering does not match that of the full CSI feedback.', '1801.04888-1-28-3': 'In Fig. [REF], we depict the probability histogram of actual order of the weaker user for angle and mean angle feedback schemes assuming a transmit SNR of [MATH] dB and [MATH].', '1801.04888-1-28-4': 'The actual order represents the order of the weaker user with respect to the full CSI, where the weaker user appears as the [MATH]th largest one when the channel qualities are ordered based on vertical or mean vertical angle information.', '1801.04888-1-28-5': 'Note that we order only the nonzero channel quantities, and zero actual order value in Fig. [REF] represents the cases where the [MATH]th user of mean vertical angle feedback is actually outside the FOV and, hence, corresponds to a full CSI value of zero.', '1801.04888-1-28-6': 'We observe that the actual order of [MATH]th user can take values [MATH], which means better channel quality as compared to [MATH] case, and is therefore the reason for a better outage and sum rate at low power regime.', '1801.04888-1-29-0': 'The respective outage probabilities are depicted in Fig. [REF] for the stronger and weaker users separately.', '1801.04888-1-29-1': 'Considering the outage for fixed receiver orientation for both users, the random vertical orientation with full CSI feedback is observed to cause a loss only in diversity order (slope of the curves) for the stronger [MATH]th user, whereas the weaker [MATH]th user experiences degradation both in diversity order and transmit SNR of as large as [MATH] dB.', '1801.04888-1-29-2': 'We observe that while the feedback of distance information results in a saturation at [MATH] for both users, the mean vertical angle causes saturation for only the weaker [MATH]th user and at a lower probability value of [MATH].', '1801.04888-1-29-3': 'Note that, any saturation in outage probabilities reflects itself as the degradation in sum rates not achieving the maximum possible value of [MATH] bit/s/Hz in our particular case, as shown in Fig. [REF].', '1801.04888-1-30-0': 'Finally, we plot sum rates for varying deviation angles [MATH] in Fig. [REF] assuming a transmit power of [MATH] dB.', '1801.04888-1-30-1': 'We observe that the feedback of vertical angle is very robust to deviation angle values since it captures the status of the receiver direction being inside or outside the FOV, which is previously referred to as FOV status, perfectly.', '1801.04888-1-30-2': 'As the receiver orientation changes more, which is represented by larger deviation angle values, the performance of mean angle feedback degrades since it becomes more likely to have a different FOV status than that of the actual receiver direction.', '1801.04888-1-30-3': 'Nevertheless, mean angle feedback is still superior to distance feedback, even at an extreme deviation of [MATH].', '1801.04888-1-30-4': 'Note that, the weaker and stronger users chosen based on distance information are more likely to have a nonzero channel value as the deviation angle [MATH] increases, which is the reason for increasing sum rate of distance feedback in Fig. [REF], although [MATH] might be mostly considered as of theoretical importance.', '1801.04888-1-31-0': '# Conclusion', '1801.04888-1-32-0': 'We consider a multiuser VLC network involving mobile users with random location and vertical orientation.', '1801.04888-1-32-1': 'In order to improve spectral efficiency, NOMA is considered with full CSI feedback as well as distance (location) and vertical angle information based limited feedback schemes.', '1801.04888-1-32-2': 'We observe that NOMA is superior to OMA while compensating loss in user rates due to the random vertical orientation, and that angle based feedback schemes can achieve a satisfactory performance.', '1801.04888-1-32-3': 'We also observe that distance based feedback scheme achieves a poor user rate performance because of random user orientation, which result in saturation at a rate much lower than the maximum achievable sum rate.'}

{'1801.04888-2-0-0': 'We consider a downlink multiuser visible light communications (VLC) network where users randomly change their location and vertical orientation.', '1801.04888-2-0-1': 'The non-orthogonal multiple access (NOMA) strategy is adopted to serve multiple users simultaneously, and, hence, to improve spectral efficiency.', '1801.04888-2-0-2': 'We propose two novel user scheduling schemes for NOMA, which are referred to as individual and group-based.', '1801.04888-2-0-3': 'In order to further reduce the computational complexity and link overhead, novel limited-feedback schemes (on channel quality) are also proposed, which basically involve mean vertical angle (instead of its instantaneous value).', '1801.04888-2-0-4': 'Moreover, a two-bit feedback scheme is proposed for group-based user scheduling, which relies on not only distance but also vertical angle (in contrast to conventional one-bit feedback with distance only).', '1801.04888-2-0-5': 'The outage probability and sum-rate expressions are derived analytically, which show a very good match with the simulation data.', '1801.04888-2-0-6': 'Numerical results verify that the practical feedback scheme with the mean vertical angle achieves a near-optimal sum-rate performance, and the two-bit feedback significantly outperforms the one-bit feedback.', '1801.04888-2-1-0': 'Non-orthogonal multiple access (NOMA), visible light communications (VLC), random receiver orientation, limited feedback, sum rates, outage probability.', '1801.04888-2-2-0': '# Introduction', '1801.04888-2-3-0': 'Visible light communications (VLC) is a promising technology for wireless 5G networks and beyond by leveraging the broad license-free optical spectrum at wavelengths of [MATH]-[MATH] nm [CITATION].', '1801.04888-2-3-1': 'Together with developments on light emitting diode (LED) as the primary illumination source, VLC networks appear as a viable solution for simultaneous illumination and communication at low power consumption and with high durability [CITATION].', '1801.04888-2-3-2': 'The non-orthogonal multiple access (NOMA) appears as a powerful technology for multiuser VLC networks, as well, which suggests to serve multiple users at the same time and frequency slot [CITATION].', '1801.04888-2-4-0': 'The NOMA strategy has been considered for VLC networks with a limited attention.', '1801.04888-2-4-1': 'In [CITATION], NOMA is considered in a VLC scenario where the respective performance is compared to that of the orthogonal frequency division multiple access (OFDMA) scheme.', '1801.04888-2-4-2': 'The performance analysis of NOMA is performed in [CITATION] for VLC networks along with lighting quality and power allocation.', '1801.04888-2-4-3': 'For a VLC NOMA system, a multiple-input multiple-output (MIMO) setting is explored in [CITATION], bit-error-rate (BER) analysis is performed in [CITATION], sum-rate maximization is conducted in [CITATION], a location based user grouping scheme is offered in [CITATION], and a phase pre-distorted symbol detection method is proposed in [CITATION].', '1801.04888-2-5-0': 'As a major drawback, VLC transmission highly relies on LOS links, which may not be readily available all the time.', '1801.04888-2-5-1': 'This problem is pronounced even more for dynamic VLC scenarios involving random receiver orientations [CITATION].', '1801.04888-2-5-2': 'In [CITATION], a new metric for the access point selection problem is proposed for receivers with random orientations.', '1801.04888-2-5-3': 'In [CITATION], a general framework for random receiver orientation is developed where the square-channel gain distribution is derived analytically.', '1801.04888-2-5-4': 'The proposed framework applies to any prior distribution for the receiver orientation, and the approach is generalized to multi-LED scenarios in [CITATION].', '1801.04888-2-5-5': 'The impact of tilting the receiver angle on the BER performance is considered in [CITATION], and further studied in [CITATION] to yield capacity bounds.', '1801.04888-2-5-6': 'Finally, the distribution of the receiver orientation in light-fidelity (LiFi) downlink networks is evaluated in [CITATION] through indoor measurements.', '1801.04888-2-6-0': 'In this paper, we consider a multiuser VLC network where mobile users having random location and vertical orientation are served simultaneously by novel limited-feedback NOMA strategies.', '1801.04888-2-6-1': 'To the best of our knowledge, this realistic VLC NOMA scenario has not been studied in the literature before.', '1801.04888-2-6-2': 'In particular, we propose two novel NOMA strategies called individual and group-based user scheduling, which are basically designed to reduce the complexity and feedback overhead.', '1801.04888-2-6-3': 'In addition, we also propose to use the mean vertical angle (instead of its instantaneous value) as a limited yet sufficient feedback scheme.', '1801.04888-2-6-4': 'Moreover, a novel two-bit feedback scheme is also proposed as a practical feedback mechanism, which employs both the distance and vertical angle information in one bit each, and differs from the conventional one-bit feedback involving distance only [CITATION].', '1801.04888-2-6-5': 'We also derive the analytical expressions for the outage probability and sum rates for each of these NOMA strategies, which show a very good match with the respective simulation data.', '1801.04888-2-7-0': 'The rest of the paper is organized as follows.', '1801.04888-2-7-1': 'Section [REF] introduces the system model.', '1801.04888-2-7-2': 'Section [REF] considers various NOMA strategies in VLC networks, where the respective outage analysis is given in Section [REF].', '1801.04888-2-7-3': 'The numerical results are presented in Section [REF], and the paper concludes with some final remarks in Section [REF].', '1801.04888-2-8-0': '# System Model', '1801.04888-2-9-0': 'We consider an indoor VLC downlink transmission scenario involving a single transmitting LED and [MATH] users.', '1801.04888-2-9-1': 'The interaction between the LED and the [MATH]th user over a LOS link is depicted in Fig. [REF], and respective optical direct current (DC) channel gain is represented as [CITATION] [EQUATION] where [MATH] is the vertical distance between the LED and the plane including all the users, [MATH] is the horizontal distance of the [MATH]th user to the LED, [MATH] and [MATH] are the corresponding angle of irradiance and incidence, respectively.', '1801.04888-2-9-2': 'The Lambertian order is [MATH] with [MATH] being the half-power beamwidth of the LED, and [MATH] and [MATH] are the detection area and half of the FOV for the photodetectors, respectively.', '1801.04888-2-9-3': 'The function [MATH] takes [MATH] whenever [MATH], and is [MATH] otherwise.', '1801.04888-2-10-0': 'We assume that the users are non-static within both the horizontal and vertical planes such that they are continuously changing their horizontal locations and vertical orientations.', '1801.04888-2-10-1': 'In particular, [MATH] is assumed to follow a Uniform distribution with [MATH] and [MATH].', '1801.04888-2-10-2': 'In addition, [MATH] is also varying around a mean vertical angle [MATH], which is picked up from a Uniform distribution with [MATH] and [MATH] [CITATION].', '1801.04888-2-10-3': "As a result, the [MATH]th user's orientation or, equivalently, the vertical angle [MATH] takes a value from a uniform distribution with [MATH] for a given value of [MATH] and a maximum deviation angle of [MATH].", '1801.04888-2-11-0': 'Considering Fig. [REF], the incidence angle [MATH] is given as [EQUATION] where [MATH] and [MATH] take values independently (in contrast to [CITATION] where they are coupled through [MATH]).', '1801.04888-2-11-1': 'Note that [MATH] in [REF] is allowed to take either positive or negative values depending on the values of [MATH] and [MATH].', '1801.04888-2-11-2': 'This definition enables a more realistic scenario, where the vertical orientation can take any value regardless of how far the [MATH]th user is away from the LED.', '1801.04888-2-11-3': 'Hence, the incidence angle [MATH] possesses independent contributions of [MATH] and [MATH].', '1801.04888-2-12-0': 'Furthermore, we assume that [MATH] and [MATH] are varying much slowly as compared to [MATH], and, hence, have relatively large coherence time, which well aligns with realistic scenarios [CITATION].', '1801.04888-2-12-1': 'In other words, each user is changing its location and mean vertical direction slowly, whereas relatively small variations happen much quickly in actual vertical direction.', '1801.04888-2-12-2': 'As a result, we consider [MATH] and [MATH] as good candidates representing the channel state information (CSI), of the [MATH]th user.', '1801.04888-2-12-3': 'Since [MATH] and [MATH] can both be tracked with less computational burden (as compared to [MATH]), we employ these parameters in limited-feedback schemes for the NOMA transmission.', '1801.04888-2-13-0': 'On the other hand, any limited-feedback scheme involving [MATH] and [MATH] (instead of [MATH]) is likely to degrade the user rates as compared to the full CSI feedback.', '1801.04888-2-13-1': 'This is, in part, because the combination of [MATH] and [MATH] is not capable of capturing the true status of the receive direction (i.e., being inside or outside the FOV), which we call FOV status and represented by [MATH] in [REF].', '1801.04888-2-13-2': 'In the subsequent sections, we consider this compromise between the limited-feedback (with lower computational burden and overhead) and the full CSI (with better user rates) mechanisms for the NOMA transmission.', '1801.04888-2-14-0': '# NOMA in VLC Downlink Channels', '1801.04888-2-15-0': 'In this section, we consider NOMA for a VLC downlink scenario with full CSI and limited-feedback schemes.', '1801.04888-2-16-0': '## Sum Rates for VLC NOMA Transmission', '1801.04888-2-17-0': 'We assume that the users are ordered in ascending order such that [MATH]th user has the [MATH]th smallest nonzero channel gain.', '1801.04888-2-17-1': 'Moreover, we assume that [MATH] users are involved in NOMA transmission out of [MATH] users with [MATH], and that [MATH] is the set including indices of these [MATH] NOMA users.', '1801.04888-2-17-2': 'The signal-to-interference-plus-noise ratio (SINR) at the [MATH]th user while decoding the message of a weaker [MATH]th user with [MATH] is [EQUATION] where [MATH] is the set involving indices of the users being stronger than the [MATH]th user, [MATH] is the optical power allocation coefficient of the [MATH]th user such that [MATH], and [MATH] is the equivalent electrical transmit signal-to-noise ratio (SNR).', '1801.04888-2-17-3': 'Note that [REF] implicitly assumes that the message of any [MATH]th user with [MATH] (i.e., having a relatively weaker channel gain) has already been decoded successfully, and subtracted from the received signal as per successive interference cancellation (SIC) approach.', '1801.04888-2-18-0': 'The SINR of the [MATH]th user while decoding its message is [EQUATION] where [MATH] for [MATH] being the index of the strongest NOMA user (i.e., [MATH] is an empty set).', '1801.04888-2-18-1': 'At any NOMA user, the overall decoding mechanism is assumed to be in outage if instantaneous user rates associated with either of [REF] or [REF] do not meet the respective target rates of the NOMA users based on their preset quality-of-service (QoS) requirements.', '1801.04888-2-19-0': 'Note that the conventional Shannon formulation does not hold for VLC links since the optical signal has certain average and peak power constraints as well as being non-negative.', '1801.04888-2-19-1': 'We therefore consider [MATH] and [MATH] as an instantaneous achievable rate pair associated with [REF] and [REF], respectively, [CITATION].', '1801.04888-2-19-2': 'The outage probability of the [MATH]th user is therefore given as [EQUATION] where [MATH] is the QoS based target rate of the [MATH]th user, and [MATH] is the set involving indices of the users being weaker than the [MATH]th user.', '1801.04888-2-19-3': 'Defining [MATH], [REF] becomes [EQUATION] and, the respective sum-rate expression is [EQUATION]', '1801.04888-2-19-4': 'For the OMA transmission, all resources are allocated to a single user being served during [MATH] of the transmission period, and hence the sum-rate expression is [EQUATION]', '1801.04888-2-20-0': '## Individual User Scheduling with Limited Feedback', '1801.04888-2-21-0': 'In this NOMA strategy, users to be served simultaneously are chosen among a total of [MATH] users based on their individual channel qualities.', '1801.04888-2-21-1': 'It is therefore vital for the NOMA transmitter to order all the potential users according to their channel qualities based on the information transmitted back from the users.', '1801.04888-2-21-2': 'The optimal strategy from this perspective is therefore to order the users based on their full CSI given as follows [EQUATION]', '1801.04888-2-22-0': 'It is, however, not practical to employ the order in [REF] as it necessitates the channel gains to be tracked continuously.', '1801.04888-2-22-1': 'Since [MATH] and the mean vertical angle [MATH] is varying slower as compared to instantaneous vertical angle [MATH], we consider a limited-feedback mechanism where [MATH] and [MATH] (not [MATH]) are sent back to the NOMA transmitter.', '1801.04888-2-22-2': 'This mechanism circumvents the necessity of continuous tracking of [MATH], and hence relieves the computational complexity.', '1801.04888-2-22-3': 'The transmitter employs the following order while choosing NOMA users [EQUATION] where [MATH] is the average DC channel gain of the user with the index [MATH], which has the [MATH]th smallest average DC channel gain among all, and is given as [EQUATION] for [MATH].', '1801.04888-2-22-4': 'This definition also implies that [MATH] does not necessarily appear as the [MATH]th record in the full CSI feedback based order of [REF].', '1801.04888-2-22-5': 'Note that this limited-feedback mechanism cannot provide the correct FOV status to the NOMA transmitter (as mean value [MATH] is sent instead of instantaneous value [MATH]), which is likely to end up with sum-rate performance degradation.', '1801.04888-2-22-6': 'Note also that this performance loss remains marginal as [MATH] (between [MATH] and the maximum value of [MATH]) gets smaller.', '1801.04888-2-23-0': '## Group-Based User Scheduling with Two-Bit Feedback', '1801.04888-2-24-0': 'We now consider a different NOMA strategy where the transmitter does not order all [MATH] potential users individually, but rather groups them based on two-bit information on their channel qualities.', '1801.04888-2-24-1': 'The feedback mechanism this time sends low-rate information being composed of two bits, which represent either 1) [MATH] and [MATH], or 2) [MATH] and [MATH].', '1801.04888-2-24-2': 'In either case, the distance and (average) incidence angle are compared to their own preset threshold values, and the result is transmitted back to the transmitter in two bits of information.', '1801.04888-2-25-0': 'More specifically, the respective feedback bits are [MATH] if [MATH] and [MATH] are both available to the user [MATH] for feedback computation, where [MATH] and [MATH] are threshold values.', '1801.04888-2-25-1': 'If the user [MATH] has the information of [MATH] and [MATH], then the feedback bits become [MATH] with [MATH].', '1801.04888-2-26-0': 'At the transmitter, all users are split into groups, where each group is composed of users having the same feedback bits.', '1801.04888-2-26-1': 'Assuming that [MATH] and [MATH] are used in feedback computations, the groups of users having weaker and stronger channel gains can be represented, respectively, as follows [EQUATION]', '1801.04888-2-26-2': 'Similarly, whenever [MATH] and [MATH] are used while computing feedback bits, these groups become [EQUATION]', '1801.04888-2-26-3': 'We confine our search for the NOMA users to the sets [MATH]) and [MATH]), which are more likely to involve good candidates for weaker and stronger NOMA users, respectively.', '1801.04888-2-26-4': 'We therefore pick up the weaker NOMA user from [MATH]), and the stronger NOMA user is similarly chosen from [MATH]).', '1801.04888-2-26-5': 'In Section [REF], we consider one-bit feedback of [MATH] only [CITATION], as well, which cannot capture the FOV status at all, and the resulting sum-rate performance is hence much worse.', '1801.04888-2-27-0': '# Outage Analysis for VLC NOMA', '1801.04888-2-28-0': 'In this section, we derive the exact outage probability expressions for the NOMA strategies in Section [REF].', '1801.04888-2-29-0': '## Outage Formulation', '1801.04888-2-30-0': 'In the VLC downlink transmission, the channel gain in [REF] can take either zero or a nonzero value, which depends on the receive direction being inside or outside the FOV.', '1801.04888-2-30-1': 'This is a major difference of the VLC transmission from its RF counterparts, and hence we force the NOMA transmitter to schedule only the users having nonzero channel gains.', '1801.04888-2-30-2': 'We designate [MATH] and [MATH] being the index of the users having weaker and stronger nonzero channel gains, respectively, with [MATH].', '1801.04888-2-30-3': 'The outage probability for the [MATH]th user is given as [EQUATION] where [MATH], and [MATH] is the number of users having nonzero channel gain.', '1801.04888-2-30-4': 'Similarly, the outage probability for the [MATH]th user is given as [EQUATION] where [MATH].', '1801.04888-2-30-5': 'Finally, employing [REF] and [REF] in [REF] gives the outage sum rates.', '1801.04888-2-30-6': 'Note that for the NOMA strategy in Section [REF], we do not have the condition [MATH] in [REF] and [REF].', '1801.04888-2-31-0': '[MATH], number of users having nonzero channel gain, follows Binomial distribution with [MATH] where [EQUATION] where [MATH] is the difference of two CDFs with [MATH] and [MATH] being the CDF of [MATH].', '1801.04888-2-31-1': 'The respective probability mass function (PMF) of [MATH] is given as [EQUATION] where [MATH], and [MATH] is the minimum number of users having nonzero channel gain to start the NOMA transmission (i.e., [MATH] in the strategy of Section [REF]).', '1801.04888-2-32-0': 'See [CITATION] for a complete proof, which we could not involve herein due to space limitations.', '1801.04888-2-33-0': '## CDF of Square-Channel for Individual User Scheduling', '1801.04888-2-34-0': 'In this section, we present the CDF of the nonzero square-channel gain for individual user-scheduling NOMA transmission.', '1801.04888-2-35-0': 'The CDF of the unordered nonzero square-channel is given as [EQUATION] where [MATH] with [MATH] and [MATH], and [MATH] is defined in Theorem [REF].', '1801.04888-2-36-0': 'See [CITATION] for a complete proof.', '1801.04888-2-37-0': 'When we choose the [MATH]th user having nonzero channel gain, we actually order a total of [MATH] users since the remaining [MATH] have all zero gain.', '1801.04888-2-37-1': 'The CDF of the ordered nonzero square-channel gain of the [MATH]th user can therefore be found using order statistics [CITATION] as follows [EQUATION]', '1801.04888-2-37-2': 'The desired outage probabilities of individual user-scheduling NOMA (i.e., [REF] and [REF]) can then be computed using [REF].', '1801.04888-2-38-0': '## CDF of Square-Channel for Group-Based User Scheduling', '1801.04888-2-39-0': 'We finally present the CDF of the nonzero square-channel gain for two-bit feedback NOMA with group-based user scheduling.', '1801.04888-2-39-1': 'We first assume that two-bit feedback is composed of [MATH] and instantaneous angle [MATH] in the following theorem.', '1801.04888-2-40-0': 'The CDF of the nonzero square-channel gain for user [MATH] is given as [EQUATION] where [MATH], the effective angle is [MATH], and [MATH] with [MATH].', '1801.04888-2-40-1': 'Similarly, the CDF of the nonzero square-channel gain for user [MATH] is given as [EQUATION]', '1801.04888-2-40-2': 'See [CITATION] for a complete proof.', '1801.04888-2-41-0': 'When the two-bit feedback is computed using the [MATH] and the mean angle [MATH], the desired distribution is given as follows.', '1801.04888-2-42-0': 'The CDF of the nonzero square-channel for user [MATH] is given as [EQUATION] where [MATH], the effective angle is [MATH], [MATH] is the same as [MATH] of Theorem [REF] except the additional given mean value [MATH] for the instantaneous angle [MATH], and [MATH] with [MATH], [MATH], [MATH], and [MATH], [MATH].', '1801.04888-2-42-1': 'Moreover, [MATH] is defined as [EQUATION] where [MATH] is the indicator function.', '1801.04888-2-42-2': 'Similarly, the CDF of the nonzero square-channel for user [MATH] is [EQUATION] where [MATH].', '1801.04888-2-43-0': 'See [CITATION] for a complete proof.', '1801.04888-2-44-0': 'As before, the desired outage probabilities of group-based user-scheduling NOMA given in [REF] and [REF] can then be computed using the respective nonzero square-channel CDFs given in Theorem [REF] and Theorem [REF].', '1801.04888-2-45-0': '# Numerical Results', '1801.04888-2-46-0': 'In this section, we present numerical results for the the sum-rate performance of the NOMA strategies and feedback schemes considered in Section [REF].', '1801.04888-2-46-1': 'In this regard, we assume a total of [MATH] users, each of which has [MATH], [MATH], [MATH], [MATH], so that the instantaneous vertical angle spans [MATH] irrespective of the particular [MATH] value.', '1801.04888-2-46-2': 'We also assume that the LED is vertically off the horizontal plane by [MATH] with [MATH], and the photodetector has [MATH] with a FOV of [MATH] (i.e., [MATH]).', '1801.04888-2-46-3': 'We choose the power allocation coefficients to be [MATH] and [MATH] along with the respective target rates of [MATH] and [MATH].', '1801.04888-2-47-0': 'In Fig. [REF], we depict the sum rates of OMA and NOMA with the individual user scheduling for [MATH], [MATH], [MATH].', '1801.04888-2-47-1': 'Note that while [MATH] corresponds to "static" receiver orientation in the vertical domain, [MATH] represents "dynamic" receiver orientation with a large variation (i.e., orientation spans as large as [MATH] in the vertical domain over time).', '1801.04888-2-47-2': 'We observe that NOMA outperforms OMA in terms sum rates, and that analytical results nicely follow the simulation data in all the cases.', '1801.04888-2-47-3': 'We also observe that the performance of the mean vertical angle based limited feedback is very close to that of the full CSI based feedback (even for a large [MATH] value of [MATH]).', '1801.04888-2-47-4': 'On the other hand, the distance only feedback cannot capture the FOV status correctly when [MATH] gets larger, and the steady-state sum-rate loss is [MATH] bit/s/Hz.', '1801.04888-2-48-0': 'We now consider the sum-rate performance of NOMA with the group-based user scheduling.', '1801.04888-2-48-1': 'We assume that [MATH] and [MATH], where [MATH] and [MATH] are the threshold coefficients to determine the values of [MATH] and [MATH], respectively.', '1801.04888-2-48-2': 'In Fig. [REF], we plot the respective sum-rate results of OMA and NOMA with varying transmit SNR, where [MATH], [MATH], and [MATH].', '1801.04888-2-48-3': 'As before, the analytical results nicely matches the experimental data, and NOMA achieves better sum-rate performance than OMA.', '1801.04888-2-48-4': 'We observe that the sum-rate performance for NOMA with two-bit feedback of [MATH] and [MATH] (referred to as Scheme I) remains the same as [MATH] increases from [MATH] to [MATH].', '1801.04888-2-48-5': 'Two-bit feedback based NOMA is therefore very robust to the random receiver orientation.', '1801.04888-2-48-6': 'When [MATH] is employed instead of [MATH] in the feedback computation (referred to as Scheme II), the degradation in NOMA sum rates is less than [MATH] bit/s/Hz at the steady state.', '1801.04888-2-48-7': 'This result underscores the power of the practical feedback scheme for two-bit feedback based NOMA involving mean vertical angle (instead of using its instantaneous value).', '1801.04888-2-49-0': 'We finally consider the impact of noisy horizontal distance and vertical angle information on the NOMA sum rates in a dynamic scenario with [MATH].', '1801.04888-2-49-1': 'To this end, we consider noisy estimates as [MATH], [MATH], and [MATH], where [MATH] and [MATH] stand for estimation error, and are assumed to be complex Gaussian with zero-mean and variance [MATH] and [MATH], respectively.', '1801.04888-2-49-2': 'We assume that [MATH] and [MATH], which correspond to [MATH] error in distance and [MATH] error in vertical angle [CITATION].', '1801.04888-2-49-3': 'In Fig. [REF], we depict the simulation results for individual user-scheduling NOMA with noisy distance and angle information assuming the same setting of Fig. [REF].', '1801.04888-2-49-4': 'We observe that performance of the mean vertical angle based limited-feedback scheme does not change while that of the instantaneous angle based scheme exhibits a marginal degradation.', '1801.04888-2-50-0': '# Conclusion', '1801.04888-2-51-0': 'We investigated a downlink multiuser VLC scenario involving mobile users with random vertical orientation.', '1801.04888-2-51-1': 'In order to increase the spectral efficiency, the NOMA transmission is employed with various user scheduling techniques and feedback mechanisms.', '1801.04888-2-51-2': 'The outage probability and sum-rate expressions are derived analytically, where the respective numerical results show a very good match with the simulation data.', '1801.04888-2-51-3': 'We observe that the practical feedback scheme with the mean vertical angle achieves a near-optimal sum-rate performance.', '1801.04888-2-51-4': 'In addition, the two-bit feedback involving both the distance and the angle information significantly outperforms the conventional one-bit feedback with the distance information only.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1801.04888

1711.01417

{'1711.01417-1-0-0': '# Introduction', '1711.01417-1-1-0': 'Filamentary structures in the interstellar medium (ISM) have been known and investigated for many years.', '1711.01417-1-1-1': 'Much effort has been invested in studying their morphology , properties , distribution within the Milky Way , and formation .', '1711.01417-1-1-2': 'More recently, it has been argued that filaments represent a crucial phase in the earliest stages of star formation, based on observational findings showing that pre-stellar cores and young stellar clusters are preferentially located within filaments or at intersections between them .', '1711.01417-1-1-3': 'This raises the question of what exactly the role of filaments is in controlling star formation .', '1711.01417-1-2-0': 'The usual model for filaments describes them as hydrostatic cylinders that fragment due to linear perturbations .', '1711.01417-1-2-1': 'However, some studies demonstrate that other environmental conditions, such as turbulence, accretion, or magnetic fields, can introduce additional fragmentation modes.', '1711.01417-1-2-2': 'Therefore, the question is whether a quasi-static description of the evolution of filaments is justified.', '1711.01417-1-2-3': 'One way to test this is to use numerical simulations that form filaments self-consistently and compare the properties of the resulting structures with those predicted by the simple model.', '1711.01417-1-2-4': 'In doing so, one can examine how each force, and combination of different forces, affects the evolution of filaments, and whether a quasi-static model can capture the essential physics involved in the process of fragmentation.', '1711.01417-1-3-0': 'Since the forces acting on filaments are not directly observable, one needs to analyse the properties of the filaments and fragments as observational diagnostics.', '1711.01417-1-3-1': 'The width of filaments and the spacing and distribution of fragments are widely-studied quantities that can be used.', '1711.01417-1-3-2': 'However, these quantities are highly degenerate in this context.', '1711.01417-1-3-3': 'For example, [CITATION] has demonstrated that, as long as self-gravity is not acting alone, the widths of the radial profiles of filaments are approximately constant over wide ranges of central density and global star formation efficiencies.', '1711.01417-1-3-4': 'Furthermore, since filaments are generally short-lived substructures within globally fragmenting molecular clouds and normally neither quiescent nor isolated, external gravitational potentials or pressures can introduce additional perturbations that either separate or compress the cores.', '1711.01417-1-4-0': "Another interesting parameter is the mass per unit length, or line mass, of a filament, which is argued to determine the filament's stability in the quasi-static model .", '1711.01417-1-4-1': 'Similar to the Jeans analysis for spheres, there is a critical line mass that marks the transition between states of equilibrium and gravitational collapse.', '1711.01417-1-4-2': 'In principle, the line mass is an easy to measure quantity since it only requires the length and the enclosed mass of the respective filament.', '1711.01417-1-4-3': 'However, there are many effects that need to be considered when measuring the line mass from observational data, such as inclination, optical depth, uncertainties in distance estimations, and overlap effects .', '1711.01417-1-4-4': 'Furthermore, and especially in the context of large filaments on galactic scales , it is not completely clear how mostly two-dimensional (2D) observational characteristics reflect the three-dimensional (3D) nature of the acting forces.', '1711.01417-1-5-0': 'The questions we address in this paper are: How do filaments evolve and fragment in numerical simulations?', '1711.01417-1-5-1': 'Is the fragmentation picture seen in simulations in agreement with the quasi-static analytic framework of gravitational fragmentation?', '1711.01417-1-5-2': 'To address these questions, we analyse a set of model clouds from 3D FLASH adaptive mesh refinement (AMR) simulations.', '1711.01417-1-5-3': 'We introduce the simulations and the analysis methods in Sect. [REF].', '1711.01417-1-5-4': 'We tested several filament finder codes and compared their performances in identifying filaments from simulations in Appendix [REF].', '1711.01417-1-5-5': 'The results of our analysis are outlined and discussed in Sect. [REF] and summarised in Sect. [REF].', '1711.01417-1-6-0': '# Methods', '1711.01417-1-7-0': '## Cloud models', '1711.01417-1-8-0': 'The filamentary structures we analyse in this paper come from numerical simulations of dense cloud formation by [CITATION].', '1711.01417-1-8-1': 'The simulations have been performed with the 3D magnetohydrodynamics (MHD), AMR FLASH code , and are of a [MATH] kpc[MATH] vertical box of the interstellar medium (ISM) in a disk galaxy, with the galactic midplane located in the middle of the box.', '1711.01417-1-8-2': 'Turbulence is driven by the injection of discrete, thermal supernova (SN) explosions.', '1711.01417-1-8-3': 'SN rates are normalized to the galactic SN rates , with 6.58 and 27.4 Myr[MATH] kpc[MATH] for Type Ia and core-collapse SNe, respectively.', '1711.01417-1-8-4': 'The positioning of SN explosions is random in the horizontal direction and exponentially decaying in the vertical direction with scale heights of 90 and 325 pc for type Ia and core-collapse SNe.', '1711.01417-1-8-5': 'SN clustering is also taken into account by assuming 3/5 of the core-collapse population are correlated in space and time.', '1711.01417-1-8-6': 'Magnetic fields are included in the simulation, with an initial field strength of 5 [MATH]G at the midplane that exponentially decays in the vertical direction.', '1711.01417-1-8-7': 'The fields are initially uniform and oriented in the horizontal direction.', '1711.01417-1-8-8': 'They are allowed to evolve self-consistently with the simulation.', '1711.01417-1-8-9': 'The simulations also include distributed photoelectric heating and radiative cooling.', '1711.01417-1-8-10': 'The photoelectric heating of dust grains is implemented with a background FUV intensity of [MATH] with a vertical scale height of 300 pc.', '1711.01417-1-8-11': 'Radiative cooling rates are appropriate for a solar metallicity, optically thin gas, with a constant ionization fraction of [MATH] for temperatures below [MATH] K , and resonant line cooling for collisionally ionized gas at higher temperatures .', '1711.01417-1-9-0': 'The simulation initially includes only a static galactic gravitational potential, accounting for a stellar disk and a spherical dark matter halo.', '1711.01417-1-9-1': 'The potential is uniform in the horizontal direction; in the vertical direction, for altitudes [MATH] kpc, it follows a parametrized model of the mass distribution of the Milky Way , and at higher altitudes, it smoothly transforms into the outer halo profile described by [CITATION] with a scale length of 20 kpc .', '1711.01417-1-9-2': 'The simulation runs for [MATH]250 Myr without gas self-gravity in order to develop a multiphase, turbulent ISM, where dense structures form self-consistently in turbulent, convergent flows.', '1711.01417-1-9-3': 'Self-gravity is then switched on to follow the formation and evolution of clouds over the next 6 Myr.', '1711.01417-1-10-0': 'Three dense clouds from the cloud population found in the simulation were selected for high-resolution re-simulation by [CITATION].', '1711.01417-1-10-1': 'Once one of these clouds was identified in the cloud catalogue, a higher-resolution refinement region was defined around the region where the cloud would form in a checkpoint prior to the onset of self-gravity.', '1711.01417-1-10-2': 'Gas self-gravity was then turned on, and the evolution and collapse of the cloud were followed.', '1711.01417-1-10-3': 'For our investigations, we focus on these three clouds and map a (40 pc)[MATH] volume enclosing each cloud with [MATH] grid cells, with an effective spatial resolution of [MATH] pc.', '1711.01417-1-10-4': 'We consider objects to be fully resolved if their local Jeans length [MATH], corresponding to a maximum resolved density at 10 K of [MATH] cm[MATH] .', '1711.01417-1-10-5': 'This means that we can trace fragmentation down to 0.4 pc, but cannot fully resolve objects that form at smaller scales.', '1711.01417-1-10-6': 'The clouds have total masses on the order of [MATH], [MATH], and [MATH] M[MATH] (hereafter denoted models M3, M4, and M8).', '1711.01417-1-10-7': 'Examples are shown in Fig. [REF].', '1711.01417-1-11-0': '## Filament Identification', '1711.01417-1-12-0': 'We use a publicly available filament finder that identifies and traces the structures within the model clouds.', '1711.01417-1-12-1': 'We choose DisPerSe as we can apply it directly to 3D volume density cubes, as well as to 2D column density maps, returning structures based on the same algorithm.', '1711.01417-1-12-2': 'Furthermore, DisPerSe identifies filaments by deriving the gradients within each grid cell and connecting maxima and saddle points with each other, which gives the detected skeletons a mathematical and physical meaning.', '1711.01417-1-13-0': 'We have compared the structures identified by DisPerSe with those detected by other codes (see Appendix [REF]).', '1711.01417-1-13-1': 'All of them return similar structures in the densest regions of the clouds, but show pronounced differences in the more diffuse envelopes.', '1711.01417-1-13-2': 'This behaviour has a significant impact on the measured properties of the filaments.', '1711.01417-1-13-3': 'The consequence is that studies that do not use the same filament finding techniques are not directly comparable unless it has been proven that the identified structures are indeed similar.', '1711.01417-1-14-0': 'In order to obtain results that can be qualitatively compared to observations we follow the approximation of [CITATION] and use two thresholds: (a) a low-density threshold at n[MATH] 100 cm[MATH] which defines the volume of the cloud, and corresponds very roughly to the density at which CO emission can first be detected; and (b) a high-density threshold at n[MATH] 5,000 cm[MATH] that points to the denser clumps within the cloud that have a high potential for forming stars .', '1711.01417-1-14-1': 'Note that we approach our number density resolution (see Sect. [REF]) when using this high-density threshold.', '1711.01417-1-14-2': 'However, we have found that using a lower threshold that is still of the same order of magnitude has no qualitative effect on the structures and properties of the filaments, or their time evolution in general (see also the analysis of the dense gas mass fraction in Sect. [REF] and Fig. [REF]).', '1711.01417-1-15-0': 'Fig. [REF] illustrates with an example how strongly the structures of the identified skeletons depend on the considered threshold.', '1711.01417-1-15-1': 'The map shows a column density map of M3 at [MATH]3.8 Myr that has been produced by projecting the volume density cube along the z-axis.', '1711.01417-1-15-2': 'The map illustrates a projection of the filament spines identified by DisPerSe in the 3D position-position-position space.', '1711.01417-1-15-3': 'Note that we only include the spines of the filaments with densities exceeding the density threshold used for their identification, i.e. 100 or 5000 cm[MATH].', '1711.01417-1-15-4': 'This restriction is necessary because of the way DisPerSe identifies filaments.', '1711.01417-1-15-5': 'DisPerSe only needs a seed point with a (column) density above the specified threshold, and then connects local maxima through tangent lines to the (column) density gradients.', '1711.01417-1-15-6': 'If no restriction to the (column) density gradients are set, DisPerSe will eventually connect all local maxima, even if they are below the specified (column) density threshold.', '1711.01417-1-15-7': 'In this work we set no restriction to the (column) density gradient for the filament identification.', '1711.01417-1-15-8': 'However, once the filamentary network was identified, we kept only those filaments whose spines are above our desired (column) density threshold.', '1711.01417-1-16-0': '## Filament Properties', '1711.01417-1-17-0': 'DisPerSe derives the skeletons of filaments based on the local gradients and user-set thresholds.', '1711.01417-1-17-1': 'It then uses the minimal spanning tree (MST) method to find the nearest neighbour for each element of the skeletons.', '1711.01417-1-17-2': 'Most other filament finders neglect this step and leave it to the user to assemble the individual points to filaments, which may not be as accurate as the algorithm DisPerSe offers.', '1711.01417-1-18-0': 'In the end, DisPerSe returns the starting points [MATH] and end points [MATH] of each segment [MATH] of an identified skeleton.', '1711.01417-1-18-1': 'With these points, we calculate the length of the filament, [EQUATION] with [MATH] being the direction vector pointing from [MATH] to [MATH].', '1711.01417-1-19-0': 'We define the volume of the filaments as the set of grid cells [MATH] that are located along the direction vector [MATH] between [MATH] and [MATH], or have any part that lies within a distance [MATH] to these cells.', '1711.01417-1-19-1': 'In Fig. [REF] we illustrate this in a two-dimensional example.', '1711.01417-1-19-2': 'The orange area represents the analytic volume of the filament while the blue area marks the grid cells that are actually used.', '1711.01417-1-20-0': 'The range of filament widths that we find is wide, partly due to the complexity of profiles (e. g., from crossing filaments) that makes it difficult to measure relevant quantities like the full-width half maximum.', '1711.01417-1-20-1': 'For simplicity, we assume that the radius of our filaments is [MATH] pc everywhere.', '1711.01417-1-20-2': 'This value agrees with average filament widths we find when fitting Gaussians to the line profiles of the filaments, although numerical resolution limits filaments in our model from collapsing to smaller extensions.', '1711.01417-1-21-0': 'The enclosed mass of a filament is given by, [EQUATION] where [MATH] is the mass density within the grid cell [MATH]; and the line mass by, [EQUATION]', '1711.01417-1-22-0': '## Fragment Identification and Properties', '1711.01417-1-23-0': 'To identify fragments, we use astrodendro.', '1711.01417-1-23-1': 'This method computes dendrogram trees that represent the hierarchical structure of the underlying matter distribution in order to unravel how sub-structures relate to each other.', '1711.01417-1-23-2': 'This way one can easily identify individual clumps, filaments, and fragments.', '1711.01417-1-24-0': 'We identify fragments as highest level leaves in the dendrograms derived with a minimal density threshold at minvalue = 5,000 cm[MATH] and a minimum of minnpix = 20 cells.', '1711.01417-1-24-1': 'We define the volume of the fragment as a sphere with a radius of [MATH] = 0.3 pc around the central position of the respective dendrogram leaf.', '1711.01417-1-25-0': '# Results Discussion', '1711.01417-1-26-0': '## Mean Properties and Evolution of 3D Filaments', '1711.01417-1-27-0': 'We begin our analysis by studying the average properties and time evolution of the filaments we find in our model clouds in 3D, and how those properties relate to the global characteristics of the surrounding clouds.', '1711.01417-1-28-0': 'In the top panel of Fig. [REF] we show the average line mass [MATH] of all the filaments identified in each of the three clouds as function of time [MATH] after self-gravity has been activated in the simulations.', '1711.01417-1-28-1': 'We see that the average line masses within all clouds increase with time and with increasing identification threshold [MATH].', '1711.01417-1-28-2': 'This is expected considering the structures we obtain when using different thresholds.', '1711.01417-1-28-3': 'When we apply a high threshold we obtain compact structures connecting the sites of first fragment formation, where the gas is more concentrated.', '1711.01417-1-28-4': 'In contrast, with a lower threshold we identify a larger number of low mass structures in more diffuse regions of the clouds (see Fig. [REF]).', '1711.01417-1-28-5': 'As a result, the average line mass of low density filaments is always lower than when using a higher threshold.', '1711.01417-1-29-0': 'The increase of average line masses over time can be understood from the fact that the clouds formed by compression of gas, due to supernova shocks and global turbulent motions, in the time before our analysis.', '1711.01417-1-29-1': 'Thus, at [MATH] Myr, when self-gravity is activated, the clouds are already highly self-gravitating and begin to collapse globally.', '1711.01417-1-29-2': 'As a result, the filaments within the clouds gain more mass with time, which increases their line masses.', '1711.01417-1-29-3': 'Only in the case of M3 does the growth of average line masses of the low-density filaments stagnate after the first megayear.', '1711.01417-1-29-4': 'However, the average line mass of the high-density filaments in all clouds steadily increases in time, suggesting that the filaments are collapsing gravitationally.', '1711.01417-1-29-5': 'Fig. [REF] shows a clearer picture of how the mass within the cloud is distributed, and its time evolution.', '1711.01417-1-29-6': 'The top panel shows the fraction of mass contained in filaments, [MATH], compared to the mass of the entire parental cloud, [MATH].', '1711.01417-1-29-7': 'In all clouds, the fraction rapidly and continuously increases in time.', '1711.01417-1-29-8': 'Since the clouds accrete mass from their environments, the growth of the filament mass fraction means that filaments are forming and growing at rates not directly correlated to the evolution of their parental clouds.', '1711.01417-1-29-9': 'A similar trend is observed in the mass fraction of the fragments with respect to the cloud mass for all three clouds (bottom panel Fig. [REF]).', '1711.01417-1-29-10': 'However, the middle panel showing the mass ratio of filaments to cores shows a steep rise when cores are formed, but then stagnates at later times suggesting that filaments and cores grow at the same rate.', '1711.01417-1-30-0': 'In order to confirm this behaviour, we take a closer look at the dense gas mass fraction (DGMF) of the clouds.', '1711.01417-1-30-1': 'We define the DGMF as the fraction of mass enclosed in cloud cells that contain gas above a given dense gas threshold, n[MATH], compared to the mass of the entire cloud: [EQUATION] with [MATH] being the number density at the time [MATH] and [EQUATION]', '1711.01417-1-30-2': 'Fig. [REF] shows the evolution of the DGMFs of the three simulated clouds using two different dense gas thresholds, namely [MATH] = 1,000 cm[MATH] and [MATH] = 5,000 cm[MATH].', '1711.01417-1-31-0': 'In all clouds we see that the DGMF continues growing until the end of the simulation, with maximal values of 55-70% for [MATH] = 1,000 cm[MATH] and 35-55% for [MATH] = 5,000 cm[MATH].', '1711.01417-1-31-1': 'These clouds continue to accrete mass from their surroundings , so we conclude that these clouds are collapsing faster than they are growing.', '1711.01417-1-32-0': 'We see a similar behaviour in Fig. [REF].', '1711.01417-1-32-1': 'In this figure, we show the fraction of gas above a given number density threshold as a function of this number density threshold, [MATH], for four individual time steps in the simulations.', '1711.01417-1-32-2': 'In all clouds we see that the maximum gas density grows with time until it reaches the limit of our numerical resolution.', '1711.01417-1-32-3': 'Note that although we reach densities of order 30,000 cm[MATH], we only resolve densities up to 8,000 cm[MATH] reliably.', '1711.01417-1-32-4': 'At the same time, the density distributions become flatter as the clouds evolve.', '1711.01417-1-32-5': 'This means that the clouds collapse in their entirety and compress the enclosed mass into denser substructures such as fragments without necessarily losing their diffuse envelopes.', '1711.01417-1-32-6': 'This likely occurs because of continuing accretion of diffuse gas from the surrounding ISM .', '1711.01417-1-33-0': 'In summary, we see that the average evolution of the filaments is influenced by the global kinematics of their parental cloud.', '1711.01417-1-33-1': 'In particular, the way that the cloud transforms the mass it accretes from the ISM into dense substructures is related to the formation of fragments within the filaments.', '1711.01417-1-33-2': 'However, we also see that the properties of the dense gas strongly depend on the parameters used to define and identify it, such as the threshold density.', '1711.01417-1-33-3': 'This might become important for our key question of how well analytic models evaluate the stability of filaments and predict their fragmentation behaviour.', '1711.01417-1-34-0': '## Properties and Evolution of individual 3D Filaments', '1711.01417-1-35-0': 'In this section we investigate the evolution of individual filaments and compare the properties of fragmenting filaments with others.', '1711.01417-1-35-1': 'We confront these properties with the predictions of analytic models.', '1711.01417-1-35-2': 'Table [REF] provides a summary of the properties of the examined filaments.', '1711.01417-1-36-0': 'In Fig. [REF] we show the line masses of individual filaments as a function of time.', '1711.01417-1-36-1': 'In the case of the low-density filaments (Fig. [REF]), the transition from filaments without embedded fragments to those with fragments occurs at very low line masses (4.0, 2.4, and 2.9 M[MATH] pc[MATH] for M3, M4, and M8, respectively).', '1711.01417-1-37-0': 'We compare the line masses at which fragmentation occurs in the model with the criterion for cylindrical filaments being in hydrostatic equilibrium.', '1711.01417-1-37-1': 'In particular, we focus on the model by [CITATION] which describes a filament as an infinitely long, isolated, isothermal cylinder filled with self-gravitating gas that is in balance between gravity and thermal pressure.', '1711.01417-1-37-2': 'In this model, the equilibrium configuration is uniquely characterised by the critical line mass [EQUATION] with [MATH] being the sound speed of the gas, [MATH] the gravitational constant, and [MATH] the gas temperature.', '1711.01417-1-38-0': 'According to [CITATION], a cylindrical filament is only thermally supported against collapse if its line mass remains below this critical value.', '1711.01417-1-38-1': 'Otherwise, initially small perturbations within the filament can grow under the influence of self-gravity, which then leads to radial collapse and fragmentation, as [CITATION], [CITATION], [CITATION], and [CITATION] describe in their fragmenting cylinder models.', '1711.01417-1-39-0': 'Thus, the cylindrical fragmentation model has become a commonly used description of a filament and set of initial conditions of its fragmentation.', '1711.01417-1-39-1': 'The key question is whether or not the evolution of the filaments in our simulations follows this criterion.', '1711.01417-1-39-2': 'Can the equilibrium configuration be regarded as a realistic initial condition for fragmentation?', '1711.01417-1-39-3': 'Is the equilibrium configuration, in fact, ever reached?', '1711.01417-1-40-0': 'To address these questions, we mark the range of the analytic values for the critical line masses for typical gas temperatures between 10 and 15 K with red areas in Fig. [REF].', '1711.01417-1-40-1': 'We see that the low-density filaments in our samples start fragmenting at line masses far below the predicted critical values, and hardly reach such high line masses even at later stages in their evolution.', '1711.01417-1-41-0': 'This suggests that the approximations of the cylindrical fragmentation model fail here.', '1711.01417-1-41-1': 'We see many differences between our filaments and those in the analytic model.', '1711.01417-1-41-2': 'In particular, our filaments are neither isolated, nor in hydrostatic equilibrium, nor cylindrically symmetric.', '1711.01417-1-41-3': 'Rather they are part of a hierarchically collapsing cloud, and interact with each other, e. g. by crossing each other or accreting gas.', '1711.01417-1-41-4': 'Thus, the filaments both are subject to external pressure and may have large density perturbations that are outside the regime of linear growth.', '1711.01417-1-42-0': 'Furthermore, studies have found that even filaments that are subcritical in terms of line mass can fragment.', '1711.01417-1-42-1': 'They show that fragmentation itself does not show any clear imprints of the forces that originally formed the fragment (e. g., shock waves, or cloud-cloud collisions).', '1711.01417-1-42-2': 'The conclusion is that there is no prediction for a threshold below which the fragmentation of filaments is prevented.', '1711.01417-1-43-0': 'The situation changes when we look at the high-density filaments (Fig. [REF]).', '1711.01417-1-43-1': 'Here we observe maximal line masses of filaments without fragments between 18-20 M[MATH] pc[MATH] in all clouds.', '1711.01417-1-43-2': 'These transitional line masses appear to be in agreement with the critical line mass of the cylindrical fragmentation model.', '1711.01417-1-43-3': 'The reason for this is that the properties of the filaments here are more likely comparable to those of the cylindrical fragmentation model since they are more likely isothermal, straighter, and closer to the sites where overdensities form compared to more diffuse filaments.', '1711.01417-1-43-4': 'Consequently, the objects we study here agree better with the properties of the analytic cylinders.', '1711.01417-1-43-5': 'However, we detect only one dense filament without embedded fragment for one single time step (in M8) and, thus, lack a statistically meaningful sample to draw final conclusions about the capability to predict the fragmentation behaviour of high-density filaments.', '1711.01417-1-43-6': 'From the analysis of the low-density filaments, though, we see that the configuration represented by the analytic model of cylindrical fragmentation is not universally part of the evolution of filaments in the simulations.', '1711.01417-1-44-0': 'We conclude that a simple cylindrical model as described by [CITATION] and the cylindrical fragmentation model does not represent the typical initial conditions for the fragmentation of a filament in our molecular cloud simulations.', '1711.01417-1-44-1': 'Therefore, it is not a complete model for evaluating the stability of filaments.', '1711.01417-1-44-2': 'To predict the fragmentation of a filament a more complex model is essential, one that not only considers the balance between internal self-gravity and thermal pressure, but also other forces, such as external hydrostatic, turbulent, or magnetic pressure.', '1711.01417-1-44-3': 'Furthermore, connecting such a model with observations needs to take into account that the properties of the filaments derived strongly depend on the parameters used to identify the filaments (as well as the filament finder code used, as discussed in Appendix [REF]).', '1711.01417-1-44-4': 'This also means that whether or not observed filaments fragment as predicted by a cylindrical fragmentation model is strongly influenced by the identification parameters, as one can always choose parameters in a way that it perfectly suits the model.', '1711.01417-1-44-5': 'In our case, this is represented by the high-density filaments that only then contain fragments after exceeding the critical line mass.', '1711.01417-1-44-6': 'Since there is no unique, universal and physically motivated definition of what filaments the fragmentation models are not universal themselves.', '1711.01417-1-45-0': '## Properties and Evolution of Fragments', '1711.01417-1-46-0': 'In this section, we discuss the properties of the fragments we have detected within our clouds using astrodendro (Sect. [REF]), including their time evolution, and connection to their parental filaments and to each other.', '1711.01417-1-46-1': 'The main properties of the fragments are summarised in Table [REF] showing that those fragments have similar properties as dense regions and condensations in star-forming regions .', '1711.01417-1-47-0': 'In order to find evidence for the fragments in our simulation forming by gravitational fragmentation following the cylindrical fragmentation criterion, we make a first estimate of how gravitationally bound our fragments are.', '1711.01417-1-47-1': 'For this, we compute their virial parameters, given by : [EQUATION] for each of the fragments at any time step.', '1711.01417-1-47-2': 'Here, [MATH] is the gravitational constant, [MATH] the enclosed mass of the fragment, [MATH] = 0.3 pc its radius and [MATH] its velocity dispersion, which we calculate following Equations (9) (10) in [CITATION]: [EQUATION] with [MATH] being the central position of the fragment, [MATH] the average gas velocity vector and [MATH] the average sound speed.', '1711.01417-1-47-3': 'We note that this estimate only takes the ratio between volumetric kinetic and gravitational energy into account.', '1711.01417-1-47-4': 'Other terms, such as magnetic fields or surface thermal or kinetic pressure , are neglected.', '1711.01417-1-47-5': 'This is a common approximation used in both observational and theoretical studies, although [CITATION] suggest that these additional terms can be as important for objects embedded within molecular clouds as the balance of volumetric kinetic and gravitational energy.', '1711.01417-1-48-0': 'In the top row of Fig. [REF] we show a histogram of virial parameters measured for the fragments at the time when they have been first detected.', '1711.01417-1-48-1': 'According to Equ.', '1711.01417-1-48-2': '([REF]), the fragments are gravitationally bound when [MATH].', '1711.01417-1-48-3': 'This criterion is fulfilled by only 20% of the fragments.', '1711.01417-1-48-4': 'The majority of fragments have virial parameters close to the mode of 3.5.', '1711.01417-1-48-5': 'We argue that these fragments may nevertheless be bound objects because of the terms not accounted for by Equ.', '1711.01417-1-48-6': '([REF]), particularly the surface terms resulting from the collapse of the surrounding cloud.', '1711.01417-1-48-7': 'Furthermore, we see that the virial parameters decrease as the fragments evolve, which means that they become more bound.', '1711.01417-1-48-8': 'The histogram in the bottom row of Fig. [REF] confirms this as it shows that most of the fragments have virial parameters around 2 in the last simulated time steps.', '1711.01417-1-49-0': 'Note that, as explained by [CITATION], we can only give lower limits on the velocity dispersions, and thus the virial parameters, since we underresolve the turbulence on small scales in the simulations.', '1711.01417-1-49-1': "Resolving the subgrid scale turbulence may increase the energy by 25% , but likely won't prevent the fragments from beginning to collapse before the criteria for cylindrical fragmentation are fulfilled.", '1711.01417-1-49-2': "This consequently means that the fragmentation must be driven by neglected terms in the virial equation that can produce significant differences in the estimation of the fragments' boundness.", '1711.01417-1-50-0': 'In the bottom row of Fig. [REF] we plot the number of identified fragments as function of time.', '1711.01417-1-50-1': 'We see that the number of fragments overall increases with time for each simulation.', '1711.01417-1-50-2': 'However, there are cases when the number of fragments drops.', '1711.01417-1-50-3': 'The missing fragments are either disrupted, for example by shock waves or intracloud turbulence, or merge with each other, meaning that they approach each other too closely ([MATH]0.4 pc) to be distinguishable with our resolution.', '1711.01417-1-51-0': "We see that the first fragments form within the first 2 Myr, corresponding to 25-50% of the clouds' free-fall times and the period when the growth of the DGMF is steepest (see Fig. [REF]).", '1711.01417-1-51-1': 'This indicates that the formation of fragments is primarily dominated by the compression of dense gas within the cloud.', '1711.01417-1-51-2': 'This is verified by the ratio of mass contained within the fragments, [MATH], relative to the mass of the entire cloud, [MATH], shown in the bottom panel of Fig. [REF].', '1711.01417-1-51-3': 'The [MATH] ratios evolve in a similar fashion as the DGMF.', '1711.01417-1-51-4': 'Compared to the more constant growth of [MATH] or [MATH], the ratio [MATH] increases rapidly before stagnating around 0.03.', '1711.01417-1-51-5': 'Interestingly, if we approximate the star formation efficiency per free-fall time of our clouds with this ratio, the value agrees with typically observed efficiencies in molecular clouds .', '1711.01417-1-51-6': 'This is consistent with the determination of star formation efficiency by the dynamics of gravitational collapse, though our relatively small sample and lack of feedback modelling does not allow definitive conclusions to be drawn.', '1711.01417-1-52-0': 'Subsequently, we follow the evolution of the mass contained in the fragments relative to the mass contained in the filaments, [MATH].', '1711.01417-1-52-1': 'We plot the ratios in the middle panel in Fig. [REF].', '1711.01417-1-52-2': 'In all clouds, we see that the [MATH] ratio increases rapidly within the first [MATH]2.5 Myr after the first fragments have formed, reaching maximal values of 15-40%.', '1711.01417-1-52-3': 'This demonstrates that the fragments accrete mass from their parental filaments efficiently as long as there is a sufficient gas reservoir accessible, as is the case at the beginning of the simulations.', '1711.01417-1-52-4': "In doing so, they take up a significant fraction of the filaments' masses.", '1711.01417-1-53-0': 'Another question that has been recently discussed in the literature is whether prestellar cores form in a regular pattern within filaments.', '1711.01417-1-53-1': 'Observations suggest that cores condense at regular intervals along their parental filaments .', '1711.01417-1-53-2': 'The mean separations between the cores appear to correlate with the properties of the respective filament, ranging from a few tenths to several parsecs. These observations seem consistent with theoretical models of periodic fragmentation .', '1711.01417-1-53-3': 'The instabilities causing fragmentation in these models have unique modes that depend on the initial conditions of the filament.', '1711.01417-1-53-4': 'The wavelengths of these modes then define the mean separations between the forming cores.', '1711.01417-1-53-5': 'However, other studies, both observational and theoretical , demonstrate that periodic fragmentation only occurs under special conditions (such as supersonic, purely compressive turbulent motions), if at all.', '1711.01417-1-53-6': 'In reality the conditions within the filaments are not as uniform as assumed by the models, so observed core patterns may also be the result of overlapping fragmentation modes.', '1711.01417-1-53-7': 'According to those studies, filaments commonly fragment in a disordered, cluster-like fashion.', '1711.01417-1-54-0': 'Using our data, we test whether the fragments in our sample form with uniform separations.', '1711.01417-1-54-1': 'Note that we can only detect separations that are larger than 0.4 pc due to the 0.1 pc resolution of the data grid and the 0.3 pc radius we assume for the fragments.', '1711.01417-1-54-2': 'We plot the separations in 3D space of the individual fragments to their individual closest neighbour within the same filament as a function of time in Fig. [REF].', '1711.01417-1-54-3': 'We see that the fragments form, on average, at distances exceeding 2 pc from their closest neighbour at the beginning of the fragmentation process, but approach each other with time down to [MATH]1 pc or even merge (below 0.4 pc).', '1711.01417-1-55-0': 'To answer the question of whether there is a typical fragmentation scale we need to consider the separations between closest neighbours at the moment the fragments form.', '1711.01417-1-55-1': 'These are summarised in Fig. [REF].', '1711.01417-1-55-2': 'If there were a typical separation we should see a significant peak at that particular scale length, or a sequence of aliased peaks with equal separations.', '1711.01417-1-55-3': 'Looking at the histograms, we might argue that we see such sequences in M3 and M8 with typical separations at 0.9 and 0.6 pc, respectively.', '1711.01417-1-55-4': 'Both numbers exceed the local Jeans length by a substantial factor ([MATH]0.1 pc), but are only a factor of 2.3-1.5 larger than our resolution limit.', '1711.01417-1-55-5': 'Furthermore, in the case of M4, we do not see any significant peak in the distribution.', '1711.01417-1-55-6': 'However, the number of fragments in the samples is too low to make a solid statement about whether there is a universal fragmentation scale length, and if it depends on the local physical conditions only.', '1711.01417-1-56-0': '## Properties and Evolution of Filaments in 2D', '1711.01417-1-57-0': 'In the previous subsections, we have studied the properties and evolution of the filaments and fragments that we identified based on the full 3D simulation data.', '1711.01417-1-57-1': 'In observations, however, such 3D data are not available .', '1711.01417-1-57-2': 'Instead, we observe the filaments projected onto the 2D plane of the sky.', '1711.01417-1-57-3': 'This raises the questions of what we would observe if we apply our methods to the projected data and how these results compare to the results from 3D data.', '1711.01417-1-58-0': 'In this section, we approach these questions by projecting our 3D volume density cubes onto 2D column density maps.', '1711.01417-1-58-1': 'We project along the three major axes [MATH], [MATH], and [MATH], in order to account for line-of-sight (LoS) specific variance.', '1711.01417-1-58-2': 'Note that these maps lack any additional observational effects, such as noise, beam, or optical depth effects, and are thus only idealised approximations to real observations.', '1711.01417-1-58-3': 'However, such maps give the best case scenario to test the 3D-2D correspondence.', '1711.01417-1-58-4': 'Since we neglect opacity, our column density maps are primarily comparable to optically thin (sub-)mm dust observations.', '1711.01417-1-58-5': 'To produce comparable synthetic spectral line emission maps, we would need to consider chemical abundances, as well as gas velocity, which is beyond the scope of this paper.', '1711.01417-1-59-0': 'Analogously to Sect. [REF], we use DisPerSe to identify 2D filaments within the column density maps.', '1711.01417-1-59-1': 'For this purpose, we convert the number density thresholds used in 3D into column density thresholds by assuming a path length of 0.1 pc, namely [MATH] cm[MATH] (corresponding to [MATH] cm[MATH]) and [MATH] cm[MATH] (corresponding to [MATH] cm[MATH]).', '1711.01417-1-59-2': 'We emphasise that the skeletons of the 2D filaments are independently identified based on the column density distributions and not the projections of the 3D filament skeletons.', '1711.01417-1-59-3': 'Fig. [REF] show an example of the structures obtained, and Table [REF] summarises their properties.', '1711.01417-1-59-4': "We see that, similar to the structures detected in 3D, the 2D filaments' properties are influenced by the identification threshold, with the lengths of the filaments becoming shorter and line masses higher with higher thresholds.", '1711.01417-1-60-0': 'Comparing the structures detected in 3D and 2D, however, reveals a more significant difference.', '1711.01417-1-60-1': 'As shown in the example in Fig. [REF], we do find 2D counterparts for all 3D filaments, but there are 2D filaments that do not have matching 3D filaments identified with the same identification threshold.', '1711.01417-1-60-2': 'This finding is a consequence of the projection: structures with high volume densities typically have high column densities; conversely other structures with lower volume densities can appear denser in column densities, amplified by projection.', '1711.01417-1-61-0': 'This also influences the measured properties of the 2D filaments, such as the 2D line masses, [MATH].', '1711.01417-1-61-1': 'Analogously to Fig. [REF], Fig. [REF] shows the value of [MATH] for the dense 2D filaments as a function of time.', '1711.01417-1-61-2': 'We see that the values lie between the values we measured based on the diffuse and dense 3D filaments.', '1711.01417-1-61-3': 'If the dense 2D filaments exclusively represented the projection of dense 3D filaments, however, we would expect the trends in average line mass to evolve similarly.', '1711.01417-1-61-4': 'This is not seen here.', '1711.01417-1-61-5': 'On the contrary, the growth of [MATH] in the dense 2D filaments correlates with the growth of [MATH] of the diffuse 3D filaments.', '1711.01417-1-61-6': 'As described before, this is because most of the dense 2D filaments are projections of diffuse 3D filaments, so the line mass of the 3D filaments is the main contributor of the average line mass of the dense 2D filaments.', '1711.01417-1-61-7': 'Additional mass comes from material along the line of sight, so [MATH].', '1711.01417-1-62-0': 'In summary, for a given identification threshold, all 3D filaments have counterparts in column density maps, but not necessarily vice-versa.', '1711.01417-1-62-1': 'Projection not only maps dense 3D filaments onto the plane of the sky, but also merges less dense structures along the same line of sight, producing structures that exceed the column density threshold.', '1711.01417-1-62-2': 'Consequently, a comparison of the properties of 3D to 2D filaments is not directly possible if the corresponding identification thresholds are used for both samples.', '1711.01417-1-62-3': 'However, if this is taken into account, the properties of 3D and 2D filaments evolve similarly, but with an offset due to the additional line of sight mass projected onto the 2D filaments.', '1711.01417-1-63-0': 'The difference in the density distributions measured in 3D and 2D is even more obvious in the overall properties of the clouds.', '1711.01417-1-63-1': 'We measure the DGMF using column density by defining clouds as coherent volumes of gas with minimal number densities of 100 cm[MATH].', '1711.01417-1-63-2': 'We choose a minimum path length of 0.6 pc based on the assumption that filaments and fragments have radii of 0.3 pc.', '1711.01417-1-63-3': 'Note that this path length is not identical to the path length we used for computing the identification threshold of the column density filaments, [MATH], before.', '1711.01417-1-63-4': 'However, by using a minimum path length of 0.6 pc ensures that the fraction of gas we hereafter consider mirrors gas of the real 3D cloud.', '1711.01417-1-63-5': 'This way, we cannot only more reliably compare the DGMFs measured based on the 2D data with those based on the 3D data, but also with DGMFs in observed molecular clouds.', '1711.01417-1-63-6': 'As a results we obtain a minimal column density of [MATH] cm[MATH].', '1711.01417-1-63-7': 'We then compute [EQUATION] with [MATH] being the column density at time [MATH] and within the pixel [MATH], [MATH] the column density above which the gas is defined as dense and [EQUATION]', '1711.01417-1-63-8': 'In Fig. [REF], the dashed lines show the evolution of the DGMF measured using this equation.', '1711.01417-1-64-0': 'Analogous to Sect. [REF], we use two thresholds for tracing the evolution of dense gas within the clouds, namely [MATH] cm[MATH] (corresponding to [MATH] = 1,000 cm[MATH]) and [MATH] cm[MATH] (corresponding to [MATH] = 5,000 cm[MATH]).', '1711.01417-1-64-1': 'We see that for most of the time the DGMF is about an order of magnitude higher than the corresponding DGMFs calculated from the volume density distributions, but maximal values (70-80% for [MATH] = 1,000 cm[MATH] and 30-50% for [MATH] = 5,000 cm[MATH]) agree with each other.', '1711.01417-1-64-2': 'In the case of a lower value of [MATH], we see that the 2D DGMFs are almost constant in time, or even slightly decreasing, in disagreement with with the steady growth of the 3D DGMFs.', '1711.01417-1-65-0': 'In summary, we see that the DGMF measured in 2D deviates from the true 3D value for most of the initial evolution of a particular cloud.', '1711.01417-1-65-1': 'The DGMF measured in column density may show a completely different temporal behaviour, particularly when a low column density threshold is used to define dense gas.', '1711.01417-1-65-2': 'However, we also see that the individual filaments we identify in the 3D and 2D data and their properties agree decently with each other if the identification threshold in column density focuses on the range of volume density one wants to study and distinguishes unassociated gas along the line of sight.', '1711.01417-1-66-0': '# Summary Conclusions', '1711.01417-1-67-0': 'In this paper we analyse the properties and fragmentation of filaments forming within 3D AMR FLASH simulations of the self-gravitating, magnetised, supernova-driven ISM by [CITATION].', '1711.01417-1-67-1': 'Our main results are as follows.', '1711.01417-1-68-0': 'Our results indicate that filament fragmentation is affected by the environment of the cloud they form in.', '1711.01417-1-68-1': 'In order to understand the onset and development of fragmentation, future theoretical studies likely need to abandon the hydrostatic initial condition and to consider the formation of filaments and their subsequent fragmentation together.', '1711.01417-1-69-0': 'Furthermore, our results demonstrate that establishing common practices for how to define filaments in 3D and 2D data from simulations and observations is crucial for studying the properties and evolution of filaments and especially for comparing different filament studies with each other.', '1711.01417-1-69-1': 'Studies using filament finders must thoroughly test applicability of the adopted algorithms to address the problem in question.', '1711.01417-1-70-0': 'The authors acknowledge the support ESO and its Studentship Programme provided.', '1711.01417-1-70-1': 'This research made use of astrodendro, a Python package to compute dendrograms of astronomical data (http://www.dendrograms.org/).', '1711.01417-1-70-2': "This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 639459 (PROMISE).", '1711.01417-1-70-3': 'JCI-M and M-MML received support from US NSF grant AST11-09395.', '1711.01417-1-70-4': 'JCI-M was additionally supported by the DFG Priority Programme 157.', '1711.01417-1-70-5': 'M-MML also thanks the A. von Humboldt-Stiftung for support.', '1711.01417-1-71-0': '# Filament finders', '1711.01417-1-72-0': '## Algorithms', '1711.01417-1-73-0': 'There are a variety of algorithms publicly available for identifying filamentary structures in molecular clouds.', '1711.01417-1-73-1': 'Naturally, there are more filament finders that work with 2D data than with 3D data, since 2D algorithms can directly be applied to observed data like intensity or column density maps, while 3D filament finders require data with a resolved third dimension, like the local standard of rest velocity observed spectroscopically.', '1711.01417-1-73-2': 'The latter is not only observationally demanding, but computationally expensive to analyse.', '1711.01417-1-74-0': 'In this Appendix, we present and compare the finder algorithms we have considered during our analysis and justify our choice of DisPerSe for the analysis performed in the main paper.', '1711.01417-1-75-0': '### DisPerSe', '1711.01417-1-76-0': 'DisPerSe extracts coherent structures by evaluating the gradients between individual grid cells and the robustness of the topological features found.', '1711.01417-1-76-1': 'It was originally written for finding structures (both over- and under-densities) in cosmological data, but can be applied to other applications.', '1711.01417-1-77-0': 'The advantage of DisPerSe is that it is independent of the content and dimension of data it receives.', '1711.01417-1-77-1': 'Therefore, it can be directly applied to volume density cubes as well as column density maps and returns filamentary structures in both based on the same algorithms.', '1711.01417-1-78-0': '### FilFinder', '1711.01417-1-79-0': 'FilFinder was written to extract filamentary structures in molecular clouds observed by the Herschel Gould Belt Survey .', '1711.01417-1-79-1': 'It does this by reducing the areas of interest (parts of molecular clouds with intensities above a specified threshold) to topological skeletons.', '1711.01417-1-79-2': 'Therefore, each element of the skeletons represents the medial position of the areas of interest within the boundaries.', '1711.01417-1-79-3': 'Unfortunately, FilFinder can only be applied to 2D maps.', '1711.01417-1-80-0': '### astrodendro', '1711.01417-1-81-0': 'astrodendro creates dendrogram trees representing the hierarchical structure of the underlying data.', '1711.01417-1-81-1': 'That means that the code reveals how individual regions are connected with each other.', '1711.01417-1-81-2': 'Those regions are then classified into trunks, branches, and leaves that represent molecular clouds, clumps and cores in our context.', '1711.01417-1-81-3': 'Thus, astrodendro does not identify filamentary structures, but it is well suited for tracing fragments within the filaments or for preparing large data sets, so the actual filament finder can focus on the regions of interest.', '1711.01417-1-82-0': '### SCIMES', '1711.01417-1-83-0': 'SCIMES works similarly to astrodendro (Sect. [REF]) using dendrograms.', '1711.01417-1-83-1': 'However, while astrodendro just detects structures, SCIMES weights the branches and leaves according to user-defined affinities (for example, minimal size, or maximal separation along position or velocity axes) and organises the dendrogram tree accordingly.', '1711.01417-1-83-2': 'It returns weighted branches as clusters that, in our case, represent individual filaments.', '1711.01417-1-83-3': 'Note that SCIMES itself is not a filament finder, but just returns regions that are likely to contain compact substructures, such as filaments.', '1711.01417-1-83-4': 'In order to obtain the filaments one additionally needs a filament finder.', '1711.01417-1-84-0': 'There are some advantages to combining SCIMES and DisPerSe.', '1711.01417-1-84-1': 'On the one hand, it is computationally more efficient to first create masks of the relevant regions before applying DisPerSe on those masks.', '1711.01417-1-84-2': 'On the other hand, most filament finders return their structures without any weighting.', '1711.01417-1-84-3': 'Hence, the users would need to distinguish the filaments from each other by hand, which is impractical for larger datasets like ours.', '1711.01417-1-84-4': 'This step can be transferred to SCIMES as well, by applying the masks on the filament finder outputs.', '1711.01417-1-85-0': '### Minimal Spanning Tree', '1711.01417-1-86-0': 'The minimal spanning trees algorithm is used for optimising costs by minimising the lengths of grids and efficiency of networks.', '1711.01417-1-86-1': 'It can also be used to find coherent, filamentary structures as, for example, [CITATION] have demonstrated.', '1711.01417-1-86-2': 'With the MST we can define filaments by connecting the leaves found by astrodendro representing pre-stellar cores in molecular clouds according to our requirements.', '1711.01417-1-87-0': 'We use this method as an alternative method for identifying filaments in 3D.', '1711.01417-1-87-1': 'DisPerSe also uses the MST algorithm to connect individual skeleton segments with each other.', '1711.01417-1-87-2': 'The difference here is that we use MSTs for connecting the fragments we identified with astrodendro and define the straight lines between them as filaments.', '1711.01417-1-87-3': "Naturally, the separation between the fragments is on average much larger than those between DisPerSe's segments, leading to less accurate curvatures.", '1711.01417-1-88-0': '## Comparison of Identified Filaments', '1711.01417-1-89-0': 'For evaluating and interpreting the results we have presented in Sect. [REF] it is essential to understand and compare the performance of the underlying filament finder algorithms.', '1711.01417-1-89-1': 'Fig. [REF] gives an example of this.', '1711.01417-1-89-2': 'The background of each panel shows the column density map of the M4 model projected along the [MATH]-axis.', '1711.01417-1-89-3': 'The purple dots represent the position of the fragments identified by astrodendro.', '1711.01417-1-89-4': 'The white lines illustrate the filaments found by DisPerSe (left), FilFinder (center), and MST (right) applied to fragments from astrodendro, when using a column density threshold of N[MATH] = 1.5 [MATH] cm[MATH].', '1711.01417-1-89-5': 'One sees that the codes return widely varying structures, although they agree well where the column density is highest.', '1711.01417-1-89-6': 'Only DisPerSe follows the filaments that connect the cloud to the ISM.', '1711.01417-1-89-7': 'This has a huge impact on the physical properties such as total length, enclosed mass and the field of interest in general.', '1711.01417-1-90-0': 'Since the different methods do not identify the same structures we cannot compare filaments individually.', '1711.01417-1-90-1': 'For evaluating the influence the underlying algorithms print on the structures, we measure the average line masses of all filaments detected within the respective cloud at a given time (analogously to Sect. [REF]).', '1711.01417-1-90-2': 'In Figs. [REF], [REF] and [REF] we compare those average line masses based on the filaments returned by the individual codes in 2D and 3D respectively.', '1711.01417-1-90-3': 'We see that not only does the morphology of the filaments differ significantly using different codes, but also the properties of the structures.', '1711.01417-1-91-0': 'We focus our further analysis on the filamentary skeletons identified by DisPerSe because we can automatically run the code on both our 2D and 3D and work with structures based on the same algorithm and parameter dependence.', '1711.01417-1-91-1': 'Furthermore, DisPerSe is the code that is least sensitive to the input parameters since it is the only code that considers gradients in the matter distribution automatically, giving the skeletons a physical meaning.'}

{'1711.01417-2-0-0': '# Introduction', '1711.01417-2-1-0': 'Filamentary structures in the interstellar medium (ISM) have been known and investigated for many years.', '1711.01417-2-1-1': 'Much effort has been invested in studying their morphology , properties , distribution within the Milky Way , and formation .', '1711.01417-2-1-2': 'More recently, it has been argued that filaments represent a crucial phase in the earliest stages of star formation, based on observational findings showing that pre-stellar cores and young stellar clusters are preferentially located within filaments or at intersections between them .', '1711.01417-2-1-3': 'This raises the question of what exactly the role of filaments is in controlling star formation .', '1711.01417-2-2-0': 'The usual model for filaments describes them as hydrostatic cylinders that fragment due to linear perturbations .', '1711.01417-2-2-1': 'However, some studies demonstrate that other environmental conditions, such as turbulence, accretion, or magnetic fields, can introduce additional fragmentation modes.', '1711.01417-2-2-2': 'Therefore, the question is whether a quasi-static description of the evolution of filaments is justified.', '1711.01417-2-2-3': 'One way to test this is to use numerical simulations that form filaments self-consistently and compare the properties of the resulting structures with those predicted by the simple model.', '1711.01417-2-2-4': 'In doing so, one can examine how each force, and combination of different forces, affects the evolution of filaments, and whether a quasi-static model can capture the essential physics involved in the process of fragmentation.', '1711.01417-2-3-0': 'Since the forces acting on filaments are not directly observable, one needs to analyse the properties of the filaments and fragments as observational diagnostics.', '1711.01417-2-3-1': 'The width of filaments and the spacing and distribution of fragments are widely-studied quantities that can be used.', '1711.01417-2-3-2': 'However, these quantities are highly degenerate in this context.', '1711.01417-2-3-3': 'For example, [CITATION] has demonstrated that, as long as self-gravity is not acting alone, the widths of the radial profiles of filaments are approximately constant over wide ranges of central density and global star formation efficiencies.', '1711.01417-2-3-4': 'Furthermore, since filaments are generally short-lived substructures within globally fragmenting molecular clouds and normally neither quiescent nor isolated, external gravitational potentials or pressures can introduce additional perturbations that either separate or compress the cores.', '1711.01417-2-4-0': "Another interesting parameter is the mass per unit length, or line mass, of a filament, which is argued to determine the filament's stability in the quasi-static model .", '1711.01417-2-4-1': 'Similar to the Jeans analysis for spheres, there is a critical line mass that marks the transition between states of equilibrium and gravitational collapse.', '1711.01417-2-4-2': 'In principle, the line mass is an easy to measure quantity since it only requires the length and the enclosed mass of the respective filament.', '1711.01417-2-4-3': 'However, there are many effects that need to be considered when measuring the line mass from observational data, such as inclination, optical depth, uncertainties in distance estimations, and overlap effects .', '1711.01417-2-4-4': 'Furthermore, and especially in the context of large filaments on galactic scales , it is not completely clear how mostly two-dimensional (2D) observational characteristics reflect the three-dimensional (3D) nature of the acting forces.', '1711.01417-2-5-0': 'The questions we address in this paper are: How do filaments evolve and fragment in numerical simulations?', '1711.01417-2-5-1': 'Is the fragmentation picture seen in simulations in agreement with the quasi-static analytic framework of gravitational fragmentation?', '1711.01417-2-5-2': 'To address these questions, we analyse a set of model clouds from 3D FLASH adaptive mesh refinement (AMR) simulations.', '1711.01417-2-5-3': 'We introduce the simulations and the analysis methods in Sect. [REF].', '1711.01417-2-5-4': 'We tested several filament finder codes and compared their performances in identifying filaments from simulations in Appendix [REF].', '1711.01417-2-5-5': 'The results of our analysis are outlined and discussed in Sect. [REF] and summarised in Sect. [REF].', '1711.01417-2-6-0': '# Methods', '1711.01417-2-7-0': '## Cloud models', '1711.01417-2-8-0': 'The filamentary structures we analyse in this paper come from numerical simulations of dense cloud formation by [CITATION].', '1711.01417-2-8-1': 'The simulations have been performed with the 3D magnetohydrodynamics (MHD), AMR FLASH code , and are of a [MATH] kpc[MATH] vertical box of the interstellar medium (ISM) in a disk galaxy, with the galactic midplane located in the middle of the box.', '1711.01417-2-8-2': 'Turbulence is driven by the injection of discrete, thermal supernova (SN) explosions.', '1711.01417-2-8-3': 'SN rates are normalized to the galactic SN rates , with 6.58 and 27.4 Myr[MATH] kpc[MATH] for Type Ia and core-collapse SNe, respectively.', '1711.01417-2-8-4': 'The positioning of SN explosions is random in the horizontal direction and exponentially decaying in the vertical direction with scale heights of 90 and 325 pc for type Ia and core-collapse SNe.', '1711.01417-2-8-5': 'SN clustering is also taken into account by assuming 3/5 of the core-collapse population are correlated in space and time.', '1711.01417-2-8-6': 'Magnetic fields are included in the simulation, with an initial field strength of 5 [MATH]G at the midplane that exponentially decays in the vertical direction.', '1711.01417-2-8-7': 'The fields are initially uniform and oriented in the horizontal direction.', '1711.01417-2-8-8': 'They are allowed to evolve self-consistently with the simulation.', '1711.01417-2-8-9': 'The simulations also include distributed photoelectric heating and radiative cooling.', '1711.01417-2-8-10': 'The photoelectric heating of dust grains is implemented with a background FUV intensity of [MATH] with a vertical scale height of 300 pc.', '1711.01417-2-8-11': 'Radiative cooling rates are appropriate for a solar metallicity, optically thin gas, with a constant ionization fraction of [MATH] for temperatures below [MATH] K , and resonant line cooling for collisionally ionized gas at higher temperatures .', '1711.01417-2-9-0': 'The simulation initially includes only a static galactic gravitational potential, accounting for a stellar disk and a spherical dark matter halo.', '1711.01417-2-9-1': 'The potential is uniform in the horizontal direction; in the vertical direction, for altitudes [MATH] kpc, it follows a parametrized model of the mass distribution of the Milky Way , and at higher altitudes, it smoothly transforms into the outer halo profile described by [CITATION] with a scale length of 20 kpc .', '1711.01417-2-9-2': 'The simulation runs for [MATH]250 Myr without gas self-gravity in order to develop a multiphase, turbulent ISM, where dense structures form self-consistently in turbulent, convergent flows.', '1711.01417-2-9-3': 'Self-gravity is then switched on to follow the formation and evolution of clouds over the next 6 Myr.', '1711.01417-2-10-0': 'Three dense clouds from the cloud population found in the simulation were selected for high-resolution re-simulation by [CITATION].', '1711.01417-2-10-1': 'Once one of these clouds was identified in the cloud catalogue, a higher-resolution refinement region was defined around the region where the cloud would form in a checkpoint prior to the onset of self-gravity.', '1711.01417-2-10-2': 'Gas self-gravity was then turned on, and the evolution and collapse of the cloud were followed.', '1711.01417-2-10-3': 'For our investigations, we focus on these three clouds and map a (40 pc)[MATH] volume enclosing each cloud with [MATH] grid cells, with an effective spatial resolution of [MATH] pc.', '1711.01417-2-10-4': 'We consider objects to be fully resolved if their local Jeans length [MATH], corresponding to a maximum resolved density at 10 K of [MATH] cm[MATH] .', '1711.01417-2-10-5': 'This means that we can trace fragmentation down to 0.4 pc, but cannot fully resolve objects that form at smaller scales.', '1711.01417-2-10-6': 'The clouds have total masses on the order of [MATH], [MATH], and [MATH] M[MATH] (hereafter denoted models M3, M4, and M8).', '1711.01417-2-10-7': 'Examples are shown in Fig. [REF].', '1711.01417-2-11-0': '## Filament Identification', '1711.01417-2-12-0': 'We use a publicly available filament finder that identifies and traces the structures within the model clouds.', '1711.01417-2-12-1': 'We choose DisPerSe as we can apply it directly to 3D volume density cubes, as well as to 2D column density maps, returning structures based on the same algorithm.', '1711.01417-2-12-2': 'Furthermore, DisPerSe identifies filaments by deriving the gradients within each grid cell and connecting maxima and saddle points with each other, which gives the detected skeletons a mathematical and physical meaning.', '1711.01417-2-13-0': 'We have compared the structures identified by DisPerSe with those detected by other codes (see Appendix [REF]).', '1711.01417-2-13-1': 'All of them return similar structures in the densest regions of the clouds, but show pronounced differences in the more diffuse envelopes.', '1711.01417-2-13-2': 'This behaviour has a significant impact on the measured properties of the filaments.', '1711.01417-2-13-3': 'The consequence is that studies that do not use the same filament finding techniques are not directly comparable unless it has been proven that the identified structures are indeed similar.', '1711.01417-2-14-0': 'In order to obtain results that can be qualitatively compared to observations we follow the approximation of [CITATION] and use two thresholds: (a) a low-density threshold at n[MATH] 100 cm[MATH] which defines the volume of the cloud, and corresponds very roughly to the density at which CO emission can first be detected; and (b) a high-density threshold at n[MATH] 5,000 cm[MATH] that points to the denser clumps within the cloud that have a high potential for forming stars .', '1711.01417-2-14-1': 'Note that we approach our number density resolution (see Sect. [REF]) when using this high-density threshold.', '1711.01417-2-14-2': 'However, we have found that using a lower threshold that is still of the same order of magnitude has no qualitative effect on the structures and properties of the filaments, or their time evolution in general (see also the analysis of the dense gas mass fraction in Sect. [REF] and Fig. [REF]).', '1711.01417-2-15-0': 'Fig. [REF] illustrates with an example how strongly the structures of the identified skeletons depend on the considered threshold.', '1711.01417-2-15-1': 'The map shows a column density map of M3 at [MATH]3.8 Myr that has been produced by projecting the volume density cube along the z-axis.', '1711.01417-2-15-2': 'The map illustrates a projection of the filament spines identified by DisPerSe in the 3D position-position-position space.', '1711.01417-2-15-3': 'Note that we only include the spines of the filaments with densities exceeding the density threshold used for their identification, i.e. 100 or 5000 cm[MATH].', '1711.01417-2-15-4': 'This restriction is necessary because of the way DisPerSe identifies filaments.', '1711.01417-2-15-5': 'DisPerSe only needs a seed point with a (column) density above the specified threshold, and then connects local maxima through tangent lines to the (column) density gradients.', '1711.01417-2-15-6': 'If no restriction to the (column) density gradients are set, DisPerSe will eventually connect all local maxima, even if they are below the specified (column) density threshold.', '1711.01417-2-15-7': 'In this work we set no restriction to the (column) density gradient for the filament identification.', '1711.01417-2-15-8': 'However, once the filamentary network was identified, we kept only those filaments whose spines are above our desired (column) density threshold.', '1711.01417-2-16-0': '## Filament Properties', '1711.01417-2-17-0': 'DisPerSe derives the skeletons of filaments based on the local gradients and user-set thresholds.', '1711.01417-2-17-1': 'It then uses the minimal spanning tree (MST) method to find the nearest neighbour for each element of the skeletons.', '1711.01417-2-17-2': 'Most other filament finders neglect this step and leave it to the user to assemble the individual points to filaments, which may not be as accurate as the algorithm DisPerSe offers.', '1711.01417-2-18-0': 'In the end, DisPerSe returns the starting points [MATH] and end points [MATH] of each segment [MATH] of an identified skeleton.', '1711.01417-2-18-1': 'With these points, we calculate the length of the filament, [EQUATION] with [MATH] being the direction vector pointing from [MATH] to [MATH].', '1711.01417-2-19-0': 'We define the volume of the filaments as the set of grid cells [MATH] that are located along the direction vector [MATH] between [MATH] and [MATH], or have any part that lies within a distance [MATH] to these cells.', '1711.01417-2-19-1': 'In Fig. [REF] we illustrate this in a two-dimensional example.', '1711.01417-2-19-2': 'The orange area represents the analytic volume of the filament while the blue area marks the grid cells that are actually used.', '1711.01417-2-20-0': 'The range of filament widths that we find is wide, partly due to the complexity of profiles (e.g., from crossing filaments) that makes it difficult to measure relevant quantities like the full-width half maximum.', '1711.01417-2-20-1': 'For simplicity, we assume that the radius of our filaments is [MATH] pc everywhere.', '1711.01417-2-20-2': 'This value agrees with average filament widths we find when fitting Gaussians to the line profiles of the filaments, although numerical resolution limits filaments in our model from collapsing to smaller extensions.', '1711.01417-2-21-0': 'The enclosed mass of a filament is given by, [EQUATION] where [MATH] is the mass density within the grid cell [MATH]; and the line mass by, [EQUATION]', '1711.01417-2-22-0': '## Fragment Identification and Properties', '1711.01417-2-23-0': 'To identify fragments, we use astrodendro.', '1711.01417-2-23-1': 'This method computes dendrogram trees that represent the hierarchical structure of the underlying matter distribution in order to unravel how sub-structures relate to each other.', '1711.01417-2-23-2': 'This way one can easily identify individual clumps, filaments, and fragments.', '1711.01417-2-24-0': 'We identify fragments as highest level leaves in the dendrograms derived with a minimal density threshold at minvalue = 5,000 cm[MATH] and a minimum of minnpix = 20 cells.', '1711.01417-2-24-1': 'We define the volume of the fragment as a sphere with a radius of [MATH] = 0.3 pc around the central position of the respective dendrogram leaf.', '1711.01417-2-25-0': '# Results Discussion', '1711.01417-2-26-0': '## Mean Properties and Evolution of 3D Filaments', '1711.01417-2-27-0': 'We begin our analysis by studying the average properties and time evolution of the filaments we find in our model clouds in 3D, and how those properties relate to the global characteristics of the surrounding clouds.', '1711.01417-2-28-0': 'In the top panel of Fig. [REF] we show the average line mass [MATH] of all the filaments identified in each of the three clouds as function of time [MATH] after self-gravity has been activated in the simulations.', '1711.01417-2-28-1': 'We see that the average line masses within all clouds increase with time and with increasing identification threshold [MATH].', '1711.01417-2-28-2': 'This is expected considering the structures we obtain when using different thresholds.', '1711.01417-2-28-3': 'When we apply a high threshold we obtain compact structures connecting the sites of first fragment formation, where the gas is more concentrated.', '1711.01417-2-28-4': 'In contrast, with a lower threshold we identify a larger number of low mass structures in more diffuse regions of the clouds (see Fig. [REF]).', '1711.01417-2-28-5': 'As a result, the average line mass of low density filaments is always lower than when using a higher threshold.', '1711.01417-2-29-0': 'The increase of average line masses over time can be understood from the fact that the clouds formed by compression of gas, due to supernova shocks and global turbulent motions, in the time before our analysis.', '1711.01417-2-29-1': 'Thus, at [MATH] Myr, when self-gravity is activated, the clouds are already highly self-gravitating and begin to collapse globally.', '1711.01417-2-29-2': 'As a result, the filaments within the clouds gain more mass with time, which increases their line masses.', '1711.01417-2-29-3': 'Only in the case of M3 does the growth of average line masses of the low-density filaments stagnate after the first megayear.', '1711.01417-2-29-4': 'However, the average line mass of the high-density filaments in all clouds steadily increases in time, suggesting that the filaments are collapsing gravitationally.', '1711.01417-2-29-5': 'Fig. [REF] shows a clearer picture of how the mass within the cloud is distributed, and its time evolution.', '1711.01417-2-29-6': 'The top panel shows the fraction of mass contained in filaments, [MATH], compared to the mass of the entire parental cloud, [MATH].', '1711.01417-2-29-7': 'In all clouds, the fraction rapidly and continuously increases in time.', '1711.01417-2-29-8': 'Since the clouds accrete mass from their environments, the growth of the filament mass fraction means that filaments are forming and growing at rates not directly correlated to the evolution of their parental clouds.', '1711.01417-2-29-9': 'A similar trend is observed in the mass fraction of the fragments with respect to the cloud mass for all three clouds (bottom panel Fig. [REF]).', '1711.01417-2-29-10': 'However, the middle panel showing the mass ratio of filaments to cores shows a steep rise when cores are formed, but then stagnates at later times suggesting that filaments and cores grow at the same rate.', '1711.01417-2-30-0': 'In order to confirm this behaviour, we take a closer look at the dense gas mass fraction (DGMF) of the clouds.', '1711.01417-2-30-1': 'We define the DGMF as the fraction of mass enclosed in cloud cells that contain gas above a given dense gas threshold, n[MATH], compared to the mass of the entire cloud: [EQUATION] with [MATH] being the number density at the time [MATH] and [EQUATION]', '1711.01417-2-30-2': 'Fig. [REF] shows the evolution of the DGMFs of the three simulated clouds using two different dense gas thresholds, namely [MATH] = 1,000 cm[MATH] and [MATH] = 5,000 cm[MATH].', '1711.01417-2-31-0': 'In all clouds we see that the DGMF continues growing until the end of the simulation, with maximal values of 55-70% for [MATH] = 1,000 cm[MATH] and 35-55% for [MATH] = 5,000 cm[MATH].', '1711.01417-2-31-1': 'These clouds continue to accrete mass from their surroundings , so we conclude that these clouds are collapsing faster than they are growing.', '1711.01417-2-32-0': 'We see a similar behaviour in Fig. [REF].', '1711.01417-2-32-1': 'In this figure, we show the fraction of gas above a given number density threshold as a function of this number density threshold, [MATH], for four individual time steps in the simulations.', '1711.01417-2-32-2': 'In all clouds we see that the maximum gas density grows with time until it reaches the limit of our numerical resolution.', '1711.01417-2-32-3': 'Note that although we reach densities of order 30,000 cm[MATH], we only resolve densities up to 8,000 cm[MATH] reliably.', '1711.01417-2-32-4': 'At the same time, the density distributions become flatter as the clouds evolve.', '1711.01417-2-32-5': 'This means that the clouds collapse in their entirety and compress the enclosed mass into denser substructures such as fragments without necessarily losing their diffuse envelopes.', '1711.01417-2-32-6': 'This likely occurs because of continuing accretion of diffuse gas from the surrounding ISM .', '1711.01417-2-33-0': 'In summary, we see that the average evolution of the filaments is influenced by the global kinematics of their parental cloud.', '1711.01417-2-33-1': 'In particular, the way that the cloud transforms the mass it accretes from the ISM into dense substructures is related to the formation of fragments within the filaments.', '1711.01417-2-33-2': 'However, we also see that the properties of the dense gas strongly depend on the parameters used to define and identify it, such as the threshold density.', '1711.01417-2-33-3': 'This might become important for our key question of how well analytic models evaluate the stability of filaments and predict their fragmentation behaviour.', '1711.01417-2-34-0': '## Properties and Evolution of individual 3D Filaments', '1711.01417-2-35-0': 'In this section we investigate the evolution of individual filaments and compare the properties of fragmenting filaments with others.', '1711.01417-2-35-1': 'We confront these properties with the predictions of analytic models.', '1711.01417-2-35-2': 'Table [REF] provides a summary of the properties of the examined filaments.', '1711.01417-2-36-0': 'In Fig. [REF] we show the line masses of individual filaments as a function of time.', '1711.01417-2-36-1': 'In the case of the low-density filaments (Fig. [REF]), the transition from filaments without embedded fragments to those with fragments occurs at very low line masses (4.0, 2.4, and 2.9 M[MATH] pc[MATH] for M3, M4, and M8, respectively).', '1711.01417-2-37-0': 'We compare the line masses at which fragmentation occurs in the model with the criterion for cylindrical filaments being in hydrostatic equilibrium.', '1711.01417-2-37-1': 'In particular, we focus on the model by [CITATION] which describes a filament as an infinitely long, isolated, isothermal cylinder filled with self-gravitating gas that is in balance between gravity and thermal pressure.', '1711.01417-2-37-2': 'In this model, the equilibrium configuration is uniquely characterised by the critical line mass [EQUATION] with [MATH] being the sound speed of the gas, [MATH] the gravitational constant, and [MATH] the gas temperature.', '1711.01417-2-38-0': 'According to [CITATION], a cylindrical filament is only thermally supported against collapse if its line mass remains below this critical value.', '1711.01417-2-38-1': 'Otherwise, initially small perturbations within the filament can grow under the influence of self-gravity, which then leads to radial collapse and fragmentation, as [CITATION], [CITATION], [CITATION], and [CITATION] describe in their fragmenting cylinder models.', '1711.01417-2-39-0': 'Thus, the cylindrical fragmentation model has become a commonly used description of a filament and set of initial conditions of its fragmentation.', '1711.01417-2-39-1': 'The key question is whether or not the evolution of the filaments in our simulations follows this criterion.', '1711.01417-2-39-2': 'Can the equilibrium configuration be regarded as a realistic initial condition for fragmentation?', '1711.01417-2-39-3': 'Is the equilibrium configuration, in fact, ever reached?', '1711.01417-2-40-0': 'To address these questions, we mark the range of the analytic values for the critical line masses for typical gas temperatures between 10 and 15 K with red areas in Fig. [REF].', '1711.01417-2-40-1': 'We see that the low-density filaments in our samples start fragmenting at line masses far below the predicted critical values, and hardly reach such high line masses even at later stages in their evolution.', '1711.01417-2-41-0': 'This suggests that the approximations of the cylindrical fragmentation model fail here.', '1711.01417-2-41-1': 'We see many differences between our filaments and those in the analytic model.', '1711.01417-2-41-2': 'In particular, our filaments are neither isolated, nor in hydrostatic equilibrium, nor cylindrically symmetric, as studies by, e.g., [CITATION] emphasise.', '1711.01417-2-41-3': 'Rather they are part of a hierarchically collapsing cloud, and interact with each other, e.g. by crossing each other or accreting gas.', '1711.01417-2-41-4': 'Thus, the filaments both are subject to external pressure and may have large density perturbations that are outside the regime of linear growth.', '1711.01417-2-42-0': 'Furthermore, studies have found that even filaments that are subcritical in terms of line mass can fragment.', '1711.01417-2-42-1': 'They show that fragmentation itself does not show any clear imprints of the forces that originally formed the fragment (e.g., shock waves, or cloud-cloud collisions).', '1711.01417-2-42-2': 'The conclusion is that there is no prediction for a threshold below which the fragmentation of filaments is prevented.', '1711.01417-2-43-0': 'The situation changes when we look at the high-density filaments (Fig. [REF]).', '1711.01417-2-43-1': 'Here we observe maximal line masses of filaments without fragments between 18-20 M[MATH] pc[MATH] in all clouds.', '1711.01417-2-43-2': 'These transitional line masses appear to be in agreement with the critical line mass of the cylindrical fragmentation model.', '1711.01417-2-43-3': 'The reason for this is that the properties of the filaments here are more likely comparable to those of the cylindrical fragmentation model since they are more likely isothermal, straighter, and closer to the sites where overdensities form compared to more diffuse filaments.', '1711.01417-2-43-4': 'Consequently, the objects we study here agree better with the properties of the analytic cylinders.', '1711.01417-2-43-5': 'However, we detect only one dense filament without embedded fragment for one single time step (in M8) and, thus, lack a statistically meaningful sample to draw final conclusions about the capability to predict the fragmentation behaviour of high-density filaments.', '1711.01417-2-43-6': 'From the analysis of the low-density filaments, though, we see that the configuration represented by the analytic model of cylindrical fragmentation is not universally part of the evolution of filaments in the simulations.', '1711.01417-2-44-0': 'We conclude that a simple cylindrical model as described by [CITATION] and the cylindrical fragmentation model does not represent the typical initial conditions for the fragmentation of a filament in our molecular cloud simulations.', '1711.01417-2-44-1': 'Therefore, it is not a complete model for evaluating the stability of filaments.', '1711.01417-2-44-2': 'To predict the fragmentation of a filament a more complex model is essential, one that not only considers the balance between internal self-gravity and thermal pressure, but also other forces, such as external hydrostatic, turbulent, or magnetic pressure.', '1711.01417-2-44-3': 'Furthermore, connecting such a model with observations needs to take into account that the properties of the filaments derived strongly depend on the parameters used to identify the filaments (as well as the filament finder code used, as discussed in Appendix [REF]).', '1711.01417-2-44-4': 'This also means that whether or not observed filaments fragment as predicted by a cylindrical fragmentation model is strongly influenced by the identification parameters, as one can always choose parameters in a way that it perfectly suits the model.', '1711.01417-2-44-5': 'In our case, this is represented by the high-density filaments that only then contain fragments after exceeding the critical line mass.', '1711.01417-2-44-6': 'Since there is no unique, universal and physically motivated definition of what filaments the fragmentation models are not universal themselves.', '1711.01417-2-45-0': '## Properties and Evolution of Fragments', '1711.01417-2-46-0': 'In this section, we discuss the properties of the fragments we have detected within our clouds using astrodendro (Sect. [REF]), including their time evolution, and connection to their parental filaments and to each other.', '1711.01417-2-46-1': 'The main properties of the fragments are summarised in Table [REF] showing that those fragments have similar properties as dense regions and condensations in star-forming regions .', '1711.01417-2-47-0': 'In order to find evidence for the fragments in our simulation forming by gravitational fragmentation following the cylindrical fragmentation criterion, we make a first estimate of how gravitationally bound our fragments are.', '1711.01417-2-47-1': 'For this, we compute their virial parameters, given by : [EQUATION] for each of the fragments at any time step.', '1711.01417-2-47-2': 'Here, [MATH] is the gravitational constant, [MATH] the enclosed mass of the fragment, [MATH] = 0.3 pc its radius and [MATH] its velocity dispersion, which we calculate following Equations (9) (10) in [CITATION]: [EQUATION] with [MATH] being the central position of the fragment, [MATH] the average gas velocity vector and [MATH] the average sound speed.', '1711.01417-2-47-3': 'We note that this estimate only takes the ratio between volumetric kinetic and gravitational energy into account.', '1711.01417-2-47-4': 'Other terms, such as magnetic fields or surface thermal or kinetic pressure , are neglected.', '1711.01417-2-47-5': 'This is a common approximation used in both observational and theoretical studies, although [CITATION] suggest that these additional terms can be as important for objects embedded within molecular clouds as the balance of volumetric kinetic and gravitational energy.', '1711.01417-2-48-0': 'In the top row of Fig. [REF] we show a histogram of virial parameters measured for the fragments at the time when they have been first detected.', '1711.01417-2-48-1': 'According to Equ.', '1711.01417-2-48-2': '([REF]), the fragments are gravitationally bound when [MATH].', '1711.01417-2-48-3': 'This criterion is fulfilled by only 20% of the fragments.', '1711.01417-2-48-4': 'The majority of fragments have virial parameters close to the mode of 3.5.', '1711.01417-2-48-5': 'We argue that these fragments may nevertheless be bound objects because of the terms not accounted for by Equ.', '1711.01417-2-48-6': '([REF]), particularly the surface terms resulting from the collapse of the surrounding cloud.', '1711.01417-2-48-7': 'Furthermore, we see that the virial parameters decrease as the fragments evolve, which means that they become more bound.', '1711.01417-2-48-8': 'The histogram in the bottom row of Fig. [REF] confirms this as it shows that most of the fragments have virial parameters around 2 in the last simulated time steps.', '1711.01417-2-49-0': 'Note that, as explained by [CITATION], we can only give lower limits on the velocity dispersions, and thus the virial parameters, since we underresolve the turbulence on small scales in the simulations.', '1711.01417-2-49-1': "Resolving the subgrid scale turbulence may increase the energy by 25% , but likely won't prevent the fragments from beginning to collapse before the criteria for cylindrical fragmentation are fulfilled.", '1711.01417-2-49-2': "This consequently means that the fragmentation must be driven by neglected terms in the virial equation that can produce significant differences in the estimation of the fragments' boundness.", '1711.01417-2-50-0': 'In the bottom row of Fig. [REF] we plot the number of identified fragments as function of time.', '1711.01417-2-50-1': 'We see that the number of fragments overall increases with time for each simulation.', '1711.01417-2-50-2': 'However, there are cases when the number of fragments drops.', '1711.01417-2-50-3': 'The missing fragments are either disrupted, for example by shock waves or intracloud turbulence, or merge with each other, meaning that they approach each other too closely ([MATH]0.4 pc) to be distinguishable with our resolution.', '1711.01417-2-51-0': "We see that the first fragments form within the first 2 Myr, corresponding to 25-50% of the clouds' free-fall times and the period when the growth of the DGMF is steepest (see Fig. [REF]).", '1711.01417-2-51-1': 'This indicates that the formation of fragments is primarily dominated by the compression of dense gas within the cloud.', '1711.01417-2-51-2': 'This is verified by the ratio of mass contained within the fragments, [MATH], relative to the mass of the entire cloud, [MATH], shown in the bottom panel of Fig. [REF].', '1711.01417-2-51-3': 'The [MATH] ratios evolve in a similar fashion as the DGMF.', '1711.01417-2-51-4': 'Compared to the more constant growth of [MATH] or [MATH], the ratio [MATH] increases rapidly before stagnating around 0.03.', '1711.01417-2-51-5': 'Interestingly, if we approximate the star formation efficiency per free-fall time of our clouds with this ratio, the value agrees with typically observed efficiencies in molecular clouds .', '1711.01417-2-51-6': 'This is consistent with the determination of star formation efficiency by the dynamics of gravitational collapse, though our relatively small sample and lack of feedback modelling does not allow definitive conclusions to be drawn.', '1711.01417-2-52-0': 'Subsequently, we follow the evolution of the mass contained in the fragments relative to the mass contained in the filaments, [MATH].', '1711.01417-2-52-1': 'We plot the ratios in the middle panel in Fig. [REF].', '1711.01417-2-52-2': 'In all clouds, we see that the [MATH] ratio increases rapidly within the first [MATH]2.5 Myr after the first fragments have formed, reaching maximal values of 15-40%.', '1711.01417-2-52-3': 'This demonstrates that the fragments accrete mass from their parental filaments efficiently as long as there is a sufficient gas reservoir accessible, as is the case at the beginning of the simulations.', '1711.01417-2-52-4': "In doing so, they take up a significant fraction of the filaments' masses.", '1711.01417-2-53-0': 'Another question that has been recently discussed in the literature is whether prestellar cores form in a regular pattern within filaments.', '1711.01417-2-53-1': 'Observations suggest that cores condense at regular intervals along their parental filaments .', '1711.01417-2-53-2': 'The mean separations between the cores appear to correlate with the properties of the respective filament, ranging from a few tenths to several parsecs. These observations seem consistent with theoretical models of periodic fragmentation .', '1711.01417-2-53-3': 'The instabilities causing fragmentation in these models have unique modes that depend on the initial conditions of the filament.', '1711.01417-2-53-4': 'The wavelengths of these modes then define the mean separations between the forming cores.', '1711.01417-2-53-5': 'However, other studies, both observational and theoretical , demonstrate that periodic fragmentation only occurs under special conditions (such as supersonic, purely compressive turbulent motions), if at all.', '1711.01417-2-53-6': 'In reality the conditions within the filaments are not as uniform as assumed by the models, so observed core patterns may also be the result of overlapping fragmentation modes.', '1711.01417-2-53-7': 'According to those studies, filaments commonly fragment in a disordered, cluster-like fashion.', '1711.01417-2-54-0': 'Using our data, we test whether the fragments in our sample form with uniform separations.', '1711.01417-2-54-1': 'Note that we can only detect separations that are larger than 0.4 pc due to the 0.1 pc resolution of the data grid and the 0.3 pc radius we assume for the fragments.', '1711.01417-2-54-2': 'We plot the separations in 3D space of the individual fragments to their individual closest neighbour within the same filament as a function of time in Fig. [REF].', '1711.01417-2-54-3': 'We see that the fragments form, on average, at distances exceeding 2 pc from their closest neighbour at the beginning of the fragmentation process, but approach each other with time down to [MATH]1 pc or even merge (below 0.4 pc).', '1711.01417-2-55-0': 'To answer the question of whether there is a typical fragmentation scale we need to consider the separations between closest neighbours at the moment the fragments form.', '1711.01417-2-55-1': 'These are summarised in Fig. [REF].', '1711.01417-2-55-2': 'If there were a typical separation we should see a significant peak at that particular scale length, or a sequence of aliased peaks with equal separations.', '1711.01417-2-55-3': 'Looking at the histograms, we might argue that we see such sequences in M3 and M8 with typical separations at 0.9 and 0.6 pc, respectively.', '1711.01417-2-55-4': 'Both numbers exceed the local Jeans length by a substantial factor ([MATH]0.1 pc), but are only a factor of 2.3-1.5 larger than our resolution limit.', '1711.01417-2-55-5': 'Furthermore, in the case of M4, we do not see any significant peak in the distribution.', '1711.01417-2-55-6': 'However, the number of fragments in the samples is too low to make a solid statement about whether there is a universal fragmentation scale length, and if it depends on the local physical conditions only.', '1711.01417-2-56-0': '## Properties and Evolution of Filaments in 2D', '1711.01417-2-57-0': 'In the previous subsections, we have studied the properties and evolution of the filaments and fragments that we identified based on the full 3D simulation data.', '1711.01417-2-57-1': 'In observations, however, such 3D data are not available .', '1711.01417-2-57-2': 'Instead, we observe the filaments projected onto the 2D plane of the sky.', '1711.01417-2-57-3': 'This raises the questions of what we would observe if we apply our methods to the projected data and how these results compare to the results from 3D data.', '1711.01417-2-58-0': 'In this section, we approach these questions by projecting our 3D volume density cubes onto 2D column density maps.', '1711.01417-2-58-1': 'We project along the three major axes [MATH], [MATH], and [MATH], in order to account for line-of-sight (LoS) specific variance.', '1711.01417-2-58-2': 'Note that these maps lack any additional observational effects, such as noise, beam, or optical depth effects, and are thus only idealised approximations to real observations.', '1711.01417-2-58-3': 'However, such maps give the best case scenario to test the 3D-2D correspondence.', '1711.01417-2-58-4': 'Since we neglect opacity, our column density maps are primarily comparable to optically thin (sub-)mm dust observations.', '1711.01417-2-58-5': 'To produce comparable synthetic spectral line emission maps, we would need to consider chemical abundances, as well as gas velocity, which is beyond the scope of this paper.', '1711.01417-2-59-0': 'Analogously to Sect. [REF], we use DisPerSe to identify 2D filaments within the column density maps.', '1711.01417-2-59-1': 'For this purpose, we convert the number density thresholds used in 3D into column density thresholds by assuming a path length of 0.1 pc, namely [MATH] cm[MATH] (corresponding to [MATH] cm[MATH]) and [MATH] cm[MATH] (corresponding to [MATH] cm[MATH]).', '1711.01417-2-59-2': 'We emphasise that the skeletons of the 2D filaments are independently identified based on the column density distributions and not the projections of the 3D filament skeletons.', '1711.01417-2-59-3': 'Fig. [REF] show an example of the structures obtained, and Table [REF] summarises their properties.', '1711.01417-2-59-4': "We see that, similar to the structures detected in 3D, the 2D filaments' properties are influenced by the identification threshold, with the lengths of the filaments becoming shorter and line masses higher with higher thresholds.", '1711.01417-2-60-0': 'Comparing the structures detected in 3D and 2D, however, reveals a more significant difference.', '1711.01417-2-60-1': 'As shown in the example in Fig. [REF], we do find 2D counterparts for all 3D filaments, but there are 2D filaments that do not have matching 3D filaments identified with the same identification threshold.', '1711.01417-2-60-2': 'This finding is a consequence of the projection: structures with high volume densities typically have high column densities; conversely other structures with lower volume densities can appear denser in column densities, amplified by projection.', '1711.01417-2-61-0': 'This also influences the measured properties of the 2D filaments, such as the 2D line masses, [MATH].', '1711.01417-2-61-1': 'Analogously to Fig. [REF], Fig. [REF] shows the value of [MATH] for the dense 2D filaments as a function of time.', '1711.01417-2-61-2': 'We see that the values lie between the values we measured based on the diffuse and dense 3D filaments.', '1711.01417-2-61-3': 'If the dense 2D filaments exclusively represented the projection of dense 3D filaments, however, we would expect the trends in average line mass to evolve similarly.', '1711.01417-2-61-4': 'This is not seen here.', '1711.01417-2-61-5': 'On the contrary, the growth of [MATH] in the dense 2D filaments correlates with the growth of [MATH] of the diffuse 3D filaments.', '1711.01417-2-61-6': 'As described before, this is because most of the dense 2D filaments are projections of diffuse 3D filaments, so the line mass of the 3D filaments is the main contributor of the average line mass of the dense 2D filaments.', '1711.01417-2-61-7': 'Additional mass comes from material along the line of sight, so [MATH].', '1711.01417-2-62-0': 'In summary, for a given identification threshold, all 3D filaments have counterparts in column density maps, but not necessarily vice-versa.', '1711.01417-2-62-1': 'Projection not only maps dense 3D filaments onto the plane of the sky, but also merges less dense structures along the same line of sight, producing structures that exceed the column density threshold.', '1711.01417-2-62-2': 'Consequently, a comparison of the properties of 3D to 2D filaments is not directly possible if the corresponding identification thresholds are used for both samples.', '1711.01417-2-62-3': 'However, if this is taken into account, the properties of 3D and 2D filaments evolve similarly, but with an offset due to the additional line of sight mass projected onto the 2D filaments.', '1711.01417-2-63-0': 'The difference in the density distributions measured in 3D and 2D is even more obvious in the overall properties of the clouds.', '1711.01417-2-63-1': 'We measure the DGMF using column density by defining clouds as coherent volumes of gas with minimal number densities of 100 cm[MATH].', '1711.01417-2-63-2': 'We choose a minimum path length of 0.6 pc based on the assumption that filaments and fragments have radii of 0.3 pc.', '1711.01417-2-63-3': 'Note that this path length is not identical to the path length we used for computing the identification threshold of the column density filaments, [MATH], before.', '1711.01417-2-63-4': 'However, by using a minimum path length of 0.6 pc ensures that the fraction of gas we hereafter consider mirrors gas of the real 3D cloud.', '1711.01417-2-63-5': 'This way, we cannot only more reliably compare the DGMFs measured based on the 2D data with those based on the 3D data, but also with DGMFs in observed molecular clouds.', '1711.01417-2-63-6': 'As a results we obtain a minimal column density of [MATH] cm[MATH].', '1711.01417-2-63-7': 'We then compute [EQUATION] with [MATH] being the column density at time [MATH] and within the pixel [MATH], [MATH] the column density above which the gas is defined as dense and [EQUATION]', '1711.01417-2-63-8': 'In Fig. [REF], the dashed lines show the evolution of the DGMF measured using this equation.', '1711.01417-2-64-0': 'Analogous to Sect. [REF], we use two thresholds for tracing the evolution of dense gas within the clouds, namely [MATH] cm[MATH] (corresponding to [MATH] = 1,000 cm[MATH]) and [MATH] cm[MATH] (corresponding to [MATH] = 5,000 cm[MATH]).', '1711.01417-2-64-1': 'We see that for most of the time the DGMF is about an order of magnitude higher than the corresponding DGMFs calculated from the volume density distributions, but maximal values (70-80% for [MATH] = 1,000 cm[MATH] and 30-50% for [MATH] = 5,000 cm[MATH]) agree with each other.', '1711.01417-2-64-2': 'In the case of a lower value of [MATH], we see that the 2D DGMFs are almost constant in time, or even slightly decreasing, in disagreement with with the steady growth of the 3D DGMFs.', '1711.01417-2-65-0': 'In summary, we see that the DGMF measured in 2D deviates from the true 3D value for most of the initial evolution of a particular cloud.', '1711.01417-2-65-1': 'The DGMF measured in column density may show a completely different temporal behaviour, particularly when a low column density threshold is used to define dense gas.', '1711.01417-2-65-2': 'However, we also see that the individual filaments we identify in the 3D and 2D data and their properties agree decently with each other if the identification threshold in column density focuses on the range of volume density one wants to study and distinguishes unassociated gas along the line of sight.', '1711.01417-2-66-0': '# Summary Conclusions', '1711.01417-2-67-0': 'In this paper we analyse the properties and fragmentation of filaments forming within 3D AMR FLASH simulations of the self-gravitating, magnetised, supernova-driven ISM by [CITATION].', '1711.01417-2-67-1': 'Our main results are as follows.', '1711.01417-2-68-0': 'Our results indicate that filament fragmentation is affected by the environment of the cloud they form in.', '1711.01417-2-68-1': 'In order to understand the onset and development of fragmentation, future theoretical studies likely need to abandon the hydrostatic initial condition and to consider the formation of filaments and their subsequent fragmentation together.', '1711.01417-2-69-0': 'Furthermore, our results demonstrate that establishing common practices for how to define filaments in 3D and 2D data from simulations and observations is crucial for studying the properties and evolution of filaments and especially for comparing different filament studies with each other.', '1711.01417-2-69-1': 'Studies using filament finders must thoroughly test applicability of the adopted algorithms to address the problem in question.', '1711.01417-2-70-0': 'The authors acknowledge the support ESO and its Studentship Programme provided.', '1711.01417-2-70-1': 'This research made use of astrodendro, a Python package to compute dendrograms of astronomical data (http://www.dendrograms.org/).', '1711.01417-2-70-2': "This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 639459 (PROMISE).", '1711.01417-2-70-3': 'JCI-M and M-MML received support from US NSF grant AST11-09395.', '1711.01417-2-70-4': 'JCI-M was additionally supported by the DFG Priority Programme 157.', '1711.01417-2-70-5': 'M-MML also thanks the A. von Humboldt-Stiftung for support.', '1711.01417-2-71-0': '# Filament finders', '1711.01417-2-72-0': '## Algorithms', '1711.01417-2-73-0': 'There are a variety of algorithms publicly available for identifying filamentary structures in molecular clouds.', '1711.01417-2-73-1': 'Naturally, there are more filament finders that work with 2D data than with 3D data, since 2D algorithms can directly be applied to observed data like intensity or column density maps, while 3D filament finders require data with a resolved third dimension, like the local standard of rest velocity observed spectroscopically.', '1711.01417-2-73-2': 'The latter is not only observationally demanding, but computationally expensive to analyse.', '1711.01417-2-74-0': 'In this Appendix, we present and compare the finder algorithms we have considered during our analysis and justify our choice of DisPerSe for the analysis performed in the main paper.', '1711.01417-2-75-0': '### DisPerSe', '1711.01417-2-76-0': 'DisPerSe extracts coherent structures by evaluating the gradients between individual grid cells and the robustness of the topological features found.', '1711.01417-2-76-1': 'It was originally written for finding structures (both over- and under-densities) in cosmological data, but can be applied to other applications.', '1711.01417-2-77-0': 'The advantage of DisPerSe is that it is independent of the content and dimension of data it receives.', '1711.01417-2-77-1': 'Therefore, it can be directly applied to volume density cubes as well as column density maps and returns filamentary structures in both based on the same algorithms.', '1711.01417-2-78-0': '### FilFinder', '1711.01417-2-79-0': 'FilFinder was written to extract filamentary structures in molecular clouds observed by the Herschel Gould Belt Survey .', '1711.01417-2-79-1': 'It does this by reducing the areas of interest (parts of molecular clouds with intensities above a specified threshold) to topological skeletons.', '1711.01417-2-79-2': 'Therefore, each element of the skeletons represents the medial position of the areas of interest within the boundaries.', '1711.01417-2-79-3': 'Unfortunately, FilFinder can only be applied to 2D maps.', '1711.01417-2-80-0': '### astrodendro', '1711.01417-2-81-0': 'astrodendro creates dendrogram trees representing the hierarchical structure of the underlying data.', '1711.01417-2-81-1': 'That means that the code reveals how individual regions are connected with each other.', '1711.01417-2-81-2': 'Those regions are then classified into trunks, branches, and leaves that represent molecular clouds, clumps and cores in our context.', '1711.01417-2-81-3': 'Thus, astrodendro does not identify filamentary structures, but it is well suited for tracing fragments within the filaments or for preparing large data sets, so the actual filament finder can focus on the regions of interest.', '1711.01417-2-82-0': '### SCIMES', '1711.01417-2-83-0': 'SCIMES works similarly to astrodendro (Sect. [REF]) using dendrograms.', '1711.01417-2-83-1': 'However, while astrodendro just detects structures, SCIMES weights the branches and leaves according to user-defined affinities (for example, minimal size, or maximal separation along position or velocity axes) and organises the dendrogram tree accordingly.', '1711.01417-2-83-2': 'It returns weighted branches as clusters that, in our case, represent individual filaments.', '1711.01417-2-83-3': 'Note that SCIMES itself is not a filament finder, but just returns regions that are likely to contain compact substructures, such as filaments.', '1711.01417-2-83-4': 'In order to obtain the filaments one additionally needs a filament finder.', '1711.01417-2-84-0': 'There are some advantages to combining SCIMES and DisPerSe.', '1711.01417-2-84-1': 'On the one hand, it is computationally more efficient to first create masks of the relevant regions before applying DisPerSe on those masks.', '1711.01417-2-84-2': 'On the other hand, most filament finders return their structures without any weighting.', '1711.01417-2-84-3': 'Hence, the users would need to distinguish the filaments from each other by hand, which is impractical for larger datasets like ours.', '1711.01417-2-84-4': 'This step can be transferred to SCIMES as well, by applying the masks on the filament finder outputs.', '1711.01417-2-85-0': '### Minimal Spanning Tree', '1711.01417-2-86-0': 'The minimal spanning trees algorithm is used for optimising costs by minimising the lengths of grids and efficiency of networks.', '1711.01417-2-86-1': 'It can also be used to find coherent, filamentary structures as, for example, [CITATION] have demonstrated.', '1711.01417-2-86-2': 'With the MST we can define filaments by connecting the leaves found by astrodendro representing pre-stellar cores in molecular clouds according to our requirements.', '1711.01417-2-87-0': 'We use this method as an alternative method for identifying filaments in 3D.', '1711.01417-2-87-1': 'DisPerSe also uses the MST algorithm to connect individual skeleton segments with each other.', '1711.01417-2-87-2': 'The difference here is that we use MSTs for connecting the fragments we identified with astrodendro and define the straight lines between them as filaments.', '1711.01417-2-87-3': "Naturally, the separation between the fragments is on average much larger than those between DisPerSe's segments, leading to less accurate curvatures.", '1711.01417-2-88-0': '## Comparison of Identified Filaments', '1711.01417-2-89-0': 'For evaluating and interpreting the results we have presented in Sect. [REF] it is essential to understand and compare the performance of the underlying filament finder algorithms.', '1711.01417-2-89-1': 'Fig. [REF] gives an example of this.', '1711.01417-2-89-2': 'The background of each panel shows the column density map of the M4 model projected along the [MATH]-axis.', '1711.01417-2-89-3': 'The purple dots represent the position of the fragments identified by astrodendro.', '1711.01417-2-89-4': 'The white lines illustrate the filaments found by DisPerSe (left), FilFinder (center), and MST (right) applied to fragments from astrodendro, when using a column density threshold of N[MATH] = 1.5 [MATH] cm[MATH].', '1711.01417-2-89-5': 'One sees that the codes return widely varying structures, although they agree well where the column density is highest.', '1711.01417-2-89-6': 'Only DisPerSe follows the filaments that connect the cloud to the ISM.', '1711.01417-2-89-7': 'This has a huge impact on the physical properties such as total length, enclosed mass and the field of interest in general.', '1711.01417-2-90-0': 'Since the different methods do not identify the same structures we cannot compare filaments individually.', '1711.01417-2-90-1': 'For evaluating the influence the underlying algorithms print on the structures, we measure the average line masses of all filaments detected within the respective cloud at a given time (analogously to Sect. [REF]).', '1711.01417-2-90-2': 'In Figs. [REF], [REF] and [REF] we compare those average line masses based on the filaments returned by the individual codes in 2D and 3D respectively.', '1711.01417-2-90-3': 'We see that not only does the morphology of the filaments differ significantly using different codes, but also the properties of the structures.', '1711.01417-2-91-0': 'We focus our further analysis on the filamentary skeletons identified by DisPerSe because we can automatically run the code on both our 2D and 3D and work with structures based on the same algorithm and parameter dependence.', '1711.01417-2-91-1': 'Furthermore, DisPerSe is the code that is least sensitive to the input parameters since it is the only code that considers gradients in the matter distribution automatically, giving the skeletons a physical meaning.'}

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[]

[['1711.01417-1-41-2', '1711.01417-2-41-2'], ['1711.01417-2-48-5', '1711.01417-3-53-4'], ['1711.01417-2-48-6', '1711.01417-3-53-4']]

[]

['1711.01417-1-48-1', '1711.01417-2-48-1']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1711.01417

{'1711.01417-3-0-0': 'The fragmentation of filaments in molecular clouds has attracted a lot of attention recently as there seems to be a close relation between the evolution of filaments and star formation.', '1711.01417-3-0-1': 'The study of the fragmentation process has been motivated by simple analytical models.', '1711.01417-3-0-2': 'However, only a few comprehensive studies have analysed the evolution of filaments using numerical simulations where the filaments form self-consistently as part of large-scale molecular cloud evolution.', '1711.01417-3-1-0': 'We address the early evolution of parsec-scale filaments that form within individual clouds.', '1711.01417-3-1-1': 'In particular, we focus on three questions: How do the line masses of filaments evolve?', '1711.01417-3-1-2': 'How and when do the filaments fragment?', '1711.01417-3-1-3': 'How does the fragmentation relate to the line masses of the filaments?', '1711.01417-3-2-0': 'We examine three simulated molecular clouds formed in kiloparsec-scale numerical simulations performed with the FLASH adaptive mesh refinement magnetohydrodynamic code.', '1711.01417-3-2-1': 'The simulations model a self-gravitating, magnetised, stratified, supernova-driven interstellar medium, including photoelectric heating and radiative cooling.', '1711.01417-3-2-2': 'We follow the evolution of the clouds for 6 Myr from the time self-gravity starts to act.', '1711.01417-3-2-3': 'We identify filaments using the DisPerSe algorithm, and compare the results to other filament-finding algorithms.', '1711.01417-3-2-4': 'We determine the properties of the identified filaments and compare them with the predictions of analytic filament stability models.', '1711.01417-3-3-0': 'The average line masses of the identified filaments, as well as the fraction of mass in filamentary structures, increases fairly continuously after the onset of self-gravity.', '1711.01417-3-3-1': "The filaments show fragmentation starting relatively early: the first fragments appear when the line masses lie well below the critical line mass of Ostriker's isolated hydrostatic equilibrium solution ([MATH]16 M[MATH] pc[MATH]), commonly used as a fragmentation criterion.", '1711.01417-3-3-2': 'The average line masses of filaments identified in three-dimensional volume density cubes increases far more quickly than those identified in two-dimensional column density maps.', '1711.01417-3-4-0': 'Our results suggest that hydrostatic or dynamic compression from the surrounding cloud has a significant impact on the early dynamical evolution of filaments.', '1711.01417-3-4-1': 'A simple model of an isolated, isothermal cylinder may not provide a good approach for fragmentation analysis.', '1711.01417-3-4-2': 'Caution must be exercised in interpreting distributions of properties of filaments identified in column density maps, especially in the case of low-mass filaments.', '1711.01417-3-4-3': 'Comparing or combining results from studies that use different filament finding techniques is strongly discouraged.', '1711.01417-3-5-0': '# Introduction', '1711.01417-3-6-0': 'Filamentary structures in the interstellar medium (ISM) have been known and investigated for many years.', '1711.01417-3-6-1': 'Much effort has been invested in studying their morphology , properties , distribution within the Milky Way , and formation .', '1711.01417-3-6-2': 'More recently, it has been argued that filaments represent a crucial phase in the earliest stages of star formation, based on observational findings showing that pre-stellar cores and young stellar clusters are preferentially located within filaments or at intersections between them .', '1711.01417-3-6-3': 'This raises the question of what exactly the role of filaments is in controlling star formation .', '1711.01417-3-7-0': 'The usual model for filaments describes them as hydrostatic cylinders that fragment due to linear perturbations .', '1711.01417-3-7-1': 'However, some studies demonstrate that other environmental conditions, such as turbulence, accretion, or magnetic fields, can introduce additional fragmentation modes.', '1711.01417-3-7-2': 'Therefore, the question is whether a quasi-static description of the evolution of filaments is justified.', '1711.01417-3-7-3': 'One way to test this is to use numerical simulations that form filaments self-consistently and compare the properties of the resulting structures with those predicted by the simple model.', '1711.01417-3-7-4': 'In doing so, one can examine how each force, and combination of different forces, affects the evolution of filaments, and whether a quasi-static model can capture the essential physics involved in the process of fragmentation.', '1711.01417-3-8-0': 'Since the forces acting on filaments are not directly observable, one needs to analyse the properties of the filaments and fragments as observational diagnostics.', '1711.01417-3-8-1': 'The width of filaments and the spacing and distribution of fragments are widely-studied quantities that can be used.', '1711.01417-3-8-2': 'However, these quantities are highly degenerate in this context.', '1711.01417-3-8-3': 'For example, [CITATION] has demonstrated that, as long as self-gravity is not acting alone, the widths of the radial profiles of filaments are approximately constant over wide ranges of central density and global star formation efficiencies.', '1711.01417-3-8-4': 'Furthermore, since filaments are generally short-lived substructures within globally fragmenting molecular clouds and normally neither quiescent nor isolated, external gravitational potentials or pressures can introduce additional perturbations that either separate or compress the cores.', '1711.01417-3-9-0': "Another interesting parameter is the mass per unit length, or line mass, of a filament, which is argued to determine the filament's stability in the quasi-static model .", '1711.01417-3-9-1': 'Similar to the Jeans analysis for spheres, there is a critical line mass that marks the transition between states of equilibrium and gravitational collapse.', '1711.01417-3-9-2': 'In principle, the line mass is an easy-to-measure quantity since it only requires the length and the enclosed mass of the respective filament.', '1711.01417-3-9-3': 'However, there are many effects that need to be considered when measuring the line mass from observational data, such as inclination, optical depth, uncertainties in distance estimations, and overlap effects .', '1711.01417-3-9-4': 'Furthermore, and especially in the context of large filaments on galactic scales , it is not completely clear how mostly two-dimensional (2D) observational characteristics reflect the three-dimensional (3D) nature of the acting forces.', '1711.01417-3-10-0': 'The questions we address in this paper are: How do filaments evolve and fragment in numerical simulations?', '1711.01417-3-10-1': 'Is the fragmentation picture seen in simulations in agreement with the quasi-static analytic framework of gravitational fragmentation?', '1711.01417-3-10-2': 'To address these questions, we analyse a set of model clouds from 3D FLASH adaptive mesh refinement (AMR) simulations.', '1711.01417-3-10-3': 'We introduce the simulations and the analysis methods in Sect. [REF].', '1711.01417-3-10-4': 'We tested several filament finder codes and compared their performances in identifying filaments from simulations in Appendix [REF].', '1711.01417-3-10-5': 'The results of our analysis are outlined and discussed in Sect. [REF] and summarised in Sect. [REF].', '1711.01417-3-11-0': '# Methods', '1711.01417-3-12-0': '## Cloud models', '1711.01417-3-13-0': 'The filamentary structures we analyse in this paper come from numerical simulations of dense cloud formation by [CITATION].', '1711.01417-3-13-1': 'The simulations have been performed with the 3D magnetohydrodynamics (MHD), AMR FLASH code , and are of a [MATH] kpc[MATH] vertical box of the ISM in a disk galaxy, with the galactic midplane located in the middle of the box.', '1711.01417-3-13-2': 'Turbulence is driven by the injection of discrete, thermal supernova (SN) explosions.', '1711.01417-3-13-3': 'SN rates are normalized to the galactic SN rates , with 6.58 and 27.4 Myr[MATH] kpc[MATH] for Type Ia and core-collapse SNe, respectively.', '1711.01417-3-13-4': 'The positioning of SN explosions is random in the horizontal direction and exponentially decaying in the vertical direction with scale heights of 90 and 325 pc for type Ia and core-collapse SNe.', '1711.01417-3-13-5': 'SN clustering is also taken into account by assuming three fifths of the core-collapse population are correlated in space and time.', '1711.01417-3-13-6': 'Magnetic fields are included in the simulation, with an initial field strength of 5 [MATH]G at the midplane that exponentially decays in the vertical direction.', '1711.01417-3-13-7': 'The fields are initially uniform and oriented in the horizontal direction.', '1711.01417-3-13-8': 'They are allowed to evolve self-consistently with the simulation.', '1711.01417-3-13-9': 'The simulations also include distributed photoelectric heating and radiative cooling.', '1711.01417-3-13-10': 'The photoelectric heating of dust grains is implemented with a background FUV intensity of [MATH] with a vertical scale height of 300 pc.', '1711.01417-3-13-11': 'Radiative cooling rates are appropriate for a solar metallicity, optically thin gas, with a constant ionization fraction of [MATH] for temperatures below [MATH] K , and resonant line cooling for collisionally ionized gas at higher temperatures .', '1711.01417-3-14-0': 'The simulation initially includes only a static galactic gravitational potential, accounting for a stellar disk and a spherical dark matter halo.', '1711.01417-3-14-1': 'The potential is uniform in the horizontal direction; in the vertical direction, for altitudes [MATH] kpc, it follows a parametrized model of the mass distribution of the Milky Way , and at higher altitudes, it smoothly transforms into the outer halo profile described by [CITATION] with a scale length of 20 kpc .', '1711.01417-3-14-2': 'The simulation runs for [MATH]250 Myr without gas self-gravity in order to develop a multiphase, turbulent ISM, where dense structures form self-consistently in turbulent, convergent flows.', '1711.01417-3-14-3': 'Self-gravity is then switched on to follow the formation and evolution of clouds over the next 6 Myr.', '1711.01417-3-15-0': 'Three dense clouds from the cloud population found in the simulation were selected for high-resolution re-simulation by [CITATION].', '1711.01417-3-15-1': 'Once one of these clouds was identified in the cloud catalogue, a higher-resolution refinement region was defined around the region where the cloud would form in a checkpoint prior to the onset of self-gravity.', '1711.01417-3-15-2': 'Gas self-gravity was then turned on, and the evolution and collapse of the cloud were followed.', '1711.01417-3-15-3': 'For our investigations, we focus on these three clouds and map a (40 pc)[MATH] volume enclosing each cloud with [MATH] grid cells, with an effective spatial resolution of [MATH] pc.', '1711.01417-3-15-4': 'We consider objects to be fully resolved if their local Jeans length [MATH], corresponding to a maximum resolved density at 10 K of [MATH] cm[MATH] .', '1711.01417-3-15-5': 'This means that we can trace fragmentation down to 0.4 pc, but cannot fully resolve objects that form at smaller scales.', '1711.01417-3-15-6': 'The clouds have total masses on the order of [MATH], [MATH], and [MATH] M[MATH] (hereafter denoted models M3, M4, and M8).', '1711.01417-3-15-7': 'Examples are shown in Fig. [REF].', '1711.01417-3-16-0': '## Filament identification', '1711.01417-3-17-0': 'We use a publicly available filament finder that identifies and traces the structures within the model clouds.', '1711.01417-3-17-1': 'We choose DisPerSe as we can apply it directly to 3D volume density cubes, as well as to 2D column density maps, returning structures based on the same algorithm.', '1711.01417-3-17-2': 'Furthermore, DisPerSe identifies filaments by deriving the gradients within each grid cell and connecting maxima and saddle points with each other, which gives the detected skeletons a mathematical and physical meaning.', '1711.01417-3-18-0': 'We have compared the structures identified by DisPerSe with those detected by other codes (see Appendix [REF]).', '1711.01417-3-18-1': 'All of them return similar structures in the densest regions of the clouds, but show pronounced differences in the more diffuse envelopes.', '1711.01417-3-18-2': 'This behaviour has a significant impact on the measured properties of the filaments.', '1711.01417-3-18-3': 'The consequence is that studies that do not use the same filament finding techniques are not directly comparable unless it has been proven that the identified structures are indeed similar.', '1711.01417-3-19-0': 'In order to obtain results that can be qualitatively compared to observations we follow the approximation of [CITATION] and use two thresholds: (a) a low-density threshold at n[MATH] 100 cm[MATH] which defines the volume of the cloud, and corresponds very roughly to the density at which CO emission can first be detected; and (b) a high-density threshold at n[MATH] 5,000 cm[MATH] that points to the denser clumps within the cloud that have a high potential for forming stars .', '1711.01417-3-19-1': 'We note that we approach our number density resolution (see Sect. [REF]) when using this high-density threshold.', '1711.01417-3-19-2': 'However, we have found that using a lower threshold that is still of the same order of magnitude has no qualitative effect on the structures and properties of the filaments, or their time evolution in general (see also the analysis of the dense gas mass fraction in Sect. [REF] and Fig. [REF]).', '1711.01417-3-20-0': 'Figure [REF] illustrates with an example how strongly the structures of the identified skeletons depend on the considered threshold.', '1711.01417-3-20-1': 'The map shows a column density map of M3 at [MATH]3.8 Myr that has been produced by projecting the volume density cube along the z-axis.', '1711.01417-3-20-2': 'The map illustrates a projection of the filament spines identified by DisPerSe in the 3D position-position-position space.', '1711.01417-3-20-3': 'We note that we only include the spines of the filaments with densities exceeding the density threshold used for their identification, that is, 100 or 5000 cm[MATH].', '1711.01417-3-20-4': 'This restriction is necessary because of the way DisPerSe identifies filaments.', '1711.01417-3-20-5': 'DisPerSe only needs a seed point with a (column) density above the specified threshold, and then connects local maxima through tangent lines to the (column) density gradients.', '1711.01417-3-20-6': 'If no restriction to the (column) density gradients is set, DisPerSe will eventually connect all local maxima, even if they are below the specified (column) density threshold.', '1711.01417-3-20-7': 'In this work we set no restriction to the (column) density gradient for the filament identification.', '1711.01417-3-20-8': 'However, once the filamentary network was identified, we kept only those filaments whose spines are above our desired (column) density threshold.', '1711.01417-3-21-0': '## Filament properties', '1711.01417-3-22-0': 'DisPerSe derives the skeletons of filaments based on the local gradients and user-set thresholds.', '1711.01417-3-22-1': 'It then uses the minimal spanning tree (MST) method to find the nearest neighbour for each element of the skeletons.', '1711.01417-3-22-2': 'Most other filament finders neglect this step and leave it to the user to assemble the individual points to filaments, which may not be as accurate as the algorithm DisPerSe offers.', '1711.01417-3-23-0': 'In the end, DisPerSe returns the starting points [MATH] and end points [MATH] of each segment [MATH] of an identified skeleton.', '1711.01417-3-23-1': 'With these points, we calculate the length of the filament, [EQUATION] with [MATH] being the direction vector pointing from [MATH] to [MATH].', '1711.01417-3-24-0': 'We define the volume of the filaments as the set of grid cells [MATH] that are located along the direction vector [MATH] between [MATH] and [MATH], or have any part that lies within a distance [MATH] to these cells.', '1711.01417-3-24-1': 'In Fig. [REF] we illustrate this in a 2D example.', '1711.01417-3-24-2': 'The orange area represents the analytic volume of the filament while the blue area marks the grid cells that are actually used.', '1711.01417-3-25-0': 'The range of filament widths that we find is wide, partly due to the complexity of profiles (e.g. from crossing filaments) that makes it difficult to measure relevant quantities like the full-width at half maximum.', '1711.01417-3-25-1': 'For simplicity, we assume that the radius of our filaments is [MATH] pc everywhere.', '1711.01417-3-25-2': 'This value agrees with average filament widths we find when fitting Gaussians to the line profiles of the filaments, although numerical resolution limits filaments in our model from collapsing to smaller extensions.', '1711.01417-3-26-0': 'The enclosed mass of a filament is given by, [EQUATION] where [MATH] is the mass density within the grid cell [MATH]; and the line mass by, [EQUATION]', '1711.01417-3-27-0': '## Fragment identification and properties', '1711.01417-3-28-0': 'To identify fragments, we use astrodendro.', '1711.01417-3-28-1': 'This method computes dendrogram trees that represent the hierarchical structure of the underlying matter distribution in order to unravel how sub-structures relate to each other.', '1711.01417-3-28-2': 'This way one can easily identify individual clumps, filaments, and fragments.', '1711.01417-3-29-0': 'We identify fragments as highest-level leaves in the dendrograms derived with a minimal density threshold at minvalue = 5,000 cm[MATH] and a minimum of minnpix = 20 cells.', '1711.01417-3-29-1': 'We define the volume of the fragment as a sphere with a radius of [MATH] = 0.3 pc around the central position of the respective dendrogram leaf.', '1711.01417-3-30-0': '# Results Discussion', '1711.01417-3-31-0': '## Mean properties and evolution of 3D filaments', '1711.01417-3-32-0': 'We begin our analysis by studying the average properties and time evolution of the filaments we find in our model clouds in 3D, and how those properties relate to the global characteristics of the surrounding clouds.', '1711.01417-3-33-0': 'In the top panel of Fig. [REF] we show the average line mass [MATH] of all the filaments identified in each of the three clouds as a function of time [MATH] after self-gravity has been activated in the simulations.', '1711.01417-3-33-1': 'We see that the average line masses within all clouds increase with time and with increasing identification threshold [MATH].', '1711.01417-3-33-2': 'This is expected considering the structures we obtain when using different thresholds.', '1711.01417-3-33-3': 'When we apply a high threshold we obtain compact structures connecting the sites of first fragment formation, where the gas is more concentrated.', '1711.01417-3-33-4': 'In contrast, with a lower threshold we identify a larger number of low-mass structures in more diffuse regions of the clouds (see Fig. [REF]).', '1711.01417-3-33-5': 'As a result, the average line mass of low-density filaments is always lower than when using a higher threshold.', '1711.01417-3-34-0': 'The increase of average line masses over time can be understood from the fact that the clouds formed by compression of gas, due to supernova shocks and global turbulent motions, in the time before our analysis.', '1711.01417-3-34-1': 'Thus, at [MATH] Myr, when self-gravity is activated, the clouds are already highly self-gravitating and begin to collapse globally.', '1711.01417-3-34-2': 'As a result, the filaments within the clouds gain more mass with time, which increases their line masses.', '1711.01417-3-34-3': 'Only in the case of M3 does the growth of average line masses of the low-density filaments stagnate after the first megayear.', '1711.01417-3-34-4': 'However, the average line mass of the high-density filaments in all clouds steadily increases in time, suggesting that the filaments are collapsing gravitationally.', '1711.01417-3-34-5': 'Figure [REF] shows a clearer picture of how the mass within the cloud is distributed, and its time evolution.', '1711.01417-3-34-6': 'The top panel shows the fraction of mass contained in filaments, [MATH], compared to the mass of the entire parental cloud, [MATH].', '1711.01417-3-34-7': 'In all clouds, the fraction rapidly and continuously increases in time.', '1711.01417-3-34-8': 'Since the clouds accrete mass from their environments, the growth of the filament mass fraction means that filaments are forming and growing at rates not directly correlated to the evolution of their parental clouds.', '1711.01417-3-34-9': 'A similar trend is observed in the mass fraction of the fragments with respect to the cloud mass for all three clouds (bottom panel Fig. [REF]).', '1711.01417-3-34-10': 'However, the middle panel showing the mass ratio of filaments to cores shows a steep rise when cores are formed, but then stagnates at later times suggesting that filaments and cores grow at the same rate.', '1711.01417-3-35-0': 'In order to confirm this behaviour, we take a closer look at the dense gas mass fraction (DGMF) of the clouds.', '1711.01417-3-35-1': 'We define the DGMF as the fraction of mass enclosed in cloud cells that contain gas above a given dense gas threshold, n[MATH], compared to the mass of the entire cloud: [EQUATION] with [MATH] being the number density at the time [MATH] and [EQUATION]', '1711.01417-3-35-2': 'Figure [REF] shows the evolution of the DGMFs of the three simulated clouds using two different dense gas thresholds, namely [MATH] = 1,000 cm[MATH] and [MATH] = 5,000 cm[MATH].', '1711.01417-3-36-0': 'In all clouds we see that the DGMF continues growing until the end of the simulation, with maximal values of 55-70% for [MATH] = 1,000 cm[MATH] and 35-55% for [MATH] = 5,000 cm[MATH].', '1711.01417-3-36-1': 'These clouds continue to accrete mass from their surroundings , so we conclude that these clouds are collapsing faster than they are growing.', '1711.01417-3-37-0': 'We see a similar behaviour in Fig. [REF].', '1711.01417-3-37-1': 'In this Figure, we show the fraction of gas above a given number density threshold as a function of this number density threshold, [MATH], for four individual time steps in the simulations.', '1711.01417-3-37-2': 'In all clouds we see that the maximum gas density grows with time until it reaches the limit of our numerical resolution.', '1711.01417-3-37-3': 'We note that although we reach densities of order 30,000 cm[MATH], we only reliably resolve densities up to 8,000 cm[MATH] .', '1711.01417-3-37-4': 'At the same time, the density distributions become flatter as the clouds evolve.', '1711.01417-3-37-5': 'This means that the clouds collapse in their entirety and compress the enclosed mass into denser substructures such as fragments without necessarily losing their diffuse envelopes.', '1711.01417-3-37-6': 'This likely occurs because of continuing accretion of diffuse gas from the surrounding ISM .', '1711.01417-3-38-0': 'In summary, we see that the average evolution of the filaments is influenced by the global kinematics of their parental cloud.', '1711.01417-3-38-1': 'In particular, the way that the cloud transforms the mass it accretes from the ISM into dense substructures is related to the formation of fragments within the filaments.', '1711.01417-3-38-2': 'However, we also see that the properties of the dense gas strongly depend on the parameters used to define and identify it, such as the threshold density.', '1711.01417-3-38-3': 'This might become important for our key question of how well analytic models evaluate the stability of filaments and predict their fragmentation behaviour.', '1711.01417-3-39-0': '## Properties and evolution of individual 3D filaments', '1711.01417-3-40-0': 'In this Section we investigate the evolution of individual filaments and compare the properties of fragmenting filaments with others.', '1711.01417-3-40-1': 'We confront these properties with the predictions of analytic models.', '1711.01417-3-40-2': 'Table [REF] provides a summary of the properties of the examined filaments.', '1711.01417-3-41-0': 'In Fig. [REF] we show the line masses of individual filaments as a function of time.', '1711.01417-3-41-1': 'In the case of the low-density filaments (Fig. [REF]), the transition from filaments without embedded fragments to those with fragments occurs at very low line masses (4.0, 2.4, and 2.9 M[MATH] pc[MATH] for M3, M4, and M8, respectively).', '1711.01417-3-42-0': 'We compare the line masses at which fragmentation occurs in the model with the criterion for cylindrical filaments being in hydrostatic equilibrium.', '1711.01417-3-42-1': 'In particular, we focus on the model by [CITATION] which describes a filament as an infinitely long, isolated, isothermal cylinder filled with self-gravitating gas that is in balance between gravity and thermal pressure.', '1711.01417-3-42-2': 'In this model, the equilibrium configuration is uniquely characterised by the critical line mass [EQUATION] with [MATH] being the sound speed of the gas, [MATH] the gravitational constant, and [MATH] the gas temperature.', '1711.01417-3-43-0': 'According to [CITATION], a cylindrical filament is only thermally supported against collapse if its line mass remains below this critical value.', '1711.01417-3-43-1': 'Otherwise, initially small perturbations within the filament can grow under the influence of self-gravity, which then leads to radial collapse and fragmentation, as [CITATION], [CITATION], [CITATION], and [CITATION] describe in their fragmenting cylinder models.', '1711.01417-3-44-0': 'Thus, the cylindrical fragmentation model has become a commonly used description of a filament and set of initial conditions of its fragmentation.', '1711.01417-3-44-1': 'The key question is whether or not the evolution of the filaments in our simulations follows this criterion.', '1711.01417-3-44-2': 'Can the equilibrium configuration be regarded as a realistic initial condition for fragmentation?', '1711.01417-3-44-3': 'Is the equilibrium configuration, in fact, ever reached?', '1711.01417-3-45-0': 'To address these questions, we mark the range of the analytic values for the critical line masses for typical gas temperatures between 10 and 15 K with red areas in Fig. [REF].', '1711.01417-3-45-1': 'We see that the low-density filaments in our samples start fragmenting at line masses far below the predicted critical values, and hardly reach such high line masses even at later stages in their evolution.', '1711.01417-3-46-0': 'This suggests that the approximations of the cylindrical fragmentation model fail here.', '1711.01417-3-46-1': 'We see many differences between our filaments and those in the analytic model.', '1711.01417-3-46-2': 'In particular, our filaments are neither isolated, nor in hydrostatic equilibrium, nor cylindrically symmetric, as studies by, for example, [CITATION] emphasise.', '1711.01417-3-46-3': 'Rather, they are part of a hierarchically collapsing cloud, and interact with each other; for example by crossing each other or accreting gas.', '1711.01417-3-46-4': 'Thus, the filaments are both subject to external pressure and may have large density perturbations that are outside the regime of linear growth.', '1711.01417-3-47-0': 'Furthermore, studies have found that even filaments that are subcritical in terms of line mass can fragment.', '1711.01417-3-47-1': 'They show that fragmentation itself does not show any clear imprints of the forces that originally formed the fragment (e.g. shock waves, or cloud-cloud collisions).', '1711.01417-3-47-2': 'The conclusion is that there is no prediction for a threshold below which the fragmentation of filaments is prevented.', '1711.01417-3-48-0': 'The situation changes when we look at the high-density filaments (Fig. [REF]).', '1711.01417-3-48-1': 'Here we observe maximal line masses of filaments without fragments between 18 and 20 M[MATH] pc[MATH] in all clouds.', '1711.01417-3-48-2': 'These transitional line masses appear to be in agreement with the critical line mass of the cylindrical fragmentation model.', '1711.01417-3-48-3': 'The reason for this is that the properties of the filaments here are more likely comparable to those of the cylindrical fragmentation model since they are more likely isothermal, straighter, and closer to the sites where overdensities form compared to more diffuse filaments.', '1711.01417-3-48-4': 'Consequently, the objects we study here agree better with the properties of the analytic cylinders.', '1711.01417-3-48-5': 'However, we detect only one dense filament without embedded fragment for one single time step (in M8) and, thus, lack a statistically meaningful sample to draw final conclusions about the capability to predict the fragmentation behaviour of high-density filaments.', '1711.01417-3-48-6': 'From the analysis of the low-density filaments, though, we see that the configuration represented by the analytic model of cylindrical fragmentation is not universally part of the evolution of filaments in the simulations.', '1711.01417-3-49-0': 'We conclude that a simple cylindrical model as described by [CITATION] and the cylindrical fragmentation model does not represent the typical initial conditions for the fragmentation of a filament in our molecular cloud simulations.', '1711.01417-3-49-1': 'Therefore, it is not a complete model for evaluating the stability of filaments.', '1711.01417-3-49-2': 'To predict the fragmentation of a filament, a more complex model is essential; one that not only considers the balance between internal self-gravity and thermal pressure, but also other forces, such as external hydrostatic, turbulent, or magnetic pressure.', '1711.01417-3-49-3': 'Furthermore, connecting such a model with observations needs to take into account that the properties of the filaments derived strongly depend on the parameters used to identify the filaments (as well as the filament-finder code used, as discussed in Appendix [REF]).', '1711.01417-3-49-4': 'This also means that whether or not observed filaments fragment as predicted by a cylindrical fragmentation model is strongly influenced by the identification parameters, as one can always choose parameters in a way that it perfectly suits the model.', '1711.01417-3-49-5': 'In our case, this is represented by the high-density filaments that only then contain fragments after exceeding the critical line mass.', '1711.01417-3-49-6': 'Since there is no unique, universal, and physically motivated definition of what filaments are, the fragmentation models are not universal themselves.', '1711.01417-3-50-0': '## Properties and evolution of fragments', '1711.01417-3-51-0': 'In this Section, we discuss the properties of the fragments we have detected within our clouds using astrodendro (Sect. [REF]), including their time evolution, and connection to their parental filaments and to each other.', '1711.01417-3-51-1': 'The main properties of the fragments are summarised in Table [REF] showing that those fragments have similar properties as dense regions and condensations in star-forming regions .', '1711.01417-3-52-0': 'In order to find evidence for the fragments in our simulation forming by gravitational fragmentation following the cylindrical fragmentation criterion, we make a first estimate of how gravitationally bound our fragments are.', '1711.01417-3-52-1': 'For this, we compute their virial parameters, given by : [EQUATION] for each of the fragments at any time step.', '1711.01417-3-52-2': 'Here, [MATH] is the gravitational constant, [MATH] the enclosed mass of the fragment, [MATH] = 0.3 pc its radius and [MATH] its velocity dispersion, which we calculate following Eqs. (9) (10) in [CITATION]: [EQUATION] with [MATH] being the central position of the fragment, [MATH] the average gas velocity vector, and [MATH] the average sound speed.', '1711.01417-3-52-3': 'We note that this estimate only takes the ratio between volumetric kinetic and gravitational energy into account.', '1711.01417-3-52-4': 'Other terms, such as magnetic fields or surface thermal or kinetic pressure , are neglected.', '1711.01417-3-52-5': 'This is a common approximation used in both observational and theoretical studies, although [CITATION] suggest that these additional terms can be as important for objects embedded within molecular clouds as the balance of volumetric kinetic and gravitational energy.', '1711.01417-3-53-0': 'In the top row of Fig. [REF] we show a histogram of virial parameters measured for the fragments at the time when they have been first detected.', '1711.01417-3-53-1': 'According to Eq. ([REF]), the fragments are gravitationally bound when [MATH].', '1711.01417-3-53-2': 'This criterion is fulfilled by only 20% of the fragments.', '1711.01417-3-53-3': 'The majority of fragments have virial parameters close to the mode of 3.5.', '1711.01417-3-53-4': 'We argue that these fragments may nevertheless be bound objects because of the terms not accounted for by Eq. ([REF]), particularly the surface terms resulting from the collapse of the surrounding cloud.', '1711.01417-3-53-5': 'Furthermore, we see that the virial parameters decrease as the fragments evolve, which means that they become more bound.', '1711.01417-3-53-6': 'The histogram in the bottom row of Fig. [REF] confirms this as it shows that most of the fragments have virial parameters around 2 in the last simulated time steps.', '1711.01417-3-54-0': 'We note that, as explained by [CITATION], we can only give lower limits on the velocity dispersions, and thus the virial parameters, since we under-resolve the turbulence on small scales in the simulations.', '1711.01417-3-54-1': "Resolving the subgrid scale turbulence may increase the energy by 25% , but likely won't prevent the fragments from beginning to collapse before the criteria for cylindrical fragmentation are fulfilled.", '1711.01417-3-54-2': "This consequently means that the fragmentation must be driven by neglected terms in the virial equation that can produce significant differences in the estimation of the fragments' boundness.", '1711.01417-3-55-0': 'In the bottom row of Fig. [REF] we plot the number of identified fragments as a function of time.', '1711.01417-3-55-1': 'We see that the number of fragments overall increases with time for each simulation.', '1711.01417-3-55-2': 'However, there are cases when the number of fragments drops.', '1711.01417-3-55-3': 'The missing fragments are either disrupted, for example by shock waves or intracloud turbulence, or merge with each other, meaning that they approach each other too closely ([MATH]0.4 pc) to be distinguishable with our resolution.', '1711.01417-3-56-0': "We see that the first fragments form within the first 2 Myr, corresponding to 25-50% of the clouds' free-fall times and the period when the growth of the DGMF is steepest (see Fig. [REF]).", '1711.01417-3-56-1': 'This indicates that the formation of fragments is primarily dominated by the compression of dense gas within the cloud.', '1711.01417-3-56-2': 'This is verified by the ratio of mass contained within the fragments, [MATH], relative to the mass of the entire cloud, [MATH], shown in the bottom panel of Fig. [REF].', '1711.01417-3-56-3': 'The [MATH] ratios evolve in a similar fashion as the DGMF.', '1711.01417-3-56-4': 'Compared to the more constant growth of [MATH] or [MATH], the ratio [MATH] increases rapidly before stagnating around 0.03.', '1711.01417-3-56-5': 'Interestingly, if we approximate the star formation efficiency per free-fall time of our clouds with this ratio, the value agrees with typically observed efficiencies in molecular clouds .', '1711.01417-3-56-6': 'This is consistent with the determination of star formation efficiency by the dynamics of gravitational collapse, though our relatively small sample and lack of feedback modelling does not allow definitive conclusions to be drawn.', '1711.01417-3-57-0': 'Subsequently, we follow the evolution of the mass contained in the fragments relative to the mass contained in the filaments, [MATH].', '1711.01417-3-57-1': 'We plot the ratios in the middle panel in Fig. [REF].', '1711.01417-3-57-2': 'In all clouds, we see that the [MATH] ratio increases rapidly within the first [MATH]2.5 Myr after the first fragments have formed, reaching maximal values of 15-40%.', '1711.01417-3-57-3': 'This demonstrates that the fragments accrete mass from their parental filaments efficiently as long as there is a sufficient gas reservoir accessible, as is the case at the beginning of the simulations.', '1711.01417-3-57-4': "In doing so, they take up a significant fraction of the filaments' masses.", '1711.01417-3-58-0': 'Another question that has been recently discussed in the literature is whether prestellar cores form in a regular pattern within filaments.', '1711.01417-3-58-1': 'Observations suggest that cores condense at regular intervals along their parental filaments .', '1711.01417-3-58-2': 'The mean separations between the cores appear to correlate with the properties of the respective filament, ranging from a few tenths to several parsecs. These observations seem consistent with theoretical models of periodic fragmentation .', '1711.01417-3-58-3': 'The instabilities causing fragmentation in these models have unique modes that depend on the initial conditions of the filament.', '1711.01417-3-58-4': 'The wavelengths of these modes then define the mean separations between the forming cores.', '1711.01417-3-58-5': 'However, other studies, both observational and theoretical , demonstrate that periodic fragmentation only occurs under special conditions (such as supersonic, purely compressive turbulent motions), if at all.', '1711.01417-3-58-6': 'In reality the conditions within the filaments are not as uniform as assumed by the models, so observed core patterns may also be the result of overlapping fragmentation modes.', '1711.01417-3-58-7': 'According to those studies, filaments commonly fragment in a disordered, cluster-like fashion.', '1711.01417-3-59-0': 'Using our data, we test whether or not the fragments in our sample form with uniform separations.', '1711.01417-3-59-1': 'We note that we can only detect separations that are larger than 0.4 pc due to the 0.1 pc resolution of the data grid and the 0.3 pc radius we assume for the fragments.', '1711.01417-3-59-2': 'We plot the separations in 3D space of the individual fragments to their individual closest neighbour within the same filament as a function of time in Fig. [REF].', '1711.01417-3-59-3': 'We see that the fragments form, on average, at distances exceeding 2 pc from their closest neighbour at the beginning of the fragmentation process, but approach each other with time down to [MATH]1 pc or even merge (below 0.4 pc).', '1711.01417-3-60-0': 'To answer the question of whether there is a typical fragmentation scale we need to consider the separations between closest neighbours at the moment the fragments form.', '1711.01417-3-60-1': 'These are summarised in Fig. [REF].', '1711.01417-3-60-2': 'If there were a typical separation we should see a significant peak at that particular scale length, or a sequence of aliased peaks with equal separations.', '1711.01417-3-60-3': 'Looking at the histograms, we might argue that we see such sequences in M3 and M8 with typical separations at 0.9 and 0.6 pc, respectively.', '1711.01417-3-60-4': 'Both numbers exceed the local Jeans length by a substantial factor ([MATH]0.1 pc), but are only a factor of 2.3-1.5 larger than our resolution limit.', '1711.01417-3-60-5': 'Furthermore, in the case of M4, we do not see any significant peak in the distribution.', '1711.01417-3-60-6': 'However, the number of fragments in the samples is too low to make a solid statement about the existence of a universal fragmentation scale length, and if it depends on the local physical conditions only.', '1711.01417-3-61-0': '## Properties and evolution of filaments in 2D', '1711.01417-3-62-0': 'In the previous Subsections, we studied the properties and evolution of the filaments and fragments that we identified based on the full 3D simulation data.', '1711.01417-3-62-1': 'In observations, however, such 3D data are not available .', '1711.01417-3-62-2': 'Instead, we observe the filaments projected onto the 2D plane of the sky.', '1711.01417-3-62-3': 'This raises the questions of what we would observe if we apply our methods to the projected data and how these results compare to the results from 3D data.', '1711.01417-3-63-0': 'In this Section, we approach these questions by projecting our 3D volume density cubes onto 2D column density maps.', '1711.01417-3-63-1': 'We project along the three major axes [MATH], [MATH], and [MATH], in order to account for line-of-sight (LoS) specific variance.', '1711.01417-3-63-2': 'We note that these maps lack any additional observational effects, such as noise, beam, or optical depth effects, and are thus only idealised approximations to real observations.', '1711.01417-3-63-3': 'However, such maps give the best case scenario to test the 3D-2D correspondence.', '1711.01417-3-63-4': 'Since we neglect opacity, our column density maps are primarily comparable to optically thin (sub-)mm dust observations.', '1711.01417-3-63-5': 'To produce comparable synthetic spectral line emission maps, we would need to consider chemical abundances, as well as gas velocity, which is beyond the scope of this paper.', '1711.01417-3-64-0': 'Analogously to Sect. [REF], we use DisPerSe to identify 2D filaments within the column density maps.', '1711.01417-3-64-1': 'For this purpose, we convert the number density thresholds used in 3D into column density thresholds by assuming a path length of 0.1 pc, namely [MATH] cm[MATH] (corresponding to [MATH] cm[MATH]) and [MATH] cm[MATH] (corresponding to [MATH] cm[MATH]).', '1711.01417-3-64-2': 'We emphasise that the skeletons of the 2D filaments are independently identified based on the column density distributions and not the projections of the 3D filament skeletons.', '1711.01417-3-64-3': 'Figure [REF] shows an example of the structures obtained, and Table [REF] summarises their properties.', '1711.01417-3-64-4': "We see that, similar to the structures detected in 3D, the 2D filaments' properties are influenced by the identification threshold, with the lengths of the filaments becoming shorter and line masses higher with higher thresholds.", '1711.01417-3-65-0': 'Comparing the structures detected in 3D and 2D, however, reveals a more significant difference.', '1711.01417-3-65-1': 'As shown in the example in Fig. [REF], we do find 2D counterparts for all 3D filaments, but there are 2D filaments that do not have matching 3D filaments identified with the same identification threshold.', '1711.01417-3-65-2': 'This finding is a consequence of the projection: structures with high volume densities typically have high column densities; conversely other structures with lower volume densities can appear denser in column densities, amplified by projection.', '1711.01417-3-66-0': 'This also influences the measured properties of the 2D filaments, such as the 2D line masses, [MATH].', '1711.01417-3-66-1': 'Analogously to Fig. [REF], Fig. [REF] shows the value of [MATH] for the dense 2D filaments as a function of time.', '1711.01417-3-66-2': 'We see that the values lie between the values we measured based on the diffuse and dense 3D filaments.', '1711.01417-3-66-3': 'If the dense 2D filaments exclusively represented the projection of dense 3D filaments, however, we would expect the trends in average line mass to evolve similarly.', '1711.01417-3-66-4': 'This is not seen here.', '1711.01417-3-66-5': 'On the contrary, the growth of [MATH] in the dense 2D filaments correlates with the growth of [MATH] of the diffuse 3D filaments.', '1711.01417-3-66-6': 'As described before, this is because most of the dense 2D filaments are projections of diffuse 3D filaments, so the line mass of the 3D filaments is the main contributor of the average line mass of the dense 2D filaments.', '1711.01417-3-66-7': 'Additional mass comes from material along the line of sight, so [MATH].', '1711.01417-3-67-0': 'In summary, for a given identification threshold, all 3D filaments have counterparts in column density maps, but not necessarily vice-versa.', '1711.01417-3-67-1': 'Projection not only maps dense 3D filaments onto the plane of the sky, but also merges less dense structures along the same line of sight, producing structures that exceed the column density threshold.', '1711.01417-3-67-2': 'Consequently, a comparison of the properties of 3D to 2D filaments is not directly possible if the corresponding identification thresholds are used for both samples.', '1711.01417-3-67-3': 'However, if this is taken into account, the properties of 3D and 2D filaments evolve similarly, but with an offset due to the additional line of sight mass projected onto the 2D filaments.', '1711.01417-3-68-0': 'The difference in the density distributions measured in 3D and 2D is even more obvious in the overall properties of the clouds.', '1711.01417-3-68-1': 'We measure the DGMF using column density by defining clouds as coherent volumes of gas with minimal number densities of 100 cm[MATH].', '1711.01417-3-68-2': 'We choose a minimum path length of 0.6 pc based on the assumption that filaments and fragments have radii of 0.3 pc.', '1711.01417-3-68-3': 'We note that this path length is not identical to the path length we used for computing the identification threshold of the column density filaments, [MATH], before.', '1711.01417-3-68-4': 'However, by using a minimum path length of 0.6 pc ensures that the fraction of gas we hereafter consider mirrors gas of the real 3D cloud.', '1711.01417-3-68-5': 'This way, we cannot only more reliably compare the DGMFs measured based on the 2D data with those based on the 3D data, but also with DGMFs in observed molecular clouds.', '1711.01417-3-68-6': 'As a results we obtain a minimal column density of [MATH] cm[MATH].', '1711.01417-3-68-7': 'We then compute [EQUATION] with [MATH] being the column density at time [MATH] and within the pixel [MATH], [MATH] the column density above which the gas is defined as dense and [EQUATION]', '1711.01417-3-68-8': 'In Fig. [REF], the dashed lines show the evolution of the DGMF measured using this equation.', '1711.01417-3-69-0': 'Analogous to Sect. [REF], we use two thresholds for tracing the evolution of dense gas within the clouds, namely [MATH] cm[MATH] (corresponding to [MATH] = 1,000 cm[MATH]) and [MATH] cm[MATH] (corresponding to [MATH] = 5,000 cm[MATH]).', '1711.01417-3-69-1': 'We see that for most of the time the DGMF is about an order of magnitude higher than the corresponding DGMFs calculated from the volume density distributions, but maximal values (70-80% for [MATH] = 1,000 cm[MATH] and 30-50% for [MATH] = 5,000 cm[MATH]) agree with each other.', '1711.01417-3-69-2': 'In the case of a lower value of [MATH], we see that the 2D DGMFs are almost constant in time, or even slightly decreasing, in disagreement with with the steady growth of the 3D DGMFs.', '1711.01417-3-70-0': 'In summary, we see that the DGMF measured in 2D deviates from the true 3D value for most of the initial evolution of a particular cloud.', '1711.01417-3-70-1': 'The DGMF measured in column density may show a completely different temporal behaviour, particularly when a low column density threshold is used to define dense gas.', '1711.01417-3-70-2': 'However, we also see that the individual filaments we identify in the 3D and 2D data and their properties agree decently with each other if the identification threshold in column density focuses on the range of volume density one wants to study and distinguishes unassociated gas along the line of sight.', '1711.01417-3-71-0': '# Summary Conclusions', '1711.01417-3-72-0': 'In this paper we analyse the properties and fragmentation of filaments forming within 3D AMR FLASH simulations of the self-gravitating, magnetised, supernova-driven ISM by [CITATION].', '1711.01417-3-72-1': 'Our main results are as follows.', '1711.01417-3-73-0': 'Our results indicate that filament fragmentation is affected by the environment of the cloud they form in.', '1711.01417-3-73-1': 'In order to understand the onset and development of fragmentation, future theoretical studies likely need to abandon the hydrostatic initial condition and to consider the formation of filaments and their subsequent fragmentation together.', '1711.01417-3-74-0': 'Furthermore, our results demonstrate that establishing common practices for how to define filaments in 3D and 2D data from simulations and observations is crucial for studying the properties and evolution of filaments and especially for comparing different filament studies with each other.', '1711.01417-3-74-1': 'Studies using filament finders must thoroughly test applicability of the adopted algorithms to address the problem in question.', '1711.01417-3-75-0': 'The authors acknowledge the support ESO and its Studentship Programme provided.', '1711.01417-3-75-1': 'This research made use of astrodendro, a Python package to compute dendrograms of astronomical data (http://www.dendrograms.org/).', '1711.01417-3-75-2': "This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 639459 (PROMISE).", '1711.01417-3-75-3': 'JCI-M and M-MML received support from US NSF grant AST11-09395.', '1711.01417-3-75-4': 'JCI-M was additionally supported by the DFG Priority Programme 157.', '1711.01417-3-75-5': 'M-MML also thanks the A. von Humboldt-Stiftung for support.', '1711.01417-3-76-0': '# Filament finders', '1711.01417-3-77-0': '## Algorithms', '1711.01417-3-78-0': 'There are a variety of algorithms publicly available for identifying filamentary structures in molecular clouds.', '1711.01417-3-78-1': 'Naturally, there are more filament finders that work with 2D data than with 3D data, since 2D algorithms can directly be applied to observed data like intensity or column density maps, while 3D filament finders require data with a resolved third dimension, like the local standard of rest velocity observed spectroscopically.', '1711.01417-3-78-2': 'The latter is not only observationally demanding, but computationally expensive to analyse.', '1711.01417-3-79-0': 'In this Appendix, we present and compare the finder algorithms we have considered during our analysis and justify our choice of DisPerSe for the analysis performed in the main paper.', '1711.01417-3-80-0': '### DisPerSe', '1711.01417-3-81-0': 'DisPerSe extracts coherent structures by evaluating the gradients between individual grid cells and the robustness of the topological features found.', '1711.01417-3-81-1': 'It was originally written for finding structures (both over- and under-densities) in cosmological data, but can be applied to other applications.', '1711.01417-3-82-0': 'The advantage of DisPerSe is that it is independent of the content and dimension of data it receives.', '1711.01417-3-82-1': 'Therefore, it can be directly applied to volume density cubes as well as column density maps and returns filamentary structures in both based on the same algorithms.', '1711.01417-3-83-0': '### FilFinder', '1711.01417-3-84-0': 'FilFinder was written to extract filamentary structures in molecular clouds observed by the Herschel Gould Belt Survey .', '1711.01417-3-84-1': 'It does this by reducing the areas of interest (parts of molecular clouds with intensities above a specified threshold) to topological skeletons.', '1711.01417-3-84-2': 'Therefore, each element of the skeletons represents the medial position of the areas of interest within the boundaries.', '1711.01417-3-84-3': 'Unfortunately, FilFinder can only be applied to 2D maps.', '1711.01417-3-85-0': '### astrodendro', '1711.01417-3-86-0': 'astrodendro creates dendrogram trees representing the hierarchical structure of the underlying data.', '1711.01417-3-86-1': 'That means that the code reveals how individual regions are connected with each other.', '1711.01417-3-86-2': 'Those regions are then classified into trunks, branches, and leaves that represent molecular clouds, clumps and cores in our context.', '1711.01417-3-86-3': 'Thus, astrodendro does not identify filamentary structures, but it is well suited for tracing fragments within the filaments or for preparing large data sets, so the actual filament finder can focus on the regions of interest.', '1711.01417-3-87-0': '### SCIMES', '1711.01417-3-88-0': 'SCIMES works similarly to astrodendro (Sect. [REF]) using dendrograms.', '1711.01417-3-88-1': 'However, while astrodendro simply detects structures, SCIMES weights the branches and leaves according to user-defined affinities (for example, minimal size, or maximal separation along position or velocity axes) and organises the dendrogram tree accordingly.', '1711.01417-3-88-2': 'It returns weighted branches as clusters that, in our case, represent individual filaments.', '1711.01417-3-88-3': 'We note that SCIMES itself is not a filament finder, but only returns regions that are likely to contain compact substructures, such as filaments.', '1711.01417-3-88-4': 'In order to obtain the filaments one additionally needs a filament finder.', '1711.01417-3-89-0': 'There are some advantages to combining SCIMES and DisPerSe.', '1711.01417-3-89-1': 'On the one hand, it is computationally more efficient to first create masks of the relevant regions before applying DisPerSe on those masks.', '1711.01417-3-89-2': 'On the other hand, most filament finders return their structures without any weighting.', '1711.01417-3-89-3': 'Hence, the users would need to distinguish the filaments from each other by hand, which is impractical for larger datasets like ours.', '1711.01417-3-89-4': 'This step can be transferred to SCIMES as well, by applying the masks on the filament finder outputs.', '1711.01417-3-90-0': '### Minimal Spanning Tree', '1711.01417-3-91-0': 'The minimal spanning trees algorithm is used for optimising costs by minimising the lengths of grids and efficiency of networks.', '1711.01417-3-91-1': 'It can also be used to find coherent, filamentary structures as, for example, [CITATION] have demonstrated.', '1711.01417-3-91-2': 'With the MST we can define filaments by connecting the leaves found by astrodendro representing pre-stellar cores in molecular clouds according to our requirements.', '1711.01417-3-92-0': 'We use this method as an alternative method for identifying filaments in 3D.', '1711.01417-3-92-1': 'DisPerSe also uses the MST algorithm to connect individual skeleton segments with each other.', '1711.01417-3-92-2': 'The difference here is that we use MSTs for connecting the fragments we identified with astrodendro and define the straight lines between them as filaments.', '1711.01417-3-92-3': "Naturally, the separation between the fragments is on average much larger than those between DisPerSe's segments, leading to less accurate curvatures.", '1711.01417-3-93-0': '## Comparison of identified filaments', '1711.01417-3-94-0': 'For evaluating and interpreting the results we have presented in Sect. [REF] it is essential to understand and compare the performance of the underlying filament finder algorithms.', '1711.01417-3-94-1': 'Figure [REF] gives an example of this.', '1711.01417-3-94-2': 'The background of each panel shows the column density map of the M4 model projected along the [MATH]-axis.', '1711.01417-3-94-3': 'The purple dots represent the position of the fragments identified by astrodendro.', '1711.01417-3-94-4': 'The white lines illustrate the filaments found by DisPerSe (left), FilFinder (center), and MST (right) applied to fragments from astrodendro, when using a column density threshold of N[MATH] = 1.5 [MATH] cm[MATH].', '1711.01417-3-94-5': 'One sees that the codes return widely varying structures, although they agree well where the column density is highest.', '1711.01417-3-94-6': 'Only DisPerSe follows the filaments that connect the cloud to the ISM.', '1711.01417-3-94-7': 'This has a huge impact on the physical properties such as total length, enclosed mass and the field of interest in general.', '1711.01417-3-95-0': 'Since the different methods do not identify the same structures we cannot compare filaments individually.', '1711.01417-3-95-1': 'For evaluating the influence the underlying algorithms print on the structures, we measure the average line masses of all filaments detected within the respective cloud at a given time (analogously to Sect. [REF]).', '1711.01417-3-95-2': 'In Figs. [REF], [REF] and [REF] we compare those average line masses based on the filaments returned by the individual codes in 2D and 3D, respectively.', '1711.01417-3-95-3': 'We see that not only does the morphology of the filaments differ significantly using different codes, but also the properties of the structures.', '1711.01417-3-96-0': 'We focus our further analysis on the filamentary skeletons identified by DisPerSe because we can automatically run the code on both our 2D and 3D and work with structures based on the same algorithm and parameter dependence.', '1711.01417-3-96-1': 'Furthermore, DisPerSe is the code that is least sensitive to the input parameters since it is the only code that considers gradients in the matter distribution automatically, giving the skeletons a physical meaning.'}

1712.04813

{'1712.04813-1-0-0': 'We develop a phase space formulation of quantum mechanics based on unitary irreducible representations of classical phase spaces.', '1712.04813-1-0-1': 'We use a quantum action functional to derive the basic equations.', '1712.04813-1-0-2': 'In principle, our formulation is equivalent to the Hilbert space formulation.', '1712.04813-1-0-3': 'However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum motions can be derived.', '1712.04813-1-0-4': 'We predict that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase spaces to establish the basic equations of the new formalism.', '1712.04813-1-1-0': '# Introduction', '1712.04813-1-2-0': 'In this article we propose a certain trajectory approach to quantum mechanics and we develop it with an emphasis on its application to simple quantum cosmological systems.', '1712.04813-1-2-1': "The essence of our approach is to reformulate the Schrodinger equation densely defined on vectors in the Hilbert space in terms of a set of Hamilton's equations in a phase space.", '1712.04813-1-2-2': 'Quantum states are represented by phase space points.', '1712.04813-1-2-3': 'The phase space variables encode both classical and nonclassical observables.', '1712.04813-1-2-4': 'By classical observables in this context we mean the expectation values of basic operators in a given quantum state.', '1712.04813-1-2-5': 'By nonclassical observables we mean all other phase space variables which encode such properties of quantum states as the dispersions of basic observables and so on.', '1712.04813-1-3-0': 'Our formalism builds on the notion of coherent states and in particular on the idea of semiclassical framework by J. Klauder [CITATION].', '1712.04813-1-3-1': 'In fact, our formalism is a natural extension of the Klauder framework which is included as a special case.', '1712.04813-1-3-2': 'The basic tools behind our formalism are, first, the variational formulation of the quantum dynamics once a quantum Hamiltonian has been provided and, second, coherent states that are constructed with a unitary and irreducible representation of a minimal canonical group in the phase space of the respective classical model.', '1712.04813-1-3-3': 'The coherent states are used to incorporate the classical observables into our formalism in a natural (more precisely, covariant) way and, as it is in the original framework, the equations of motion of classical observables, in general, include terms with nonvanishing [MATH].', '1712.04813-1-3-4': 'Nevertheless, our extension of the framework by nonclassical observables brings in some new advantages that are not present in the original framework.', '1712.04813-1-4-0': 'In principle, our approach is based on an infinite-dimensional phase space.', '1712.04813-1-4-1': 'However, its main advantage is that it includes consistent truncations to finite-dimensional phase spaces.', '1712.04813-1-4-2': 'Specifically, the dimensionality can be as large as to reproduce the exact Schrodinger equation in the form of phase space trajectories in the case of infinitely many dimensions, or as small as to give the roughest quantum corrections to the classical equations of motion in the case of the classical phase space dimensionality.', '1712.04813-1-4-3': 'The latter case corresponds, in fact, to the Klauder framework.', '1712.04813-1-4-4': 'Our approach can be adjusted to any specific quantum system under consideration and, in particular, used to deal with those quantum dynamics which are too complex to be solved explicitly.', '1712.04813-1-5-0': 'Since we are mainly interested in quantum cosmological systems, we will develop our ideas for the case of the classical phase space which is a half-plane rather than a plane.', '1712.04813-1-5-1': 'For this particular phase space, a well-suited minimal canonical group is the affine group.', '1712.04813-1-5-2': 'Nevertheless, we wish to emphasize that our approach can be easily developed for the case of Weyl-Heisenberg (W-H) and other groups.', '1712.04813-1-6-0': 'The great advantage of our approach is that it allows for addressing physical questions which are not possible in the simplest semiclassical framework.', '1712.04813-1-6-1': 'For example, as a classical phase space is now extended by nonclassical observables one can investigate in detail the energy transfers between classical and nonclassical observables.', '1712.04813-1-6-2': 'It is worth mentioning that that the very existence of such a phenomenon could be mistakenly interpreted in terms of energy transfers from or to hidden spatial dimensions in the universe, which are introduced by the brane-world theories and alike [CITATION].', '1712.04813-1-6-3': 'In fact, the energy carried by a wavefunction naturally spreads into nonclassical observables when a system leaves the classical regime and enters the quantum one.', '1712.04813-1-6-4': 'This is in particular expected of cosmological systems when they approach the big-bang singularity.', '1712.04813-1-6-5': 'As it was already shown in [CITATION], in this case a quantum repulsive potential may halt the contraction and bring the universe to expansion, that is, prevent the universe form collapsing into the singularity by making it bounce.', '1712.04813-1-6-6': 'But then, since the full quantum dynamics is not expected to be simply reverted in time, the dynamics near the bounce is not expected to be fully symmetric in time either.', '1712.04813-1-6-7': 'How this exactly happens can be captured within our extended approach by inspecting the evolution of nonclassical observables.', '1712.04813-1-6-8': 'Other associated questions could include: How the classical universe emerges from a quantum state?', '1712.04813-1-6-9': 'Had been the universe classical before the bounce?', '1712.04813-1-6-10': 'And, what is a good measure of classicality?', '1712.04813-1-7-0': 'To finish these introductory remarks, let us notice that our formulation offers an alternative way of looking at quantum mechanics, it is a kind of the hidden-variable theory.', '1712.04813-1-7-1': 'We believe that our approach can be a useful tool to those who investigate hidden-variable theories or are simply interested in fundamental principles of quantum mechanics.', '1712.04813-1-8-0': 'The outline of the article is as follows.', '1712.04813-1-8-1': 'In Sec II we recall the semiclassical framework of Klauder.', '1712.04813-1-8-2': 'In Sec III we discuss the variational formulation of quantum dynamics.', '1712.04813-1-8-3': 'In Sec IV we develop our formalism.', '1712.04813-1-8-4': 'We apply it to two examples in Sec V.', '1712.04813-1-8-5': 'In Sec VI we revisit the quantum flat Friedmann model with our approach.', '1712.04813-1-8-6': 'We conclude in Sec VII.', '1712.04813-1-8-7': 'The appendices deal with some technicalities (self-adjointness and the numerical code) which have been omitted from the main text.', '1712.04813-1-9-0': "# Coherent states and Klauder's framework", '1712.04813-1-10-0': '## Coherent states', '1712.04813-1-11-0': 'By coherent states (see e.g. [CITATION]) we mean a continuous mapping from a set of labels, denoted by [MATH] and equipped with a measure [MATH], into unit vectors in Hilbert space, [EQUATION] such that it resolves the identity, [EQUATION]', '1712.04813-1-11-1': 'This basic property was used by Klauder [CITATION] in his definition of what he calls an overcomplete family of states (OFS).', '1712.04813-1-11-2': 'They provide a bridge between the abstract quantum formalism and the continuous label-space that can be further identified with classical observables.', '1712.04813-1-11-3': 'Suppose there exists a unitary irreducible representation on [MATH], [MATH], of a minimal group of canonical transformations in a phase space [MATH] of a physical system.', '1712.04813-1-11-4': 'Then, the mapping [EQUATION] defines a family of coherent states whose labels describe the classical states of that system [CITATION].', '1712.04813-1-11-5': 'The normalized vector [MATH] is called the fiducial vector.', '1712.04813-1-12-0': '## Framework', '1712.04813-1-13-0': 'First, let us recall that the quantum dynamics can be obtained via variation of the quantum action, [EQUATION] with respect to the normalized [MATH], where [MATH] is the quantum Hamiltonian that corresponds to a certain classical Hamiltonian, [MATH].', '1712.04813-1-13-1': 'The stationary points of the action ([REF]) are found to satisfy the Schrodinger equation, [EQUATION]', '1712.04813-1-13-2': 'The idea of the semiclassical framework based on the coherent states was introduced by Klauder in [CITATION].', '1712.04813-1-13-3': 'Initially, he applied it to the case of the phase space [MATH] and the Weyl-Heisenberg coherent states.', '1712.04813-1-13-4': 'The W-H coherent states, [MATH], are defined as follows [EQUATION] where the displacement operator [MATH] satisfies [EQUATION] and where [MATH] are the position and momentum operators [CITATION].', '1712.04813-1-13-5': 'The fiducial vector [MATH] is fixed for all [MATH] and its choice is almost arbitrary.', '1712.04813-1-13-6': 'The only condition that one often imposes is the so called physical centering which relates the classical observables and the expectation values by demanding [EQUATION] where [MATH] and [MATH] are the position and momentum operator, respectively.', '1712.04813-1-14-0': 'We find that [EQUATION] where [MATH] is the exterior derivative.', '1712.04813-1-14-1': 'This result can be guessed (up to the irrelevant total derivative) from the fact that the above one-form must be invariant with respect to the action of the W-H group.', '1712.04813-1-14-2': 'The same reasoning applies to all other canonical groups.', '1712.04813-1-14-3': 'Now the quantum action functional ([REF]) can be evaluated on the family of coherent states, [EQUATION] where [MATH].', '1712.04813-1-14-4': 'Thus, the variation of the quantum action with respect to the [MATH]-labelled coherent states yields the Hamilton equations for [MATH] and [MATH], [EQUATION]', '1712.04813-1-14-5': 'On the one hand, the above equation provides an approximation to the exact quantum motion, [EQUATION] and on the other hand, it establishes a very appealing interpretation of the classical observables and their dynamics within the more fundamental quantum framework.', '1712.04813-1-14-6': 'We notice that these equations, in general, differ from the classical equations as the semiclassical Hamiltonian [MATH] may include [MATH]-corrections, [EQUATION]', '1712.04813-1-14-7': 'Since in the real world [MATH] never vanishes, the possibility of modeling the dynamics of the classical observables with nonvanishing [MATH] could be very useful.', '1712.04813-1-14-8': 'Indeed, this possibility becomes particularly important for improving the dynamics of classically singular cosmological models as we show below.', '1712.04813-1-15-0': '## Affine group', '1712.04813-1-16-0': 'As we are concerned with gravitational systems, we shall turn to an important example of the phase space that appears in cosmology, namely the half-plane [MATH].', '1712.04813-1-16-1': 'The basic observables form a canonical pair, [EQUATION] where [MATH] is the volume of the universe and [MATH] is a rate of its expansion.', '1712.04813-1-16-2': 'Clearly, the W-H group is not applicable to the present case as one of the canonical variables, [MATH], is confined to the half-line.', '1712.04813-1-16-3': 'Instead, we shall employ the affine group [CITATION], [MATH], that is defined by the multiplication law, [EQUATION] and preserves the symplectic structure of the half-plane phase space, [EQUATION] where [MATH] is a fixed element of the affine group.', '1712.04813-1-16-4': 'There exists a unique (up to sign) unitary irreducible representation of [MATH], which in [MATH] takes the form [EQUATION]', '1712.04813-1-16-5': 'Thus, we define the affine coherent states as [EQUATION] where [MATH] is the fiducial vector that is subject to the constraint [EQUATION] (which follows from the group integrability condition).', '1712.04813-1-16-6': 'To tighten the connection between quantum and classical observables we demand [EQUATION] where [MATH] and [MATH].', '1712.04813-1-16-7': 'This is equivalent to [EQUATION]', '1712.04813-1-16-8': 'One finds that [EQUATION] where [MATH] is the dilation operator and which confirms the general statement given below Eq. ([REF]).', '1712.04813-1-16-9': 'Now, provided a quantum Hamiltonian [MATH], the quantum action functional evaluated on the affine coherent states reads [EQUATION] where [MATH].', '1712.04813-1-16-10': 'Hence, the variation of the quantum action with respect to the classical labels [MATH] yields the Hamilton equations, [EQUATION] for the stationary trajectories.', '1712.04813-1-17-0': '## Free particle dynamics on [MATH]', '1712.04813-1-18-0': 'In this article we are going to study a quantum free motion of a particle on the half-line, [MATH].', '1712.04813-1-18-1': 'It is a very important example as it formally describes the dynamics of the flat Friedmann universe with a perfect fluid-source [CITATION].', '1712.04813-1-18-2': 'The big-bang singularity is represented by the end-point, [MATH].', '1712.04813-1-18-3': 'The variable [MATH] describes the volume and the variable [MATH] describes the expansion of the universe (see Sec VI).', '1712.04813-1-18-4': 'Let the classical system be defined as follows, [EQUATION] and the quantum Hamiltonian read [EQUATION]', '1712.04813-1-18-5': 'We discuss the technical issue of extending the above symmetric operator to a self-adjoint one in Appendix A. For a fiducial vector [MATH], we obtain [EQUATION] where [EQUATION] where [MATH].', '1712.04813-1-18-6': 'The respective Hamilton equations ([REF]) include an [MATH]-correction that resolves the singularity at [MATH].', '1712.04813-1-18-7': 'The particle is repelled away from the singularity by the quantum potential [MATH] and this produces a bounce in its dynamics.', '1712.04813-1-18-8': 'See Fig. [REF].', '1712.04813-1-19-0': '# Theory of the restricted quantum action', '1712.04813-1-20-0': 'The idea of this work is to extend the phase space description of quantum mechanics due to J. Klauder to a much broader framework that we shall call the trajectory approach to quantum dynamics.', '1712.04813-1-20-1': 'For this purpose it is useful to discuss the quantum action formulation of quantum dynamics in somewhat more detail.', '1712.04813-1-21-0': '## Variation of the quantum action', '1712.04813-1-22-0': 'The quantum action is defined on trajectories in the Hilbert space and reads [EQUATION]', '1712.04813-1-22-1': 'Its variation with respect to [MATH] such that [MATH] at each [MATH], yields [EQUATION]', '1712.04813-1-22-2': 'Provided that the variations vanish at the endpoints, [MATH], the stationary points of the quantum action [MATH] satisfy the Schrodinger equation, [EQUATION]', '1712.04813-1-22-3': 'We conclude that for each [MATH] there exists a unique stationary trajectory in the Hilbert space [MATH] such that [MATH].', '1712.04813-1-23-0': '## Variation of the reduced quantum action', '1712.04813-1-24-0': 'Now, suppose we confine the quantum action functional to trajectories in a subspace [MATH] that is parametrized by real parameters.', '1712.04813-1-24-1': 'More precisely, we assume a differentiable map [EQUATION]', '1712.04813-1-24-2': 'We will consider the reduced action [EQUATION]', '1712.04813-1-24-3': 'Its variation yields [EQUATION] where [MATH] and [MATH].', '1712.04813-1-24-4': "The stationary points [MATH] satisfy [EQUATION] for any variations [MATH]'s.", '1712.04813-1-24-5': 'In other words, the Schrodinger equation [MATH] holds only in the tangent space to [MATH], namely [EQUATION] and, in general, [MATH].', '1712.04813-1-24-6': 'Given an orthonormal basis, [MATH], in the tangent space [MATH], the equation of motion ([REF]) reads [EQUATION]', '1712.04813-1-24-7': 'Suppose we gradually enlarge the subspace [MATH] and its tangent space [MATH] by increasing the number of real parameters [MATH].', '1712.04813-1-24-8': 'Then, the orthonormal basis is enlarged accordingly [MATH].', '1712.04813-1-24-9': "Thus, for a fixed [MATH], its time derivative [MATH] becomes progressively a better and better approximation to the exact [MATH] as the series [EQUATION] converges by the virtue of Parseval's identity.", '1712.04813-1-24-10': 'Notice that the convergence is defined for each point separately.', '1712.04813-1-25-0': '## How to confine quantum motion?', '1712.04813-1-26-0': 'The Klauder semiclassical framework is based on a fixed family of coherent states.', '1712.04813-1-26-1': 'Each element of a given family satisfies the constraints, [EQUATION] which tighten the relation between classical observables and their quantum counterparts.', '1712.04813-1-26-2': 'We may view the families of coherent states as sections of a certain fiber bundle [CITATION].', '1712.04813-1-26-3': 'Namely, the total space is the Hilbert space (or, its dense subspace), the base space is the space of all possible expectation values of the basic operators, and the fibers are made of state vectors that give equal expectation values, [EQUATION]', '1712.04813-1-26-4': 'The "coherent" sections are defined by fixing a fiducial vector [MATH] in the fibre [MATH] and then by transporting it to all the other fibers via the unitary group action, [EQUATION]', '1712.04813-1-26-5': 'In other words, the orbits of the group define the "coherent" sections.', '1712.04813-1-26-6': 'There are as many families of coherent states as fiducial vectors, [MATH], and the particular choice of the fiducial vector fixes purely quantum characteristics of the coherent states such as dispersions of the basic observables.', '1712.04813-1-26-7': 'They are nonclassical parameters that are completely fixed by the fiducial vector and are not allowed to evolve as they normally would do.', '1712.04813-1-26-8': 'Thus, the dynamical contribution from nonclassical observables is completely neglected in the Klauder framework and the only dynamical observables are the expectation values [MATH] whose dynamics can be very rough.', '1712.04813-1-27-0': 'A way to improve this framework is to consider a fiducial space rather than a fiducial vector.', '1712.04813-1-27-1': 'It translates into confining the quantum motion to families of families of coherent states instead of a single family of coherent states.', '1712.04813-1-27-2': 'Such a framework allows the quantum motion to take place along the fibers of fixed expectation values of the basic observables.', '1712.04813-1-27-3': 'This idea is presented in Fig. [REF].', '1712.04813-1-27-4': 'Such a framework would keep the connection between quantum states and classical observables while adding more dimensions to the phase space, which would describe purely quantum features of quantum states.', '1712.04813-1-27-5': 'The number of the extra features would be controlled by the dimensionality of the fiducial space.', '1712.04813-1-27-6': 'Moreover, as we showed above, one expects that as the fiducial space is enlarged, the accuracy of this description is increased and it converges to a fully quantum mechanics expressed in terms of trajectories in a phase space of infinite dimension.', '1712.04813-1-28-0': "# Extension of Klauder's framework", '1712.04813-1-29-0': '## Quantum action', '1712.04813-1-30-0': 'We will now extend the Klauder semiclassical framework based on the affine coherent states.', '1712.04813-1-30-1': 'Instead of fixing a fiducial vector [MATH], we shall consider a fiducial space, [MATH], which contains the vectors labelled by [MATH].', '1712.04813-1-30-2': 'Hence, we obtain a family of families of affine coherent states, [EQUATION] which are labelled by [MATH].', '1712.04813-1-30-3': 'The quantum action restricted to those families of affine coherent states reads [EQUATION] where [EQUATION] where we assume that [MATH] and that the conditions of normalization and for the expectation values for the basic observables hold, [EQUATION]', '1712.04813-1-30-4': 'We assume the fiducial space to be linear and consist of the fiducial vectors of the form: [EQUATION] such that [MATH].', '1712.04813-1-30-5': 'Then, [EQUATION] where [MATH], [MATH] and [MATH] are hermitian.', '1712.04813-1-30-6': 'The quantum action reads now (after removing total time derivatives) [EQUATION] where [MATH] reads [EQUATION]', '1712.04813-1-30-7': 'From the above action one derives the Hamiltonian formalism [EQUATION] with the quadratic constraints [EQUATION]', '1712.04813-1-30-8': 'Note that the action ([REF]) yields the symplectic structure for both the classical and nonclassical observables.', '1712.04813-1-30-9': 'We follow the Dirac procedure [CITATION] and define the total Hamiltonian [EQUATION] where [MATH] are to be determined with the use of the consistency conditions, [EQUATION]', '1712.04813-1-31-0': '## Dynamics', '1712.04813-1-32-0': 'Since [MATH] is the identity operator and [MATH] is a hermitian operator, they can be simultaneously diagonalized.', '1712.04813-1-32-1': 'Suppose that they are diagonal, i.e. [MATH], [MATH].', '1712.04813-1-32-2': 'Then, [EQUATION]', '1712.04813-1-32-3': 'We introduce [MATH] and find [EQUATION]', '1712.04813-1-32-4': 'Thus, we may turn to a canonically equivalent formalism in which [EQUATION] and [EQUATION]', '1712.04813-1-32-5': 'Let us define [EQUATION]', '1712.04813-1-32-6': 'Now, the consistency relations yield [EQUATION]', '1712.04813-1-32-7': 'It follows that the normalization condition is a first-class constraint that generates a pure gauge transformation (an overall phase-shift) and thus, the coefficient [MATH] is arbitrary.', '1712.04813-1-32-8': 'On the other hand, the physical centering conditions are second-class and the vaules of the coefficients [MATH] and [MATH] are determined.', '1712.04813-1-32-9': 'The equations of motion take the form [EQUATION]', '1712.04813-1-33-0': '## A basis for the fiducial space', '1712.04813-1-34-0': 'In what follows we propose a set of orthonormal vectors [MATH] that diagonalize the dilation operator [MATH].', '1712.04813-1-34-1': 'Observe the following unitary transformation: [EQUATION]', '1712.04813-1-34-2': 'It transforms the dilation, position and momentum operator as follows [EQUATION]', '1712.04813-1-34-3': 'Notice that the dilation operator is the momentum operator on [MATH].', '1712.04813-1-34-4': 'Let us take the harmonic oscillator eigenvectors: [EQUATION] where [MATH] are the Hermite polynomials.', '1712.04813-1-34-5': 'If we restrict the considerations to the even eigenvectors, i.e. [EQUATION] we obtain [EQUATION]', '1712.04813-1-34-6': 'In this case the dynamical analysis becomes very simple.', '1712.04813-1-34-7': 'Indeed, the equations of motion ([REF]) become [EQUATION] where [MATH] is arbitrary and [EQUATION]', '1712.04813-1-35-0': '# Numerical examples', '1712.04813-1-36-0': 'In what follows we consider two simple examples.', '1712.04813-1-36-1': 'In the first example, we set the fiducial space to be two-dimensional, [EQUATION] where the vectors [MATH] are defined below Eq. ([REF]) and [MATH].', '1712.04813-1-36-2': 'We find that the absolute values [MATH] are constant in time while the respective phases are dynamical.', '1712.04813-1-36-3': 'The classical observables [MATH] and [MATH] undergo a simple bounce as in the case of Eq. ([REF]) to which solutions are presented in Fig. [REF].', '1712.04813-1-36-4': 'This result is not surprising as there is, in fact, no extra degrees of freedom.', '1712.04813-1-36-5': 'The counting of the extra degrees of freedom gives: 4 (two complex parameters) - 2 (two second-class constraints from the physical centering) -2 (a first-class constraint and the respective gauge transformation from the normalization condition) = 0.', '1712.04813-1-37-0': 'In the second example, we set the fiducial space to be three-dimensional, [EQUATION]', '1712.04813-1-37-1': 'In this case neither the absolute values [MATH] nor the respective phases are preserved during the evolution.', '1712.04813-1-37-2': 'In Fig. [REF] we compare the dynamics of the classical observables and of the extra parameters between the two- and three-parameter cases.', '1712.04813-1-38-0': '# Quantum dynamics of the Friedmann universe', '1712.04813-1-39-0': 'Let us see how one can apply the formalism developed herein to a quantum cosmological model, the quantum radiation-filled flat Friedmann universe with a bounce.', '1712.04813-1-39-1': 'For more details on the framework we refer to [CITATION].', '1712.04813-1-39-2': 'The metric of the classical model reads: [EQUATION] where [MATH] is a nonvanishing and otherwise arbitrary lapse function.', '1712.04813-1-39-3': 'The Hamiltonian constraint reads [EQUATION] where [MATH] and [MATH] are canonical variables that describe the radiation and [EQUATION] are canonical variables that describe the geometry, the scale factor [MATH] and the Hubble rate [MATH] times the scale factor squared, respectively.', '1712.04813-1-39-4': 'We solve the Hamiltonian constraint with respect to [MATH], set the lapse function [MATH] and employ the variable [MATH] as the internal clock.', '1712.04813-1-39-5': 'Then, the reduced phase space is given just by the canonical pair [MATH] and the physical Hamiltonian reads [EQUATION]', '1712.04813-1-39-6': 'The above Hamiltonian can be promoted to the quantum Hamiltonian of Eq. ([REF]).', '1712.04813-1-39-7': 'Then, we can use our approach to determine the quantum dynamics of the Friedmann universe in terms of a trajectory.', '1712.04813-1-39-8': 'In Fig. [REF] we will plot the dynamics of the classical variables [MATH] and [MATH] and their dispersions.', '1712.04813-1-39-9': 'Note the following relations, [EQUATION]', '1712.04813-1-39-10': 'In this section we just present one example of the extended phase space formulation of a quantum cosmological model.', '1712.04813-1-39-11': 'In our future work [CITATION] we investigate the quantum Friedmann model much more thoroughly.', '1712.04813-1-40-0': '# Conclusion', '1712.04813-1-41-0': 'In this article we present a phase space formulation of quantum mechanics.', '1712.04813-1-41-1': 'We start from the semiclassical framework introduced by J. Klauder many years ago and we extend it by inclusion of nonclassical observables that are equipped with a symplectic form.', '1712.04813-1-41-2': 'The obtained infinite-dimensional phase space trajectories are, in principle, equivalent to the exact solutions of the Schrodinger equation, though it is the possibility for consistent truncations to finite phase spaces that makes our approach attractive.', '1712.04813-1-41-3': 'We show that the respective Hamilton equations are not too complicated and can be successfully used for numerically integrating the dynamics.', '1712.04813-1-42-0': 'Our trajectory approach is a tool that opens many new possibilities in the studies of quantum cosmological systems.', '1712.04813-1-42-1': 'In the present article we test our approach with two simple examples.', '1712.04813-1-42-2': 'We postpone a detailed study of cosmological systems to our next papers.', '1712.04813-1-42-3': 'We believe that our approach can be helpful in establishing a definition of the classicality of cosmological systems.', '1712.04813-1-42-4': 'Given such a definition, we may be able to explain the classicality of the universe and probe the effects of the lack of classicality in the past of the universe on the present-day cosmological observables.', '1712.04813-1-42-5': 'We may learn if the universe can move back and forth between the classical and quantum phases.', '1712.04813-1-42-6': 'Finally, we could verify whether the universe could had been classical before the bounce.', '1712.04813-1-42-7': 'We investigate these and other issues in the forthcoming paper [CITATION].', '1712.04813-1-43-0': 'As a final remark, let us make a brief comparison of our approach to the Bohm-de Broglie (BdB) approach used in quantum cosmology [CITATION].', '1712.04813-1-43-1': 'According to the BdB formulation, a given solution to the Schrodinger equation, the so-called pilot-wave, is a source of an extra quantum term in the classical equations of motion which determine a complete set (i.e., for arbitrary initial data) of quantum trajectories in the classical phase space.', '1712.04813-1-43-2': 'Furthermore, it is the technology rather than physics that disables to predict which particular trajectory a physical system is going to follow.', '1712.04813-1-43-3': 'Interestingly, it is claimed that the technology-induced uncertainties generically evolve into the wavefunction-induced uncertainties and thus, the two become indistinguishable.', '1712.04813-1-43-4': 'Whereas in our approach a given solution to the Schrodinger equation is represented by a unique trajectory in an infinite dimensional phase space that includes both "classical" and "nonclassical" observables and a system follows a predictable trajectory.', '1712.04813-1-43-5': 'However, according to the Copenhagen interpretation, measuring devices are incompatible with the observables used in this description and necessarily lead to uncertainties.', '1712.04813-1-43-6': 'Whether our approach can stimulate new ways of thinking about the measurement is an open issue.'}

{'1712.04813-2-0-0': 'We develop an approach to quantum dynamics based on quantum phase space trajectories.', '1712.04813-2-0-1': 'The latter are built from a unitary irreducible representation of the symmetry group of the respective classical phase space.', '1712.04813-2-0-2': 'We use a quantum action functional to derive the basic equations.', '1712.04813-2-0-3': 'In principle, our formulation is equivalent to the Hilbert space formulation.', '1712.04813-2-0-4': 'However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum dynamics can be derived.', '1712.04813-2-0-5': 'We believe that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase space example to establish the basic equations of this formalism.', '1712.04813-2-1-0': '# Introduction', '1712.04813-2-2-0': 'In this article we propose a certain trajectory approach to quantum mechanics and we develop it with an emphasis on its application to simple quantum cosmological systems.', '1712.04813-2-2-1': "The essence of our approach is to reformulate the Schrodinger equation densely defined on vectors in the Hilbert space in terms of a set of Hamilton's equations in a phase space.", '1712.04813-2-2-2': 'The quantum states are represented by phase space points.', '1712.04813-2-2-3': 'The phase space variables encode both classical and nonclassical observables.', '1712.04813-2-2-4': 'By classical observables in this context we mean the expectation values of basic operators in a given quantum state.', '1712.04813-2-2-5': 'By nonclassical observables we mean all other phase space variables which encode such properties as the dispersions of basic observables and so on.', '1712.04813-2-3-0': 'Our formalism builds on the notion of coherent states and in particular on the idea of semiclassical framework by J. Klauder [CITATION].', '1712.04813-2-3-1': 'In fact, our formalism is a natural extension of the Klauder framework which is included as a special case.', '1712.04813-2-3-2': 'The basic tools behind our formalism are, first, the variational formulation of the quantum dynamics once a quantum Hamiltonian has been provided and, second, coherent states that are constructed with a unitary and irreducible representation of a minimal canonical group in the phase space of the respective classical model.', '1712.04813-2-3-3': 'The coherent states are used to incorporate the classical observables into our formalism in a natural (more precisely, covariant) way and, as it is in the original framework, the equations of motion for the classical observables include in general terms with nonvanishing [MATH].', '1712.04813-2-3-4': 'Nevertheless, our extension of the framework by nonclassical observables brings in some new advantages that are not present in the original framework.', '1712.04813-2-4-0': 'In principle, our approach is based on an infinite-dimensional phase space representing the quantum Hilbert space.', '1712.04813-2-4-1': 'However, its main advantage is that it includes consistent truncations to finite-dimensional phase spaces.', '1712.04813-2-4-2': 'Specifically, the dimensionality can be as large as to reproduce the exact Schrodinger equation in the form of phase space trajectories in the case of infinitely many dimensions, or as small as to give the roughest quantum corrections to the classical equations of motion in the case of the classical phase space dimensionality.', '1712.04813-2-4-3': 'The latter case corresponds, in fact, to the Klauder framework.', '1712.04813-2-4-4': 'Our approach allows to control the level of detail and can be adjusted to any specific quantum system, in particular, it can be used to deal with those quantum dynamics which are too complex to be solved explicitly.', '1712.04813-2-5-0': 'Since we are mainly interested in quantum cosmological systems, we will develop our ideas for the case of the classical phase space which is a half-plane rather than a plane.', '1712.04813-2-5-1': 'For this particular phase space, a well-suited minimal canonical group is the affine group.', '1712.04813-2-5-2': 'Nevertheless, we wish to emphasize that our approach can be easily developed for the case of Weyl-Heisenberg (W-H) and other groups.', '1712.04813-2-6-0': 'The great advantage of our approach is that it allows for addressing physical questions which are not possible in the simplest semiclassical framework.', '1712.04813-2-6-1': 'For example, as usual classical phase space is now extended by nonclassical observables (such as spreading variables which describe specific quantum effects) one can investigate the dynamical properties of nonclassical observables in pseudo-classical terms of energy transfers between classical and nonclassical observables.', '1712.04813-2-6-2': 'It is worth mentioning that the appearance of such phenomenons could be misinterpreted as "real" energy transfers from/to hidden spatial dimensions in the universe, which are introduced by the brane-world theories and alike [CITATION].', '1712.04813-2-6-3': 'In fact, a wave-function naturally spreads when a system leaves a classical-like regime and enters a more quantum one.', '1712.04813-2-6-4': 'The dynamical spreading is in particular expected for cosmological systems when they approach the big-bang singularity.', '1712.04813-2-6-5': 'As it was already shown in [CITATION], in this case a quantum repulsive potential may halt the contraction preventing the universe form collapsing into the singularity and make it bounce and re-expand.', '1712.04813-2-6-6': 'Despite the fact that the expectation values of the basic observables such as the volume or the Hubble rate evolve symmetrically on both sides of the bounce (see [CITATION] or almost any other work on the semiclassical dynamics of bouncing Friedmann models), on the fully quantum level the bounce does not simply revert the evolution.', '1712.04813-2-6-7': 'Thus, the evolution of some of quantum features is expected to be asymmetric with respect to the bounce.', '1712.04813-2-6-8': 'The detailed behavior can be captured within our extended approach by inspecting the evolution of nonclassical observables.', '1712.04813-2-7-0': 'Other associated questions that can be potentially examined within our approach include: How to specify the degree of "classicality" (i.e. a measure of classicality) of a quantum behavior or, put differently in the present context, what is the "classical universe"?', '1712.04813-2-7-1': 'How this classical universe emerges from a quantum state?', '1712.04813-2-7-2': 'Had the universe been classical before the bounce?', '1712.04813-2-7-3': 'Note, however, that our approach concerns only the deterministic dynamical behavior of objects in quantum mechanics, which includes the wave-function itself and the expectation values of observables.', '1712.04813-2-7-4': 'Other deep questions involved in quantum mechanics such its interpretation or the measurement problem are not addressed within this framework.', '1712.04813-2-8-0': 'To finish these introductory remarks, let us notice that our formulation offers an alternative way of looking at quantum dynamics.', '1712.04813-2-8-1': 'To some extend it could be also viewed as a kind of a hidden-variable theory.', '1712.04813-2-8-2': 'Our approach is based on the time-dependent variational principle and therefore, it bears some resemblance to the formalism developed in [CITATION].', '1712.04813-2-8-3': 'However, the latter lacks the physical interpretation that we obtain thanks to the use of coherent states.', '1712.04813-2-9-0': 'The outline of the article is as follows.', '1712.04813-2-9-1': 'We begin by recalling in Sec II some basic properties of the expectation values in quantum mechanics, which sets a broad context for our framework.', '1712.04813-2-9-2': 'In Sec III we recall the semiclassical framework of Klauder.', '1712.04813-2-9-3': 'In Sec IV we discuss the variational formulation of quantum dynamics.', '1712.04813-2-9-4': 'In Sec V we develop our formalism.', '1712.04813-2-9-5': 'We apply it to two examples in Sec VI.', '1712.04813-2-9-6': 'In Sec VII we revisit the quantum flat Friedmann model with our approach.', '1712.04813-2-9-7': 'We conclude in Sec VIII.', '1712.04813-2-9-8': 'The appendices deal with some technicalities (self-adjointness and the numerical code) which have been omitted from the main text.', '1712.04813-2-10-0': '# States and expectation values in quantum mechanics', '1712.04813-2-11-0': 'Let us assume a quantum system described by a normalized state [MATH], or rather by the corresponding projector (ray) [MATH] in a finite dimensional Hilbert space [MATH] of dimension [MATH].', '1712.04813-2-11-1': '[MATH] belongs to the complex projective space [MATH] and depends on [MATH] real parameters.', '1712.04813-2-11-2': 'For a quantum observable represented by a self-adjoint operator [MATH] on [MATH] the expectation value in a state [MATH] is given by [EQUATION]', '1712.04813-2-11-3': 'The Lie algebra of self-adjoint operators on [MATH] is a real vector space of dimension [MATH], or [MATH] if we exclude the identity.', '1712.04813-2-11-4': 'Notice that [MATH] for [MATH].', '1712.04813-2-11-5': 'Therefore, if we choose appropriately [MATH] independent observables [MATH], the mapping [EQUATION] is locally invertible.', '1712.04813-2-11-6': 'Hence, the set of rays [MATH] can be seen as a manifold locally parametrized by an array of expectation values [MATH].', '1712.04813-2-11-7': 'This mapping gives a natural physical picture of a quantum state: a quantum state is a complete set of statistical properties specified by a family of expectation values.', '1712.04813-2-11-8': 'The inverse mapping: [MATH], allows to define any expectation value of any quantum observable [MATH] as a function [EQUATION]', '1712.04813-2-11-9': 'Hence, the set of quantum expectation values looks like a set of classical observables defined on a classical phase space represented here by the set of [MATH].', '1712.04813-2-11-10': 'This picture is enhanced by the Ehrenfest theorem stipulating that expectation values have a deterministic behavior through equations similar to the Hamilton equations.', '1712.04813-2-11-11': 'Notice, however, that any function of [MATH] is not an expectation value of a quantum observable.', '1712.04813-2-11-12': 'This is different from the usual classical framework.', '1712.04813-2-12-0': 'Notice that this picture obscures those quantum aspects of single systems that are addressed by the so-called measurement axioms of quantum mechanics.', '1712.04813-2-12-1': 'Nevertheless, the usual stochastic quantum reasoning remains in principle accessible since the quantum probabilities yielded by the Born rule, [EQUATION] are included in the framework through Eq. ([REF]) for [MATH] and [MATH].', '1712.04813-2-13-0': 'The above picture is very attractive for establishing a bridge between classical and quantum calculations.', '1712.04813-2-13-1': 'Indeed, if we ignore the quantum stochastic origin of the picture, we recover a classical-like formalism.', '1712.04813-2-13-2': 'The presented construction is valid only for finite dimensional Hilbert spaces, though, the idea of using expectation values is obviously attractive for the infinite dimensional spaces as well.', '1712.04813-2-13-3': 'A desired extension can be established if one finds a way to truncate the principally infinite sequence of expectation values needed to specify a quantum state belonging to an infinite dimensional Hilbert space.', '1712.04813-2-13-4': 'Herein we propose a suitable framework.', '1712.04813-2-13-5': 'Let us emphasize that this framework bears little resemblance with the usual "phase space formulation of quantum mechanics" based on Wigner functions, Weyl-Wigner transformation, the star product, etc.', '1712.04813-2-14-0': "# Coherent states and Klauder's framework", '1712.04813-2-15-0': '## Coherent states', '1712.04813-2-16-0': 'By coherent states (see e.g. [CITATION]) we mean a continuous mapping from a set of labels, collectively denoted by [MATH] and equipped with a measure [MATH], into unit vectors in Hilbert space, [EQUATION] such that it resolves the identity, [EQUATION]', '1712.04813-2-16-1': 'Hence, the coherent states [MATH] form an overcomplete basis in [MATH].', '1712.04813-2-16-2': 'The above property was first used by Klauder [CITATION] in his definition of what he called an overcomplete family of states (OFS).', '1712.04813-2-16-3': 'They provide a bridge between the abstract quantum formalism and the continuous label-space, [EQUATION] where [MATH] is a normalized (with respect to [MATH]) probability distribution on the space [MATH] that can be further identified with some classical observables.', '1712.04813-2-16-4': 'Suppose that there exists a unitary irreducible representation on [MATH], [MATH], of a minimal group of canonical transformations in a phase space [MATH].', '1712.04813-2-16-5': 'Then, the mapping [EQUATION] defines a family of coherent states whose labels describe the classical states of that system [CITATION].', '1712.04813-2-16-6': 'The fixed normalized vector [MATH] is called the fiducial vector.', '1712.04813-2-17-0': '## Framework', '1712.04813-2-18-0': 'First, let us recall that the quantum dynamics can be obtained via the variation of the quantum action, [EQUATION] with respect to the normalized [MATH], where [MATH] is the quantum Hamiltonian that corresponds to a certain classical Hamiltonian, [MATH].', '1712.04813-2-18-1': 'The stationary points of the action ([REF]) are found to satisfy the Schrodinger equation, [EQUATION]', '1712.04813-2-18-2': 'The idea of the semiclassical framework based on the coherent states was introduced by Klauder in [CITATION].', '1712.04813-2-18-3': 'Initially, he applied it to the case of the phase space [MATH] and the Weyl-Heisenberg coherent states.', '1712.04813-2-18-4': 'The W-H coherent states, [MATH], are defined as follows [EQUATION] where the displacement operator [MATH] satisfies [EQUATION] and where [MATH] are the position and momentum operators [CITATION].', '1712.04813-2-18-5': 'The fiducial vector [MATH] is fixed and its choice is almost arbitrary.', '1712.04813-2-18-6': 'The only condition that one imposes on the fiducial is the so called physical centering condition.', '1712.04813-2-18-7': 'It relates the classical observables and the expectation values of the respective operators by demanding [EQUATION] and leads to the constraint, [EQUATION]', '1712.04813-2-18-8': 'We find that [EQUATION] where [MATH] is the exterior derivative.', '1712.04813-2-18-9': 'This result can be guessed (up to the irrelevant total derivative) from the fact that the above one-form must be invariant with respect to the action of the W-H group.', '1712.04813-2-18-10': 'The same reasoning applies to all other canonical groups.', '1712.04813-2-18-11': 'Now the quantum action functional ([REF]) can be evaluated on the family of coherent states, [EQUATION] where [MATH].', '1712.04813-2-18-12': 'Thus, the variation of the quantum action with respect to the [MATH]-labelled coherent states yields the Hamilton equations for [MATH] and [MATH], [EQUATION]', '1712.04813-2-18-13': 'On the one hand, the above equation provides an approximation to the exact quantum motion via the coherent states, [EQUATION] and on the other hand, it establishes a very appealing interpretation of the classical observables and their dynamics within the more fundamental quantum framework.', '1712.04813-2-18-14': 'We notice that these equations, in general, differ from the classical equations as the semiclassical Hamiltonian [MATH] may include [MATH]-corrections, [EQUATION]', '1712.04813-2-18-15': 'Since in the real world [MATH] never vanishes, the possibility of modeling the dynamics of the classical observables with nonvanishing [MATH] could be very useful.', '1712.04813-2-18-16': 'Indeed, this possibility becomes particularly important for improving the dynamics of classically singular cosmological models as we show later.', '1712.04813-2-19-0': '## Affine group', '1712.04813-2-20-0': 'As we are concerned with gravitational systems, we shall turn to the important example of the phase space that appears in cosmology, namely the half-plane [MATH].', '1712.04813-2-20-1': 'The basic observables form a canonical pair, [EQUATION] where [MATH] is the volume of the universe and [MATH] is a rate of its expansion.', '1712.04813-2-20-2': 'Clearly, the W-H group is not applicable to the present case as one of the canonical variables, [MATH], is confined to the half-line.', '1712.04813-2-20-3': 'Instead, we shall employ the affine group [CITATION], [MATH], that is defined by the multiplication law, [EQUATION] and preserves the symplectic structure of the half-plane phase space, [EQUATION] where [MATH] is a fixed element of the affine group.', '1712.04813-2-20-4': 'There exists a unique (up to sign) unitary irreducible representation of [MATH], which in [MATH] takes the form [EQUATION]', '1712.04813-2-20-5': 'Thus, we define the affine coherent states as [EQUATION] where [MATH] is the fiducial vector that is subject to the constraint [EQUATION] (which follows from the group integrability condition).', '1712.04813-2-20-6': 'To tighten the connection between quantum and classical observables we demand [EQUATION] where [MATH] and [MATH].', '1712.04813-2-20-7': 'This is equivalent to [EQUATION]', '1712.04813-2-20-8': 'One finds that [EQUATION] where [MATH] is the dilation operator and which confirms the general statement given below Eq. ([REF]).', '1712.04813-2-20-9': 'Now, provided a quantum Hamiltonian [MATH], the quantum action functional evaluated on the affine coherent states reads [EQUATION] where [MATH].', '1712.04813-2-20-10': 'Hence, the variation of the quantum action with respect to the classical labels [MATH] yields the Hamilton equations, [EQUATION] for the stationary trajectories.', '1712.04813-2-21-0': '## Free particle dynamics on [MATH]', '1712.04813-2-22-0': 'In this article we are going to study a quantum free motion of a particle on the half-line, [MATH].', '1712.04813-2-22-1': 'It is a very important example as it formally describes the dynamics of the flat Friedmann universe with a perfect fluid-source [CITATION].', '1712.04813-2-22-2': 'The big-bang singularity is represented by the end-point, [MATH].', '1712.04813-2-22-3': 'The variable [MATH] describes the volume and the variable [MATH] describes the expansion of the universe (see Sec VI for more details).', '1712.04813-2-22-4': 'Let the classical system be defined as follows, [EQUATION] and the quantum Hamiltonian read [EQUATION]', '1712.04813-2-22-5': 'We discuss the technical issue of extending the above symmetric operator to a self-adjoint one in Appendix A. For a fiducial vector [MATH], we obtain [EQUATION] where [EQUATION] where [MATH].', '1712.04813-2-22-6': 'The respective Hamilton equations ([REF]) include an [MATH]-correction that resolves the singularity at [MATH].', '1712.04813-2-22-7': 'The particle is repelled away from the singularity by the quantum potential [MATH] and this produces a bounce in its dynamics.', '1712.04813-2-22-8': 'See Fig. [REF].', '1712.04813-2-23-0': '# Theory of the restricted quantum action', '1712.04813-2-24-0': 'The idea of this work is to extend the phase space description of quantum mechanics due to J. Klauder to a significantly broader framework.', '1712.04813-2-24-1': 'For this purpose it is useful to discuss the quantum action formulation of quantum dynamics in somewhat more detail.', '1712.04813-2-25-0': '## Variation of the quantum action', '1712.04813-2-26-0': 'The quantum action is defined on trajectories in the Hilbert space and reads [EQUATION]', '1712.04813-2-26-1': 'Its variation with respect to [MATH] such that [MATH] at each [MATH], yields [EQUATION]', '1712.04813-2-26-2': 'Provided that the variations vanish at the endpoints, [MATH], the stationary points of the quantum action [MATH] satisfy the Schrodinger equation, [EQUATION]', '1712.04813-2-26-3': 'We conclude that for each [MATH] there exists a unique stationary trajectory in the Hilbert space [MATH] such that [MATH].', '1712.04813-2-27-0': '## Variation of the reduced quantum action', '1712.04813-2-28-0': 'Now, suppose we confine the quantum action functional to trajectories in a subspace [MATH] that is parametrized by real parameters.', '1712.04813-2-28-1': 'More precisely, we assume a differentiable map [EQUATION]', '1712.04813-2-28-2': 'We will consider the reduced action [EQUATION]', '1712.04813-2-28-3': 'Its variation yields [EQUATION] where [MATH] and [MATH].', '1712.04813-2-28-4': "The stationary trajectories [MATH] satisfy [EQUATION] for any variations [MATH]'s.", '1712.04813-2-28-5': 'In other words, the Schrodinger equation [MATH] holds only in the tangent space to [MATH], namely [EQUATION] and, in general, [MATH].', '1712.04813-2-28-6': 'Given an orthonormal basis, [MATH], in the tangent space [MATH], the equation of motion ([REF]) reads [EQUATION]', '1712.04813-2-28-7': 'Suppose we gradually enlarge the subspace [MATH] and its tangent space [MATH] by increasing the number of real parameters [MATH].', '1712.04813-2-28-8': 'Then, the orthonormal basis is enlarged accordingly [MATH].', '1712.04813-2-28-9': "Thus, for a fixed [MATH], its time derivative [MATH] becomes progressively a better and better approximation to the exact [MATH] as the series [EQUATION] converges by the virtue of Parseval's identity.", '1712.04813-2-28-10': 'Notice that the convergence is defined for each point separately.', '1712.04813-2-29-0': '## How to confine quantum motion?', '1712.04813-2-30-0': 'The Klauder semiclassical framework is based on a fixed family of coherent states.', '1712.04813-2-30-1': 'Each element of a given family satisfies the constraints, [EQUATION] which tighten the relation between classical observables and their quantum counterparts.', '1712.04813-2-30-2': 'We may view the families of coherent states as sections of a certain fiber bundle [CITATION].', '1712.04813-2-30-3': 'Namely, the total space is the Hilbert space (or, its dense subspace), the base space is the space of all possible expectation values of the basic operators, and the fibers are made of state vectors that give equal expectation values, [EQUATION]', '1712.04813-2-30-4': 'The "coherent" sections are defined by fixing a fiducial vector [MATH] in the fibre [MATH] and then by transporting it to all the other fibers via the unitary group action, [EQUATION]', '1712.04813-2-30-5': 'In other words, the orbits of the group define the "coherent" sections.', '1712.04813-2-30-6': 'There are as many families of coherent states as fiducial vectors, [MATH], and the particular choice of the fiducial vector fixes purely quantum characteristics of the coherent states such as dispersions of the basic observables.', '1712.04813-2-30-7': 'They are nonclassical parameters that are completely fixed by the fiducial vector and are not allowed to evolve as they normally would do.', '1712.04813-2-30-8': 'Thus, the dynamical contribution from nonclassical observables is completely neglected in the Klauder framework and the only dynamical observables are the expectation values [MATH] whose approximate dynamics could be for some purposes too rough.', '1712.04813-2-31-0': 'A way to improve this framework is to consider a fiducial space rather than a fiducial vector.', '1712.04813-2-31-1': 'It translates into confining the quantum motion to families of families of coherent states instead of a single family of coherent states.', '1712.04813-2-31-2': 'Such a framework allows the quantum motion to take place along the fibers of fixed expectation values of the basic observables.', '1712.04813-2-31-3': 'This idea is presented in Fig. [REF].', '1712.04813-2-31-4': 'Such a framework would keep the connection between quantum states and classical observables while adding more dimensions to the phase space, which would describe purely quantum features of the quantum states.', '1712.04813-2-31-5': 'The number of the extra features would be controlled by the dimensionality of the fiducial space.', '1712.04813-2-31-6': 'Moreover, as we showed above, one expects that as the fiducial space is enlarged, the accuracy of this description is increased and it converges to a fully quantum mechanics expressed in terms of trajectories in a phase space of infinite dimension.', '1712.04813-2-32-0': "# Extension of Klauder's framework", '1712.04813-2-33-0': '## Quantum action', '1712.04813-2-34-0': 'We will now extend the Klauder semiclassical framework based on the affine coherent states.', '1712.04813-2-34-1': 'Instead of fixing a fiducial vector [MATH], we shall consider a fiducial space, [MATH], which contains the vectors labelled by [MATH].', '1712.04813-2-34-2': 'Hence, we obtain a family of families of the affine coherent states, [EQUATION] which are labelled by [MATH].', '1712.04813-2-34-3': 'The quantum action functional ([REF]) restricted to those families of the affine coherent states reads [EQUATION] where [EQUATION] where we assume that [MATH] and that the conditions of normalization and for the expectation values for the basic observables hold, [EQUATION]', '1712.04813-2-34-4': 'We assume the fiducial space to be linear and consist of the fiducial vectors of the form: [EQUATION] such that [MATH].', '1712.04813-2-34-5': 'Then, [EQUATION] where [MATH], [MATH] and [MATH] are hermitian.', '1712.04813-2-34-6': 'The quantum action reads now (after removing total time derivatives) [EQUATION] where [MATH] reads [EQUATION]', '1712.04813-2-34-7': 'From the above action one derives the Hamiltonian formalism [EQUATION] with the quadratic constraints [EQUATION]', '1712.04813-2-34-8': 'Note that the action ([REF]) yields the symplectic structure for both the classical and nonclassical observables.', '1712.04813-2-34-9': 'We follow the Dirac procedure [CITATION] and define the total Hamiltonian [EQUATION] where [MATH] are to be determined with the use of the consistency conditions, [EQUATION]', '1712.04813-2-35-0': '## Dynamics', '1712.04813-2-36-0': 'Since [MATH] is the identity operator and [MATH] is a hermitian operator, they can be simultaneously diagonalized.', '1712.04813-2-36-1': 'Suppose that they are diagonal, i.e. [MATH], [MATH].', '1712.04813-2-36-2': 'Then, [EQUATION]', '1712.04813-2-36-3': 'We introduce [MATH] and find [EQUATION]', '1712.04813-2-36-4': 'Thus, we may turn to a canonically equivalent formalism in which [EQUATION] and [EQUATION]', '1712.04813-2-36-5': 'Let us define [EQUATION]', '1712.04813-2-36-6': 'Now, the consistency relations yield [EQUATION]', '1712.04813-2-36-7': 'It follows that the normalization condition is a first-class constraint that generates a pure gauge transformation (an overall phase-shift) and thus, the coefficient [MATH] is arbitrary.', '1712.04813-2-36-8': 'On the other hand, the physical centering conditions are second-class and the vaules of the coefficients [MATH] and [MATH] are determined.', '1712.04813-2-36-9': 'The equations of motion take the form [EQUATION]', '1712.04813-2-37-0': '## A basis for the fiducial space', '1712.04813-2-38-0': 'In what follows we propose a set of orthonormal vectors [MATH] that diagonalize the dilation operator [MATH].', '1712.04813-2-38-1': 'Observe the following unitary transformation: [EQUATION]', '1712.04813-2-38-2': 'It transforms the dilation, position and momentum operator as follows [EQUATION]', '1712.04813-2-38-3': 'Notice that the dilation operator is the momentum operator on [MATH].', '1712.04813-2-38-4': 'Let us take the harmonic oscillator eigenvectors: [EQUATION] where [MATH] are the Hermite polynomials.', '1712.04813-2-38-5': 'If we restrict the considerations to the even eigenvectors, i.e. [EQUATION] we obtain [EQUATION]', '1712.04813-2-38-6': 'In this case the dynamical analysis becomes very simple.', '1712.04813-2-38-7': 'Indeed, the equations of motion ([REF],[REF],[REF]) become [EQUATION] where [MATH] is arbitrary and [EQUATION]', '1712.04813-2-39-0': '# Numerical examples', '1712.04813-2-40-0': 'In what follows we consider two simple examples.', '1712.04813-2-40-1': 'In the first example, we set the fiducial space to be two-dimensional, [EQUATION] where the vectors [MATH] are defined by Eq. ([REF]) and [MATH].', '1712.04813-2-40-2': 'We find that the absolute values [MATH] are constant in time while the respective phases are dynamical.', '1712.04813-2-40-3': 'The classical observables [MATH] and [MATH] undergo a simple bounce as in the case of Eq. ([REF]) to which solutions are presented in Fig. [REF].', '1712.04813-2-40-4': 'This result is not surprising as there is, in fact, no extra degree of freedom.', '1712.04813-2-40-5': 'The counting of the extra degrees of freedom gives: 4 (two complex parameters) - 2 (two second-class constraints from the physical centering) - 2 (a first-class constraint and the respective gauge transformation from the normalization condition) = 0.', '1712.04813-2-41-0': 'In the second example, we set the fiducial space to be three-dimensional, [EQUATION]', '1712.04813-2-41-1': 'In this case neither the absolute values [MATH] nor the respective phases are preserved during the evolution.', '1712.04813-2-41-2': 'In Fig. [REF] we compare the dynamics of the classical observables and of the extra parameters between the two- and three-parameter cases.', '1712.04813-2-42-0': '# Quantum dynamics of the Friedmann universe', '1712.04813-2-43-0': 'Let us see how one can apply the formalism developed above to a quantum cosmological model, namely the quantum radiation-filled flat Friedmann universe with a bounce.', '1712.04813-2-43-1': 'For more details on the framework we refer to [CITATION].', '1712.04813-2-43-2': 'The metric of the classical model reads: [EQUATION] where [MATH] is a nonvanishing and otherwise arbitrary lapse function.', '1712.04813-2-43-3': 'The Hamiltonian constraint reads [EQUATION] where [MATH] and [MATH] are canonical variables that describe the radiation and [EQUATION] are canonical variables that describe the geometry, the scale factor [MATH] and the Hubble rate [MATH] times the scale factor squared, respectively.', '1712.04813-2-43-4': 'We solve the Hamiltonian constraint with respect to [MATH], set the lapse function [MATH] and employ the variable [MATH] as the internal clock.', '1712.04813-2-43-5': 'Then, the reduced phase space is given just by the canonical pair [MATH] and the physical Hamiltonian reads [EQUATION]', '1712.04813-2-43-6': 'The above Hamiltonian can be promoted to the quantum Hamiltonian of Eq. ([REF]).', '1712.04813-2-43-7': 'Then, we can use our approach to determine the quantum dynamics of the Friedmann universe in terms of a trajectory.', '1712.04813-2-43-8': 'In Fig. [REF] we will plot the dynamics of the classical variables [MATH] and [MATH] and their dispersions.', '1712.04813-2-43-9': 'Note the following relations, [EQUATION]', '1712.04813-2-43-10': 'In this section we present just one example of the extended phase space formulation of a quantum cosmological model.', '1712.04813-2-43-11': 'In our future work [CITATION] we investigate the quantum Friedmann model much more thoroughly.', '1712.04813-2-44-0': '# Conclusion', '1712.04813-2-45-0': 'In this article we present a quantum phase space trajectory approach to quantum dynamics.', '1712.04813-2-45-1': 'We start from the semiclassical framework introduced by J. Klauder many years ago and we extend it by inclusion of nonclassical observables that are equipped with a symplectic form.', '1712.04813-2-45-2': 'The obtained infinite-dimensional phase space trajectories are, in principle, equivalent to the exact solutions of the Schrodinger equation, though it is the possibility for consistent truncations to finite phase spaces that makes our approach attractive.', '1712.04813-2-45-3': 'We show that the respective Hamilton equations are not too complicated and can be successfully used for numerically integrating the dynamics.', '1712.04813-2-46-0': 'Our trajectory approach is a tool that opens new possibilities in the studies of quantum cosmological systems.', '1712.04813-2-46-1': 'In the present article we test our approach with two simple examples.', '1712.04813-2-46-2': 'We postpone a detailed study of cosmological systems to our next papers.', '1712.04813-2-46-3': 'We believe that our approach can be helpful in establishing a definition of the "degree of classicality" of cosmological systems.', '1712.04813-2-46-4': 'If the universe is quantum by nature, it is never really classical or, put differently, quantum mechanics cannot "disappear".', '1712.04813-2-46-5': 'Therefore, "classicality" must correspond to a special quantum behavior or, more precisely, to a special behavior in a particular picture of quantum dynamics.', '1712.04813-2-46-6': 'Given such a definition, we may be able to "explain", or "recover", the supposed classicality of the present universe and probe the effects of the lack of classicality on the past of the universe.', '1712.04813-2-46-7': 'We may learn if the universe can move back and forth between the classical and quantum phases.', '1712.04813-2-46-8': 'Finally, we could verify whether the universe could had been classical before the bounce.', '1712.04813-2-46-9': 'We investigate these and other related issues in the forthcoming paper [CITATION].', '1712.04813-2-47-0': 'Since the main purpose of developing this framework was to study quantum cosmological systems, we are led to ask to what extent a framework based on expectation values can reasonably describe a single system, namely the universe.', '1712.04813-2-47-1': 'There are two possible attitudes.', '1712.04813-2-47-2': 'The first attitude is to focus on the mathematical structure.', '1712.04813-2-47-3': 'Since our framework includes the complete time behavior of the wave-function, it is physically equivalent to the Schrodinger equation.', '1712.04813-2-47-4': 'If our main purpose is to construct a picture of quantum dynamics that allows for a direct comparison with the corresponding classical equations of motion, we simply state that our framework is a very good candidate.', '1712.04813-2-47-5': 'The second attitude starts with the observation that the question of the interpretation of our formulation has been ignored.', '1712.04813-2-47-6': 'However, a similar interpretational issue arises in statistical physics where it is addressed with the so-called thermodynamical limit.', '1712.04813-2-47-7': 'Namely, statistical physics is designed to describe ensembles of systems in terms of probabilities.', '1712.04813-2-47-8': 'However, the expectation values obtained from this theory are able to describe individual large systems: this is the thermodynamical limit.', '1712.04813-2-47-9': 'We can make an analogy and view a homogeneous cosmological system as made of an infinite number of "copies" of the same system localized at different points of space and therefore, describable as a large system made of "small identical systems".', '1712.04813-2-47-10': 'Provided that the property valid in the framework of statistical physics can somehow be applied to the quantum cosmological context, the set of quantum expectation values in the cosmological framework becomes a relevant description of the universe.', '1712.04813-2-48-0': 'As a final remark, let us make a brief comparison of our approach to the Bohm-de Broglie (BdB) approach used in quantum cosmology [CITATION].', '1712.04813-2-48-1': 'In the BdB formulation, a given solution to the Schrodinger equation plays the role of the so-called pilot-wave which is a source of an extra quantum term in the classical equations of motion.', '1712.04813-2-48-2': 'The latter determine a complete set (i.e., for arbitrary initial data) of quantum trajectories in the classical phase space.', '1712.04813-2-48-3': 'Whereas in our approach, a given solution to the Schrodinger equation is represented by a unique trajectory of quantum expectation values in an infinite dimensional phase space that includes both classical and nonclassical variables.', '1712.04813-2-48-4': 'Thus, in our approach the system follows a unique and predictable, though, higher dimensional trajectory.'}

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'1712.04813-2-30-2'], ['1712.04813-1-26-3', '1712.04813-2-30-3'], ['1712.04813-1-26-4', '1712.04813-2-30-4'], ['1712.04813-1-26-5', '1712.04813-2-30-5'], ['1712.04813-1-26-6', '1712.04813-2-30-6'], ['1712.04813-1-26-7', '1712.04813-2-30-7'], ['1712.04813-1-43-0', '1712.04813-2-48-0'], ['1712.04813-1-34-0', '1712.04813-2-38-0'], ['1712.04813-1-34-1', '1712.04813-2-38-1'], ['1712.04813-1-34-2', '1712.04813-2-38-2'], ['1712.04813-1-34-3', '1712.04813-2-38-3'], ['1712.04813-1-34-4', '1712.04813-2-38-4'], ['1712.04813-1-34-5', '1712.04813-2-38-5'], ['1712.04813-1-34-6', '1712.04813-2-38-6'], ['1712.04813-1-39-1', '1712.04813-2-43-1'], ['1712.04813-1-39-2', '1712.04813-2-43-2'], ['1712.04813-1-39-3', '1712.04813-2-43-3'], ['1712.04813-1-39-4', '1712.04813-2-43-4'], ['1712.04813-1-39-5', '1712.04813-2-43-5'], ['1712.04813-1-39-6', '1712.04813-2-43-6'], ['1712.04813-1-39-7', '1712.04813-2-43-7'], ['1712.04813-1-39-8', '1712.04813-2-43-8'], ['1712.04813-1-39-9', '1712.04813-2-43-9'], ['1712.04813-1-39-11', '1712.04813-2-43-11'], ['1712.04813-1-4-1', '1712.04813-2-4-1'], ['1712.04813-1-4-2', '1712.04813-2-4-2'], ['1712.04813-1-4-3', '1712.04813-2-4-3'], ['1712.04813-1-0-1', '1712.04813-2-0-2'], ['1712.04813-1-0-2', '1712.04813-2-0-3'], ['1712.04813-1-16-1', '1712.04813-2-20-1'], ['1712.04813-1-16-2', '1712.04813-2-20-2'], ['1712.04813-1-16-3', '1712.04813-2-20-3'], ['1712.04813-1-16-4', '1712.04813-2-20-4'], ['1712.04813-1-16-5', '1712.04813-2-20-5'], ['1712.04813-1-16-6', '1712.04813-2-20-6'], ['1712.04813-1-16-7', '1712.04813-2-20-7'], ['1712.04813-1-16-8', '1712.04813-2-20-8'], ['1712.04813-1-16-9', '1712.04813-2-20-9'], ['1712.04813-1-16-10', '1712.04813-2-20-10'], ['1712.04813-1-18-0', '1712.04813-2-22-0'], ['1712.04813-1-18-1', '1712.04813-2-22-1'], ['1712.04813-1-18-2', '1712.04813-2-22-2'], ['1712.04813-1-18-4', '1712.04813-2-22-4'], ['1712.04813-1-18-5', '1712.04813-2-22-5'], ['1712.04813-1-18-6', '1712.04813-2-22-6'], ['1712.04813-1-18-7', '1712.04813-2-22-7'], ['1712.04813-1-32-0', '1712.04813-2-36-0'], ['1712.04813-1-32-1', '1712.04813-2-36-1'], ['1712.04813-1-32-3', '1712.04813-2-36-3'], ['1712.04813-1-32-4', '1712.04813-2-36-4'], ['1712.04813-1-32-5', '1712.04813-2-36-5'], ['1712.04813-1-32-6', '1712.04813-2-36-6'], ['1712.04813-1-32-7', '1712.04813-2-36-7'], ['1712.04813-1-32-8', '1712.04813-2-36-8'], ['1712.04813-1-32-9', '1712.04813-2-36-9'], ['1712.04813-1-11-4', '1712.04813-2-16-5'], ['1712.04813-1-42-1', '1712.04813-2-46-1'], ['1712.04813-1-42-2', '1712.04813-2-46-2'], ['1712.04813-1-42-5', '1712.04813-2-46-7'], ['1712.04813-1-42-6', '1712.04813-2-46-8'], ['1712.04813-1-37-0', '1712.04813-2-41-0'], ['1712.04813-1-37-1', '1712.04813-2-41-1'], ['1712.04813-1-37-2', '1712.04813-2-41-2'], ['1712.04813-1-30-0', '1712.04813-2-34-0'], ['1712.04813-1-30-1', '1712.04813-2-34-1'], ['1712.04813-1-30-4', '1712.04813-2-34-4'], ['1712.04813-1-30-5', '1712.04813-2-34-5'], ['1712.04813-1-30-6', '1712.04813-2-34-6'], ['1712.04813-1-30-7', '1712.04813-2-34-7'], ['1712.04813-1-30-8', '1712.04813-2-34-8'], ['1712.04813-1-30-9', '1712.04813-2-34-9'], ['1712.04813-1-2-0', '1712.04813-2-2-0'], ['1712.04813-1-2-1', '1712.04813-2-2-1'], ['1712.04813-1-2-3', '1712.04813-2-2-3'], ['1712.04813-1-2-4', '1712.04813-2-2-4'], ['1712.04813-1-24-0', '1712.04813-2-28-0'], ['1712.04813-1-24-1', '1712.04813-2-28-1'], ['1712.04813-1-24-2', '1712.04813-2-28-2'], ['1712.04813-1-24-3', '1712.04813-2-28-3'], ['1712.04813-1-24-5', '1712.04813-2-28-5'], ['1712.04813-1-24-6', '1712.04813-2-28-6'], ['1712.04813-1-24-7', '1712.04813-2-28-7'], ['1712.04813-1-24-8', '1712.04813-2-28-8'], ['1712.04813-1-24-9', '1712.04813-2-28-9'], ['1712.04813-1-24-10', '1712.04813-2-28-10'], ['1712.04813-1-8-0', '1712.04813-2-9-0'], ['1712.04813-1-8-7', '1712.04813-2-9-8'], ['1712.04813-1-3-0', '1712.04813-2-3-0'], ['1712.04813-1-3-1', '1712.04813-2-3-1'], ['1712.04813-1-3-2', '1712.04813-2-3-2'], ['1712.04813-1-3-4', '1712.04813-2-3-4'], ['1712.04813-1-6-0', '1712.04813-2-6-0'], ['1712.04813-1-13-1', '1712.04813-2-18-1'], ['1712.04813-1-13-2', '1712.04813-2-18-2'], ['1712.04813-1-13-3', '1712.04813-2-18-3'], ['1712.04813-1-13-4', '1712.04813-2-18-4'], ['1712.04813-1-14-0', '1712.04813-2-18-8'], ['1712.04813-1-14-1', '1712.04813-2-18-9'], ['1712.04813-1-14-2', '1712.04813-2-18-10'], ['1712.04813-1-14-3', '1712.04813-2-18-11'], ['1712.04813-1-14-4', '1712.04813-2-18-12'], ['1712.04813-1-14-6', '1712.04813-2-18-14'], ['1712.04813-1-14-7', '1712.04813-2-18-15']]

[['1712.04813-1-27-4', '1712.04813-2-31-4'], ['1712.04813-1-36-1', '1712.04813-2-40-1'], ['1712.04813-1-36-4', '1712.04813-2-40-4'], ['1712.04813-1-36-5', '1712.04813-2-40-5'], ['1712.04813-1-26-8', '1712.04813-2-30-8'], ['1712.04813-1-34-7', '1712.04813-2-38-7'], ['1712.04813-1-39-0', '1712.04813-2-43-0'], ['1712.04813-1-39-10', '1712.04813-2-43-10'], ['1712.04813-1-4-0', '1712.04813-2-4-0'], ['1712.04813-1-0-3', '1712.04813-2-0-4'], ['1712.04813-1-0-4', '1712.04813-2-0-5'], ['1712.04813-1-16-0', '1712.04813-2-20-0'], ['1712.04813-1-11-0', '1712.04813-2-16-0'], ['1712.04813-1-11-1', '1712.04813-2-16-2'], ['1712.04813-1-11-3', '1712.04813-2-16-4'], ['1712.04813-1-11-5', '1712.04813-2-16-6'], ['1712.04813-1-42-0', '1712.04813-2-46-0'], ['1712.04813-1-42-3', '1712.04813-2-46-3'], ['1712.04813-1-42-7', '1712.04813-2-46-9'], ['1712.04813-1-30-2', '1712.04813-2-34-2'], ['1712.04813-1-30-3', '1712.04813-2-34-3'], ['1712.04813-1-2-2', '1712.04813-2-2-2'], ['1712.04813-1-2-5', '1712.04813-2-2-5'], ['1712.04813-1-24-4', '1712.04813-2-28-4'], ['1712.04813-1-8-1', '1712.04813-2-9-2'], ['1712.04813-1-8-2', '1712.04813-2-9-3'], ['1712.04813-1-8-3', '1712.04813-2-9-4'], ['1712.04813-1-8-4', '1712.04813-2-9-5'], ['1712.04813-1-8-5', '1712.04813-2-9-6'], ['1712.04813-1-3-3', '1712.04813-2-3-3'], ['1712.04813-1-6-4', '1712.04813-2-6-4'], ['1712.04813-1-6-5', '1712.04813-2-6-5'], ['1712.04813-1-6-9', '1712.04813-2-7-2'], ['1712.04813-1-13-0', '1712.04813-2-18-0'], ['1712.04813-1-13-5', '1712.04813-2-18-5'], ['1712.04813-1-14-5', '1712.04813-2-18-13'], ['1712.04813-1-14-8', '1712.04813-2-18-16']]

[]

[['1712.04813-1-20-0', '1712.04813-2-24-0'], ['1712.04813-1-7-0', '1712.04813-2-8-0'], ['1712.04813-1-7-0', '1712.04813-2-8-1'], ['1712.04813-1-41-0', '1712.04813-2-45-0'], ['1712.04813-1-43-1', '1712.04813-2-48-1'], ['1712.04813-1-43-1', '1712.04813-2-48-2'], ['1712.04813-1-43-4', '1712.04813-2-48-3'], ['1712.04813-1-4-4', '1712.04813-2-4-4'], ['1712.04813-1-0-0', '1712.04813-2-0-0'], ['1712.04813-1-18-3', '1712.04813-2-22-3'], ['1712.04813-1-11-2', '1712.04813-2-16-3'], ['1712.04813-1-42-4', '1712.04813-2-46-6'], ['1712.04813-1-8-6', '1712.04813-2-9-7'], ['1712.04813-1-6-1', '1712.04813-2-6-1'], ['1712.04813-1-6-2', '1712.04813-2-6-2'], ['1712.04813-1-6-3', '1712.04813-2-6-3'], ['1712.04813-1-6-7', '1712.04813-2-6-8'], ['1712.04813-1-6-8', '1712.04813-2-7-1'], ['1712.04813-1-13-6', '1712.04813-2-18-6']]

[]

['1712.04813-1-18-8', '1712.04813-1-32-2', '1712.04813-2-22-8', '1712.04813-2-36-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1712.04813

1305.1209

{'1305.1209-1-0-0': 'We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics.', '1305.1209-1-0-1': 'In fact our scheme is more general and would cover also other one-dimensional hamiltonian systems, for example classical and quantum fluids.', '1305.1209-1-0-2': 'The only input parameters required are the average equilibrium currents and the static susceptibilities of the conserved fields.', '1305.1209-1-0-3': 'In our context fluctuating hydrodynamics is a nonlinear system of conservation laws with noise.', '1305.1209-1-0-4': 'For a single mode it is equivalent to the noisy Burgers equation, for which explicit solutions are available.', '1305.1209-1-0-5': 'Our focus is the case of several modes.', '1305.1209-1-0-6': 'No exact solutions have been found so far and we rely on a one-loop approximation.', '1305.1209-1-0-7': 'The resulting mode-coupling equation has a quadratic memory kernel and describes the time evolving correlator matrix of all locally conserved fields.', '1305.1209-1-0-8': 'Long time asymptotics is computed analytically and finite time properties are obtained through a numerical simulation of the mode-coupling equations, which provide predictions still to be compared with data from molecular dynamics.', '1305.1209-1-1-0': 'To predict the dynamic correlator of anharmonic chains is still a theoretical challenge.', '1305.1209-1-1-1': 'In higher dimensions fluctuating hydrodynamics serves as a convenient starting point [CITATION].', '1305.1209-1-1-2': 'But, as recognized already in the 1970ies [CITATION], in one dimension, while the static correlations are of short range, the dynamic current-current correlations generically have an anomalously slow decay.', '1305.1209-1-1-3': 'In particular the transport coefficients, required as an input for fluctuating hydrodynamics, are no longer properly defined.', '1305.1209-1-1-4': 'There have been huge efforts, both through theoretical investigations and numerical simulations, to more precisely characterize this "anomalous" behavior (for a partial list on FPU chains only, see [CITATION]).', '1305.1209-1-1-5': 'Here we argue that, in one dimension, linear fluctuating hydrodynamics has to be extended to a nonlinear version, which will be outlined below.', '1305.1209-1-1-6': 'There have been related attempts before [CITATION].', '1305.1209-1-1-7': 'Our main advance is to treat the full system of coupled conserved modes and to run time-dependent numerical simulations of the respective mode-coupling equations.', '1305.1209-1-1-8': 'In the simulation we use the exact, microscopically computed parameters for the particular anharmonic chain under consideration and thereby provide time-resolved predictions which can be compared directly with the corresponding molecular dynamics.', '1305.1209-1-2-0': 'As a start-up, let us recall the case of a single conserved field, say [MATH], space [MATH], time [MATH], which satisfies the conservation law [EQUATION] with given current function [MATH].', '1305.1209-1-2-1': 'We want to study the fluctuations relative to a uniform background [MATH], i.e. [MATH], hence expand [REF] to second order and add dissipation and noise, [EQUATION] where [MATH] is normalized space-time white noise.', '1305.1209-1-2-2': 'We consider the mean zero, space-time stationary process [MATH] governed by [REF].', '1305.1209-1-2-3': 'Then the spatial statistics at fixed time [MATH] is white noise, [MATH], [MATH], which reflects that the static correlations of an underlying microscopic model decay exponentially fast.', '1305.1209-1-2-4': 'Of particular interest is the correlator [MATH], [MATH].', '1305.1209-1-2-5': 'Its large scale behavior will be dominated by the nonlinearity, but dissipation and noise is required to maintain the proper steady state.', '1305.1209-1-2-6': '[REF] is the noisy Burgers equation, equivalently the spatial derivative of the one-dimensional KPZ equation [CITATION].', '1305.1209-1-2-7': 'There is an exact computation of [MATH] using replica [CITATION].', '1305.1209-1-2-8': 'In particular one knows the universal long time limit, [EQUATION] valid for large [MATH], with [MATH].', '1305.1209-1-2-9': 'Identical scaling properties have been derived also for stochastic lattice gases [CITATION].', '1305.1209-1-2-10': 'The universal scaling function [MATH] can be written in terms of a Fredholm determinant and has been computed with great precision [CITATION].', '1305.1209-1-2-11': 'Because of the nonlinearity the spreading is faster than diffusive.', '1305.1209-1-2-12': 'On the other hand [MATH] appear in [REF] only through the static covariance [MATH].', '1305.1209-1-2-13': 'This quantity will have to come from the microscopic model, while dissipation and noise by themselves are phenomenological, not directly identifiable with microscopic properties.', '1305.1209-1-2-14': '[REF] and similar type of predictions based on [REF] have been confirmed experimentally using a thin film of turbulent liquid crystal [CITATION] and in Monte Carlo simulations of Eden cluster growth [CITATION].', '1305.1209-1-2-15': 'These results provide an indirect confirmation that higher order nonlinearities, like [MATH], will not change the scaling asymptotics.', '1305.1209-1-3-0': 'Anharmonic chains have three conservation laws, compression (or elongation), momentum, and energy.', '1305.1209-1-3-1': 'Thus we have to extend the previous considerations to several modes (or components).', '1305.1209-1-3-2': 'We use [MATH] as mode index.', '1305.1209-1-3-3': 'Then [REF] generalizes to [EQUATION] [MATH].', '1305.1209-1-3-4': 'Expanding as [MATH], the coefficients of the linearized equation are [EQUATION] and coefficients of the quadratic part are given by the Hessians [EQUATION]', '1305.1209-1-3-5': 'Since the background [MATH] is fixed, it will be suppressed in our notation.', '1305.1209-1-3-6': 'The equal time equilibrium correlations decay rapidly.', '1305.1209-1-3-7': 'Hence [MATH] at fixed [MATH] is modeled as white noise with covariance [MATH].', '1305.1209-1-3-8': '[MATH] and [MATH] as a matrix.', '1305.1209-1-3-9': 'As only microscopic input, nonlinear hydrodynamics requires the currents [MATH], more precisely [MATH], and the static correlator [MATH].', '1305.1209-1-3-10': "As can be verified directly, for anharmonic chains it holds [EQUATION] with 't' denoting transpose.", '1305.1209-1-3-11': 'Hence [MATH] has a system of right and left eigenvectors, [MATH], and it is convenient to switch to normal modes.', '1305.1209-1-3-12': 'We form the matrices [MATH].', '1305.1209-1-3-13': 'Up to a global sign the eigenvectors are unique when normalized such that [MATH].', '1305.1209-1-3-14': 'Then [MATH] and the normal modes, [MATH], are defined through [MATH].', '1305.1209-1-3-15': 'While not strictly necessary, the linear transformation to normal modes is of advantage, since normal modes have a definite propagation velocity and turn out to be essentially uncorrelated.', '1305.1209-1-4-0': 'We now expand [REF] to second order in [MATH], transform to normal modes, and add dissipation and noise with the result [EQUATION] where [MATH].', '1305.1209-1-4-1': 'The diffusion matrix [MATH] is positive definite.', '1305.1209-1-4-2': '[MATH] is space-time white noise with strength [EQUATION]', '1305.1209-1-4-3': 'As before, we consider the mean zero, stationary process [MATH] governed by ([REF]).', '1305.1209-1-4-4': 'In the linear case, [MATH], [MATH] is a Gaussian process, which for fixed [MATH] has white noise statistics with independent components of unit strength as required by [MATH].', '1305.1209-1-4-5': 'It can be checked that, for a suitable spatial discretization of [REF], this measure remains invariant under the full nonlinear evolution, provided the couplings [MATH] are symmetric in all indices [CITATION].', '1305.1209-1-4-6': 'In general, we assume that the steady state of ([REF]) has short range correlations.', '1305.1209-1-5-0': 'On the basis of [REF] one would like to compute the dynamic correlator [EQUATION] which would serve then as an approximation to the true dynamic correlator of the microscopic model.', '1305.1209-1-5-1': 'Because of the nonlinearity, this is a difficult program and we rely here on a mode-coupling equation resulting from the one-loop expansion of [REF].', '1305.1209-1-5-2': 'The precise details of the computation can be found elsewhere [CITATION].', '1305.1209-1-5-3': 'In essence, one writes down the evolution equation for [MATH], once iterated in time, which yields a time integral over the Gaussian propagator and an expectation of the form [MATH] for the intermediate time [MATH], [MATH].', '1305.1209-1-5-4': 'The Gaussian propagator is replaced by the full propagator and the 4-point correlation is replaced by its Gaussian pairings, where the equal time pairing vanishes because of the conservation law and the remaining two pairings yield the same contribution.', '1305.1209-1-5-5': 'The result is a cubic memory equation.', '1305.1209-1-5-6': 'Anticipating already the application to anharmonic chains with integer particle index, numerically we use a unit spatial grid.', '1305.1209-1-5-7': 'Adopting the standard conventions for discrete Fourier transforms, in Fourier space the mode-coupling equation reads [EQUATION] [MATH], with the quadratic memory kernel [EQUATION] which has to be solved with the initial condition [EQUATION] [REF] is a lowest order approximation.', '1305.1209-1-6-0': 'However, for [MATH], first written down in [CITATION], the exact solution [REF] can be compared with the numerical solution of [REF], [REF].', '1305.1209-1-6-1': 'In Fig. [REF] we display a time sequence for a single mode, [MATH], with [MATH].', '1305.1209-1-6-2': 'For [MATH] the scaled solution remains stationary.', '1305.1209-1-6-3': 'We compare with the universal KPZ scaling function, thereby reproducing Fig. 3 of [CITATION].', '1305.1209-1-6-4': 'The asymptotic scaling function differs from [MATH] by a few percent only.', '1305.1209-1-6-5': 'We expect that such a precision extends to several modes.', '1305.1209-1-7-0': 'Numerically one observes that, while off-diagonal elements of [MATH] develop immediately, they decay fairly rapidly.', '1305.1209-1-7-1': 'Thus we may invoke the diagonal approximation, [MATH] in position space, and the memory kernel becomes [EQUATION]', '1305.1209-1-7-2': 'Let us assume that all velocities [MATH] are distinct.', '1305.1209-1-7-3': 'Then, ignoring nonlinearities, the mode [MATH] is sharply peaked at [MATH] and the modes separate from each other linearly in [MATH].', '1305.1209-1-7-4': 'Hence the product [MATH] decays rapidly, unless [MATH], and the off-diagonal terms in the sum [REF] can be discarded.', '1305.1209-1-7-5': 'In general, two distinct scenarios may occur.', '1305.1209-1-7-6': '(i) [MATH].', '1305.1209-1-7-7': 'Then [MATH] travels with the same velocity as [MATH] and dominates the interaction with the other modes.', '1305.1209-1-7-8': 'The dynamics of the mode [MATH] reduces to the case [MATH] with scaling [REF] and shift [MATH].', '1305.1209-1-7-9': 'This is the KPZ dominated behavior.', '1305.1209-1-7-10': '(ii) [MATH].', '1305.1209-1-7-11': 'The leading term vanishes and long time behavior is dominated by [MATH].', '1305.1209-1-7-12': 'If also [MATH], then the coupling of mode [MATH] to the pair of modes [MATH] results in a scaling function [MATH] with computable coefficient [MATH], see [CITATION].', '1305.1209-1-8-0': 'There are more cases to be considered, but equipped with this nonlinear extension of fluctuating hydrodynamics it is more interesting to understand how one obtains a prediction for the structure factor of anharmonic chains.', '1305.1209-1-8-1': 'We first have to figure out the conserved fields and their macroscopic Euler equations.', '1305.1209-1-8-2': 'The chain consists of [MATH] particles, position [MATH], momentum [MATH], [MATH], unit mass, and is governed by the hamiltonian [EQUATION] where periodic boundary conditions, [MATH], are imposed.', '1305.1209-1-8-3': 'A prototypical potential is the FPU choice [MATH].', '1305.1209-1-8-4': 'The locally conserved microscopic fields are compression [MATH], momentum [MATH], and energy [MATH].', '1305.1209-1-8-5': 'In a microcanonical simulation one fixes the length (compression) per particle, [MATH], as [MATH], the momentum per particle, [MATH], as [MATH], and the energy per particle, [MATH], as [MATH].', '1305.1209-1-8-6': 'Computationally, it is convenient to switch to the grand canonical pressure ensemble.', '1305.1209-1-8-7': 'Then [MATH] is paired with the pressure [MATH] and the internal energy [MATH] with the inverse temperature [MATH].', '1305.1209-1-8-8': 'In the grand canonical ensemble, [MATH] become independent random variables.', '1305.1209-1-8-9': 'The distribution of [MATH] is a Maxwellian shifted by [MATH], and the distribution of [MATH] is given by [MATH] with partition function [MATH].', '1305.1209-1-8-10': 'Clearly the pressure is just the average force acting on a specified particle.', '1305.1209-1-8-11': 'The microcanonical and grand canonical parameters are related through [EQUATION]', '1305.1209-1-8-12': 'The hydrodynamic currents are [EQUATION] which inserted in [REF] result in the Euler hydrodynamics of the anharmonic chain.', '1305.1209-1-8-13': 'Note that variable [MATH] corresponds to the continuum approximation of the particle index.', '1305.1209-1-9-0': 'Without loss of generality we expand at [MATH].', '1305.1209-1-9-1': 'All coupling coefficients appearing in [REF] can be computed in terms of at most third order cumulants involving [MATH].', '1305.1209-1-9-2': 'These integrals and the somewhat unwieldy substitutions are easily performed using Mathematica.', '1305.1209-1-9-3': 'There are three modes: the heat mode, [MATH], with velocity [MATH] and two sound modes, [MATH], with velocity [MATH], [MATH], where [MATH] is the sound speed, [EQUATION]', '1305.1209-1-9-4': 'Since momentum is conserved, [MATH].', '1305.1209-1-9-5': 'Most importantly [MATH], always.', '1305.1209-1-9-6': 'Hence the heat mode is predicted to have non-KPZ scaling.', '1305.1209-1-9-7': 'A particular and much studied case is an even potential, [MATH], at [MATH].', '1305.1209-1-9-8': 'Then many cumulants vanish.', '1305.1209-1-9-9': 'In fact the only non-zero coefficients are [MATH] and [MATH].', '1305.1209-1-9-10': 'On this basis one expects the sound modes to be diffusive and the feed-back on the heat mode to lead to the scaling function [MATH].', '1305.1209-1-10-0': 'To go beyond the large [MATH] asymptotics we turn to the simulation of the mode-coupling equations, which is based on the Fourier space representation [REF], in such a way that the values of the memory kernel [MATH] for [MATH] can be stored and re-used.', '1305.1209-1-10-1': 'The time and momentum variables are discretized by a uniform grid.', '1305.1209-1-10-2': 'The parameters of the chain are picked somewhat arbitrarily but related to the parameters used in molecular dynamics [CITATION].', '1305.1209-1-10-3': 'We stress that simulations can be performed for any choice of the potential and thermodynamic parameters at minimal numerical efforts.', '1305.1209-1-10-4': 'Fig. [REF] is obtained for the FPU potential with [MATH] and [MATH], [MATH], [MATH], resulting in [MATH].', '1305.1209-1-10-5': '[MATH] has a single critical point at [MATH].', '1305.1209-1-10-6': 'The grey vertical lines at [MATH] indicate the predicted position of the sound mode peaks.', '1305.1209-1-10-7': 'In the time sequence we display the superimposed normal mode correlations (area [MATH] under each curve).', '1305.1209-1-10-8': 'More details are provided in the blow-up.', '1305.1209-1-10-9': 'For the heat mode one observes oscillations which move away from the center and eventually die out.', '1305.1209-1-10-10': 'The tail of the heat mode is cut off at the location of the sound mode, but in-between the two sound mode peaks the theoretically predicted curve is approached.', '1305.1209-1-10-11': 'At the longest available time the sound modes are still asymmetric and have not yet reached their asymptotic shape.', '1305.1209-1-11-0': 'Based on these and further simulations of the mode-coupling equations for anharmonic chains, the following qualitative picture for the motion of the peaks in index number space seems to emerge.', '1305.1209-1-11-1': 'The sound modes "rapidly" decay to a shape function which is centered at [MATH] and varies on the scale [MATH].', '1305.1209-1-11-2': 'The shape function itself is still slowly varying.', '1305.1209-1-11-3': 'The coupling [MATH] determines the scaling of the heat mode.', '1305.1209-1-11-4': 'Since only the integral over the square of the shape function is involved, the heat mode rapidly achieves its asymptotic form in the range [MATH] and with a still slowly varying non-universal constant.', '1305.1209-1-11-5': 'The slow motion of the sound modes is governed by [MATH] and [MATH] .', '1305.1209-1-11-6': 'Assuming already the overall scaling picture, the size of these corrections is estimated to be of the order [MATH], resp. [MATH], relative to the leading term which indicates that [MATH] is approached rather slowly.', '1305.1209-1-11-7': 'Of course, only a guideline can be presented.', '1305.1209-1-11-8': 'For the precise dynamics all couplings have to be used.', '1305.1209-1-12-0': 'Conclusions.', '1305.1209-1-12-1': 'We developed a nonlinear extension of fluctuating hydrodynamics applicable to one-dimensional systems, in principle including classical fluids, quantum fluids [CITATION], and quantum spin chains.', '1305.1209-1-12-2': 'Already at the level of the one-loop approximation it is crucial to maintain the couplings between all conserved modes.', '1305.1209-1-12-3': 'Numerical solutions of the mode-coupling equations provide a realistic picture of the correlation dynamics.', '1305.1209-1-12-4': 'On the limited time scale available, the numerical solutions are in agreement with the analytical computations.', '1305.1209-1-12-5': 'But the theory still has to pass the key test, namely a quantitative comparison with molecular dynamics.', '1305.1209-1-12-6': 'In this test one has to keep in mind that nonlinear fluctuating hydrodynamics is itself an approximation at best valid only for long times and wavelengths.', '1305.1209-1-12-7': 'In addition mode-coupling is an approximation to nonlinear fluctuating hydrodynamics, albeit likely to be fairly accurate, as demonstrated in Fig. [REF]'}

{'1305.1209-2-0-0': 'We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics.', '1305.1209-2-0-1': 'In fact our scheme is more general and would cover also other one-dimensional hamiltonian systems, for example classical and quantum fluids.', '1305.1209-2-0-2': 'Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise.', '1305.1209-2-0-3': 'For a single mode it is equivalent to the noisy Burgers equation, for which explicit solutions are available.', '1305.1209-2-0-4': 'Our focus is the case of several modes.', '1305.1209-2-0-5': 'No exact solutions have been found so far and we rely on a one-loop approximation.', '1305.1209-2-0-6': 'The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving [MATH] correlator matrix of the locally conserved fields.', '1305.1209-2-0-7': 'Long time asymptotics is computed analytically and finite time properties are obtained through a numerical simulation of the mode-coupling equations.', '1305.1209-2-1-0': 'To predict the dynamic correlator of anharmonic chains is still a theoretical challenge.', '1305.1209-2-1-1': 'In higher dimensions fluctuating hydrodynamics serves as a convenient starting point [CITATION].', '1305.1209-2-1-2': 'But, as recognized already in the 1970ies [CITATION], in one dimension, while the static correlations are of short range, the dynamic current-current correlations generically have an anomalously slow decay.', '1305.1209-2-1-3': 'In particular the transport coefficients, required as an input for fluctuating hydrodynamics, are divergent formal expressions.', '1305.1209-2-1-4': 'There have been huge efforts, both through theoretical investigations and numerical simulations, to more precisely characterize this "anomalous" behavior (for a partial list on FPU chains only, see [CITATION]).', '1305.1209-2-1-5': 'Here we argue that, in one dimension, linear fluctuating hydrodynamics has to be extended to a nonlinear version, which will be outlined below.', '1305.1209-2-1-6': 'There have been earlier attempts [CITATION].', '1305.1209-2-1-7': 'Our main advance is to treat the full system of coupled conserved modes and to run time-dependent numerical simulations of the respective mode-coupling equations.', '1305.1209-2-1-8': 'In our simulations we use the exact, microscopically computed parameters for the particular anharmonic chain under consideration.', '1305.1209-2-1-9': 'Thereby time-resolved predictions are provided which can be tested against molecular dynamics.', '1305.1209-2-2-0': 'For anharmonic chains, the locally conserved fields are compression (or elongation), momentum, and energy.', '1305.1209-2-2-1': 'Fluctuating hydrodynamics provides a mesoscopic description of the dynamics of these fields.', '1305.1209-2-2-2': 'To illustrate the general framework it is instructive to first recall the simpler case of a single conserved field, here denoted by [MATH], space [MATH], time [MATH].', '1305.1209-2-2-3': 'On the macroscopic scale it satisfies the conservation law [EQUATION] with given current function [MATH].', '1305.1209-2-2-4': 'We want to study the fluctuations relative to a uniform background [MATH], i.e. [MATH], hence expand [REF] to second order in [MATH] and add dissipation plus noise, resulting in the Langevin equation [EQUATION] where [MATH] is space-time white noise of strength [MATH].', '1305.1209-2-2-5': 'Since [MATH] models the deviations from uniformity, we consider the mean zero, space-time stationary process [MATH] governed by [REF].', '1305.1209-2-2-6': 'Then the spatial statistics at fixed time [MATH] is white noise, [MATH], [MATH], which reflects that the static correlations of an underlying microscopic model decay exponentially fast.', '1305.1209-2-2-7': 'Of particular interest is the correlator [MATH], [MATH].', '1305.1209-2-2-8': 'Its large scale behavior will be dominated by the nonlinearity, but dissipation and noise are required to maintain the proper steady state.', '1305.1209-2-2-9': 'Eq. [REF] is the noisy Burgers equation, equivalently the spatial derivative of the one-dimensional KPZ equation [CITATION].', '1305.1209-2-2-10': 'There is an exact computation of [MATH] using replica [CITATION].', '1305.1209-2-2-11': 'In particular one knows the universal long time limit, [EQUATION] valid for large [MATH] with [MATH].', '1305.1209-2-2-12': 'Because of the nonlinearity the spreading is faster than diffusive.', '1305.1209-2-2-13': 'Note that [MATH] appear in Eq. [REF] only through the static susceptibility [MATH].', '1305.1209-2-2-14': 'Identical scaling properties have been derived also for stochastic lattice gases [CITATION].', '1305.1209-2-2-15': 'The universal scaling function [MATH] can be written in terms of a Fredholm determinant and has been computed with great precision [CITATION].', '1305.1209-2-2-16': 'Interpreting the [MATH]-field as the slope of a moving front, Eq. [REF] and related predictions have been confirmed for growth processes in the plane, both in experiments on slow combustions fronts [CITATION] and on turbulent liquid crystals [CITATION] and numerically through Monte Carlo simulations of Eden cluster growth [CITATION].', '1305.1209-2-3-0': 'To handle anharmonic chains, we have to extend the above scheme to several components.', '1305.1209-2-3-1': 'We use [MATH] as mode index.', '1305.1209-2-3-2': 'Then Eq. [REF] generalizes to [EQUATION] [MATH].', '1305.1209-2-3-3': 'Expanding as [MATH], the coefficients of the linearized equation are [EQUATION] and the coefficients of the quadratic part are given by the Hessians [EQUATION]', '1305.1209-2-3-4': 'Since the background [MATH] is already prescribed, we will suppress it in our notation.', '1305.1209-2-3-5': 'It is assumed that the equal time correlations decay rapidly.', '1305.1209-2-3-6': 'Hence [MATH] at fixed [MATH] is modeled as white noise with covariance [MATH], where [MATH] is the [MATH] susceptibility matrix, [MATH].', '1305.1209-2-3-7': 'As valid in great generality [CITATION], and as can be verified directly for anharmonic chains, it holds [EQUATION] transpose denoted by [MATH], which implies that [MATH] has real eigenvalues.', '1305.1209-2-3-8': 'It turns out to be crucial to switch to normal modes, [MATH], defined by [MATH].', '1305.1209-2-3-9': 'We require the normal mode [MATH] to have a definite propagation velocity, [MATH], and the modes to be statistically uncorrelated.', '1305.1209-2-3-10': 'Hence the similarity matrix [MATH] has to satisfy [MATH] and [MATH], where both properties together determine [MATH] uniquely up to an overall sign.', '1305.1209-2-4-0': 'We now expand in Eq. [REF] to second order in [MATH], transform to normal modes, and add dissipation plus noise, resulting in the statistical field theory [EQUATION] [MATH], where [EQUATION]', '1305.1209-2-4-1': 'The diffusion matrix [MATH] is positive definite.', '1305.1209-2-4-2': '[MATH] is space-time white noise with strength [EQUATION]', '1305.1209-2-4-3': 'As before, since [MATH] models the deviation from uniformity, we consider the mean zero, stationary process [MATH] governed by [REF].', '1305.1209-2-4-4': 'In the linear case, [MATH], [MATH] is a Gaussian process, which for fixed [MATH] has white noise statistics with independent components of unit strength, as imposed by [MATH].', '1305.1209-2-4-5': 'Note that nonlinear fluctuating hydrodynamics requires as only microscopic input the average currents [MATH], more precisely [MATH], and the susceptibility [MATH].', '1305.1209-2-5-0': 'Coupled Langevin equations of the form [REF] have been proposed and studied before in disguise.', '1305.1209-2-5-1': 'Introducing the height [MATH] through [MATH], Eq. [REF] turns into the coupled KPZ equations in one dimension, which describe dynamic roughening of directed lines [CITATION], sedimenting colloidal suspensions [CITATION], stochastic lattice gases [CITATION], and magnetohydrodynamics [CITATION].', '1305.1209-2-6-0': 'Equipped with the above frame, let us turn to anharmonic chains, for which purpose we first have to figure out the conserved fields and their macroscopic Euler equations.', '1305.1209-2-6-1': 'The chain consists of [MATH] particles, position [MATH], momentum [MATH], [MATH], unit mass, and is governed by the Hamiltonian [EQUATION] where periodic boundary conditions of the form [MATH] are imposed.', '1305.1209-2-6-2': 'A prototypical potential is the FPU choice [MATH].', '1305.1209-2-6-3': 'The locally conserved microscopic fields are compression [MATH], momentum [MATH], and energy [MATH].', '1305.1209-2-6-4': 'Following our blueprint we collect them as the three-vector [MATH] with [MATH], [MATH], and [MATH].', '1305.1209-2-6-5': 'In a microcanonical simulation one fixes the length (compression) per particle, [MATH], as [MATH], the momentum per particle, [MATH], as [MATH], and the energy per particle, [MATH], as [MATH].', '1305.1209-2-6-6': 'Computationally, it is convenient to switch to the grand canonical pressure ensemble.', '1305.1209-2-6-7': 'Then [MATH] is conjugate to the pressure [MATH] and the internal energy [MATH] to the inverse temperature [MATH].', '1305.1209-2-6-8': 'In the grand canonical ensemble, [MATH] become independent random variables.', '1305.1209-2-6-9': 'The distribution of [MATH] is a Maxwellian shifted by [MATH], and the distribution of [MATH] is given by [MATH] with partition function [MATH].', '1305.1209-2-6-10': 'Clearly the pressure equals the average force acting on a specified particle.', '1305.1209-2-6-11': 'The microcanonical and grand canonical parameters are related through [EQUATION]', '1305.1209-2-6-12': 'On the hydrodynamic scale the average conserved fields [MATH] are slowly varying and approximated by the continuum fields [MATH], where [MATH] stands for the continuum approximation of the particle index [MATH].', '1305.1209-2-6-13': 'From the microscopic conservation laws one deduces the hydrodynamic currents [EQUATION] which, when inserted in Eq. [REF], result in the Euler hydrodynamics of the anharmonic chain.', '1305.1209-2-7-0': 'Without loss of generality the equilibrium state of the chain is taken at [MATH].', '1305.1209-2-7-1': 'The similarity transformation [MATH], the velocities [MATH], and the coupling coefficients [MATH] are computed in terms of at most third order cumulants involving [MATH] with average [MATH].', '1305.1209-2-7-2': 'These integrals and the somewhat unwieldy required substitutions are easily performed using Mathematica.', '1305.1209-2-7-3': 'There are three modes: the heat mode, [MATH], with velocity [MATH] and two sound modes, [MATH], with velocity [MATH], [MATH], where [MATH] is the sound speed, [EQUATION]', '1305.1209-2-7-4': 'The microscopic equilibrium time correlations of the conserved fields in normal mode representation are defined by ([MATH] signals normal mode) [EQUATION] the index [MATH] standing for connected.', '1305.1209-2-7-5': 'Our central claim is that the normal mode correlations of the chain are approximated for large [MATH] as [EQUATION]', '1305.1209-2-7-6': 'This leaves us with the task to work out the correlator [MATH] for the stochastic field theory [REF].', '1305.1209-2-7-7': 'With no exact solution at hand, we rely on the mode-coupling equations in one-loop approximation.', '1305.1209-2-7-8': 'But before, since the three modes travel with distinct velocities, for long times only the self-coupling will contribute.', '1305.1209-2-7-9': 'Since [MATH] and generically [MATH], the two sound modes are expected to satisfy the KPZ scaling [REF] with [MATH], [MATH] and [MATH], [MATH].', '1305.1209-2-7-10': 'The decoupling of modes is convincingly confirmed in a two-component lattice gas [CITATION].', '1305.1209-2-7-11': 'For the heat mode our argument fails, since [MATH] always.', '1305.1209-2-7-12': 'Note that for the popular case of an even potential, [MATH], at [MATH] also [MATH] implying that all three modes are non-KPZ.', '1305.1209-2-8-0': 'The derivation of the mode-coupling equations is presented in [CITATION].', '1305.1209-2-8-1': 'Numerically one observes that, while off-diagonal elements of [MATH] develop immediately, they decay fairly rapidly.', '1305.1209-2-8-2': 'Thus we may invoke the diagonal approximation, [MATH] in position space.', '1305.1209-2-8-3': 'Switching to Fourier space and adopting the standard conventions for discrete Fourier transforms, the mode-coupling equations then read [EQUATION] [MATH], with memory kernel [EQUATION]', '1305.1209-2-9-0': 'The special case [MATH] is discussed already in [CITATION], see [CITATION] for a first numerical integration.', '1305.1209-2-10-0': 'In Fig. [REF] we display a time sequence for a single mode with [MATH].', '1305.1209-2-10-1': 'For [MATH] the scaled solution remains stationary.', '1305.1209-2-10-2': 'The asymptotic scaling function differs from [MATH] by a few percent only.', '1305.1209-2-10-3': 'On this basis we expect that such a precision extends to several modes.', '1305.1209-2-11-0': 'As a representative example for anharmonic chains we choose the FPU potential with [MATH] and [MATH], [MATH], [MATH], resulting in [MATH], which are commonly used parameters in molecular dynamics simulations.', '1305.1209-2-11-1': '[MATH] has a single minimum at [MATH].', '1305.1209-2-11-2': 'We stress that simulations can be performed for any choice of the potential and thermodynamic parameters at minimal numerical efforts.', '1305.1209-2-11-3': 'With the theoretically determined velocities and couplings, the mode-coupling equations are iterated in time, using Fourier space representation as in [REF], in such a way that the values of the memory kernel [MATH] for [MATH] can be stored and re-used.', '1305.1209-2-11-4': 'No diagonal approximation is invoked.', '1305.1209-2-11-5': 'The time and momentum variables are discretized by a uniform grid.', '1305.1209-2-11-6': 'In Fig. [REF] the grey vertical lines at [MATH] indicate the predicted position of the sound mode peaks.', '1305.1209-2-11-7': 'The off-diagonal elements of [MATH] are essentially zero.', '1305.1209-2-11-8': 'In the time sequence we display the superimposed diagonal normal mode correlations (area [MATH] under each curve).', '1305.1209-2-11-9': 'More details are provided in the blow-up.', '1305.1209-2-11-10': 'For the heat mode one observes oscillations which move away from the center and eventually die out.', '1305.1209-2-11-11': 'The tail of the heat mode is cut at the location of the sound mode.', '1305.1209-2-11-12': 'At the longest available time the sound modes are still asymmetric and have not yet reached their asymptotic shape.', '1305.1209-2-12-0': 'Theoretically the scaling function for the heat mode can be obtained by inserting the known asymptotic form of [MATH] in [REF] with [MATH].', '1305.1209-2-12-1': 'Solving the then linear memory equation [REF] results in the symmetric Levy [MATH] distribution, [MATH] with computable non-universal coefficient [MATH].', '1305.1209-2-12-2': 'The Levy [MATH] for the heat mode peak has been observed in molecular dynamics for the hard point potential, [MATH] for [MATH] and [MATH] otherwise, with a suitable choice of thermodynamic parameters [CITATION].', '1305.1209-2-12-3': 'Note that the slow tails of the Levy distribution are cut by the sound mode.', '1305.1209-2-13-0': 'Based on these and further simulations of the mode-coupling equations for anharmonic chains, the following qualitative picture for the motion of the normal mode peaks in index number space seems to emerge.', '1305.1209-2-13-1': 'The sound modes "rapidly" decay to a shape function which is centered at [MATH] and varies on the scale [MATH].', '1305.1209-2-13-2': 'The shape function itself is still slowly varying.', '1305.1209-2-13-3': 'The couplings [MATH] determine the scaling of the heat mode.', '1305.1209-2-13-4': 'Since only the integral over the square of the shape function is involved, the heat mode rapidly achieves its asymptotic shape in the range [MATH] and with a still slowly varying non-universal constant.', '1305.1209-2-13-5': 'The slow motion of the sound modes is governed by [MATH] and [MATH].', '1305.1209-2-13-6': 'Assuming already the validity of overall scaling picture, the size of these corrections is estimated to be of the order [MATH], resp. [MATH], relative to the leading term which signals that [MATH] is approached rather slowly.', '1305.1209-2-13-7': 'Of course, only a qualitative guideline is presented.', '1305.1209-2-13-8': 'For the precise dynamics all velocities and couplings have to be used.', '1305.1209-2-14-0': 'Conclusions.', '1305.1209-2-14-1': 'We developed a nonlinear extension of fluctuating hydrodynamics applicable to one-dimensional systems, in principle including classical fluids, quantum fluids [CITATION], and quantum spin chains.', '1305.1209-2-14-2': 'Already at the level of the one-loop approximation it is crucial to maintain the couplings between all conserved modes.', '1305.1209-2-14-3': 'As applied to anharmonic chains, the numerical solutions of the mode-coupling equations provide a realistic picture of the correlation dynamics and, on the limited time scale simulated, are consistent with the analytical computations and also with molecular dynamics, as far as available.', '1305.1209-2-15-0': 'We thank Henk van Beijeren, Patrik Ferrari, Tomohiro Sasamoto, and Hong Zhao for most useful comments.', '1305.1209-2-15-1': 'The research is supported by the DFG project SP 181/29-1.'}

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[]

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['1305.1209-1-7-6', '1305.1209-1-7-10', '1305.1209-1-12-0', '1305.1209-2-14-0', '1305.1209-3-15-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1305.1209

{'1305.1209-3-0-0': 'We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics.', '1305.1209-3-0-1': 'In fact our scheme is more general and would cover also other one-dimensional hamiltonian systems, for example classical and quantum fluids.', '1305.1209-3-0-2': 'Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise.', '1305.1209-3-0-3': 'For a single mode it is equivalent to the noisy Burgers equation, for which explicit solutions are available.', '1305.1209-3-0-4': 'Our focus is the case of several modes.', '1305.1209-3-0-5': 'No exact solutions have been found so far and we rely on a one-loop approximation.', '1305.1209-3-0-6': 'The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving [MATH] correlator matrix of the locally conserved fields.', '1305.1209-3-0-7': 'Long time asymptotics is computed analytically and finite time properties are obtained through a numerical simulation of the mode-coupling equations.', '1305.1209-3-1-0': 'To predict the dynamic correlator of anharmonic chains is still a theoretical challenge.', '1305.1209-3-1-1': 'In higher dimensions fluctuating hydrodynamics serves as a convenient starting point [CITATION].', '1305.1209-3-1-2': 'But, as recognized already in the 1970ies [CITATION], in one dimension, while the static correlations are of short range, the dynamic current-current correlations generically have an anomalously slow decay.', '1305.1209-3-1-3': 'In particular the transport coefficients, required as an input for fluctuating hydrodynamics, are divergent formal expressions.', '1305.1209-3-1-4': 'There have been huge efforts, both through theoretical investigations and numerical simulations, to more precisely characterize this anomalous behavior ( [CITATION] for a partial list on FPU chains only, a detailed and commented discussion of the literature can be found in Sect. 7 of [CITATION]).', '1305.1209-3-1-5': 'Here we argue that, in one dimension, linear fluctuating hydrodynamics has to be extended to a nonlinear version, which will be outlined below.', '1305.1209-3-1-6': 'Compared to related contributions [CITATION], our main advance is to treat the full system of coupled conserved modes and to run time-dependent numerical simulations of the respective mode-coupling equations.', '1305.1209-3-1-7': 'In our simulations we use effective coupling constants, which are computed exactly for the particular anharmonic chain under consideration.', '1305.1209-3-1-8': 'Thereby time-resolved predictions are provided which can be tested against molecular dynamics.', '1305.1209-3-2-0': 'For anharmonic chains, the locally conserved fields are elongation, momentum, and energy.', '1305.1209-3-2-1': 'Fluctuating hydrodynamics provides a mesoscopic description of the dynamics of these fields.', '1305.1209-3-2-2': 'To illustrate the general framework it is instructive to first recall the simpler case of a single conserved field, here denoted by [MATH], space [MATH], time [MATH].', '1305.1209-3-2-3': 'On the macroscopic scale it satisfies the conservation law [EQUATION] with given current function [MATH].', '1305.1209-3-2-4': 'We want to study the fluctuations relative to a uniform background [MATH], i.e. [MATH], hence expand [REF] to second order in [MATH] and add dissipation plus noise, resulting in the Langevin equation [EQUATION] where [MATH] is space-time white noise of strength [MATH].', '1305.1209-3-2-5': 'Since [MATH] models the deviations from uniformity, we consider the mean zero, space-time stationary process [MATH] governed by [REF].', '1305.1209-3-2-6': 'Then, at fixed time [MATH], the spatial statistics is white noise, [MATH], [MATH], which reflects that the static correlations of an underlying microscopic model decay exponentially fast.', '1305.1209-3-2-7': 'Of particular interest is the correlator [MATH], [MATH].', '1305.1209-3-2-8': 'Its large scale behavior will be dominated by the nonlinearity, but dissipation and noise are required to maintain the proper steady state.', '1305.1209-3-2-9': 'Eq. [REF] is the noisy Burgers equation, equivalently the spatial derivative of the one-dimensional KPZ equation [CITATION].', '1305.1209-3-2-10': 'There is an exact computation of [MATH] using replica [CITATION].', '1305.1209-3-2-11': 'In particular one knows the universal long time limit, [EQUATION] valid for large [MATH] with [MATH].', '1305.1209-3-2-12': 'Because of the nonlinearity the spreading is faster than diffusive.', '1305.1209-3-2-13': 'Note that [MATH] appear in Eq. [REF] only through the static susceptibility [MATH].', '1305.1209-3-2-14': 'Identical scaling properties have been derived also for stochastic lattice gases [CITATION].', '1305.1209-3-2-15': 'The universal scaling function [MATH] can be written in terms of a Fredholm determinant and has been computed with great precision [CITATION].', '1305.1209-3-2-16': 'Interpreting the [MATH]-field as the slope of a moving front, Eq. [REF] and related predictions have been confirmed for growth processes in the plane, both in experiments on slow combustions fronts [CITATION] and on turbulent liquid crystals [CITATION] and numerically through Monte Carlo simulations of Eden cluster growth [CITATION].', '1305.1209-3-3-0': 'To handle anharmonic chains, we have to extend the above scheme to several components.', '1305.1209-3-3-1': 'We use [MATH] as component index.', '1305.1209-3-3-2': 'Then Eq. [REF] generalizes to [EQUATION] [MATH].', '1305.1209-3-3-3': 'Expanding as [MATH], the coefficients of the linearized equation are [EQUATION] and the coefficients of the quadratic part are given by the Hessians [EQUATION]', '1305.1209-3-3-4': 'Since the background [MATH] is already prescribed, we will suppress it in our notation.', '1305.1209-3-3-5': 'For anharmonic chains the static correlations of the conserved fields are [MATH]-correlated in space.', '1305.1209-3-3-6': 'More generally, rapid decay of static correlations is assumed and hence, at fixed [MATH], [MATH] is modeled as white noise with covariance [MATH], where [MATH] is the [MATH] susceptibility matrix with [MATH].', '1305.1209-3-3-7': 'As discussed in [CITATION], [MATH] and [MATH] are related through [EQUATION] transpose denoted by [MATH], which implies that [MATH] has real eigenvalues.', '1305.1209-3-3-8': 'We use the eigenvectors of [MATH] to construct a linear transformation in component space such that the [MATH]-th component of the new field travels with a definite velocity, say [MATH], at the linearized level.', '1305.1209-3-3-9': 'The transformed field components are commonly called normal modes, denoted here by [MATH].', '1305.1209-3-3-10': 'Then [MATH] with the [MATH] matrix [MATH] acting in component space only.', '1305.1209-3-3-11': 'In addition we require that initially the normal modes are statistically uncorrelated.', '1305.1209-3-3-12': 'Hence [MATH] has to satisfy [MATH] and [MATH], where both properties together determine [MATH] uniquely up to an overall sign.', '1305.1209-3-4-0': 'We now expand in Eq. [REF] to second order in [MATH], transform to normal modes, and add dissipation plus noise, resulting in the statistical field theory [EQUATION] [MATH], where [EQUATION]', '1305.1209-3-4-1': 'The diffusion matrix [MATH] is positive definite.', '1305.1209-3-4-2': '[MATH] is space-time white noise with strength [EQUATION]', '1305.1209-3-4-3': 'As before, since [MATH] models the deviation from uniformity, we consider the mean zero, stationary process [MATH] governed by [REF].', '1305.1209-3-4-4': 'In the linear case, [MATH], [MATH] is a Gaussian process, which for fixed [MATH] has white noise statistics with independent components of unit strength, as imposed by [MATH].', '1305.1209-3-4-5': 'Note that nonlinear fluctuating hydrodynamics requires as microscopic input only the average currents [MATH], more precisely [MATH], and the susceptibility [MATH].', '1305.1209-3-5-0': 'Coupled Langevin equations of the form [REF] have been proposed and studied before in disguise.', '1305.1209-3-5-1': 'Introducing the height [MATH] through [MATH], Eq. [REF] turns into the coupled KPZ equations in one dimension, which describe dynamic roughening of directed lines [CITATION], sedimenting colloidal suspensions [CITATION], stochastic lattice gases [CITATION], and magnetohydrodynamics [CITATION].', '1305.1209-3-5-2': 'The application to one-dimensional Hamiltonian systems is novel, however.', '1305.1209-3-6-0': 'Equipped with the above frame, let us turn to anharmonic chains, for which purpose we first have to figure out the conserved fields and their macroscopic Euler equations.', '1305.1209-3-6-1': 'The chain consists of [MATH] particles, position [MATH], momentum [MATH], [MATH], unit mass, and is governed by the Hamiltonian [EQUATION] where periodic boundary conditions of the form [MATH] are imposed.', '1305.1209-3-6-2': 'A prototypical potential is the FPU choice [MATH].', '1305.1209-3-6-3': 'The locally conserved microscopic fields are elongation [MATH], momentum [MATH], and energy [MATH].', '1305.1209-3-6-4': 'Following our blueprint we collect them as the three-vector [MATH] with [MATH], [MATH], and [MATH].', '1305.1209-3-6-5': 'In a microcanonical simulation one fixes the elongation per particle, [MATH], as [MATH], the momentum per particle, [MATH], as [MATH], and the energy per particle, [MATH], as [MATH].', '1305.1209-3-6-6': 'Computationally, it is convenient to switch to the canonical pressure ensemble.', '1305.1209-3-6-7': 'Then [MATH] is conjugate to the pressure [MATH] and the internal energy [MATH] to the inverse temperature [MATH].', '1305.1209-3-6-8': 'In the canonical ensemble, [MATH] are independent random variables.', '1305.1209-3-6-9': 'The distribution of [MATH] is a Maxwellian shifted by [MATH], and the distribution of [MATH] is given by [MATH] with partition function [MATH].', '1305.1209-3-6-10': 'Clearly the pressure equals the average force acting on a specified particle.', '1305.1209-3-6-11': 'The microcanonical and canonical parameters are related through [EQUATION]', '1305.1209-3-6-12': 'On the hydrodynamic scale the average conserved fields [MATH] are slowly varying and approximated by the continuum fields [MATH], where [MATH] stands for the continuum approximation of the particle index [MATH].', '1305.1209-3-6-13': 'From the microscopic conservation laws together with local equilibrium one deduces the hydrodynamic currents [EQUATION] which, when inserted in Eq. [REF], result in the Euler hydrodynamics of the anharmonic chain.', '1305.1209-3-7-0': 'Without loss of generality the equilibrium state of the chain is taken at [MATH].', '1305.1209-3-7-1': 'The linear transformation [MATH], the velocities [MATH], and the coupling coefficients [MATH] are computed in terms of at most third order cumulants involving [MATH] with average [MATH].', '1305.1209-3-7-2': 'These integrals and the somewhat unwieldy required substitutions are easily performed using Mathematica.', '1305.1209-3-7-3': 'There are three modes: the heat mode, [MATH], with velocity [MATH] and two sound modes, [MATH], with velocity [MATH], [MATH], where [MATH] is the sound speed, [EQUATION]', '1305.1209-3-7-4': 'The microscopic equilibrium time correlations of the conserved fields in normal mode representation are defined by ([MATH] signals normal mode) [EQUATION] the index [MATH] standing for second cumulant.', '1305.1209-3-7-5': 'Our central claim is that the normal mode correlations of the chain are approximated for large [MATH] as [EQUATION]', '1305.1209-3-7-6': 'This leaves us with the task to work out the correlator [MATH] for the stochastic field theory [REF].', '1305.1209-3-7-7': 'With no exact solution at hand, we rely on the mode-coupling equations in one-loop approximation.', '1305.1209-3-7-8': 'But before, since the three modes travel with distinct velocities, they decouple for long times and only the self-interaction proportional to [MATH] will contribute.', '1305.1209-3-7-9': 'Since generically [MATH], and [MATH], the two sound modes are expected to satisfy the KPZ scaling [REF] with the substitutions [MATH], [MATH] and [MATH], [MATH].', '1305.1209-3-7-10': 'The decoupling of modes is convincingly confirmed in a two-component lattice gas [CITATION].', '1305.1209-3-7-11': 'For the heat mode our argument fails, since [MATH] always.', '1305.1209-3-7-12': 'Note that for the popular case of an even potential, [MATH], at [MATH] also [MATH] implying that all three modes are non-KPZ.', '1305.1209-3-8-0': 'The derivation of the mode-coupling equations is explained in [CITATION].', '1305.1209-3-8-1': 'To be concise we only display the diagonal approximation, for which [MATH] is assumed.', '1305.1209-3-8-2': 'Switching to Fourier space and adopting the standard conventions for discrete Fourier transforms, the mode-coupling equations then simplify to [EQUATION] [MATH], with memory kernel [EQUATION]', '1305.1209-3-9-0': 'Numerically we always simulate the dynamics of the full correlator matrix.', '1305.1209-3-9-1': 'For a wide range of parameters, after some transient time the off-diagonal matrix elements decay and are always by an order of magnitude smaller than the diagonal ones.', '1305.1209-3-10-0': 'The special case [MATH] is discussed already in [CITATION], see [CITATION] for a first numerical integration.', '1305.1209-3-11-0': 'In Fig. [REF] we display a time sequence for a single mode with [MATH].', '1305.1209-3-11-1': 'For [MATH] the scaled solution remains stationary.', '1305.1209-3-11-2': 'The asymptotic scaling function differs from [MATH] by a few percent only.', '1305.1209-3-11-3': 'On this basis we expect that such a precision extends to several modes.', '1305.1209-3-12-0': 'As a representative example for anharmonic chains we choose the FPU potential with [MATH] and [MATH], [MATH], [MATH], resulting in [MATH], which are commonly used parameters in molecular dynamics simulations.', '1305.1209-3-12-1': '[MATH] has a single minimum at [MATH].', '1305.1209-3-12-2': 'We stress that simulations can be performed for any choice of the potential and thermodynamic parameters at minimal numerical efforts.', '1305.1209-3-12-3': 'With the theoretically determined velocities and couplings, the mode-coupling equations are iterated in time, using Fourier space representation as in [REF], in such a way that the values of the memory kernel [MATH] for [MATH] can be stored and re-used.', '1305.1209-3-12-4': 'The time and momentum variables are discretized by a uniform grid.', '1305.1209-3-12-5': 'In Fig. [REF] the grey vertical lines at [MATH] indicate the predicted position of the sound mode peaks.', '1305.1209-3-12-6': 'The off-diagonal elements of [MATH] are essentially zero.', '1305.1209-3-12-7': 'In the time sequence we display the superimposed diagonal normal mode correlations (area [MATH] under each curve).', '1305.1209-3-12-8': 'More details are provided in the blow-up.', '1305.1209-3-12-9': 'For the heat mode peak one observes oscillations which move away from the center and eventually die out.', '1305.1209-3-12-10': 'The tail of the heat mode peak is cut at the location of the sound mode.', '1305.1209-3-12-11': 'At the longest available time the sound mode peaks are still asymmetric and have not yet reached their asymptotic shape.', '1305.1209-3-13-0': 'Theoretically the scaling function for the heat mode peak is obtained by inserting the known asymptotic form of [MATH] in [REF] with [MATH].', '1305.1209-3-13-1': 'Solving the then linear memory equation [REF] results in the symmetric Levy [MATH] distribution, [MATH] with computed non-universal coefficient [MATH].', '1305.1209-3-13-2': 'The Fourier transform of [MATH] is plotted as green curve in Fig. [REF] (d).', '1305.1209-3-13-3': 'The Levy [MATH] distribution for the heat mode peak has been guessed earlier based on the molecular dynamics of the hard-point gas with alternating masses [CITATION], reconfirmed in case a maximal distance between the hard-point particles is imposed [CITATION], and also for FPU chains [CITATION].', '1305.1209-3-13-4': 'Currently the Levy distribution is the strongest numerical support of nonlinear fluctuating hydrodynamics.', '1305.1209-3-13-5': 'Note that no signal propagates outside the sound cone.', '1305.1209-3-14-0': 'Based on these and further simulations of the mode-coupling equations for anharmonic chains, the following qualitative picture for the motion of the normal mode peaks in index number space emerges.', '1305.1209-3-14-1': 'The sound mode peaks "rapidly" decay to a shape function which is centered at [MATH] and varies on the scale [MATH].', '1305.1209-3-14-2': 'The shape function itself is still slowly varying.', '1305.1209-3-14-3': 'The couplings [MATH] determine the scaling of the heat mode peak.', '1305.1209-3-14-4': 'Since only the integral over the square of the shape function is involved, the heat mode peak rapidly achieves its asymptotic shape in the range [MATH] with a still slowly varying non-universal constant.', '1305.1209-3-14-5': 'The intermediate time motion of the sound mode peaks is dominated by [MATH] and [MATH].', '1305.1209-3-14-6': 'Assuming already the validity of overall scaling picture, the size of these finite time corrections is estimated to be of the order [MATH], resp. [MATH], relative to the leading term which signals that [MATH] is approached rather slowly.', '1305.1209-3-14-7': 'Of course, only a qualitative guideline is presented.', '1305.1209-3-14-8': 'For the precise dynamics all velocities and couplings have to be taken into account.', '1305.1209-3-15-0': 'Conclusions.', '1305.1209-3-15-1': 'We developed a nonlinear extension of fluctuating hydrodynamics applicable to one-dimensional systems, in principle including classical fluids, quantum fluids [CITATION], and quantum spin chains.', '1305.1209-3-15-2': 'Already at the level of the one-loop approximation it is crucial to maintain the couplings between all conserved modes.', '1305.1209-3-15-3': 'As applied to anharmonic chains, the numerical solutions of the mode-coupling equations provide a realistic picture of the correlation dynamics and, on the limited space-time scale simulated, are consistent with the analytical computations and also with molecular dynamics, as far as available.', '1305.1209-3-16-0': 'We thank Henk van Beijeren, Patrik Ferrari, Tomohiro Sasamoto, and Hong Zhao for most useful comments.', '1305.1209-3-16-1': 'The research is supported by DFG project SP 181/29-1.'}

1410.3634

{'1410.3634-1-0-0': 'We discover a transition from extended to localized quasi-modes for light in a gas of immobile two-level atoms in a magnetic field.', '1410.3634-1-0-1': 'The transition takes place either upon increasing the number density of atoms in a strong field or upon increasing the field at a high enough density.', '1410.3634-1-0-2': 'It has many characteristic features of a disorder-driven (Anderson) transition but is strongly influenced by near-field interactions between atoms and the anisotropy of the atomic medium induced by the magnetic field.', '1410.3634-1-1-0': 'The transition from extended to localized eigenstates upon increasing disorder in a quantum or wave system is called after Philip Anderson who was the first to predict it for electrons in disordered solids [CITATION].', '1410.3634-1-1-1': 'More recently, this transition was studied for various types of quantum particles (cold atoms [CITATION], Bose-Einstein condensates [CITATION]) as well as for classical waves (light [CITATION], ultrasound [CITATION]).', '1410.3634-1-1-2': 'In the most common case of time-reversal symmetric systems invariant under spin rotation Anderson transition takes place for a three-dimensional (3D) disorder only, eigenstates of low-dimensional systems being always localized [CITATION].', '1410.3634-1-1-3': 'Anderson localization of light may find applications in the design of future quantum-information devices [CITATION], miniature lasers [CITATION] and solar cells [CITATION].', '1410.3634-1-1-4': 'However, no undisputable experimental observation of optical Anderson transition in 3D exists to date since alternative explanations were proposed for all published reports of it [CITATION].', '1410.3634-1-1-5': 'Moreover, we have recently shown that the simplest theoretical model in which light is scattered by point scatterers (atoms) does not predict Anderson localization of light at all [CITATION].', '1410.3634-1-2-0': 'In the present Letter we show that an external magnetic field may induce a transition between extended and localized states for light in a gas of cold atoms.', '1410.3634-1-2-1': 'This result may seem counterintuitive from the perspective of condensed matter physics where a magnetic field leads to a breakdown of the time-reversal invariance and thus suppresses the lowest-order interference effects leading to the weak localization-a precursor of Anderson localization [CITATION].', '1410.3634-1-2-2': 'In atomic systems, however, the main effect of the magnetic field is to lift the degeneracy of states involved in the interaction of an individual atom with light-the Zeeman effect.', '1410.3634-1-2-3': 'For light that is resonant with an atomic transition with a degenerate ground state the Zeeman effect can actually amplify interferences leading to an enhancement of the coherent backscattering peak [CITATION].', '1410.3634-1-2-4': 'On the other hand, the Zeeman effect reduces the strength of resonant dipole-dipole interactions between nearby atoms [CITATION].', '1410.3634-1-2-5': 'These interactions suppress light scattering [CITATION] and prevent Anderson localization [CITATION].', '1410.3634-1-2-6': 'We show below that the partial suppression of resonant dipole-dipole interactions by the magnetic field is sufficient to induce a transition from extended to localized states in the atomic system.', '1410.3634-1-2-7': 'This critical phenomenon stands out from other magneto-optical effects that take place in disordered media (such as, e.g., the photonic Hall effect [CITATION] or Hanle effect in coherent backscattering [CITATION]) which only give rise to weak corrections to wave transport.', '1410.3634-1-3-0': 'We consider an ensemble of [MATH] identical two-level atoms at random position [MATH] inside a spherical volume [MATH] of radius [MATH].', '1410.3634-1-3-1': 'The resonant frequency [MATH] of atoms defines the natural length scale [MATH], where [MATH] is the vacuum speed of light.', '1410.3634-1-3-2': 'The ground state [MATH] of an isolated atom [MATH] is nondegenerate with the total angular momentum [MATH], whereas the excited states [MATH] is three-fold degenerate with [MATH].', '1410.3634-1-3-3': 'The three degenerate substates [MATH] correspond to the three possible projections [MATH], [MATH] of the total angular momentum [MATH] on the quantization axis [MATH].', '1410.3634-1-3-4': 'The natural lifetime [MATH] of the excited state sets the time scale of the problem.', '1410.3634-1-3-5': 'The atoms are subject to a uniform magnetic field [MATH] and interact with the free electromagnetic field surrounding them.', '1410.3634-1-3-6': 'The system "atoms [MATH] field" is described by the following Hamiltonian [CITATION]: [EQUATION]', '1410.3634-1-3-7': "Here [MATH] is the Planck's constant divided by [MATH], [MATH] and [MATH] are the wave and the polarization vectors of the modes of the free electromagnetic field, [MATH]) are the corresponding creation (annihilation) operators, [MATH] are the atomic dipole operators, [MATH] is the electric displacement vector divided by the vacuum permittivity [MATH], [MATH] is the Bohr magneton, and [MATH] is the Lande factor of the excited state.", '1410.3634-1-4-0': 'Previous work [CITATION] demonstrated that in the absence of magnetic field ([MATH]), the degrees of freedom corresponding to the electromagnetic field can be traced out leading to an effective Hamiltonian describing the dynamics of [MATH] atoms coupled by the electromagnetic field.', '1410.3634-1-4-1': "This effective Hamiltonian takes the form of a [MATH] Green's matrix [MATH] describing the propagation of light between the atoms [CITATION].", '1410.3634-1-4-2': "The same approach can be used when [MATH] leading to the following Green's matrix: [EQUATION] where where [MATH] is the Zeeman shift in units of the natural line width, [MATH], and [MATH].", '1410.3634-1-5-0': "In the absence of magnetic field ([MATH]) the eigenvalues of the Green's matrix [MATH] concentrate in a roughly circular domain on the complex plane roughly symmetric with respect to the vertical axis [MATH] and almost touching the horizontal axis [MATH] [CITATION].", '1410.3634-1-5-1': 'The field splits the eigenvalues into three equal groups centered around [MATH], [MATH]) [CITATION], see Fig. [REF].', '1410.3634-1-5-2': 'The three groups of eigenvalues become well separated in the limit of strong magnetic field [MATH] to which we will restrict our consideration in the present Letter.', '1410.3634-1-5-3': 'Although at a low density [MATH] the three groups of eigenvalues are similar [Fig. [REF](a)], the groups corresponding to [MATH] start to differ significantly from the [MATH] group at higher densities [Fig. [REF](b)].', '1410.3634-1-5-4': 'In particular, the [MATH] groups of eigenvalues develop "holes" that were previously associated with Anderson localization in the framework of the scalar model of wave scattering [CITATION].', '1410.3634-1-6-0': "To see whether localized states indeed appear at high densities of atoms, we analyze the inverse participation ratios (IPRs) of eignevectors [MATH] of the Green's matrix [MATH], [MATH], where [MATH] is the square of the length of the vector [MATH].", '1410.3634-1-6-1': 'Low IPR [MATH] corresponds to an extended state whereas IPR [MATH] signals a state localized on [MATH] atoms.', '1410.3634-1-6-2': 'Figure [REF](a) shows that at low density of atoms most of the eigenvectors have low IPRs with the eigenvectors localized on pairs of closely located atoms being an exception.', '1410.3634-1-6-3': 'These "subradiant" states exist at any density and should be distinguished from localized states that are due to the multiple scattering of light on many atoms and that appear at higher densities in relatively narrow bands of frequencies [MATH] on the left from the resonances [MATH] [see Fig. [REF](b)].', '1410.3634-1-6-4': 'These states may have smaller IPRs than the subradiant states but they have significantly longer lifetimes (i.e. smaller [MATH]).', '1410.3634-1-7-0': 'The appearance of states localized on large clusters of atoms in a magnetic field is due to the removal of degeneracy of the excited states [MATH] by the field.', '1410.3634-1-7-1': 'As a result, the transitions [MATH] effectively decouple for different [MATH] since photons scattered on these transitions have frequencies discrepant by [MATH].', '1410.3634-1-7-2': 'As a consequence, a behavior similar to the scalar case may be expected for a given [MATH] with, in particular, localized states appearing at high densities of atoms as found in the scalar model [CITATION].', '1410.3634-1-7-3': 'However, as follows from Fig. [REF], this naive picture is largely oversimplified because it does not explain the absence of localized states near [MATH] corresponding to [MATH].', '1410.3634-1-7-4': "A more detailed study shows that indeed, the full vector problem can be reduced to an effective scalar one in the limit of strong magnetic field, but the effective Green's matrix following from this analysis is different from the one corresponding to scalar waves.", '1410.3634-1-7-5': "We have found that for [MATH], the group of eigenvalues corresponding to a given [MATH] can be approximately found by diagonalizing the effective [MATH] Green's matrix [EQUATION] where [MATH] and [MATH] is the angle between [MATH] and the [MATH] axis.", '1410.3634-1-8-0': 'Equation ([REF]) explains the differences between [MATH] and [MATH] seen in Figs. [REF] and [REF].', '1410.3634-1-8-1': 'First, the far-field contribution to [MATH] given by the second line of Eq. ([REF]) varies from 0 to 1 for [MATH] and from [MATH] to 1 for [MATH] as a function of [MATH].', '1410.3634-1-8-2': 'It is thus closer to its scalar-wave value of 1 in the former case, suggesting that the case of [MATH] may be better approximated by the scalar model than the case of [MATH].', '1410.3634-1-8-3': 'Second, the near-field term [the third line of Eq. ([REF])] is a factor of two smaller for [MATH] than for [MATH].', '1410.3634-1-8-4': 'Because near-field terms responsible for resonant dipole-dipole interactions between nearby atoms were shown to suppress the localization transition [CITATION], the weakness of these terms for [MATH] is an advantage.', '1410.3634-1-8-5': 'We see therefore that both far- and near-field features of Eq. ([REF]) are closer to its scalar approximation for [MATH] than for [MATH].', '1410.3634-1-8-6': 'This explains the appearance of localized states for [MATH] rather than for [MATH] transitions.', '1410.3634-1-9-0': 'To have a more quantitative characterization of the localization transition demonstrated in Fig. [REF], we compute the Thouless parameter [MATH] that we define as a ratio of the inverse of the average lifetime of eigenstates [MATH] to the average eigenvalue spacing along the horizontal axis [MATH], [MATH] [CITATION].', '1410.3634-1-9-1': 'This quantity is calculated as a function of [MATH] with the averaging performed over all eigenvalues in a unit interval around [MATH].', '1410.3634-1-9-2': 'As can be seen from Fig. [REF], [MATH] reaches small values [MATH] expected for localized states only at large densities corresponding to [MATH] and only for the values [MATH] corresponding to [MATH], in full agreement with Fig. [REF].', '1410.3634-1-10-0': 'The simple fact that [MATH] becomes smaller than 1 does not necessary signals a transition from extended to localized eigenstates.', '1410.3634-1-10-1': 'It is much more important that curves corresponding to different [MATH] (and thus to different sample sizes at a given [MATH]) cross in Figs. [REF](a) and (c).', '1410.3634-1-10-2': 'We reproduce [MATH] for [MATH] slightly shifted to the left of the single-atom resonant frequencies [MATH] in Fig. [REF].', '1410.3634-1-10-3': 'The independence of [MATH] from the sample size at the point where curves corresponding to different [MATH] cross is a hallmark of critical point and confirms localization transition for light at frequencies [MATH] with [MATH].', '1410.3634-1-10-4': 'This transition is also evidenced by the scaling function [MATH] [CITATION] shown in the insets of Fig. [REF].', '1410.3634-1-10-5': '[MATH] changes sign for [MATH] and 1998 but not for [MATH] proving that the localization transition takes place at large frequency shift [MATH] but not around the fundamental resonance [MATH].', '1410.3634-1-11-0': 'In conclusion, we have found that the magnetic field can induce a transition from extended to localized states for light in an ensemble of identical immobile two-level atoms.', '1410.3634-1-11-1': 'This is due to the removal of degeneracy of the excited atomic state by the field and the resulting partial suppression of resonant dipole-dipole interactions between nearby atoms.', '1410.3634-1-11-2': 'However, the discovered localization transition is different from that in the scalar model with a nondegenerate excited state.', '1410.3634-1-11-3': 'In particular, higher atomic density is required to reach localization and the values of Thouless number [MATH] that can be realized in the localized regime are not as small as in the scalar case.', '1410.3634-1-11-4': 'These differences will be the subject of further studies.', '1410.3634-1-12-0': 'The work of SES is supported by the French National Research Agency ANR (project ANR-14-CE26-0032 LOVE).'}

{'1410.3634-2-0-0': 'We discover a transition from extended to localized quasi-modes for light in a gas of immobile two-level atoms in a magnetic field.', '1410.3634-2-0-1': 'The transition takes place either upon increasing the number density of atoms in a strong field or upon increasing the field at a high enough density.', '1410.3634-2-0-2': 'It has many characteristic features of a disorder-driven (Anderson) transition but is strongly influenced by near-field interactions between atoms and the anisotropy of the atomic medium induced by the magnetic field.', '1410.3634-2-1-0': 'The transition from extended to localized eigenstates upon increasing disorder in a quantum or wave system is called after Philip Anderson who was the first to predict it for electrons in disordered solids [CITATION].', '1410.3634-2-1-1': 'More recently, this transition was studied for various types of quantum particles (cold atoms [CITATION], Bose-Einstein condensates [CITATION]) as well as for classical waves (light [CITATION], ultrasound [CITATION]).', '1410.3634-2-1-2': 'In the most common case of time-reversal symmetric systems invariant under spin rotation Anderson transition takes place for a three-dimensional (3D) disorder only, eigenstates of low-dimensional systems being always localized [CITATION].', '1410.3634-2-1-3': 'Anderson localization of light may find applications in the design of future quantum-information devices [CITATION], miniature lasers [CITATION] and solar cells [CITATION].', '1410.3634-2-1-4': 'However, no undisputable experimental observation of optical Anderson transition in 3D exists to date since alternative explanations were proposed for all published reports of it [CITATION].', '1410.3634-2-1-5': 'Moreover, we have recently shown that the simplest theoretical model in which light is scattered by point scatterers (atoms) does not predict Anderson localization of light at all [CITATION].', '1410.3634-2-2-0': 'In the present Letter we show that an external magnetic field may induce a transition between extended and localized states for light in a gas of cold atoms.', '1410.3634-2-2-1': 'Magnetic field is an important and unique means of controlling wave propagation in disordered media.', '1410.3634-2-2-2': 'On the one hand, it breaks down the time-reversal invariance leading to a suppression of weak localization in electronic [CITATION] and optical [CITATION] systems and to metal-insulator transitions in topological insulators [CITATION].', '1410.3634-2-2-3': 'On the other hand, by profoundly modifying the scattering properties of individual scatterers the magnetic field can produce an enhancement of the coherent backscattering peak for light scattered by atoms with a degenerate ground state [CITATION].', '1410.3634-2-2-4': 'Our work adds a new element in the mosaic of magnetic-field-induced phenomena in disordered systems by demonstrating that the removal of degeneracy of the excited atomic state due to the Zeeman effect and the resulting reduction of the strength of resonant dipole-dipole interactions between nearby atoms [CITATION] are sufficient to induce a transition from extended to localized states in a dense atomic system where Anderson localization does not take place in the absence of the field [CITATION].', '1410.3634-2-2-5': 'This critical phenomenon stands out from other magneto-optical effects that take place in disordered media (including also the photonic Hall effect [CITATION] and Hanle effect in coherent backscattering [CITATION]) which only give rise to weak corrections to wave transport.', '1410.3634-2-3-0': 'We consider an ensemble of [MATH] identical two-level atoms at random position [MATH] inside a spherical volume [MATH] of radius [MATH].', '1410.3634-2-3-1': 'The resonant frequency [MATH] of atoms defines the natural length scale [MATH], where [MATH] is the vacuum speed of light.', '1410.3634-2-3-2': 'The ground state [MATH] of an isolated atom [MATH] is nondegenerate with the total angular momentum [MATH], whereas the excited states [MATH] is three-fold degenerate with [MATH].', '1410.3634-2-3-3': 'The three degenerate substates [MATH] correspond to the three possible projections [MATH], [MATH] of the total angular momentum [MATH] on the quantization axis [MATH].', '1410.3634-2-3-4': 'The natural lifetime [MATH] of the excited state sets the time scale of the problem.', '1410.3634-2-3-5': 'The atoms are subject to a uniform magnetic field [MATH] and interact with the free electromagnetic field surrounding them.', '1410.3634-2-3-6': 'The system "atoms [MATH] field" is described by the following Hamiltonian [CITATION]: [EQUATION]', '1410.3634-2-3-7': "Here [MATH] is the Planck's constant divided by [MATH], [MATH] and [MATH] are the wave and the polarization vectors of the modes of the free electromagnetic field, [MATH]) are the corresponding creation (annihilation) operators, [MATH] are the atomic dipole operators, [MATH] is the electric displacement vector divided by the vacuum permittivity [MATH], [MATH] is the Bohr magneton, and [MATH] is the Lande factor of the excited state.", '1410.3634-2-4-0': 'Previous work [CITATION] demonstrated that in the absence of magnetic field ([MATH]), the degrees of freedom corresponding to the electromagnetic field can be traced out leading to an effective Hamiltonian describing the dynamics of [MATH] atoms coupled by the electromagnetic field.', '1410.3634-2-4-1': "This effective Hamiltonian takes the form of a [MATH] Green's matrix [MATH] describing the propagation of light between the atoms [CITATION].", '1410.3634-2-4-2': "The same approach can be used when [MATH] leading to the following Green's matrix: [EQUATION] where [MATH] is the Zeeman shift in units of the natural line width, [MATH], and [MATH].", '1410.3634-2-5-0': "In the absence of magnetic field ([MATH]) the eigenvalues of the Green's matrix [MATH] concentrate in a roughly circular domain on the complex plane roughly symmetric with respect to the vertical axis [MATH] and almost touching the horizontal axis [MATH] [CITATION].", '1410.3634-2-5-1': 'The field splits the eigenvalues into three equal groups centered around [MATH], [MATH]) [CITATION], see Fig. [REF].', '1410.3634-2-5-2': 'The three groups of eigenvalues become well separated in the limit of strong magnetic field [MATH] to which we will restrict our consideration in the present Letter.', '1410.3634-2-5-3': 'Although at a low density [MATH] the three groups of eigenvalues are similar [Fig. [REF](a)], the groups corresponding to [MATH] start to differ significantly from the [MATH] group at higher densities [Fig. [REF](b)].', '1410.3634-2-5-4': 'In particular, the [MATH] groups of eigenvalues develop "holes" that were previously associated with Anderson localization in the framework of the scalar model of wave scattering [CITATION].', '1410.3634-2-6-0': "To see whether localized states indeed appear at high densities of atoms, we analyze the inverse participation ratios (IPRs) of eigenvectors [MATH] of the Green's matrix [MATH], [MATH], where [MATH] is the square of the length of the vector [MATH].", '1410.3634-2-6-1': 'Low IPR [MATH] corresponds to an extended state whereas IPR [MATH] signals a state localized on [MATH] atoms.', '1410.3634-2-6-2': 'Figure [REF](a) shows that at low density of atoms most of the eigenvectors have low IPRs with the eigenvectors localized on pairs of closely located atoms being an exception.', '1410.3634-2-6-3': 'These "subradiant" states exist at any density and should be distinguished from localized states that are due to the multiple scattering of light on many atoms and that appear at higher densities in relatively narrow bands of frequencies [MATH] on the left from the resonances [MATH] [see Fig. [REF](b)].', '1410.3634-2-6-4': 'These states may have smaller IPRs than the subradiant states but they have significantly longer lifetimes (i.e. smaller [MATH]).', '1410.3634-2-7-0': 'The appearance of states localized on large clusters of atoms in a magnetic field is due to the removal of degeneracy of the excited states [MATH] by the field.', '1410.3634-2-7-1': 'As a result, the transitions [MATH] effectively decouple for different [MATH] since photons scattered on these transitions have frequencies discrepant by [MATH].', '1410.3634-2-7-2': 'As a consequence, a behavior similar to the scalar case may be expected for a given [MATH] with, in particular, localized states appearing at high densities of atoms as found in the scalar model [CITATION].', '1410.3634-2-7-3': 'However, as follows from Fig. [REF], this naive picture is largely oversimplified because it does not explain the absence of localized states near [MATH] corresponding to [MATH].', '1410.3634-2-7-4': "A more detailed study shows that indeed, the full vector problem can be reduced to an effective scalar one in the limit of strong magnetic field, but the effective Green's matrix following from this analysis is different from the one corresponding to scalar waves.", '1410.3634-2-7-5': "We have found that for [MATH], the group of eigenvalues corresponding to a given [MATH] can be approximately found by diagonalizing the effective [MATH] Green's matrix [EQUATION] where [MATH] and [MATH] is the angle between [MATH] and the [MATH] axis.", '1410.3634-2-8-0': 'Equation ([REF]) explains the differences between [MATH] and [MATH] seen in Figs. [REF] and [REF].', '1410.3634-2-8-1': 'First, the far-field contribution to [MATH] given by the second line of Eq. ([REF]) varies from 0 to 1 for [MATH] and from [MATH] to 1 for [MATH] as a function of [MATH].', '1410.3634-2-8-2': 'It is thus closer to its scalar-wave value of 1 in the former case, suggesting that the case of [MATH] may be better approximated by the scalar model than the case of [MATH].', '1410.3634-2-8-3': 'Second, the near-field term [the third line of Eq. ([REF])] is a factor of two smaller for [MATH] than for [MATH].', '1410.3634-2-8-4': 'Because near-field terms responsible for resonant dipole-dipole interactions between nearby atoms suppress light scattering [CITATION] and prevent Anderson localization [CITATION], their weakness for [MATH] is an advantage.', '1410.3634-2-8-5': 'We see therefore that both far- and near-field features of Eq. ([REF]) are closer to its scalar approximation for [MATH] than for [MATH].', '1410.3634-2-8-6': 'This explains the appearance of localized states for [MATH] rather than for [MATH] transitions.', '1410.3634-2-9-0': 'To have a quantitative characterization of the localization transition demonstrated in Fig. [REF], we compute the Thouless number [MATH] that we define as a ratio of the inverse of the average lifetime of eigenstates [MATH] to the average eigenvalue spacing along the horizontal axis [MATH] [CITATION].', '1410.3634-2-9-1': 'At a given [MATH], [MATH] reaches small values [MATH] expected for localized states only at large densities corresponding to [MATH] and only for [MATH] corresponding to [MATH] (see Fig. [REF]), in agreement with Fig. [REF].', '1410.3634-2-9-2': 'The independence of [MATH] from the sample size at the points where curves corresponding to the same value of [MATH] but different [MATH] cross-a hallmark of critical behavior-is further illustrated in Fig. [REF] where we reproduce [MATH] for [MATH] slightly shifted to the left of the single-atom resonances [MATH].', '1410.3634-2-9-3': 'The localization transition is also evidenced by the scaling function [MATH] [CITATION] shown in the insets of Fig. [REF].', '1410.3634-2-9-4': '[MATH] changes sign for [MATH] and 1998 but not for [MATH] proving that the localization transition takes place at large frequency shifts [MATH] but not around the fundamental resonance [MATH].', '1410.3634-2-10-0': 'The localization transition reported here takes place under conditions when not only the disorder-induced multiple scattering of photons is strong but cooperative effects leading to Dicke super- and sub-radiance [CITATION] and resonant dipole-dipole interactions between neighboring atoms [CITATION] are important as well.', '1410.3634-2-10-1': 'Therefore, despite its overall similarity with the Anderson transition (see also Ref. [CITATION]), it remains to be seen if this transition can be classified as such.', '1410.3634-2-10-2': 'Nonetheless, for light of a given frequency [MATH] the localized regime [MATH] is realized only in the intermediate range of [MATH] (e.g., [MATH]-[MATH] for [MATH] in Fig. [REF]) which corresponds to large sizes of the atomic cloud [MATH] (e.g., [MATH]-[MATH] at [MATH]).', '1410.3634-2-10-3': 'Hence, the localized states disappear in the Dicke limit [MATH] when the cooperative effects dominate.', '1410.3634-2-10-4': 'This suggests that the latter are not the main driving force behind the reported localization transition.', '1410.3634-2-11-0': 'In conclusion, we have found that the magnetic field can induce a transition from extended to localized states for light in an ensemble of identical immobile two-level atoms.', '1410.3634-2-11-1': 'This is due to the removal of degeneracy of the excited atomic state by the field and the resulting partial suppression of resonant dipole-dipole interactions between nearby atoms.', '1410.3634-2-11-2': 'Our theoretical predictions can be directly verified in experiments with, e.g., Sr atoms that have a nondegenerate ground state and were already used to study multiple scattering of light [CITATION].', '1410.3634-2-11-3': 'Theoretical analysis of light scattering in dense clouds of alkali atoms (such as, e.g., Rb[MATH]) is, however, much more involved [CITATION] and our results cannot be trivially extended to this case.', '1410.3634-2-12-0': 'SES thanks the Agence Nationale de la Recherche for financial support under grant ANR-14-CE26-0032 LOVE.', '1410.3634-2-13-0': 'Supplemental Material', '1410.3634-2-14-0': 'We present additional evidence for the localization transition reported in the main text.', '1410.3634-2-14-1': 'In particular, we provide more details about the evolution of IPR maps shown in Fig. 2 with the number density of atoms, analyze the minimum value of the decay rate of quasi-modes, and study the suppression of eigenvalue repulsion due to the appearance of localized quasi-modes.', '1410.3634-2-14-2': 'In addition, we discuss some of the subtle details of light scattering in magnetic field and show that "escape channels" that appear in the atomic medium in the magnetic field due to photons with certain combinations of polarization and propagation direction do not shorten the lifetime of excited atomic states.', '1410.3634-2-15-0': 'The primary purpose of this Supplemental Material is to characterize the localization transition reported in the main text in more detail and to show that, in many aspects, it exhibits the behavior expected for the Anderson transition driven by disorder (section [REF]).', '1410.3634-2-15-1': "We start by analyzing the density dependence of IPR of eigenvectors of the Green's matrix (2) complementing the results presented in Fig. 2 (section [REF]).", '1410.3634-2-15-2': 'Then we study the behavior of the minimum decay rate of quasi-modes (section [REF]).', '1410.3634-2-15-3': 'Finally, the statistics of nearest eigenvalue spacings and the phenomenon of eigenvalue repulsion are addressed in section [REF].', '1410.3634-2-15-4': 'In section [REF] we discuss some of the subtleties of light scattering by atoms in a magnetic field and, in particular, show by analyzing emission and absorption diagrams of individual atoms that photons that can propagate in the atomic medium without scattering due to a special relation between their polarization and propagation direction do not influence the lifetime of the excited atomic states.', '1410.3634-2-16-0': '# Evidence for the localization transition', '1410.3634-2-17-0': '## Inverse participation ratio', '1410.3634-2-18-0': "It is instructive to analyze how IPR (see the main text) of eigenvectors of the Green's matrix (2) evolves when the number density of atoms increases [CITATION].", '1410.3634-2-18-1': 'In addition to gray-scale IPR maps presented in Fig. 2, we show false color maps of the logarithm of average IPR at several additional densities in Figs. [REF] and [REF].', '1410.3634-2-18-2': 'The color scale of these figures allows distinguishing between the subradiant states localized on pairs of closely located atoms that appear in red and states localized on larger clusters of atoms that show up in other colors from the orange to the light blue.', '1410.3634-2-18-3': 'Figure [REF] clearly shows that for the states that correspond to eigenvalues with [MATH] close to 0 (the cloud of eigenvalues corresponding to [MATH], see the main text), IPR maps do not change qualitatively when the density increases from [MATH] to 1.5.', '1410.3634-2-18-4': 'The cloud of eigenvalues grows in size, states localized on pairs of atoms ([MATH]) show up along the two-atom subradiant branches shown by dashed lines in Fig. 2, but no localized states with very small decay rates [MATH] appear upon increasing the density.', '1410.3634-2-18-5': 'States with significant IPRs in the middle of the eigenvalue cloud, near [MATH] (visible in white and yellow for [MATH]) are not sufficient to reach [MATH] and, most importantly, do not show critical behavior [the curves [MATH] corresponding to different [MATH] do not cross and [MATH] does not change sign in Figs. 3 and 4].', '1410.3634-2-18-6': 'In contrast, for the eigenvalues with [MATH] close to [MATH], IPR maps change qualitatively with increasing density and a band of localized states with very small decay rates [MATH] appears for densities [MATH] (see Fig. [REF] for [MATH], a similar scenario takes place around [MATH] for [MATH]).', '1410.3634-2-18-7': 'The band widens and starts to disappear by merging with the two-atom subradiant branch for [MATH].', '1410.3634-2-19-0': "Figures [REF] and [REF] clearly demonstrate the advantage of analyzing properties of eigenvectors of the Green's matrix as a function of both the real and imaginary parts of the corresponding eigenvalues instead of projecting on one of the axes ([MATH] or [MATH]).", '1410.3634-2-19-1': 'The two-atom subradiant states typically have large frequency shifts with respect to single-atom resonances [MATH] whereas the nontrivial states localized on larger clusters of atoms appear for near-resonant [MATH].', '1410.3634-2-19-2': 'Hence, the two types of states are clearly separated on the complex plane and can be distinguished.', '1410.3634-2-19-3': 'This advantage is lost when a projection on one of the axes, and, in particular, on the imaginary axis, is performed.', '1410.3634-2-19-4': 'In this case one observes a dependence that is due to both the two-atom subradiant states and the states localized on larger atomic clusters.', '1410.3634-2-19-5': 'The two cannot be disentangled anymore.', '1410.3634-2-20-0': 'Before closing up the discussion of IPR maps shown in Figs. [REF] and [REF] we would like to attract the attention of the reader to the fact that the correspondence between small decay rates [MATH] and the localized nature of the corresponding states should be used with care.', '1410.3634-2-20-1': 'Indeed, even if it might seem natural that localized states should have small decay rates and, vice versa, that small decay rates are likely to correspond to states localized in space, we see from Figs. [REF] and [REF] that states with a significant IPR (e.g., white squares) systematically appear at large [MATH] and even at [MATH] which corresponds to lifetimes which are even shorter than the lifetime of the excited state of an isolated atom.', '1410.3634-2-20-2': "Therefore, making any conclusions about the spatial structure of eigenvectors of the Green's matrix based uniquely on the analysis of eigenvalues [MATH] (as it was done in Refs. [CITATION], for example) is dangerous and should necessarily be supported by the analysis of eigenvectors themselves, similarly to the analysis that we present in Fig. 2 and Figs. [REF] and [REF].", '1410.3634-2-21-0': '## Minimum decay rate', '1410.3634-2-22-0': "Let us first remind that the complex eigenvalues [MATH] of the Green's matrix defined by Eq. (2) of the main text yield eigenfrequencies [MATH] and decay rates [MATH] of the quasi-modes [MATH] of the atomic system.", '1410.3634-2-22-1': 'The temporal evolution of the quasi-mode [MATH] obeys [MATH].', '1410.3634-2-22-2': 'The probability distribution [MATH] of dimensionless decay rates [MATH] contains important information about the nature of quasi-modes in the system although it cannot be considered as the only proof of any statement concerning the spatial structure of the modes (extended, localized, etc., see the discussion at the end of the previous subsection).', '1410.3634-2-22-3': 'However, once the existence of spatially localized modes is demonstrated by studying the eigenvectors of the Green\'s matrix (see Fig. 2 and Figs. [REF] and [REF]), for a mode [MATH] that is weakly coupled to the exterior of the disordered sample one can assume [MATH], where [MATH] is the "typical" point at the open surface of the medium [CITATION].', '1410.3634-2-22-4': 'Because for a mode localized deep inside a disordered sample [MATH], where [MATH] is the size of the sample, the exponential decrease of [MATH] with [MATH] may be considered as a signature of exponential localization of the mode [MATH] in space.', '1410.3634-2-23-0': 'In order to focus our attention on the modes that have the smallest decay rates, we analyze the average value of the minimum dimensionless decay rate [MATH].', '1410.3634-2-23-1': 'In the absence of magnetic field, a scaling [EQUATION] was derived for this quantity in the low-density regime in the scalar approximation [CITATION] and in the full vector model [CITATION].', '1410.3634-2-23-2': 'This scaling was further confirmed by independent numerical simulations [CITATION].', '1410.3634-2-23-3': 'Moreover, Eq. ([REF]) that is due to subradiant states localized on pairs of closely located atoms, was shown to hold at any density for the vector case [CITATION].', '1410.3634-2-23-4': 'In the presence of a strong magnetic field, however, Fig. [REF] shows a breakdown of Eq. ([REF]) at sufficiently high densities [MATH].', '1410.3634-2-23-5': 'In the region [MATH] the numerical data for all [MATH] can be fit by a phenomenological formula [EQUATION] with a density-dependent localization length [MATH].', '1410.3634-2-23-6': 'Figure [REF] shows fits obtained with reasonable values [MATH] and [MATH].', '1410.3634-2-23-7': 'It is important to note that these fits do not allow determining the critical density [MATH] and the critical exponent [MATH] with any acceptable precision because fits of similar quality can be obtained for a range of values [MATH]-[MATH] and [MATH]-[MATH].', '1410.3634-2-23-8': 'They witness, however, that our numerical data are compatible with the exponential decay of [MATH] with sample size [MATH] beyond a certain critical value of density, which is one of the signatures of exponential localization of eigenstates expected for the disorder-induced (Anderson) localization mechanism.', '1410.3634-2-24-0': 'It is also instructive to see at which value of [MATH] the minimum values of [MATH] shown in Fig. [REF] are attained.', '1410.3634-2-24-1': 'As we show in Fig. [REF], the average value of [MATH] at which [MATH] is minimized first grows with density to very large values [MATH] before abruptly dropping to an [MATH]-independent value [MATH] at [MATH].', '1410.3634-2-24-2': 'The fast initial growth of [MATH] with density takes place in the regime in which [MATH] is due to subradiant states localized on pairs of closely located atoms and decays as a power-law with [MATH] (see Fig. [REF]).', '1410.3634-2-24-3': 'These subradiant states typically have large frequency shifts corresponding to large values of [MATH].', '1410.3634-2-24-4': 'However, we clearly see that for [MATH] the nature of states that have minimum decay rates changes abruptly.', '1410.3634-2-24-5': 'Now [MATH] is dominated by the states localized on larger clusters of atoms (see Fig. 2 and Figs. [REF] and [REF]) that have frequency shifts [MATH] [CITATION].', '1410.3634-2-24-6': 'This situation is preserved until a sufficiently high density ([MATH] in Fig. [REF]) at which the localized states at [MATH] start to disappear and [MATH] becomes dominated by the same subradiant states as in the low-density limit.', '1410.3634-2-24-7': 'This change of regime is also witnessed by [MATH] that grows with density and approaches the line [MATH] for [MATH] (see Fig. [REF]).', '1410.3634-2-25-0': '## Eigenvalue repulsion', '1410.3634-2-26-0': 'A well-known impact of localized states on the spectrum of a random matrix (or, more generally, of a disordered system) is the suppression of the so-called eigenvalue (or level) repulsion phenomenon.', '1410.3634-2-26-1': 'In brief, an Hermitian matrix (or a closed disordered system) with extended eigenstates is expected to exhibit eigenvalue (level) repulsion: the probability density function of spacings between nearest eigenvalues [MATH] and [MATH] [CITATION], [MATH], goes to zero for [MATH] [CITATION].', '1410.3634-2-26-2': 'This is due to the mutual orthogonality of eigenvectors of an Hermitian matrix that forbids that two extended eigenvectors correspond to the same eigenvalue.', '1410.3634-2-26-3': 'The appearance of localized states leads to the suppression of the eigenvalue repulsion because two states localized far from each other can now correspond to the same eigenvalue and [MATH] can become arbitrary small.', '1410.3634-2-27-0': 'The concept of eigenvalue repulsion can be generalized to non-Hermitian matrices [CITATION].', '1410.3634-2-27-1': 'The eigenvalues [MATH] are now complex but for each [MATH] the nearest eigenvalue [MATH] can still be identified as the eigenvalue that minimizes the distance between the two eigenvalues on the complex plane [MATH].', '1410.3634-2-27-2': "For the Ginibre's ensemble of random matrices [CITATION], the probability density function of normalized eigenvalue spacings [MATH] is found to be [CITATION] [EQUATION]", '1410.3634-2-27-3': 'We see that [MATH] for [MATH] which means that two complex eigenvalues avoid being too close to each other on the complex plane.', '1410.3634-2-27-4': 'Importantly, the mutual repulsion of eigenvalues is stronger than it would be for the Poissonian random process on the plane that would lead to [MATH] [CITATION].', '1410.3634-2-28-0': "We now compute the probability density of normalized nearest eigenvalue spacings [MATH] for the complex eigenvalues [MATH] of the Green's matrix (2) and compare the results obtained for unit bands of [MATH] centered at [MATH] (where no localization transition is expected according to Fig. 2 and Fig. [REF]) and at [MATH] (where we do expect a localization transition).", '1410.3634-2-28-1': 'To avoid the influence of irrelevant superradiant states with [MATH], we limit our analysis to eigenvalues with [MATH].', '1410.3634-2-28-2': 'As can be seen from Figs. [REF] and [REF], at a low density [MATH] the probability density [MATH] is close to the behavior predicted by Eq. ([REF]) independent of the frequency interval under consideration.', '1410.3634-2-28-3': "This can be understood from the fact that at small densities the atomic positions [MATH] are far apart so that most of the elements of the Green's matrix (2) become effectively uncorrelated due to the large phases [MATH].", '1410.3634-2-28-4': "Statistical properties of the ensemble of Green's matrices thus approach those of the Ginibre's ensemble for which Eq. ([REF]) was derived.", '1410.3634-2-28-5': 'In particular, the eigenvalue repulsion is clearly observed in both frequency intervals.', '1410.3634-2-28-6': 'At larger densities, however, the behaviors of [MATH] computed for [MATH] around [MATH] and around [MATH] differ significantly.', '1410.3634-2-28-7': 'As we see from Fig. [REF], for [MATH] around [MATH] where no localization transition is expected, [MATH] attains a roughly universal shape already for [MATH].', '1410.3634-2-28-8': 'This shape hardly changes when the density is further increased.', '1410.3634-2-28-9': 'In contrast, for [MATH] around [MATH] where a localization transition takes place, we clearly observe that [MATH] acquires an increasingly important weight at small [MATH] when the density is increased up to [MATH] (see Fig. [REF]).', '1410.3634-2-28-10': 'We interpret this increase of [MATH] for small [MATH] as a suppression of eigenvalue repulsion due to the appearance of localized states.', '1410.3634-2-28-11': 'Further increase of density restores level repulsion signaling that the system leaves the localized regime and returns to the situation in which the eigenstates are extended.', '1410.3634-2-28-12': 'This picture becomes even more obvious if one looks at the cumulative distribution function [MATH], i.e. at the probability for the normalized nearest eigenvalue spacing to fall below a given [MATH], [EQUATION]', '1410.3634-2-28-13': 'In Fig. [REF] we show [MATH] for a small value of [MATH] as a function of atomic number density.', '1410.3634-2-28-14': 'The difference between the two intervals of [MATH] under study as well as an important suppression of eigenvalue repulsion leading to a more than ten-fold increase of [MATH] around [MATH] are obvious.', '1410.3634-2-29-0': '# Remarks on light scattering by atoms in a magnetic field', '1410.3634-2-30-0': "The results obtained in the main text of the Letter follow from the analysis of the exact [MATH] Green's matrix (2) that take into account all the complex physical phenomena that take place in light scattering by atoms and relies on quite a limited number of reasonable approximations (see the main text and Refs. [17], [28], [29]).", '1410.3634-2-30-1': 'However, when trying to understand these results qualitatively, a number of legitimate questions may arise concerning the microscopic details of physical processes at work and their interplay.', '1410.3634-2-30-2': 'In this section, we provide elements of answers to a couple of such questions that we anticipate.', '1410.3634-2-31-0': 'One might argue that given the decoupling of transitions corresponding to different [MATH], [MATH] in a strong magnetic field,-the decoupling that is rigorously confirmed by the existence of the effective scalar Green\'s matrix (3) that describes scattering on each transition separately from the others,-light may escape from the medium along some special directions in which it has "wrong" polarization and cannot be scattered by the transition corresponding to its frequency.', '1410.3634-2-31-1': 'Indeed, imagine a photon emitted by an atomic excited state corresponding, say, to [MATH].', '1410.3634-2-31-2': 'The photon will be resonant with the same transition of the next atom on its way, but can be scattered by this atoms only if its helicity has a projection on the direction of the dipole moment of the atomic transition.', '1410.3634-2-31-3': 'The latter projection depending on the direction in which the photon propagates, a special direction will exist for which scattering vanishes, and the photon will be able to propagate ballistically until it reaches a boundary of the atomic cloud and escapes to the free space.', '1410.3634-2-31-4': 'Such "escape channels" induced by the magnetic field would decrease the lifetime of collective states in the atomic cloud and should not favor localized states.', '1410.3634-2-31-5': 'How can this picture be reconciled with our main conclusion about the appearance of localized states in the magnetic field?', '1410.3634-2-32-0': 'To answer this question, let us consider the problem a little bit more rigorously.', '1410.3634-2-32-1': 'The spontaneous decay of an initially excited atom [MATH] (initial state [MATH], [MATH]) leads to the emission of a photon propagating along a direction determined by angles [MATH], [MATH] of the spherical coordinate system (see Fig. [REF]).', '1410.3634-2-32-2': 'To describe the polarization properties of radiation we will use the so-called spiral basis although the subsequent discussion can be repeated for the linear basis as well.', '1410.3634-2-32-3': 'The unit vectors of the spiral basis are denoted by [MATH], [MATH] and [MATH].', '1410.3634-2-32-4': '[MATH] correspond to the right ([MATH]) or left ([MATH]) helicity of the photon whereas [MATH] is parallel to the direction of its propagation.', '1410.3634-2-32-5': 'In the rotating wave approximation the probability of emission of a photon with polarization [MATH] is determined by the scalar product of [MATH] and the dipole moment of transition [MATH] [CITATION].', '1410.3634-2-32-6': 'The three atomic transitions from excited states [MATH], [MATH]; [MATH], [MATH]; [MATH], [MATH] to the ground state [MATH] are characterized by three different vectors of the transition dipole moment [MATH].', '1410.3634-2-32-7': 'The latter are parallel to unit vectors of the cyclic coordinate system [MATH], [MATH] and [MATH], respectively, with the quantization axis [MATH] chosen parallel to the external magnetic field [MATH].', '1410.3634-2-33-0': 'Because for a given atomic transition the probability of emission and of subsequent interaction of a photon with other atoms is determined by projections of spiral unit vectors on cyclic ones, a relation between the two bases is necessary to proceed.', '1410.3634-2-33-1': 'Such a relation can be found, for example, in Ref. [CITATION] and involves the direction of propagation of the photon determined by the angles [MATH], [MATH]: [EQUATION]', '1410.3634-2-33-2': 'Projecting cyclic unit vectors onto spiral ones we can determine the angular dependence of the probability amplitude of emission of a photon with a given helicity.', '1410.3634-2-34-0': 'To start with, consider a photon emitted on the transition [MATH], [MATH].', '1410.3634-2-34-1': 'In this case, the transition dipole moment [MATH] is parallel to [MATH].', '1410.3634-2-34-2': 'From Eq. ([REF]) we see that probability amplitude [MATH] for the photon to have right helicity is proportional to [MATH] whereas the probability amplitude of having left helicity [MATH] will be [MATH].', '1410.3634-2-34-3': 'Photons with different helicities have different mean free paths.', '1410.3634-2-34-4': 'To estimate the latter, let us consider the probability [MATH] of absorption of these photons by a second atom (atom [MATH] in Fig. [REF]) initially in the ground state [MATH].', '1410.3634-2-34-5': 'We assume that the photons are resonant with the transition [MATH], [MATH] and neglect the existence of two other transitions which is justified in a strong magnetic field when the three transitions corresponding to different [MATH] have very different frequencies.', '1410.3634-2-35-0': 'Similarly to the emission probability amplitude, the absorption probability amplitude for a photon of right (left) helicity is determined by the scalar product of the unit vector [MATH]) and the dipole moment [MATH].', '1410.3634-2-35-1': 'Because [MATH], the required scalar product can be calculated using the complex conjugate of Eqs. ([REF]-[REF]).', '1410.3634-2-35-2': 'This yields the absorption probability amplitudes [MATH] and [MATH] for photons of right and left helicities, respectively.', '1410.3634-2-35-3': 'On the one hand, these absorption amplitudes are smaller than those in the absence of magnetic field, when the photon emitted by the first atom is resonant with all three transitions of the second one.', '1410.3634-2-35-4': 'Moreover, for certain combinations of helicity and propagation direction (e.g., for the right helicity and [MATH] or for the left helicity and [MATH]), the absorption amplitude vanishes which corresponds to propagation without scattering and an infinitely large scattering mean free path for photons of these precise helicities in these precise directions.', '1410.3634-2-35-5': 'However, on the other hand, for the particular directions in which the probability amplitude of photon absorption by the atom [MATH], [MATH], vanishes, the probability amplitude of photon emission by the atom [MATH], [MATH], vanishes as well.', '1410.3634-2-35-6': 'In other words, there exist indeed certain combinations of helicity and propagation direction in which scattering is absent such that the corresponding photon would leave the atomic medium ballistically.', '1410.3634-2-35-7': 'However, the probability of emission of such a photon by an atomic excited state is zero and these particular combinations of helicity and propagation direction cannot serve as decay channels for the excited atomic states.', '1410.3634-2-36-0': 'An extension of the above reasoning to photons emitted not exactly but close to the "critical" directions identified above shows that the scattering mean free path of these photons will be large but finite, whereas the probability of their emission will be small.', '1410.3634-2-36-1': 'As a result, they do not have any special influence on the quantities studied in the main text of the Letter (although, as we will see below, they are fully taken into account in our analysis).', '1410.3634-2-36-2': "For example, no atomic states with anomalously short lifetimes appear due to these photons as can be verified by comparing eigenvalues of the random Green's matrix (2) with and without the external magnetic field [MATH].", '1410.3634-2-36-3': 'Moreover, the average lifetime of collective atomic states remains equal to [MATH] (which follows from [MATH]) independent of [MATH].', '1410.3634-2-37-0': 'The arguments presented above can be repeated for the two other transitions with the conclusion remaining exactly the same.', '1410.3634-2-37-1': 'For a photon emitted on the transition [MATH], [MATH], for example, the angular dependencies of emission and absorption probability amplitudes are given by the same equations as for the photon emitted on the [MATH], [MATH] transition with right and left helicities interchanged.', '1410.3634-2-37-2': 'For a photon emitted on the transition [MATH], [MATH] the emission and absorption amplitudes are proportional to [MATH] for the right ([MATH]) and left ([MATH]) helicities, respectively.', '1410.3634-2-37-3': 'Once again, the directions [MATH], [MATH] with a vanishing absorption probability correspond to a vanishing emission probability as well.', '1410.3634-2-37-4': 'We believe that the difference in the dependencies of emission/absorption probabilities for different helicities on the propagation direction for [MATH], [MATH] may be at the origin of the peculiar difference observed between the localized states appearing near [MATH] (states resonant with the transitions [MATH], [MATH]) and extended states near [MATH] (states resonant with the transitions [MATH], [MATH]) although more work is required to establish a precise link between the two phenomena.', '1410.3634-2-38-0': 'Finally, we would like to note that the analysis presented above allows for calculating the total probability amplitude [MATH] for an atom to get excited by the photon emitted by another atom (see Fig. [REF]).', '1410.3634-2-38-1': 'Important for us is the dependence of this amplitude on the angles [MATH] and [MATH] that can be obtained by multiplying the probability of emission [MATH] by the probability of absorption [MATH] and then summing over the two helicities.', '1410.3634-2-38-2': 'We obtain [EQUATION]', '1410.3634-2-38-3': "These results coincide with those that can be obtained from the Green's matrix (2).", '1410.3634-2-38-4': 'Indeed, for two atoms [MATH] and [MATH] separated by a distance [MATH] much exceeding the wavelength, the probability amplitude of photon scattering on the resonant transition following from Eq. (2) is [EQUATION]', '1410.3634-2-38-5': 'Here [MATH] and [MATH] are the cyclic components of the vectors [MATH] and [MATH], respectively.', '1410.3634-2-38-6': 'Equation ([REF]) reduces to [MATH] and [MATH] and coincides exactly with the predictions of Eqs. ([REF]) and ([REF]).', '1410.3634-2-39-0': '# Conclusions', '1410.3634-2-40-0': 'The analysis presented in section [REF] clearly shows that the localization transition discovered in the main text exhibits a number of features expected for the disorder-driven, Anderson localization transition.', '1410.3634-2-40-1': 'In particular, the minimum decay rate of quasi-modes decreases exponentially with sample size beyond a certain critical number density of atoms (section [REF]) and the mutual repulsion of complex eigenvalues is suppressed (section [REF]).', '1410.3634-2-40-2': 'In addition, the localized states under study can be clearly distinguished from subradiant states localized on pairs of closely located atoms due to very different values of the corresponding eigenvalues [MATH] near the resonances [MATH] for the new discovered localized states but [MATH] far from resonances for the two-atom subradiant states, see section [REF]).', '1410.3634-2-40-3': 'Despite this, however, it would be premature to claim that the discovered transition is a standard Anderson transition because the role of cooperative phenomena and of dipole-dipole interactions between atoms still remains to be clarified.', '1410.3634-2-41-0': 'In section [REF] we have demonstrated that escape channels-special combinations of polarization and propagation direction for which a photon is not scattered by the atoms,-do not affect the lifetime of the excited atomic states and are thus compatible with the localization transition discovered in the main text of the Letter.', '1410.3634-2-41-1': 'This is due to the relation that exists between the emission and absorption diagrams of two-level atoms in the magnetic field.', '1410.3634-2-41-2': 'The latter relation leads to a vanishing emission probability for photons that would have a vanishing absorption probability.', '1410.3634-2-41-3': 'Therefore, such photons are not emitted by the excited atomic states and thus the escape channels cannot serve as efficient decay channels for the excited atomic states.'}

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[]

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['1410.3634-2-13-0', '1410.3634-2-38-2']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1410.3634

0710.3057

{'0710.3057-1-0-0': '# Introduction', '0710.3057-1-1-0': 'Among the problems still left open by the Standard Model (SM) of particle physics, the understanding of the exact mechanism that leads to the breaking of the electroweak symmetry at low energies is of particular importance.', '0710.3057-1-1-1': 'Besides the SM basic Higgs mechanism, still to be verified by experiments, possible alternative solutions to the problem are offered by extensions of the old technicolor (TC) theories, where the Higgs is realized as a composite state of strongly interacting fermions.', '0710.3057-1-2-0': 'These theories have recently received a renewed attention starting from higher dimensional Lagrangians; effective chiral Lagrangians in four dimensions containing new resonance states can be obtained by the deconstruction technique [CITATION] or as holographic versions of 5-dimensional (5D) theories in warped background [CITATION].', '0710.3057-1-3-0': 'Models have been proposed, working in the framework suggested by the AdS/CFT correspondence, which assume a [MATH] gauge group in the 5D bulk, [CITATION], or also simpler with a [MATH] in the bulk [CITATION].', '0710.3057-1-3-1': 'One of the main challenges of these models is the value of the [MATH] parameter [CITATION] or the related [MATH] [CITATION].', '0710.3057-1-3-2': 'Indeed, the experimental value of [MATH] is of the order of [MATH] [CITATION], whereas the value naturally expected in TC theories is an order of magnitude bigger.', '0710.3057-1-4-0': 'A delocalization of the fermionic fields into the bulk as in [CITATION], realized in the deconstructed version by allowing standard fermions to have direct couplings to all the moose gauge fields as in [CITATION], leads to direct contributions to the electroweak parameters that can correct the bad behavior of the [MATH] parameter.', '0710.3057-1-4-1': 'The fine tuning which cancels out the oblique and direct contributions to [MATH] in each bulk point, that is from each internal moose gauge group, corresponds to the so called ideal delocalisation of fermions, [CITATION].', '0710.3057-1-4-2': 'Other solutions to get a suppressed contribution to [MATH] have been investigated, like the one suggested by holographic QCD, assuming that different five dimensional metrics are felt by the axial and vector states [CITATION].', '0710.3057-1-4-3': 'However it has been shown recently that the backgrounds that allow a negative oblique contribution to [MATH] are pathological, since they require unphysical Higgs profile or higher dimensional operators [CITATION].', '0710.3057-1-5-0': 'An alternative solution to the [MATH] problem was proposed in [CITATION] (see also [CITATION]).', '0710.3057-1-5-1': 'The solution was realized in terms of an effective TC theory of non linear [MATH]-model scalars and massive gauge fields.', '0710.3057-1-5-2': 'The model is a four site model with three sigma fields.', '0710.3057-1-5-3': 'The physical spectrum consists of three massless scalar fields (the Goldstone bosons giving masses to the gauge vector particles) and two triplets of massive vector fields degenerate in mass and couplings.', '0710.3057-1-5-4': 'This model, named degenerate BESS model (D-BESS), has an enhanced custodial symmetry such as to allow [MATH] at the lowest order in the electroweak interactions.', '0710.3057-1-5-5': 'This idea has been also recently used in phenomenological analysis of low scale technicolor theories with vector and axial vector resonances very close in mass [CITATION].', '0710.3057-1-6-0': 'A generalization of D-BESS was studied in [CITATION].', '0710.3057-1-6-1': 'This extended model is a moose model, with a replicated [MATH] gauge symmetry, that maintains the most useful feature of D-BESS, namely the custodial symmetry which guarantees the vanishing of [MATH].', '0710.3057-1-7-0': 'Two other quantities, [MATH] and [MATH] [CITATION] or equivalently [MATH] and [MATH] [CITATION], are customarily used to parameterize the electroweak precision observables.', '0710.3057-1-7-1': 'More recently, an alternative parametrization was proposed in terms of seven parameters [CITATION] which describe in a very general way the effects of so-called "universal" extensions of the SM, that is theories whose deviations from the SM are all contained in the vector boson self-energies.', '0710.3057-1-8-0': 'In the present paper, we wish to extend the calculation made in [CITATION] by deriving the seven parameters of ref. [CITATION], and from them the [MATH] parameters, to the next-to-leading order in the weak interactions; this means that we will consider terms up to order [MATH], where [MATH] is the mass scale of the new bosons.', '0710.3057-1-8-1': 'Also, we will calculate, at the same order, the trilinear gauge boson vertex anomalous contributions due to the new physics.', '0710.3057-1-9-0': 'In Section [REF] we review the notations and the main constitutive elements of the model.', '0710.3057-1-9-1': 'In Section [REF] we derive the low energy effective Lagrangian by eliminating the fields of the internal moose.', '0710.3057-1-9-2': 'In Section [REF] we compute the effective gauge boson correlation functions, and from them the parameters of ref. [CITATION] and the [MATH] parameters, to the next-to-leading order in the weak interactions; we then derive bounds on the model parameter space from experimental data.', '0710.3057-1-9-3': 'In Section [REF] we obtain the effective trilinear gauge couplings to order [MATH].', '0710.3057-1-9-4': 'Finally, in Section [REF] we present our conclusions.', '0710.3057-1-10-0': '# A linear moose model for the electroweak symmetry breaking', '0710.3057-1-11-0': 'Our model is based on the idea of dimensional deconstruction [CITATION] and on the hidden gauge symmetry approach, historically applied both to strong interactions [CITATION] and to electroweak symmetry breaking [CITATION].', '0710.3057-1-12-0': 'Consider [MATH] non linear [MATH]-model scalar fields [MATH], [MATH] and [MATH] gauge groups, [MATH], [MATH] with global symmetry [MATH].', '0710.3057-1-12-1': 'Since we are interested in studying the electroweak symmetry breaking mechanism, we will assume [MATH], [MATH].', '0710.3057-1-12-2': 'The transformation properties of the fields are [EQUATION] with [MATH], [MATH]; [MATH], [MATH]; the Lagrangian is given by [EQUATION] where [MATH] are [MATH] free constants ("link" coupling constants).', '0710.3057-1-12-3': 'The covariant derivatives are defined as follows: [EQUATION] as implied by the transformation rules ([REF]), where [MATH] and [MATH] are the gauge fields and gauge coupling constants associated to the groups [MATH], [MATH].', '0710.3057-1-12-4': '[MATH] has the standard definition [EQUATION] with [EQUATION]', '0710.3057-1-12-5': 'Notice that one could introduce an additional field, [EQUATION] which transforms just like the usual chiral field of the Higgsless SM: [MATH].', '0710.3057-1-12-6': 'The field [MATH] is an invariant under the [MATH] transformations (which are then effectively "hidden" to [MATH]).', '0710.3057-1-13-0': 'As shown in [CITATION], in this model, due to the presence of a custodial [MATH] symmetry, we get no corrections to [MATH].', '0710.3057-1-13-1': 'Also, if we put one (and only one) of the link coupling constants [MATH] equal to zero, we effectively enlarge the global symmetry to [MATH], getting [MATH] too.', '0710.3057-1-13-2': 'We want to explore this particular case; also, for simplicity, we impose an extra left-right symmetry of the moose which identifies the two ends:', '0710.3057-1-14-0': 'f_i f_K+2-i,', '0710.3057-1-15-0': 'g_i g_K+1-i.', '0710.3057-1-16-0': 'The reflection symmetry, together with the condition that just one of the link coupling constants must vanish, implies that the number of moose links has to be odd (and hence the number of gauge fields has to be even), and that the vanishing link has to be the central one.', '0710.3057-1-16-1': 'So we will put:', '0710.3057-1-17-0': 'K 2N,', '0710.3057-1-18-0': 'f_N+1 = 0.', '0710.3057-1-19-0': 'It is instructive to count out the number of degrees of freedom.', '0710.3057-1-19-1': 'Before cutting the central link, we had [MATH] matrices of scalar fields, for [MATH] degrees of freedom, and [MATH] massless vector fields, for [MATH] degrees of freedom; of these, only 3 scalar fields are physical, the others disappearing (in the unitary gauge) to give mass to all the gauge bosons via the Higgs mechanism.', '0710.3057-1-19-2': 'After the cutting, we only get [MATH] scalar degrees of freedom to start with, so that no one survives the Higgsing of the gauge bosons.', '0710.3057-1-20-0': 'It will be useful for further considerations to look at the form of the gauge boson mass matrix, which can be obtained by putting [MATH] in eq. ([REF]).', '0710.3057-1-20-1': 'We find [EQUATION] with [EQUATION]', '0710.3057-1-20-2': 'Thanks to the condition [MATH] and to the reflection symmetry, the matrix will be block diagonal with two degenerate blocks.', '0710.3057-1-20-3': 'Each block can be independently diagonalized through an orthogonal transformation [MATH].', '0710.3057-1-20-4': 'By calling [MATH], [MATH] the mass eigenstates, and [MATH] the squared mass eigenvalues, we have [EQUATION] and [EQUATION] and an analogous result holds for [MATH].', '0710.3057-1-20-5': 'We will assume [MATH], otherwise the model describes an unphysical situation.', '0710.3057-1-21-0': '# Calculation of the effective Lagrangian', '0710.3057-1-22-0': 'We will now switch on the electroweak interactions by gauging the [MATH] subgroup of the global [MATH].', '0710.3057-1-22-1': 'For simplicity, we will consider only standard fermion couplings; in this way, there will be no extra contribution to [MATH] from the fermions.', '0710.3057-1-22-2': 'Then, considering the limit of heavy mass for the extra gauge bosons, we will integrate them out in order to obtain an effective description in terms of the electroweak and the fermion fields only.', '0710.3057-1-23-0': 'Note that just promoting part of the global symmetry to a gauge symmetry is not enough to describe realistic [MATH] and [MATH] bosons: we also need to provide suitable mass terms for three out of four of the newly added gauge fields.', '0710.3057-1-23-1': 'In the model as it is, however, there is not any scalar degree of freedom left to trigger a Higgs mechanism, as we have seen.', '0710.3057-1-23-2': 'A natural way out is to add to the Lagrangian an additional term containing the chiral field [MATH], which obeys the transformation rule [MATH]: [EQUATION] with [EQUATION]', '0710.3057-1-23-3': 'The [MATH] field gives us the additional three degrees of freedom we need and provides a SM-like symmetry breaking term for the gauge bosons [MATH] and [MATH] associated to [MATH].', '0710.3057-1-24-0': 'Summing up, the Lagrangian of the bosonic sector of the generalized D-BESS (GD-BESS) model is [EQUATION] where: [EQUATION] and the covariant derivatives of [MATH] and [MATH] are modified as follows: [EQUATION] due to the gauging of [MATH].', '0710.3057-1-25-0': 'The fermion interactions will be given by SM-like terms: [EQUATION] where [MATH] is a generic fermion doublet, and [MATH], [MATH] are the barion and lepton numbers respectively.', '0710.3057-1-25-1': 'In this way, the new gauge bosons are coupled to the fermions only through their mixing with the SM ones.', '0710.3057-1-26-0': 'By expanding eq. ([REF]) in the unitary gauge [MATH], and separating the kinetic term contribution from that of the terms containing the link coupling constants, we get [EQUATION] where we have made the identifications: [EQUATION] and defined: [EQUATION]', '0710.3057-1-26-1': 'The model field content is summarized in Fig. [REF].', '0710.3057-1-26-2': 'For [MATH] the model reduces to the D-BESS model [CITATION].', '0710.3057-1-27-0': 'file=figure_5old.', '0710.3057-1-27-1': 'eps,width= Moose diagram for GD-BESS; it consists of a linear moose with the central link cut and a nonlocal connection (the [MATH] field) between the two end points of the moose.', '0710.3057-1-27-2': '[MATH] is the global symmetry group after the gauging of the electroweak subgroup.', '0710.3057-1-28-0': 'From the Lagrangian ([REF]), we can derive the classical equations of motion for the [MATH] fields: [EQUATION] where again we have identified [EQUATION]', '0710.3057-1-28-1': 'If this model is to be consistent with the existing experimental data, the masses of the [MATH] fields must be significantly larger than those of the SM gauge bosons.', '0710.3057-1-28-2': "Let's call [MATH] the common mass scale of the heavy gauge bosons.", '0710.3057-1-28-3': 'Here we will be concerned only about the low energy predictions of the model, in which the new particles are not directly produced, but rather manifest themselves only by small modifications of the SM gauge boson propagators.', '0710.3057-1-28-4': 'So we will use an effective Lagrangian approach and work in the limit [MATH], where [MATH] represents the typical momentum scale of the processes we wish our effective theory to describe.', '0710.3057-1-28-5': 'Since the mass spectrum cannot be determined analytically in the general case, we do not have an exact expression for [MATH] but we can give an estimate of it by looking at the mass matrix ([REF]).', '0710.3057-1-28-6': 'We see that all the terms in ([REF]) are a sum of contributions which are proportional to [MATH] for some [MATH]; so we can assume that the typical mass scale will be of order [MATH] (we will get a more explicit estimate for the mass scale in the following).', '0710.3057-1-28-7': 'This assumption can actually be checked in the simplest case [MATH], as shown by the direct analysis made in ref. [CITATION].', '0710.3057-1-28-8': 'In this case, we have four massive eigenstates; their masses are, to lowest order: [MATH], [MATH] for the two lightest eigenstates, which can be identified with the SM gauge bosons, and [MATH] for the two heaviest states, which are degenerate.', '0710.3057-1-29-0': 'As a consequence, the limit we will study is [EQUATION]', '0710.3057-1-29-1': 'If we now rewrite eq. ([REF]) as [EQUATION] we can see that the quantities on the left-hand side are of higher order with respect to those on the right-hand side.', '0710.3057-1-29-2': 'Keeping only leading order terms, the equations reduce to [EQUATION] which imply [EQUATION] and [EQUATION]', '0710.3057-1-29-3': 'Substituting this leading order expressions in the unitary gauge Lagrangian given in eqs. ([REF]),([REF]), limiting us for the moment to the bilinear terms, we get: [EQUATION] where [EQUATION] and we have introduced the charged gauge fields [MATH].', '0710.3057-1-29-4': 'This expression exactly reproduces the SM electroweak gauge Lagrangian, provided we rescale the fields [MATH], [MATH]: [MATH], [MATH], and identify [EQUATION] with [EQUATION] in the limit [MATH].', '0710.3057-1-29-5': 'This means that any deviation from the SM at low energy will be suppressed at least by a factor [MATH].', '0710.3057-1-30-0': 'We can now get the next-to-leading order expression for the [MATH] iteratively, by substituting the leading order solutions ([REF])-([REF]) in the left-hand side of eq. ([REF]).', '0710.3057-1-30-1': 'We get', '0710.3057-1-31-0': 'g_i A_^i = g W_- c_i K_, i=1, , N;', '0710.3057-1-32-0': "g_i A_^i = g' Y_- c_i H_, i = N+1, , 2N;", '0710.3057-1-33-0': 'where we have introduced: [EQUATION]', '0710.3057-1-33-1': 'Notice that the [MATH] are positive definite and of order [MATH], and the reflection symmetry implies [MATH].', '0710.3057-1-34-0': 'Let us make the substitution for the [MATH].', '0710.3057-1-34-1': 'Limiting us to the quadratic part of the Lagrangian and using eqs. ([REF])-([REF]), we get: [EQUATION] where', '0710.3057-1-35-0': '1M^2= CG^2, C _i = 1^N c_ig_i^2 _i = N+1^2N c_ig_i^2.', '0710.3057-1-36-0': '[MATH] can be used as an explicit estimate for the scale [MATH], and recalling the definition of the [MATH] in ([REF]), we can see that it is indeed of order [MATH].', '0710.3057-1-37-0': '# Effective gauge boson correlation functions and [MATH] parameters', '0710.3057-1-38-0': 'From eq. ([REF]), it is straightforward to calculate the correlators for the fields [MATH] and [MATH].', '0710.3057-1-38-1': 'Up to the fourth power of the momentum, we get [EQUATION]', '0710.3057-1-38-2': 'It is immediate to verify that, as it should, [EQUATION] where the derivatives of the [MATH] are taken with respect to [MATH].', '0710.3057-1-39-0': 'Following ref. [CITATION], one can consider seven form factors, encoding the corrections of new physics to the electroweak precision observables: [EQUATION]', '0710.3057-1-39-1': 'Notice that the analysis of ref. [CITATION] only applies to "universal" theories; the GD-BESS model belongs to this class since the couplings with the fermions are of the standard form.', '0710.3057-1-39-2': 'In our model, it follows from the equality of [MATH] and [MATH] and the expression for [MATH] in eq. ([REF]) that [MATH]; so we have only two non vanishing form factors, namely [MATH] and [MATH]: [EQUATION]', '0710.3057-1-39-3': 'We can now compare the GD-BESS model predictions to experimental results.', '0710.3057-1-39-4': 'For this purpose, it is convenient to consider the [MATH] parameters, since they are better constrained by the data and more widely used in the literature.', '0710.3057-1-39-5': 'From the definition of the [MATH] in terms of the form factors [CITATION], we get, as contributions from new physics, [EQUATION] where [MATH].', '0710.3057-1-39-6': 'For the GD-BESS model we find: [EQUATION] with [MATH] given by [EQUATION]', '0710.3057-1-39-7': 'As we can see, after the gauging of the electroweak interactions, the new physics contribution to the [MATH] parameters is no longer equal to zero, but the leading non vanishing order of the correction is [MATH].', '0710.3057-1-39-8': 'This contribution can be understood as follows: the weak interactions explicitly break the custodial [MATH], which protects [MATH], down to the standard [MATH].', '0710.3057-1-39-9': 'The case with no weak interactions can be re-obtained in the limit [MATH], which represents a zeroth-order approximation of the model.', '0710.3057-1-39-10': 'More generally, all of the SM electroweak sector will be modified by [MATH] contributions due to the new physics.', '0710.3057-1-39-11': 'As an explicit example of this, in the following Section [REF] we will calculate the effective contribution to the trilinear gauge boson couplings.', '0710.3057-1-40-0': 'The new physics contribution to the [MATH] parameters can be tested against experimental data.', '0710.3057-1-40-1': 'In Figure [REF] we show a [MATH] contour plot in the plane [MATH] at [MATH] C.L.; the contour is obtained by considering the following experimental values for the [MATH] parameters: [EQUATION] and adding to the present model contributions the SM values: [EQUATION] given for [MATH] GeV and assuming an effective [MATH] GeV (experimental data are taken from [CITATION] for the [MATH] parameters and from the Tevatron EWWG web site for the top mass, while the SM radiative corrections are obtained as a linear interpolation from the values listed in [CITATION]).', '0710.3057-1-40-2': 'Notice that a relatively low scale [MATH] is still allowed by present data.', '0710.3057-1-40-3': 'This is due to the double suppression factor present in [MATH], eq. ([REF]).', '0710.3057-1-40-4': 'Notice also that the GD-BESS model in the limit [MATH] reproduces the SM to all orders in [MATH] (decoupling).', '0710.3057-1-41-0': 'file=chi2M.eps,width=0.6 [MATH] C.L. allowed region (the darker one), in the parameter space ([MATH]) by comparison of GD-BESS model predictions to electroweak precision parameters [MATH], [MATH] and [MATH].', '0710.3057-1-41-1': 'Predictions include radiative corrections as in the SM with [MATH] GeV and [MATH] GeV.', '0710.3057-1-42-0': 'The result can be made much more explicit in the simplest case [MATH] e [MATH].', '0710.3057-1-42-1': 'Remembering that [EQUATION] we have, in this case: [EQUATION] and, from [MATH], we immediately derive the [MATH] parameters from eq. ([REF]).', '0710.3057-1-42-2': 'We can verify that with the substitutions [MATH] and [MATH], with [MATH], eqs. ([REF]), ([REF]) coincide with the D-BESS result [CITATION].', '0710.3057-1-42-3': 'Indeed, in this case we find that [EQUATION] since [MATH].', '0710.3057-1-43-0': '# Trilinear couplings to the next-to-leading order', '0710.3057-1-44-0': 'We will now calculate, still at next-to-leading order in the weak interactions, the contributions of GD-BESS model to the SM trilinear couplings.', '0710.3057-1-44-1': 'These can be read out of the effective trilinear Lagrangian, which is again obtained by substituting eqs. ([REF]) and ([REF]) in eqs. ([REF]), ([REF]).', '0710.3057-1-44-2': 'We get [EQUATION] where [MATH] operate by convention just on the fields which bear the same index.', '0710.3057-1-45-0': 'In order to make explicit the new physics anomalous contributions to the trilinear couplings, we will rewrite eq. ([REF]) in terms of the mass eigenstates.', '0710.3057-1-45-1': 'First of all, we introduce [MATH] and [MATH] fields from [MATH] and [MATH] in the usual way: [EQUATION]', '0710.3057-1-45-2': 'Substituting in eq. ([REF]) we get [EQUATION] where [EQUATION]', '0710.3057-1-45-3': 'We then introduce the field rescaling:', '0710.3057-1-46-0': 'W_^ = (1+ z_w2(M^2 - M_W^2M^2) )W_^,', '0710.3057-1-47-0': 'Z_= (1 + z_z2(M^2 - M_Z^2M^2) )Z_,', '0710.3057-1-48-0': 'A_= (1 + z_2M^2 )A_+ z_z M^2 Z_, which allows us to get rid of the anomalous "[MATH]" terms in the quadratic part of the Lagrangian.', '0710.3057-1-48-1': 'We then obtain [EQUATION] where: [EQUATION]', '0710.3057-1-48-2': 'This is just the SM electroweak gauge boson bilinear Lagrangian; however, the rescaling ([REF]) of the fields will affect both the couplings with fermions and the trilinear bosonic couplings.', '0710.3057-1-48-3': "Let's shift to the [MATH] basis in eq. ([REF]), then rescale the fields according to ([REF]).", '0710.3057-1-48-4': 'We get: [EQUATION] where again we use the convention that [MATH] does only operate on [MATH] and [MATH].', '0710.3057-1-49-0': 'We see that the photon-fermion interaction at zero momentum correctly predicts [MATH] as the physical value of the electric charge.', '0710.3057-1-49-1': 'The Fermi constant [MATH] can be measured from the [MATH] decay, still at zero momentum.', '0710.3057-1-49-2': 'We have: [EQUATION] where we have substituted the physical masses [MATH] and [MATH] to [MATH] and [MATH] since they only differ by terms of [MATH], which are negligible in a term which is already of the same order.', '0710.3057-1-49-3': 'From eq. ([REF]) we can define the effective Weinberg angle (see [CITATION]): [EQUATION] that is [EQUATION]', '0710.3057-1-49-4': 'Notice that the [MATH] parameters, which we obtained from the correlators ([REF]), can be also derived as in [CITATION] by using the rescaling of the fields given in ([REF]) and by evaluating its effects on the [MATH], [MATH] and [MATH] parameters [CITATION].', '0710.3057-1-49-5': 'The results precisely agree with those obtained in Section [REF].', '0710.3057-1-50-0': 'Now, to get the corrections to the trilinear gauge boson couplings, it is sufficient to substitute eq. ([REF]) in eq. ([REF]).', '0710.3057-1-50-1': 'The general expression we get is quite long, so we will not report it.', '0710.3057-1-50-2': 'We will instead specialize to the study of a particular physical process, which will allow us to simplify eq. ([REF]) slightly, in order to get a more readable result.', '0710.3057-1-50-3': 'The process we will consider as an example is [MATH] scattering; in this case, the [MATH] are on-shell, so that we have [EQUATION] thanks to the Ward identity.', '0710.3057-1-50-4': 'Limiting the study to tree level, we can either have a virtual [MATH] or a virtual [MATH] as an intermediate state (besides the neutrino exchange which is not relevant to the study of the trilinear gauge couplings).', '0710.3057-1-50-5': 'The 4-divergence of the virtual [MATH] also vanishes due to the Ward identity; while the virtual [MATH] has an approximately vanishing transverse contribution thanks to the Dirac equation, since it is coupled to external fermions of negligible mass (compared to the center of mass energy which is of order [MATH]).', '0710.3057-1-50-6': 'So we can take: [EQUATION]', '0710.3057-1-50-7': 'As a consequence all the divergence-proportional terms in the effective Lagrangian (the last three lines in eq. ([REF])) can be safely dropped out.', '0710.3057-1-50-8': 'In this way, taking into account eq. ([REF]), we get the following expression for the trilinear gauge boson couplings, relevant for the [MATH] process: [EQUATION]', '0710.3057-1-50-9': 'We see that the tensor structure of the anomalous terms is identical to that of the SM, while the coefficients of the various operators contain derivative terms.', '0710.3057-1-50-10': 'Due to the presence of these nontrivial form factors and to the fact that the fermion-gauge boson couplings are also modified, as shown in eq. ([REF]), the comparison of the predictions of eq. ([REF]) to the experimental data is not direct, but rather requires a full calculation of the [MATH] cross-section in the GD-BESS model.', '0710.3057-1-50-11': 'We have not yet put this calculation through; it represents one of the future developments of this work.', '0710.3057-1-51-0': '# Conclusions', '0710.3057-1-52-0': 'We have considered a linear moose model based on the extended gauge symmetry [MATH].', '0710.3057-1-52-1': 'The model has the central link missing and a left-right symmetry along the moose.', '0710.3057-1-52-2': 'As a consequence of the missing link the [MATH] parameter is zero at the leading order in [MATH], where [MATH] is the mass scale of the new resonances.', '0710.3057-1-52-3': 'This result can also be understood in the following way: since the model describes [MATH] pairs of new gauge boson triplets degenerate in mass, the [MATH] contribution of the vector resonances is canceled by the axial vector one.', '0710.3057-1-53-0': 'We have computed the low energy effective Lagrangian by eliminating the internal moose gauge fields and analyzed the electroweak precision parameters and the trilinear anomalous couplings.', '0710.3057-1-53-1': 'Since these parameters turn out to be of order [MATH] and since the new effective coupling [MATH] could well be much larger than [MATH], the possibility of a low scale [MATH] is left open.', '0710.3057-1-54-0': 'It would be interesting, among the possible future developments, to study the unitarity limits of the model.', '0710.3057-1-54-1': 'However, we expect these to be of the same order of the Higgsless SM ones; in fact the new bosons only have a small coupling to those mediating the electroweak interactions, so that we do not expect to be able to compensate the amplitudes of longitudinal electroweak gauge boson scattering which are responsible of unitarity violation in the SM without a Higgs boson.', '0710.3057-1-55-0': 'This model is a direct extension to [MATH] sites of the previously proposed D-BESS model [CITATION].'}

{'0710.3057-2-0-0': 'The possibility of a strongly interacting electroweak symmetry breaking sector, as opposed to the weakly interacting light Higgs of the Standard Model, is not yet ruled out by experiments.', '0710.3057-2-0-1': 'In this paper we make an extensive study of a deconstructed model (or "moose" model) providing an effective description of such a strong symmetry breaking sector, and show its compatibility with experimental data for a wide portion of the model parameter space.', '0710.3057-2-0-2': 'The model is a direct generalization of the previously proposed D-BESS model.', '0710.3057-2-1-0': '# Introduction', '0710.3057-2-2-0': 'Among the problems still left open by the Standard Model (SM) of particle physics, the understanding of the exact mechanism that leads to the breaking of the electroweak symmetry at low energies is of particular importance.', '0710.3057-2-2-1': 'Besides the SM basic Higgs mechanism, still to be verified by experiments, possible alternative solutions to the problem are offered by extensions of the old technicolor (TC) theories, where the Higgs boson is realized as a composite state of strongly interacting fermions.', '0710.3057-2-3-0': 'These theories have recently received a renewed attention starting from higher dimensional Lagrangians; effective chiral Lagrangians in four dimensions containing new resonance states can be obtained by the deconstruction technique [CITATION] or as holographic versions of 5-dimensional (5D) theories in warped background [CITATION].', '0710.3057-2-4-0': 'Models have been proposed, working in the framework suggested by the AdS/CFT correspondence, which assume a [MATH] gauge group in the 5D bulk, [CITATION], or also simpler with a [MATH] in the bulk [CITATION].', '0710.3057-2-4-1': 'One of the main challenges of these models is the value of the [MATH] parameter [CITATION] or the related [MATH] [CITATION].', '0710.3057-2-4-2': 'Indeed, the experimental value of [MATH] is of the order of [MATH] [CITATION], whereas the value naturally expected in TC theories is an order of magnitude bigger.', '0710.3057-2-5-0': 'A delocalization of the fermionic fields into the bulk as in [CITATION], realized in the deconstructed version by allowing standard fermions to have direct couplings to all the moose gauge fields as in [CITATION], leads to direct contributions to the electroweak parameters that can correct the bad behavior of the [MATH] parameter.', '0710.3057-2-5-1': 'The fine tuning which cancels out the oblique and direct contributions to [MATH] in each bulk point, that is from each internal moose gauge group, corresponds to the so called ideal delocalisation of fermions, [CITATION].', '0710.3057-2-5-2': 'Other solutions to get a suppressed contribution to [MATH] have been investigated, like the one suggested by holographic QCD, assuming that different five dimensional metrics are felt by the axial and vector states [CITATION].', '0710.3057-2-5-3': 'However it has been shown recently that the backgrounds that allow a negative oblique contribution to [MATH] are pathological, since they require unphysical Higgs profile or higher dimensional operators [CITATION].', '0710.3057-2-6-0': 'An alternative solution to the [MATH] problem was proposed in [CITATION] (see also [CITATION]).', '0710.3057-2-6-1': 'The solution was realized in terms of an effective TC theory of non linear [MATH]-model scalars and massive gauge fields.', '0710.3057-2-6-2': 'The model is a four site model with three sigma fields.', '0710.3057-2-6-3': 'The physical spectrum consists of three massless scalar fields (the Goldstone bosons giving masses to the gauge vector particles) and two triplets of massive vector fields degenerate in mass and couplings.', '0710.3057-2-6-4': 'This model, named degenerate BESS (Breaking Electroweak Symmetry Strongly) model (D-BESS), has an enhanced custodial symmetry such as to allow [MATH] at the lowest order in the electroweak interactions.', '0710.3057-2-6-5': 'This idea has been also recently used in phenomenological analysis of low scale technicolor theories with vector and axial vector resonances very close in mass [CITATION].', '0710.3057-2-7-0': 'A generalization of D-BESS was studied in [CITATION].', '0710.3057-2-7-1': 'This extended model is a moose model, with a replicated [MATH] gauge symmetry, that maintains the most useful feature of D-BESS, namely the custodial symmetry which guarantees the vanishing of [MATH].', '0710.3057-2-8-0': 'Two other quantities, [MATH] and [MATH] [CITATION] or equivalently [MATH] and [MATH] [CITATION], are customarily used to parameterize the electroweak precision observables.', '0710.3057-2-8-1': 'More recently, an alternative parametrization was proposed in terms of seven parameters [CITATION] which describe in a very general way the effects of so-called "universal" extensions of the SM, that is theories whose deviations from the SM are all contained in the vector boson self-energies.', '0710.3057-2-9-0': 'In the present paper, we wish to extend the calculation made in [CITATION] by deriving the seven parameters of ref. [CITATION], and from them the [MATH] parameters, to the next-to-leading order in [MATH], where [MATH] is the mass scale of the new bosons, and without any expansion in their gauge couplings.', '0710.3057-2-9-1': 'Also, we will calculate, at the same order, the trilinear gauge boson vertex anomalous contributions due to the new physics.', '0710.3057-2-10-0': 'In Section [REF] we review the notations and the main constitutive elements of the model.', '0710.3057-2-10-1': 'In Section [REF] we derive the low energy effective Lagrangian by eliminating the fields of the internal moose and show that the model decouples in the limit [MATH].', '0710.3057-2-10-2': 'In Section [REF] we compute the effective gauge boson correlation functions, and from them the parameters of ref. [CITATION] and the [MATH] parameters, to the next-to-leading order in [MATH]; we then derive bounds on the model parameter space from experimental data.', '0710.3057-2-10-3': 'In Section [REF] we obtain the effective trilinear gauge couplings to order [MATH].', '0710.3057-2-10-4': 'Finally, in Section [REF] we present our conclusions.', '0710.3057-2-11-0': '# A linear moose model for the electroweak symmetry breaking', '0710.3057-2-12-0': 'Our model is based on the idea of dimensional deconstruction [CITATION] and on the hidden gauge symmetry approach, historically applied both to strong interactions [CITATION] and to electroweak symmetry breaking [CITATION].', '0710.3057-2-13-0': 'Consider [MATH] non linear [MATH]-model scalar fields [MATH], [MATH] and [MATH] gauge groups, [MATH], [MATH] with global symmetry [MATH].', '0710.3057-2-13-1': 'Since we are interested in studying the electroweak symmetry breaking mechanism, we will assume [MATH], [MATH].', '0710.3057-2-13-2': 'The transformation properties of the fields are [EQUATION] with [MATH], [MATH]; [MATH], [MATH]; the Lagrangian is given by [EQUATION] where [MATH] are [MATH] free constants ("link" coupling constants).', '0710.3057-2-13-3': 'The covariant derivatives are defined as follows: [EQUATION] as implied by the transformation rules ([REF]), where [MATH] and [MATH] are the gauge fields and gauge coupling constants associated to the groups [MATH], [MATH].', '0710.3057-2-13-4': '[MATH] has the standard definition [EQUATION] with [EQUATION]', '0710.3057-2-13-5': 'Notice that one could introduce an additional field, [EQUATION] which transforms just like the usual chiral field of the Higgsless SM: [MATH].', '0710.3057-2-13-6': 'The field [MATH] is an invariant under the [MATH] transformations (which are then effectively "hidden" to [MATH]).', '0710.3057-2-14-0': 'As shown in [CITATION], in this model, due to the presence of a custodial [MATH] symmetry, we get no corrections to [MATH].', '0710.3057-2-14-1': 'Also, if we put one (and only one) of the link coupling constants [MATH] equal to zero, we effectively enlarge the global symmetry to [MATH], getting [MATH] too.', '0710.3057-2-14-2': 'We want to explore this particular case; also, for simplicity, we impose an extra left-right symmetry of the moose which identifies the two ends:', '0710.3057-2-15-0': 'f_i f_K+2-i,', '0710.3057-2-16-0': 'g_i g_K+1-i.', '0710.3057-2-17-0': 'The reflection symmetry, together with the condition that just one of the link coupling constants must vanish, implies that the number of moose links has to be odd (and hence the number of gauge fields has to be even), and that the vanishing link has to be the central one.', '0710.3057-2-17-1': 'So we will consider:', '0710.3057-2-18-0': 'K = 2N,', '0710.3057-2-19-0': 'f_N+1 = 0.', '0710.3057-2-20-0': 'It is instructive to count out the number of degrees of freedom.', '0710.3057-2-20-1': 'Before cutting the central link, we had [MATH] matrices of scalar fields, for [MATH] degrees of freedom, and [MATH] massless vector fields, for [MATH] degrees of freedom; of these, only 3 scalar fields are physical, the others disappearing (in the unitary gauge) to give mass to all the gauge bosons via the Higgs mechanism.', '0710.3057-2-20-2': 'After the cutting, we only get [MATH] scalar degrees of freedom to start with, so that no one survives the Higgsing of the gauge bosons.', '0710.3057-2-21-0': 'It will be useful for further considerations to look at the form of the gauge boson mass matrix, which can be obtained by putting [MATH] in eq. ([REF]).', '0710.3057-2-21-1': 'We find [EQUATION] with [EQUATION]', '0710.3057-2-21-2': 'Thanks to the condition [MATH] and to the reflection symmetry, the matrix [MATH] is block diagonal with two degenerate blocks.', '0710.3057-2-21-3': 'Each block can be independently diagonalized through an orthogonal transformation [MATH].', '0710.3057-2-21-4': 'By calling [MATH], [MATH] the mass eigenstates, and [MATH] the squared mass eigenvalues, we have [EQUATION] with [EQUATION] and an analogous result holds for [MATH].', '0710.3057-2-21-5': 'We will assume [MATH], otherwise the model describes an unphysical situation.', '0710.3057-2-22-0': '# Calculation of the effective Lagrangian', '0710.3057-2-23-0': 'We will now switch on the electroweak interactions by gauging the [MATH] subgroup of the global [MATH].', '0710.3057-2-23-1': 'We will include in the model only standard fermions coupled to [MATH].', '0710.3057-2-23-2': 'Then, considering the limit of heavy mass for the extra gauge bosons, we will integrate them out in order to obtain an effective description in terms of the electroweak and the fermion fields only.', '0710.3057-2-24-0': 'Note that just promoting part of the global symmetry to a gauge symmetry is not enough to describe realistic [MATH] and [MATH] bosons: we also need to provide suitable mass terms for three out of four of the newly added gauge fields.', '0710.3057-2-24-1': 'In the model as it is, however, there is not any scalar degree of freedom left to trigger a Higgs mechanism, as we have seen.', '0710.3057-2-24-2': 'A natural way out is to add to the Lagrangian an additional term containing the chiral field [MATH], which obeys the transformation rule [MATH]: [EQUATION] with [EQUATION]', '0710.3057-2-24-3': 'The [MATH] field gives us the additional three degrees of freedom we need and provides a SM-like symmetry breaking term for the gauge bosons [MATH] and [MATH] associated to [MATH].', '0710.3057-2-25-0': 'Summing up, the Lagrangian of the bosonic sector of the generalized D-BESS (GD-BESS) model is [EQUATION] where: [EQUATION] and the covariant derivatives of [MATH] and [MATH] are modified as follows: [EQUATION] due to the gauging of [MATH].', '0710.3057-2-26-0': 'The fermion interactions will be given by SM-like terms: [EQUATION] where [MATH] is a generic fermion doublet, and [MATH], [MATH] are the barion and lepton numbers respectively.', '0710.3057-2-26-1': 'In this way, the new gauge bosons are coupled to the fermions only through their mixing with the SM ones.', '0710.3057-2-27-0': 'By expanding eq. ([REF]) in the unitary gauge [MATH], and separating the kinetic term contribution from that of the terms containing the link coupling constants, we get [EQUATION] where we have made the identifications: [EQUATION] and defined: [EQUATION]', '0710.3057-2-27-1': 'The model field content is summarized in Fig. [REF].', '0710.3057-2-27-2': 'For [MATH] the model reduces to the D-BESS model [CITATION].', '0710.3057-2-28-0': 'From the Lagrangian ([REF]), we can derive the classical equations of motion for the [MATH] fields: [EQUATION] where again we have identified [EQUATION]', '0710.3057-2-28-1': 'If this model is to be consistent with the existing experimental data, the masses of the [MATH] fields must be significantly larger than those of the SM gauge bosons.', '0710.3057-2-28-2': "Let's call [MATH] the common mass scale of the heavy gauge bosons.", '0710.3057-2-28-3': 'Here we will be concerned only about the low energy predictions of the model, in which the new particles are not directly produced, but rather manifest themselves only by small modifications of the SM gauge boson propagators.', '0710.3057-2-28-4': 'So we will use an effective Lagrangian approach and work in the limit [MATH], where [MATH] represents the typical momentum scale of the processes we wish our effective theory to describe.', '0710.3057-2-28-5': 'Since the mass spectrum cannot be determined analytically in the general case, we do not have an exact expression for [MATH] but we can give an estimate of it by looking at the mass matrix ([REF]).', '0710.3057-2-28-6': 'We see that all the terms in ([REF]) are a sum of contributions which are proportional to [MATH] for some [MATH]; so we can assume that the typical mass scale will be of order [MATH] (we will get a more explicit estimate for the mass scale in the following).', '0710.3057-2-28-7': 'This assumption can actually be checked in the simplest case [MATH], as shown by the direct analysis made in ref. [CITATION].', '0710.3057-2-28-8': 'In this case, we have four massive eigenstates; their masses are, to lowest order: [MATH], [MATH] for the two lightest eigenstates, which can be identified with the SM gauge bosons, and [MATH] for the two heaviest states, which are degenerate.', '0710.3057-2-29-0': 'As a consequence, the limit we will study is [EQUATION]', '0710.3057-2-29-1': 'If we now rewrite eq. ([REF]) as [EQUATION] we can see that the quantities on the left-hand side are of higher order with respect to those on the right-hand side.', '0710.3057-2-29-2': 'Keeping only leading order terms, the equations reduce to [EQUATION] which imply [EQUATION] and [EQUATION]', '0710.3057-2-29-3': 'Substituting this leading order expressions in the unitary gauge Lagrangian given in eqs. ([REF]),([REF]), limiting us for the moment to the bilinear terms, we get: [EQUATION] where [EQUATION] and we have introduced the charged gauge fields [MATH].', '0710.3057-2-29-4': 'This expression exactly reproduces the SM electroweak gauge Lagrangian, provided we rescale the fields [MATH], [MATH]: [MATH], [MATH], and identify [EQUATION] with [EQUATION] in the limit [MATH].', '0710.3057-2-29-5': 'This means that any deviation from the SM at low energy will be suppressed at least by a factor [MATH].', '0710.3057-2-30-0': 'We can now get the next-to-leading order expression for the [MATH] iteratively, by substituting the leading order solutions ([REF])-([REF]) in the left-hand side of eq. ([REF]).', '0710.3057-2-30-1': 'We get', '0710.3057-2-31-0': 'g_i A_^i = g W_- c_i K_, i=1, , N;', '0710.3057-2-32-0': "g_i A_^i = g' Y_- c_i H_, i = N+1, , 2N;", '0710.3057-2-33-0': 'where we have introduced: [EQUATION]', '0710.3057-2-33-1': 'Notice that the [MATH] are positive definite and of order [MATH], and the reflection symmetry implies [MATH].', '0710.3057-2-34-0': 'Let us make the substitution for the [MATH].', '0710.3057-2-34-1': 'Limiting us to the quadratic part of the Lagrangian and using eqs. ([REF])-([REF]), we get: [EQUATION] where', '0710.3057-2-35-0': '1M^2= CG^2, C _i = 1^N c_ig_i^2 _i = N+1^2N c_ig_i^2.', '0710.3057-2-36-0': '[MATH] can be used as an explicit estimate for the scale [MATH] (from the definition of the [MATH] in ([REF]), we see that [MATH] is indeed of order [MATH]).', '0710.3057-2-37-0': '# Effective gauge boson correlation functions and [MATH] parameters', '0710.3057-2-38-0': 'From eq. ([REF]), it is straightforward to calculate the correlators for the fields [MATH] and [MATH].', '0710.3057-2-38-1': 'Up to the fourth power of the momentum, we get [EQUATION]', '0710.3057-2-38-2': 'It is immediate to verify that, as it should, [EQUATION] where the derivatives of the [MATH] are taken with respect to [MATH].', '0710.3057-2-39-0': 'Following ref. [CITATION], one can consider seven form factors, encoding the corrections of new physics to the electroweak precision observables: [EQUATION]', '0710.3057-2-39-1': 'Notice that the analysis of ref. [CITATION] only applies to "universal" theories; the GD-BESS model belongs to this class since the couplings with the fermions are of the standard form.', '0710.3057-2-39-2': 'In our model, from the equality of [MATH] and [MATH] and the expression for [MATH] in eq. ([REF]) it follows that [MATH]; so we have only two non vanishing form factors, namely [MATH] and [MATH]: [EQUATION]', '0710.3057-2-39-3': 'We can now compare the GD-BESS model predictions to experimental results.', '0710.3057-2-39-4': 'For this purpose, it is convenient to consider the [MATH] parameters, since they are better constrained by the data and more widely used in the literature.', '0710.3057-2-39-5': 'From the definition of the [MATH] in terms of the form factors [CITATION], we get, as contributions from new physics, [EQUATION] where [MATH].', '0710.3057-2-39-6': 'For the GD-BESS model we find: [EQUATION] with [MATH] given by [EQUATION]', '0710.3057-2-39-7': 'As we can see, after the gauging of the electroweak interactions, the new physics contribution to the [MATH] parameters is no longer equal to zero, but the leading non vanishing order of the correction is [MATH].', '0710.3057-2-39-8': 'This contribution can be understood as follows: the weak interactions explicitly break the custodial [MATH], which protects [MATH], down to the standard [MATH].', '0710.3057-2-39-9': 'The case with no weak interactions can be re-obtained in the limit [MATH], which represents a zeroth-order approximation of the model.', '0710.3057-2-39-10': 'More generally, all of the SM electroweak sector will be modified by [MATH] contributions due to the new physics.', '0710.3057-2-39-11': 'As an explicit example of this, in the following Section [REF] we will calculate the effective contribution to the trilinear gauge boson couplings.', '0710.3057-2-40-0': 'The new physics contribution to the [MATH] parameters can be tested against experimental data.', '0710.3057-2-40-1': 'In Figure [REF] we show a [MATH] contour plot in the plane [MATH] at [MATH] C.L.; the contour is obtained by considering the following experimental values for the [MATH] parameters: [EQUATION] and adding to the present model contributions the SM values: [EQUATION] given for [MATH] GeV and assuming an effective [MATH] GeV (experimental data are taken from [CITATION] for the [MATH] parameters and from the Tevatron EWWG web site for the top mass, while the SM radiative corrections are obtained as a linear interpolation from the values listed in [CITATION]).', '0710.3057-2-40-2': 'Notice that a relatively low scale [MATH] is still allowed by present data.', '0710.3057-2-40-3': 'This is due to the double suppression factor present in [MATH], eq. ([REF]).', '0710.3057-2-40-4': 'Notice also that the GD-BESS model in the limit [MATH] reproduces the SM to all orders in [MATH] (decoupling).', '0710.3057-2-41-0': 'The result can be made much more explicit in the simplest case [MATH] e [MATH].', '0710.3057-2-41-1': 'Recalling that [EQUATION] we have, in this case: [EQUATION] and, from [MATH], we immediately derive the [MATH] parameters from eq. ([REF]).', '0710.3057-2-41-2': 'We can verify that with the substitutions [MATH] and [MATH], with [MATH], eqs. ([REF]), ([REF]) coincide with the D-BESS result [CITATION].', '0710.3057-2-41-3': 'Indeed, in this case we find that [EQUATION] where [MATH] coincides with [MATH] for [MATH] and, as shown in [CITATION], represents the mass of the two degenerate new resonances.', '0710.3057-2-41-4': 'Therefore for [MATH] the limit shown in Fig. [REF] can be interpreted as a bound on the degenerate masses of the new gauge vectors.', '0710.3057-2-42-0': 'In general in order to get limitations from the electroweak precision data on the mass spectrum, one needs to perform the mass diagonalization which depends on the specific value of [MATH] and also on the particular choices of [MATH] and [MATH].', '0710.3057-2-42-1': 'However, for any [MATH], [MATH] gives the typical mass scale of the lowest resonance.', '0710.3057-2-42-2': 'For example, for [MATH], [MATH], using the result for the spectrum given in [CITATION], neglecting the electroweak interactions, the relation between the lightest charged resonance mass and [MATH] is [EQUATION]', '0710.3057-2-42-3': 'Let us conclude this Section with a comment on the partial wave unitarity violation.', '0710.3057-2-42-4': 'A special feature of GD-BESS model is that the unitarity bound is completely determined by the [MATH] term given in eq. ([REF]).', '0710.3057-2-42-5': "In fact one can verify explicilty that the scattering amplitudes for the longitudinal electroweak vector bosons (using the equivalence theorem) are equal to the ones obtained for the Higgsless SM (for [MATH] see [CITATION]) and that all the amplitudes for the longitudinal [MATH]'s can be always arranged to have a higher unitarity bound [CITATION].", '0710.3057-2-42-6': 'Therefore this model is expected to become strongly interacting around a scale [MATH] 1.7 TeV.', '0710.3057-2-42-7': 'One possible way to unitarize the GD-BESS model is to include also scalars associated to the [MATH] fields on each site and to the [MATH] field.', '0710.3057-2-42-8': 'In the simple case of [MATH] we have shown that the resulting theory is renormalizable (and unitary) and decoupling holds [CITATION].', '0710.3057-2-42-9': 'The generalization to generic [MATH] is under study.', '0710.3057-2-43-0': '# Trilinear couplings to the next-to-leading order', '0710.3057-2-44-0': 'We will now calculate, still to the next-to-leading order in the weak interactions, the contributions of GD-BESS model to the SM trilinear couplings.', '0710.3057-2-44-1': 'These can be read out of the effective trilinear Lagrangian, which is again obtained by substituting eqs. ([REF]) and ([REF]) in eqs. ([REF]), ([REF]).', '0710.3057-2-44-2': 'We get [EQUATION] where [MATH] operate only on the fields which bear the same index.', '0710.3057-2-45-0': 'In order to make explicit the new physics anomalous contributions to the trilinear couplings, we will rewrite eq. ([REF]) in terms of the mass eigenstates.', '0710.3057-2-45-1': 'First of all, we introduce [MATH] and [MATH] fields from [MATH] and [MATH] in the usual way: [EQUATION]', '0710.3057-2-45-2': 'Substituting in eq. ([REF]) we get [EQUATION] where [EQUATION]', '0710.3057-2-45-3': 'We then introduce the field rescaling:', '0710.3057-2-46-0': 'W_^ = (1+ z_w2(M^2 - M_W^2M^2) )W_^,', '0710.3057-2-47-0': 'Z_= (1 + z_z2(M^2 - M_Z^2M^2) )Z_,', '0710.3057-2-48-0': 'A_= (1 + z_2M^2 )A_+ z_z M^2 Z_, which allows us to get rid of the anomalous "[MATH]" terms in the quadratic part of the Lagrangian.', '0710.3057-2-48-1': 'We then obtain [EQUATION] where: [EQUATION]', '0710.3057-2-48-2': 'This is just the SM electroweak gauge boson bilinear Lagrangian; however, the rescaling ([REF]) of the fields will affect both the couplings with fermions and the trilinear bosonic couplings.', '0710.3057-2-48-3': "Let's shift to the [MATH] basis in eq. ([REF]), then rescale the fields according to ([REF]).", '0710.3057-2-48-4': 'We get: [EQUATION] where again we use the convention that [MATH] does only operate on [MATH] and [MATH].', '0710.3057-2-49-0': 'We see that the photon-fermion interaction at zero momentum correctly predicts [MATH] as the physical value of the electric charge.', '0710.3057-2-49-1': 'The Fermi constant [MATH] can be measured from the [MATH] decay, still at zero momentum.', '0710.3057-2-49-2': 'We have: [EQUATION] where we have substituted the physical masses [MATH] and [MATH] to [MATH] and [MATH] since they only differ by terms of [MATH], which are negligible in a term which is already of the same order.', '0710.3057-2-49-3': 'From eq. ([REF]) we can define the effective Weinberg angle (see [CITATION]): [EQUATION] that is [EQUATION]', '0710.3057-2-49-4': 'Notice that the [MATH] parameters, which we obtained from the correlators ([REF]), can be also derived as in [CITATION] by using the rescaling of the fields given in ([REF]) and by evaluating the [MATH], [MATH] and [MATH] parameters [CITATION].', '0710.3057-2-49-5': 'The results precisely agree with those obtained in Section [REF].', '0710.3057-2-50-0': 'Now, to get the corrections to the trilinear gauge boson couplings, it is sufficient to substitute eq. ([REF]) in eq. ([REF]).', '0710.3057-2-50-1': 'The general expression we get is quite long, so we will not report it.', '0710.3057-2-50-2': 'We will instead specialize to the study of a particular physical process, which will allow us to simplify eq. ([REF]) slightly, in order to get a more readable result.', '0710.3057-2-50-3': 'The process we will consider as an example is [MATH] scattering; in this case, the [MATH] are on-shell, so that we have [EQUATION] thanks to the Ward identity.', '0710.3057-2-50-4': 'Limiting the study to tree level, we can either have a virtual [MATH] or a virtual [MATH] as an intermediate state (besides the neutrino exchange which is not relevant to the study of the trilinear gauge couplings).', '0710.3057-2-50-5': 'The 4-divergence of the virtual [MATH] also vanishes due to the Ward identity; while the virtual [MATH] has an approximately vanishing transverse contribution thanks to the Dirac equation, since it is coupled to external fermions of negligible mass (compared to the center of mass energy which is of order [MATH]).', '0710.3057-2-50-6': 'So we can take: [EQUATION]', '0710.3057-2-50-7': 'As a consequence all the divergence-proportional terms in the effective Lagrangian (the last three lines in eq. ([REF])) can be safely dropped out.', '0710.3057-2-50-8': 'In this way, taking into account eq. ([REF]), we get the following expression for the trilinear gauge boson couplings, relevant for the [MATH] process: [EQUATION]', '0710.3057-2-50-9': 'We see that the tensor structure of the anomalous terms is identical to that of the SM, while the coefficients of the various operators contain derivative terms.', '0710.3057-2-50-10': 'Due to the presence of these nontrivial form factors and to the fact that the fermion-gauge boson couplings are also modified, as shown in eq. ([REF]), the comparison of the predictions of eq. ([REF]) to the experimental data is not direct, but requires a full calculation of the [MATH] cross-section in the GD-BESS model.', '0710.3057-2-50-11': 'However the present experimental bounds from LEP2 on the anomalous trilinear couplings [CITATION] have errors of the order of a few percent.', '0710.3057-2-50-12': 'Since the determination of the new physics parameters entering in eq. ([REF]) is at the level of a few permil from LEP/Tevatron, it is clear that a higher precision will be necessary in order to achieve the same accuracy.', '0710.3057-2-50-13': 'We have nevertheless checked that, taking for example the expression of [MATH] extracted from eq. ([REF]), and comparing with the present experimental value for [MATH], we get bounds on the plane [MATH] which are not relevant with respect to the ones shown in Fig. [REF].', '0710.3057-2-50-14': 'We have also checked that, in order to have comparable bounds, one would need an extimation of [MATH] at the permil level.', '0710.3057-2-51-0': '# Conclusions', '0710.3057-2-52-0': 'We have considered a linear moose model based on the extended gauge symmetry [MATH].', '0710.3057-2-52-1': 'The model has the central link missing and a left-right symmetry along the moose.', '0710.3057-2-52-2': 'As a consequence of the missing link the [MATH] parameter is zero at the leading order in [MATH], where [MATH] is the mass scale of the new resonances.', '0710.3057-2-52-3': 'This result can also be understood in the following way: since the model describes [MATH] pairs of new gauge boson triplets degenerate in mass, the [MATH] contribution of the vector resonances is canceled by the axial vector one.', '0710.3057-2-53-0': 'We have computed the low energy effective Lagrangian by eliminating the internal moose gauge fields and extracted the electroweak precision parameters and the trilinear anomalous couplings.', '0710.3057-2-53-1': 'Since these parameters turn out to be of order [MATH] and since the new effective coupling [MATH] could well be much larger than [MATH], the possibility of a low scale [MATH] is left open.', '0710.3057-2-53-2': 'We expect the GD-BESS model in the present formulation to become a strongly interacting theory at energies of the order of 1.7 TeV independently of the values of the model parameters as a consequence of the perturbative unitarity violation, so it is interesting, among the future developments, to study how to unitarize it.', '0710.3057-2-53-3': 'We are currently investigating the possibility to include scalars associated to the non linear [MATH]-model fields.', '0710.3057-2-53-4': 'However, in this unitarized extension, we expect corrections to the [MATH] parameters of the same order of magnitude as the ones evaluated here so not spoiling our overall conclusions.'}

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/0710.3057

gr-qc-0211074

{'gr-qc-0211074-1-0-0': 'We contrast features of simple varying speed of light (VSL) cosmologies with inflationary universe models.', 'gr-qc-0211074-1-0-1': 'We present new features of VSL cosmologies and show that they cannot explain the cosmological isotropy problem.', 'gr-qc-0211074-1-0-2': 'We find that if [MATH] falls fast enough to solve the flatness and horizon problems then the quantum wavelengths of massive particle states and the radii of primordial black holes will grow to exceed the scale of the particle horizon.', 'gr-qc-0211074-1-0-3': 'This may provide VSL cosmologies with a self-reproduction property.The constraint of entropy increase is also discussed.', 'gr-qc-0211074-1-1-0': '# Introduction', 'gr-qc-0211074-1-2-0': 'We have explored a naive class of models in which the speed of light varies in time [CITATION], [CITATION].', 'gr-qc-0211074-1-2-1': "These can be reformulated more generally and recast as theories in which other dimensional 'constants' carry the operationally meaningful space or time variation of a dimensionless constant - in this case the fine structure 'constant', [MATH] [CITATION].", 'gr-qc-0211074-1-2-2': 'Theories in which this variation is carried explicitly by the electron change have been extensively investigated recently [CITATION].', 'gr-qc-0211074-1-2-3': "Such theories are a 'small' perturbation of standard physics in the sense that even though the speed of light falls there still exists a maximum signal propagation velocity that is achieved by gravitational waves.", 'gr-qc-0211074-1-2-4': "The motivation for a careful consideration of these cosmological models was their possible viability as alternatives to inflation as explanations for a number of unusual properties of the universe and the consistency of quasar absorption spectra with a variation in the value of the fine structure 'constant at [MATH] [CITATION].", 'gr-qc-0211074-1-2-5': 'In particular, it was shown by Moffat [CITATION], Albrecht and Magueijo [CITATION] and by Barrow [CITATION] that during the very early universe a finite period of time during which the speed of light falls at an appropriate rate can lead to a solution of the flatness, horizon, and monopole problems.', 'gr-qc-0211074-1-2-6': 'However, unlike inflation it can also provide a solution for the cosmological constant problem.', 'gr-qc-0211074-1-2-7': 'Barrow and Magueijo also showed that for some ranges of variation in the speed of light these theories can also naturally create long-lived universe in which the dynamics are almost flat or in which the cosmological constant is almost zero.', 'gr-qc-0211074-1-2-8': 'These problems of explaining a universe in which the present value of the matter density parameter, [MATH], or the cosmological constant energy density parameter, [MATH], are [MATH] we called the quasi-flatness and quasi-lambda problems.', 'gr-qc-0211074-1-2-9': 'Inflation does not seem to offer a natural explanation of a universe which is almost flat or has a dynamically significant cosmological constant today, as current observations imply.', 'gr-qc-0211074-1-3-0': 'In this paper, we want to discuss some other cosmological problems in the light of varying speed of light (VSL) theories of the simple sort discussed by Albrecht, Magueijo, and Barrow (AMB).', 'gr-qc-0211074-1-3-1': 'Although these problems are articulated in the context of this simple AMB theory we believe that are more general challenges to any theory which manifests itself in a particular coordinate system as a VSL theory.', 'gr-qc-0211074-1-3-2': 'In particular, we shall show that in general AMB theories cannot solve the isotropy problem and are unable to generate a spectrum of almost constant curvature fluctuations in the standard way.', 'gr-qc-0211074-1-3-3': 'We shall also show that they have a number of very unusual consequences for the quantum states of massive particles and for primordial black holes after they enter the particle horizon in the early universe.', 'gr-qc-0211074-1-3-4': 'It is not clear whether these features are a reductio ad absurdum for these VSL theories or whether they provide an exotic counterpart to the self-reproducing property of inflationary universe models that provides the basis for eternal inflationary universes.', 'gr-qc-0211074-1-4-0': '# Naive VSL Theories', 'gr-qc-0211074-1-5-0': '## Solving the flatness, horizon and lambda problems', 'gr-qc-0211074-1-6-0': 'The simple VSL model introduced by Albrecht and Magueijo and solved by Barrow is based upon the simplest possible premises.', 'gr-qc-0211074-1-6-1': "We assume that the Friedmann cosmology in the presence of time-dependent speed of light [MATH] is described by the Friedmann and Raychaudhuri equations (although the original formulation allowed the Newtonian gravitation 'constant' [MATH] to vary also, we shall ignore this possibility as it makes no contribution to the essential conclusions): [EQUATION] where [MATH] is the expansion scale factor of the Friedmann metric, [MATH] is the fluid pressure, [MATH] is the fluid density, [MATH] is the curvature parameter, [MATH] is the cosmological constant, and all derivatives are with respect to [MATH], the comoving proper time.", 'gr-qc-0211074-1-6-2': 'From these equations we can derive the modified matter conservation which incorporates the effects of [MATH] [EQUATION]', 'gr-qc-0211074-1-6-3': 'Of course, in general relativity, where [MATH] the right-hand side of ( [REF]) is zero.', 'gr-qc-0211074-1-6-4': 'Notice that it also vanishes in a VSL theory when [MATH].', 'gr-qc-0211074-1-6-5': 'For concreteness, now consider the universe to contain a perfect fluid with equation of state [EQUATION] where [MATH] is constant and we shall assume that the speed of light varies as some power of the expansion scale factor: [EQUATION] where [MATH] and [MATH] are constants.', 'gr-qc-0211074-1-6-6': 'These assumptions are sufficient to solve the equations completely.', 'gr-qc-0211074-1-6-7': 'Integrating ([REF]) with [MATH], we find [EQUATION]', 'gr-qc-0211074-1-6-8': 'Now by inspection of ([REF]) we see that at large [MATH] the curvature term falls off faster than the [MATH]term whenever', 'gr-qc-0211074-1-7-0': '[EQUATION]', 'gr-qc-0211074-1-7-1': 'Thus the flatness problem can be solved in a radiation-dominated early universe by an interval of VSL evolution if', 'gr-qc-0211074-1-8-0': '[EQUATION]', 'gr-qc-0211074-1-8-1': 'It is easy to show that, as in the case of inflation, this is the same condition required to solve the monopole and horizon problems.', 'gr-qc-0211074-1-8-2': 'However, the nature of the solution differs from that provided by a period of inflation.', 'gr-qc-0211074-1-8-3': 'Inflation requires the dynamics to be dominated for a finite period by an unusual fluid ([MATH]), which is therefore gravitationally repulsive, so that the expansion will accelerate ([MATH]) and the [MATH]term will grow to dominate the [MATH] term in ([REF]) as [MATH] grows large.', 'gr-qc-0211074-1-8-4': 'By contrast, in the VSL model, no special fluid is required and in a radiation universe the flatness problem is solved because the [MATH] term falls off faster than the [MATH] term because of the fall in [MATH] as [MATH] increases.', 'gr-qc-0211074-1-9-0': 'This behaviour also permits a solution of the classical cosmological constant problem in just the same way if', 'gr-qc-0211074-1-10-0': '[EQUATION]', 'gr-qc-0211074-1-10-1': 'When [MATH] we can solve the system of equations to obtain the density:', 'gr-qc-0211074-1-11-0': '[EQUATION]', 'gr-qc-0211074-1-11-1': 'In this case the [MATH] term falls off faster with increasing time than the [MATH]term and the dynamics naturally approach those of the [MATH] Friedmann model at late times.', 'gr-qc-0211074-1-11-2': 'A period of radiation-dominated evolution with', 'gr-qc-0211074-1-12-0': '[EQUATION] can therefore solve an initial classical lambda problem.', 'gr-qc-0211074-1-13-0': 'Inflationary universe models do not offer a solution of this problem.', 'gr-qc-0211074-1-13-1': 'It is interesting that the flatness and cosmological constant problems can be solved by these VSL theories within any fine tuning of [MATH].', 'gr-qc-0211074-1-13-2': 'But note also that even in the VSL models there is no solution of the non-classical [MATH] problem because a sequence of phase transitions are able to reinstate the value of the cosmological constant at successive cosmological epochs even if its value is set equal to zero at some arbitrarily early time.', 'gr-qc-0211074-1-14-0': 'The simple mathematical model for VSL considered here is oversimplified in many respects but it can easily be put on a former foundation.', 'gr-qc-0211074-1-14-1': 'If we define a scalar field [MATH] then a simple action to describe its coupling to gravity is the choice', 'gr-qc-0211074-1-15-0': '[EQUATION] parametrized by the constants [MATH] and [MATH].', 'gr-qc-0211074-1-15-1': 'Similar properties of the resulting cosmological solutions are found to those displayed by the simple theory above.', 'gr-qc-0211074-1-16-0': '## The Quasi-flatness and Quasi-lambda Problems', 'gr-qc-0211074-1-17-0': 'However, although string theories seem to predict that the cosmological constant should be zero, there is solid observational evidence that it is small and positive and dominating the expansion dynamics of the universe today ([MATH]).', 'gr-qc-0211074-1-17-1': "These observations lead us to ask whether there can be 'natural' explanations for universes which are 'almost' flat or have 'almost' zero cosmological constant.", 'gr-qc-0211074-1-17-2': 'Barrow and Magueijo [CITATION] have shown that these simple VSL models can provide solutions to both problems if the range of [MATH] is narrowed.', 'gr-qc-0211074-1-17-3': 'Thus, when the speed of light falls as ([REF]) with', 'gr-qc-0211074-1-18-0': '[EQUATION] we find that at late times we have a quasi-flat universe and', 'gr-qc-0211074-1-19-0': '[EQUATION]', 'gr-qc-0211074-1-19-1': 'Similarly, when', 'gr-qc-0211074-1-20-0': '[EQUATION] we have a significant cosmological constant contribution to the expansion at late times with', 'gr-qc-0211074-1-21-0': '[EQUATION]', 'gr-qc-0211074-1-22-0': '# Other Consequences of VSL Theories', 'gr-qc-0211074-1-23-0': '## The Isotropy Problem', 'gr-qc-0211074-1-24-0': 'The isotropy problem is solved by inflationary universe models with [MATH] because the dominant anisotropy modes in expanding anisotropic universes fall off no slower than [MATH] and equation ([REF]) is modified in the presence of anisotropies by the addition of a [MATH] term to its right-hand side.', 'gr-qc-0211074-1-24-1': 'Thus we see that at late times in an inflationary expansion the [MATH]term falls off more slowly than [MATH] and dominates the dynamics, driving the expansion away from isotropy.', 'gr-qc-0211074-1-24-2': 'Note however that in generic ever-expanding universes the source of this dominant [MATH] or [MATH] late-time anisotropy is the anisotropy in the 3-curvature of space.', 'gr-qc-0211074-1-24-3': 'The simpler, much-studied anisotropy characteristic of Bianchi I or Kasner universes, arises from simple expansion rate anisotropy with isotropic spatial 3-curvature is sub-dominant at late times, with [MATH].', 'gr-qc-0211074-1-24-4': 'This behaviour is of measure zero in the space of all anisotropic cosmological models.', 'gr-qc-0211074-1-24-5': 'A solution of the anisotropy problem must explain why the [MATH] anisotropy mode has not come to dominate the expansion.', 'gr-qc-0211074-1-24-6': 'This is possible for inflationary expansion dominated by a [MATH] effective stress since this falls off more slowly than [MATH] as [MATH] and the influence of the anisotropy can be made arbitrarily small for a sufficiently long period of accelerated expansion.', 'gr-qc-0211074-1-25-0': 'In the VSL theories described here it is not clear that the isotropy problem can be solved in general.', 'gr-qc-0211074-1-25-1': 'This is a consequence of the fact that VSL theories solve the flatness problem by making the [MATH] term falls off faster than the [MATH]term in ([REF]).', 'gr-qc-0211074-1-25-2': 'The rapid decrease in the curvature term required to solve the flatness problem has no effect on the [MATH] or [MATH] anisotropy term and so the VSL dynamics will not provide a new solution of the isotropy problem.', 'gr-qc-0211074-1-25-3': 'Only if there is a solution of the isotropy problem in the context of non-inflationary general relativistic cosmological models it will be possible to solve it in VSL theories.', 'gr-qc-0211074-1-25-4': 'The anisotropising mode is well understood in the context of homogeneous and anisotropic cosmologies.', 'gr-qc-0211074-1-25-5': 'It is a Bianchi type [MATH] plane gravitational wave described by an exact solution found by Lukash [CITATION].', 'gr-qc-0211074-1-25-6': 'One can show that a particular family of known exact vacuum solutions of type [MATH]are stable as [MATH] [CITATION].', 'gr-qc-0211074-1-25-7': 'In particular, the isotropic open Friedmann universe is stable (but not asymptotically stable) with respect to these anisotropic curvature modes: that is, as [MATH], the ratio of the shear [MATH] to mean Hubble expansion rate, [MATH] approaches a non-zero constant value.', 'gr-qc-0211074-1-25-8': 'Inflation can make this constant arbitrarily close to zero; VSL evolution cannot.', 'gr-qc-0211074-1-26-0': '## The Inhomogeneity Problem', 'gr-qc-0211074-1-27-0': 'The spectrum of primordial fluctuations is the most interesting prediction that any VSL model can make because the observational evidence from the microwave background temperature fluctuations is potentially the most decisive of observational tests.', 'gr-qc-0211074-1-27-1': 'Inflationary theories have been able to provide a natural explanation for our observations of an almost constant curvature spectrum of inhomogeneous fluctuations in the universe.', 'gr-qc-0211074-1-27-2': 'Other competing predictions have been made by colliding brane models [CITATION].', 'gr-qc-0211074-1-28-0': 'In inflationary universes the approximately constant curvature spectrum of inhomogeneities arises from the near de Sitter behaviour of the expansion dynamics during an inflationary epoch where [MATH].', 'gr-qc-0211074-1-28-1': 'In a finite time interval of [MATH] variation in a VSL theory that solves the flatness or lambda problems the expansion dynamics will be dominated by the usual [MATH]term.', 'gr-qc-0211074-1-28-2': 'If the expansion is radiation dominated then no special inhomogeneity spectrum of the constant curvature form will be created by the VSL evolution.', 'gr-qc-0211074-1-28-3': 'One way of imprinting a characteristic spectrum could be via the sudden phase transition model of VSL favoured by Moffat and Albrecht and Magueijo.', 'gr-qc-0211074-1-28-4': 'Here, it would need to be shown that a constant curvature spectrum results.', 'gr-qc-0211074-1-28-5': 'Again, this is a major challenge because the phase transition models of inflation achieve a constant curvature spectrum of fluctuations by virtue of their proximity to a de Sitter state in the vacuum state where the scalar field stops rolling.', 'gr-qc-0211074-1-28-6': 'Recently, one attempt along these lines for the creation of a constant curvature spectrum has been made by Moffat [CITATION], and Magueijo and Pogosian [CITATION] have investigated the possibility that thermal fluctuations might give rise to an interesting primordial inhomogeneity spectrum as a result of modified dispersion relations or cosmological bounce at high energy.', 'gr-qc-0211074-1-29-0': '## The Massive Particle Problem', 'gr-qc-0211074-1-30-0': 'VSL theories have two further strange properties that appear to have dramatic consequences.', 'gr-qc-0211074-1-30-1': 'Suppose that the universe starts to experience a period of VSL evolution of the form ([REF]).', 'gr-qc-0211074-1-30-2': 'Let us consider the fate of a particle state of mass [MATH] that exists on a scale smaller than the particle horizon [MATH].', 'gr-qc-0211074-1-30-3': 'Now if it has a de Broglie wavelength', 'gr-qc-0211074-1-31-0': '[EQUATION] then if [MATH] falls as [MATH] in a radiation-dominated universe we have for the ratio of [MATH] to the proper size of the particle horizon', 'gr-qc-0211074-1-32-0': '[EQUATION] is given by', 'gr-qc-0211074-1-33-0': '[EQUATION] as [MATH] if [MATH].', 'gr-qc-0211074-1-33-1': 'So if we take [MATH] as required to solve the flatness problem, massive particles evolve to become acausal separate universes!', 'gr-qc-0211074-1-33-2': 'How should be interpret this.', 'gr-qc-0211074-1-33-3': 'Is is a reductio ad absurdum of the VSL or should we take a lesson from new inflation and regard our observable universe as the interior of the single particle state.', 'gr-qc-0211074-1-33-4': 'This might provide a natural explanation for some of its coordinated properties and provide a reason for a special irregularity spectrum to be formed.', 'gr-qc-0211074-1-33-5': 'Note that this behaviour of massive particle states ceases when the VSL evolution ends and will not be going on in the universe today.', 'gr-qc-0211074-1-33-6': 'Since the early universe may contain a population of massive states inside the horizon scale when VSL evolution begins, the ensuing evolution is not dissimilar to the self-reproducing inflationary universe.', 'gr-qc-0211074-1-33-7': 'There small regions create eternal inflation due to the quantum evolution dominating the classical slow roll under fairly general conditions.', 'gr-qc-0211074-1-33-8': 'The rapid expansion of many VSL regions would produce collisions that might lad to unacceptable levels of inhomogeneity when they subsequently re-enter the horizon if the amount of expansion was small.', 'gr-qc-0211074-1-33-9': 'But if it was large then we could find ourselves inhabiting a single VSL pre-expanded domain.', 'gr-qc-0211074-1-33-10': 'This scenario may repay further detailed analysis.', 'gr-qc-0211074-1-33-11': 'There are many complexities that have been ignored here, in particular relating to the growth of dimensionless couplings like [MATH] as [MATH] falls.', 'gr-qc-0211074-1-34-0': '## The Primordial Black Hole Problem', 'gr-qc-0211074-1-35-0': 'A similar fate awaits any small primordial black hole of mass [MATH] that forms on sub-horizon scales.', 'gr-qc-0211074-1-35-1': 'Let us ignore Hawking evaporation and consider the ratio of the Schwarzschild radius of the black hole, [MATH]to the horizon scale:', 'gr-qc-0211074-1-36-0': '[EQUATION]', 'gr-qc-0211074-1-36-1': 'So if [MATH] the black hole horizon grows faster than the particle horizon and the black hole becomes an acausal separate universe.', 'gr-qc-0211074-1-36-2': 'Much of the discussion regarding the expansion of massive particle states applies to this situation also.', 'gr-qc-0211074-1-36-3': 'Here there is an added interpretational uncertainty in that it is not clear what happens to the black hole when [MATH] changes.', 'gr-qc-0211074-1-37-0': '## An Entropy Problem', 'gr-qc-0211074-1-38-0': 'If we believe that we can apply the Bekenstein-Hawking entropy formula to the particle horizon of an expanding universe then the entropy inside the VSL horizon is', 'gr-qc-0211074-1-39-0': '[EQUATION] and this entropy increases in VSL theories only if [MATH] in the radiation era[MATH].', 'gr-qc-0211074-1-39-1': 'Thus the solution of the flatness an lambda problems requires this entropy measure to decrease with increasing time.', 'gr-qc-0211074-1-39-2': "The Bekenstein-Hawking entropy technically applies only to event horizons but many analogous measures of 'gravitational entropy' have been proposed and it is possible that the argument framed here will have application to any better-motivated masure of gravitational entropy.", 'gr-qc-0211074-1-39-3': 'This worry also besets arguments like those of Davies et al [CITATION] which use black hole thermodynamics to assess whether the variation of certain constants are in accord with the second law of thermodynamics.', 'gr-qc-0211074-1-39-4': 'The difficulty is that the required black holes and their thermodynamic will only exist as particular solutions of a theory with varying constants - particular solutions in which those varying constants take constant values - and so the argument cannot be carried through.', 'gr-qc-0211074-1-39-5': 'A specific case arises in Brans-Dicke theory.', 'gr-qc-0211074-1-39-6': 'The Schwarzschild solution is a particular solution of Brans-Dicke theory and its Bekenstein-Hawking entropy is [MATH].', 'gr-qc-0211074-1-39-7': 'One might be tempted therefore to think that any solution of Brans-Dicke gravity in which [MATH] decreases with time would therefore violate a second law of black hole thermodynamics.', 'gr-qc-0211074-1-39-8': 'However, the Schwarzschild black hole is only a solution of Brans-Dicke theory when [MATH] is constant.', 'gr-qc-0211074-1-39-9': 'If [MATH] is allowed to vary (as in the thought [MATH]experiment designed to violate the second law) the static spherically symmetric solution of the Brans-Dicke equations is not a black hole.', 'gr-qc-0211074-1-40-0': '# Discussion', 'gr-qc-0211074-1-41-0': 'We have considered further properties of naive VSL cosmologies, in addition to those related to the flatness and horizon problems.', 'gr-qc-0211074-1-41-1': 'There are problems explaining the isotropy of the universe in general and unusual consequences of varying [MATH] on particle de Broglie wavelengths and black holes.', 'gr-qc-0211074-1-41-2': 'It is not clear whether the massive particle and black hole problems are fatal to the conception of VSL theories.', 'gr-qc-0211074-1-41-3': 'At first sight they appear to create an unsatisfactory state of affairs.', 'gr-qc-0211074-1-41-4': 'However, it may be that they are in effect counterparts of the self-reproduction property of inflationary universes that gives rise to the quasi-stationary eternal inflationary universe scenario.', 'gr-qc-0211074-1-41-5': 'The rapid enlargement of single particle states to super-horizon scales simply provides a way of producing large scale regions that possess the coherent properties possessed by single particle states when they were on sub-horizon scales and to generate fluctuations on super-horizon scales.', 'gr-qc-0211074-1-41-6': 'However, although states grow faster than the horizon they do not necessarily grow fast enough to create regions which could encompass our entire visible universe.', 'gr-qc-0211074-1-41-7': 'Whether or not these features provide a reductio ad absurdum for these cosmologies remains to be explored in greater detail.'}

{'gr-qc-0211074-2-0-0': 'We contrast features of simple varying speed of light (VSL) cosmologies with inflationary universe models.', 'gr-qc-0211074-2-0-1': 'We present new features of VSL cosmologies and show that they face problems explaining the cosmological isotropy problem.', 'gr-qc-0211074-2-0-2': 'We also find that if [MATH] falls fast enough to solve the flatness and horizon problems then the quantum wavelengths of massive particle states and the radii of primordial black holes can grow to exceed the scale of the particle horizon.', 'gr-qc-0211074-2-0-3': 'This may provide VSL cosmologies with a self-reproduction property.', 'gr-qc-0211074-2-0-4': 'The constraint of entropy increase is also discussed.', 'gr-qc-0211074-2-0-5': 'The new problems described in the this letter provide a set of bench tests for more sophisticated VSL theories to pass.', 'gr-qc-0211074-2-1-0': '# Introduction', 'gr-qc-0211074-2-2-0': 'We have explored a naive class of models in which the speed of light varies in time [CITATION], [CITATION].', 'gr-qc-0211074-2-2-1': "These can be reformulated more generally and recast as theories in which other dimensional 'constants' carry the operationally meaningful space or time variation of a dimensionless constant - in this case the fine structure 'constant', [MATH] [CITATION].", 'gr-qc-0211074-2-2-2': 'Theories in which this variation is carried explicitly by the electron change have been extensively investigated recently [CITATION].', 'gr-qc-0211074-2-2-3': "Such theories are a 'small' perturbation of standard physics in the sense that even though the speed of light falls there still exists a maximum signal propagation velocity that is achieved by gravitational waves.", 'gr-qc-0211074-2-2-4': "The motivation for a careful consideration of these cosmological models was their possible viability as alternatives to inflation as explanations for a number of unusual properties of the universe and the consistency of quasar absorption spectra with a variation in the value of the fine structure 'constant at [MATH] [CITATION].", 'gr-qc-0211074-2-2-5': 'In particular, it was shown by Moffat [CITATION], Albrecht and Magueijo [CITATION] and by Barrow [CITATION] that during the very early universe a finite period of time during which the speed of light falls at an appropriate rate can lead to a solution of the flatness, horizon, and monopole problems.', 'gr-qc-0211074-2-2-6': 'However, unlike inflation it can also provide a solution for the cosmological constant problem.', 'gr-qc-0211074-2-2-7': 'Barrow and Magueijo also showed that for some ranges of variation in the speed of light these theories can also naturally create long-lived universe in which the dynamics are almost flat or in which the cosmological constant is almost zero.', 'gr-qc-0211074-2-2-8': 'These problems of explaining a universe in which the present value of the matter density parameter, [MATH], or the cosmological constant energy density parameter, [MATH], are [MATH] we called the quasi-flatness and quasi-lambda problems.', 'gr-qc-0211074-2-2-9': 'Inflation does not seem to offer a natural explanation of a universe which is almost flat or has a dynamically significant cosmological constant today, as current observations imply.', 'gr-qc-0211074-2-3-0': 'In this paper, we want to discuss some other cosmological problems in the light of varying speed of light (VSL) theories of the simple sort discussed by Albrecht, Magueijo, and Barrow (AMB).', 'gr-qc-0211074-2-3-1': 'Although these problems are articulated in the context of this simple AMB theory we believe that are more general challenges to any theory which manifests itself in a particular coordinate system as a VSL theory.', 'gr-qc-0211074-2-3-2': 'In particular, we shall show that in general AMB theories cannot solve the isotropy problem and are unable to generate a spectrum of almost constant curvature fluctuations in the standard way.', 'gr-qc-0211074-2-3-3': 'We shall also show that they have a number of very unusual consequences for the quantum states of massive particles and for primordial black holes after they enter the particle horizon in the early universe.', 'gr-qc-0211074-2-3-4': 'It is not clear whether these features are a reductio ad absurdum for these VSL theories or whether they provide an exotic counterpart to the self-reproducing property of inflationary universe models that provides the basis for eternal inflationary universes.', 'gr-qc-0211074-2-4-0': '# Naive VSL Theories', 'gr-qc-0211074-2-5-0': '## Solving the flatness, horizon and lambda problems', 'gr-qc-0211074-2-6-0': 'The simple VSL model introduced by Albrecht and Magueijo and solved by Barrow is based upon the simplest possible premises.', 'gr-qc-0211074-2-6-1': "We assume that the Friedmann cosmology in the presence of time-dependent speed of light [MATH] is described by the Friedmann and Raychaudhuri equations (although the original formulation allowed the Newtonian gravitation 'constant' [MATH] to vary also, we shall ignore this possibility as it makes no contribution to the essential conclusions): [EQUATION] where [MATH] is the expansion scale factor of the Friedmann metric, [MATH] is the fluid pressure, [MATH] is the fluid density, [MATH] is the curvature parameter, [MATH] is the cosmological constant, and all derivatives are with respect to [MATH], the comoving proper time.", 'gr-qc-0211074-2-6-2': 'From these equations we can derive the modified matter conservation which incorporates the effects of [MATH] [EQUATION]', 'gr-qc-0211074-2-6-3': 'Of course, in general relativity, where [MATH] the right-hand side of ( [REF]) is zero.', 'gr-qc-0211074-2-6-4': 'Notice that it also vanishes in a VSL theory when [MATH].', 'gr-qc-0211074-2-6-5': 'For concreteness, now consider the universe to contain a perfect fluid with equation of state [EQUATION] where [MATH] is constant and we shall assume that the speed of light varies as some power of the expansion scale factor: [EQUATION] where [MATH] and [MATH] are constants.', 'gr-qc-0211074-2-6-6': 'These assumptions are sufficient to solve the equations completely.', 'gr-qc-0211074-2-6-7': 'Integrating ([REF]) with [MATH], we find [EQUATION]', 'gr-qc-0211074-2-6-8': 'Now by inspection of ([REF]) we see that at large [MATH] the curvature term falls off faster than the [MATH]term whenever [EQUATION]', 'gr-qc-0211074-2-6-9': 'Thus the flatness problem can be solved in a radiation-dominated early universe by an interval of VSL evolution if', 'gr-qc-0211074-2-7-0': '[EQUATION]', 'gr-qc-0211074-2-7-1': 'It is easy to show that, as in the case of inflation, this is the same condition required to solve the monopole and horizon problems.', 'gr-qc-0211074-2-7-2': 'However, the nature of the solution differs from that provided by a period of inflation.', 'gr-qc-0211074-2-7-3': 'Inflation requires the dynamics to be dominated for a finite period by an unusual fluid ([MATH]), which is therefore gravitationally repulsive, so that the expansion will accelerate ([MATH]) and the [MATH]term will grow to dominate the [MATH] term in ([REF]) as [MATH] grows large.', 'gr-qc-0211074-2-7-4': 'By contrast, in the VSL model, no special fluid is required and in a radiation universe the flatness problem is solved because the [MATH] term falls off faster than the [MATH] term because of the fall in [MATH] as [MATH] increases.', 'gr-qc-0211074-2-8-0': 'This behaviour also permits a solution of the classical cosmological constant problem in just the same way if', 'gr-qc-0211074-2-9-0': '[EQUATION]', 'gr-qc-0211074-2-9-1': 'When [MATH] we can solve the system of equations to obtain the density:', 'gr-qc-0211074-2-10-0': '[EQUATION]', 'gr-qc-0211074-2-10-1': 'In this case the [MATH] term falls off faster with increasing time than the [MATH]term and the dynamics naturally approach those of the [MATH] Friedmann model at late times.', 'gr-qc-0211074-2-10-2': 'A period of radiation-dominated evolution with', 'gr-qc-0211074-2-11-0': '[EQUATION] can therefore solve an initial classical lambda problem.', 'gr-qc-0211074-2-12-0': 'Inflationary universe models do not offer a solution of this problem.', 'gr-qc-0211074-2-12-1': 'It is interesting that the flatness and cosmological constant problems can be solved by these VSL theories within any fine tuning of [MATH].', 'gr-qc-0211074-2-12-2': 'But note also that even in the VSL models there is no solution of the non-classical [MATH] problem because a sequence of phase transitions are able to reinstate the value of the cosmological constant at successive cosmological epochs even if its value is set equal to zero at some arbitrarily early time.', 'gr-qc-0211074-2-13-0': 'The simple mathematical model for VSL considered here is oversimplified in many respects but it can easily be put on a former foundation.', 'gr-qc-0211074-2-13-1': 'If we define a scalar field [MATH] then a simple action to describe its coupling to gravity is the choice', 'gr-qc-0211074-2-14-0': '[EQUATION] parametrized by the constants [MATH] and [MATH].', 'gr-qc-0211074-2-14-1': 'Similar properties of the resulting cosmological solutions are found to those displayed by the simple theory above.', 'gr-qc-0211074-2-14-2': 'Many of the potential problems raised in the paper need to be readdressed in the context of a specific theory, defined by a given lagrangian.', 'gr-qc-0211074-2-14-3': 'At present there is no unique VSL theory.', 'gr-qc-0211074-2-14-4': 'However, the naive VSL theory described above plays an important role in generating hypotheses to be tested in more sophisticated theories.', 'gr-qc-0211074-2-15-0': '## The Quasi-flatness and Quasi-lambda Problems', 'gr-qc-0211074-2-16-0': 'However, although string theories seem to predict that the cosmological constant should be zero, there is solid observational evidence that it is small and positive and dominating the expansion dynamics of the universe today ([MATH]).', 'gr-qc-0211074-2-16-1': "These observations lead us to ask whether there can be 'natural' explanations for universes which are 'almost' flat or have 'almost' zero cosmological constant.", 'gr-qc-0211074-2-16-2': 'Barrow and Magueijo [CITATION] have shown that these simple VSL models can provide solutions to both problems if the range of [MATH] is narrowed.', 'gr-qc-0211074-2-16-3': 'Thus, when the speed of light falls as ([REF]) with [EQUATION] we find that at late times we have a quasi-flat universe and', 'gr-qc-0211074-2-17-0': '[EQUATION]', 'gr-qc-0211074-2-17-1': 'Similarly, when', 'gr-qc-0211074-2-18-0': '[EQUATION] we have a significant cosmological constant contribution to the expansion at late times with', 'gr-qc-0211074-2-19-0': '[EQUATION]', 'gr-qc-0211074-2-20-0': '# Other Consequences of VSL Theories', 'gr-qc-0211074-2-21-0': '## The Isotropy Problem', 'gr-qc-0211074-2-22-0': 'The isotropy problem is solved by inflationary universe models with [MATH] because the dominant anisotropy modes in expanding anisotropic universes fall off no slower than [MATH] and equation ([REF]) is modified in the presence of anisotropies by the addition of a [MATH] term to its right-hand side.', 'gr-qc-0211074-2-22-1': 'Thus we see that at late times in an inflationary expansion the [MATH]term falls off more slowly than [MATH] and dominates the dynamics, driving the expansion away from isotropy.', 'gr-qc-0211074-2-22-2': 'Note however that in generic ever-expanding universes the source of this dominant [MATH] or [MATH] late-time anisotropy is the anisotropy in the 3-curvature of space.', 'gr-qc-0211074-2-22-3': 'The simpler, much-studied anisotropy characteristic of Bianchi I or Kasner universes, arises from simple expansion rate anisotropy with isotropic spatial 3-curvature is sub-dominant at late times, with [MATH].', 'gr-qc-0211074-2-22-4': 'This behaviour is of measure zero in the space of all anisotropic cosmological models.', 'gr-qc-0211074-2-22-5': 'A solution of the anisotropy problem must explain why the [MATH] anisotropy mode has not come to dominate the expansion.', 'gr-qc-0211074-2-22-6': 'This is possible for inflationary expansion dominated by a [MATH] effective stress since this falls off more slowly than [MATH] as [MATH] and the influence of the anisotropy can be made arbitrarily small for a sufficiently long period of accelerated expansion.', 'gr-qc-0211074-2-23-0': 'In the situation, with constant [MATH], where inflation does not occur and isotropisation is not generic.', 'gr-qc-0211074-2-23-1': 'The anisotropising mode is well understood in the context of homogeneous and anisotropic cosmologies.', 'gr-qc-0211074-2-23-2': 'It is a Bianchi type [MATH] plane gravitational wave described by an exact solution found by Lukash [CITATION].', 'gr-qc-0211074-2-23-3': 'It is interesting to note that they can be excluded if an open universe has a finite topology [CITATION][MATH]In the case with the natural [MATH] topology one can show that a particular family of known exact vacuum solutions of type [MATH]are stable as [MATH] [CITATION].', 'gr-qc-0211074-2-23-4': 'In particular, the isotropic open Friedmann universe is stable (but not asymptotically stable) with respect to these anisotropic curvature modes: that is, as [MATH], the ratio of the shear [MATH] to mean Hubble expansion rate, [MATH] approaches a non-zero constant value.', 'gr-qc-0211074-2-23-5': 'Inflation can make this constant arbitrarily close to zero.', 'gr-qc-0211074-2-24-0': 'In the simplest anisotropic universes with comoving perfect-fluid matter and isotropic 3-curvature and VSL we can repeat the approach taken in the case of isotropic universes, using the Raychaudhuri acceleration equation, its first integral, and the shear propagation equation - with [MATH] assumed - to derive a generalised matter conservation equation.', 'gr-qc-0211074-2-24-1': 'Thus', 'gr-qc-0211074-2-25-0': '[EQUATION]', 'gr-qc-0211074-2-25-1': 'Hence we see that [MATH] and if we were to write [MATH] then', 'gr-qc-0211074-2-26-0': '[EQUATION]', 'gr-qc-0211074-2-26-1': 'Using eqns. ([REF]) and ([REF]) we can solve for [MATH] and we find that the flatness problem is solved as usual when ([REF]) holds.', 'gr-qc-0211074-2-26-2': 'We can see immediately that the shear effects are negligible with respect to the curvature and', 'gr-qc-0211074-2-27-0': '[EQUATION].', 'gr-qc-0211074-2-27-1': 'Hence, the shear term is always insignificant and the condition for solution of the flatness problem is the same as in an isotropic universe, ([REF] ).', 'gr-qc-0211074-2-28-0': 'In the general case we have to consider the contribution made the anisotropic 3-curvature terms and anisotropic fluid pressure and not just an anisotropic Hubble flow ([MATH]), as considered above.', 'gr-qc-0211074-2-28-1': 'For simplicity, we neglect the anisotropic pressure effects but consider the role of anisotropic 3-curvature.', 'gr-qc-0211074-2-28-2': 'The shear propagation equation becomes [CITATION], :', 'gr-qc-0211074-2-29-0': '[EQUATION] where [MATH] is the anisotropic part of the spatial 3-curvature and [MATH]is the projection tensor.', 'gr-qc-0211074-2-29-1': 'Generic evolution at late times for ever-expanding anisotropic universes close to isotropy with [MATH] will have the scaling form [MATH].', 'gr-qc-0211074-2-29-2': 'Hence we [MATH] as in the case with constant [MATH].', 'gr-qc-0211074-2-29-3': 'However, we see that [MATH] in the generalised Friedmann equation ([REF]) whereas the isotropic matter terms go as [MATH]Thus the shear terms will dominate at late times close the Friedmann expansion, [MATH] whenever [MATH] and a falling speed of light does not solve the anisotropy problem.', 'gr-qc-0211074-2-29-4': 'There are several ways in which the VSL effects on anisotropy can be understood more physically.', 'gr-qc-0211074-2-29-5': 'The naive VSL theories preserve the metric structure of spacetime so that gravitational-wave propagation still occurs at a maximum possible propagation speed.', 'gr-qc-0211074-2-29-6': 'But light propagates with a variable speed that is less than or equal to the gravitational-wave propagation speed.', 'gr-qc-0211074-2-29-7': 'Thus one can see that anisotropies that are carried by long-wavelength gravitational waves can avoid being made innocuous by a fall in the speed of light.', 'gr-qc-0211074-2-29-8': 'The generic anisotropies at late time in ever-expanding Bianchi types that contain isotropic Friedmann universes are of this type.', 'gr-qc-0211074-2-30-0': '## The Inhomogeneity Problem', 'gr-qc-0211074-2-31-0': 'The spectrum of primordial fluctuations is the most interesting prediction that any VSL model can make because the observational evidence from the microwave background temperature fluctuations is potentially the most decisive of observational tests.', 'gr-qc-0211074-2-31-1': 'Inflationary theories have been able to provide a natural explanation for our observations of an almost constant curvature spectrum of inhomogeneous fluctuations in the universe.', 'gr-qc-0211074-2-31-2': 'Other competing predictions have been made by colliding brane models [CITATION].', 'gr-qc-0211074-2-32-0': 'In inflationary universes the approximately constant curvature spectrum of inhomogeneities arises from the near de Sitter behaviour of the expansion dynamics during an inflationary epoch where [MATH].', 'gr-qc-0211074-2-32-1': 'In a finite time interval of [MATH] variation in a VSL theory that solves the flatness or lambda problems the expansion dynamics will be dominated by the usual [MATH]term.', 'gr-qc-0211074-2-32-2': 'If the expansion is radiation dominated then no special inhomogeneity spectrum of the constant curvature form will be created by the VSL evolution.', 'gr-qc-0211074-2-32-3': 'One way of imprinting a characteristic spectrum could be via the sudden phase transition model of VSL favoured by Moffat and Albrecht and Magueijo.', 'gr-qc-0211074-2-32-4': 'Here, it would need to be shown that a constant curvature spectrum results.', 'gr-qc-0211074-2-32-5': 'Again, this is a major challenge because the phase transition models of inflation achieve a constant curvature spectrum of fluctuations by virtue of their proximity to a de Sitter state in the vacuum state where the scalar field stops rolling.', 'gr-qc-0211074-2-32-6': 'Recently, one attempt along these lines for the creation of a constant curvature spectrum has been made by Moffat [CITATION], and Magueijo and Pogosian [CITATION] have investigated the possibility that thermal fluctuations might give rise to an interesting primordial inhomogeneity spectrum as a result of modified dispersion relations or cosmological bounce at high energy.', 'gr-qc-0211074-2-33-0': '## The Massive Particle Problem', 'gr-qc-0211074-2-34-0': 'VSL theories have two further strange properties that appear to have dramatic consequences.', 'gr-qc-0211074-2-34-1': 'Suppose that the universe starts to experience a period of VSL evolution of the form ([REF]).', 'gr-qc-0211074-2-34-2': 'Let us consider the fate of a particle state of mass [MATH] that exists on a scale smaller than the particle horizon [MATH].', 'gr-qc-0211074-2-34-3': 'Now if it has a de Broglie wavelength', 'gr-qc-0211074-2-35-0': '[EQUATION] then if [MATH] falls as [MATH] in a radiation-dominated universe we have for the ratio of [MATH] to the proper size of the particle horizon', 'gr-qc-0211074-2-36-0': '[EQUATION] is given by', 'gr-qc-0211074-2-37-0': '[EQUATION] as [MATH] if [MATH].', 'gr-qc-0211074-2-37-1': 'So if we take [MATH] as required to solve the flatness problem, massive particles evolve to become acausal separate universes!', 'gr-qc-0211074-2-37-2': 'Note that in particular theories [MATH] may vary as well.', 'gr-qc-0211074-2-37-3': 'For example in the naive VSL theory of refs [CITATION], [CITATION] [MATH] and so [MATH] and a massive particle problem can arise when [MATH] as required to solve the classical cosmological constant problem.', 'gr-qc-0211074-2-38-0': 'How should be interpret this.', 'gr-qc-0211074-2-38-1': 'Is is a reductio ad absurdum of the VSL or should we take a lesson from new inflation and regard our observable universe as the interior of the single particle state.', 'gr-qc-0211074-2-38-2': 'This might provide a natural explanation for some of its coordinated properties and provide a reason for a special irregularity spectrum to be formed.', 'gr-qc-0211074-2-38-3': 'Note that this behaviour of massive particle states ceases when the VSL evolution ends and will not be going on in the universe today.', 'gr-qc-0211074-2-38-4': 'Since the early universe may contain a population of massive states inside the horizon scale when VSL evolution begins, the ensuing evolution is not dissimilar to the self-reproducing inflationary universe.', 'gr-qc-0211074-2-38-5': 'There small regions create eternal inflation due to the quantum evolution dominating the classical slow roll under fairly general conditions.', 'gr-qc-0211074-2-38-6': 'The rapid expansion of many VSL regions would produce collisions that might lad to unacceptable levels of inhomogeneity when they subsequently re-enter the horizon if the amount of expansion was small.', 'gr-qc-0211074-2-38-7': 'But if it was large then we could find ourselves inhabiting a single VSL pre-expanded domain.', 'gr-qc-0211074-2-38-8': 'This scenario may repay further detailed analysis.', 'gr-qc-0211074-2-38-9': 'There are many complexities that have been ignored here, in particular relating to the rapid growth of dimensionless couplings like [MATH] as [MATH] falls, rendering all interactions strong.', 'gr-qc-0211074-2-38-10': 'It may be that the avoidance of this problem is a constraint that needs to be placed on sophisticated VSL theories or it could be exploited as a new mechanism for making small local regions become large.', 'gr-qc-0211074-2-39-0': '## The Primordial Black Hole Problem', 'gr-qc-0211074-2-40-0': 'A similar fate awaits any small primordial black hole of mass [MATH] that forms on sub-horizon scales.', 'gr-qc-0211074-2-40-1': 'Let us ignore Hawking evaporation and consider the ratio of the Schwarzschild radius of the black hole, [MATH]to the horizon scale:', 'gr-qc-0211074-2-41-0': '[EQUATION]', 'gr-qc-0211074-2-41-1': 'So if [MATH] the black hole horizon grows faster than the particle horizon and the black hole becomes an acausal separate universe.', 'gr-qc-0211074-2-41-2': 'Much of the discussion regarding the expansion of massive particle states applies to this situation also.', 'gr-qc-0211074-2-41-3': 'Here there is an added interpretational uncertainty in that it is not clear what happens to the black hole when [MATH] changes.', 'gr-qc-0211074-2-41-4': 'Changing [MATH] may be sufficient to stop a black hole forming.', 'gr-qc-0211074-2-41-5': 'But if a Schwarzschild black hole formed when [MATH] did not change then it might be that it had to remain constant on the horizon even while [MATH] changed in the background.', 'gr-qc-0211074-2-42-0': '## An Entropy Problem', 'gr-qc-0211074-2-43-0': 'If we believe that we can apply the Bekenstein-Hawking entropy formula to the particle horizon of an expanding universe then the entropy inside the VSL horizon is', 'gr-qc-0211074-2-44-0': '[EQUATION] and this entropy evolves as [MATH]in VSL theories and hence increases only if [MATH] in the radiation era when [MATH] is constant.', 'gr-qc-0211074-2-44-1': 'If [MATH], so [MATH] then the entropy increases only if [MATH].', 'gr-qc-0211074-2-44-2': 'In general we might consider a suite of theories in which [MATH] so [MATH] so [MATH] and entropy increases during the radiation era when [MATH].', 'gr-qc-0211074-2-44-3': 'Thus the solution of the flatness an lambda problems can require this entropy measure to decrease with increasing time.', 'gr-qc-0211074-2-44-4': "The Bekenstein-Hawking entropy technically applies only to event horizons but many analogous measures of 'gravitational entropy' have been proposed and it is possible that the argument framed here will have application to any better-motivated masure of gravitational entropy.", 'gr-qc-0211074-2-44-5': 'This worry also besets arguments like those of Davies et al [CITATION] which use black hole thermodynamics to assess whether the variation of certain constants are in accord with the second law of thermodynamics.', 'gr-qc-0211074-2-44-6': 'The difficulty is that the required black holes and their thermodynamic will only exist as particular solutions of a theory with varying constants - particular solutions in which those varying constants take constant values - and so the argument cannot be carried through.', 'gr-qc-0211074-2-44-7': 'A specific case arises in Brans-Dicke theory.', 'gr-qc-0211074-2-44-8': 'The Schwarzschild solution is a particular solution of Brans-Dicke theory and its Bekenstein-Hawking entropy is [MATH] .', 'gr-qc-0211074-2-44-9': 'One might be tempted therefore to think that any solution of Brans-Dicke gravity in which [MATH] decreases with time would therefore violate a second law of black hole thermodynamics.', 'gr-qc-0211074-2-44-10': 'However, the Schwarzschild black hole is only a solution of Brans-Dicke theory when [MATH] is constant.', 'gr-qc-0211074-2-44-11': 'If [MATH] is allowed to vary (as in the thought [MATH]experiment designed to violate the second law) the static spherically symmetric solution of the Brans-Dicke equations is not a black hole.', 'gr-qc-0211074-2-45-0': '# Discussion', 'gr-qc-0211074-2-46-0': 'We have considered further properties of naive VSL cosmologies, in addition to those related to the flatness and horizon problems.', 'gr-qc-0211074-2-46-1': 'There are problems explaining the isotropy of the universe in general and unusual consequences of varying [MATH] on particle de Broglie wavelengths and black holes.', 'gr-qc-0211074-2-46-2': 'It is not clear whether the massive particle and black hole problems are fatal to the conception of VSL theories.', 'gr-qc-0211074-2-46-3': 'At first sight they appear to create an unsatisfactory state of affairs.', 'gr-qc-0211074-2-46-4': 'However, it may be that they are in effect counterparts of the self-reproduction property of inflationary universes that gives rise to the quasi-stationary eternal inflationary universe scenario.', 'gr-qc-0211074-2-46-5': 'The rapid enlargement of single particle states to super-horizon scales simply provides a way of producing large scale regions that possess the coherent properties possessed by single particle states when they were on sub-horizon scales and to generate fluctuations on super-horizon scales.', 'gr-qc-0211074-2-46-6': 'However, although states grow faster than the horizon they do not necessarily grow fast enough to create regions which could encompass our entire visible universe.', 'gr-qc-0211074-2-46-7': 'Whether or not these features provide a reductio ad absurdum for these cosmologies remains to be explored in greater detail.', 'gr-qc-0211074-2-46-8': 'Any more completely specified VSL theory will need to be judged both by its ability to solve the standard problems that inflation can resolve and by its ability to circumvent the difficulties described in this letter.'}

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[['gr-qc-0211074-1-39-1', 'gr-qc-0211074-2-44-3'], ['gr-qc-0211074-1-39-6', 'gr-qc-0211074-2-44-8'], ['gr-qc-0211074-1-0-1', 'gr-qc-0211074-2-0-1'], ['gr-qc-0211074-1-0-2', 'gr-qc-0211074-2-0-2'], ['gr-qc-0211074-1-33-11', 'gr-qc-0211074-2-38-9'], ['gr-qc-0211074-1-6-8', 'gr-qc-0211074-2-6-8']]

[]

[['gr-qc-0211074-1-39-0', 'gr-qc-0211074-2-44-0'], ['gr-qc-0211074-1-39-0', 'gr-qc-0211074-2-44-1'], ['gr-qc-0211074-1-0-3', 'gr-qc-0211074-2-0-3'], ['gr-qc-0211074-1-0-3', 'gr-qc-0211074-2-0-4'], ['gr-qc-0211074-1-25-6', 'gr-qc-0211074-2-23-3'], ['gr-qc-0211074-1-25-8', 'gr-qc-0211074-2-23-5'], ['gr-qc-0211074-1-17-3', 'gr-qc-0211074-2-16-3'], ['gr-qc-0211074-1-18-0', 'gr-qc-0211074-2-16-3']]

[]

['gr-qc-0211074-1-7-0', 'gr-qc-0211074-1-8-0', 'gr-qc-0211074-1-10-0', 'gr-qc-0211074-1-10-1', 'gr-qc-0211074-1-11-0', 'gr-qc-0211074-1-12-0', 'gr-qc-0211074-1-19-0', 'gr-qc-0211074-1-19-1', 'gr-qc-0211074-1-21-0', 'gr-qc-0211074-1-32-0', 'gr-qc-0211074-1-33-0', 'gr-qc-0211074-1-35-1', 'gr-qc-0211074-1-36-0', 'gr-qc-0211074-2-7-0', 'gr-qc-0211074-2-9-0', 'gr-qc-0211074-2-9-1', 'gr-qc-0211074-2-10-0', 'gr-qc-0211074-2-11-0', 'gr-qc-0211074-2-17-0', 'gr-qc-0211074-2-17-1', 'gr-qc-0211074-2-19-0', 'gr-qc-0211074-2-24-1', 'gr-qc-0211074-2-25-0', 'gr-qc-0211074-2-26-0', 'gr-qc-0211074-2-27-0', 'gr-qc-0211074-2-28-2', 'gr-qc-0211074-2-36-0', 'gr-qc-0211074-2-37-0', 'gr-qc-0211074-2-40-1', 'gr-qc-0211074-2-41-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/gr-qc/0211074

hep-lat-0001001

{'hep-lat-0001001-1-0-0': '# Introduction', 'hep-lat-0001001-1-1-0': 'If we construct a lattice fermion formulation, there are a number of goals to be considered: doubling should be avoided; even at finite lattice spacing [MATH], we want to represent chiral symmetry in a sound way; and we are seeking a good scaling behavior.', 'hep-lat-0001001-1-1-1': 'Conceptually we have to require locality (the lattice Dirac operator [MATH] has to decay at least exponentially in [MATH] ).', 'hep-lat-0001001-1-1-2': 'In addition, for practical purposes we desire a high level of locality, i.e. a fast exponential decay or even ultralocality (which means that the couplings in [MATH] drop to zero beyond a finite number of lattice spacings).', 'hep-lat-0001001-1-1-3': 'A further issue is a good approximation to rotation invariance.', 'hep-lat-0001001-1-1-4': 'Last but not least, the formulation should be simple enough to allow for efficient simulations.', 'hep-lat-0001001-1-1-5': 'Here we report on a construction, which is designed to do justice to all of these goals.', 'hep-lat-0001001-1-2-0': '# Ginsparg-Wilson fermions (an unconventional introduction)', 'hep-lat-0001001-1-3-0': 'For a lattice Dirac operator [MATH], full chiral invariance ( [MATH] ) is incompatible with other basic requirements (Hermiticity, locality, absence of doublers, discrete translation invariance) [CITATION].', 'hep-lat-0001001-1-3-1': 'Therefore we only implement a modified chiral symmetry, which does allow [MATH] to fulfill those requirements.', 'hep-lat-0001001-1-3-2': 'For such a modified chiral transformation we start from the ansatz [EQUATION] [MATH] being an infinitesimal transformation parameter.', 'hep-lat-0001001-1-3-3': 'The transformation, and therefore [MATH] and [MATH] should be local, and [MATH], so that we reproduce the full chiral symmetry in the (naive) continuum limit.', 'hep-lat-0001001-1-4-0': 'Invariance of the Lagrangian [MATH] holds to [MATH] if [EQUATION]', 'hep-lat-0001001-1-4-1': 'This implies a continuous modified chiral symmetry, which has the full number of generators.', 'hep-lat-0001001-1-4-2': 'It may be compared to the remnant chiral symmetry of staggered fermions: there the doubling problem is not solved, and one is only left with a [MATH] symmetry, which does, however, protect the mass from additive renormalization.', 'hep-lat-0001001-1-4-3': 'The same can be shown here if we assume "[MATH]-Hermiticity", [MATH], and we choose [MATH], [MATH], where [MATH] is local again, non-trivial and [MATH] (this generalizes Ref. [CITATION]).', 'hep-lat-0001001-1-4-4': 'Then eq. ([REF]) turns into the Ginsparg-Wilson relation (GWR) [CITATION] [EQUATION] and it implies the absence of additive mass renormalization (see also Ref. [CITATION]) since [EQUATION]', 'hep-lat-0001001-1-4-5': 'As another illustration we can write the GWR as [MATH], and we see that a local term [MATH] does not shift the poles in [MATH] (in contrast to the cases where [MATH] is local, such as a mass or a Wilson term).', 'hep-lat-0001001-1-5-0': '[MATH]-Hermiticity implies [MATH].', 'hep-lat-0001001-1-5-1': 'If we now start from some lattice Dirac operator [MATH] (obeying the assumption of the Nielsen-Ninomiya theorem such as absence of doublers, but otherwise quite arbitrary), we can construct a Ginsparg-Wilson operator [MATH] from it by enforcing eq. ([REF]) as [EQUATION]', 'hep-lat-0001001-1-5-2': 'This is the generalization [CITATION] of the "overlap formula", which uses the "standard GW kernel" [EQUATION] and which leads from the Wilson fermion [MATH] to the Neuberger fermion [MATH] [CITATION].', 'hep-lat-0001001-1-6-0': 'Any solution of the GWR is related to a fully chirally invariant Dirac operator [MATH], which is, however, non-local (in the free case, [MATH] has poles, cf. eq. ([REF])).', 'hep-lat-0001001-1-6-1': 'Vice versa, if we start from a [MATH] with this type of non-locality (such as the Rebbi fermion [CITATION], for example) we can construct a GW solution [CITATION] [MATH], which is local, at least in the free and weakly interacting case.', 'hep-lat-0001001-1-6-2': 'The mechanism of providing locality by inserting a local term [MATH] is known from the framework of perfect actions, where the factor [MATH] occurs in a Gaussian block variable renormalization group transformation term of the fermions [CITATION].', 'hep-lat-0001001-1-6-3': 'Hence [MATH] corresponds to a [MATH] function block variable transformation, and the corresponding perfect action has a Rebbi-type non-locality.', 'hep-lat-0001001-1-6-4': 'The transition to locality requires the superficial breaking of the full chiral symmetry, [MATH]: chirality is manifest in the action only in the sense of the GWR, but it is fully present in the physical observables [CITATION].', 'hep-lat-0001001-1-7-0': 'In contrast to the Rebbi fermion [CITATION], the axial anomaly is correctly reproduced [CITATION] for the perfect action at any local term [MATH], including the perfect [MATH] (for [MATH]).', 'hep-lat-0001001-1-7-1': 'This should also be checked if one generally wants to use [MATH] in an indirect way [CITATION], by measuring the right-hand side of [MATH].', 'hep-lat-0001001-1-8-0': 'By introducing a non-trivial kernel [MATH] we have relaxed the condition of chiral symmetry somewhat - without doing harm to the physical properties related to chirality [CITATION] - and this allows for locality of [MATH] (as well as the absence of doublers etc.)', 'hep-lat-0001001-1-9-0': 'without contradiction to the Nielsen-Ninomiya theorem.', 'hep-lat-0001001-1-9-1': 'In the case of the Neuberger fermion [MATH], locality has been proven in a smooth gauge background.', 'hep-lat-0001001-1-9-2': 'In particular, zero eigenvalues in [MATH] are excluded if the inequality (in [MATH] dimensions) [EQUATION] holds for any plaquette variable [MATH] and a suitable bound [MATH].', 'hep-lat-0001001-1-9-3': 'From Ref. [CITATION] we obtain [MATH], which has recently been improved to [MATH] [CITATION].', 'hep-lat-0001001-1-9-4': 'Still, this constraint is somewhat inconvenient; for instance, at least one eigenvalue of [MATH] has to cross zero if we want to change the topological sector.', 'hep-lat-0001001-1-10-0': 'Furthermore, the GWR seems to allow for locality only in the sense that the couplings in [MATH] decay exponentially, but not for ultralocality [CITATION].', 'hep-lat-0001001-1-10-1': 'To demonstrate this No-Go rule for GW fermions, it is sufficient to show it for the 2d free fermion.', 'hep-lat-0001001-1-10-2': 'In fact, a proof has been given specifically for [MATH] [CITATION], and it has later been generalized to all the cases, where [MATH] is ultralocal, which includes all ultralocal terms [MATH] [CITATION].', 'hep-lat-0001001-1-10-3': 'But the extension to all local [MATH] is not proven yet.', 'hep-lat-0001001-1-10-4': 'In that context, it is amusing to reconsider the ordinary Wilson fermion.', 'hep-lat-0001001-1-10-5': 'From the mass shift we know that it is certainly not a GW fermion in general, but how about the free fermion only ?', 'hep-lat-0001001-1-10-6': 'If we insert the free [MATH] into the GWR and solve for [MATH], we find that it decays [MATH] in [MATH], and [MATH] in [MATH].', 'hep-lat-0001001-1-10-7': 'Since this is nonlocal, not even the free [MATH] obeys a GWR.', 'hep-lat-0001001-1-11-0': 'Indeed, all known GW operators [MATH] decay exponentially even in the free case.', 'hep-lat-0001001-1-11-1': 'This is satisfactory from the conceptual point of view, but the existence of couplings over an infinite range is a problem for practical purposes.', 'hep-lat-0001001-1-11-2': 'One would hope for a high degree of locality at least, i.e. for a fast exponential decay.', 'hep-lat-0001001-1-11-3': 'Almost all the literature on overlap fermions solely deals with the Neuberger fermion, but it turns out that the couplings in [MATH] do not decay as fast as one would wish, see Fig. [REF].', 'hep-lat-0001001-1-11-4': 'Moreover, also other properties listed in Sec. 1 - most importantly scaling, but also approximate rotational invariance - are unfortunately rather poor.', 'hep-lat-0001001-1-11-5': 'This is obvious even from the free fermion, see Figs. [REF], [REF].', 'hep-lat-0001001-1-11-6': 'On the level of the action, there are generally no [MATH] artifacts for GW fermions, because any additional clover term is incompatible with the GWR [CITATION], but from the free case we see already that the [MATH] scaling artifacts in [MATH] are large.', 'hep-lat-0001001-1-11-7': 'And yet its simulation is tedious, allowing only for the use of small lattices.', 'hep-lat-0001001-1-12-0': 'However, [MATH] arises only from a very special choice in a large class of GWR solutions described by eq. ([REF]), namely [MATH] and [MATH].', 'hep-lat-0001001-1-12-1': 'The message of this report is that there are better options, and we are going to show in Sec. 3, 4 how improved overlap fermions can be constructed, tested and applied.', 'hep-lat-0001001-1-12-2': 'This generalization and the improvement concept for overlap fermions was introduced in Ref. [CITATION].', 'hep-lat-0001001-1-12-3': 'It was extensively tested in the framework of the Schwinger model [CITATION].', 'hep-lat-0001001-1-13-0': '## THE CONCEPT OF IMPROVING OVERLAP FERMIONS', 'hep-lat-0001001-1-14-0': 'We first summarize the main idea: if [MATH] happens to be a GW operator already (with respect to a fixed term [MATH]), then eq. ([REF]) yields [MATH]; the operator reproduces itself.', 'hep-lat-0001001-1-14-1': 'Therefore, any GW fermion - such as the perfect [CITATION] or the classically perfect [CITATION] fermion - is automatically an overlap fermion too.', 'hep-lat-0001001-1-14-2': 'The construction of a classically perfect action for asymptotically free models only requires minimization - and no (numeric) functional integral, in contrast to the perfect action - and it still has the additional virtues of excellent scaling and rotation invariance.', 'hep-lat-0001001-1-14-3': 'Unfortunately, a really powerful quasi-perfect action is not available so far for interacting fermions in [MATH].', 'hep-lat-0001001-1-15-0': 'However, if we manage to construct at least an approximate GW fermion, then we can expect it to change only modestly if we insert it as [MATH] in the overlap formula (for the corresponding [MATH]), [MATH].', 'hep-lat-0001001-1-15-1': 'If our approximate GW fermion is in addition short-ranged, then we can expect [MATH] to have a high degree of locality, since the long distance couplings are turned on just a little in [MATH].', 'hep-lat-0001001-1-15-2': '(Also [MATH] is short-ranged, but since this is far from a GW fermion, it changes a lot in the overlap formula, and those long distance couplings cannot be predicted to be tiny).', 'hep-lat-0001001-1-15-3': 'Similarly, if [MATH] scales well, then we can expect this quality to be essentially preserved in [MATH] if [MATH], and the same argument applies to the approximate rotation invariance.', 'hep-lat-0001001-1-15-4': 'In Sec. 3 we discuss examples for promising approximate GW operators.', 'hep-lat-0001001-1-15-5': 'In Sec. 4 they are transformed into exact GW fermions, and the above predictions are verified.', 'hep-lat-0001001-1-16-0': '# Short-ranged approximate Ginsparg-Wilson fermions', 'hep-lat-0001001-1-17-0': 'Perfect free fermions can be constructed and parameterized in c-space explicitly [CITATION].', 'hep-lat-0001001-1-17-1': 'If we choose the term [MATH] such that the locality is optimal at mass zero, then we arrive at the standard form [MATH].', 'hep-lat-0001001-1-17-2': 'To make such a fermion tractable, first of all its couplings have to be truncated to a short range.', 'hep-lat-0001001-1-17-3': 'A truncation to couplings inside a unit hypercube ("hypercube fermion", HF) was performed in Ref. [CITATION] by means of periodic boundary conditions.', 'hep-lat-0001001-1-17-4': '(Alternative 4d truncated perfect HFs can be found in Refs. [CITATION].)', 'hep-lat-0001001-1-17-5': 'Truncation causes some scaling artifacts, but they are small, so that the free HF is still strongly improved over the Wilson fermion.', 'hep-lat-0001001-1-17-6': 'This can be observed from the dispersion relation as well as thermodynamic scaling ratios [CITATION].', 'hep-lat-0001001-1-17-7': 'At the same time, truncation also implies a (small) violation of the GWR, in agreement with the absence of ultralocal GW fermions.', 'hep-lat-0001001-1-17-8': 'The spectrum of a GW fermion with [MATH], ([MATH]) is situated on a circle in [MATH] with center and radius [MATH] (GW circle), see eq. ([REF]).', 'hep-lat-0001001-1-17-9': 'Hence we can test the quality of our approximate GW fermion (with respect to [MATH]) by checking how close its spectrum comes to a unit circle.', 'hep-lat-0001001-1-17-10': 'This is shown in Fig. [REF] for the 4d HF of Ref. [CITATION] on a [MATH] lattice.', 'hep-lat-0001001-1-17-11': 'We see that we have a good approximation, especially in the physically important regime of eigenvalues close to 0.', 'hep-lat-0001001-1-18-0': 'In [MATH] we start off from a similar massless HF, which is optimized for its scaling behavior; its set of couplings is given in Ref. [CITATION], and its strong improvement over the Wilson fermion is visible in Fig. [REF].', 'hep-lat-0001001-1-18-1': 'The free spectrum of the "scaling optimal hypercube fermion" (SO-HF) is again close to a unit circle, on the same level as the 4d spectrum in Fig. [REF].', 'hep-lat-0001001-1-18-2': 'There are also other ways to see that we are in the vicinity of a GW fermion, for instance by summing over its violation (squared) in each site [CITATION], or by inserting the HF into the GWR and solving for [MATH]: for the SO-HF, [MATH] decays about 6 times faster than the corresponding (pseudo-)[MATH] for [MATH].', 'hep-lat-0001001-1-19-0': 'We now proceed to the 2-flavor Schwinger model, and we gauge the SO-HF by attaching the couplings to the shortest lattice paths only.', 'hep-lat-0001001-1-19-1': 'When there are several shortest paths, the coupling is split and attached to them in equal parts.', 'hep-lat-0001001-1-19-2': 'Moreover we add a clover term with coefficient 1.', 'hep-lat-0001001-1-19-3': 'For the gauge part we use the standard plaquette action (which is perfect for 2d pure [MATH] gauge theory [CITATION]).', 'hep-lat-0001001-1-20-0': 'This simple "gauging by hand" causes a further deviation from the GWR, which is increasingly manifest if the gauge background becomes rougher.', 'hep-lat-0001001-1-20-1': 'In Fig. [REF] we show the spectra for typical configurations at [MATH] and at [MATH].', 'hep-lat-0001001-1-21-0': 'It turns out, however, that the SO-HF is indeed an approximate GW fermion up to a considerable couplings strength.', 'hep-lat-0001001-1-21-1': 'Next we test the scaling behavior in the presence of gauge interaction.', 'hep-lat-0001001-1-21-2': 'Our simulation results (here and below) were obtained in collaboration with I. Hip.', 'hep-lat-0001001-1-21-3': 'They are based on 5000 quenched configurations on a [MATH] lattice at [MATH] (we use the same set of configurations for all types of fermions), but the evaluation does include the fermion determinant, following Ref. [CITATION].', 'hep-lat-0001001-1-21-4': 'Fig. [REF] shows the dispersion relations for the two meson-type states, a massless triplet and one massive mode [CITATION], which we denote as [MATH] and [MATH] (by analogy).', 'hep-lat-0001001-1-21-5': 'Again the SO-HF is drastically improved over the Wilson fermion (at [MATH], [CITATION]).', 'hep-lat-0001001-1-21-6': 'It even reaches the same level as a (very mildly truncated) classically perfect action [CITATION], which was parameterized by 123 independent couplings per site, whereas only 6 such couplings are used for the SO-HF.', 'hep-lat-0001001-1-21-7': 'Therefore, it is realistic to extend the HF formulation to QCD, and in fact it has been shown already that minimally gauged 4d HFs can indeed be applied in QCD simulations [CITATION].', 'hep-lat-0001001-1-21-8': 'However, in QCD as well as in the Schwinger model, we observed as an unpleasant feature of the directly applied HF a strong additive mass renormalization.', 'hep-lat-0001001-1-21-9': 'Using the SO-HF, even at [MATH] the [MATH] mass is renormalized from 0 to 0.13, as Fig. [REF] shows.', 'hep-lat-0001001-1-21-10': 'This corresponds to a lowest real eigenvalue around 0.03, cf. Fig. [REF].', 'hep-lat-0001001-1-21-11': 'As an even more striking example, the 4d massless HF, minimally gauged by hand and applied to QCD at [MATH], leads to a "pion mass" of 3.0 [CITATION], and at [MATH] the critical bare HF mass amounts to [MATH] [CITATION].', 'hep-lat-0001001-1-21-12': 'We overcome this problem in the next section by inserting the SO-HF into the overlap formula.', 'hep-lat-0001001-1-22-0': '# Improved overlap fermions', 'hep-lat-0001001-1-23-0': 'We now perform the second step in our program and insert the SO-HF - which is an approximate GW fermion with an excellent scaling behavior - into the overlap formula ([REF]) with [MATH] given in eq. ([REF]).', 'hep-lat-0001001-1-23-1': 'This leads to an exact GW fermion and therefore to all the nice properties related to chirality, which are extensively discussed in the recent literature on GW fermions: correct anomalies; no renormalization of mass zero, vector current and flavor non-singlet axial vector current; no mixing of weak matrix elements; no exceptional configurations [CITATION]; correctly reproduced chiral symmetry breaking [CITATION] etc.', 'hep-lat-0001001-1-24-0': 'Our first prediction was that the level of locality should be improved over the Neuberger fermion, and this is clearly confirmed, both, in the free and in the interacting case.', 'hep-lat-0001001-1-24-1': 'Fig. [REF] compares the decay of the couplings of the free fermion (left) and the decay of the "maximal correlation" [MATH] over a certain distance [MATH] - as suggested in Ref. [CITATION] - at [MATH] (right).', 'hep-lat-0001001-1-25-0': 'Next we want to verify if the good scaling quality survives the modification due to the overlap formula.', 'hep-lat-0001001-1-25-1': 'For the free fermions, this is confirmed in an impressive way, see Fig. [REF].', 'hep-lat-0001001-1-26-0': 'On fine lattices, the scaling is practically identical before and after the use of the overlap formula; only as the lattice becomes really coarse, the overlap does some harm to the scaling quality at some point.', 'hep-lat-0001001-1-27-0': 'In the presence of gauge interaction, we consider again the "meson" dispersions, and we observe again the persistence of the improvement, Fig. [REF].', 'hep-lat-0001001-1-28-0': 'Furthermore, continuous rotation invariance is approximated much better for the HF than for the Wilson fermion (after all, also this holds exactly for the perfect fermion [CITATION]).', 'hep-lat-0001001-1-28-1': 'In Ref. [CITATION] this was also tested in the interacting case by measuring how smoothly correlations decay with the Euclidean distance, and we observed that the SO-HF is again by far superior over the Wilson fermion; once more it reaches the same level as the classically perfect action.', 'hep-lat-0001001-1-28-2': 'The overlap formula ([REF]) suggests that this property is essentially inherited for the overlap fermions, and indeed we observed a similar level for the Wilson fermion and the Neuberger fermion on one hand, and for the SO-HF and the overlap SO-HF on the other hand.', 'hep-lat-0001001-1-28-3': 'This confirms that also the strongly improved approximate rotational invariance of the SO-HF survives if it is turned into an overlap fermion.', 'hep-lat-0001001-1-28-4': 'For the free fermion, this progress can also been observed in Fig. [REF] (left).', 'hep-lat-0001001-1-29-0': '## Chiral correction in terms of a power series', 'hep-lat-0001001-1-30-0': 'In [MATH] we can afford an exact evaluation of the notorious square root in the overlap formula, but in QCD this is not feasible any more.', 'hep-lat-0001001-1-30-1': 'In the recent literature, a number of iterative procedures have been suggested for the Neuberger fermion.', 'hep-lat-0001001-1-30-2': 'In Ref. [CITATION] we recommend a new method, which is very simple and robust, and which is especially designed for the case where [MATH] is an approximate GW fermion already.', 'hep-lat-0001001-1-30-3': 'We evaluate the square root in [EQUATION] as a power series in [MATH].', 'hep-lat-0001001-1-30-4': 'For [MATH] we have [MATH] (if the configuration is not extremely rough), hence the expansion converges quickly.', 'hep-lat-0001001-1-30-5': 'On the other hand, for the case of the Neuberger fermion, i.e. for [MATH], this expansion fails to converge even in the free case, which is presumably the reason why it had not been considered in the earlier literature.', 'hep-lat-0001001-1-30-6': 'We call this method a "perturbative chiral correction", where the perturbative expansion refers to the GWR violation [MATH] (and not to the coupling [MATH]).', 'hep-lat-0001001-1-30-7': 'It yields a Dirac operator of the form [EQUATION]', 'hep-lat-0001001-1-30-8': 'For the chiral correction to [MATH], [MATH] is a polynomial in [MATH] of order [MATH] (for instance, [MATH] for [MATH], [MATH] for [MATH], etc.).', 'hep-lat-0001001-1-30-9': 'Hence the computational effort amounts roughly to [MATH] matrix-vector multiplication (the matrix being [MATH] or [MATH]), i.e. it increases only linearly.', 'hep-lat-0001001-1-30-10': 'Actually this represents a fermion with couplings of range [MATH] in each component, and it would be very tedious to implement it explicitly even for [MATH].', 'hep-lat-0001001-1-30-11': 'However, due to its specific form we never need to do so; [MATH] can always be evaluated by iteration of the above matrix-vector products.', 'hep-lat-0001001-1-31-0': 'The crucial question now is if the first few orders are sufficient already to do most of the chiral correction.', 'hep-lat-0001001-1-31-1': 'We first look at the free SO-HF, and Fig. [REF] (left) shows that the first order alone does practically the full job.', 'hep-lat-0001001-1-31-2': 'In this context, we also obtain a geometric picture of the effect of the overlap formula: for [MATH] it can be viewed as a projection of the eigenvalues onto the circle with center and radius [MATH].', 'hep-lat-0001001-1-31-3': 'This projection is often close to radial.', 'hep-lat-0001001-1-32-0': 'In the interacting case, the convergence to the circle under iteration is slowest in the arc around 0.', 'hep-lat-0001001-1-32-1': 'As an example, we show in Fig. [REF] (right) a histogram of the small real eigenvalues at [MATH].', 'hep-lat-0001001-1-32-2': 'We see that the mass renormalization is removed almost completely if we proceed to [MATH].', 'hep-lat-0001001-1-33-0': '## The behavior at extremely rough configurations', 'hep-lat-0001001-1-34-0': 'For smooth configurations, it is easy to find a parameter [MATH] - and hence a GW circle where the spectrum is mapped on - such that the small (large) real eigenvalues are mapped on 0 ([MATH]).', 'hep-lat-0001001-1-34-1': 'This leaves the index unchanged, and it provides also a sensible definition of the topological charge via the index theorem [CITATION].', 'hep-lat-0001001-1-34-2': 'It also means that the doubling problem is safely avoided for all typical configurations at moderate or large [MATH].', 'hep-lat-0001001-1-35-0': 'However, if we dare proceeding to extremely strong coupling, then it is not possible any more to find such a center [MATH] of a GW circle, which does the right mapping for all typical configurations.', 'hep-lat-0001001-1-35-1': 'In fact, for extremely rough QCD test configurations (on very small lattices) it was observed explicitly that all the eigenvalues of the minimally gauged [MATH] are close to the arc of the GW circle, which is opposite to 0 [CITATION].', 'hep-lat-0001001-1-35-2': 'For [MATH] the eigenvalues are scattered over a wide area with a large real part.', 'hep-lat-0001001-1-35-3': 'Examples for such spectra of an extremely rough configuration are shown in Fig. [REF], which was provided by N. Eicker, I. Hip and Th.', 'hep-lat-0001001-1-35-4': 'Lippert.', 'hep-lat-0001001-1-35-5': 'At a coupling strength where such configurations are frequent, the doubling problem is back for those overlap fermions, which are constructed from some simple [MATH] (for the Schwinger model, this problem sets in around [MATH] [CITATION]).', 'hep-lat-0001001-1-35-6': 'This agrees with the result of a strong coupling expansion (in the Hamiltonian formulation) which applies to [MATH] [CITATION].', 'hep-lat-0001001-1-35-7': 'As we mentioned earlier, also locality is in danger in that regime [CITATION], and we should therefore keep away from it.', 'hep-lat-0001001-1-35-8': 'As one more advantage of choosing [MATH] to be an approximate GW fermion, the regime of [MATH] where we are (statistically) on safe grounds is enlarged compared to the Neuberger fermion.', 'hep-lat-0001001-1-36-0': 'Of course, in the safe regime where [MATH] is large enough (in QCD this includes [MATH] for sure [CITATION]) the chiral correction of [MATH] can also be carried out by iteration methods different from the one described in this subsection, for alternative experiments see Ref. [CITATION].', 'hep-lat-0001001-1-36-1': 'The efficiency in QCD is still to be compared, but for sure in any method the convergence will be much faster for [MATH] than for [MATH].', 'hep-lat-0001001-1-37-0': '# Conclusions', 'hep-lat-0001001-1-38-0': 'Our program outlined in Sec. 2.1 has been realized in the Schwinger model, and the properties of a resulting improved overlap fermion have been tested extensively.', 'hep-lat-0001001-1-38-1': 'They are all clearly superior over the Neuberger fermion, confirming our prediction: the overlap SO-HF scales much better, it is more local and it comes much closer to rotation invariance.', 'hep-lat-0001001-1-39-0': 'The question now is the applicability of this program in [MATH].', 'hep-lat-0001001-1-39-1': 'The 4d HF formulation is worked out already, and the corresponding improved overlap fermion is currently under investigation in QCD by the SESAM collaboration in Julich and Wuppertal.'}

{'hep-lat-0001001-2-0-0': '# Introduction', 'hep-lat-0001001-2-1-0': 'If we construct a lattice fermion formulation, there are a number of goals to be considered: doubling should be avoided; even at finite lattice spacing [MATH], we want to represent chiral symmetry in a sound way; and we are seeking a good scaling behavior.', 'hep-lat-0001001-2-1-1': 'Conceptually we have to require locality (the lattice Dirac operator [MATH] has to decay at least exponentially in [MATH] ).', 'hep-lat-0001001-2-1-2': 'In addition, for practical purposes we desire a high level of locality, i.e. a fast exponential decay or even ultralocality (which means that the couplings in [MATH] drop to zero beyond a finite number of lattice spacings).', 'hep-lat-0001001-2-1-3': 'A further issue is a good approximation to rotation invariance.', 'hep-lat-0001001-2-1-4': 'Last but not least, the formulation should be simple enough to allow for efficient simulations.', 'hep-lat-0001001-2-1-5': 'Here we report on a construction, which is designed to do justice to all of these goals.', 'hep-lat-0001001-2-2-0': '# Ginsparg-Wilson fermions (an unconventional introduction)', 'hep-lat-0001001-2-3-0': 'For a lattice Dirac operator [MATH], full chiral invariance ( [MATH] ) is incompatible with other basic requirements (Hermiticity, locality, absence of doublers, discrete translation invariance) [CITATION].', 'hep-lat-0001001-2-3-1': 'Therefore we only implement a modified chiral symmetry, which does allow [MATH] to fulfill those requirements.', 'hep-lat-0001001-2-3-2': 'For such a modified chiral transformation we start from the ansatz [EQUATION] [MATH] being an infinitesimal transformation parameter.', 'hep-lat-0001001-2-3-3': 'The transformation, and therefore [MATH] and [MATH] should be local, and [MATH], so that we reproduce the full chiral symmetry in the (naive) continuum limit.', 'hep-lat-0001001-2-4-0': 'Invariance of the Lagrangian [MATH] holds to [MATH] if [EQUATION]', 'hep-lat-0001001-2-4-1': 'This implies a continuous modified chiral symmetry, which has the full number of generators.', 'hep-lat-0001001-2-4-2': 'It may be compared to the remnant chiral symmetry of staggered fermions: there the doubling problem is not solved, and one is only left with a [MATH] symmetry, which does, however, protect the mass from additive renormalization.', 'hep-lat-0001001-2-4-3': 'The same can be shown here if we assume "[MATH]-Hermiticity", [MATH], and we choose [MATH], [MATH], where [MATH] is local again, non-trivial and [MATH] (this generalizes Ref. [CITATION]).', 'hep-lat-0001001-2-4-4': 'Then eq. ([REF]) turns into the Ginsparg-Wilson relation (GWR) [CITATION] [EQUATION] and it implies the absence of additive mass renormalization (see also Ref. [CITATION]) since [EQUATION]', 'hep-lat-0001001-2-4-5': 'As another illustration we can write the GWR as [MATH], and we see that a local term [MATH] does not shift the poles in [MATH] (in contrast to the cases where [MATH] is local, such as a mass or a Wilson term).', 'hep-lat-0001001-2-5-0': '[With respect to the general ansatz, we have to require the right-hand side of [MATH] to be local.]', 'hep-lat-0001001-2-6-0': '[MATH]-Hermiticity implies [MATH].', 'hep-lat-0001001-2-6-1': 'If we now start from some lattice Dirac operator [MATH] (obeying the assumption of the Nielsen-Ninomiya theorem such as absence of doublers, but otherwise quite arbitrary), we can construct a Ginsparg-Wilson operator [MATH] from it by enforcing eq. ([REF]) as [EQUATION]', 'hep-lat-0001001-2-6-2': 'This is the generalization [CITATION] of the "overlap formula", which uses the "standard GW kernel" [EQUATION] and which leads from the Wilson fermion [MATH] to the Neuberger fermion [MATH] [CITATION].', 'hep-lat-0001001-2-6-3': 'In the general solution of the GWR, eq. ([REF]), we can obviously vary the parameters in [MATH] or in [MATH] (or in both) in many ways, without violating their required properties.', 'hep-lat-0001001-2-6-4': 'This shows that there exists a continuous set of GWR solutions [MATH] in the space of coupling parameters.', 'hep-lat-0001001-2-7-0': 'Any solution of the GWR is related to a fully chirally invariant Dirac operator [MATH], which is, however, non-local (in the free case, [MATH] has poles, cf. eq. ([REF])).', 'hep-lat-0001001-2-7-1': 'Vice versa, if we start from some [MATH] with this type of non-locality (such as the Rebbi fermion [CITATION], for example) we can construct a GW solution [CITATION] [MATH], which is local, at least in the free and weakly interacting case.', 'hep-lat-0001001-2-7-2': 'The mechanism of providing locality by inserting a local term [MATH] is known from the framework of perfect actions, where the factor [MATH] occurs in a Gaussian block variable renormalization group transformation term of the fermions [CITATION].', 'hep-lat-0001001-2-7-3': 'Hence [MATH] corresponds to a [MATH] function block variable transformation, and the corresponding perfect action has a Rebbi-type non-locality.', 'hep-lat-0001001-2-7-4': 'The transition to locality requires the superficial breaking of the full chiral symmetry, [MATH]: chirality is manifest in the action only in the sense of the GWR, but it is fully present in the physical observables [CITATION].', 'hep-lat-0001001-2-8-0': 'In contrast to the Rebbi fermion [CITATION], the axial anomaly is correctly reproduced [CITATION] for the perfect action at any local term [MATH], including the perfect [MATH] (for [MATH]).', 'hep-lat-0001001-2-8-1': 'This should also be checked if one generally wants to use [MATH] in an indirect way [CITATION], by measuring the right-hand side of [MATH].', 'hep-lat-0001001-2-9-0': 'By introducing a non-trivial kernel [MATH] we have relaxed the condition of chiral symmetry somewhat - without doing harm to the physical properties related to chirality [CITATION] - and this allows for locality of [MATH] (as well as the absence of doublers etc.)', 'hep-lat-0001001-2-10-0': 'without contradiction to the Nielsen-Ninomiya theorem.', 'hep-lat-0001001-2-10-1': 'In the case of the Neuberger fermion [MATH], locality has been demonstrated in a smooth gauge background.', 'hep-lat-0001001-2-10-2': 'In particular, zero eigenvalues in [MATH] are excluded if the inequality (in [MATH] dimensions) [EQUATION] holds for any plaquette variable [MATH] and a suitable bound [MATH].', 'hep-lat-0001001-2-10-3': 'From Ref. [CITATION] we obtain [MATH], which has recently been improved to [MATH] [CITATION].', 'hep-lat-0001001-2-10-4': 'Still, this constraint is somewhat inconvenient; for instance, at least one eigenvalue of [MATH] has to cross zero if we want to change the topological sector.', 'hep-lat-0001001-2-11-0': 'Furthermore, the GWR allows for locality only in the sense that the couplings in [MATH] decay exponentially, but not for ultralocality.', 'hep-lat-0001001-2-11-1': 'This was conjectured intuitively in Ref. [CITATION].', 'hep-lat-0001001-2-11-2': 'To demonstrate this No-Go rule for GW fermions, it is sufficient to show it for the free fermion.', 'hep-lat-0001001-2-11-3': 'A proof, which was specifically restricted to [MATH], has been given in Ref. [CITATION].', 'hep-lat-0001001-2-11-4': 'By now, a complete proof covering all (local) GW kernels [MATH] has been added, hence this rule is completely general [CITATION].', 'hep-lat-0001001-2-12-0': 'In that context, it is amusing to reconsider the ordinary Wilson fermion.', 'hep-lat-0001001-2-12-1': 'From the mass shift we know that it is certainly not a GW fermion in general, and according to our No-Go rule not even the free Wilson fermions can obey any GWR.', 'hep-lat-0001001-2-12-2': 'Indeed, if we insert the free [MATH] into the GWR and solve for [MATH], we find that it decays [MATH] in [MATH], and [MATH] in [MATH], which is nonlocal and therefore not a GW kernel.', 'hep-lat-0001001-2-13-0': 'The exponential decay of [MATH] is satisfactory from the conceptual point of view, but the existence of couplings over an infinite range is a problem for practical purposes.', 'hep-lat-0001001-2-13-1': 'One would hope for a high degree of locality at least, i.e. for a fast exponential decay.', 'hep-lat-0001001-2-13-2': 'Almost all the literature on overlap fermions solely deals with the Neuberger fermion, but it turns out that the couplings in [MATH] do not decay as fast as one would wish, see Fig. [REF].', 'hep-lat-0001001-2-13-3': 'Moreover, also other properties listed in Sec. 1 - most importantly scaling, but also approximate rotational invariance - are unfortunately rather poor.', 'hep-lat-0001001-2-13-4': 'This is obvious even from the free fermion, see Figs. [REF], [REF].', 'hep-lat-0001001-2-13-5': 'On the level of the action, there are generally no [MATH] artifacts for GW fermions, because any additional clover term violates the GWR [CITATION], but from the free case we see already that the [MATH] scaling artifacts in [MATH] are large.', 'hep-lat-0001001-2-13-6': 'And yet its simulation is tedious (the quenched case requires already a similar effort as simulating [MATH] with dynamical fermions [CITATION]), allowing only for the use of small lattices.', 'hep-lat-0001001-2-14-0': 'However, [MATH] arises only from a very special choice in a large class of GWR solutions described by eq. ([REF]), namely [MATH] and [MATH].', 'hep-lat-0001001-2-14-1': 'The message of this report is that there are better options, and we are going to show in Sec. 3, 4 how improved overlap fermions can be constructed, tested and applied.', 'hep-lat-0001001-2-14-2': 'This generalization and the improvement concept for overlap fermions was introduced in Ref. [CITATION].', 'hep-lat-0001001-2-14-3': 'It was extensively tested in the framework of the Schwinger model [CITATION].', 'hep-lat-0001001-2-15-0': '## THE CONCEPT OF IMPROVING OVERLAP FERMIONS', 'hep-lat-0001001-2-16-0': 'We first summarize the main idea: if [MATH] happens to be a GW operator already (with respect to a fixed term [MATH]), then eq. ([REF]) yields [MATH]; the operator reproduces itself.', 'hep-lat-0001001-2-16-1': 'Therefore, any GW fermion - such as the perfect [CITATION] or the classically perfect [CITATION] fermion - is automatically an overlap fermion too.', 'hep-lat-0001001-2-16-2': 'The construction of a classically perfect action for asymptotically free models only requires minimization - no (numeric) functional integral, in contrast to the perfect action - and it still has the additional virtues of excellent scaling and rotation invariance.', 'hep-lat-0001001-2-16-3': 'Unfortunately, a really powerful quasi-perfect action is not available so far for interacting fermions in [MATH].', 'hep-lat-0001001-2-17-0': 'However, if we manage to construct at least an approximate GW fermion, then we can expect it to change only modestly if we insert it as [MATH] in the overlap formula (for the corresponding [MATH]), [MATH].', 'hep-lat-0001001-2-17-1': 'If our approximate GW fermion is in addition short-ranged, then we can expect [MATH] to have a high degree of locality, since the long distance couplings are turned on just a little in [MATH].', 'hep-lat-0001001-2-17-2': '(Also [MATH] is short-ranged, but since this is far from a GW fermion, it changes a lot in the overlap formula, and those long distance couplings cannot be predicted to be tiny).', 'hep-lat-0001001-2-17-3': 'Similarly, if [MATH] scales well, then we can expect this quality to be essentially preserved in [MATH] if [MATH], and the same argument applies to the approximate rotation invariance.', 'hep-lat-0001001-2-17-4': 'In Sec. 3 we discuss examples for promising approximate GW operators.', 'hep-lat-0001001-2-17-5': 'In Sec. 4 they are transformed into exact GW fermions, and the above predictions are verified.', 'hep-lat-0001001-2-18-0': '# Short-ranged approximate Ginsparg-Wilson fermions', 'hep-lat-0001001-2-19-0': 'Perfect free fermions can be constructed and parameterized in c-space explicitly [CITATION].', 'hep-lat-0001001-2-19-1': 'If we choose the term [MATH] such that the locality is optimal at mass zero, then we arrive at the standard form [MATH].', 'hep-lat-0001001-2-19-2': 'To make such a fermion tractable, first of all its couplings have to be truncated to a short range.', 'hep-lat-0001001-2-19-3': 'A truncation to couplings inside a unit hypercube ("hypercube fermion", HF) was performed in Ref. [CITATION] by means of periodic boundary conditions.', 'hep-lat-0001001-2-19-4': '(Alternative 4d truncated perfect HFs can be found in Refs. [CITATION].)', 'hep-lat-0001001-2-19-5': 'Truncation causes some scaling artifacts, but they are small, so that the free HF is still strongly improved over the Wilson fermion.', 'hep-lat-0001001-2-19-6': 'This can be observed from the dispersion relation as well as thermodynamic scaling ratios [CITATION].', 'hep-lat-0001001-2-19-7': 'At the same time, truncation also implies a (small) violation of the GWR, in agreement with the absence of ultralocal GW fermions.', 'hep-lat-0001001-2-19-8': 'The spectrum of a GW fermion with [MATH], ([MATH]) is situated on a circle in [MATH] with center and radius [MATH] (GW circle), see eq. ([REF]).', 'hep-lat-0001001-2-19-9': 'Hence we can test the quality of our approximate GW fermion (with respect to [MATH]) by checking how close its spectrum comes to a unit circle.', 'hep-lat-0001001-2-19-10': 'This is shown in Fig. [REF] for the 4d HF of Ref. [CITATION] on a [MATH] lattice.', 'hep-lat-0001001-2-19-11': 'We see that we have a good approximation, especially in the physically important regime of eigenvalues close to 0.', 'hep-lat-0001001-2-20-0': 'In [MATH] we start off from a similar massless HF, which is optimized for its scaling behavior; its set of couplings is given in Ref. [CITATION], and its strong improvement over the Wilson fermion is visible in Fig. [REF].', 'hep-lat-0001001-2-21-0': 'The free spectrum of the "scaling optimal hypercube fermion" (SO-HF) is again close to a unit circle, on the same level as the 4d spectrum in Fig. [REF].', 'hep-lat-0001001-2-21-1': 'There are also other ways to see that we are in the vicinity of a GW fermion, for instance by summing over its violation (squared) in each site [CITATION], or by inserting the HF into the GWR and solving for [MATH]: for the SO-HF, [MATH] decays about 6 times faster than the corresponding (pseudo-)[MATH] for [MATH].', 'hep-lat-0001001-2-21-2': 'If we compare the truncated perfect HF to [MATH], then this factor even amounts to [MATH] in [MATH], and to [MATH] in [MATH].', 'hep-lat-0001001-2-22-0': 'We now proceed to the 2-flavor Schwinger model, and we gauge the SO-HF by attaching the couplings to the shortest lattice paths only.', 'hep-lat-0001001-2-22-1': 'When there are several shortest paths, the coupling is split and attached to them in equal parts.', 'hep-lat-0001001-2-22-2': 'Moreover we add a clover term with coefficient 1.', 'hep-lat-0001001-2-22-3': 'For the gauge part we use the standard plaquette action (which is perfect for 2d pure [MATH] gauge theory [CITATION]).', 'hep-lat-0001001-2-23-0': 'This simple "gauging by hand" causes a further deviation from the GWR, which is increasingly manifest if the gauge background becomes rougher.', 'hep-lat-0001001-2-23-1': 'In Fig. [REF] we show the spectra for typical configurations at [MATH] and at [MATH].', 'hep-lat-0001001-2-24-0': 'It turns out, however, that the SO-HF is indeed an approximate GW fermion up to a considerable couplings strength.', 'hep-lat-0001001-2-24-1': 'Next we test the scaling behavior in the presence of gauge interaction.', 'hep-lat-0001001-2-24-2': 'Our simulation results (here and below) were obtained in collaboration with I. Hip.', 'hep-lat-0001001-2-24-3': 'They are based on 5000 quenched configurations on a [MATH] lattice at [MATH] (we use the same set of configurations for all types of fermions), but the evaluation does include the fermion determinant, following Ref. [CITATION].', 'hep-lat-0001001-2-24-4': 'Fig. [REF] shows the dispersion relations for the two meson-type states, a massless triplet and one massive mode [CITATION], which we denote as [MATH] and [MATH] (by analogy).', 'hep-lat-0001001-2-24-5': 'Again the SO-HF is drastically improved over the Wilson fermion (at [MATH], from Ref. [CITATION]).', 'hep-lat-0001001-2-24-6': 'It even reaches the same level as a (very mildly truncated) classically perfect action [CITATION], which was parameterized by 123 independent couplings per site, whereas only 6 such couplings are used for the SO-HF.', 'hep-lat-0001001-2-24-7': 'Therefore, it is realistic to extend the HF formulation to QCD, and in fact it has been shown already that minimally gauged 4d HFs can indeed be applied in QCD simulations [CITATION].', 'hep-lat-0001001-2-24-8': 'However, in QCD as well as in the Schwinger model, we observed as an unpleasant feature of the directly applied HF a strong additive mass renormalization.', 'hep-lat-0001001-2-24-9': 'Using the SO-HF, even at [MATH] the [MATH] mass is renormalized from 0 to 0.13, as Fig. [REF] shows.', 'hep-lat-0001001-2-24-10': 'This corresponds to a lowest real eigenvalue around 0.03, cf. Fig. [REF].', 'hep-lat-0001001-2-24-11': 'As an even more striking example, the 4d massless HF, minimally gauged by hand and applied to QCD at [MATH], leads to a "pion mass" of 3.0 [CITATION], and at [MATH] the critical bare HF mass amounts to [MATH] [CITATION] (which is unfavorable for the level of locality before truncation, and hence for the magnitude of the truncation effects).', 'hep-lat-0001001-2-24-12': 'We overcome this problem in the next section by inserting the SO-HF into the overlap formula.', 'hep-lat-0001001-2-25-0': '[As a further alternative to negative bare mass and overlap, we can reach the exact chiral limit solely by the use of fat links [CITATION].', 'hep-lat-0001001-2-25-1': 'This is essentially equivalent to gauging the HF such that the interacting GWR is just violated modestly.', 'hep-lat-0001001-2-25-2': 'As yet another possibility one may attach an amplification factor [MATH] to each link [CITATION].]', 'hep-lat-0001001-2-26-0': '# Improved overlap fermions', 'hep-lat-0001001-2-27-0': 'We now perform the second step in our program and insert the SO-HF - which is an approximate GW fermion with an excellent scaling behavior - into the overlap formula ([REF]) with [MATH] given in eq. ([REF]).', 'hep-lat-0001001-2-27-1': 'This leads to an exact GW fermion and therefore to all the nice properties related to chirality, which are extensively discussed in the recent literature on GW fermions: correct anomalies; no renormalization of mass zero, vector current and flavor non-singlet axial vector current; no mixing of weak matrix elements; no exceptional configurations [CITATION]; correctly reproduced chiral symmetry breaking [CITATION] etc.', 'hep-lat-0001001-2-28-0': 'Our first prediction was that the level of locality should be improved over the Neuberger fermion, and this is clearly confirmed, both, in the free and in the interacting case.', 'hep-lat-0001001-2-28-1': 'Fig. [REF] compares the decay of the couplings of the free fermion (left) and the decay of the "maximal correlation" [MATH] over a certain distance [MATH] - as suggested in Ref. [CITATION] - at [MATH] (right).', 'hep-lat-0001001-2-29-0': '(In the latter figure one could still try to improve the locality in both cases by deviating from [MATH] and tuning [MATH] for optimal locality.)', 'hep-lat-0001001-2-30-0': 'Next we want to verify if the good scaling quality survives the modification due to the overlap formula.', 'hep-lat-0001001-2-30-1': 'For the free fermions, this is confirmed in an impressive way, see Fig. [REF].', 'hep-lat-0001001-2-31-0': 'On fine lattices, the scaling is practically identical before and after the use of the overlap formula; only as the lattice becomes really coarse, the overlap does some harm to the scaling quality at some point.', 'hep-lat-0001001-2-32-0': 'In the presence of gauge interaction, we consider again the "meson" dispersions, and we observe again the persistence of the improvement, see Fig. [REF].', 'hep-lat-0001001-2-33-0': 'Furthermore, continuous rotation invariance is approximated much better for the HF than for the Wilson fermion (after all, also this holds exactly for the perfect fermion [CITATION]).', 'hep-lat-0001001-2-33-1': 'In Ref. [CITATION] this was also tested in the interacting case by measuring how smoothly correlations decay with the Euclidean distance, and we observed that the SO-HF is again by far superior over the Wilson fermion; once more it reaches the same level as the classically perfect action.', 'hep-lat-0001001-2-33-2': 'The overlap formula ([REF]) suggests that this property is essentially inherited for the overlap fermions, and indeed we observed a similar level for the Wilson fermion and the Neuberger fermion on one hand, and for the SO-HF and the overlap SO-HF on the other hand.', 'hep-lat-0001001-2-33-3': 'This confirms that also the strongly improved approximate rotational invariance of the SO-HF survives if it is turned into an overlap fermion.', 'hep-lat-0001001-2-33-4': 'For the free fermion, this progress can also been observed in Fig. [REF] (left) from the width of the "cones".', 'hep-lat-0001001-2-34-0': '## Chiral correction in terms of a power series', 'hep-lat-0001001-2-35-0': 'In [MATH] we can afford an exact evaluation of the notorious square root in the overlap formula, but in QCD this is not feasible any more.', 'hep-lat-0001001-2-35-1': 'In the recent literature, a number of iterative procedures have been suggested for the Neuberger fermion [CITATION].', 'hep-lat-0001001-2-35-2': 'In Ref. [CITATION] we presented a new method, which is very simple and robust, and which is especially designed for the case where [MATH] is an approximate GW fermion already.', 'hep-lat-0001001-2-35-3': 'We evaluate the square root in [EQUATION] as a power series in [MATH].', 'hep-lat-0001001-2-35-4': 'For [MATH] we have [MATH] (if the configuration is not extremely rough), hence the expansion converges rapidly.', 'hep-lat-0001001-2-35-5': 'On the other hand, for the case of the Neuberger fermion, i.e. for [MATH], this expansion fails to converge even in the free case, which is presumably the reason why it had not been considered in the earlier literature.', 'hep-lat-0001001-2-35-6': 'We call this method a "perturbative chiral correction", where the perturbative expansion refers to the GWR violation [MATH] (and not to the coupling [MATH]).', 'hep-lat-0001001-2-35-7': 'It yields a Dirac operator of the form [EQUATION]', 'hep-lat-0001001-2-35-8': 'For the chiral correction to [MATH], [MATH] is a polynomial in [MATH] of order [MATH], for instance, [EQUATION]', 'hep-lat-0001001-2-35-9': 'Hence the computational effort amounts roughly to [MATH] matrix-vector multiplication (the matrix being [MATH] or [MATH]), i.e. it increases only linearly (though the convergence is also just linear).', 'hep-lat-0001001-2-35-10': 'Actually this represents a fermion with couplings of range [MATH] in each component, and it would be very tedious to implement it explicitly, even for [MATH].', 'hep-lat-0001001-2-35-11': 'However, due to its specific form we never need to do so; [MATH] can always be evaluated by iteration of the above matrix-vector products.', 'hep-lat-0001001-2-36-0': 'The crucial question now is if the first few orders are sufficient already to do most of the chiral correction.', 'hep-lat-0001001-2-36-1': 'We first look at the free SO-HF, and Fig. [REF] (left) shows that the first order alone does practically the full job.', 'hep-lat-0001001-2-36-2': 'In this context, we also obtain a geometric picture of the effect of the overlap formula: for [MATH] it can be viewed as a projection of the eigenvalues onto the circle with center and radius [MATH].', 'hep-lat-0001001-2-36-3': 'This projection is often close to radial.', 'hep-lat-0001001-2-37-0': 'In the interacting case, the convergence to the circle under iteration is slowest in the arc around 0.', 'hep-lat-0001001-2-37-1': 'As an example, we show in Fig. [REF] (right) a histogram of the small real eigenvalues at [MATH].', 'hep-lat-0001001-2-37-2': 'We see that the mass renormalization is removed almost completely if we proceed to [MATH].', 'hep-lat-0001001-2-38-0': '## Behavior in extremely rough gauge configurations', 'hep-lat-0001001-2-39-0': 'For smooth configurations, it is easy to find a parameter [MATH] - and hence a GW circle where the spectrum is mapped on - such that the small (large) real eigenvalues are mapped on 0 ([MATH]).', 'hep-lat-0001001-2-39-1': 'This leaves the index unchanged (though it is defined by exact zero modes now) and it provides a sensible definition of the topological charge via the index theorem [CITATION].', 'hep-lat-0001001-2-39-2': 'It also means that the doubling problem is safely avoided for all typical configurations at moderate or large [MATH].', 'hep-lat-0001001-2-40-0': 'However, if we dare proceeding to extremely strong coupling, then it is not possible any more to find such a center [MATH] of a GW circle, which does the right mapping for all typical configurations.', 'hep-lat-0001001-2-40-1': 'In fact, for extremely rough QCD test configurations (on very small lattices) it was observed explicitly that all the eigenvalues of the minimally gauged [MATH] are close to the arc of the GW circle, which is opposite to 0 [CITATION].', 'hep-lat-0001001-2-40-2': 'For [MATH] the eigenvalues are densely scattered over a wide area with a large real part.', 'hep-lat-0001001-2-40-3': 'Examples for such spectra of an extremely rough configuration are shown in Fig. [REF], which was provided by N. Eicker, I. Hip and Th.', 'hep-lat-0001001-2-40-4': 'Lippert.', 'hep-lat-0001001-2-40-5': 'At a coupling strength where such configurations are frequent, the doubling problem is back for those overlap fermions, which are constructed from some simple [MATH] (for the Schwinger model, this problem sets in around [MATH] [CITATION]).', 'hep-lat-0001001-2-40-6': 'This agrees with the result of a strong coupling expansion (in the Hamiltonian formulation) which applies to [MATH] [CITATION].', 'hep-lat-0001001-2-41-0': 'In such cases, the construction of [MATH] maps all (almost) real eigenvalues onto the arc close to 2, hence additive mass renormalization is back as well - again in agreement with Ref. [CITATION].', 'hep-lat-0001001-2-41-1': 'In view of the latter point one could be tempted to just use a large mass parameter [MATH]; this does not help, however, with respect to the doubling problem.', 'hep-lat-0001001-2-41-2': 'At very strong coupling, only the (classically) perfect action would help.', 'hep-lat-0001001-2-42-0': 'As we mentioned earlier, also locality is in danger in that regime [CITATION], and we should therefore keep away from it.', 'hep-lat-0001001-2-42-1': 'As one more advantage of choosing [MATH] to be an approximate GW fermion, the regime of [MATH] where we are (statistically) on safe grounds is enlarged compared to the Neuberger fermion.', 'hep-lat-0001001-2-43-0': 'Of course, in the safe regime where [MATH] is large enough (in QCD this includes [MATH] for sure [CITATION]) the chiral correction of [MATH] can also be carried out by iteration methods different from the one described in the previous Subsection, for alternative experiments see Ref. [CITATION].', 'hep-lat-0001001-2-43-1': 'The efficiency in QCD is still to be compared, but for sure in any method the convergence will be much faster for [MATH] than for [MATH].', 'hep-lat-0001001-2-44-0': '# Conclusions', 'hep-lat-0001001-2-45-0': 'Our program outlined in Sec. 2.1 has been realized in the Schwinger model, and the properties of a resulting improved overlap fermion have been tested extensively.', 'hep-lat-0001001-2-45-1': 'They are all clearly superior over the Neuberger fermion, confirming our prediction: the overlap SO-HF scales much better, it is more local and it comes much closer to rotation invariance.', 'hep-lat-0001001-2-46-0': 'The question now is the applicability of this program in [MATH].', 'hep-lat-0001001-2-46-1': 'The 4d HF formulation is worked out already, and the corresponding improved overlap fermion is currently under investigation in QCD by the SESAM collaboration in Julich and Wuppertal.'}

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[['hep-lat-0001001-1-38-0', 'hep-lat-0001001-2-45-0'], ['hep-lat-0001001-1-38-1', 'hep-lat-0001001-2-45-1'], ['hep-lat-0001001-1-4-0', 'hep-lat-0001001-2-4-0'], ['hep-lat-0001001-1-4-1', 'hep-lat-0001001-2-4-1'], ['hep-lat-0001001-1-4-2', 'hep-lat-0001001-2-4-2'], ['hep-lat-0001001-1-4-3', 'hep-lat-0001001-2-4-3'], ['hep-lat-0001001-1-4-4', 'hep-lat-0001001-2-4-4'], ['hep-lat-0001001-1-4-5', 'hep-lat-0001001-2-4-5'], ['hep-lat-0001001-1-15-0', 'hep-lat-0001001-2-17-0'], ['hep-lat-0001001-1-15-1', 'hep-lat-0001001-2-17-1'], ['hep-lat-0001001-1-15-2', 'hep-lat-0001001-2-17-2'], ['hep-lat-0001001-1-15-3', 'hep-lat-0001001-2-17-3'], ['hep-lat-0001001-1-15-4', 'hep-lat-0001001-2-17-4'], ['hep-lat-0001001-1-15-5', 'hep-lat-0001001-2-17-5'], ['hep-lat-0001001-1-31-0', 'hep-lat-0001001-2-36-0'], ['hep-lat-0001001-1-31-1', 'hep-lat-0001001-2-36-1'], ['hep-lat-0001001-1-31-2', 'hep-lat-0001001-2-36-2'], ['hep-lat-0001001-1-31-3', 'hep-lat-0001001-2-36-3'], ['hep-lat-0001001-1-39-0', 'hep-lat-0001001-2-46-0'], ['hep-lat-0001001-1-39-1', 'hep-lat-0001001-2-46-1'], ['hep-lat-0001001-1-9-0', 'hep-lat-0001001-2-10-0'], ['hep-lat-0001001-1-9-2', 'hep-lat-0001001-2-10-2'], ['hep-lat-0001001-1-9-3', 'hep-lat-0001001-2-10-3'], ['hep-lat-0001001-1-9-4', 'hep-lat-0001001-2-10-4'], ['hep-lat-0001001-1-7-0', 'hep-lat-0001001-2-8-0'], ['hep-lat-0001001-1-7-1', 'hep-lat-0001001-2-8-1'], ['hep-lat-0001001-1-30-0', 'hep-lat-0001001-2-35-0'], ['hep-lat-0001001-1-30-3', 'hep-lat-0001001-2-35-3'], ['hep-lat-0001001-1-30-5', 'hep-lat-0001001-2-35-5'], ['hep-lat-0001001-1-30-6', 'hep-lat-0001001-2-35-6'], ['hep-lat-0001001-1-30-7', 'hep-lat-0001001-2-35-7'], ['hep-lat-0001001-1-30-11', 'hep-lat-0001001-2-35-11'], ['hep-lat-0001001-1-11-2', 'hep-lat-0001001-2-13-1'], ['hep-lat-0001001-1-11-3', 'hep-lat-0001001-2-13-2'], ['hep-lat-0001001-1-11-4', 'hep-lat-0001001-2-13-3'], ['hep-lat-0001001-1-11-5', 'hep-lat-0001001-2-13-4'], ['hep-lat-0001001-1-34-0', 'hep-lat-0001001-2-39-0'], 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'hep-lat-0001001-2-14-0'], ['hep-lat-0001001-1-12-1', 'hep-lat-0001001-2-14-1'], ['hep-lat-0001001-1-12-2', 'hep-lat-0001001-2-14-2'], ['hep-lat-0001001-1-12-3', 'hep-lat-0001001-2-14-3'], ['hep-lat-0001001-1-28-0', 'hep-lat-0001001-2-33-0'], ['hep-lat-0001001-1-28-1', 'hep-lat-0001001-2-33-1'], ['hep-lat-0001001-1-28-2', 'hep-lat-0001001-2-33-2'], ['hep-lat-0001001-1-28-3', 'hep-lat-0001001-2-33-3'], ['hep-lat-0001001-1-21-0', 'hep-lat-0001001-2-24-0'], ['hep-lat-0001001-1-21-1', 'hep-lat-0001001-2-24-1'], ['hep-lat-0001001-1-21-2', 'hep-lat-0001001-2-24-2'], ['hep-lat-0001001-1-21-3', 'hep-lat-0001001-2-24-3'], ['hep-lat-0001001-1-21-4', 'hep-lat-0001001-2-24-4'], ['hep-lat-0001001-1-21-6', 'hep-lat-0001001-2-24-6'], ['hep-lat-0001001-1-21-7', 'hep-lat-0001001-2-24-7'], ['hep-lat-0001001-1-21-8', 'hep-lat-0001001-2-24-8'], ['hep-lat-0001001-1-21-9', 'hep-lat-0001001-2-24-9'], ['hep-lat-0001001-1-21-10', 'hep-lat-0001001-2-24-10'], ['hep-lat-0001001-1-21-12', 'hep-lat-0001001-2-24-12'], ['hep-lat-0001001-1-6-0', 'hep-lat-0001001-2-7-0'], ['hep-lat-0001001-1-6-2', 'hep-lat-0001001-2-7-2'], ['hep-lat-0001001-1-6-3', 'hep-lat-0001001-2-7-3'], ['hep-lat-0001001-1-6-4', 'hep-lat-0001001-2-7-4'], ['hep-lat-0001001-1-20-0', 'hep-lat-0001001-2-23-0'], ['hep-lat-0001001-1-20-1', 'hep-lat-0001001-2-23-1'], ['hep-lat-0001001-1-17-0', 'hep-lat-0001001-2-19-0'], ['hep-lat-0001001-1-17-1', 'hep-lat-0001001-2-19-1'], ['hep-lat-0001001-1-17-2', 'hep-lat-0001001-2-19-2'], ['hep-lat-0001001-1-17-3', 'hep-lat-0001001-2-19-3'], ['hep-lat-0001001-1-17-4', 'hep-lat-0001001-2-19-4'], ['hep-lat-0001001-1-17-5', 'hep-lat-0001001-2-19-5'], ['hep-lat-0001001-1-17-6', 'hep-lat-0001001-2-19-6'], ['hep-lat-0001001-1-17-7', 'hep-lat-0001001-2-19-7'], ['hep-lat-0001001-1-17-8', 'hep-lat-0001001-2-19-8'], ['hep-lat-0001001-1-17-9', 'hep-lat-0001001-2-19-9'], ['hep-lat-0001001-1-17-10', 'hep-lat-0001001-2-19-10'], ['hep-lat-0001001-1-17-11', 'hep-lat-0001001-2-19-11'], ['hep-lat-0001001-1-25-0', 'hep-lat-0001001-2-30-0'], ['hep-lat-0001001-1-25-1', 'hep-lat-0001001-2-30-1'], ['hep-lat-0001001-1-1-0', 'hep-lat-0001001-2-1-0'], ['hep-lat-0001001-1-1-1', 'hep-lat-0001001-2-1-1'], ['hep-lat-0001001-1-1-2', 'hep-lat-0001001-2-1-2'], ['hep-lat-0001001-1-1-3', 'hep-lat-0001001-2-1-3'], ['hep-lat-0001001-1-1-4', 'hep-lat-0001001-2-1-4'], ['hep-lat-0001001-1-1-5', 'hep-lat-0001001-2-1-5'], ['hep-lat-0001001-1-23-0', 'hep-lat-0001001-2-27-0'], ['hep-lat-0001001-1-23-1', 'hep-lat-0001001-2-27-1'], ['hep-lat-0001001-1-36-1', 'hep-lat-0001001-2-43-1'], ['hep-lat-0001001-1-5-1', 'hep-lat-0001001-2-6-1'], ['hep-lat-0001001-1-5-2', 'hep-lat-0001001-2-6-2'], ['hep-lat-0001001-1-26-0', 'hep-lat-0001001-2-31-0'], ['hep-lat-0001001-1-18-0', 'hep-lat-0001001-2-20-0'], ['hep-lat-0001001-1-18-1', 'hep-lat-0001001-2-21-0'], ['hep-lat-0001001-1-18-2', 'hep-lat-0001001-2-21-1'], ['hep-lat-0001001-1-35-0', 'hep-lat-0001001-2-40-0'], ['hep-lat-0001001-1-35-1', 'hep-lat-0001001-2-40-1'], ['hep-lat-0001001-1-35-3', 'hep-lat-0001001-2-40-3'], ['hep-lat-0001001-1-35-5', 'hep-lat-0001001-2-40-5'], ['hep-lat-0001001-1-35-6', 'hep-lat-0001001-2-40-6'], ['hep-lat-0001001-1-35-7', 'hep-lat-0001001-2-42-0'], ['hep-lat-0001001-1-35-8', 'hep-lat-0001001-2-42-1'], ['hep-lat-0001001-1-10-4', 'hep-lat-0001001-2-12-0']]

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[]

[['hep-lat-0001001-1-11-7', 'hep-lat-0001001-2-13-6'], ['hep-lat-0001001-1-34-1', 'hep-lat-0001001-2-39-1'], ['hep-lat-0001001-1-28-4', 'hep-lat-0001001-2-33-4'], ['hep-lat-0001001-1-21-11', 'hep-lat-0001001-2-24-11'], ['hep-lat-0001001-1-10-5', 'hep-lat-0001001-2-12-1'], ['hep-lat-0001001-1-10-6', 'hep-lat-0001001-2-12-2']]

[]

['hep-lat-0001001-1-5-0', 'hep-lat-0001001-1-35-4', 'hep-lat-0001001-2-6-0', 'hep-lat-0001001-2-40-4']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-lat/0001001

0910.2978

{'0910.2978-1-0-0': 'The second and third order coherence functions [MATH] of an exciton-polariton condensate is measured and compared to the theory.', '0910.2978-1-0-1': 'Contrary to an ideal photon laser, deviation from unity in the second and third order coherence functions is observed, thus showing a bunching effect, but not the characteristics of a standard thermal state with [MATH].', '0910.2978-1-0-2': 'The increase of bunching with the order of the coherence function, [MATH], indicates that the polariton condensate is different from coherent state, number state and thermal state.', '0910.2978-1-0-3': 'The experimental results are in agreement with the theoretical model where polariton-polariton and polariton-phonon interactions are responsible for the loss of temporal coherence.', '0910.2978-1-1-0': 'In a seminal paper by R. J. Glauber marking the beginning of quantum optics, coherent states were studied as unique states, whose normalized coherence functions [MATH] are unity for all orders [MATH] [CITATION].', '0910.2978-1-1-1': 'It has been shown theoretically and experimentally that a photon laser is well characterized by coherence functions [MATH] close to one at pump intensities well above threshold, thus showing all order coherence, except for a random walk phase diffusion.', '0910.2978-1-1-2': 'Bose-Einstein condensation (BEC) of polaritons has recently been the subject of detailed investigations [CITATION] and the question of its coherence properties arises.', '0910.2978-1-1-3': 'BEC is characterized by a large number of particles sharing the same quantum state.', '0910.2978-1-1-4': 'Therefore it is expected that the state of polariton condensates shows temporal coherence properties leading to a coherence function of coherent state [MATH] or [MATH]-particle number state [MATH].', '0910.2978-1-1-5': 'However, there are important differences between photon laser and polariton BEC, one of which is that the former is based on stimulated emission of photons from an excited level and population inversion is necessary to reach a threshold, while the latter is based on stimulated cooling of polaritons to the ground state from excited states and BEC threshold condition is given by [MATH], where [MATH], [MATH] and [MATH] are the polariton density, thermal de Broglie wavelength and system size, respectively.', '0910.2978-1-1-6': 'Thus, when condensation occurs in a lower polariton (LP) ground state, the population inversion is not created yet in quantum wells [CITATION].', '0910.2978-1-1-7': 'Furthermore, scattering processes to the non-condensed polariton modes may influence the coherence properties of the polariton condensate.', '0910.2978-1-1-8': 'This last effect is absent in a photon laser.', '0910.2978-1-1-9': 'Contrary to a photon laser, deviation from unity in the second order coherence function [MATH] for the exciton-polariton condensate has been observed [CITATION] showing a bunching effect.', '0910.2978-1-1-10': 'The origin of the deviation from unity has not been well identified so far.', '0910.2978-1-1-11': 'However, in recent theoretical papers, the deviation from full coherence in higher order correlations have been shown [CITATION].', '0910.2978-1-1-12': 'The model presented in [CITATION], considers both polariton-polariton and polariton-phonon scattering.', '0910.2978-1-1-13': 'This scattering processes lead to two competing effects that involve the condensed polaritons: resonant scattering into and out of the polariton ground state involving one polariton in the ground state and parametric scattering of two polaritons in the ground state into two states of opposite momentum.', '0910.2978-1-1-14': 'Below threshold, the statistics of the polaritons is determined by the thermal phonon bath, which is responsible for polariton relaxation, and the effect of polariton-polariton scattering is negligible.', '0910.2978-1-1-15': 'In this case, the corresponding [MATH] is given by that of a thermal state [MATH].', '0910.2978-1-1-16': 'Above threshold, resonant polariton-polariton and polariton-phonon interactions compete.', '0910.2978-1-1-17': 'The parametric scattering between ground state and excited states induces a depletion of the ground state population.', '0910.2978-1-1-18': 'This effect induces fluctuations in the population and manifests itself in a loss of coherence in the polariton condensate [CITATION].', '0910.2978-1-1-19': 'Indeed, when higher order coherence functions are calculated, a deviation from full coherence and bunching effect above threshold are obtained.', '0910.2978-1-2-0': 'Let us discuss this point in more detail.', '0910.2978-1-2-1': 'The particle statistics of a thermal state are characterized by the geometrical distribution [MATH], where [MATH] is a probability of [MATH] polaritons and [MATH] is mean polariton number, and lead to a normalized coherence function [MATH], which grows with [MATH].', '0910.2978-1-2-2': 'In this case, we expect a larger bunching behavior in the higher order coherence functions, such as [MATH].', '0910.2978-1-2-3': 'This behavior is indeed found for polaritons in the ground state below threshold [CITATION].', '0910.2978-1-2-4': 'Above threshold, the calculated particle distribution has a shape close to but different from a Poissonian distribution.', '0910.2978-1-2-5': 'The second order coherence function is larger than one and the higher order coherence functions are even larger [CITATION].', '0910.2978-1-2-6': 'In this paper, we present the measurement results of the second and third order normalized coherence functions.', '0910.2978-1-2-7': 'In particular, we show that the observed behavior of growing bunching with the order [MATH] agrees well with the theory.', '0910.2978-1-3-0': 'The second order coherence function for exciton-polariton condensates has been measured for GaAs quantum well (QW) microcavity [CITATION] and CdTe QW microcavity [CITATION].', '0910.2978-1-3-1': 'Below condensation threshold, [MATH] takes almost unity since both polariton emission decay time below threshold and detector response time are much longer than the intensity correlation time of polaritons.', '0910.2978-1-3-2': 'Above threshold [MATH] increases steeply and then gradually saturates as the pump intensity goes up.', '0910.2978-1-3-3': 'A full coherence [MATH] has not been observed even at far above threshold.', '0910.2978-1-4-0': 'In the present work, we used 12 GaAs QWs embedded in GaAlAs/AlAs distributed Bragg reflector (DBR) microcavity that was need in our previous experiments [CITATION].', '0910.2978-1-4-1': 'The sample has three stacks of four GaAs QWs which are embedded at the central three antinode positions of a DBR planar microcavity.', '0910.2978-1-4-2': 'Lateral trapping potential is provided by a hole surrounded by a thin metal (Ti/Au) film which pushes photon-field amplitude antinode position at GaAlAs/air interface inside GaAlAs layer, resulting in the blue shift of the cavity resonance and hence LP energy under the metallic layer.', '0910.2978-1-4-3': 'By using this metallic hole structure, we could obtain the confinement of polaritons under a hole, which allows us to access to a single spatial mode in second and third order coherence measurements.', '0910.2978-1-4-4': 'An optical measurement system together with electronics for third order coherence measurement is shown in Fig. [REF].', '0910.2978-1-5-0': 'A mode-locked Ti:sa laser with 4 ps pulse width is utilized as a pump laser.', '0910.2978-1-5-1': 'It is focused on the sample surface with diameter of about 50 [MATH]m.', '0910.2978-1-5-2': 'An objective lens collimates the output luminescence and near field image is made at the position of the pinhole by the second lens.', '0910.2978-1-5-3': 'Here we can spatially filter the metallic hole area of about 5 [MATH]m diameter.', '0910.2978-1-5-4': 'Then at the entrance slit of the spectrometer, far field is imaged by the third lens.', '0910.2978-1-5-5': 'Dispersion relation can be measured by the spectrometer and an attached nitrogen-cooled CCD camera', '0910.2978-1-6-0': 'The left-hand side of Figure [REF] is an example of observed dispersion at [MATH] is threshold power ).', '0910.2978-1-6-1': 'In case we use the side exit of the monochromator, the output signal enters Hanbury-Brown Twiss (HBT) setup consisting of two non-polarizing beam splitters and photon detectors.', '0910.2978-1-6-2': 'We can measure [MATH] and [MATH] with this setup.', '0910.2978-1-6-3': '@Signal entering HBT setup is split into three paths by a 2:1 splitting ratio beam splitter and a 1:1 ratio beam splitter.', '0910.2978-1-6-4': 'We used single photon counting modules (Perkin Elmer, SPCM AQR series) for this coherence function measurement.', '0910.2978-1-6-5': "The detectors' response time is relatively slow (about 300 ps timing resolution), but its slow response does not limit our capability of measuring [MATH] and [MATH] since we used a pulsed pump laser of 4 ps pulse width and the emission time becomes shorter than the correlation time at well above threshold.", '0910.2978-1-6-6': 'This setup allows us to pick up arbitrary part of condensates at LP ground state by setting appropriate wavelength filter of spectrometer.', '0910.2978-1-6-7': 'The in-plane momentum region is determined by collecting optics and actual CCD image size.', '0910.2978-1-6-8': 'Area surrounded by dotted line in Figure [REF] is the detected region.', '0910.2978-1-7-0': 'For [MATH] measurement, we need to take 3-fold coincidence of three detectors.', '0910.2978-1-7-1': 'For normalization of coherence function, 3-fold coincidence rate of simultaneous three photon detection for one pump pulse must be divided by accidental signal (product of independent signal rates of three detectors).', '0910.2978-1-7-2': 'In the actual measurement, we used a time interval analyzer (TIA) which measures time difference between start and stop inputs.', '0910.2978-1-7-3': '[MATH] can be directly measured by TIA but [MATH] cannot be since our TIA has just 2 input channels (start and stop).', '0910.2978-1-7-4': '[MATH] is instead measured by the following method.', '0910.2978-1-7-5': 'At first, we took 2-fold coincidence between detector 1 and detector 2 by 2-fold coincidence unit.', '0910.2978-1-7-6': 'Then the output signal is input to start port of the TIA (Fig. [REF]).', '0910.2978-1-7-7': 'Right-hand side of Figure [REF] is an example of raw histogram taken by TIA.', '0910.2978-1-7-8': 'The horizontal axis corresponds to delay time between start signal and stop signal.', '0910.2978-1-7-9': 'The interval between next pulses is 13 ns which corresponds to repetition period of the pump laser.', '0910.2978-1-7-10': 'The highest peak corresponds to 3-fold coincidence of simultaneously detected three photons for one pump pulse and surrounding peaks correspond to 2-fold coincidence by simultaneously detected two photons for one pump pulse and accidental third photon from different time slot.', '0910.2978-1-7-11': 'So the surrounding peaks correspond to [MATH], i.e. just detector 1 and detector 2 are fired by the same pulse.', '0910.2978-1-7-12': 'In this configuration, we need to preliminary know [MATH] to obtain [MATH] since taking the ratio between central peak and average of surrounding peaks just gives [MATH].', '0910.2978-1-7-13': 'So [MATH] can be obtained by multiplying [MATH] to the ratio of central peak to surrounding peak.', '0910.2978-1-8-0': 'We measured [MATH] and [MATH] at various pump powers (Fig. [REF]).', '0910.2978-1-9-0': 'In Fig [REF], theoretical results are drawn together with experimental data.', '0910.2978-1-9-1': 'The details of the theoretical model are given in Ref. [CITATION].', '0910.2978-1-9-2': 'Below threshold, though they are not shown in Fig [REF], the statistics obeys thermal distribution, hence, the theory predicts [MATH].', '0910.2978-1-9-3': 'Just above threshold, they begin to gain coherence and rapidly decrease towards unity.', '0910.2978-1-9-4': 'However, the decreases stop around [MATH] due to increasing effect of polariton scattering.', '0910.2978-1-9-5': 'As the pump intensity increases, they converge into certain values ([MATH] and [MATH]).', '0910.2978-1-9-6': 'The experimental data shows [MATH] and [MATH] are still close to unity just above threshold.', '0910.2978-1-9-7': 'However, they increase and become closer to theoretical values as the pump intensity increases.', '0910.2978-1-9-8': 'After gradual increase, they reach a flat area at around [MATH].', '0910.2978-1-10-0': 'To understand the discrepancies between theory and experiment at the pumping regime of [MATH], we need to consider intensity correlation time and decay time of the condensate emission.', '0910.2978-1-11-0': 'The ground state photoluminescence decay time measured by a streak camera preceded by a spectrometer is shown in Fig. [REF].', '0910.2978-1-11-1': 'Since the streak camera was set after a spectrometer, we could avoid the contamination by the non-condensates and the wavelength range giving maximum PL intensities at each pump power was picked up.', '0910.2978-1-12-0': 'Below threshold where photon statistics is expected to obey thermal distribution, the observed statistics is always [MATH] since decay time [MATH] of photoluminescence is much longer than the intensity correlation time [MATH].', '0910.2978-1-12-1': 'In the case of longer [MATH], [MATH] [CITATION].', '0910.2978-1-12-2': 'The measured [MATH] is the integration of [MATH] over an integration time window which is given by an emission time [MATH] in our case, , hence, it is close to unity after averaged over the whole emission lifetime.', '0910.2978-1-12-3': 'Just above threshold, the condition is still the same since the emission pulse width is longer than the intensity correlation time until the pump rate reaches [MATH].', '0910.2978-1-12-4': 'Finally far above threshold, the intensity correlation time becomes closer to the pulse width, and then the intrinsic noise property of the condensates begins to be detected properly.', '0910.2978-1-13-0': 'In conclusion, we have experimentally measured the second and third order coherence functions.', '0910.2978-1-13-1': 'The observed bunching effect is the experimental evidence for the relatively strong thermal and quantum depletion of the LP condensate.', '0910.2978-1-13-2': 'This higher order coherence function measurement technique may contribute to a further investigation of coherence property of an exciton-polariton condensate.', '0910.2978-1-14-0': 'T.H. thanks G. Roumpos, N. Y. Kim, and S. Utsunomiya for their help.', '0910.2978-1-14-1': 'And T.H. and Y. Y. acknowledge financial support from NICT, Special Coordination Funds for Promoting Science and Technology, MEXT and DARPA.'}

{'0910.2978-2-0-0': 'The second and third order coherence functions [MATH] of an exciton-polariton condensate is measured and compared to the theory.', '0910.2978-2-0-1': 'Contrary to an ideal photon laser, deviation from unity in the second and third order coherence functions is observed, thus showing a bunching effect, but not the characteristics of a standard thermal state with [MATH].', '0910.2978-2-0-2': 'The increase of bunching with the order of the coherence function, [MATH], indicates that the polariton condensate is different from coherent state, number state and thermal state.', '0910.2978-2-0-3': 'The experimental results are in agreement with the theoretical model where polariton-polariton and polariton-phonon interactions are responsible for the loss of temporal coherence.', '0910.2978-2-1-0': 'In a seminal paper by R. J. Glauber marking the beginning of quantum optics, coherent states were studied as unique states, whose normalized coherence functions [MATH] are unity for all orders [MATH] [CITATION].', '0910.2978-2-1-1': 'It has been shown theoretically and experimentally that a photon laser is well characterized by coherence functions [MATH] close to one at pump intensities well above threshold, thus showing all order coherence, except for a random walk phase diffusion.', '0910.2978-2-1-2': 'Bose-Einstein condensation (BEC) of polaritons has recently been the subject of detailed investigations [CITATION] and the question of its coherence properties arises.', '0910.2978-2-1-3': 'BEC is characterized by a large number of particles sharing the same quantum state.', '0910.2978-2-1-4': 'Therefore it is expected that the state of polariton condensates shows temporal coherence properties leading to a coherence function of coherent state [MATH] or [MATH]-particle number state [MATH].', '0910.2978-2-1-5': 'However, there are important differences between photon laser and polariton BEC, one of which is that the former is based on stimulated emission of photons from an excited level and population inversion is necessary to reach a threshold, while the latter is based on stimulated cooling of polaritons to the ground state from excited states and BEC threshold condition is given by [MATH], where [MATH], [MATH] and [MATH] are the polariton density, thermal de Broglie wavelength and system size, respectively.', '0910.2978-2-1-6': 'Thus, when condensation occurs in a lower polariton (LP) ground state, the population inversion is not created yet in quantum wells [CITATION].', '0910.2978-2-1-7': 'Furthermore, scattering processes to the non-condensed polariton modes may influence the coherence properties of the polariton condensate.', '0910.2978-2-1-8': 'This last effect is absent in a photon laser.', '0910.2978-2-1-9': 'Contrary to a photon laser, deviation from unity in the second order coherence function [MATH] for the exciton-polariton condensate has been observed [CITATION] showing a bunching effect.', '0910.2978-2-1-10': 'The origin of the deviation from unity has not been well identified so far.', '0910.2978-2-1-11': 'However, in recent theoretical papers, the deviation from full coherence in higher order correlations have been shown [CITATION].', '0910.2978-2-1-12': 'The model presented in [CITATION], considers both polariton-polariton and polariton-phonon scattering.', '0910.2978-2-1-13': 'This scattering processes lead to two competing effects that involve the condensed polaritons: resonant scattering into and out of the polariton ground state involving one polariton in the ground state and parametric scattering of two polaritons in the ground state into two states of opposite momentum.', '0910.2978-2-1-14': 'Below threshold, the statistics of the polaritons is determined by the thermal phonon bath, which is responsible for polariton relaxation, and the effect of polariton-polariton scattering is negligible.', '0910.2978-2-1-15': 'In this case, the corresponding [MATH] is given by that of a thermal state [MATH].', '0910.2978-2-1-16': 'Above threshold, resonant polariton-polariton and polariton-phonon interactions compete.', '0910.2978-2-1-17': 'The parametric scattering between ground state and excited states induces a depletion of the ground state population.', '0910.2978-2-1-18': 'This effect induces fluctuations in the population and manifests itself in a loss of coherence in the polariton condensate [CITATION].', '0910.2978-2-1-19': 'Indeed, when higher order coherence functions are calculated, a deviation from full coherence and bunching effect above threshold are obtained.', '0910.2978-2-2-0': 'Let us discuss this point in more detail.', '0910.2978-2-2-1': 'The particle statistics of a thermal state are characterized by the geometrical distribution [MATH], where [MATH] is a probability of [MATH] polaritons and [MATH] is mean polariton number, and lead to a normalized coherence function [MATH], which grows with [MATH].', '0910.2978-2-2-2': 'In this case, we expect a larger bunching behavior in the higher order coherence functions, such as [MATH].', '0910.2978-2-2-3': 'This behavior is indeed found for polaritons in the ground state below threshold [CITATION].', '0910.2978-2-2-4': 'Above threshold, the calculated particle distribution has a shape close to but different from a Poissonian distribution.', '0910.2978-2-2-5': 'The second order coherence function is larger than one and the higher order coherence functions are even larger [CITATION].', '0910.2978-2-2-6': 'In this paper, we present the measurement results of the second and third order normalized coherence functions.', '0910.2978-2-2-7': 'In particular, we show that the observed behavior of growing bunching with the order [MATH] agrees well with the theory.', '0910.2978-2-3-0': 'The second order coherence function for exciton-polariton condensates has been measured for GaAs quantum well (QW) microcavity [CITATION] and CdTe QW microcavity [CITATION].', '0910.2978-2-3-1': 'Below condensation threshold, [MATH] takes almost unity since both polariton emission decay time below threshold and detector response time are much longer than the intensity correlation time of polaritons.', '0910.2978-2-3-2': 'Above threshold [MATH] increases steeply and then gradually saturates as the pump intensity goes up.', '0910.2978-2-3-3': 'A full coherence [MATH] has not been observed even at far above threshold.', '0910.2978-2-4-0': 'In the present work, we used 12 GaAs QWs embedded in GaAlAs/AlAs distributed Bragg reflector (DBR) microcavity that was need in our previous experiments [CITATION].', '0910.2978-2-4-1': 'The sample has three stacks of four GaAs QWs which are embedded at the central three antinode positions of a DBR planar microcavity.', '0910.2978-2-4-2': 'Lateral trapping potential is provided by a hole surrounded by a thin metal (Ti/Au) film which pushes photon-field amplitude antinode position at GaAlAs/air interface inside GaAlAs layer, resulting in the blue shift of the cavity resonance and hence LP energy under the metallic layer.', '0910.2978-2-4-3': 'By using this metallic hole structure, we could obtain the confinement of polaritons under a hole, which allows us to access to a single spatial mode in second and third order coherence measurements.', '0910.2978-2-4-4': 'An optical measurement system together with electronics for third order coherence measurement is shown in Fig. [REF].', '0910.2978-2-5-0': 'A mode-locked Ti:sa laser with 4 ps pulse width is utilized as a pump laser.', '0910.2978-2-5-1': 'It is focused on the sample surface with diameter of about 50 [MATH]m.', '0910.2978-2-5-2': 'An objective lens collimates the output luminescence and near field image is made at the position of the pinhole by the second lens.', '0910.2978-2-5-3': 'Here we can spatially filter the metallic hole area of about 5 [MATH]m diameter.', '0910.2978-2-5-4': 'Then at the entrance slit of the spectrometer, far field is imaged by the third lens.', '0910.2978-2-5-5': 'Dispersion relation can be measured by the spectrometer and an attached nitrogen-cooled CCD camera', '0910.2978-2-6-0': 'The left-hand side of Figure [REF] is an example of observed dispersion at [MATH] is threshold power ).', '0910.2978-2-6-1': 'In case we use the side exit of the monochromator, the output signal enters Hanbury-Brown Twiss (HBT) setup consisting of two non-polarizing beam splitters and photon detectors.', '0910.2978-2-6-2': 'We can measure [MATH] and [MATH] with this setup.', '0910.2978-2-6-3': '@Signal entering HBT setup is split into three paths by a 2:1 splitting ratio beam splitter and a 1:1 ratio beam splitter.', '0910.2978-2-6-4': 'We used single photon counting modules (Perkin Elmer, SPCM AQR series) for this coherence function measurement.', '0910.2978-2-6-5': "The detectors' response time is relatively slow (about 300 ps timing resolution), but its slow response does not limit our capability of measuring [MATH] and [MATH] since we used a pulsed pump laser of 4 ps pulse width and the emission time becomes shorter than the correlation time at well above threshold.", '0910.2978-2-6-6': 'This setup allows us to pick up arbitrary part of condensates at LP ground state by setting appropriate wavelength filter of spectrometer.', '0910.2978-2-6-7': 'The in-plane momentum region is determined by collecting optics and actual CCD image size.', '0910.2978-2-6-8': 'Area surrounded by dotted line in Figure [REF] is the detected region.', '0910.2978-2-7-0': 'For [MATH] measurement, we need to take 3-fold coincidence of three detectors.', '0910.2978-2-7-1': 'For normalization of coherence function, 3-fold coincidence rate of simultaneous three photon detection for one pump pulse must be divided by accidental signal (product of independent signal rates of three detectors).', '0910.2978-2-7-2': 'In the actual measurement, we used a time interval analyzer (TIA) which measures time difference between start and stop inputs.', '0910.2978-2-7-3': '[MATH] can be directly measured by TIA but [MATH] cannot be since our TIA has just 2 input channels (start and stop).', '0910.2978-2-7-4': '[MATH] is instead measured by the following method.', '0910.2978-2-7-5': 'At first, we took 2-fold coincidence between detector 1 and detector 2 by 2-fold coincidence unit.', '0910.2978-2-7-6': 'Then the output signal is input to start port of the TIA (Fig. [REF]).', '0910.2978-2-7-7': 'Right-hand side of Figure [REF] is an example of raw histogram taken by TIA.', '0910.2978-2-7-8': 'The horizontal axis corresponds to delay time between start signal and stop signal.', '0910.2978-2-7-9': 'The interval between next pulses is 13 ns which corresponds to repetition period of the pump laser.', '0910.2978-2-7-10': 'The highest peak corresponds to 3-fold coincidence of simultaneously detected three photons for one pump pulse and surrounding peaks correspond to 2-fold coincidence by simultaneously detected two photons for one pump pulse and accidental third photon from different time slot.', '0910.2978-2-7-11': 'So the surrounding peaks correspond to [MATH], i.e. just detector 1 and detector 2 are fired by the same pulse.', '0910.2978-2-7-12': 'In this configuration, we need to preliminary know [MATH] to obtain [MATH] since taking the ratio between central peak and average of surrounding peaks just gives [MATH].', '0910.2978-2-7-13': 'So [MATH] can be obtained by multiplying [MATH] to the ratio of central peak to surrounding peak.', '0910.2978-2-8-0': 'We measured [MATH] and [MATH] at various pump powers (Fig. [REF]).', '0910.2978-2-9-0': 'In Fig [REF], theoretical results are drawn together with experimental data.', '0910.2978-2-9-1': 'The details of the theoretical model are given in Ref. [CITATION].', '0910.2978-2-9-2': 'Below threshold, though they are not shown in Fig [REF], the statistics obeys thermal distribution, hence, the theory predicts [MATH].', '0910.2978-2-9-3': 'Just above threshold, they begin to gain coherence and rapidly decrease towards unity.', '0910.2978-2-9-4': 'However, the decreases stop around [MATH] due to increasing effect of polariton scattering.', '0910.2978-2-9-5': 'As the pump intensity increases, they converge into certain values ([MATH] and [MATH]).', '0910.2978-2-9-6': 'The experimental data shows [MATH] and [MATH] are still close to unity just above threshold.', '0910.2978-2-9-7': 'However, they increase and become closer to theoretical values as the pump intensity increases.', '0910.2978-2-9-8': 'After gradual increase, they reach a flat area at around [MATH].', '0910.2978-2-10-0': 'To understand the discrepancies between theory and experiment at the pumping regime of [MATH], we need to consider intensity correlation time and decay time of the condensate emission.', '0910.2978-2-11-0': 'The ground state photoluminescence decay time measured by a streak camera preceded by a spectrometer is shown in Fig. [REF].', '0910.2978-2-11-1': 'Since the streak camera was set after a spectrometer, we could avoid the contamination by the non-condensates and the wavelength range giving maximum PL intensities at each pump power was picked up.', '0910.2978-2-12-0': 'Below threshold where photon statistics is expected to obey thermal distribution, the observed statistics is always [MATH] since decay time [MATH] of photoluminescence is much longer than the intensity correlation time [MATH].', '0910.2978-2-12-1': 'In the case of longer [MATH], [MATH] [CITATION].', '0910.2978-2-12-2': 'The measured [MATH] is the integration of [MATH] over an integration time window which is given by an emission time [MATH] in our case, , hence, it is close to unity after averaged over the whole emission lifetime.', '0910.2978-2-12-3': 'Just above threshold, the condition is still the same since the emission pulse width is longer than the intensity correlation time until the pump rate reaches [MATH].', '0910.2978-2-12-4': 'Finally far above threshold, the intensity correlation time becomes closer to the pulse width, and then the intrinsic noise property of the condensates begins to be detected properly.', '0910.2978-2-13-0': 'In conclusion, we have experimentally measured the second and third order coherence functions.', '0910.2978-2-13-1': 'The observed bunching effect is the experimental evidence for the relatively strong thermal and quantum depletion of the LP condensate.', '0910.2978-2-13-2': 'This higher order coherence function measurement technique may contribute to a further investigation of coherence property of an exciton-polariton condensate.', '0910.2978-2-14-0': 'T.H. thanks G. Roumpos, N. Y. Kim, and S. Utsunomiya for their help.', '0910.2978-2-14-1': 'And T.H. and Y. Y. acknowledge financial support from NICT, Special Coordination Funds for Promoting Science and Technology, MEXT and DARPA.'}

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/0910.2978

hep-ph-0012110

{'hep-ph-0012110-1-0-0': 'We consider a general two-Higgs-doublet model with CP violation in the scalar sector.', 'hep-ph-0012110-1-0-1': 'Three neutral Higgs fields of the model all mix and the resulting physical Higgs bosons have no definite CP properties.', 'hep-ph-0012110-1-0-2': 'That leads, at the one-loop level of the perturbation expansion, to CP-violating form factors for [MATH], [MATH] and [MATH] interaction vertices.', 'hep-ph-0012110-1-0-3': 'We discuss asymmetries sensitive to CP violation induced by the form factors for the process [MATH] and [MATH] at future linear [MATH] colliders.', 'hep-ph-0012110-1-1-0': '# Introduction', 'hep-ph-0012110-1-2-0': 'Even though the top quark has been already discovered several years ago [CITATION], its interactions are still weakly constrained.', 'hep-ph-0012110-1-2-1': 'It remains an open question if top-quark couplings obey the Standard Model (SM) scheme of the electroweak forces or there exists a contribution from physics beyond the SM.', 'hep-ph-0012110-1-2-2': 'In particular, CP violation in the top-quark interactions has not been verified.', 'hep-ph-0012110-1-2-3': 'The classical method for incorporating CP violation into the SM is to make the Yukawa couplings of the Higgs boson to quarks explicitly complex, as built into the Kobayashi-Maskawa mixing matrix [CITATION] proposed more than two decades ago.', 'hep-ph-0012110-1-2-4': 'However, CP violation could equally well be partially or wholly due to other mechanisms.', 'hep-ph-0012110-1-2-5': 'The possibility that CP violation derives largely from the Higgs sector itself is particularly appealing in the context of the observed baryon asymmetry, since its explanation requires more CP violation [CITATION] then is provided by the SM.', 'hep-ph-0012110-1-2-6': 'Even the simple two-Higgs-doublet model (2HDM) extension of the one-doublet SM Higgs sector provides a much richer framework for describing CP violation; in the 2HDM, spontaneous and/or explicit CP violation is possible in the scalar sector [CITATION].', 'hep-ph-0012110-1-2-7': 'The model, besides CP violation, offers many other appealing phenomena, for a review see Ref. [CITATION].', 'hep-ph-0012110-1-3-0': 'For our analysis, the most relevant part of the interaction Lagrangian takes the following form : L= -vh(a+i_5 b)t + C ^2v h Z_Z^,', 'hep-ph-0012110-1-4-0': 'where [MATH] is the lowest mass scalar, [MATH] is the SU(2) coupling constant, [MATH] is the Higgs boson vacuum expectation value (with the normalization adopted here such that [MATH]), [MATH], [MATH] and [MATH] are real parameters which account for deviations from the SM, [MATH], [MATH] and [MATH] reproduce the SM Lagrangian.', 'hep-ph-0012110-1-4-1': 'Since under CP, [MATH] and [MATH], one can observe that terms in the cross section proportional to [MATH] or [MATH] would indicate CP violation.', 'hep-ph-0012110-1-4-2': 'The 2HDM is the minimal extension of the SM that provides non-zero [MATH] and/or [MATH].', 'hep-ph-0012110-1-5-0': 'In this paper we will focus on CP-violating contributions to the process [MATH] and [MATH] induced within 2HDM.', 'hep-ph-0012110-1-5-1': 'However the fundamental goal is seeking for the ultimate theory of electroweak interactions.', 'hep-ph-0012110-1-5-2': 'There are several reasons to utilize CP violation in the top physics while looking for physics beyond the SM: The top quark decays immediately after being produced as its huge mass [MATH] [CITATION] leads to a decay width [MATH] much larger than [MATH].', 'hep-ph-0012110-1-5-3': 'Therefore the decay process is not contaminated by any fragmentation effects [CITATION] and decay products may provide useful information on top-quark properties.', 'hep-ph-0012110-1-5-4': 'Since the top quark is heavy, its Yukawa coupling is large and therefore its interactions could be sensitive to a Higgs sector of the electroweak theory.', 'hep-ph-0012110-1-5-5': 'At the same time, the TESLA collider design is supposed to offer an integrated luminosity of the order of [MATH] at [MATH].', 'hep-ph-0012110-1-5-6': 'Therefore expected number of [MATH] events per year could reach [MATH] even for [MATH] tagging efficiency [MATH].', 'hep-ph-0012110-1-5-7': 'That should allow to study subtle properties of the top quark, which could e.g. lead to CP-sensitive asymmetries of the order of [MATH].', 'hep-ph-0012110-1-5-8': 'Since the top quark is that heavy and the third family of quarks effectively decouples from the first two, any CP-violating observables within the SM are expected to be tiny, e.g.: i) non-zero electric dipole moment of fermions is generated at the three-loop approximation of the perturbation expansion [CITATION], or ii) the decay rate asymmetry (being a one-loop effect) is strongly GIM suppressed reaching at most a value [MATH] [CITATION].', 'hep-ph-0012110-1-5-9': 'So, one can expect that for CP-violating asymmetries any SM background could be safely neglected.', 'hep-ph-0012110-1-5-10': 'Therefore it seems to be justified to look for CP-violating Higgs effects in the process of [MATH] production and its subsequent decay at future linear [MATH] colliders.', 'hep-ph-0012110-1-5-11': 'Even though 2HDM contributions to various CP-sensitive asymmetries has been already presented in the existing literature, see Refs. [CITATION], here we are providing a consistent treatment of CP violation both in the production, [MATH], and in the top-quark decay, [MATH].', 'hep-ph-0012110-1-5-12': 'For an extensive review of CP violation in top-quark interactions see Ref. [CITATION].', 'hep-ph-0012110-1-6-0': 'The paper is organized as follows.', 'hep-ph-0012110-1-6-1': 'In Section [REF], we briefly outline the mechanism of CP violation in the 2HDM, introduce the mixing matrix for neutral scalars and derive necessary couplings.', 'hep-ph-0012110-1-6-2': 'In section [REF], we present results for CP-violating form factors both for the [MATH] production process and for [MATH] and [MATH] decays.', 'hep-ph-0012110-1-6-3': 'In Section [REF], we recall current experimental constraints relevant for the CP-violating observables considered in this paper.', 'hep-ph-0012110-1-6-4': 'In Section [REF], we collect results for various energy and angular CP-violating asymmetries.', 'hep-ph-0012110-1-6-5': 'Concluding remarks are given in Section [REF].', 'hep-ph-0012110-1-7-0': '# The two-Higgs-doublet model with CP violation', 'hep-ph-0012110-1-8-0': 'The 2HDM of electroweak interactions contains two SU(2) Higgs doublets denoted by [MATH] and [MATH].', 'hep-ph-0012110-1-8-1': 'It is well known [CITATION] that the model allows both for spontaneous and explicit CP violation.', 'hep-ph-0012110-1-9-0': 'After SU(2)[MATH]U(1) gauge symmetry breaking, one combination of neutral Higgs fields, [MATH], becomes a would-be Goldstone boson which is absorbed while giving mass to the [MATH] gauge boson.', 'hep-ph-0012110-1-9-1': '(Here, we use the notation [MATH], [MATH], where [MATH].)', 'hep-ph-0012110-1-9-2': 'The same mixing angle, [MATH], also diagonalizes the mass matrix in the charged Higgs sector.', 'hep-ph-0012110-1-9-3': 'If either explicit or spontaneous CP violation is present, the remaining three neutral degrees of freedom, [EQUATION] are not mass eigenstates.', 'hep-ph-0012110-1-9-4': 'The physical neutral Higgs bosons [MATH]) are obtained by an orthogonal transformation, [MATH], where the rotation matrix is given in terms of three Euler angles ([MATH]) by [EQUATION] where [MATH] and [MATH].', 'hep-ph-0012110-1-10-0': 'As a result of the mixing between real and imaginary parts of neutral Higgs fields, the Yukawa interactions of the [MATH] mass-eigenstates are not invariant under CP.', 'hep-ph-0012110-1-10-1': 'They are given by: [EQUATION] where the scalar ([MATH]) and pseudoscalar ([MATH]) couplings are functions of the mixing angles.', 'hep-ph-0012110-1-10-2': 'For up-type quarks we have [EQUATION] and for down-type quarks: [EQUATION] and similarly for charged leptons.', 'hep-ph-0012110-1-10-3': 'For large [MATH], the couplings to down-type fermions are typically enhanced over the couplings to up-type fermions.', 'hep-ph-0012110-1-11-0': 'In the following analysis we will also need the couplings of neutral Higgs and [MATH] bosons; they are given by [EQUATION]', 'hep-ph-0012110-1-11-1': 'Hereafter we shall denote the lightest Higgs boson by [MATH] and its [MATH]-matrix index by [MATH].', 'hep-ph-0012110-1-12-0': '# Form Factors', 'hep-ph-0012110-1-13-0': '## [MATH] Production', 'hep-ph-0012110-1-14-0': 'width=0.65phd-diag-eett6.epsDiagrams contributing to CP-violating form factors [MATH].', 'hep-ph-0012110-1-14-1': 'fig:diag:eett', 'hep-ph-0012110-1-15-0': 'The effective [MATH] and [MATH] vertices will be parameterized by the following form factors: [EQUATION] where [MATH] denotes the SU(2) gauge coupling constant, [MATH], and [EQUATION] denote the SM contributions to the vertices for [EQUATION].', 'hep-ph-0012110-1-15-1': 'The terms multiplied by [MATH], [MATH], and [MATH] form factors have the same properties under [MATH] while the term multiplied by [MATH] has a relative [MATH] sign.', 'hep-ph-0012110-1-15-2': 'Therefore the presence of both [MATH] and any of the former is a sign of CP-violation.', 'hep-ph-0012110-1-15-3': 'As only [MATH] an [MATH] are present on the tree-level in the SM, it is customary to call [MATH] the CP-violating form factor while the others CP-conserving ones.', 'hep-ph-0012110-1-16-0': 'Further in this paper the following parameters will be adopted: [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012110-1-17-0': 'Since in this paper we are focusing on CP-violating asymmetries, the only relevant form factors are [MATH] and [MATH].', 'hep-ph-0012110-1-17-1': 'Direct calculation of diagrams shown in Fig. [REF] leads to the following result in terms of 3-point Passarino-Veltman [CITATION] functions defined in the appendix [REF]:', 'hep-ph-0012110-1-18-0': 'D_&=& 2^2 A_m_t^2v^2 b_i^t a_i^t m_t^2 12(p_t, p_t, m_t^2, m_h^2, m_t^2),', 'hep-ph-0012110-1-19-0': 'D_Z&=& 2^2 A_Z m_t^2v^2 b_i^t [ a_i^t m_t^2 12(p_t, p_t, m_t^2, m_h^2, m_t^2).', 'hep-ph-0012110-1-20-0': '&& .', 'hep-ph-0012110-1-20-1': '- C_i m_Z^2 12(p_t,p_t, m_h^2, m_t^2, m_Z^2) ].', 'hep-ph-0012110-1-20-2': 'Since the asymmetries we are going to discuss are generated by real parts of the above form factors, let us decompose (using eqs. ([REF], [REF])) [MATH] and [MATH] in the following way: D_&=& R_i2 R_i3 f_23^(a)', 'hep-ph-0012110-1-21-0': 'Since the asymmetries we are going to discuss are generated by real parts of the above form factors, let us decompose (using eqs.([REF], [REF])) [MATH] and [MATH] in the following way: D_&=& R_i2 R_i3 f_23^(a)', 'hep-ph-0012110-1-22-0': 'where superscripts indicate graphs that generate the contribution according to the notation of Fig.[REF].', 'hep-ph-0012110-1-22-1': 'Since in the case of the photon only Yukawa couplings [MATH] and [MATH] contribute, any signal of CP violation must be proportional to [MATH].', 'hep-ph-0012110-1-22-2': 'However, for the [MATH] boson vertex there is also other "source" of CP violation, namely [MATH].', 'hep-ph-0012110-1-23-0': 'It is useful to discuss [MATH] dependence of the functions [MATH] first.', 'hep-ph-0012110-1-23-1': 'From eq.([REF]) and eqs.([REF], [REF]) one can find out that all contributions to the form factors [MATH], [MATH] are enhanced for small [MATH]: [MATH], [MATH] and [MATH].', 'hep-ph-0012110-1-23-2': 'Therefore, for [MATH], the contributions from diagrams c)-f) are expected to be suppressed relatively to those generated by diagrams a) or b) in Fig.[REF].', 'hep-ph-0012110-1-24-0': 'Hereafter we assume that there exists only one light Higgs boson [MATH] and possible effects of the heavier scalar degrees of freedom decouple.', 'hep-ph-0012110-1-24-1': 'In Fig. [REF] we illustrate dependence of the functions [MATH] on the lightest Higgs boson mass, [MATH].', 'hep-ph-0012110-1-24-2': 'In order to amplify possible contributions we have chosen [MATH].', 'hep-ph-0012110-1-24-3': 'As it is seen from the figure, the dominant CP-violating effects will be generated by [MATH] and [MATH], which are generated by the Yukawa-coupling contribution, [MATH], from diagrams a) and b) in Fig. [REF].', 'hep-ph-0012110-1-24-4': 'As one could have anticipated, we observe an enhancement of the contributions generated by the diagrams a) and b) for low Higgs boson mass.', 'hep-ph-0012110-1-24-5': 'The growth of [MATH] and [MATH] is more pronounced for lower [MATH] (closer to the [MATH] production threshold), it is a consequence of the Coulomb-like singularity generated by graph a) and b) in the limit [MATH] at the [MATH] production threshold.', 'hep-ph-0012110-1-24-6': 'Similar behavior have been also noted in the case of CP-conserving form factors [CITATION].', 'hep-ph-0012110-1-24-7': 'One also observes a typical threshold behavior at [MATH] for [MATH] and [MATH] as a non-trivial absorptive part of diagrams c)-f) is necessary for nonzero contribution to [MATH].', 'hep-ph-0012110-1-25-0': 'If all three Higgs bosons of the model have the same mass, then the orthogonality of the mixing matrix [MATH] guarantees [CITATION] vanishing CP violation while summed over all the scalars.', 'hep-ph-0012110-1-25-1': 'Therefore, the leading contribution to [MATH] and [MATH] originating from an exchange of the lightest Higgs boson [MATH] could be partially cancelled by heavier scalars [MATH].', 'hep-ph-0012110-1-25-2': 'However here we will assume that masses of the heavier scalars are above the production threshold for [MATH], therefore, as observed (for the lightest Higgs boson mass below [MATH]) from Fig. [REF], the cancellation by heavier scalars for [MATH] and [MATH] could reach at most [MATH] and [MATH].', 'hep-ph-0012110-1-26-0': 'Leading asymptotic formulae for small and large Higgs mass are presented in the appendix [REF].', 'hep-ph-0012110-1-26-1': 'It is worth to notice that for non-zero [MATH], in the limit [MATH] both [MATH] and [MATH] are finite, whereas a typical decoupling limit, [MATH], is observed for large [MATH].', 'hep-ph-0012110-1-27-0': 'Fig. [REF] shows the functions [MATH] for [MATH] and two different Higgs boson masses [MATH] and [MATH], as a function of [MATH].', 'hep-ph-0012110-1-27-1': 'It is seen that it is not desirable to choose too high beam energy, as the size of the functions drops.', 'hep-ph-0012110-1-27-2': 'Again, [MATH] dominate over [MATH] by 1-2 orders of magnitude.', 'hep-ph-0012110-1-28-0': 'width=.65plot-eett-prodre-ffmhdep.eps The functions [MATH] defined in eq. ([REF]) as a function of [MATH] for [MATH], [MATH] (left), [MATH] (middle) and [MATH] (right).', 'hep-ph-0012110-1-28-1': 'fig:eett:prodre:hdep', 'hep-ph-0012110-1-29-0': 'width=.65plot-eett-prodre-ffsdep.eps The functions [MATH] defined in eq. ([REF]) as a function of [MATH] for [MATH], [MATH] and [MATH]fig:eett:prodre:sdep', 'hep-ph-0012110-1-30-0': '## Top Decay', 'hep-ph-0012110-1-31-0': 'We will adopt the following parameterization of the [MATH] vertex suitable for the [MATH] and [MATH] decays: ^&=&-g2V_tb [^(f_1^L P_L +f_1^R P_R) -i^k_M_W (f_2^L P_L +f_2^R P_R)],', 'hep-ph-0012110-1-32-0': '^&=&-g2V_tb^* [^(f_1^L P_L +f_1^R P_R) -i^k_M_W (f_2^L P_L +f_2^R P_R)],', 'hep-ph-0012110-1-33-0': 'where [MATH], [MATH] is the [MATH] element of the Kobayashi-Maskawa matrix and [MATH] is the momentum of [MATH].', 'hep-ph-0012110-1-33-1': 'In the SM [MATH] and all the other form factors vanish.', 'hep-ph-0012110-1-33-2': 'It turns out that in the limit of massless bottom quarks the only form factors that interfere with the SM are [MATH] and [MATH] for the top and anti-top decays, respectively.', 'hep-ph-0012110-1-33-3': 'Currently, there is no relevant experimental bound on those form factors.', 'hep-ph-0012110-1-34-0': 'One can show that the CP-violating and CP-conserving parts of the form factors for [MATH] and [MATH] are not independent: f_1^L,R=f_1^L,R and f_2^L,R=f_2^R,L,', 'hep-ph-0012110-1-35-0': 'where upper (lower) signs are those for [MATH]-conserving (-violating) contributions [CITATION].', 'hep-ph-0012110-1-35-1': 'Therefore any [MATH]-violating observable defined for the top-quark decay must be proportional to [MATH] or [MATH].', 'hep-ph-0012110-1-36-0': 'width=0.65phd-diag-tbw.epsDiagrams contributing to [MATH].', 'hep-ph-0012110-1-36-1': 'fig:diag:tbw', 'hep-ph-0012110-1-37-0': 'Diagrams contributing to CP violation in the decay process are shown in Fig. [REF].', 'hep-ph-0012110-1-37-1': 'Direct calculation leads to the following result for the CP-violating part of [MATH] : [EQUATION] where ija &=& ij(p_t, -p_W, m_h^2, m_t^2, m_b^2)', 'hep-ph-0012110-1-38-0': 'ijbd&=& ij(p_W, p_b, m_W^2, m_h^2, m_b^2)', 'hep-ph-0012110-1-39-0': 'ijce&=& ij(p_W, p_b, m_h^2, m_W^2, m_t^2)', 'hep-ph-0012110-1-40-0': 'ijf &=& ij(p_W, -p_t, m_h^2, m_H^2, m_b^2)', 'hep-ph-0012110-1-41-0': 'ijg &=& ij(p_W, p_b, m_h^2, m_H^2, m_t^2) and [EQUATION] where [MATH] and [MATH].', 'hep-ph-0012110-1-41-1': 'The asymmetries that will be discussed here depends on real parts of [MATH] and [MATH].', 'hep-ph-0012110-1-41-2': 'It is easy to note from eq. ([REF]) that imaginary parts of [MATH] functions contribute to the real part of [MATH].', 'hep-ph-0012110-1-41-3': 'It is seen that only diagrams b), d) and f) will contribute to [MATH].', 'hep-ph-0012110-1-41-4': 'However, [MATH] strongly suggest [CITATION] that for 2HDM [MATH], therefore eventually (adopting the relation ([REF])) one gets the following result (from graphs b) and d) only) for CP-violating contribution to [MATH]: [EQUATION]', 'hep-ph-0012110-1-41-5': 'As will be discussed in Section [REF] there is a strong experimental bound on [MATH] for [MATH].', 'hep-ph-0012110-1-41-6': 'Taking into account the limit on [MATH] and choosing [MATH] (in order to illustrate a possible enhancement) we plot [MATH] in Fig. [REF] as a function of [MATH].', 'hep-ph-0012110-1-41-7': 'It is seen that [MATH] is by [MATH] orders of magnitude below [MATH] or [MATH] even for large b-quark Yukawa coupling, compare Fig. [REF] and [REF].', 'hep-ph-0012110-1-41-8': 'The suppression is caused both by the experimental limit on [MATH] (for [MATH]) and by an extra suppression factor of [MATH] (relative to [MATH]).', 'hep-ph-0012110-1-42-0': 'There is a comment in order here; since in the 2HDM the real part of CP-violating form factors in the top decay is much smaller then in the production process, it is interesting to look closer at the suppression mechanism and find class of possible extensions of the SM that provide large [MATH].', 'hep-ph-0012110-1-42-1': 'Since an absorptive part is needed, the only graphs that may contribute are those denoted by b), d) and f) in Fig. [REF] (allowing for the neutral scalar to be replaced by a neutral vector).', 'hep-ph-0012110-1-42-2': 'One source of the suppression is the bottom-quark mass that originate from the propagator while the second one comes from the Yukawa vertex.', 'hep-ph-0012110-1-42-3': 'The latter one could be easy amplified by large [MATH]: for [MATH] the bottom-quark Yukawa coupling is as strong as the SU(2) gauge coupling.', 'hep-ph-0012110-1-42-4': 'Therefore the suppression to overcome is [MATH] from the bottom-quark propagator.', 'hep-ph-0012110-1-42-5': 'A possible solution [CITATION] seems to be a multi-doublet-Higgs model that could evade the stringent restriction from the [MATH] decay and also overcome (through a contribution from the graph type f) in Fig. [REF]) the limit on [MATH] that comes from the LEP limit on [MATH].', 'hep-ph-0012110-1-42-6': 'Let us notice that it is much easier to develop large [MATH], see e.g. Refs. [CITATION] and [CITATION].', 'hep-ph-0012110-1-43-0': 'eps [MATH] as a function of [MATH] for [MATH].', 'hep-ph-0012110-1-43-1': 'It has been assumed (according to constraints from [MATH]) that the charged Higgs boson is heavy, [MATH], therefore there is no absorptive part necessary to develop a non-zero contribution from diagram f) in Fig.[REF].', 'hep-ph-0012110-1-43-2': 'Additionally, we assumed [MATH] (what is a very good approximation for [MATH]) and selected such values for [MATH] and [MATH] that are consistent both with [MATH] orthogonality and [MATH] LEP bound, and provide a maximal value of [MATH].', 'hep-ph-0012110-1-43-3': 'fig:eett:dec:ref2mhdep', 'hep-ph-0012110-1-44-0': 'It is worth to mention that even though in the SM there exists one-loop contribution to [MATH], it turns out to be strongly GIM suppressed [CITATION].', 'hep-ph-0012110-1-44-1': 'Therefore, although the 2HDM prediction for [MATH] is smaller than for CP violating from factors in the production mechanism, it is still by a factor [MATH] larger then the SM result.', 'hep-ph-0012110-1-45-0': '# Experimental Constraints', 'hep-ph-0012110-1-46-0': 'Hereafter we will focus on Higgs boson masses in the region, [MATH].', 'hep-ph-0012110-1-46-1': 'As it has been shown in the literature [CITATION] the existing LEP data are perfectly consistent with one light Higgs boson within the 2HDM.', 'hep-ph-0012110-1-46-2': 'It turns out that even precision electroweak tests allow for light Higgs bosons [CITATION].', 'hep-ph-0012110-1-47-0': 'In order to amplify the form factors calculated in this paper we have adopted for an illustration [MATH].', 'hep-ph-0012110-1-47-1': 'However, there exist experimental constraints on [MATH] from [MATH] and [MATH] mixing [CITATION], [MATH] decay [CITATION] and [MATH] decay [CITATION].', 'hep-ph-0012110-1-47-2': 'Since small [MATH] enhances [MATH] coupling, in order to maintain [MATH] we have to decouple charged Higgs effects and therefore we assume that [MATH]>[MATH][MATH].', 'hep-ph-0012110-1-48-0': 'The constraints on the mixing angles [MATH] that should be imposed in our numerical analysis are as follows:', 'hep-ph-0012110-1-49-0': 'It turns out that the restriction on the [MATH] coupling from its contribution to the total [MATH]-width is always weaker then the one from [MATH] production if [MATH]>[MATH][MATH].', 'hep-ph-0012110-1-50-0': 'The LEP constraints on the [MATH] coupling restrict the following entries of the mixing matrix [MATH]: [EQUATION] where [MATH] stands for the upper limit for the relative strength of [MATH] coupling determined experimentally in Ref. [CITATION] up to the Higgs mass [MATH].', 'hep-ph-0012110-1-50-1': 'As we have concluded in the previous section, CP-violating phenomena we are considering are enhanced by small [MATH], in that case one can see from eq. ([REF]) that the LEP constraints mostly restrict [MATH].', 'hep-ph-0012110-1-50-2': 'Through the orthogonality the restriction on [MATH] is being transfered to constrain [MATH] which multiplies leading contributions to all CP-violating asymmetries considered here.', 'hep-ph-0012110-1-50-3': 'The final result for upper limit on [MATH] as a function of [MATH] is shown in Fig. [REF].', 'hep-ph-0012110-1-50-4': 'In fact the bound on [MATH] depends on the Higgs mass, however, in order to be conservative, we have assumed [MATH] that is the most restrictive experimental limit (obtained for [MATH]).', 'hep-ph-0012110-1-51-0': 'epsMaximal value of [MATH] allowed by the LEP constraints on [MATH] coupling as a function of [MATH].f', 'hep-ph-0012110-1-52-0': 'As it is seen from Fig.[REF] the constraints for [MATH] are weak for small [MATH].', 'hep-ph-0012110-1-52-1': 'Therefore for [MATH] it should be legitimate to assume [MATH] which is the maximal value consistent with orthogonality.', 'hep-ph-0012110-1-53-0': 'Using the maximal value of [MATH] allowed by the orthogonality and the LEP constraints for small [MATH], we may discuss a possibility for an experimental determination of the calculated form factors at future [MATH] colliders.', 'hep-ph-0012110-1-53-1': 'A detailed discussion of expected statistical uncertainties for a measurement of the form factors has been performed in Ref. [CITATION].', 'hep-ph-0012110-1-53-2': 'It has been shown that adjusting an optimal [MATH] beam polarizations, using the energy and angular double distribution of final leptons and fitting all 9 form factors leads to the following statistical errors for the determination of CP-violating form factors: [MATH] and [MATH] for [MATH].', 'hep-ph-0012110-1-53-3': 'It is seen that only [MATH], could be measured with a high precision.', 'hep-ph-0012110-1-53-4': 'We have observed in Figs. [REF],[REF] that [MATH] may reach at most a value of [MATH], therefore one shall conclude that several years of running with yearly integrated luminosity [MATH] should allow for an observation of [MATH] generated within 2HDM, provided the lightest Higgs boson mass is not too large.', 'hep-ph-0012110-1-53-5': 'On the other hand, the expected [CITATION] precision for the determination of the decay form factors is much more promising: [MATH].', 'hep-ph-0012110-1-53-6': 'However, as we have seen in Fig. [REF], the maximal expected size of [MATH] is [MATH] (for [MATH]), therefore either an unrealistic growth of the luminosity, or other observables (besides the energy and angular double distribution of final leptons) are required in order to observe CP-violating from factors in the top-quark decay process.', 'hep-ph-0012110-1-53-7': 'The results of Ref. [CITATION] assumed simultaneous determination of all 9 form factors, therefore another chance to reduce of [MATH] is to have some extra independent constraints on the top-quark coupling coming from other colliders, like the Fermilab Tevatron or LHC.', 'hep-ph-0012110-1-54-0': '# CP-Violating Asymmetries', 'hep-ph-0012110-1-55-0': 'Looking for CP violation one can directly measure [CITATION] all the form factors including those which are odd under CP.', 'hep-ph-0012110-1-55-1': 'However another possible attitude is to construct certain asymmetries sensitive to CP violation.', 'hep-ph-0012110-1-55-2': 'In this section we will discuss several asymmetries that could probe CP violation in the process [MATH].', 'hep-ph-0012110-1-55-3': 'We will systematically drop all contributions quadratic in non-standard form factors and calculate various asymmetries keeping only interference between the SM and [MATH], [MATH] or [MATH].', 'hep-ph-0012110-1-56-0': '## Lepton-Energy Asymmetry', 'hep-ph-0012110-1-57-0': 'Let us introduce the rescaled lepton energy, [MATH], by x2 E_l(1-_t1+_t)^1/2,', 'hep-ph-0012110-1-58-0': 'where [MATH] is the energy of [MATH] in [MATH] c.m. frame and [MATH].', 'hep-ph-0012110-1-58-1': 'Using lepton energy distribution [MATH] calculated [CITATION] for the general form factors given in eqs. ([REF],[REF]) one can define the following energy asymmetry: [EQUATION]', 'hep-ph-0012110-1-58-2': 'Direct calculation leads to the following result in terms of the CP-violating form factors: A_CP^l(x)=2g(x) +(f_2^R-f_2^L) [ f(x)+ g(x)]2[f(x)+ g(x)].', 'hep-ph-0012110-1-59-0': 'eps The coefficient functions [MATH] defined by eq.([REF]) for the energy-asymmetry as function of [MATH] for [MATH] 360 (left), 500 (middle) and 1000 GeV (right) for [MATH].', 'hep-ph-0012110-1-59-1': 'The solid curve represents the coefficient [MATH], dashed [MATH] and dotted [MATH].f', 'hep-ph-0012110-1-60-0': 'In order to estimate a relative strength of various sources of CP violation it is worth to decompose the asymmetry as follows: [EQUATION]', 'hep-ph-0012110-1-60-1': 'As one can see from Fig. [REF] the CP-violating effects that originate from the decay are substantially enhanced in the soft energy region.', 'hep-ph-0012110-1-60-2': 'It is worth to notice that the minimal lepton energy for [MATH] is [MATH] that is large enough to detect the lepton.', 'hep-ph-0012110-1-60-3': 'Therefore the region of soft leptons should be carefully studied experimentally.', 'hep-ph-0012110-1-60-4': 'The enhancement is a consequence of particular behavior of [MATH], [MATH] and [MATH] that causes the relative amplification of the decay effects, see Fig.1 in Ref. [CITATION].', 'hep-ph-0012110-1-60-5': 'The same figure explains the observed smallness of the [MATH]-boson contribution.', 'hep-ph-0012110-1-60-6': 'For hard leptons both [MATH] and [MATH] are enhanced and they are raising with the c.m. energy.', 'hep-ph-0012110-1-61-0': 'The energy-asymmetry could be decomposed into the leading contribution proportional to [MATH] and the remaining piece proportional to [MATH].', 'hep-ph-0012110-1-61-1': 'The former one (that provides the leading contribution) is plotted in Fig. [REF] for a fixed energy, [MATH], as a function of [MATH].', 'hep-ph-0012110-1-61-2': 'One can observe that the largest asymmetry for the chosen energy corresponds to [MATH] and [MATH].', 'hep-ph-0012110-1-62-0': 'width=.65eett-asy9604r-mhdep.epsThe Higgs mass dependence of the coefficient of [MATH] for the asymmetry given by eq. ([REF]) for [MATH]=360 (solid), 500 (dashed), 1000 GeV (dotted) for a fixed energy, [MATH], and [MATH].', 'hep-ph-0012110-1-62-1': 'fig:eett:asy9604r:mhdep', 'hep-ph-0012110-1-63-0': '## Integrated Lepton-Energy Asymmetry', 'hep-ph-0012110-1-64-0': 'CP symmetry could be also tested using the following leptonic double energy distribution [CITATION]: 1d^2dxd =_i=1^3c_i f_i(x,),', 'hep-ph-0012110-1-65-0': 'where [MATH] and [MATH] are for [MATH] and [MATH] respectively, for [EQUATION] and [EQUATION] where [EQUATION].', 'hep-ph-0012110-1-66-0': 'The following asymmetry could be a measure of CP violation: [EQUATION]', 'hep-ph-0012110-1-66-1': 'As before, it is useful to separate contributions from various form factors: [EQUATION]', 'hep-ph-0012110-1-66-2': 'In Table [REF] we show the coefficients [MATH] for various c.m. energies.', 'hep-ph-0012110-1-66-3': 'Firstly, is clear that for any given [MATH] the coefficient [MATH] is the smallest one.', 'hep-ph-0012110-1-66-4': 'Secondly, it is seen that just above the threshold for [MATH] production there is an enhancement of relative contributions from the decay, however that still not sufficient to overcome the suppression of [MATH] that we have observed in Fig. [REF].', 'hep-ph-0012110-1-66-5': 'Therefore we can conclude that the leading contribution is provided by CP violation in the [MATH] vertex.', 'hep-ph-0012110-1-67-0': 'Fig. [REF] illustrates the Higgs-mass dependence of the leading (proportional to [MATH]) contribution to the integrated lepton-energy asymmetry.', 'hep-ph-0012110-1-67-1': 'It turns out that [MATH] provides the largest asymmetry.', 'hep-ph-0012110-1-68-0': 'width=.65eett-asy9608r-mhdep.eps Higgs mass dependence of the coefficient of [MATH] for the asymmetry given by eq. ([REF]) for [MATH]=360 (solid), 500 (dashed), 1000 GeV (dotted) for [MATH].', 'hep-ph-0012110-1-68-1': 'fig:eett:asy9608r:mhdep', 'hep-ph-0012110-1-69-0': 'Using results of Ref. [CITATION] one can find out an expected statistical error for the determination of [MATH] at any given [MATH] collider.', 'hep-ph-0012110-1-69-1': 'Assuming [MATH], [MATH] and lepton tagging efficiency, [MATH] we get [MATH].', 'hep-ph-0012110-1-69-2': 'As it is seen from Fig. [REF] an observation of the asymmetry would require several years of running at the assumed luminosity.', 'hep-ph-0012110-1-70-0': '## Angular Asymmetry', 'hep-ph-0012110-1-71-0': 'Another CP-violating asymmetry could be constructed using the angular distributions of the bottom quarks or leptons originating from the top-quark decay: dd_f= 3_EM^22sB_f (_0^f+_1^f _f+_2^f^2_fright),', 'hep-ph-0012110-1-72-0': 'where [MATH], [MATH] is an appropriate top-quark branching ratio, [MATH] is the angle between the [MATH] beam direction and the direction of [MATH] momentum in the [MATH] c.m. frame and [MATH] are coefficients calculable in terms of the form factors, see Ref. [CITATION].', 'hep-ph-0012110-1-72-1': 'The following asymmetry provides a signal of CP violation: A_CP^f(_f)= d^+(_f)d_f- d^-(f)d_f d^+(_f)d_f+ d^-(f)d_f,', 'hep-ph-0012110-1-73-0': 'where [MATH] is referring to [MATH] and [MATH] distributions, respectively.', 'hep-ph-0012110-1-73-1': 'Since [MATH] under [MATH], the asymmetry defined above is a true measure of [MATH] violation.', 'hep-ph-0012110-1-74-0': 'Adopting general formulas for the asymmetry from Ref. [CITATION] and inserting form factors calculated here we plot the asymmetry in Figs. [REF], [REF] as a function of [MATH] for bottom quarks and leptons, respectively.', 'hep-ph-0012110-1-74-1': 'As before, the asymmetry can be decomposed into [MATH], [MATH] and [MATH] vertex contributions: [EQUATION]', 'hep-ph-0012110-1-74-2': 'It is seen that forward-backward directions are favored, however an experimental cut [MATH] should be imposed in the realistic experimental environment.', 'hep-ph-0012110-1-75-0': 'width=.65eett-asy9911bff-thdep.eps The coefficients functions [MATH] defined by eq. ([REF]) for the angular asymmetry, for final bottom quarks ([MATH]) as a function of [MATH] for [MATH] 360 (left), 500 (middle) and 1000 GeV (right) for [MATH].', 'hep-ph-0012110-1-75-1': 'The solid curve represents the coefficient [MATH], dashed [MATH] and dotted [MATH]fig:eett:asy9911bff-thdep', 'hep-ph-0012110-1-76-0': 'In order to illustrate the Higgs mass dependence we plot in Fig.[REF] the angular asymmetry both for [MATH] and [MATH] for chosen polar angle [MATH].', 'hep-ph-0012110-1-76-1': 'As expected the maximal effect could be reached for minimal Higgs mass [MATH], [MATH] is the most suitable energy.', 'hep-ph-0012110-1-77-0': 'width=.65eett-asy9911r-mhdep.epsThe Higgs mass dependence of the coefficient of [MATH] for the angular asymmetry defined by eq. ([REF]) for [MATH]=360 (solid), 500 (dashed), 1000 (dotted) GeV for a fixed polar angle [MATH] and [MATH].', 'hep-ph-0012110-1-77-1': 'fig:eett:asy9911br-mhdep', 'hep-ph-0012110-1-78-0': '## Integrated Angular Asymmetry', 'hep-ph-0012110-1-79-0': 'The angular distribution given in eq. ([REF]) could be adopted to define an integrated version [CITATION] of the angular asymmetry [MATH]: [EQUATION] where [MATH] and [MATH] are the polarizations of [MATH] and [MATH] beams, [MATH] is referring to [MATH] and [MATH] distributions respectively, and [MATH] expresses the experimental polar-angle cut.', 'hep-ph-0012110-1-79-1': 'In order to discuss possible advantages of polarized initial beams we are considering here dependence of the asymmetry on the polarization.', 'hep-ph-0012110-1-79-2': 'Hereafter we will discuss the same polarization for [MATH] and [MATH]: [MATH].', 'hep-ph-0012110-1-80-0': 'Again we decompose the asymmetry as follows: [EQUATION]', 'hep-ph-0012110-1-80-1': 'In Table [REF] we show the coefficient functions [MATH] calculated for various energy and polarization choices assuming the polar angle cut [MATH], i.e. [MATH] in eq. ([REF]), both for leptons and bottom quarks .', 'hep-ph-0012110-1-80-2': 'It could be seen that a positive polarization leads to higher coefficients [MATH] and [MATH].', 'hep-ph-0012110-1-80-3': 'Since [MATH] that implies that maximal asymmetry could be reached for [MATH] and the dominant contribution is originating from [MATH].', 'hep-ph-0012110-1-80-4': 'Since the number of events does not drop drastically when going from unpolarized beams to [MATH], it turns out that the positive polarization is the most suitable for testing the integrated angular asymmetry.', 'hep-ph-0012110-1-80-5': 'It is clear from the table that the asymmetry for final leptons should be larger by a factor [MATH] then the one for bottom quarks and their signs should be reversed.', 'hep-ph-0012110-1-81-0': 'Using the general formula for the asymmetry from Ref. [CITATION] and adopting results for the CP-violating form factors we plot [MATH] in Fig. [REF] as a function of the Higgs mass both for bottom quarks and leptons.', 'hep-ph-0012110-1-81-1': 'It is clear that the largest asymmetry could be expected for [MATH] for final leptons at [MATH].', 'hep-ph-0012110-1-81-2': 'With the maximal mixing, [MATH] the [MATH] asymmetry could be expected for the Higgs boson with mass [MATH].', 'hep-ph-0012110-1-81-3': 'Since the statistical error expected [CITATION] for the asymmetry is of the order of [MATH], we can conclude that the asymmetry [MATH] is the most promising one, leading to [MATH] effect for light Higgs mass and [MATH].', 'hep-ph-0012110-1-81-4': 'As it is seen form Fig.[REF] it is relevant to have polarized [MATH] beams.', 'hep-ph-0012110-1-82-0': 'epsThe Higgs mass dependence of the coefficient of [MATH] for the angular asymmetry defined by eq.([REF]) for bottom quarks (upper) and leptons (lower) at [MATH]=360 (solid), 500 (dashed), 1000 GeV (dotted) with unpolarized beams (left), [MATH] (middle) and [MATH] (right) for [MATH].f', 'hep-ph-0012110-1-83-0': '# Summary and Conclusions', 'hep-ph-0012110-1-84-0': 'We have considered a general two-Higgs-doublet model with CP violation in the scalar sector.', 'hep-ph-0012110-1-84-1': 'Mixing of the three neutral Higgs fields of the model leads to CP-violating Yukawa couplings of the physical Higgs bosons.', 'hep-ph-0012110-1-84-2': 'CP-asymmetric form factors generated at the one-loop level of perturbation theory has been calculated within the model.', 'hep-ph-0012110-1-84-3': 'Although in general the existing experimental data from LEP1 and LEP2 constraint the mixing angles of the three neutral Higgs fields, their combination relevant for CP violation is not bounded for small [MATH] which is the region of our interest.', 'hep-ph-0012110-1-84-4': 'We have shown that the decay form factors are typically smaller then the production ones by 2-3 orders of magnitude.', 'hep-ph-0012110-1-84-5': 'The dominant contribution to CP violation in the production is coming from [MATH] coupling.', 'hep-ph-0012110-1-84-6': 'Several energy and angular CP-violating asymmetries for the process [MATH] and [MATH] has been considered using the form factors calculated within the two-Higgs-doublet model.', 'hep-ph-0012110-1-84-7': 'It turned out that the best test of CP invariance would be provided by the integrated angular asymmetry [MATH] for positive polarizations of [MATH] beams.', 'hep-ph-0012110-1-84-8': 'For one year of running at TESLA collider with the integrated luminosity [MATH] one could expect [MATH] effect for the asymmetry for light Higgs boson and [MATH].'}

{'hep-ph-0012110-2-0-0': 'We consider a general two-Higgs-doublet model with CP violation in the scalar sector.', 'hep-ph-0012110-2-0-1': 'Three neutral Higgs fields of the model all mix and the resulting physical Higgs bosons have no definite CP properties.', 'hep-ph-0012110-2-0-2': 'That leads, at the one-loop level of the perturbation expansion, to CP-violating form factors for [MATH], [MATH] and [MATH] interaction vertices.', 'hep-ph-0012110-2-0-3': 'We discuss asymmetries sensitive to CP violation induced by the form factors for the process [MATH] and [MATH] at future linear [MATH] colliders.', 'hep-ph-0012110-2-1-0': '# Introduction', 'hep-ph-0012110-2-2-0': 'Even though the top quark has been already discovered several years ago [CITATION], its interactions are still weakly constrained.', 'hep-ph-0012110-2-2-1': 'It remains an open question if top-quark couplings obey the Standard Model (SM) scheme of the electroweak forces or there exists a contribution from physics beyond the SM.', 'hep-ph-0012110-2-2-2': 'In particular, CP violation in the top-quark interactions has not been verified.', 'hep-ph-0012110-2-2-3': 'The classical method for incorporating CP violation into the SM is to make the Yukawa couplings of the Higgs boson to quarks explicitly complex, as built into the Kobayashi-Maskawa mixing matrix [CITATION] proposed more than two decades ago.', 'hep-ph-0012110-2-2-4': 'However, CP violation could equally well be partially or wholly due to other mechanisms.', 'hep-ph-0012110-2-2-5': 'The possibility that CP violation derives largely from the Higgs sector itself is particularly appealing in the context of the observed baryon asymmetry, since its explanation requires more CP violation [CITATION] then is provided by the SM.', 'hep-ph-0012110-2-2-6': 'Even the simple two-Higgs-doublet model (2HDM) extension of the one-doublet SM Higgs sector provides a much richer framework for describing CP violation; in the 2HDM, spontaneous and/or explicit CP violation is possible in the scalar sector [CITATION].', 'hep-ph-0012110-2-2-7': 'The model, besides CP violation, offers many other appealing phenomena, for a review see Ref. [CITATION].', 'hep-ph-0012110-2-3-0': 'For our analysis, the most relevant part of the interaction Lagrangian takes the following form : L= -vh(a+i_5 b)t + C hv (^2 Z_Z^+2 ^2 W_W^),', 'hep-ph-0012110-2-4-0': 'where [MATH] is the lowest mass scalar, [MATH] is the SU(2) coupling constant, [MATH] is the Higgs boson vacuum expectation value (with the normalization adopted here such that [MATH]), [MATH], [MATH] and [MATH] are real parameters which account for deviations from the SM, [MATH], [MATH] and [MATH] reproduce the SM Lagrangian.', 'hep-ph-0012110-2-4-1': 'Since under CP, [MATH] and [MATH], one can observe that terms in the cross section proportional to [MATH] or [MATH] would indicate CP violation.', 'hep-ph-0012110-2-4-2': 'The 2HDM is the minimal extension of the SM that provides non-zero [MATH] and/or [MATH].', 'hep-ph-0012110-2-5-0': 'In this paper we will focus on CP-violating contributions to the process [MATH] and [MATH] induced within 2HDM.', 'hep-ph-0012110-2-5-1': 'However the fundamental goal is seeking for the ultimate theory of electroweak interactions.', 'hep-ph-0012110-2-5-2': 'There are several reasons to utilize CP violation in the top physics while looking for physics beyond the SM: The top quark decays immediately after being produced as its huge mass [MATH] [CITATION] leads to a decay width [MATH] much larger than [MATH].', 'hep-ph-0012110-2-5-3': 'Therefore the decay process is not contaminated by any fragmentation effects [CITATION] and decay products may provide useful information on top-quark properties.', 'hep-ph-0012110-2-5-4': 'Since the top quark is heavy, its Yukawa coupling is large and therefore its interactions could be sensitive to a Higgs sector of the electroweak theory.', 'hep-ph-0012110-2-5-5': 'At the same time, the TESLA collider design is supposed to offer an integrated luminosity of the order of [MATH] at [MATH].', 'hep-ph-0012110-2-5-6': 'Therefore expected number of [MATH] events per year could reach [MATH] even for [MATH] tagging efficiency [MATH].', 'hep-ph-0012110-2-5-7': 'That should allow to study subtle properties of the top quark, which could e.g. lead to CP-sensitive asymmetries of the order of [MATH].', 'hep-ph-0012110-2-5-8': 'Since the top quark is that heavy and the third family of quarks effectively decouples from the first two, any CP-violating observables within the SM are expected to be tiny, e.g.: i) non-zero electric dipole moment of fermions is generated at the three-loop approximation of the perturbation expansion [CITATION], or ii) the decay rate asymmetry (being a one-loop effect) is strongly GIM suppressed reaching at most a value [MATH] [CITATION].', 'hep-ph-0012110-2-5-9': 'So, one can expect that for CP-violating asymmetries any SM background could be safely neglected.', 'hep-ph-0012110-2-5-10': 'Therefore it seems to be justified to look for CP-violating Higgs effects in the process of [MATH] production and its subsequent decay at future linear [MATH] colliders.', 'hep-ph-0012110-2-5-11': 'Even though 2HDM contributions to various CP-sensitive asymmetries has been already presented in the existing literature, see Refs. [CITATION], here we are providing a consistent treatment of CP violation both in the production, [MATH], and in the top-quark decay, [MATH].', 'hep-ph-0012110-2-5-12': 'For an extensive review of CP violation in top-quark interactions see Ref. [CITATION].', 'hep-ph-0012110-2-6-0': 'The paper is organized as follows.', 'hep-ph-0012110-2-6-1': 'In Section [REF], we briefly outline the mechanism of CP violation in the 2HDM, introduce the mixing matrix for neutral scalars and derive necessary couplings.', 'hep-ph-0012110-2-6-2': 'In section [REF], we present results for CP-violating form factors both for the [MATH] production process and for [MATH] and [MATH] decays.', 'hep-ph-0012110-2-6-3': 'In Section [REF], we recall current experimental constraints relevant for the CP-violating observables considered in this paper.', 'hep-ph-0012110-2-6-4': 'In Section [REF], we collect results for various energy and angular CP-violating asymmetries.', 'hep-ph-0012110-2-6-5': 'Concluding remarks are given in Section [REF].', 'hep-ph-0012110-2-7-0': '# The two-Higgs-doublet model with CP violation', 'hep-ph-0012110-2-8-0': 'The 2HDM of electroweak interactions contains two SU(2) Higgs doublets denoted by [MATH] and [MATH].', 'hep-ph-0012110-2-8-1': 'It is well known [CITATION] that the model allows both for spontaneous and explicit CP violation.', 'hep-ph-0012110-2-9-0': 'After SU(2)[MATH]U(1) gauge symmetry breaking, one combination of neutral Higgs fields, [MATH], becomes a would-be Goldstone boson which is absorbed while giving mass to the [MATH] gauge boson.', 'hep-ph-0012110-2-9-1': '(Here, we use the notation [MATH], [MATH], where [MATH].)', 'hep-ph-0012110-2-9-2': 'The same mixing angle, [MATH], also diagonalizes the mass matrix in the charged Higgs sector.', 'hep-ph-0012110-2-9-3': 'If either explicit or spontaneous CP violation is present, the remaining three neutral degrees of freedom, [EQUATION] are not mass eigenstates.', 'hep-ph-0012110-2-9-4': 'The physical neutral Higgs bosons [MATH]) are obtained by an orthogonal transformation, [MATH], where the rotation matrix is given in terms of three Euler angles ([MATH]) by [EQUATION] where [MATH] and [MATH].', 'hep-ph-0012110-2-10-0': 'As a result of the mixing between real and imaginary parts of neutral Higgs fields, the Yukawa interactions of the [MATH] mass-eigenstates are not invariant under CP.', 'hep-ph-0012110-2-10-1': 'They are given by: [EQUATION] where the scalar ([MATH]) and pseudoscalar ([MATH]) couplings are functions of the mixing angles.', 'hep-ph-0012110-2-10-2': 'For up-type quarks we have [EQUATION] and for down-type quarks: [EQUATION] and similarly for charged leptons.', 'hep-ph-0012110-2-10-3': 'For large [MATH], the couplings to down-type fermions are typically enhanced over the couplings to up-type fermions.', 'hep-ph-0012110-2-11-0': 'In the following analysis we will also need the couplings of neutral Higgs and vector bosons, they are given by [EQUATION] for [MATH].', 'hep-ph-0012110-2-11-1': 'Hereafter we shall denote the lightest Higgs boson by [MATH] and its [MATH]-matrix index by [MATH].', 'hep-ph-0012110-2-12-0': '# Form Factors', 'hep-ph-0012110-2-13-0': '## [MATH] Production', 'hep-ph-0012110-2-14-0': 'width=0.65phd-diag-eett6.epsDiagrams contributing to CP-violating form factors [MATH].', 'hep-ph-0012110-2-14-1': 'fig:diag:eett', 'hep-ph-0012110-2-15-0': 'The effective [MATH] and [MATH] vertices will be parameterized by the following form factors: [EQUATION] where [MATH] denotes the SU(2) gauge coupling constant, [MATH], and [EQUATION] denote the SM contributions to the vertices for [EQUATION].', 'hep-ph-0012110-2-15-1': 'The form factors [MATH], [MATH], [MATH] describe [MATH]-conserving while [MATH] parameterizes [MATH]-violating contributions.', 'hep-ph-0012110-2-16-0': 'Further in this paper the following parameters will be adopted: [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0012110-2-17-0': 'Since in this paper we are focusing on CP-violating asymmetries, the only relevant form factors are [MATH] and [MATH].', 'hep-ph-0012110-2-17-1': 'Direct calculation of diagrams shown in Fig. [REF] leads to the following result in terms of 3-point Passarino-Veltman [CITATION] functions defined in the appendix [REF]:', 'hep-ph-0012110-2-18-0': 'D_&=& 2^2 A_m_t^2v^2 b_i^t a_i^t m_t^2 12(p_t, p_t, m_t^2, m_h^2, m_t^2),', 'hep-ph-0012110-2-19-0': 'D_Z&=& 2^2 A_Z m_t^2v^2 b_i^t [ a_i^t m_t^2 12(p_t, p_t, m_t^2, m_h^2, m_t^2).', 'hep-ph-0012110-2-20-0': '&& .', 'hep-ph-0012110-2-20-1': '- C_i m_Z^2 12(p_t,p_t, m_h^2, m_t^2, m_Z^2) ].', 'hep-ph-0012110-2-20-2': 'Since the asymmetries we are going to discuss are generated by real parts of the above form factors, let us decompose (using eqs. ([REF], [REF])) [MATH] and [MATH] in the following way: D_&=& R_i2 R_i3 f_23^(a)', 'hep-ph-0012110-2-21-0': 'Since the asymmetries we are going to discuss are generated by real parts of the above form factors, let us decompose (using eqs.([REF], [REF])) [MATH] and [MATH] in the following way: D_&=& R_i2 R_i3 f_23^(a)', 'hep-ph-0012110-2-22-0': 'where superscripts indicate graphs that generate the contribution according to the notation of Fig.[REF].', 'hep-ph-0012110-2-22-1': 'Since in the case of the photon only Yukawa couplings [MATH] and [MATH] contribute, any signal of CP violation must be proportional to [MATH].', 'hep-ph-0012110-2-22-2': 'However, for the [MATH] boson vertex there is also other "source" of CP violation, namely [MATH].', 'hep-ph-0012110-2-23-0': 'It is useful to discuss [MATH] dependence of the functions [MATH] first.', 'hep-ph-0012110-2-23-1': 'From eq.([REF]) and eqs.([REF], [REF]) one can find out that all contributions to the form factors [MATH], [MATH] are enhanced for small [MATH]: [MATH], [MATH] and [MATH].', 'hep-ph-0012110-2-23-2': 'Therefore, for [MATH], the contributions from diagrams c)-f) are expected to be suppressed relatively to those generated by diagrams a) or b) in Fig.[REF].', 'hep-ph-0012110-2-24-0': 'Hereafter we assume that there exists only one light Higgs boson [MATH] and possible effects of the heavier scalar degrees of freedom decouple.', 'hep-ph-0012110-2-24-1': 'In Fig. [REF] we illustrate dependence of the functions [MATH] on the lightest Higgs boson mass, [MATH].', 'hep-ph-0012110-2-24-2': 'In order to amplify possible contributions we have chosen [MATH].', 'hep-ph-0012110-2-24-3': 'As it is seen from the figure, the dominant CP-violating effects will be generated by [MATH] and [MATH], which are generated by the Yukawa-coupling contribution, [MATH], from diagrams a) and b) in Fig. [REF].', 'hep-ph-0012110-2-24-4': 'As one could have anticipated, we observe an enhancement of the contributions generated by the diagrams a) and b) for low Higgs boson mass.', 'hep-ph-0012110-2-24-5': 'The growth of [MATH] and [MATH] is more pronounced for lower [MATH] (closer to the [MATH] production threshold), it is a consequence of the Coulomb-like singularity generated by graph a) and b) in the limit [MATH] at the [MATH] production threshold.', 'hep-ph-0012110-2-24-6': 'Similar behavior have been also noted in the case of CP-conserving form factors [CITATION].', 'hep-ph-0012110-2-24-7': 'One also observes a typical threshold behavior at [MATH] for [MATH] and [MATH] as a non-trivial absorptive part of diagrams c)-f) is necessary for nonzero contribution to [MATH].', 'hep-ph-0012110-2-25-0': 'If all three Higgs bosons of the model have the same mass, then the orthogonality of the mixing matrix [MATH] guarantees [CITATION] vanishing CP violation while summed over all the scalars.', 'hep-ph-0012110-2-25-1': 'Therefore, the leading contribution to [MATH] and [MATH] originating from an exchange of the lightest Higgs boson [MATH] could be partially cancelled by heavier scalars [MATH].', 'hep-ph-0012110-2-25-2': 'However here we will assume that masses of the heavier scalars are above the production threshold for [MATH], therefore, as observed (for the lightest Higgs boson mass below [MATH]) from Fig. [REF], the cancellation by heavier scalars for [MATH] and [MATH] could reach at most [MATH] and [MATH].', 'hep-ph-0012110-2-26-0': 'Leading asymptotic formulae for small and large Higgs mass are presented in the appendix [REF].', 'hep-ph-0012110-2-26-1': 'It is worth to notice that for non-zero [MATH], in the limit [MATH] both [MATH] and [MATH] are finite, whereas a typical decoupling limit, [MATH], is observed for large [MATH].', 'hep-ph-0012110-2-27-0': 'Fig. [REF] shows the functions [MATH] for [MATH] and two different Higgs boson masses [MATH] and [MATH], as a function of [MATH].', 'hep-ph-0012110-2-27-1': 'It is seen that it is not desirable to choose too high beam energy, as the size of the functions drops.', 'hep-ph-0012110-2-27-2': 'Again, [MATH] dominate over [MATH] by 1-2 orders of magnitude.', 'hep-ph-0012110-2-28-0': 'width=.65plot-eett-prodre-ffmhdep.eps The functions [MATH] defined in eq. ([REF]) as a function of [MATH] for [MATH], [MATH] (left), [MATH] (middle) and [MATH] (right).', 'hep-ph-0012110-2-28-1': 'fig:eett:prodre:hdep', 'hep-ph-0012110-2-29-0': 'width=.65plot-eett-prodre-ffsdep.eps The functions [MATH] defined in eq. ([REF]) as a function of [MATH] for [MATH], [MATH] and [MATH]fig:eett:prodre:sdep', 'hep-ph-0012110-2-30-0': '## Top Decay', 'hep-ph-0012110-2-31-0': 'We will adopt the following parameterization of the [MATH] vertex suitable for the [MATH] and [MATH] decays: ^&=&-g2V_tb [^(f_1^L P_L +f_1^R P_R) -i^k_M_W (f_2^L P_L +f_2^R P_R)],', 'hep-ph-0012110-2-32-0': '^&=&-g2V_tb^* [^(f_1^L P_L +f_1^R P_R) -i^k_M_W (f_2^L P_L +f_2^R P_R)],', 'hep-ph-0012110-2-33-0': 'where [MATH], [MATH] is the [MATH] element of the Kobayashi-Maskawa matrix and [MATH] is the momentum of [MATH].', 'hep-ph-0012110-2-33-1': 'In the SM [MATH] and all the other form factors vanish.', 'hep-ph-0012110-2-33-2': 'It turns out that in the limit of massless bottom quarks the only form factors that interfere with the SM are [MATH] and [MATH] for the top and anti-top decays, respectively.', 'hep-ph-0012110-2-33-3': 'Currently, there is no relevant experimental bound on those form factors.', 'hep-ph-0012110-2-34-0': 'One can show that the CP-violating and CP-conserving parts of the form factors for [MATH] and [MATH] are not independent: f_1^L,R=f_1^L,R and f_2^L,R=f_2^R,L,', 'hep-ph-0012110-2-35-0': 'where upper (lower) signs are those for [MATH]-conserving (-violating) contributions [CITATION].', 'hep-ph-0012110-2-35-1': 'Therefore any [MATH]-violating observable defined for the top-quark decay must be proportional to [MATH] or [MATH].', 'hep-ph-0012110-2-36-0': 'width=0.65phd-diag-tbw.epsDiagrams contributing to [MATH].', 'hep-ph-0012110-2-36-1': 'fig:diag:tbw', 'hep-ph-0012110-2-37-0': 'Diagrams contributing to CP violation in the decay process are shown in Fig. [REF].', 'hep-ph-0012110-2-37-1': 'Direct calculation leads to the following result for the CP-violating part of [MATH] : [EQUATION] where ija &=& ij(p_t, -p_W, m_h^2, m_t^2, m_b^2)', 'hep-ph-0012110-2-38-0': 'ijbd&=& ij(p_W, p_b, m_W^2, m_h^2, m_b^2)', 'hep-ph-0012110-2-39-0': 'ijce&=& ij(p_W, p_b, m_h^2, m_W^2, m_t^2)', 'hep-ph-0012110-2-40-0': 'ijf &=& ij(p_W, -p_t, m_h^2, m_H^2, m_b^2)', 'hep-ph-0012110-2-41-0': 'ijg &=& ij(p_W, p_b, m_h^2, m_H^2, m_t^2) and [EQUATION] where [MATH] and [MATH].', 'hep-ph-0012110-2-41-1': 'The asymmetries that will be discussed here depends on real parts of [MATH] and [MATH].', 'hep-ph-0012110-2-41-2': 'It is easy to note from eq. ([REF]) that imaginary parts of [MATH] functions contribute to the real part of [MATH].', 'hep-ph-0012110-2-41-3': 'It is seen that only diagrams b), d) and f) will contribute to [MATH].', 'hep-ph-0012110-2-41-4': 'However, [MATH] strongly suggest [CITATION] that for 2HDM [MATH], therefore eventually (adopting the relation ([REF])) one gets the following result (from graphs b) and d) only) for CP-violating contribution to [MATH]: [EQUATION]', 'hep-ph-0012110-2-41-5': 'As will be discussed in Section [REF] there is a strong experimental bound on [MATH] for [MATH].', 'hep-ph-0012110-2-41-6': 'Taking into account the limit on [MATH] and choosing [MATH] (in order to illustrate a possible enhancement) we plot [MATH] in Fig. [REF] as a function of [MATH].', 'hep-ph-0012110-2-41-7': 'It is seen that [MATH] is by [MATH] orders of magnitude below [MATH] or [MATH] even for large b-quark Yukawa coupling, compare Fig. [REF] and [REF].', 'hep-ph-0012110-2-41-8': 'The suppression is caused both by the experimental limit on [MATH] (for [MATH]) and by an extra suppression factor of [MATH] (relative to [MATH]).', 'hep-ph-0012110-2-42-0': 'There is a comment in order here; since in the 2HDM the real part of CP-violating form factors in the top decay is much smaller then in the production process, it is interesting to look closer at the suppression mechanism and find class of possible extensions of the SM that provide large [MATH].', 'hep-ph-0012110-2-42-1': 'Since an absorptive part is needed, the only graphs that may contribute are those denoted by b), d) and f) in Fig. [REF] (allowing for the neutral scalar to be replaced by a neutral vector).', 'hep-ph-0012110-2-42-2': 'One source of the suppression is the bottom-quark mass that originate from the propagator while the second one comes from the Yukawa vertex.', 'hep-ph-0012110-2-42-3': 'The latter one could be easy amplified by large [MATH]: for [MATH] the bottom-quark Yukawa coupling is as strong as the SU(2) gauge coupling.', 'hep-ph-0012110-2-42-4': 'Therefore the suppression to overcome is [MATH] from the bottom-quark propagator.', 'hep-ph-0012110-2-42-5': 'A possible solution [CITATION] seems to be a multi-doublet-Higgs model that could evade the stringent restriction from the [MATH] decay and also overcome (through a contribution from the graph type f) in Fig. [REF]) the limit on [MATH] that comes from the LEP limit on [MATH].', 'hep-ph-0012110-2-42-6': 'Let us notice that it is much easier to develop large [MATH], see e.g. Refs. [CITATION] and [CITATION].', 'hep-ph-0012110-2-43-0': 'eps [MATH] as a function of [MATH] for [MATH].', 'hep-ph-0012110-2-43-1': 'It has been assumed (according to constraints from [MATH]) that the charged Higgs boson is heavy, [MATH], therefore there is no absorptive part necessary to develop a non-zero contribution from diagram f) in Fig.[REF].', 'hep-ph-0012110-2-43-2': 'Additionally, we assumed [MATH] (what is a very good approximation for [MATH]) and selected such values for [MATH] and [MATH] that are consistent both with [MATH] orthogonality and [MATH] LEP bound, and provide a maximal value of [MATH].', 'hep-ph-0012110-2-43-3': 'fig:eett:dec:ref2mhdep', 'hep-ph-0012110-2-44-0': 'It is worth to mention that even though in the SM there exists one-loop contribution to [MATH], it turns out to be strongly GIM suppressed [CITATION].', 'hep-ph-0012110-2-44-1': 'Therefore, although the 2HDM prediction for [MATH] is smaller than for CP violating from factors in the production mechanism, it is still by a factor [MATH] larger then the SM result.', 'hep-ph-0012110-2-45-0': '# Experimental Constraints', 'hep-ph-0012110-2-46-0': 'Hereafter we will focus on Higgs boson masses in the region, [MATH].', 'hep-ph-0012110-2-46-1': 'As it has been shown in the literature [CITATION] the existing LEP data are perfectly consistent with one light Higgs boson within the 2HDM.', 'hep-ph-0012110-2-46-2': 'It turns out that even precision electroweak tests allow for light Higgs bosons [CITATION].', 'hep-ph-0012110-2-47-0': 'In order to amplify the form factors calculated in this paper we have adopted for an illustration [MATH].', 'hep-ph-0012110-2-47-1': 'However, there exist experimental constraints on [MATH] from [MATH] and [MATH] mixing [CITATION], [MATH] decay [CITATION] and [MATH] decay [CITATION].', 'hep-ph-0012110-2-47-2': 'Since small [MATH] enhances [MATH] coupling, in order to maintain [MATH] we have to decouple charged Higgs effects and therefore we assume that [MATH]>[MATH][MATH].', 'hep-ph-0012110-2-48-0': 'The constraints on the mixing angles [MATH] that should be imposed in our numerical analysis are as follows:', 'hep-ph-0012110-2-49-0': 'It turns out that the restriction on the [MATH] coupling from its contribution to the total [MATH]-width is always weaker then the one from [MATH] production if [MATH]>[MATH][MATH].', 'hep-ph-0012110-2-50-0': 'The LEP constraints on the [MATH] coupling restrict the following entries of the mixing matrix [MATH]: [EQUATION] where [MATH] stands for the upper limit for the relative strength of [MATH] coupling determined experimentally in Ref. [CITATION] up to the Higgs mass [MATH].', 'hep-ph-0012110-2-50-1': 'As we have concluded in the previous section, CP-violating phenomena we are considering are enhanced by small [MATH], in that case one can see from eq. ([REF]) that the LEP constraints mostly restrict [MATH].', 'hep-ph-0012110-2-50-2': 'Through the orthogonality the restriction on [MATH] is being transfered to constrain [MATH] which multiplies leading contributions to all CP-violating asymmetries considered here.', 'hep-ph-0012110-2-50-3': 'The final result for upper limit on [MATH] as a function of [MATH] is shown in Fig. [REF].', 'hep-ph-0012110-2-50-4': 'In fact the bound on [MATH] depends on the Higgs mass, however, in order to be conservative, we have assumed [MATH] that is the most restrictive experimental limit (obtained for [MATH]).', 'hep-ph-0012110-2-51-0': 'epsMaximal value of [MATH] allowed by the LEP constraints on [MATH] coupling as a function of [MATH].f', 'hep-ph-0012110-2-52-0': 'As it is seen from Fig.[REF] the constraints for [MATH] are weak for small [MATH].', 'hep-ph-0012110-2-52-1': 'Therefore for [MATH] it should be legitimate to assume [MATH] which is the maximal value consistent with orthogonality.', 'hep-ph-0012110-2-53-0': 'Using the maximal value of [MATH] allowed by the orthogonality and the LEP constraints for small [MATH], we may discuss a possibility for an experimental determination of the calculated form factors at future [MATH] colliders.', 'hep-ph-0012110-2-53-1': 'A detailed discussion of expected statistical uncertainties for a measurement of the form factors has been performed in Ref. [CITATION].', 'hep-ph-0012110-2-53-2': 'It has been shown that adjusting an optimal [MATH] beam polarizations, using the energy and angular double distribution of final leptons and fitting all 9 form factors leads to the following statistical errors for the determination of CP-violating form factors: [MATH] and [MATH] for [MATH].', 'hep-ph-0012110-2-53-3': 'It is seen that only [MATH], could be measured with a high precision.', 'hep-ph-0012110-2-53-4': 'We have observed in Figs. [REF],[REF] that [MATH] may reach at most a value of [MATH], therefore one shall conclude that several years of running with yearly integrated luminosity [MATH] should allow for an observation of [MATH] generated within 2HDM, provided the lightest Higgs boson mass is not too large.', 'hep-ph-0012110-2-53-5': 'On the other hand, the expected [CITATION] precision for the determination of the decay form factors is much more promising: [MATH].', 'hep-ph-0012110-2-53-6': 'However, as we have seen in Fig. [REF], the maximal expected size of [MATH] is [MATH] (for [MATH]), therefore either an unrealistic growth of the luminosity, or other observables (besides the energy and angular double distribution of final leptons) are required in order to observe CP-violating from factors in the top-quark decay process.', 'hep-ph-0012110-2-53-7': 'The results of Ref. [CITATION] assumed simultaneous determination of all 9 form factors, therefore another chance to reduce of [MATH] is to have some extra independent constraints on the top-quark coupling coming from other colliders, like the Fermilab Tevatron or LHC.', 'hep-ph-0012110-2-54-0': '# CP-Violating Asymmetries', 'hep-ph-0012110-2-55-0': 'Looking for CP violation one can directly measure [CITATION] all the form factors including those which are odd under CP.', 'hep-ph-0012110-2-55-1': 'However another possible attitude is to construct certain asymmetries sensitive to CP violation.', 'hep-ph-0012110-2-55-2': 'In this section we will discuss several asymmetries that could probe CP violation in the process [MATH].', 'hep-ph-0012110-2-55-3': 'We will systematically drop all contributions quadratic in non-standard form factors and calculate various asymmetries keeping only interference between the SM and [MATH], [MATH] or [MATH].', 'hep-ph-0012110-2-56-0': '## Lepton-Energy Asymmetry', 'hep-ph-0012110-2-57-0': 'Let us introduce the rescaled lepton energy, [MATH], by x2 E_l(1-_t1+_t)^1/2,', 'hep-ph-0012110-2-58-0': 'where [MATH] is the energy of [MATH] in [MATH] c.m. frame and [MATH].', 'hep-ph-0012110-2-58-1': 'Using lepton energy distribution [MATH] calculated [CITATION] for the general form factors given in eqs. ([REF],[REF]) one can define the following energy asymmetry: [EQUATION]', 'hep-ph-0012110-2-58-2': 'Direct calculation leads to the following result in terms of the CP-violating form factors: A_CP^l(x)=2g(x) +(f_2^R-f_2^L) [ f(x)+ g(x)]2[f(x)+ g(x)].', 'hep-ph-0012110-2-59-0': 'eps The coefficient functions [MATH] defined by eq.([REF]) for the energy-asymmetry as function of [MATH] for [MATH] 360 (left), 500 (middle) and 1000 GeV (right) for [MATH].', 'hep-ph-0012110-2-59-1': 'The solid curve represents the coefficient [MATH], dashed [MATH] and dotted [MATH].f', 'hep-ph-0012110-2-60-0': 'In order to estimate a relative strength of various sources of CP violation it is worth to decompose the asymmetry as follows: [EQUATION]', 'hep-ph-0012110-2-60-1': 'As one can see from Fig. [REF] the CP-violating effects that originate from the decay are substantially enhanced in the soft energy region.', 'hep-ph-0012110-2-60-2': 'It is worth to notice that the minimal lepton energy for [MATH] is [MATH] that is large enough to detect the lepton.', 'hep-ph-0012110-2-60-3': 'Therefore the region of soft leptons should be carefully studied experimentally.', 'hep-ph-0012110-2-60-4': 'The enhancement is a consequence of particular behavior of [MATH], [MATH] and [MATH] that causes the relative amplification of the decay effects, see Fig.1 in Ref. [CITATION].', 'hep-ph-0012110-2-60-5': 'The same figure explains the observed smallness of the [MATH]-boson contribution.', 'hep-ph-0012110-2-60-6': 'For hard leptons both [MATH] and [MATH] are enhanced and they are raising with the c.m. energy.', 'hep-ph-0012110-2-61-0': 'The energy-asymmetry could be decomposed into the leading contribution proportional to [MATH] and the remaining piece proportional to [MATH].', 'hep-ph-0012110-2-61-1': 'The former one (that provides the leading contribution) is plotted in Fig. [REF] for a fixed energy, [MATH], as a function of [MATH].', 'hep-ph-0012110-2-61-2': 'One can observe that the largest asymmetry for the chosen energy corresponds to [MATH] and [MATH].', 'hep-ph-0012110-2-62-0': 'width=.65eett-asy9604r-mhdep.epsThe Higgs mass dependence of the coefficient of [MATH] for the asymmetry given by eq. ([REF]) for [MATH]=360 (solid), 500 (dashed), 1000 GeV (dotted) for a fixed energy, [MATH], and [MATH].', 'hep-ph-0012110-2-62-1': 'fig:eett:asy9604r:mhdep', 'hep-ph-0012110-2-63-0': '## Integrated Lepton-Energy Asymmetry', 'hep-ph-0012110-2-64-0': 'CP symmetry could be also tested using the following leptonic double energy distribution [CITATION]: 1d^2dxd =_i=1^3c_i f_i(x,),', 'hep-ph-0012110-2-65-0': 'where [MATH] and [MATH] are for [MATH] and [MATH] respectively, for [EQUATION] and [EQUATION] where [EQUATION].', 'hep-ph-0012110-2-66-0': 'The following asymmetry could be a measure of CP violation: [EQUATION]', 'hep-ph-0012110-2-66-1': 'As before, it is useful to separate contributions from various form factors: [EQUATION]', 'hep-ph-0012110-2-66-2': 'In Table [REF] we show the coefficients [MATH] for various c.m. energies.', 'hep-ph-0012110-2-66-3': 'Firstly, is clear that for any given [MATH] the coefficient [MATH] is the smallest one.', 'hep-ph-0012110-2-66-4': 'Secondly, it is seen that just above the threshold for [MATH] production there is an enhancement of relative contributions from the decay, however that still not sufficient to overcome the suppression of [MATH] that we have observed in Fig. [REF].', 'hep-ph-0012110-2-66-5': 'Therefore we can conclude that the leading contribution is provided by CP violation in the [MATH] vertex.', 'hep-ph-0012110-2-67-0': 'Fig. [REF] illustrates the Higgs-mass dependence of the leading (proportional to [MATH]) contribution to the integrated lepton-energy asymmetry.', 'hep-ph-0012110-2-67-1': 'It turns out that [MATH] provides the largest asymmetry.', 'hep-ph-0012110-2-68-0': 'width=.65eett-asy9608r-mhdep.eps Higgs mass dependence of the coefficient of [MATH] for the asymmetry given by eq. ([REF]) for [MATH]=360 (solid), 500 (dashed), 1000 GeV (dotted) for [MATH].', 'hep-ph-0012110-2-68-1': 'fig:eett:asy9608r:mhdep', 'hep-ph-0012110-2-69-0': 'Using results of Ref. [CITATION] one can find out an expected statistical error for the determination of [MATH] at any given [MATH] collider.', 'hep-ph-0012110-2-69-1': 'Assuming [MATH], [MATH] and lepton tagging efficiency, [MATH] we get [MATH].', 'hep-ph-0012110-2-69-2': 'As it is seen from Fig. [REF] an observation of the asymmetry would require several years of running at the assumed luminosity.', 'hep-ph-0012110-2-70-0': '## Angular Asymmetry', 'hep-ph-0012110-2-71-0': 'Another CP-violating asymmetry could be constructed using the angular distributions of the bottom quarks or leptons originating from the top-quark decay: dd_f= 3_EM^22sB_f (_0^f+_1^f _f+_2^f^2_fright),', 'hep-ph-0012110-2-72-0': 'where [MATH], [MATH] is an appropriate top-quark branching ratio, [MATH] is the angle between the [MATH] beam direction and the direction of [MATH] momentum in the [MATH] c.m. frame and [MATH] are coefficients calculable in terms of the form factors, see Ref. [CITATION].', 'hep-ph-0012110-2-72-1': 'The following asymmetry provides a signal of CP violation: A_CP^f(_f)= d^+(_f)d_f- d^-(f)d_f d^+(_f)d_f+ d^-(f)d_f,', 'hep-ph-0012110-2-73-0': 'where [MATH] is referring to [MATH] and [MATH] distributions, respectively.', 'hep-ph-0012110-2-73-1': 'Since [MATH] under [MATH], the asymmetry defined above is a true measure of [MATH] violation.', 'hep-ph-0012110-2-74-0': 'Adopting general formulas for the asymmetry from Ref. [CITATION] and inserting form factors calculated here we plot the asymmetry in Figs. [REF], [REF] as a function of [MATH] for bottom quarks and leptons, respectively.', 'hep-ph-0012110-2-74-1': 'As before, the asymmetry can be decomposed into [MATH], [MATH] and [MATH] vertex contributions: [EQUATION]', 'hep-ph-0012110-2-74-2': 'It is seen that forward-backward directions are favored, however an experimental cut [MATH] should be imposed in the realistic experimental environment.', 'hep-ph-0012110-2-75-0': 'width=.65eett-asy9911bff-thdep.eps The coefficients functions [MATH] defined by eq. ([REF]) for the angular asymmetry, for final bottom quarks ([MATH]) as a function of [MATH] for [MATH] 360 (left), 500 (middle) and 1000 GeV (right) for [MATH].', 'hep-ph-0012110-2-75-1': 'The solid curve represents the coefficient [MATH], dashed [MATH] and dotted [MATH]fig:eett:asy9911bff-thdep', 'hep-ph-0012110-2-76-0': 'In order to illustrate the Higgs mass dependence we plot in Fig.[REF] the angular asymmetry both for [MATH] and [MATH] for chosen polar angle [MATH].', 'hep-ph-0012110-2-76-1': 'As expected the maximal effect could be reached for minimal Higgs mass [MATH], [MATH] is the most suitable energy.', 'hep-ph-0012110-2-77-0': 'width=.65eett-asy9911r-mhdep.epsThe Higgs mass dependence of the coefficient of [MATH] for the angular asymmetry defined by eq. ([REF]) for [MATH]=360 (solid), 500 (dashed), 1000 (dotted) GeV for a fixed polar angle [MATH] and [MATH].', 'hep-ph-0012110-2-77-1': 'fig:eett:asy9911br-mhdep', 'hep-ph-0012110-2-78-0': '## Integrated Angular Asymmetry', 'hep-ph-0012110-2-79-0': 'The angular distribution given in eq. ([REF]) could be adopted to define an integrated version [CITATION] of the angular asymmetry [MATH]: [EQUATION] where [MATH] and [MATH] are the polarizations of [MATH] and [MATH] beams, [MATH] is referring to [MATH] and [MATH] distributions respectively, and [MATH] expresses the experimental polar-angle cut.', 'hep-ph-0012110-2-79-1': 'In order to discuss possible advantages of polarized initial beams we are considering here dependence of the asymmetry on the polarization.', 'hep-ph-0012110-2-79-2': 'Hereafter we will discuss the same polarization for [MATH] and [MATH]: [MATH].', 'hep-ph-0012110-2-80-0': 'Again we decompose the asymmetry as follows: [EQUATION]', 'hep-ph-0012110-2-80-1': 'In Table [REF] we show the coefficient functions [MATH] calculated for various energy and polarization choices assuming the polar angle cut [MATH], i.e. [MATH] in eq. ([REF]), both for leptons and bottom quarks .', 'hep-ph-0012110-2-80-2': 'It could be seen that a positive polarization leads to higher coefficients [MATH] and [MATH].', 'hep-ph-0012110-2-80-3': 'Since [MATH] that implies that maximal asymmetry could be reached for [MATH] and the dominant contribution is originating from [MATH].', 'hep-ph-0012110-2-80-4': 'Since the number of events does not drop drastically when going from unpolarized beams to [MATH], it turns out that the positive polarization is the most suitable for testing the integrated angular asymmetry.', 'hep-ph-0012110-2-80-5': 'It is clear from the table that the asymmetry for final leptons should be larger by a factor [MATH] then the one for bottom quarks and their signs should be reversed.', 'hep-ph-0012110-2-81-0': 'Using the general formula for the asymmetry from Ref. [CITATION] and adopting results for the CP-violating form factors we plot [MATH] in Fig. [REF] as a function of the Higgs mass both for bottom quarks and leptons.', 'hep-ph-0012110-2-81-1': 'It is clear that the largest asymmetry could be expected for [MATH] for final leptons at [MATH].', 'hep-ph-0012110-2-81-2': 'With the maximal mixing, [MATH] the [MATH] asymmetry could be expected for the Higgs boson with mass [MATH].', 'hep-ph-0012110-2-81-3': 'Since the statistical error expected [CITATION] for the asymmetry is of the order of [MATH], we can conclude that the asymmetry [MATH] is the most promising one, leading to [MATH] effect for light Higgs mass and [MATH].', 'hep-ph-0012110-2-81-4': 'As it is seen form Fig.[REF] it is relevant to have polarized [MATH] beams.', 'hep-ph-0012110-2-82-0': 'epsThe Higgs mass dependence of the coefficient of [MATH] for the angular asymmetry defined by eq.([REF]) for bottom quarks (upper) and leptons (lower) at [MATH]=360 (solid), 500 (dashed), 1000 GeV (dotted) with unpolarized beams (left), [MATH] (middle) and [MATH] (right) for [MATH].f', 'hep-ph-0012110-2-83-0': '# Summary and Conclusions', 'hep-ph-0012110-2-84-0': 'We have considered a general two-Higgs-doublet model with CP violation in the scalar sector.', 'hep-ph-0012110-2-84-1': 'Mixing of the three neutral Higgs fields of the model leads to CP-violating Yukawa couplings of the physical Higgs bosons.', 'hep-ph-0012110-2-84-2': 'CP-asymmetric form factors generated at the one-loop level of perturbation theory has been calculated within the model.', 'hep-ph-0012110-2-84-3': 'Although in general the existing experimental data from LEP1 and LEP2 constraint the mixing angles of the three neutral Higgs fields, their combination relevant for CP violation is not bounded for small [MATH] which is the region of our interest.', 'hep-ph-0012110-2-84-4': 'We have shown that the decay form factors are typically smaller then the production ones by 2-3 orders of magnitude.', 'hep-ph-0012110-2-84-5': 'The dominant contribution to CP violation in the production is coming from [MATH] coupling.', 'hep-ph-0012110-2-84-6': 'Several energy and angular CP-violating asymmetries for the process [MATH] and [MATH] has been considered using the form factors calculated within the two-Higgs-doublet model.', 'hep-ph-0012110-2-84-7': 'It turned out that the best test of CP invariance would be provided by the integrated angular asymmetry [MATH] for positive polarizations of [MATH] beams.', 'hep-ph-0012110-2-84-8': 'For one year of running at TESLA collider with the integrated luminosity [MATH] one could expect [MATH] effect for the asymmetry for light Higgs boson and [MATH].'}

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'hep-ph-0012110-2-10-3'], ['hep-ph-0012110-1-81-0', 'hep-ph-0012110-2-81-0'], ['hep-ph-0012110-1-81-1', 'hep-ph-0012110-2-81-1'], ['hep-ph-0012110-1-81-2', 'hep-ph-0012110-2-81-2'], ['hep-ph-0012110-1-81-3', 'hep-ph-0012110-2-81-3'], ['hep-ph-0012110-1-81-4', 'hep-ph-0012110-2-81-4'], ['hep-ph-0012110-1-75-0', 'hep-ph-0012110-2-75-0'], ['hep-ph-0012110-1-75-1', 'hep-ph-0012110-2-75-1'], ['hep-ph-0012110-1-52-0', 'hep-ph-0012110-2-52-0'], ['hep-ph-0012110-1-52-1', 'hep-ph-0012110-2-52-1'], ['hep-ph-0012110-1-76-0', 'hep-ph-0012110-2-76-0'], ['hep-ph-0012110-1-76-1', 'hep-ph-0012110-2-76-1'], ['hep-ph-0012110-1-42-0', 'hep-ph-0012110-2-42-0'], ['hep-ph-0012110-1-42-1', 'hep-ph-0012110-2-42-1'], ['hep-ph-0012110-1-42-2', 'hep-ph-0012110-2-42-2'], ['hep-ph-0012110-1-42-3', 'hep-ph-0012110-2-42-3'], ['hep-ph-0012110-1-42-4', 'hep-ph-0012110-2-42-4'], ['hep-ph-0012110-1-42-5', 'hep-ph-0012110-2-42-5'], ['hep-ph-0012110-1-42-6', 'hep-ph-0012110-2-42-6'], ['hep-ph-0012110-1-72-0', 'hep-ph-0012110-2-72-0'], ['hep-ph-0012110-1-24-0', 'hep-ph-0012110-2-24-0'], ['hep-ph-0012110-1-24-1', 'hep-ph-0012110-2-24-1'], ['hep-ph-0012110-1-24-2', 'hep-ph-0012110-2-24-2'], ['hep-ph-0012110-1-24-3', 'hep-ph-0012110-2-24-3'], ['hep-ph-0012110-1-24-4', 'hep-ph-0012110-2-24-4'], ['hep-ph-0012110-1-24-5', 'hep-ph-0012110-2-24-5'], ['hep-ph-0012110-1-24-6', 'hep-ph-0012110-2-24-6'], ['hep-ph-0012110-1-24-7', 'hep-ph-0012110-2-24-7'], ['hep-ph-0012110-1-41-1', 'hep-ph-0012110-2-41-1'], ['hep-ph-0012110-1-41-2', 'hep-ph-0012110-2-41-2'], ['hep-ph-0012110-1-41-3', 'hep-ph-0012110-2-41-3'], ['hep-ph-0012110-1-41-4', 'hep-ph-0012110-2-41-4'], ['hep-ph-0012110-1-41-5', 'hep-ph-0012110-2-41-5'], ['hep-ph-0012110-1-41-6', 'hep-ph-0012110-2-41-6'], ['hep-ph-0012110-1-41-7', 'hep-ph-0012110-2-41-7'], ['hep-ph-0012110-1-41-8', 'hep-ph-0012110-2-41-8'], ['hep-ph-0012110-1-11-0', 'hep-ph-0012110-2-11-0']]

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[['hep-ph-0012110-1-11-0', 'hep-ph-0012110-2-11-0']]

[]

[]

[]

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{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/hep-ph/0012110

1701.06154

{'1701.06154-1-0-0': 'We study a consistent deformation of the cubic open bosonic string theory in such a way that the non-planar world sheet diagrams of the perturbative string theory are mapped onto their equivalent planar diagrams of the light-cone string field theory with some length parameters fixed.', '1701.06154-1-0-1': 'An explicit evaluation of the cubic string vertex in the zero-slope limit yields the correct relationship between the string coupling constant and the Yang-Mills coupling constant.', '1701.06154-1-0-2': 'The deformed cubic open string field theory is shown to produce the non-Abelian Yang-Mills action in the zero-slope limit if it is defined on multiple D-branes.', '1701.06154-1-0-3': 'Applying the consistent deformation systematically to multi-string world sheet diagrams, we may be able to calculate scattering amplitudes with an arbitrary number of external open strings.', '1701.06154-1-1-0': '# Introduction', '1701.06154-1-2-0': 'If its perturbation theory is correctly defined, the covariant string field theory is expected to replace eventually the quantum field theory which has not been successful to describe quantum particles with spin two and higher spins.', '1701.06154-1-2-1': 'However, in practice, it is rather difficult to make use of the covariant cubic string field theory [CITATION] to calculate the particle scattering amplitudes.', '1701.06154-1-2-2': 'The main reason is that the world sheet diagrams of cubic open string field theory are non-planar unlike those of the light-cone string field theory [CITATION].', '1701.06154-1-2-3': 'Witten [CITATION] introduced an associative product between the open string field operators which represents the mid-point overlapping interaction.', '1701.06154-1-2-4': 'With the associative star product, the string field action takes the form of the Chern-Simons three-form which is invariant under the BRST gauge transformation.', '1701.06154-1-2-5': 'The cubic open string field theory has a merit of the BRST gauge invariance due to the associative algebra of the string field operators.', '1701.06154-1-2-6': 'But at the same time the mid-point overlapping interaction renders the world-sheet diagrams non-planar so that it becomes a difficult task to get the Fock space representations of the multi-string vertices.', '1701.06154-1-3-0': 'The Fock space representation of the three-string vertex of the cubic open string field theory has been obtained by Gross and Jevicki in Refs. [CITATION] and [CITATION] by mapping the world-sheet diagram of six strings onto a circular disk and imposing an orbifold condition.', '1701.06154-1-3-1': 'The conformal mapping of the four-string world sheet to the upper half complex plane with branch cuts has been constructed by Giddings [CITATION].', '1701.06154-1-3-2': 'The Neumann functions of the three-string vertex have been calculated in Refs. [CITATION] and the Neumann functions of the four-string vertex has been computed by Samuel in Ref. [CITATION].', '1701.06154-1-3-3': 'However, there seems to be no similarity between the conformal mappings for the three-string vertex and that of the four-string vertex.', '1701.06154-1-3-4': 'It seems also difficult to apply those constructions of the conformal mappings to more complex world sheet diagrams of multi-string vertices.', '1701.06154-1-3-5': 'Thus, it is desirable to develop a more systematic technique which could be applied to string scattering diagrams with an arbitrary number of external strings.', '1701.06154-1-3-6': 'In the present work, we propose a consistent deformation of the world sheet diagrams which transforms the non-planar diagrams of multi-string scattering into planar diagrams.', '1701.06154-1-3-7': 'Once having obtained the planar diagrams of the multi-string vertices, we can make use of the light-cone string field theory technique by mapping the world sheet diagrams onto the upper half complex plane.', '1701.06154-1-3-8': 'For the three-string vertex and the four-string vertex, it is enough to choose external string states such that physical string states are encoded only on the halves of the external strings.', '1701.06154-1-3-9': 'By an explicit calculation, we shall show that the deformed cubic string vertex yields the three-gauge field vertex with the correct Yang-Mills coupling constant in the zero-slope limit.', '1701.06154-1-3-10': 'The four-gauge field vertex of the Yang-Mills action shall be also evaluated by using the deformed world sheet diagram of the four-string vertex which is effectively generated by two cubic string vertices and an intermediate string propagator.', '1701.06154-1-4-0': "# Deformation of the Witten's Open String Field Theory Diagrams", '1701.06154-1-5-0': "We shall begin the Witten's cubic open string field theory action [CITATION] on muti-D-branes which is given as [EQUATION] where [MATH] is the BRST operator and the sring field [MATH] is [MATH] matrix valued [EQUATION]", '1701.06154-1-5-1': 'The star product of between the string field operators is defined as follows [EQUATION]', '1701.06154-1-5-2': 'In terms of the normal modes, the string coordinates [MATH], [MATH] are expanded as [EQUATION]', '1701.06154-1-5-3': 'It is the associativity of the star product algebra [EQUATION] that ensures invariance of the cubic string field theory action under the gauge transformation of the string field [EQUATION]', '1701.06154-1-5-4': 'In order to discuss the deformation of the cubic string field theory we extend the range of the world sheet coordinate [MATH] firstly as [EQUATION]', '1701.06154-1-5-5': 'The mid-point is now located at [MATH].', '1701.06154-1-5-6': 'Accordingly, the star product Eq. ([REF]) and the normal mode expansion Eq. ([REF]) should be appropriately redefined [EQUATION] as shown in Fig. [REF].', '1701.06154-1-6-0': 'Fig. [REF] depicts the world sheet diagram of three-string scattering.', '1701.06154-1-6-1': 'We observe that during the scattering process, physical information encoded on the left half of the first string and physical information encoded on the right half of the second string are not carried over to the third string.', '1701.06154-1-6-2': 'In view of scattering process roles of the left half of the first string and the right half of the second string are auxiliary.', '1701.06154-1-6-3': 'Note that the strings satisfy the Neumann boundary condition on the boundary [MATH] in Fig. [REF] .', '1701.06154-1-6-4': 'We may separate the path, corresponding to the world sheet trajectory of the left half of the first string and the right half of the second string from the rest part of the world sheet of three-string scattering.', '1701.06154-1-6-5': 'On the patch as we redefine the world sheet local coordinates by interchanging [MATH], the boundary condition on [MATH] becomes [EQUATION] (See fig. [REF].)', '1701.06154-1-6-6': 'On the patch we also define new string coordinates [EQUATION]', '1701.06154-1-6-7': 'The Neumann condition on the boundary [MATH] may be written as [EQUATION] where [MATH] is a normalization constant for the Neumann state.', '1701.06154-1-6-8': '[MATH] is the open string analogue of the Neumann boundary state of the closed string theory.', '1701.06154-1-6-9': 'The open string on the boundary [MATH] may propagates freely to the line [MATH] if the endpoints of the open string on the patch satisfy the Neumann condition [EQUATION]', '1701.06154-1-6-10': 'Then the open string state on [MATH] turns out to be the Neumann state again [EQUATION] (The extra phase factor [MATH] may be absorbed into the normalization constant of the Neumann state.)', '1701.06154-1-6-11': 'Hence, if we choose the Neumann condition for the left half of the first string and for the right half of the second string at the initial time, we may remove the patch which consists of the world sheets of the left half of the first string and the right half of the second string.', '1701.06154-1-6-12': 'The string path integral over the patch to scattering amplitude is simply [EQUATION]', '1701.06154-1-6-13': 'Thus, the string path integral over the patch does not contribute to the scattering amplitude', '1701.06154-1-7-0': 'To be consistent with this scheme we may encode the initial states of the first and the second string states onto the right half of the first string and the left half of the second string respectively as depicted in Fig. [REF] and Fig. [REF]: [EQUATION]', '1701.06154-1-7-1': 'Fig. [REF] depicts the deformed world sheet diagram of the three-string scattering after the auxiliary patch is completely removed.', '1701.06154-1-7-2': 'Because the world sheet diagram is not deformed uniformly, the associativity of the star product is not preserved.', '1701.06154-1-7-3': 'Consequently, the BRST gauge invariance is not manifest in the string field action with the deformed cubic interaction.', '1701.06154-1-7-4': 'But if we formally keep the auxiliary patch, the associativity of the star product, hence the gauge invariance can be kept intact.', '1701.06154-1-7-5': 'As we remove the auxiliary patch, the world sheet diagram of the three-string becomes planar, which then can be mapped onto the upper half complex plane without any additional condition.', '1701.06154-1-8-0': '# Three-Gauge Field Vertex from the Deformed Three-String Vertex', '1701.06154-1-9-0': 'The planar diagram of the deformed three-string scattering is equivalent to that of the covariantized light-cone string field theory of HIKKO [CITATION] with length parameters fixed as [EQUATION]', '1701.06154-1-9-1': "Unlike the HIKKO's open covariant string field theory, we do not need to integrate over the unphysical length parameters to make the string field action invariant under the BRST gauge transformation.", '1701.06154-1-9-2': 'Simply reattaching the auxiliary patch would restore the BRST gauge invariant form.', '1701.06154-1-9-3': 'On the planar world sheet we may introduce a global coordinate [MATH] of which real part is the proper time [MATH].', '1701.06154-1-9-4': 'The planar world sheet may be mapped onto the upper half complex plane by the Schwarz-Christoffel transformation given as [EQUATION]', '1701.06154-1-9-5': 'The temporal boundaries of the world sheet (labeled as [MATH] in Fig. [REF]) are mapped onto the real ine.', '1701.06154-1-9-6': 'On each string world sheet patch we may define local coordinates [MATH], [MATH] which are related to [MATH] as follows [EQUATION] where [MATH].', '1701.06154-1-10-0': 'The Fock space representation of the three-string vertex in terms of the Neumann funcitons [MATH] follows from the light-cone string theory with length parameters fixed: [EQUATION]', '1701.06154-1-10-1': 'The interaction term of three-string field may be written as [EQUATION]', '1701.06154-1-10-2': 'The three-gauge interaction term may be obtained by choosing the external state as [EQUATION] in the zero-slope limit: [EQUATION] where the Yang-Mills coupling constant [MATH] is related to the string interaction coupling [MATH] as [EQUATION]', '1701.06154-1-10-3': 'Making use of the explicit expressions of the Neumann functions [EQUATION] we find the three-gauge interaction term [EQUATION]', '1701.06154-1-11-0': '# Four-Gauge Field Vertex from the Deformed Four-String Vertex', '1701.06154-1-12-0': 'The four-gauge field interaction term of the Yang-Mills gauge field theory is obtained from the four-string scattering diagram which is perturbatively generated by the cubic interaction.', '1701.06154-1-12-1': 'Fig. [REF] depicts the effective four-string vertex of the cubic open string field theory.', '1701.06154-1-12-2': 'Choosing the external string states such that the physical information is encoded only on halves of external strings, we may effectively remove the auxiliary patches as in the case of three-string scattering diagram.', '1701.06154-1-12-3': 'This deformation process results in choosing the length parameters of the four strings as [EQUATION]', '1701.06154-1-12-4': 'The resultant planar world sheet diagram of the deformed four-string scattering is described by Fig. [REF]', '1701.06154-1-13-0': 'Now we shall discuss the reduction of the four-string vertex to the four-gauge field vertex in the zero-slope limit.', '1701.06154-1-13-1': "The Witten's cubic open string field theory action does not contain a four-string interaction term in contrast to the light-cone string field theory and the covariantized light-cone string field theory of HIKKO [CITATION].", '1701.06154-1-13-2': 'Thus, the four-gauge field interaction term of the Yang-Mills gauge field theory should be derived solely from the effective four-string interaction, perturbatively generated by the three-string interaction.', '1701.06154-1-13-3': 'Having deformed the four-string world sheet diagram into the planar diagram, we may map it onto the upper half complex plane as shown in Fig. [REF] by the following Schwarz-Christoffel transformation [EQUATION]', '1701.06154-1-13-4': 'If we choose the external four-string state as [EQUATION] we may find that the four-string scattering amplitude yields in the zero-slope limit to following effective four-gauge field action: [EQUATION]', '1701.06154-1-13-5': 'Using [MATH], [EQUATION] and [MATH] in the zero-slope limit, we get the effective four-gauge field action as follows [EQUATION]', '1701.06154-1-13-6': 'Here we define the Mandelstam variables as [EQUATION]', '1701.06154-1-13-7': 'In the zero-slope limit [EQUATION]', '1701.06154-1-13-8': 'The resultant effective four-gauge interaction term [MATH] does not only contain the contact four-gauge field interaction but also contribution of the effective four-gauge interaction generated perturbatively by the three-gauge field interaction of the Yang-Mills field theory.', '1701.06154-1-13-9': 'Substracting the effective four-gauge field interaction of the Yang-Mills theory [MATH] from [MATH] [CITATION], we get the four-gauge field contact interaction of the Yang-Mills theory [EQUATION]', '1701.06154-1-13-10': 'Putting together the guage field interaction terms [MATH], Eq. ([REF]) and [MATH], Eq. ([REF]) as well as the free field action [MATH] which may be derived easily from the kinetic term of the string field action [MATH] in the zero-slope limit, yields the covariant Yang-Mills field action [EQUATION]', '1701.06154-1-14-0': '# Conclusions', '1701.06154-1-15-0': "The Witten's cubic open string field theory possesses a number of advantages over the light-cone string field theory [CITATION] and the covariantized light-cone string field theory [CITATION]: 1.", '1701.06154-1-15-1': 'The theory is covariant and invariant under the BRST gauge transformation.', '1701.06154-1-15-2': '2.', '1701.06154-1-15-3': 'The theory does not contain any other unphysical parameter like the length parameters except for the string coupling [MATH].', '1701.06154-1-15-4': '3.', '1701.06154-1-15-5': "In contrast to two other string field theories, the Witten's open string field theory does not have a quartic interaction term besides the cubic interaction term.", '1701.06154-1-15-6': 'However, despite those advantages, it has not been fully utilized to calculate particle scattering amplitudes except in a few cases.', '1701.06154-1-15-7': 'The main reason is that the world sheet diagrams generated by the cubic string field theory are non-planar: It is difficult to find a conformal mapping by which the world sheet is mapped onto simple complex planes such as the upper half plane or a circular disk without any additional conditions or structures.', '1701.06154-1-15-8': 'One needs to impose an orbifold condition to map the world sheet diagrams of the three-string vertex onto a circular disk [CITATION] and has to introduce branch cuts to map the four-string vertex to the upper half plane [CITATION].', '1701.06154-1-15-9': 'However, even if we found maps of the world sheets to the complex planes in the cases of the three-string and the four-string scatterings, it is difficult to extend those mappings systematically to evaluate general multi-string amplitudes.', '1701.06154-1-15-10': 'It is also difficult to fix the relative strengths of the cubic gauge field interaction term and the quartic gauge field interaction term because there is no analog of the Cremmer-Gervais identity [CITATION] which relates the three-string scattering amplitude to the four-string scattering amplitude.', '1701.06154-1-16-0': 'In this work, we proposed a consistent deformation of the cubic string field theory by which the world sheet diagrams of the multi-string scattering are effectively transformed into planar diagrams.', '1701.06154-1-16-1': 'Having obtained planar diagrams representing the string scattering amplitudes, we can adopt the light-cone field theory technique to construct the Fock space representations of multi-string vertices systematically.', '1701.06154-1-16-2': 'By explicit calculations, we show that the three-string amplitude and the four-string amplitude in the zero-slope limit yield the cubic and quartic gauge interaction terms of the Yang-Mills theory if the external string states are chosen to be the massless gauge particles.', '1701.06154-1-16-3': 'The deformation process is applicable to multi-string scattering with an arbitrary number of strings.', '1701.06154-1-16-4': 'This work may be also regarded as a proof that the string field theory in the proper time gauge [CITATION] is invariant under the BRST gauge transformation.', '1701.06154-1-16-5': 'Applications of the deformed cubic string field theory to various scattering processes [CITATION] will be given elsewhere.', '1701.06154-1-17-0': 'This work was supported by Kangwon National University.', '1701.06154-1-17-1': 'The author benefited from discussions with Soo-Jong Rey, Yi Yang, Jen-Chi Lee, Yuji Okawa, Yu-tin Huang and participants of IBS string workshop 2016.', '1701.06154-1-17-2': "Part of this work was done during author's visit to IBS (Korea) and NCTU (Taiwan)."}

{'1701.06154-2-0-0': 'We study a consistent deformation of the cubic open bosonic string theory in such a way that the non-planar world sheet diagrams of the perturbative string theory are mapped onto their equivalent planar diagrams of the light-cone string field theory with some length parameters fixed.', '1701.06154-2-0-1': 'An explicit evaluation of the cubic string vertex in the zero-slope limit yields the correct relationship between the string coupling constant and the Yang-Mills coupling constant.', '1701.06154-2-0-2': 'The deformed cubic open string field theory is shown to produce the non-Abelian Yang-Mills action in the zero-slope limit if it is defined on multiple D-branes.', '1701.06154-2-0-3': 'Applying the consistent deformation systematically to multi-string world sheet diagrams, we may be able to calculate scattering amplitudes with an arbitrary number of external open strings.', '1701.06154-2-1-0': '# Introduction', '1701.06154-2-2-0': 'If its perturbation theory is correctly defined, the covariant string field theory is expected to replace eventually the quantum field theory which has not been successful to describe quantum particles with spin two and higher spins.', '1701.06154-2-2-1': 'However, in practice, it is rather difficult to make use of the covariant cubic string field theory [CITATION] to calculate the particle scattering amplitudes.', '1701.06154-2-2-2': 'The main reason is that the world sheet diagrams of cubic open string field theory are non-planar unlike those of the light-cone string field theory [CITATION].', '1701.06154-2-2-3': 'Witten [CITATION] introduced an associative product between the open string field operators which represents the mid-point overlapping interaction.', '1701.06154-2-2-4': 'With the associative star product, the string field action takes the form of the Chern-Simons three-form which is invariant under the BRST gauge transformation.', '1701.06154-2-2-5': 'The cubic open string field theory has a merit of the BRST gauge invariance due to the associative algebra of the string field operators.', '1701.06154-2-2-6': 'But at the same time the mid-point overlapping interaction renders the world-sheet diagrams non-planar so that it becomes a difficult task to get the Fock space representations of the multi-string vertices.', '1701.06154-2-3-0': 'The Fock space representation of the three-string vertex of the cubic open string field theory has been obtained by Gross and Jevicki in Refs. [CITATION] and [CITATION] by mapping the world-sheet diagram of six strings onto a circular disk and imposing an orbifold condition.', '1701.06154-2-3-1': 'The conformal mapping of the four-string world sheet to the upper half complex plane with branch cuts has been constructed by Giddings [CITATION].', '1701.06154-2-3-2': 'The Neumann functions of the three-string vertex have been calculated in Refs. [CITATION] and the Neumann functions of the four-string vertex has been computed by Samuel in Ref. [CITATION].', '1701.06154-2-3-3': 'However, there seems to be no similarity between the conformal mappings for the three-string vertex and that of the four-string vertex.', '1701.06154-2-3-4': 'It seems also difficult to apply those constructions of the conformal mappings to more complex world sheet diagrams of multi-string vertices.', '1701.06154-2-3-5': 'Thus, it is desirable to develop a more systematic technique which could be applied to string scattering diagrams with an arbitrary number of external strings.', '1701.06154-2-3-6': 'In the present work, we propose a consistent deformation of the world sheet diagrams which transforms the non-planar diagrams of multi-string scattering into planar diagrams.', '1701.06154-2-3-7': 'Once having obtained the planar diagrams of the multi-string vertices, we can make use of the light-cone string field theory technique by mapping the world sheet diagrams onto the upper half complex plane.', '1701.06154-2-3-8': 'For the three-string vertex and the four-string vertex, it is enough to choose external string states such that physical string states are encoded only on the halves of the external strings.', '1701.06154-2-3-9': 'By an explicit calculation, we shall show that the deformed cubic string vertex yields the three-gauge field vertex with the correct Yang-Mills coupling constant in the zero-slope limit.', '1701.06154-2-3-10': 'The four-gauge field vertex of the Yang-Mills action shall be also evaluated by using the deformed world sheet diagram of the four-string vertex which is effectively generated by two cubic string vertices and an intermediate string propagator.', '1701.06154-2-4-0': "# Deformation of the Witten's Open String Field Theory Diagrams", '1701.06154-2-5-0': "We shall begin the Witten's cubic open string field theory action [CITATION] on muti-D-branes which is given as [EQUATION] where [MATH] is the BRST operator and the sring field [MATH] is [MATH] matrix valued [EQUATION]", '1701.06154-2-5-1': 'The star product of between the string field operators is defined as follows [EQUATION]', '1701.06154-2-5-2': 'In terms of the normal modes, the string coordinates [MATH], [MATH] are expanded as [EQUATION]', '1701.06154-2-5-3': 'It is the associativity of the star product algebra [EQUATION] that ensures invariance of the cubic string field theory action under the gauge transformation of the string field [EQUATION]', '1701.06154-2-5-4': 'In order to discuss the deformation of the cubic string field theory we extend the range of the world sheet coordinate [MATH] firstly as [EQUATION]', '1701.06154-2-5-5': 'The mid-point is now located at [MATH].', '1701.06154-2-5-6': 'Accordingly, the star product Eq. ([REF]) and the normal mode expansion Eq. ([REF]) should be appropriately redefined [EQUATION] as shown in Fig. [REF].', '1701.06154-2-6-0': 'Fig. [REF] depicts the world sheet diagram of three-string scattering.', '1701.06154-2-6-1': 'We observe that during the scattering process, physical information encoded on the left half of the first string and physical information encoded on the right half of the second string are not carried over to the third string.', '1701.06154-2-6-2': 'In view of scattering process roles of the left half of the first string and the right half of the second string are auxiliary.', '1701.06154-2-6-3': 'Note that the strings satisfy the Neumann boundary condition on the boundary [MATH] in Fig. [REF] .', '1701.06154-2-6-4': 'We may separate the path, corresponding to the world sheet trajectory of the left half of the first string and the right half of the second string from the rest part of the world sheet of three-string scattering.', '1701.06154-2-6-5': 'On the patch as we redefine the world sheet local coordinates by interchanging [MATH], the boundary condition on [MATH] becomes [EQUATION] (See fig. [REF].)', '1701.06154-2-6-6': 'On the patch we also define new string coordinates [EQUATION]', '1701.06154-2-6-7': 'The Neumann condition on the boundary [MATH] may be written as [EQUATION] where [MATH] is a normalization constant for the Neumann state.', '1701.06154-2-6-8': '[MATH] is the open string analogue of the Neumann boundary state of the closed string theory.', '1701.06154-2-6-9': 'The open string on the boundary [MATH] may propagates freely to the line [MATH] if the endpoints of the open string on the patch satisfy the Neumann condition [EQUATION]', '1701.06154-2-6-10': 'Then the open string state on [MATH] turns out to be the Neumann state again [EQUATION]', '1701.06154-2-6-11': 'The extra phase factor [MATH] may be absorbed into the normalization constant of the Neumann state.', '1701.06154-2-6-12': '(We may also extend the range of [MATH] as [MATH]: It result in removing the factor [MATH] because the open string state on [MATH] becomes [MATH].)', '1701.06154-2-6-13': 'Hence, if we choose the Neumann condition for the left half of the first string and for the right half of the second string at the initial time, we may remove the patch which consists of the world sheets of the left half of the first string and the right half of the second string.', '1701.06154-2-6-14': 'The string path integral over the patch to scattering amplitude is simply [EQUATION]', '1701.06154-2-6-15': 'Thus, the string path integral over the patch does not contribute to the scattering amplitude', '1701.06154-2-7-0': 'To be consistent with this scheme we may encode the initial states of the first and the second string states onto the right half of the first string and the left half of the second string respectively as depicted in Fig. [REF] and Fig. [REF]: [EQUATION]', '1701.06154-2-7-1': 'Fig. [REF] depicts the deformed world sheet diagram of the three-string scattering after the auxiliary patch is completely removed.', '1701.06154-2-7-2': 'Because the world sheet diagram is not deformed uniformly, the associativity of the star product is not preserved.', '1701.06154-2-7-3': 'Consequently, the BRST gauge invariance is not manifest in the string field action with the deformed cubic interaction.', '1701.06154-2-7-4': 'But if we formally keep the auxiliary patch, the associativity of the star product, hence the gauge invariance can be kept intact.', '1701.06154-2-7-5': 'As we remove the auxiliary patch, the world sheet diagram of the three-string becomes planar, which then can be mapped onto the upper half complex plane without any additional condition.', '1701.06154-2-8-0': '# Three-Gauge Field Vertex from the Deformed Three-String Vertex', '1701.06154-2-9-0': 'The planar diagram of the deformed three-string scattering is equivalent to that of the covariantized light-cone string field theory of HIKKO [CITATION] with length parameters fixed as [EQUATION]', '1701.06154-2-9-1': "Unlike the HIKKO's open covariant string field theory, we do not need to integrate over the unphysical length parameters to make the string field action invariant under the BRST gauge transformation.", '1701.06154-2-9-2': 'Simply reattaching the auxiliary patch would restore the BRST gauge invariant form.', '1701.06154-2-9-3': 'On the planar world sheet we may introduce a global coordinate [MATH] of which real part is the proper time [MATH].', '1701.06154-2-9-4': 'The planar world sheet may be mapped onto the upper half complex plane by the Schwarz-Christoffel transformation given as [EQUATION]', '1701.06154-2-9-5': 'The temporal boundaries of the world sheet (labeled as [MATH] in Fig. [REF]) are mapped onto the real ine.', '1701.06154-2-9-6': 'On individual string world sheet patches we may define local coordinates [MATH], [MATH] which are related to [MATH] as follows [EQUATION] where [MATH].', '1701.06154-2-10-0': 'The Fock space representation of the three-string vertex in terms of the Neumann funcitons [MATH] follows from the light-cone string theory with length parameters fixed: [EQUATION]', '1701.06154-2-10-1': 'The interaction term of three-string field may be written as [EQUATION]', '1701.06154-2-10-2': 'The three-gauge interaction term may be obtained by choosing the external state as [EQUATION] in the zero-slope limit: [EQUATION] where the Yang-Mills coupling constant [MATH] is related to the string interaction coupling [MATH] as [EQUATION]', '1701.06154-2-10-3': 'Making use of the explicit expressions of the Neumann functions [EQUATION] we find the three-gauge interaction term [EQUATION]', '1701.06154-2-11-0': '# Four-Gauge Field Vertex from the Deformed Four-String Vertex', '1701.06154-2-12-0': 'The four-gauge field interaction term of the Yang-Mills gauge field theory is obtained from the four-string scattering diagram which is perturbatively generated by the cubic interaction.', '1701.06154-2-12-1': 'Fig. [REF] depicts the effective four-string vertex of the cubic open string field theory.', '1701.06154-2-12-2': 'Choosing the external string states such that the physical information is encoded only on halves of external strings, we may effectively remove the auxiliary patches as in the case of three-string scattering diagram.', '1701.06154-2-12-3': 'This deformation process results in choosing the length parameters of the four strings as [EQUATION]', '1701.06154-2-12-4': 'The resultant planar world sheet diagram of the deformed four-string scattering is described by Fig. [REF]', '1701.06154-2-13-0': 'Now we shall discuss the reduction of the four-string vertex to the four-gauge field vertex in the zero-slope limit.', '1701.06154-2-13-1': "The Witten's cubic open string field theory action does not contain a four-string interaction term in contrast to the light-cone string field theory and the covariantized light-cone string field theory of HIKKO [CITATION].", '1701.06154-2-13-2': 'Thus, the four-gauge field interaction term of the Yang-Mills gauge field theory should be derived solely from the effective four-string interaction, perturbatively generated by the three-string interaction.', '1701.06154-2-13-3': 'Having deformed the four-string world sheet diagram into the planar diagram, we may map it onto the upper half complex plane as shown in Fig. [REF] by the following Schwarz-Christoffel transformation [EQUATION] with [MATH].', '1701.06154-2-13-4': 'The parameter [MATH] is identified as the Koba-Nielsen variable of the four-string scattering.', '1701.06154-2-14-0': 'We may derive the [MATH] invariant momentum dependent four-string scattering amplitude from the three-string interaction by using the Cremmer-Gervais identity [CITATION] as follows [EQUATION]', '1701.06154-2-14-1': 'If we choose the external four-string state as [EQUATION] we may find that the four-string scattering amplitude yields in the zero-slope limit to following effective four-gauge field action: [EQUATION]', '1701.06154-2-14-2': 'Using [MATH], [EQUATION] and [MATH] in the zero-slope limit, we get the effective four-gauge field action as follows [EQUATION]', '1701.06154-2-14-3': 'Here we define the Mandelstam variables as [EQUATION]', '1701.06154-2-14-4': 'In the zero-slope limit [EQUATION]', '1701.06154-2-14-5': 'The resultant effective four-gauge interaction term [MATH] does not only contain the contact four-gauge field interaction but also contribution of the effective four-gauge interaction generated perturbatively by the three-gauge field interaction of the Yang-Mills field theory.', '1701.06154-2-14-6': 'Substracting the effective four-gauge field interaction of the Yang-Mills theory [MATH] from [MATH] [CITATION], we get the four-gauge field contact interaction of the Yang-Mills theory [EQUATION]', '1701.06154-2-14-7': 'Putting together the guage field interaction terms [MATH], Eq. ([REF]) and [MATH], Eq. ([REF]) as well as the free field action [MATH] which may be derived easily from the kinetic term of the string field action [MATH] in the zero-slope limit, yields the covariant Yang-Mills field action [EQUATION]', '1701.06154-2-15-0': '# Conclusions', '1701.06154-2-16-0': "The Witten's cubic open string field theory possesses a number of advantages over the light-cone string field theory [CITATION] and the covariantized light-cone string field theory [CITATION]: 1.", '1701.06154-2-16-1': 'The theory is covariant and invariant under the BRST gauge transformation.', '1701.06154-2-16-2': '2.', '1701.06154-2-16-3': 'The theory does not contain any other unphysical parameter like the length parameters except for the string coupling [MATH].', '1701.06154-2-16-4': '3.', '1701.06154-2-16-5': "In contrast to two other string field theories, the Witten's open string field theory does not have a quartic interaction term besides the cubic interaction term.", '1701.06154-2-16-6': 'However, despite those advantages, it has not been fully utilized to calculate particle scattering amplitudes except in a few cases.', '1701.06154-2-16-7': 'The main reason is that the world sheet diagrams generated by the cubic string field theory are non-planar: It is difficult to find a conformal mapping by which the world sheet is mapped onto simple complex planes such as the upper half plane or a circular disk without any additional conditions or structures.', '1701.06154-2-16-8': 'One needs to impose an orbifold condition to map the world sheet diagrams of the three-string vertex onto a circular disk [CITATION] and has to introduce branch cuts to map the four-string vertex to the upper half plane [CITATION].', '1701.06154-2-16-9': 'However, even if we found maps of the world sheets to the complex planes in the cases of the three-string and the four-string scatterings, it is difficult to extend those mappings systematically to evaluate general multi-string amplitudes.', '1701.06154-2-16-10': 'It is also difficult to fix the relative strengths of the cubic gauge field interaction term and the quartic gauge field interaction term because there is no analog of the Cremmer-Gervais identity [CITATION] which relates the three-string scattering amplitude to the four-string scattering amplitude.', '1701.06154-2-17-0': 'In this work, we proposed a consistent deformation of the cubic string field theory by which the world sheet diagrams of the multi-string scattering are effectively transformed into planar diagrams.', '1701.06154-2-17-1': 'Having obtained planar diagrams representing the string scattering amplitudes, we can adopt the light-cone field theory technique to construct the Fock space representations of multi-string vertices systematically.', '1701.06154-2-17-2': 'By explicit calculations, we show that the three-string amplitude and the four-string amplitude in the zero-slope limit yield the cubic and quartic gauge interaction terms of the Yang-Mills theory if the external string states are chosen to be the massless gauge particles.', '1701.06154-2-17-3': 'The deformation process is applicable to multi-string scattering with an arbitrary number of strings.', '1701.06154-2-17-4': 'This work may be also regarded as a proof that the string field theory in the proper time gauge [CITATION] is invariant under the BRST gauge transformation.', '1701.06154-2-17-5': 'Applications of the deformed cubic string field theory to various scattering processes [CITATION] will be given elsewhere.', '1701.06154-2-18-0': 'This work was supported by Kangwon National University.', '1701.06154-2-18-1': 'The author benefited from discussions with Soo-Jong Rey, Yi Yang, Jen-Chi Lee, Yuji Okawa, Yu-tin Huang and participants of IBS string workshop 2016.', '1701.06154-2-18-2': "Part of this work was done during author's visit to IBS (Korea) and NCTU (Taiwan)."}

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[]

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[]

['1701.06154-1-15-2', '1701.06154-1-15-4', '1701.06154-2-16-2', '1701.06154-2-16-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1701.06154

1404.0762

{'1404.0762-1-0-0': 'Let [MATH] be an algebraic variety of characteristic zero.', '1404.0762-1-0-1': 'Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over [MATH].', '1404.0762-1-0-2': 'We prove that every terminal valuation over [MATH] is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of [MATH].', '1404.0762-1-0-3': 'In dimension two, this result gives a new proof of the theorem of Fernandez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.', '1404.0762-1-1-0': '# Introduction', '1404.0762-1-2-0': 'Working in characteristic zero, the space of formal arcs passing through the singular points of an algebraic variety [MATH] decomposes into finitely many irreducible families, and carries some of the essential information encoded in a resolution of singularities.', '1404.0762-1-2-1': 'The Nash map associates a divisorial valuation to every maximal irreducible family of arcs through the singular locus of [MATH] [CITATION].', '1404.0762-1-2-2': 'In this paper, we will refer to these valuations as the Nash valuations over [MATH].', '1404.0762-1-3-0': 'The Nash problem asks for a geometric characterization of Nash valuations in terms of resolutions of [MATH].', '1404.0762-1-3-1': 'To this end, Nash introduced the notion of essential valuations as those divisorial valuations whose center on every resolution is an irreducible component of the inverse image of the singular locus of [MATH].', '1404.0762-1-3-2': 'It is easy to see that every Nash valuation is essential, and Nash asked whether the converse is also true.', '1404.0762-1-3-3': 'Regarded as a function to the set of essential valuations, the question is whether the Nash map is surjective.', '1404.0762-1-4-0': 'The Nash problem was successfully settled in dimension two in [CITATION] using topological arguments (we refer to their paper for a comprehensive list of references on previous results).', '1404.0762-1-4-1': 'In higher dimensions, the characterization of Nash valuations as essential valuations is known to hold for toric singularities and in some other special cases [CITATION].', '1404.0762-1-4-2': 'However, examples showing that the Nash map is not always surjective were found in all dimensions [MATH] [CITATION].', '1404.0762-1-4-3': 'In view of these examples, one should rephrase the problem by asking whether there is some other way to characterize the image of the Nash map.', '1404.0762-1-5-0': 'In this paper, we approach this problem from the point of view of the minimal model program.', '1404.0762-1-5-1': 'Recall that a minimal model over [MATH] is a projective birational morphism [MATH] from a normal variety [MATH] with terminal singularities such that the canonical class [MATH] is relatively nef over [MATH].', '1404.0762-1-5-2': 'We say that a divisorial valuation [MATH] on [MATH] is terminal with respect to the minimal model program over [MATH], or simply that it is a terminal valuation over [MATH], if there exists a prime exceptional divisor [MATH] on a minimal model [MATH] such that [MATH].', '1404.0762-1-6-0': 'Our main result provides a characterization of a subset of the image of the Nash map as the one consisting of terminal valuations.', '1404.0762-1-7-0': 'Since, in dimension two, terminal valuations and essential valuations are clearly the same (both are the valuations defined by the exceptional divisors on the minimal resolution), we obtain a new, purely algebro-geometric proof of the main theorem of [CITATION].', '1404.0762-1-8-0': '[[CITATION]]', '1404.0762-1-9-0': 'The Nash map is a bijection in dimension two.', '1404.0762-1-10-0': 'Theorem [REF] is the natural generalization of this result to higher dimensions.', '1404.0762-1-10-1': 'It implies, for instance, that the Nash map is surjective whenever there exists a nonsingular minimal model over [MATH] with exceptional locus of pure codimension one.', '1404.0762-1-10-2': 'Since every exceptional divisor over a variety with terminal singularities is uniruled [CITATION], Theorem [REF] also implies, as a by-product, that every divisorial valuation defined by a divisor that is not uniruled is necessarily a Nash valuation.', '1404.0762-1-10-3': 'This recovers, in particular, the main result of [CITATION], whose proof is however simpler and more direct.', '1404.0762-1-11-0': 'At a first sight, one may wonder whether all Nash valuations are terminal valuations.', '1404.0762-1-11-1': 'While this is the case in dimension two, it fails for simple reasons in higher dimensions.', '1404.0762-1-11-2': 'For instance, it is clear that there are no terminal valuations over a variety with terminal singularities, or over a variety which admits a small resolution.', '1404.0762-1-11-3': 'On the other hand, there are always Nash valuations over any singular variety.', '1404.0762-1-12-0': 'More examples showing that not all Nash valuations are terminal valuations can be constructed using toric geometry.', '1404.0762-1-12-1': 'In [REF], we give a toric description of terminal valuations over a toric variety and compare it with the description of Nash valuations given in [CITATION].', '1404.0762-1-13-0': 'In conclusion, Theorem [REF] should be viewed as complementing the fact that Nash valuations are essential.', '1404.0762-1-13-1': 'We have inclusions [EQUATION] but either inclusion can be strict.', '1404.0762-1-14-0': '## Outline of the proof', '1404.0762-1-15-0': 'We were led to consider minimal models (and define, accordingly, terminal valuations) as a result of our attempt to understand, from an algebro-geometric standpoint, some of the topological computations carried out in [CITATION].', '1404.0762-1-15-1': 'The idea of looking at divisors on minimal models in connection to the Nash problem was also suggested by Fernandez de Bobadilla.', '1404.0762-1-16-0': 'For simplicity, let us focus on the two-dimensional case, as the main ideas of the proof are already there.', '1404.0762-1-16-1': 'So, let [MATH] be a surface.', '1404.0762-1-16-2': 'Let [MATH] be the minimal resolution, and let [MATH] be the map induced on arc spaces.', '1404.0762-1-16-3': 'For every irreducible component [MATH] of the exceptional locus [MATH], let [MATH] denote the largest open set disjoint from the other irreducible components of [MATH].', '1404.0762-1-16-4': 'Then let [MATH] where [MATH] is the natural projection, and let [MATH] be the closure of [MATH].', '1404.0762-1-17-0': 'Assuming that the Nash problem fails for [MATH], one can find two irreducible components [MATH] and [MATH] of [MATH] such that [MATH].', '1404.0762-1-17-1': 'The idea, due to Lejeune-Jalabert [CITATION], is to detect this by producing a morphism [MATH], for some field extension [MATH], such that [EQUATION].', '1404.0762-1-17-2': 'Here [MATH] is the closed point of [MATH], and [MATH] is its generic point.', '1404.0762-1-17-3': 'Such a map is called a wedge.', '1404.0762-1-17-4': 'The existence of the wedge [MATH] is a delicate issue, and is established, in different forms, in [CITATION].', '1404.0762-1-17-5': 'Note that [MATH] does not factor through [MATH].', '1404.0762-1-18-0': 'Let us assume that [MATH].', '1404.0762-1-18-1': 'The wedge [MATH] can be regarded as a morphism [EQUATION] that does not lift to [MATH].', '1404.0762-1-19-0': 'At this point, the approach in [CITATION] is roughly the following.', '1404.0762-1-19-1': 'Reducing to work over [MATH], and using a suitable approximation theorem, one can assume that [MATH] is locally given by power series with positive radius of convergence, and thus assume without loss of generality that [MATH] is the product of two open disks of radius 1 [CITATION].', '1404.0762-1-19-2': 'The curves [MATH], for [MATH], lift to [MATH] and degenerate, as [MATH], to a cycle [MATH] that is supported on the union of the exceptional locus [MATH] of [MATH] and the image [MATH] of the [MATH]-axis [MATH].', '1404.0762-1-19-3': 'The contradiction is then reached by computing the Euler characteristic of these curves in two ways, as images of small disks, and as they degenerate inside a small tubular neighborhood of [MATH].', '1404.0762-1-19-4': 'The contradiction, resulting from the computation, stems from the fact [MATH] does not contain any rational curve with self-intersection [MATH].', '1404.0762-1-20-0': 'In order to translate this into algebro-geometric language, we take a resolution of indeterminacy of the rational map [MATH].', '1404.0762-1-20-1': 'We construct the resolution by taking a minimal sequence of blow-ups of maximal ideals.', '1404.0762-1-20-2': 'This gives us a diagram [EQUATION] where [MATH] is a smooth two-dimensional scheme.', '1404.0762-1-20-3': 'Then we shift the computation from [MATH] to [MATH].', '1404.0762-1-21-0': 'This reduction has several advantages.', '1404.0762-1-21-1': 'First, it allows us to bypass the use of approximation theorems and to work directly in the formal setting.', '1404.0762-1-21-2': 'Furthermore, we avoid having to deal with the singularities of [MATH] and can work in fact with partial resolutions [MATH].', '1404.0762-1-21-3': 'Finally, working on [MATH] allows us to extend the computation to all higher dimensions, by taking wedges defined over suitable field extensions [MATH].', '1404.0762-1-22-0': 'The proof relies on the analysis of the ramification of the map [MATH] at the generic point of the component [MATH] of [MATH] intersecting the proper transform [MATH] of the [MATH]-axis [MATH].', '1404.0762-1-23-0': 'We consider the contraction [MATH] of all the irreducible components of [MATH] that are contracted by [MATH].', '1404.0762-1-23-1': 'The map [MATH] factors through [MATH] and a morphism [MATH].', '1404.0762-1-23-2': 'We look at the relative canonical divisor [MATH] of [MATH], which we decompose as [MATH] by separating those components that exceptional over [MATH] from those that are not.', '1404.0762-1-23-3': 'By a local computation in codimension one (like in Hurwitz formula), we check that [EQUATION].', '1404.0762-1-23-4': 'On the other hand, a negativity lemma (essentially the Hodge index theorem) implies that [EQUATION].', '1404.0762-1-23-5': 'This is the step where we use the assumption that [MATH] is a minimal model.', '1404.0762-1-23-6': 'To conclude, we just observe that [MATH] since [MATH] is smooth, and [MATH] since [MATH] maps to an arc on [MATH] with order of contact one along [MATH].', '1404.0762-1-23-7': 'This gives the contradiction we wanted.', '1404.0762-1-24-0': 'It would be interesting to try to extend this approach to positive characteristics.', '1404.0762-1-24-1': 'There are some difficulties to overcome.', '1404.0762-1-24-2': "First of all, the proof of Theorem [REF] relies on Reguera's curve selection lemma which requires, in the construction of the wedge, to take a field extension which may in principle be inseparable.", '1404.0762-1-24-3': 'This creates problems for the definition of [MATH].', '1404.0762-1-24-4': 'Notice that this issue does not occur in dimension two, since in this case we are led to work with wedges defined over finite algebraic extensions of [MATH], and if [MATH] is algebraically closed then there is only the trivial one.', '1404.0762-1-24-5': 'However, one runs into another difficulty: the local computation leading to the first inequality breaks down if [MATH] is wildly ramified.', '1404.0762-1-24-6': 'In order to extend the result of this paper to positive characteristics, it seems that one would need to gain control on the construction of the wedge to avoid these problems.'}

{'1404.0762-2-0-0': 'Let [MATH] be an algebraic variety of characteristic zero.', '1404.0762-2-0-1': 'Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over [MATH].', '1404.0762-2-0-2': 'We prove that every terminal valuation over [MATH] is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of [MATH].', '1404.0762-2-0-3': 'In dimension two, this result gives a new proof of the theorem of Fernandez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.', '1404.0762-2-1-0': '# Introduction', '1404.0762-2-2-0': 'Working in characteristic zero, the space of formal arcs passing through the singular points of an algebraic variety [MATH] decomposes into finitely many irreducible families, and carries some of the essential information encoded in a resolution of singularities.', '1404.0762-2-2-1': 'The Nash map associates a divisorial valuation to every maximal irreducible family of arcs through the singular locus of [MATH] [CITATION].', '1404.0762-2-2-2': 'In this paper, we will refer to these valuations as the Nash valuations over [MATH].', '1404.0762-2-3-0': 'The Nash problem asks for a geometric characterization of Nash valuations in terms of resolutions of [MATH].', '1404.0762-2-3-1': 'To this end, Nash introduced the notion of essential valuations as those divisorial valuations whose center on every resolution is an irreducible component of the inverse image of the singular locus of [MATH].', '1404.0762-2-3-2': 'It is easy to see that every Nash valuation is essential, and Nash asked whether the converse is also true.', '1404.0762-2-3-3': 'Regarded as a function to the set of essential valuations, the question is whether the Nash map is surjective.', '1404.0762-2-4-0': 'The Nash problem was successfully settled in dimension two in [CITATION] using topological arguments (we refer to their paper for a comprehensive list of references on previous results).', '1404.0762-2-4-1': 'In higher dimensions, the characterization of Nash valuations as essential valuations is known to hold for toric singularities and in some other special cases [CITATION].', '1404.0762-2-4-2': 'However, examples showing that the Nash map is not always surjective were found in all dimensions [MATH] [CITATION].', '1404.0762-2-4-3': 'In view of these examples, one should rephrase the problem by asking whether there is some other way to characterize the image of the Nash map.', '1404.0762-2-5-0': 'In this paper, we approach this problem from the point of view of the minimal model program.', '1404.0762-2-5-1': 'Recall that a minimal model over [MATH] is a projective birational morphism [MATH] from a normal variety [MATH] with terminal singularities such that the canonical class [MATH] is relatively nef over [MATH].', '1404.0762-2-5-2': 'We say that a divisorial valuation [MATH] on [MATH] is terminal with respect to the minimal model program over [MATH], or simply that it is a terminal valuation over [MATH], if there exists a prime exceptional divisor [MATH] on a minimal model [MATH] such that [MATH].', '1404.0762-2-6-0': 'Our main result provides a characterization of a subset of the image of the Nash map as the one consisting of terminal valuations.', '1404.0762-2-7-0': 'Since, in dimension two, terminal valuations and essential valuations are clearly the same (both are the valuations defined by the exceptional divisors on the minimal resolution), we obtain a new, purely algebro-geometric proof of the main theorem of [CITATION].', '1404.0762-2-8-0': '[[CITATION]]', '1404.0762-2-9-0': 'The Nash map is a bijection in dimension two.', '1404.0762-2-10-0': 'Theorem [REF] is the natural generalization of this result to higher dimensions.', '1404.0762-2-10-1': 'It implies, for instance, that the Nash map is surjective whenever there exists a nonsingular minimal model over [MATH] with exceptional locus of pure codimension one.', '1404.0762-2-10-2': 'Since every exceptional divisor over a variety with terminal singularities is uniruled [CITATION], Theorem [REF] also implies, as a by-product, that every divisorial valuation defined by a divisor that is not uniruled is necessarily a Nash valuation.', '1404.0762-2-10-3': 'This recovers, in particular, the main result of [CITATION], whose proof is however simpler and more direct.', '1404.0762-2-11-0': 'At a first sight, one may wonder whether all Nash valuations are terminal valuations.', '1404.0762-2-11-1': 'While this is the case in dimension two, it fails for simple reasons in higher dimensions.', '1404.0762-2-11-2': 'For instance, it is clear that there are no terminal valuations over a variety with terminal singularities, or over a variety which admits a small resolution.', '1404.0762-2-11-3': 'On the other hand, there are always Nash valuations over any singular variety.', '1404.0762-2-12-0': 'More examples showing that not all Nash valuations are terminal valuations can be constructed using toric geometry.', '1404.0762-2-12-1': 'In [REF], we give a toric description of terminal valuations over a toric variety and compare it with the description of Nash valuations given in [CITATION].', '1404.0762-2-13-0': 'In conclusion, Theorem [REF] should be viewed as complementing the fact that Nash valuations are essential.', '1404.0762-2-13-1': 'We have inclusions [EQUATION] but either inclusion can be strict.', '1404.0762-2-14-0': '## Outline of the proof', '1404.0762-2-15-0': 'We were led to consider minimal models (and define, accordingly, terminal valuations) as a result of our attempt to understand, from an algebro-geometric standpoint, some of the topological computations carried out in [CITATION].', '1404.0762-2-15-1': 'The idea of looking at divisors on minimal models in connection to the Nash problem was also suggested by Fernandez de Bobadilla.', '1404.0762-2-16-0': 'For simplicity, let us focus on the two-dimensional case, as the main ideas of the proof are already there.', '1404.0762-2-16-1': 'So, let [MATH] be a surface.', '1404.0762-2-16-2': 'Let [MATH] be the minimal resolution, and let [MATH] be the map induced on arc spaces.', '1404.0762-2-16-3': 'For every irreducible component [MATH] of the exceptional locus [MATH], let [MATH] denote the largest open set disjoint from the other irreducible components of [MATH].', '1404.0762-2-16-4': 'Then let [MATH] where [MATH] is the natural projection, and let [MATH] be the closure of [MATH].', '1404.0762-2-17-0': 'Assuming that the Nash problem fails for [MATH], one can find two irreducible components [MATH] and [MATH] of [MATH] such that [MATH].', '1404.0762-2-17-1': 'The idea, due to Lejeune-Jalabert [CITATION], is to detect this by producing a morphism [MATH], for some field extension [MATH], such that [EQUATION].', '1404.0762-2-17-2': 'Here [MATH] is the closed point of [MATH], and [MATH] is its generic point.', '1404.0762-2-17-3': 'Such a map is called a wedge.', '1404.0762-2-17-4': 'The existence of the wedge [MATH] is a delicate issue, and is established in [CITATION].', '1404.0762-2-17-5': 'Note that [MATH] does not factor through [MATH].', '1404.0762-2-18-0': 'Let us assume that [MATH].', '1404.0762-2-18-1': 'The wedge [MATH] can be regarded as a morphism [EQUATION] that does not lift to [MATH].', '1404.0762-2-19-0': 'At this point, the approach in [CITATION] is roughly the following.', '1404.0762-2-19-1': 'Reducing to work over [MATH], and using a suitable approximation theorem, one can assume that [MATH] is locally given by power series with positive radius of convergence, and thus assume without loss of generality that [MATH] is the product of two open disks of radius 1 [CITATION].', '1404.0762-2-19-2': 'The curves [MATH], for [MATH], lift to [MATH] and degenerate, as [MATH], to a cycle [MATH] that is supported on the union of the exceptional locus [MATH] of [MATH] and the image [MATH] of the [MATH]-axis [MATH].', '1404.0762-2-19-3': 'The contradiction is then reached by computing the Euler characteristic of these curves in two ways, as images of small disks, and as they degenerate inside a small tubular neighborhood of [MATH].', '1404.0762-2-19-4': 'The contradiction, resulting from the computation, stems from the fact [MATH] does not contain any rational curve with self-intersection [MATH].', '1404.0762-2-20-0': 'In order to translate this into algebro-geometric language, we take a resolution of indeterminacy of the rational map [MATH].', '1404.0762-2-20-1': 'We construct the resolution by taking a minimal sequence of blow-ups of maximal ideals.', '1404.0762-2-20-2': 'This gives us a diagram [EQUATION] where [MATH] is a regular two-dimensional scheme.', '1404.0762-2-20-3': 'Then we shift the computation from [MATH] to [MATH].', '1404.0762-2-21-0': 'This reduction has several advantages.', '1404.0762-2-21-1': 'First, it allows us to bypass the use of approximation theorems and to work directly in the formal setting.', '1404.0762-2-21-2': 'Furthermore, we avoid having to deal with the singularities of [MATH] and can work in fact with partial resolutions [MATH].', '1404.0762-2-21-3': 'Finally, working on [MATH] allows us to extend the computation to all higher dimensions, by taking wedges defined over suitable field extensions [MATH].', '1404.0762-2-22-0': 'The proof relies on the analysis of the ramification of the map [MATH] at the generic point of the component [MATH] of [MATH] intersecting the proper transform [MATH] of the [MATH]-axis [MATH].', '1404.0762-2-23-0': 'We consider the contraction [MATH] of all the irreducible components of [MATH] that are contracted by [MATH].', '1404.0762-2-23-1': 'The map [MATH] factors through [MATH] and a morphism [MATH].', '1404.0762-2-23-2': 'We look at the relative canonical divisor [MATH] of [MATH], which we decompose as [MATH] by separating those components that exceptional over [MATH] from those that are not.', '1404.0762-2-23-3': 'By a local computation in codimension one (like in Hurwitz formula), we check that [EQUATION].', '1404.0762-2-23-4': 'On the other hand, a negativity lemma (essentially the Hodge index theorem) implies that [EQUATION].', '1404.0762-2-23-5': 'This is the step where we use the assumption that [MATH] is a minimal model.', '1404.0762-2-23-6': 'To conclude, we just observe that [MATH] since [MATH] is smooth, and [MATH] since [MATH] maps to an arc on [MATH] with order of contact one along [MATH].', '1404.0762-2-23-7': 'This gives the contradiction we wanted.', '1404.0762-2-24-0': 'The computation at the core of the proof becomes particularly transparent if [MATH] (and [MATH] is smooth).', '1404.0762-2-24-1': 'In this special case, [MATH] is the effective divisor defined by the vanishing of the Jacobian of [MATH], and its coefficient at [MATH] is zero because otherwise the special arc [MATH] of the wedge could not meet transversally [MATH] in [MATH].', '1404.0762-2-24-2': 'The Jacobian of [MATH] gives a homomorphism from [MATH] to [MATH], and hence we have the linear equivalence [MATH].', '1404.0762-2-24-3': 'Note, on the other hand, that [MATH] is linear equivalent to the exceptional divisor [MATH], which is defined by the Jacobian of [MATH].', '1404.0762-2-24-4': 'Combining these linear equivalences, we obtain the equivalence [MATH].', '1404.0762-2-24-5': 'The divisor in the right hand side is [MATH]-nef because [MATH] is horizontal and effective and [MATH] is [MATH]-nef, and the divisor in the left hand side is [MATH]-exceptional.', '1404.0762-2-24-6': 'It follows by the negative definitness of the intersection form of the exceptional divisors of [MATH] that [MATH] is anti-effective.', '1404.0762-2-24-7': 'This is impossible, however, since the coefficient of [MATH] in [MATH] is positive and its coefficient in [MATH] is zero.', '1404.0762-2-25-0': 'It would be interesting to try to extend the proof to positive characteristics.', '1404.0762-2-25-1': 'There are some difficulties to overcome.', '1404.0762-2-25-2': "First of all, the proof of Theorem [REF] relies on Reguera's curve selection lemma which requires, in the construction of the wedge, to take a field extension which may in principle be inseparable.", '1404.0762-2-25-3': 'This creates problems for the definition of [MATH].', '1404.0762-2-25-4': 'Notice that this issue does not occur in dimension two, since in this case we are led to work with wedges defined over finite algebraic extensions of [MATH], and if [MATH] is algebraically closed then there is only the trivial one.', '1404.0762-2-25-5': 'However, one runs into another difficulty: the local computation leading to the first inequality displayed above breaks down if [MATH] is wildly ramified.', '1404.0762-2-25-6': 'In order to extend the result of this paper to positive characteristics, it seems that one would need to gain control on the construction of the wedge to avoid these problems.'}

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[]

[['1404.0762-1-24-0', '1404.0762-2-25-0']]

[]

['1404.0762-1-8-0', '1404.0762-1-9-0', '1404.0762-2-8-0', '1404.0762-2-9-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1404.0762

1707.05535

{'1707.05535-1-0-0': 'In this paper, we attempt to implement the neutrino [MATH]-[MATH] reflection symmetry (which predicts [MATH] and [MATH] as well as trivial Majorana phases) in the minimal seesaw (which enables us to fix the neutrino masses).', '1707.05535-1-0-1': 'For some direct (the preliminary experimental hints towards [MATH] and [MATH]) and indirect (inclusion of the renormalization group equation effect and implementation of the leptogenesis mechanism) reasons, we particularly study the breakings of this symmetry and their phenomenological consequences.', '1707.05535-1-1-0': '# Introduction', '1707.05535-1-2-0': 'As is known, the discovery of neutrino oscillations indicates that neutrinos are massive and mixing among different flavors [CITATION].', '1707.05535-1-2-1': 'The most popular way of generating neutrino masses is to invoke the seesaw mechanism which allows for a natural explanation of their smallness.', '1707.05535-1-2-2': 'In the type-1@ seesaw mechanism [CITATION], one usually introduces three right-handed neutrino fields [MATH] (for [MATH]).', '1707.05535-1-2-3': 'They not only take part in Yukawa interactions with the left-handed neutrino fields which lead to a Dirac mass matrix [MATH], but also have a Majorana mass matrix [MATH] of their own.', '1707.05535-1-2-4': 'Under the condition of [MATH], the effective neutrino mass matrix for light neutrinos appears as [CITATION] [EQUATION]', '1707.05535-1-2-5': 'The overall minus sign here is of no physical meaning and will be neglected in the following discussions.', '1707.05535-1-2-6': 'Consequently, the neutrino mixing arises from a mismatch between their mass and flavor eigenstates and is described by a [MATH] unitary matrix [MATH] [CITATION] with [MATH] and [MATH] being respectively the unitary matrix for diagonalizing the charged-lepton mass matrix [MATH] and [MATH].', '1707.05535-1-2-7': 'In the standard parametrization, [MATH] reads [CITATION] [EQUATION]', '1707.05535-1-2-8': 'Here [MATH] consists of three unphysical phases that can be removed via the rephasing of charged-lepton fields, while [MATH] contains two Majorana phases.', '1707.05535-1-2-9': 'Similar to the CKM matrix, [MATH] possesses a Dirac phase [MATH] and three mixing angles [MATH] (for [MATH]).', '1707.05535-1-2-10': 'For convenience, the abbreviations [MATH] and [MATH] have been used.', '1707.05535-1-2-11': 'Besides the mixing parameters, neutrino oscillations are also regulated by two mass-squared differences [MATH] (for [MATH]) [CITATION] [EQUATION]', '1707.05535-1-2-12': 'In comparison, the absolute neutrino mass scale has to be inferred from non-oscillatory experiments [CITATION], and a definite result is still missing.', '1707.05535-1-2-13': 'Note that the sign of [MATH] has not yet been determined, thereby allowing for two possible mass orderings [MATH] (the normal hierarchy, NH for short) and [MATH] (the inverted hierarchy, IH for short).', '1707.05535-1-2-14': 'It turns out that the fitted values of mixing parameters depends on the mass ordering in a certain way: [MATH], [MATH] and [MATH] take the values [EQUATION] in the NH case, or [EQUATION] in the IH case, whereas [MATH] takes the value [MATH] in either case [CITATION].', '1707.05535-1-3-0': 'How to understand the observed neutrino mixing poses an interesting question.', '1707.05535-1-3-1': 'As one can see, the measured [MATH], [MATH] and [MATH] are close to some special values [EQUATION]', '1707.05535-1-3-2': 'These remarkable coincidences invite us to speculate that some flavor symmetry has played an important role in shaping the neutrino mixing [CITATION].', '1707.05535-1-3-3': 'In this connection, the [MATH]-[MATH] reflection symmetry [CITATION] serves as a unique candidate: In the basis of [MATH] being diagonal, [MATH] should stay invariant with respect to the transformations [EQUATION] and have its elements obeying the conditions [EQUATION]', '1707.05535-1-3-4': 'We point out that these conditions and thus the [MATH]-[MATH] reflection symmetry still hold when the left-handed neutrino fields undergo the rephasing [EQUATION]', '1707.05535-1-3-5': 'An [MATH] respecting such a symmetry can be diagonalized by one [MATH] that features [CITATION] [EQUATION]', '1707.05535-1-3-6': 'Note that in the basis under study, [MATH] is identical to [MATH] up to the rephasing of charged-lepton fields.', '1707.05535-1-3-7': 'Thanks to these interesting consequences, this symmetry has been attracting a lot of attention recently [CITATION].', '1707.05535-1-4-0': 'Nevertheless, the neutrino masses receive no clues from the [MATH]-[MATH] reflection symmetry.', '1707.05535-1-4-1': 'But they will be fixed if the smallest one ([MATH] in the NH case, [MATH] in the IH case) is to vanish, a possibility that is experimentally allowed and can be accommodated by the minimal seesaw [CITATION] where only two right-handed neutrino fields [MATH] are involved.', '1707.05535-1-4-2': 'In this paper, we just attempt to combine the [MATH]-[MATH] reflection symmetry with the minimal seesaw so that both the neutrino mixing and mass spectrum can be settled.', '1707.05535-1-4-3': 'The implications of such a marriage are discussed in the next section.', '1707.05535-1-4-4': 'In consideration of the preliminary experimental results towards [MATH] and [MATH], we particularly explore the possible symmetry breakings and their impacts on the mixing parameters in section 3.', '1707.05535-1-4-5': 'Section 4 is devoted to the specific symmetry breaking triggered by the renormalization group equation (RGE) effect.', '1707.05535-1-4-6': 'In section 5, the operation of leptogenesis in this framework will be studied.', '1707.05535-1-4-7': 'Finally, our main results are summarized in section 6.', '1707.05535-1-5-0': '# [MATH]-[MATH] reflection symmetry in the minimal seesaw', '1707.05535-1-6-0': 'If an [MATH] of the [MATH]-[MATH] reflection symmetry has one vanishing eigenvalue (i.e., [MATH]), then its elements [MATH] (for [MATH]) can be expressed in terms of the neutrino masses and mixing parameters as [EQUATION] in the NH case, or [EQUATION] in the IH case, where [MATH] and [MATH] have been defined.', '1707.05535-1-6-1': 'In the minimal seesaw, there is only one effective Majorana phase which we assign for [MATH] (i.e., [MATH]).', '1707.05535-1-6-2': 'Here the free phase [MATH] results from the maintenance of [MATH]-[MATH] reflection symmetry against the neutrino-field rephasing in Eq. ([REF]).', '1707.05535-1-6-3': 'Because of the condition [MATH] , only five out of the six real components of these elements are independent.', '1707.05535-1-6-4': 'Taking the best-fit results for [MATH], [MATH], [MATH] and [MATH] as input, we give the values of [MATH] for various combinations of [MATH] and [MATH] (i.e., [MATH] and [MATH]) in the NH (in Table 1) and IH (in Table 2) cases.', '1707.05535-1-6-5': 'As is well known, the size of [MATH] which governs the rate of neutrino-less double beta decays [CITATION] is much greater in the IH case than in the NH case.', '1707.05535-1-6-6': 'In the NH case, the elements exhibit a hierarchical structure as [MATH], implying that they might have received contributions from different levels [CITATION].', '1707.05535-1-6-7': 'In the IH case, [MATH] (so does [MATH] for [MATH]) becomes comparable to [MATH] and [MATH].', '1707.05535-1-6-8': 'But for [MATH], [MATH] will have a magnitude much smaller than the other ones as a result of the heavy cancellation between its two components respectively associated with [MATH] and [MATH].', '1707.05535-1-6-9': 'By choosing the value of [MATH] in such a way that the phase of [MATH] or [MATH] cancels out, one may convert [MATH] to the form as given by Tables 3-4.', '1707.05535-1-7-0': 'This kind of [MATH] may originate from the minimal seesaw: The Dirac mass matrix connecting [MATH] with the left-handed neutrino fields is assumed to take a form as [CITATION] [EQUATION] with [MATH], [MATH], [MATH] and [MATH] being real parameters.', '1707.05535-1-7-1': 'Apparently, its elements satisfy the conditions [MATH] and [MATH] being real (for [MATH]).', '1707.05535-1-7-2': 'It is easy to see that these conditions will maintain if the right-handed neutrino fields experience an orthogonal basis transformation.', '1707.05535-1-7-3': 'So, without loss of generality, we choose to work in the basis where the Majorana mass matrix for [MATH] is diagonal [MATH].', '1707.05535-1-7-4': 'By virtue of the seesaw formula in Eq. ([REF]), we arrive at an effective neutrino mass matrix [EQUATION]', '1707.05535-1-7-5': 'Diagonalizing this [MATH] with a [MATH] prescribed by Eq. ([REF]) yields the following mixing parameters and neutrino masses [EQUATION] where [MATH], [MATH] (for [MATH] or [MATH]) and [EQUATION]', '1707.05535-1-7-6': 'For any given values of [MATH], [MATH] and [MATH], one mass will necessarily vanish as promised by the minimal seesaw .', '1707.05535-1-7-7': 'The resulting [MATH], [MATH], [MATH] and two non-zero masses can be calculated by means of the other five equations.', '1707.05535-1-8-0': 'If we are to derive the allowed values of [MATH], [MATH] and [MATH] from the measured [MATH], [MATH], [MATH] and [MATH], one just needs to confront the [MATH] in Eq. ([REF]) with the results given by Tables 1-2.', '1707.05535-1-8-1': 'Above all, we note that the [MATH] and [MATH] elements of such an [MATH] are both positive.', '1707.05535-1-8-2': 'In order for them to fit in with the corresponding results in Tables 1-2, one must have [MATH] (or 0) and [MATH] in the NH (or IH) case.', '1707.05535-1-8-3': 'In light of the unphysical nature of [MATH], [MATH] rather than [MATH] will be treated as effective independent parameters.', '1707.05535-1-8-4': 'Recall that only five real components of the neutrino mass matrix elements are independent.', '1707.05535-1-8-5': 'So the free parameters are more than the constraint equations by one.', '1707.05535-1-8-6': 'For this reason, we choose to present the results for [MATH], [MATH] and [MATH] as functions of [MATH] (which stands in an equivalent position as [MATH]) in Fig. 1.', '1707.05535-1-8-7': 'In the numerical calculations here and in the following, the best-fit results for [MATH], [MATH], [MATH] and [MATH] are input, whereas [MATH] is specified as [MATH].', '1707.05535-1-8-8': 'We have only shown the results in the case of both [MATH] and [MATH] being positive.', '1707.05535-1-8-9': 'The results in the case of [MATH] or (and) [MATH] being negative can be achieved by simply making the replacement [MATH] combined with [MATH] or (and) [MATH] combined with [MATH].', '1707.05535-1-8-10': 'This is because the [MATH] in Eq. ([REF]) keeps invariant under this kind of transformations.', '1707.05535-1-8-11': 'It is interesting to find that the possibility [MATH] is allowed.', '1707.05535-1-8-12': 'The possible values of [MATH], [MATH] and [MATH] in such a particular case are listed in Table 5.', '1707.05535-1-8-13': 'If we further let one of [MATH] vanish by giving [MATH] an appropriate value, then we will reach the simplest [MATH].', '1707.05535-1-8-14': 'In the NH case, for instance, a value of [MATH] or [MATH] for [MATH] allows us to have [MATH].', '1707.05535-1-8-15': '(In the meantime, [MATH] is fixed to [MATH] or [MATH].)', '1707.05535-1-9-0': '# Breaking of the [MATH]-[MATH] reflection symmetry', '1707.05535-1-10-0': 'In this section, we study the possible breakings of [MATH]-[MATH] reflection symmetry and their impacts on the mixing parameters [CITATION].', '1707.05535-1-10-1': 'The most general perturbation to [MATH] [EQUATION] can be decomposed into a symmetry-conserving part and a symmetry-violating part: [EQUATION]', '1707.05535-1-10-2': 'The full Dirac mass matrix can be reparameterized as [EQUATION] with [EQUATION] and [EQUATION]', '1707.05535-1-10-3': 'The dimensionless quantities [MATH] measure the strength of [MATH]-[MATH] reflection symmetry breaking.', '1707.05535-1-10-4': 'They should be small (e.g., [MATH]) in order for [MATH] to keep an approximate [MATH]-[MATH] reflection symmetry.', '1707.05535-1-11-0': 'Since [MATH] and [MATH] plays equivalent roles as [MATH] and [MATH], they will be assumed to vanish in the following discussions.', '1707.05535-1-11-1': 'For the sake of simplicity, the hat symbols on [MATH], [MATH] and [MATH] will also be neglected.', '1707.05535-1-11-2': 'Accordingly, we are led to an effective neutrino mass matrix [MATH] as [EQUATION]', '1707.05535-1-11-3': 'The unitary matrix [MATH] for diagonalizing it is expected to have some mixing parameters around the special values given by Eq. ([REF]), with the corresponding deviations [EQUATION] presumably being some small quantities.', '1707.05535-1-11-4': 'By making series expansions for these deviations in the diagonalization process, at the leading order we acquire the following relations connecting them with the symmetry-breaking parameters [MATH] [EQUATION] where [MATH] (or [MATH]) in the NH (or IH) case, the values of [MATH], [MATH] and [MATH] are the same as those presented in Fig. 1.', '1707.05535-1-11-5': 'The sign [MATH] which takes [MATH] (or [MATH]) in the NH (or IH) case arises from the aforementioned fact that [MATH] might call for an overall sign change so as to fit in with the numerical results.', '1707.05535-1-12-0': 'By solving these equations in a straightforward way, one will obtain the mixing-parameter deviations as some linear functions of [MATH], [MATH] and [MATH].', '1707.05535-1-12-1': 'For illustration, in Fig. 2 we present the [MATH], [MATH] and [MATH] (as functions of [MATH]) from [MATH], [MATH] and [MATH] in the NH and IH cases.', '1707.05535-1-12-2': 'Provided that the linear approximation holds to a good degree (i.e., the expected small quantities are really [MATH]), the [MATH], [MATH] and [MATH] generated by other values of [MATH], [MATH] and [MATH] can be inferred by rescaling these results (according to the linear dependence of mixing-parameter deviations on symmetry-breaking parameters).', '1707.05535-1-12-3': 'The results in Fig. 2 tell us: (a) In the NH case, [MATH] may give rise to a [MATH] as large as 0.1.', '1707.05535-1-12-4': 'But the resulting [MATH] and [MATH] are desperately small.', '1707.05535-1-12-5': '(b) In the IH case, the [MATH], [MATH] and [MATH] from [MATH] are [MATH].', '1707.05535-1-12-6': '(c) In the NH case, [MATH] likely leads to some considerable ([MATH] or so) [MATH], [MATH] and [MATH].', '1707.05535-1-12-7': '(d) In the IH case, the [MATH] and [MATH] arising from [MATH] may reach 0.1 and 0.35 (but for distinct values of [MATH]), while [MATH] is rather small.', '1707.05535-1-12-8': '(e)-(f) In both the NH and IH cases, [MATH] can result in considerable [MATH] and [MATH] but relatively small [MATH].', '1707.05535-1-12-9': 'In the particular case of [MATH], [MATH] contributes [MATH], [MATH] and [MATH] for NH (or IH), while [MATH] contributes [MATH], [MATH] and [MATH].', '1707.05535-1-12-10': 'To summarize, [MATH] is unlikely to induce some considerable mixing-parameter deviations, while [MATH] is likely.', '1707.05535-1-12-11': 'Inversely, a considerable [MATH] can be ascribed to [MATH], while a considerable [MATH] may result from any symmetry-breaking parameter.', '1707.05535-1-13-0': '# RGE triggered symmetry breaking', '1707.05535-1-14-0': 'A flavor symmetry (FS) such as the [MATH]-[MATH] reflection one discussed here is usually introduced at an energy scale [MATH] much higher than the electroweak (EW) one [MATH] GeV [CITATION].', '1707.05535-1-14-1': 'Therefore, the RGE effects should be taken into account when a flavor-symmetry model is confronted with the low-energy experimental data [CITATION].', '1707.05535-1-14-2': 'During the evolution process, the significant difference between the Yukawa coupling of [MATH] and that of [MATH] may provide a unique source for symmetry breaking.', '1707.05535-1-14-3': 'This section is just devoted to such a specific breaking of the [MATH]-[MATH] reflection symmetry.', '1707.05535-1-14-4': 'At the one-loop level, the energy dependence of [MATH] is described by [CITATION] [EQUATION] where [MATH] with [MATH] denoting the renormalization scale, [MATH] and [MATH] read [EQUATION]', '1707.05535-1-14-5': 'In the basis of [MATH] being diagonal, the Yukawa coupling matrix for charged leptons appears as [MATH].', '1707.05535-1-14-6': 'Due to [MATH], it is reasonable to neglect the contributions of [MATH] and [MATH].', '1707.05535-1-14-7': 'In Eq. ([REF]), the [MATH]-term is flavor universal and only contributes an overall rescaling factor [MATH] for the neutrino mass matrix, while the other two terms are able to modify its structure.', '1707.05535-1-14-8': 'Given an [MATH] of the form given by Eq. ([REF]) at [MATH], integration of Eq. ([REF]) enables us to derive the RGE-corrected neutrino mass matrix at [MATH] [CITATION] [EQUATION] with [MATH] and [EQUATION]', '1707.05535-1-14-9': 'Obviously, [MATH] measures the strength of symmetry breaking.', '1707.05535-1-14-10': 'Owing to the smallness of [MATH] and consequently [MATH], the RGE effect is negligible in the SM.', '1707.05535-1-14-11': 'But in the MSSM, [MATH] (with [MATH] GeV being the Higgs VEV) can be greatly enhanced by a large [MATH].', '1707.05535-1-14-12': 'To be explicit, the value of [MATH] depends on [MATH] in a way as [EQUATION] for [MATH] GeV.', '1707.05535-1-15-0': 'Following the same approach as in the previous section, one obtains some relations connecting the mixing-parameter deviations with [MATH] as given by [EQUATION] with [MATH] (or [MATH]) in the NH (or IH) case.', '1707.05535-1-15-1': 'Solving these equations numerically gives [EQUATION] in the NH case, or [EQUATION] in the IH case.', '1707.05535-1-15-2': 'We subsequently show the dependence of these mixing-parameter deviations on the value of [MATH] (which varies from 10 to 50) in Fig. 3.', '1707.05535-1-15-3': 'One can see that the mixing parameters are pretty stable against the RGE corrections.', '1707.05535-1-15-4': 'Even for [MATH], one merely has [MATH], [MATH] and [MATH] in the NH (or IH) case.', '1707.05535-1-16-0': '# Leptogenesis and the symmetry breaking', '1707.05535-1-17-0': 'Besides accommodating the smallness of neutrino masses, the seesaw mechanism can also explain the observed baryon-antibaryon asymmetry of the Universe via the leptogenesis mechanism [CITATION]: The CP-violating, lepton-number-violating and out-of-equilibrium decays of [MATH] may generate a lepton-antilepton asymmetry which is eventually converted to the baryon-antibaryon asymmetry through the sphaleron process [CITATION].', '1707.05535-1-17-1': 'The essential CP violation is provided by the complex Yukawa couplings [MATH] which connect [MATH] with the left-handed neutrino fields, while the lepton-number violation by the Majorana mass terms of [MATH].', '1707.05535-1-17-2': 'And the departure from thermal equilibrium can occur if [MATH] decay in a rate smaller than the expansion rate of the Universe when the temperature [MATH] drops to their mass scale.', '1707.05535-1-17-3': 'The resulting amount of lepton-antilepton asymmetry is crucially dependent on the CP-violating asymmetries between the decays of [MATH] and their CP conjugate processes.', '1707.05535-1-17-4': 'In the minimal seesaw under study, we assume [MATH] to have a strong mass hierarchy [MATH], in which case only the decay of [MATH] will be relevant for leptogenesis.', '1707.05535-1-17-5': 'The flavored CP-violating asymmetries [MATH] are approximately given by [CITATION] [EQUATION] for [MATH].', '1707.05535-1-17-6': 'A [MATH] corresponding to the [MATH] in Eq. ([REF]) immediately yields [CITATION] [EQUATION] rendering the total CP-violating asymmetry [MATH] vanishing.', '1707.05535-1-17-7': 'However, a successful leptogenesis is possible when the [MATH]-[MATH] reflection symmetry is broken [CITATION] or (and) the flavor effects become relevant [CITATION].', '1707.05535-1-18-0': 'In the situation of [MATH] GeV, the flavor effects are negligible.', '1707.05535-1-18-1': 'In order to achieve [MATH], one has to break the [MATH]-[MATH] reflection symmetry.', '1707.05535-1-18-2': 'A [MATH] corresponding to the [MATH] in Eq. ([REF]) (with [MATH] as assumed before) gives [EQUATION] which states that [MATH] is proportional to [MATH] and a linear function of [MATH], [MATH] and [MATH].', '1707.05535-1-18-3': 'The final baryon-to-entropy ratio can be written as [CITATION] [EQUATION]', '1707.05535-1-18-4': 'Here [MATH] is the efficiency factor of converting the lepton-antilepton asymmetry to the baryon-antibaryon asymmetry, whereas [MATH] is the effective number of relativistic degrees of freedom at [MATH] in the SM.', '1707.05535-1-18-5': 'In particular, [MATH] is the washout factor and can be parametrized as [CITATION] [EQUATION] with [MATH].', '1707.05535-1-18-6': 'In the present epoch where we have [MATH], the baryon-to-photon ratio is given by [EQUATION] which has an observed value of [MATH] [CITATION].', '1707.05535-1-18-7': 'To figure out what kind of [MATH] and symmetry-breaking parameters may give rise to the observed baryon-antibaryon asymmetry, we present the [MATH] (as functions of [MATH]) from some example values of them in Fig. 4.', '1707.05535-1-18-8': 'The results show that a combination of [MATH] GeV, [MATH] GeV or [MATH] GeV is qualified for this job in the NH (or IH) case.', '1707.05535-1-18-9': 'Clearly, it is much easier to gain the observed [MATH] in the NH case than in the IH case.', '1707.05535-1-18-10': 'In particular, a [MATH] (or smaller, depending on the value of [MATH]) is sufficient for our purpose in the NH case.', '1707.05535-1-19-0': 'If [MATH] turns out to be smaller than [MATH] GeV, then the Yukawa interactions of charged leptons will enter in thermal equilibrium, making different lepton flavors distinguishable.', '1707.05535-1-19-1': 'In such a case, both the CP-violating asymmetries and washout factors associated with different flavors should be treated separately.', '1707.05535-1-19-2': 'Accordingly, the final baryon-to-entropy ratio can be rewritten as [CITATION] [EQUATION] where [MATH] is given by [EQUATION] with [EQUATION]', '1707.05535-1-19-3': 'As illustration, we show the [MATH] (as functions of [MATH]) from some example values of [MATH] and symmetry-breaking parameters in the situation of [MATH] in Fig. 5.', '1707.05535-1-19-4': 'For comparison, the contributions from pure flavor effects (without symmetry-breaking effects) are also explicitly shown.', '1707.05535-1-19-5': 'It is easy to see that the flavor effects are more significant than the symmetry-breaking effects in the NH case, while the contrary is the case in the IH case.', '1707.05535-1-19-6': 'In the NH case the flavor effects themselves are competent for generating the observed [MATH].', '1707.05535-1-19-7': 'But in the IH case the symmetry-breaking effects have to be invoked.', '1707.05535-1-19-8': 'And the symmetry-breaking parameters should take some values at least [MATH].', '1707.05535-1-20-0': '# Summary', '1707.05535-1-21-0': 'The [MATH]-[MATH] reflection symmetry is powerful in shaping the neutrino mixing: it leads to the interesting results [MATH] and [MATH] (which are close to the current experimental results) as well as trivial Majorana phases.', '1707.05535-1-21-1': 'On the other hand, the minimal seesaw has strong predictive power in settling the neutrino mass spectrum: it enforces the condition of [MATH] or [MATH].', '1707.05535-1-21-2': 'In this paper, the attempt of implementing the [MATH]-[MATH] reflection symmetry in the minimal seesaw has been made.', '1707.05535-1-21-3': 'Such a marriage results in [MATH] (or 0) and thus [MATH] (or 4.86) eV in the NH (or IH) case.', '1707.05535-1-21-4': 'Motivated by the preliminary experimental hints towards [MATH] and [MATH], we particularly explore the possible symmetry breakings and their phenomenological consequences.', '1707.05535-1-21-5': 'Some parameters characterizing the breaking of [MATH]-[MATH] refection symmetry are first defined and their implications for the mixing-parameter deviations then analyzed.', '1707.05535-1-21-6': 'One finds that [MATH] is difficult to produce some considerable mixing-parameter deviations, while [MATH] is relatively easy.', '1707.05535-1-21-7': 'Inversely, a considerable [MATH] can be attributed to [MATH], while a considerable [MATH] may arise from any symmetry-breaking parameter.', '1707.05535-1-22-0': 'As a unique example, the symmetry breaking triggered by the RGE effects is studied in some detail.', '1707.05535-1-22-1': 'It turns out that the mixing parameters are pretty stable against the RGE corrections.', '1707.05535-1-22-2': 'Even for [MATH] in the MSSM, the RGE-induced mixing-parameter deviations are only of [MATH].', '1707.05535-1-22-3': 'Finally, the operation of leptogenesis in the framework under study is discussed.', '1707.05535-1-22-4': 'In the situation of [MATH] GeV where one has a vanishing [MATH], the [MATH]-[MATH] reflection symmetry must be broken to make the leptogenesis mechanism work.', '1707.05535-1-22-5': 'For illustration, we give some example values of [MATH] and symmetry-breaking parameters that may give rise to the observed value of [MATH].', '1707.05535-1-22-6': 'In the situation of [MATH] GeV, it is possible that the flavor effects themselves (without symmetry-breaking effects) are sufficient for our purpose in the NH case.', '1707.05535-1-22-7': 'But in the IH case, one has to turn to the symmetry-breaking effects for help.', '1707.05535-1-22-8': 'In both situations, it is easier to achieve a realistic value of [MATH] in the NH case than in the IH case.', '1707.05535-1-22-9': 'Last but not least, we point out that the mixing-parameter deviations may be bridged to the leptogenesis, considering that they may originate from the same symmetry breaking after all.', '1707.05535-1-22-10': 'This will be another motivation for us to determine the neutrino mixing parameters as precisely as possible.'}

{'1707.05535-2-0-0': 'In this paper, we attempt to implement the neutrino [MATH]-[MATH] reflection symmetry (which predicts [MATH] and [MATH] as well as trivial Majorana phases) in the minimal seesaw (which enables us to fix the neutrino masses).', '1707.05535-2-0-1': 'For some direct (the preliminary experimental hints towards [MATH] and [MATH]) and indirect (inclusion of the renormalization group equation effect and implementation of the leptogenesis mechanism) reasons, we particularly study the breakings of this symmetry and their phenomenological consequences.', '1707.05535-2-1-0': '# Introduction', '1707.05535-2-2-0': 'As is known, the discovery of neutrino oscillations indicates that neutrinos are massive and mixed [CITATION].', '1707.05535-2-2-1': 'On the one hand, the most popular way of generating neutrino masses is to invoke the seesaw mechanism which allows for a natural explanation of their smallness.', '1707.05535-2-2-2': 'In the type-1@ seesaw mechanism [CITATION], one usually introduces three right-handed neutrino fields [MATH] (for [MATH]).', '1707.05535-2-2-3': 'They not only take part in Yukawa interactions with the left-handed neutrino fields which lead to a Dirac mass matrix [MATH], but also have a Majorana mass matrix [MATH] of themselves.', '1707.05535-2-2-4': 'Under the assumption of [MATH], the effective mass matrix for light neutrinos is obtained as [CITATION] [EQUATION]', '1707.05535-2-2-5': 'The overall minus sign here is of no physical meaning and will be neglected in the following discussions.', '1707.05535-2-2-6': 'On the other hand, the neutrino mixing arises from a mismatch between their mass and flavor eigenstates and is described by a [MATH] unitary matrix [MATH] [CITATION] with [MATH] and [MATH] being respectively the unitary matrix for diagonalizing the charged-lepton mass matrix [MATH] and [MATH].', '1707.05535-2-2-7': 'In the standard parametrization, [MATH] reads [CITATION] [EQUATION]', '1707.05535-2-2-8': 'Here [MATH] consists of three unphysical phases that can be removed via the rephasing of charged-lepton fields, while [MATH] contains two Majorana phases.', '1707.05535-2-2-9': 'Similar to the CKM matrix, [MATH] possesses a Dirac phase [MATH] and three mixing angles [MATH] (for [MATH]).', '1707.05535-2-2-10': 'For convenience, the abbreviations [MATH] and [MATH] have been used.', '1707.05535-2-2-11': 'Besides the mixing parameters, neutrino oscillations are also regulated by two neutrino mass-squared differences [MATH] (for [MATH]) [CITATION] [EQUATION]', '1707.05535-2-2-12': 'In comparison, the absolute neutrino mass scale has to be inferred from non-oscillatory experiments [CITATION], and a definite result is still missing.', '1707.05535-2-2-13': 'Note that the sign of [MATH] has not yet been determined, thereby allowing for two possible mass orderings [MATH] (the normal hierarchy, NH for short) and [MATH] (the inverted hierarchy, IH for short).', '1707.05535-2-2-14': 'It turns out that the fitted values of mixing parameters depends on the mass ordering in a certain way: [MATH], [MATH] and [MATH] take the values [EQUATION] in the NH case, or [EQUATION] in the IH case, whereas [MATH] takes the value [MATH] in either case [CITATION].', '1707.05535-2-3-0': 'How to understand the observed neutrino mixing poses an interesting question.', '1707.05535-2-3-1': 'As one can see, the measured [MATH], [MATH] and [MATH] are close to some special values [EQUATION]', '1707.05535-2-3-2': 'These remarkable coincidences invite us to speculate that some flavor symmetry has played an important role in shaping the neutrino mixing [CITATION].', '1707.05535-2-3-3': 'In this connection, the [MATH]-[MATH] reflection symmetry [CITATION] serves as a unique candidate: In the basis of [MATH] being diagonal, [MATH] should stay invariant with respect to the transformations [EQUATION] and have its elements [MATH] (for [MATH]) obeying the conditions [EQUATION]', '1707.05535-2-3-4': 'Consequently, [MATH] is a diagonal phase matrix of no physical meaning [MATH] where [MATH] can take arbitrary values.', '1707.05535-2-3-5': 'On the other hand, as a result of the six real conditions for [MATH] given by Eq. (1), there are six predictions [CITATION] [EQUATION] for the mixing parameters of [MATH] which itself takes a form as given by Eq. ([REF]).', '1707.05535-2-3-6': 'Furthermore, unlike the [MATH]-[MATH] interchange symmetry [CITATION] which predicts [MATH], the [MATH]-[MATH] reflection symmetry allows for an arbitrary [MATH].', '1707.05535-2-3-7': 'Thanks to these interesting consequences, this symmetry has been attracting a lot of attention recently [CITATION].', '1707.05535-2-4-0': 'Nevertheless, the neutrino masses receive no clues from the [MATH]-[MATH] reflection symmetry.', '1707.05535-2-4-1': 'But they will be fixed if the smallest one ([MATH] in the NH case, [MATH] in the IH case) is to vanish, a possibility that is experimentally allowed and can be accommodated by the minimal seesaw [CITATION] where only two right-handed neutrino fields [MATH] are involved.', '1707.05535-2-4-2': 'In this paper, we just attempt to combine the [MATH]-[MATH] reflection symmetry with the minimal seesaw so that both the neutrino mixing and mass spectrum can be settled.', '1707.05535-2-4-3': 'The implications of such a marriage are discussed in the next section.', '1707.05535-2-4-4': 'In consideration of the preliminary experimental results towards [MATH] and [MATH], we particularly explore the possible symmetry breakings and their impacts on the mixing parameters in section 3.', '1707.05535-2-4-5': 'Section 4 is devoted to the specific symmetry breaking triggered by the renormalization group equation (RGE) effect.', '1707.05535-2-4-6': 'In section 5, the operation of leptogenesis in this framework will be studied.', '1707.05535-2-4-7': 'Finally, our main results are summarized in section 6.', '1707.05535-2-5-0': '# [MATH]-[MATH] reflection symmetry in the minimal seesaw', '1707.05535-2-6-0': 'Conversely, one can reconstruct an [MATH] of the [MATH]-[MATH] reflection symmetry in terms of the [MATH] characterized by Eq. (2) and neutrino masses by means of the relation [MATH].', '1707.05535-2-6-1': 'In the situation of one neutrino mass being vanishing, the reconstructed [MATH] in such a way will take a form as [EQUATION] in the NH case, or [EQUATION] in the IH case, where [MATH] and [MATH] have been defined.', '1707.05535-2-6-2': 'Note that [MATH] and [MATH] depend on the unphysical phase [MATH] whose value can be chosen freely without affecting the physical results.', '1707.05535-2-6-3': 'And there is only one effective Majorana phase which we assign for [MATH] (i.e., [MATH]).', '1707.05535-2-6-4': 'Because of the condition [MATH] , only five out of the six real components of these elements are independent.', '1707.05535-2-6-5': 'Taking the best-fit values for [MATH], [MATH], [MATH] and [MATH] as input, we give the values of [MATH] for various combinations of [MATH] and [MATH] (i.e., [MATH] and [MATH]) in the NH (Table 1) and IH (Table 2) cases.', '1707.05535-2-6-6': 'As is well known, the size of [MATH] which governs the rate of neutrino-less double beta decays [CITATION] is much larger in the IH case than in the NH case.', '1707.05535-2-6-7': 'In the NH case, the elements exhibit a hierarchical structure as [MATH], implying that they might have received contributions from different levels [CITATION].', '1707.05535-2-6-8': 'In the IH case, [MATH] (so does [MATH] for [MATH]) becomes comparable to [MATH] and [MATH].', '1707.05535-2-6-9': 'But for [MATH], [MATH] will have a magnitude much smaller than the other ones as a result of the heavy cancellation between its two components respectively associated with [MATH] and [MATH].', '1707.05535-2-6-10': 'By choosing the value of [MATH] in such a way that the phase of [MATH] or [MATH] cancels out, one may convert [MATH] to a simpler form as given by Tables 3-4.', '1707.05535-2-7-0': 'The [MATH] given by Eqs. ([REF]-[REF]) can be viewed as a result of the minimal seesaw: The Dirac mass matrix coupling [MATH] with the left-handed neutrino fields is assumed to appear as [CITATION] [EQUATION] with [MATH], [MATH], [MATH] and [MATH] being real parameters.', '1707.05535-2-7-1': 'Apparently, its elements satisfy the conditions of [MATH] and [MATH] being real (for [MATH]).', '1707.05535-2-7-2': 'It is easy to see that these conditions still hold when the right-handed neutrino fields experience an orthogonal basis transformation.', '1707.05535-2-7-3': 'So, without loss of generality, we choose to work in the basis where the Majorana mass matrix for [MATH] is diagonal [MATH].', '1707.05535-2-7-4': 'By virtue of the seesaw formula in Eq. ([REF]), we arrive at an effective neutrino mass matrix [EQUATION]', '1707.05535-2-7-5': 'Diagonalizing this [MATH] with a [MATH] characterized by Eq. ([REF]) yields the mixing parameters [EQUATION] and neutrino masses [EQUATION] where [MATH], [MATH] (for [MATH] or [MATH]) and [EQUATION]', '1707.05535-2-7-6': 'For any given values of [MATH], [MATH] and [MATH], one mass will necessarily vanish as promised by the minimal seesaw .', '1707.05535-2-7-7': 'The resulting [MATH], [MATH], [MATH] and two non-zero masses can be calculated with the help of the other five equations.', '1707.05535-2-8-0': 'If we are to derive the allowed values of [MATH], [MATH] and [MATH] from the measured [MATH], [MATH], [MATH] and [MATH], one just needs to confront the [MATH] in Eq. ([REF]) with the results given by Tables 1-2.', '1707.05535-2-8-1': 'Above all, it should be noted that both the [MATH] and [MATH] elements of this [MATH] are positive.', '1707.05535-2-8-2': 'In order for them to fit in with the corresponding results in Tables 1-2, one must have [MATH] (or 0) and [MATH] in the NH (or IH) case.', '1707.05535-2-8-3': 'In light of the unphysical nature of [MATH], [MATH] rather than [MATH] will be treated as effective independent parameters.', '1707.05535-2-8-4': 'Recall that only five real components of the neutrino mass matrix elements are independent, so the free parameters are more than the constraint equations by one.', '1707.05535-2-8-5': 'For this reason, in Fig. 1 we choose to present the results for [MATH], [MATH] and [MATH] as functions of [MATH] (which stands in an equivalent position as [MATH]).', '1707.05535-2-8-6': 'In the numerical calculations here and in the following, the best-fit values for [MATH], [MATH], [MATH] and [MATH] are input, whereas [MATH] is specified as [MATH].', '1707.05535-2-8-7': 'We have only shown the results in the case of both [MATH] and [MATH] being positive.', '1707.05535-2-8-8': 'The results in the case of [MATH] or (and) [MATH] being negative can be obtained by simply making the replacement [MATH] combined with [MATH] or (and) [MATH] combined with [MATH].', '1707.05535-2-8-9': 'This is because the [MATH] in Eq. ([REF]) keeps invariant under this kind of transformations.', '1707.05535-2-8-10': 'It is interesting to find that the possibility of [MATH] is allowed.', '1707.05535-2-8-11': 'The possible values of [MATH], [MATH] and [MATH] in such a particular case are listed in Table 5.', '1707.05535-2-8-12': 'If we further make one of [MATH] vanish by giving [MATH] an appropriate value, then we will reach the simplest [MATH].', '1707.05535-2-8-13': 'In the NH case, for instance, a value of [MATH] or [MATH] for [MATH] allows us to have [MATH].', '1707.05535-2-8-14': '(In the meantime, [MATH] is fixed to [MATH] or [MATH].)', '1707.05535-2-9-0': '# Breaking of the [MATH]-[MATH] reflection symmetry', '1707.05535-2-10-0': 'In this section we study the possible breakings of [MATH]-[MATH] reflection symmetry and their impacts on the mixing parameters [CITATION].', '1707.05535-2-10-1': 'The most general perturbation to [MATH] [EQUATION] can be decomposed into a symmetry-conserving part and a symmetry-violating part: [EQUATION]', '1707.05535-2-10-2': 'And the full Dirac mass matrix can be reparameterized as [EQUATION] with [EQUATION] and [EQUATION]', '1707.05535-2-10-3': 'The dimensionless quantities [MATH] measure the strength of symmetry breaking.', '1707.05535-2-10-4': 'They should be small (e.g., [MATH]) in order for [MATH] to assume an approximate [MATH]-[MATH] reflection symmetry.', '1707.05535-2-11-0': 'Since [MATH] and [MATH] play equivalent roles as [MATH] and [MATH], they will be assumed to vanish in the following discussions.', '1707.05535-2-11-1': 'For the sake of simplicity, the hat symbols on [MATH], [MATH] and [MATH] will also be neglected.', '1707.05535-2-11-2': 'Consequently, we are led to an effective neutrino mass matrix [MATH] of the form [EQUATION] at the leading order.', '1707.05535-2-11-3': 'The unitary matrix [MATH] for diagonalizing [MATH] is expected to have some mixing parameters around the special values given by Eq. ([REF]), with the corresponding deviations [EQUATION] being some small quantities.', '1707.05535-2-11-4': 'By making series expansions for these mixing-parameter deviations in the diagonalization process, at the leading order we acquire the following relations connecting them with the symmetry-breaking parameters [MATH] [EQUATION] where [MATH] (or [MATH]) in the NH (or IH) case and the values of [MATH], [MATH] and [MATH] are the same as those presented in Fig. 1.', '1707.05535-2-11-5': 'The sign [MATH] which takes [MATH] (or [MATH]) in the NH (or IH) case arises from the aforementioned fact that [MATH] might need an overall sign change so as to fit in with the numerical results.', '1707.05535-2-12-0': 'By solving these equations in a straightforward way, one will obtain the mixing-parameter deviations as some linear functions of [MATH], [MATH] and [MATH].', '1707.05535-2-12-1': 'For illustration, in Fig. 2 we present the [MATH], [MATH] and [MATH] (as functions of [MATH]) arising from [MATH], [MATH] and [MATH] in the NH and IH cases.', '1707.05535-2-12-2': 'Provided that the linear approximation holds to a good degree (i.e., the expected small quantities are really [MATH]), the [MATH], [MATH] and [MATH] generated by other values of [MATH], [MATH] and [MATH] can be inferred by rescaling these results (according to the linear dependence of mixing-parameter deviations on symmetry-breaking parameters).', '1707.05535-2-12-3': 'The results in Fig. 2 tell us: (a) In the NH case, [MATH] may give rise to a [MATH] as large as 0.1.', '1707.05535-2-12-4': 'But the resulting [MATH] and [MATH] are desperately small.', '1707.05535-2-12-5': '(b) In the IH case, the [MATH], [MATH] and [MATH] from [MATH] are [MATH].', '1707.05535-2-12-6': '(c) In the NH case, [MATH] likely leads to some considerable ([MATH] or so) [MATH], [MATH] and [MATH].', '1707.05535-2-12-7': '(d) In the IH case, the [MATH] and [MATH] induced by [MATH] may reach 0.1 and 0.35 (but for distinct values of [MATH]), while [MATH] is rather small.', '1707.05535-2-12-8': '(e)-(f) In both the NH and IH cases, [MATH] can result in considerable [MATH] and [MATH] but relatively small [MATH].', '1707.05535-2-12-9': 'In the particular case of [MATH], [MATH] contributes [MATH], [MATH] and [MATH] for NH (or IH), while [MATH] contributes [MATH], [MATH] and [MATH].', '1707.05535-2-12-10': 'To summarize, [MATH] is unlikely to induce considerable mixing-parameter deviations, while [MATH] is likely.', '1707.05535-2-12-11': 'Inversely, a considerable [MATH] can be ascribed to [MATH], while a considerable [MATH] may result from any symmetry-breaking parameter.', '1707.05535-2-13-0': '# RGE triggered symmetry breaking', '1707.05535-2-14-0': 'A flavor symmetry (FS) such as the [MATH]-[MATH] reflection symmetry under study is usually introduced at an energy scale [MATH] much higher than the electroweak (EW) one [MATH] GeV [CITATION].', '1707.05535-2-14-1': 'Therefore, the RGE effects should be taken into account when the flavor-symmetry model is confronted with the low-energy experimental data [CITATION].', '1707.05535-2-14-2': 'During the evolution process, the significant difference between the Yukawa coupling of [MATH] and that of [MATH] may provide a unique source for symmetry breaking.', '1707.05535-2-14-3': 'This section is just devoted to such a specific breaking of the [MATH]-[MATH] reflection symmetry.', '1707.05535-2-14-4': 'At the one-loop level, the energy dependence of [MATH] is described by [CITATION] [EQUATION] where [MATH] with [MATH] denoting the renormalization scale, [MATH] and [MATH] read [EQUATION]', '1707.05535-2-14-5': 'In the basis of [MATH] being diagonal, the Yukawa coupling matrix for charged leptons is given by [MATH].', '1707.05535-2-14-6': 'Due to [MATH], it is reasonable to neglect the contributions of [MATH] and [MATH].', '1707.05535-2-14-7': 'In Eq. ([REF]), the [MATH]-term is flavor universal and only contributes an overall rescaling factor [MATH] for the neutrino mass matrix, while the other two terms are able to modify its structure.', '1707.05535-2-14-8': 'Given an [MATH] of the form given by Eq. ([REF]) at [MATH], integration of Eq. ([REF]) enables us to derive the RGE-corrected neutrino mass matrix at [MATH] [CITATION] [EQUATION] with [MATH] and [EQUATION]', '1707.05535-2-14-9': 'Obviously, [MATH] measures the strength of symmetry breaking.', '1707.05535-2-14-10': 'Owing to the smallness of [MATH] which gives [MATH], the RGE effect is negligible in the SM.', '1707.05535-2-14-11': 'But in the MSSM, [MATH] (with [MATH] GeV being the Higgs VEV) can be greatly enhanced by a large [MATH].', '1707.05535-2-14-12': 'To be explicit, the value of [MATH] depends on [MATH] in a way as [EQUATION] if we take [MATH] GeV as an example.', '1707.05535-2-15-0': 'Following the same approach as in the previous section, one can obtain the following relations connecting the mixing-parameter deviations with [MATH] [EQUATION] with [MATH] (or [MATH]) in the NH (or IH) case.', '1707.05535-2-15-1': 'Solving these equations gives [EQUATION] in the NH case, or [EQUATION] in the IH case.', '1707.05535-2-15-2': 'We subsequently show the dependence of these mixing-parameter deviations on the value of [MATH] (which varies from 10 to 50) in Fig. 3.', '1707.05535-2-15-3': 'One can see that the mixing parameters are pretty stable against the RGE corrections.', '1707.05535-2-15-4': 'Even for [MATH], one merely has [MATH], [MATH] and [MATH] in the NH (or IH) case.', '1707.05535-2-16-0': '# Leptogenesis and the symmetry breaking', '1707.05535-2-17-0': 'Besides accommodating the smallness of neutrino masses, the seesaw mechanism can also explain the observed baryon-antibaryon asymmetry of the Universe via the leptogenesis mechanism [CITATION]: The CP-violating, lepton-number-violating and out-of-equilibrium decays of [MATH] may generate a lepton-antilepton asymmetry which is eventually converted to the baryon-antibaryon asymmetry through the sphaleron process [CITATION].', '1707.05535-2-17-1': 'The essential CP violation is provided by the complex Yukawa couplings [MATH] which couple [MATH] with the left-handed neutrino fields, while the lepton-number violation originates from the Majorana mass terms of [MATH].', '1707.05535-2-17-2': 'And the departure from thermal equilibrium can occur if [MATH] decay in a rate smaller than the expansion rate of the Universe when the temperature [MATH] drops to the mass scale of [MATH].', '1707.05535-2-17-3': 'The produced amount of lepton-antilepton asymmetry is crucially dependent on the CP-violating asymmetries between the decays of [MATH] and their CP conjugate processes.', '1707.05535-2-17-4': 'In the minimal seesaw under study, we assume [MATH] to have a strong mass hierarchy [MATH], in which case only the decay of [MATH] is relevant for leptogenesis.', '1707.05535-2-17-5': 'The flavored CP-violating asymmetries [MATH] are given by [CITATION] [EQUATION] for [MATH].', '1707.05535-2-17-6': 'A [MATH] corresponding to the [MATH] in Eq. ([REF]) immediately yields [CITATION] [EQUATION] rendering the total CP-violating asymmetry [MATH] vanishing.', '1707.05535-2-17-7': 'However, a successful leptogenesis is possible when the [MATH]-[MATH] reflection symmetry is broken [CITATION] or (and) the flavor effects become relevant [CITATION].', '1707.05535-2-18-0': 'For the situation of [MATH] GeV, the flavor effects are negligible.', '1707.05535-2-18-1': 'In order to achieve [MATH], one has to break the [MATH]-[MATH] reflection symmetry.', '1707.05535-2-18-2': 'A [MATH] corresponding to the [MATH] in Eq. ([REF]) (with [MATH] as assumed before) gives [EQUATION] which means that [MATH] is proportional to [MATH] and a linear function of [MATH], [MATH] and [MATH].', '1707.05535-2-18-3': 'The final baryon-to-entropy ratio can be written as [CITATION] [EQUATION]', '1707.05535-2-18-4': 'Here [MATH] is the efficiency factor of converting the lepton-antilepton asymmetry to the baryon-antibaryon asymmetry, whereas [MATH] is the effective number of relativistic degrees of freedom at [MATH] in the SM.', '1707.05535-2-18-5': 'In particular, [MATH] is the washout factor and can be parametrized as [CITATION] [EQUATION] with [MATH].', '1707.05535-2-18-6': 'In the present epoch (for [MATH]), the baryon-to-photon ratio is given by [EQUATION] which has an observed value of [MATH] [CITATION].', '1707.05535-2-18-7': 'To figure out what kind of [MATH] and symmetry-breaking parameters may give rise to the observed baryon-antibaryon asymmetry, in Fig. 4 we present the [MATH] (as functions of [MATH]) arising from some example values of them.', '1707.05535-2-18-8': 'The results show that a combination of [MATH] GeV, [MATH] GeV or [MATH] GeV can give successful leptogenesis in the NH (or IH) case.', '1707.05535-2-18-9': 'Clearly, it is much easier to gain the observed [MATH] in the NH case than in the IH case.', '1707.05535-2-18-10': 'In particular, a [MATH] (or smaller if [MATH] takes a value larger than [MATH] GeV) is sufficient for generating the observed [MATH] in the NH case.', '1707.05535-2-19-0': 'If [MATH] turns out to be smaller than [MATH] GeV, then the Yukawa interactions of charged leptons will enter in thermal equilibrium, making different lepton flavors distinguishable.', '1707.05535-2-19-1': 'In such a case, both the CP-violating asymmetries and washout factors associated with different flavors should be treated separately.', '1707.05535-2-19-2': 'Accordingly, the final baryon-to-entropy ratio can be rewritten as [CITATION] [EQUATION] where the parameter [MATH] is defined as [EQUATION] with [EQUATION]', '1707.05535-2-19-3': 'For illustration, in Fig. 5 we show the [MATH] (as functions of [MATH]) arising from some example values of [MATH] and symmetry-breaking parameters for the situation of [MATH].', '1707.05535-2-19-4': 'For comparison, the contributions from pure flavor effects (without symmetry-breaking effects) are also shown.', '1707.05535-2-19-5': 'It is easy to see that the flavor effects are more significant than the symmetry-breaking effects in the NH case, while the contrary is the case in the IH case.', '1707.05535-2-19-6': 'In the NH case the flavor effects themselves are competent for generating the observed [MATH].', '1707.05535-2-19-7': 'But in the IH case the symmetry-breaking effects have to be invoked and the symmetry-breaking parameters should take some values at least [MATH].', '1707.05535-2-20-0': 'In order to see whether there exists a choice of basic parameters which can lead to sizable [MATH], [MATH] and successful leptogenesis, in Figs. 6-7 we present the resulting [MATH] and [MATH] from the parameter choices shown in Figs. 4-5 that can lead to successful leptogenesis.', '1707.05535-2-20-1': 'From Fig. 6 one finds that for the situation of [MATH] GeV a sizable [MATH] can be generated in association with a realistic [MATH] from [MATH] in the NH case (see the sub-figure labelled as (a)), [MATH] in the IH case (see the sub-figure labelled as (d)) or [MATH] in the IH case (see the sub-figure labelled as (f)), but a sizable [MATH] has no chance to arise along with a successful leptogenesis.', '1707.05535-2-20-2': 'The results in Fig. 7 tell us that for the situation of [MATH] all the parameter choices except for that shown in the sub-figure labelled as (b) may give rise to a sizable [MATH], while only the parameter choice shown in the sub-figure labelled as (c) is capable of producing a sizable [MATH].', '1707.05535-2-21-0': '# Summary', '1707.05535-2-22-0': 'The [MATH]-[MATH] reflection symmetry is powerful in shaping the neutrino mixing: it leads to the interesting results [MATH] and [MATH] (which are close to the current experimental results) as well as trivial Majorana phases.', '1707.05535-2-22-1': 'On the other hand, the minimal seesaw has strong predictive power in settling the neutrino mass spectrum: it enforces the condition of [MATH] or [MATH].', '1707.05535-2-22-2': 'In this paper, an attempt of implementing the [MATH]-[MATH] reflection symmetry in the minimal seesaw has been made.', '1707.05535-2-22-3': 'Such a marriage results in [MATH] (or 0) and thus [MATH] (or 4.86) eV in the NH (or IH) case.', '1707.05535-2-22-4': 'Motivated by the preliminary experimental hints towards [MATH] and [MATH], we particularly explore the possible symmetry breakings and their phenomenological consequences.', '1707.05535-2-22-5': 'Some parameters characterizing the breaking of [MATH]-[MATH] refection symmetry are first defined and their implications for the mixing-parameter deviations then analyzed.', '1707.05535-2-22-6': 'It is found that [MATH] is difficult to produce considerable mixing-parameter deviations, while [MATH] is relatively easy.', '1707.05535-2-22-7': 'Inversely, a considerable [MATH] can be attributed to [MATH], while a considerable [MATH] may arise from any symmetry-breaking parameter.', '1707.05535-2-23-0': 'As a unique example, the symmetry breaking triggered by the RGE effects is studied.', '1707.05535-2-23-1': 'It turns out that the mixing parameters are rather stable against the RGE corrections.', '1707.05535-2-23-2': 'Even for [MATH] in the MSSM, the RGE-induced mixing-parameter deviations are only of [MATH].', '1707.05535-2-23-3': 'Finally, the operation of leptogenesis in the framework under study is discussed.', '1707.05535-2-23-4': 'For the situation of [MATH] GeV where one has a vanishing [MATH], the [MATH]-[MATH] reflection symmetry must be broken to make the leptogenesis mechanism work.', '1707.05535-2-23-5': 'For illustration, we give some example values of [MATH] and symmetry-breaking parameters that may give rise to the observed value of [MATH].', '1707.05535-2-23-6': 'For the situation of [MATH] GeV, it is possible that the flavor effects themselves (without symmetry-breaking effects) are sufficient for producing the observed [MATH] in the NH case.', '1707.05535-2-23-7': 'But in the IH case, one has to turn to the symmetry-breaking effects for help.', '1707.05535-2-23-8': 'For both situations, it is easier to achieve a realistic value of [MATH] in the NH case than in the IH case.', '1707.05535-2-23-9': 'Last but not least, we point out that the mixing-parameter deviations can be connected to the implementation of leptogenesis, considering that they may originate from the same symmetry breaking.', '1707.05535-2-23-10': 'The results show that a sizable [MATH] can be generated in association with the observed [MATH] in many cases, but a sizable [MATH] can only arise along with a successful leptogenesis from [MATH] in the NH case for the situation of [MATH].'}

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['1707.05535-1-8-2', '1707.05535-2-8-2'], ['1707.05535-1-8-3', '1707.05535-2-8-3'], ['1707.05535-1-8-8', '1707.05535-2-8-7'], ['1707.05535-1-8-10', '1707.05535-2-8-9'], ['1707.05535-1-8-12', '1707.05535-2-8-11'], ['1707.05535-1-8-14', '1707.05535-2-8-13'], ['1707.05535-1-8-15', '1707.05535-2-8-14'], ['1707.05535-1-11-1', '1707.05535-2-11-1'], ['1707.05535-1-15-2', '1707.05535-2-15-2'], ['1707.05535-1-15-3', '1707.05535-2-15-3'], ['1707.05535-1-15-4', '1707.05535-2-15-4'], ['1707.05535-1-0-0', '1707.05535-2-0-0'], ['1707.05535-1-0-1', '1707.05535-2-0-1'], ['1707.05535-1-14-2', '1707.05535-2-14-2'], ['1707.05535-1-14-3', '1707.05535-2-14-3'], ['1707.05535-1-14-4', '1707.05535-2-14-4'], ['1707.05535-1-14-6', '1707.05535-2-14-6'], ['1707.05535-1-14-7', '1707.05535-2-14-7'], ['1707.05535-1-14-8', '1707.05535-2-14-8'], ['1707.05535-1-14-9', '1707.05535-2-14-9'], ['1707.05535-1-14-11', '1707.05535-2-14-11'], ['1707.05535-1-22-2', '1707.05535-2-23-2'], ['1707.05535-1-22-3', '1707.05535-2-23-3'], ['1707.05535-1-22-5', '1707.05535-2-23-5'], ['1707.05535-1-22-7', '1707.05535-2-23-7'], ['1707.05535-1-21-0', '1707.05535-2-22-0'], ['1707.05535-1-21-1', '1707.05535-2-22-1'], ['1707.05535-1-21-3', '1707.05535-2-22-3'], ['1707.05535-1-21-4', '1707.05535-2-22-4'], ['1707.05535-1-21-5', '1707.05535-2-22-5'], ['1707.05535-1-21-7', '1707.05535-2-22-7'], ['1707.05535-1-3-0', '1707.05535-2-3-0'], ['1707.05535-1-3-1', '1707.05535-2-3-1'], ['1707.05535-1-3-2', '1707.05535-2-3-2'], ['1707.05535-1-3-7', '1707.05535-2-3-7'], ['1707.05535-1-19-0', '1707.05535-2-19-0'], ['1707.05535-1-19-1', '1707.05535-2-19-1'], ['1707.05535-1-19-5', '1707.05535-2-19-5'], ['1707.05535-1-19-6', '1707.05535-2-19-6'], ['1707.05535-1-7-3', '1707.05535-2-7-3'], ['1707.05535-1-7-4', '1707.05535-2-7-4'], ['1707.05535-1-7-6', '1707.05535-2-7-6'], ['1707.05535-1-4-0', '1707.05535-2-4-0'], ['1707.05535-1-4-1', '1707.05535-2-4-1'], ['1707.05535-1-4-2', '1707.05535-2-4-2'], ['1707.05535-1-4-3', 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[]

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[]

[]

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1707.05535

1212.5079

{'1212.5079-1-0-0': 'The chiral partner structure of heavy baryons is studied using the bound state approach by binding the heavy-light mesons to nucleon as soliton in an effective Lagrangian for the pseudoscalar and vector mesons based on the hidden local symmetry.', '1212.5079-1-0-1': 'In the heavy-light meson sector, we regard the [MATH]-doublet and [MATH]-doublet as chiral partners and couple them to light meson with minimal derivative.', '1212.5079-1-0-2': 'We find that the chiral partner of [MATH] is the [MATH] heavy quark doublet with the mass of about [MATH]GeV but not ([MATH], [MATH]) which might be interpreted as an orbital excitation of the [MATH].', '1212.5079-1-0-3': 'The same model is applied to the bottom sector, and the chiral partner of [MATH] is shown to have the mass of about [MATH]GeV.', '1212.5079-1-0-4': 'The chiral partner structures for the isospin vector heavy baryons are also discussed.', '1212.5079-1-0-5': 'For the pentaquark states, we find that the masses of the pentaquark state made of ground state heavy-light meson and its chiral partner are similar, and both of them are below the [MATH] threshold, which therefore cannot be ruled out by the present data.', '1212.5079-1-1-0': '# Introduction', '1212.5079-1-2-0': 'Chiral symmetry plays an important role in hadron physics.', '1212.5079-1-2-1': 'When we set [MATH] flavor light quarks to be massless, QCD has an [MATH] chiral symmetry at Lagrangian level.', '1212.5079-1-2-2': 'The chiral symmetry is not preserved by QCD vacuum but broken dynamically to its vector component, i.e., [MATH].', '1212.5079-1-2-3': 'This dynamical chiral symmetry breaking splits the degeneracy of the chiral parters, which are supposed to be degenerate when the chiral symmetry is restored.', '1212.5079-1-2-4': 'So that the study of the chiral partner structure of hadrons can help us to reveal the magnitude of the chiral symmetry breaking, i.e., the order parameter.', '1212.5079-1-3-0': 'In the light meson sector, the chiral partner structure might be complicated since two of light quarks are involved (see e.g. Refs. [CITATION]).', '1212.5079-1-3-1': 'On the other hand, studying the chiral partner structure in the heavy-light meson sector would be easier since they include only one light quark, the dynamics of which is controlled by the chiral symmetry.', '1212.5079-1-3-2': 'In addition to the chiral symmetry, the dynamics of the heavy-light mesons is also controlled by spin-flavor symmetry due to their heavy quark constituent [CITATION].', '1212.5079-1-3-3': 'Based on this heavy quark symmetry, the ground states form a doublet with spin-parity quantum numbers [MATH], and the first excited states belong to [MATH] doublet.', '1212.5079-1-3-4': 'In Ref. [CITATION], it was proposed that these two doublets are chiral partner to each other in the QCD-like models.', '1212.5079-1-3-5': 'As a signature of the chiral partner structure, the mass splitting of them is induced by the dynamical breaking of the chiral symmetry, so that the mass difference is about the constituent quark mass.', '1212.5079-1-3-6': 'This was confirmed by the spectrum of the relevant particles, [MATH]MeV is at the same order of [MATH]MeV (see e.g. Refs. [CITATION]).', '1212.5079-1-4-0': 'In Ref. [CITATION], the analysis on the chiral partner structure of heavy baryons was made based on the bound state picture [CITATION] together with the heavy quark symmetry in which the heavy baryon is introduced [CITATION] as the heavy mesons bound with the nucleon as the soliton.', '1212.5079-1-4-1': 'In the analysis, the excited heavy baryon [MATH] with [MATH] is regarded as the chiral partner to the ground state baryon [MATH]).', '1212.5079-1-5-0': 'In this paper, we revisit the chiral partner structure of the heavy baryons in the bound state approach based on our recent progress in the soliton property [CITATION] and the effective Lagrangian for the heavy-light meson chiral partner structure [CITATION].', '1212.5079-1-5-1': 'In the light meson sector, we considered all the [MATH] terms of hidden local symmetry (HLS), all the [MATH], [MATH] and homogeneous Wess-Zumino (hWZ) terms [CITATION].', '1212.5079-1-5-2': 'In the heavy and light meson interaction sector, we start with the interaction Lagrangian for heavy-light meson and light mesons analyzed in Ref. [CITATION], where the chiral partner is introduced in the framework of a linear sigma model.', '1212.5079-1-5-3': 'We integrate out the scalar mesons and integrate in the vector mesons to construct an effective Lagrangian for heavy mesons interacting with the pseudoscalar mesons and vector mesons based on the hidden local symmety (HLS) [CITATION] (See, e.g. Refs. [CITATION] for alternative approaches.)', '1212.5079-1-5-4': ', and the heavy quark symmetry.', '1212.5079-1-5-5': 'Then, we consider that the static soliton couples to the heavy-light meson and study their spectrum.', '1212.5079-1-5-6': 'After the derivation of the heavy baryon spectrum in the static case, we consider the collective coordinate quantization to make states definite quantum numbers.', '1212.5079-1-5-7': 'Our explicit calculation shows that, in the heavy quark limit and large [MATH] limit, up to the [MATH] terms of HLS, the chiral partner of [MATH] is predicted to be the [MATH] heavy quark doublet with the mass of about [MATH]GeV but not ([MATH], [MATH]) listed in the PDG tabel [CITATION], which might be interpreted as an orbital excitation of the [MATH].', '1212.5079-1-5-8': 'Extending our approach to bottom sector we predicted the chiral partner structure of bottom baryons.', '1212.5079-1-5-9': 'We finally studied the pentaquark spectrum using our framework and found that the masses of the pentaquark states made of ground state heavy-light meson and its chiral partner are similar, and both of them are below the [MATH] threshold, which therefore cannot be ruled out by the present data [CITATION].', '1212.5079-1-6-0': 'This paper is organized as follows: In Sec. [REF], the chiral partner structure of heavy baryons is studied.', '1212.5079-1-6-1': 'We derive the analytic forms of the heavy baryons masses.', '1212.5079-1-6-2': 'The heavy baryon spectrum with chiral partner structure is estimated in Sec. [REF].', '1212.5079-1-6-3': 'And the pentaquark states spectrum is estimated in Sec. [REF].', '1212.5079-1-6-4': 'The last section is for a summary and discussions.', '1212.5079-1-6-5': 'Some useful explicit derivations are given in Appendix.', '1212.5079-1-7-0': '# Heavy Baryons in the Effective Lagrangian for Heavy-light Mesons with Chiral Doubling', '1212.5079-1-8-0': '## Effective Lagrangian for heavy-light mesons with chiral doubling', '1212.5079-1-9-0': 'Here, we construct the effective Lagrangian describing the interaction between the heavy-light mesons and the light mesons.', '1212.5079-1-9-1': 'With respect to the chiral transformation property of the light quarks in the heavy-light mesons, the heavy-light meson field can be decomposed into a right-handed component [MATH] and left-handed one [MATH] [CITATION].', '1212.5079-1-9-2': 'They, under chiral [MATH] symmetry, transform as [EQUATION]', '1212.5079-1-9-3': 'In Ref. [CITATION] these fields are used to construct a Lagrangian where the chiral symmetry are realized linearly.', '1212.5079-1-9-4': 'In the present paper for the study of the heavy baryon spectrum in the bound state approach, we adopt the non-linear realization of the chiral symmetry.', '1212.5079-1-9-5': 'Then, by replacing [MATH] in Eq. (29) of Ref. [CITATION] with [MATH] where [MATH], we obtain [EQUATION] where [MATH] and [MATH] are parameters.', '1212.5079-1-9-6': 'In the present work, we include the vector mesons using the HLS [CITATION] by introducing the matrix valued variables [MATH] and [MATH] as [MATH].', '1212.5079-1-9-7': 'Then, similarly to Ref. [CITATION] we convert the heavy meson fields as [EQUATION] which, under the full symmetry transformation [MATH], transform as [EQUATION]', '1212.5079-1-9-8': 'Associated with the field redefinitions in Eq. ([REF]), it is convenient to use the following quantities for the [MATH] fields: [EQUATION] where the covariant derivative [MATH] is given by [MATH] with [MATH] being the gauge field of the HLS.', '1212.5079-1-9-9': 'By using these quantities, the above Lagrangian is extended to include the vector mesons as [EQUATION] where [MATH] is defined as [MATH] with [MATH] being a real parameter measuring the magnitude of the violation of the vector meson dominance.', '1212.5079-1-10-0': 'To study the chiral partner structure of the heavy baryons, we rewrite the Lagrangian ([REF]) in terms of the heavy-light meson doublets [MATH] and [MATH] with quantum numbers [MATH] and [MATH], specifically, make the substitution [EQUATION]', '1212.5079-1-10-1': 'In terms of the physical states, the [MATH] and [MATH] doublets can be explicitly expressed as [EQUATION]', '1212.5079-1-10-2': 'Substituting Eq. ([REF]) into Eq. ([REF]), we obtain [EQUATION]', '1212.5079-1-10-3': 'This expression explicitly shows that the [MATH] term splits the spectrum of [MATH] and [MATH] doublets while the [MATH] term shifts the masses of these two doublets toward the same direction.', '1212.5079-1-10-4': 'In the present paper, we use the physical masses of heavy mesons as inputs to calculate the heavy baryon masses, so that we drop the [MATH] term and the [MATH] term in the following calculation of the masses of heavy baryons.', '1212.5079-1-10-5': 'Note that, due to the chiral partner structure adopted here, the magnitudes of the coupling constants in the last two terms of Eq. ([REF]) are the same therefore the chiral partner spectrum is predictable.', '1212.5079-1-11-0': '## Heavy baryon masses from the bound state approach', '1212.5079-1-12-0': 'In this subsection we derive the heavy baryon masses based on the bound state approach [CITATION].', '1212.5079-1-13-0': 'To make the mesonic theory to be a baryonic one, we follow the standard procedure to take the Hedgehog ans[MATH]tze for a classical soliton [CITATION] [EQUATION] with [MATH] as the Pauli matrices and the subscript [MATH] standing for the classical solution.', '1212.5079-1-13-1': 'From the Hedgehog ans[MATH]tze ([REF]) one can easily see that [MATH] transforms under separate spatial rotation and isospin rotation but is invariant under the combined rotation, i.e., the Hedgehog profile correlates the angular momentum and the isospin.', '1212.5079-1-13-2': 'For the vector mesons, their profile functions can be parameterized as [CITATION] [EQUATION]', '1212.5079-1-13-3': 'From the hedgehog ans[MATH]tze ([REF]) and profile functions ([REF]) we express the quantities [MATH] and [MATH] as [EQUATION] where [EQUATION]', '1212.5079-1-13-4': 'In the rest frame of the heavy-light meson, i.e., [MATH], the [MATH] doublet has nonvanishing elements only in the upper-right [MATH] subblock while the [MATH] doublet has nonvanishing elements only in the upper-left [MATH] subblock.', '1212.5079-1-13-5': 'The matrix forms of [MATH] and [MATH] doublets become [EQUATION]', '1212.5079-1-13-6': 'Then, the Lagrangian ([REF]) is reduced to [EQUATION]', '1212.5079-1-13-7': 'Since the Hedgehog ans[MATH]tz for the Skyrme soliton correlates the angular momentum and isospin, the bound states should be invariant under the "grand spin" rotation with the operator defined as [EQUATION] where [MATH] and [MATH] are the ordinary orbital angular momentum between the soliton and heavy-light meson, heavy meson spin, and the heavy meson isospin operators.', '1212.5079-1-13-8': 'Taking into account that the heavy quark spin is conserved in the heavy quark limit, one simply defines the "light quark grand spin" operator [EQUATION] with [MATH] as the spin operator of the light degree of freedom of the heavy-light meson, and in both [MATH] and [MATH] doublets, the eigenvalue of the operator [MATH] is [MATH].', '1212.5079-1-13-9': 'So that the eigenmodes of the heavy baryons can be classified by the third component of heavy quark spin [MATH] and the light quark grand spin [MATH] and the parity [MATH].', '1212.5079-1-14-0': 'Taking into account the isospin, light quark spin and heavy quark spin indices that the heavy-light meson has, one can write the static wave functions of the heavy-light mesons as [CITATION] [EQUATION] where [MATH] and [MATH] denote the indices for the isospin of the heavy-light meson, the spin of the light degree of freedom, and the heavy quark spin, respectively.', '1212.5079-1-14-1': '[MATH] is the eigenvalue of the operator [EQUATION] with [MATH] as its third component.', '1212.5079-1-14-2': '[MATH] is factorized out due to the conservation of the heavy quark spin.', '1212.5079-1-14-3': '[MATH] is a radial function which is strongly peaked at the origin and normalized as [MATH] [CITATION].', '1212.5079-1-14-4': 'This implies that the relevant matrix elements are independent of the quantum number [MATH] [CITATION].', '1212.5079-1-14-5': 'The generalized "angular" wave function [MATH] can be expanded as [CITATION] [EQUATION] where [MATH] stands for the standard spherical harmonic representing orbital angular momentum [MATH] while [MATH] denotes the ordinary Clebsch-Gordan coefficients.', '1212.5079-1-14-6': '[MATH] represents a wave function in which the "light spin" and "light isospin" referring to the "light cloud" component of the heavy meson are added vectorially to give [MATH] with eigenvalues [MATH].', '1212.5079-1-14-7': 'Note that, in the present analysis [MATH] is a good quantum number since the relevant matrix elements are independent of the quantum number [MATH].', '1212.5079-1-14-8': 'Furtheremore, both the quantum numbers for the "light spin" and "light isospin" are given by [MATH], so that the possible values of [MATH] are either [MATH] or [MATH].', '1212.5079-1-14-9': 'The normalization of the eigenstate [MATH] is [EQUATION] where [MATH] is the solid angle integration.', '1212.5079-1-15-0': 'From Lagrangian ([REF]), we parameterize the potential as [EQUATION]', '1212.5079-1-15-1': 'By substituting the ans[MATH]tz ([REF]), [MATH] and [MATH] are obtained as [EQUATION]', '1212.5079-1-15-2': 'Next, we make a quantization by a time dependent [MATH] rotation of the fields in the HLS Lagrangian in the light-quark sector as [EQUATION] where [MATH] is a time dependent unitary matrix satisfying [MATH].', '1212.5079-1-15-3': 'Accordingly, the heavy-light meson fields are rotated as [EQUATION]', '1212.5079-1-15-4': 'This collective rotation gives an additional contribution to the Lagrangian [EQUATION] where the angular velocity [MATH] corresponding to the collective coordinate rotation is defined as [EQUATION] [MATH] is the moment of the inertia of the soliton configuration.', '1212.5079-1-15-5': 'By using [MATH], the light baryon masses are expressed as [EQUATION] where [MATH] is the soliton mass and [MATH] is the spin of light baryon.', '1212.5079-1-15-6': 'Using the nucleon mass [MATH] and the Delta mass [MATH] as inputs, [MATH] and [MATH] are given as [EQUATION]', '1212.5079-1-15-7': 'From Eq. ([REF]) one obtains [CITATION] [EQUATION] where the coefficient [MATH] is calculated as [EQUATION] with [MATH] being the spin quantum number of the light degree of freedom of heavy-light meson.', '1212.5079-1-15-8': 'For convenience we show the derivation in Appendix.', '1212.5079-1-15-9': '[REF].', '1212.5079-1-15-10': 'The Hamiltonian of the collective rotated system is obtained by the standard Legendre transformation as [EQUATION] where [MATH] is the canonical momentum conjugating to the collective variable [MATH]: [EQUATION]', '1212.5079-1-15-11': 'The first term [MATH] is the isospin operator for the light baryon sector while the second term [MATH] is the isospin operator for the heavy-light mesons interacting with the light baryon, so that [MATH] is identical to the isospin operator for the heavy baryon, [MATH]: [EQUATION]', '1212.5079-1-15-12': 'After the collective coordinate rotation, the total spin of the light degrees of freedom in the heavy baryon is defined as [EQUATION]', '1212.5079-1-15-13': 'By including the heavy quark spin, the spin operator for the heavy baryon is expressed as [MATH] with eigenvalues [MATH] (in the case of [MATH], only [MATH] exists).', '1212.5079-1-15-14': 'Then, we can express the collectively rotated Hamiltonian as [EQUATION]', '1212.5079-1-15-15': 'Gathering all the above contributions, we finally obtain the heavy baryon mass as [EQUATION] where [MATH]) is the binding energy corresponding to the heavy meson [MATH]).', '1212.5079-1-15-16': '[MATH] are the weight-averaged heavy meson masses with [MATH] and [MATH].', '1212.5079-1-15-17': 'Note that each combination of [MATH] generates a pair of degenerate states with [MATH].', '1212.5079-1-16-0': '# Chiral Doubling of Heavy Baryons', '1212.5079-1-17-0': 'In this section, we study the chiral doubling structure using the formulas obtained in the previous section.', '1212.5079-1-17-1': 'In the following analysis, we restrict ourselves to the case with [MATH], and use the following values of heavy meson masses as inputs: [EQUATION] which lead to [EQUATION]', '1212.5079-1-17-2': 'Furthermore, to simulate the profile functions [MATH] and [MATH] which are necessary to evaluate the binding energy [MATH] and [MATH] expressed in Eq. ([REF]) we use the following inputs: [EQUATION] which yields soliton mass [MATH] GeV and the inverse of the moment of inertia [MATH] GeV.', '1212.5079-1-17-3': 'Then, using the relevant expressions from HLS up to [MATH] including the hWZ terms given in Refs. [CITATION], by taking [MATH] and [MATH] as free parameters to fit the inputs ([REF]) we obtain [MATH]MeV and [MATH]MeV and the values of the profile functions at origin as [EQUATION]', '1212.5079-1-17-4': 'Moreover, we take the parameter [MATH] [CITATION] and fix the universal coupling constant [MATH] in HLS through [EQUATION]', '1212.5079-1-17-5': 'Let us first consider the binding energy in order to determine which channel can form bound state and calculate the spectra of the chiral partners.', '1212.5079-1-17-6': 'Using Eq. ([REF]) we obtain the binding energy between the heavy-light mesons in [MATH] doublet and soliton as [EQUATION]', '1212.5079-1-17-7': 'The value of [MATH] is determined through the [MATH] decay as [MATH] [CITATION].', '1212.5079-1-17-8': 'It does not seem possible to determine the [MATH] coupling constant [MATH] from the available experimental data for heavy meson decay.', '1212.5079-1-17-9': 'In the case of the vector meson dominance we have [MATH], therefore it is reasonable to regard [MATH].', '1212.5079-1-17-10': 'Then we conclude that the [MATH] channel gives a bound state when [MATH].', '1212.5079-1-17-11': 'This bound state can be naturally identified with [MATH] from the quantum number, so that we assume [MATH] in the following analysis.', '1212.5079-1-17-12': 'Since the collective energy is zero for the [MATH] state, we use the experimental value of the mass of [MATH] as an input to determine the value of [MATH].', '1212.5079-1-17-13': 'Using [MATH]GeV, we obtain [MATH].', '1212.5079-1-18-0': 'Next, we consider the bound states made from [MATH] doublet.', '1212.5079-1-18-1': 'The binding energy is expressed as [EQUATION]', '1212.5079-1-18-2': 'As we can see easily, [MATH] for [MATH]), so that there is no bound state in the [MATH] channel.', '1212.5079-1-18-3': 'For the [MATH] channel, using [MATH] determined above, we obtain [MATH]GeV, which implies that the [MATH] channel is actually bound.', '1212.5079-1-19-0': 'From Eq. ([REF]) the total spin of the light degrees of freedom becomes [MATH], so that the resultant heavy baryons form a heavy-quark doublet consisting [MATH] and [MATH].', '1212.5079-1-19-1': 'Combined with the collective energy, the mass of the bound state is expressed as [EQUATION] which leads to [EQUATION]', '1212.5079-1-19-2': 'This value is much larger than the experimental values for the masses of negative parity baryons; [MATH]GeV and [MATH]GeV.', '1212.5079-1-19-3': 'So, we conclude that the negative parity baryons found by experiments [MATH] and [MATH] are not the chiral partner to the ground state baryon [MATH].', '1212.5079-1-19-4': 'In the present bound state approach, they should be regarded as [MATH] state made from the [MATH] doublet and nucleon.', '1212.5079-1-19-5': 'Then, we expect to have a doublet for the chiral partner around [MATH]GeV region.', '1212.5079-1-20-0': 'We next study the [MATH] baryons.', '1212.5079-1-20-1': 'In the positive parity baryon sector, [MATH] channel is bound, so that the total spin of the light degrees of freedom in the heavy baryon becomes one.', '1212.5079-1-20-2': 'As a result, the spin of [MATH] baryons with positive parity is either [MATH] or [MATH].', '1212.5079-1-20-3': 'In the mass formula in Eq. ([REF]), only [MATH] changes its value depending on the isospin of baryons.', '1212.5079-1-20-4': 'Since [MATH] for [MATH], the mass difference between the [MATH] and the [MATH] in the positive negative parity is obtained as [EQUATION]', '1212.5079-1-20-5': 'In the negative parity baryon sector, [MATH] channel is bound, then the eigenvalue for [MATH] is either [MATH], [MATH] or [MATH].', '1212.5079-1-20-6': 'We summarize our predicted results for the charm baryon spectrum in Table. [REF].', '1212.5079-1-21-0': 'We next study the mass spectrum of bottom baryon by substituting the bottom meson masses into the charm meson masses in Eq. ([REF]).', '1212.5079-1-21-1': 'In the bottom meson spectrum, the masses of the ground states [MATH] and [MATH] are well measured but masses of the mesons in the [MATH] doublets are not well established.', '1212.5079-1-21-2': 'Here, we naively estimate them using [MATH]MeV and [MATH]MeV which lead to [MATH]MeV and [MATH]MeV.', '1212.5079-1-21-3': 'Our numerical results for the masses of the heavy baryons including bottom quark with the corresponding quantum numbers are given in Table [REF].', '1212.5079-1-22-0': '# Pentaquarks with heavy quark', '1212.5079-1-23-0': 'We next consider the pentaquark channel.', '1212.5079-1-23-1': 'Although the existence of these kind of states still needs the experimental confirmation, theoretical study of them is meaningful.', '1212.5079-1-23-2': 'For a pentaquark state, the large component of the anti-heavy quark can be projected out with the projection operator [MATH].', '1212.5079-1-23-3': 'So that, in case the heavy meson is at rest, the [MATH] doublet has nonvanishing elements only in the lower-left [MATH] subblock while the [MATH] doublet has nonvanishing elements only in the lower-right [MATH] subblock, i.e., [EQUATION]', '1212.5079-1-23-4': 'Then, substituting [MATH] with [MATH], following the above derivation, one can see that both the binding energies given by Eqs. ([REF]) change a sign.', '1212.5079-1-23-5': 'As a result, the binding energies for the pentaquark states made from anti-[MATH] doublet and the pentaquark states made from anti-[MATH] doublet are expressed as [EQUATION]', '1212.5079-1-23-6': 'Therefore, for anti-[MATH] doublet, the [MATH] channel gives the bound states with binding energy [MATH] MeV, while for anti-[MATH] doublet, the [MATH] channel gives the bound states with binding energy [MATH] MeV.', '1212.5079-1-24-0': 'Substituting relevant numerical results, we obtain the spectrum of the pentaquark states.', '1212.5079-1-24-1': 'We list our results in Table. [REF] for pentaquark states consisting anti-charm quark and Table. [REF] for pentaquark states consisting anti-bottom quark.', '1212.5079-1-25-0': 'The results in Table. [REF] show that the lightest charmed pentaquark states made of soliton and heavy-light mesons in the anti-[MATH] doublet have masses about [MATH]GeV and their chiral partner made of the anti-[MATH] doublet have the masses of about [MATH]GeV.', '1212.5079-1-25-1': 'Both of them are below the [MATH] threshold.', '1212.5079-1-25-2': 'The reason that the pentaquark states from anti-[MATH] doublet and anti-[MATH] doublet have the similar masses is that the binding energy of the anti-[MATH] doublet is about [MATH] MeV smaller than that of the anti-[MATH] doublet and collective rotation energy which is about [MATH] MeV does not contribute to the latter.', '1212.5079-1-25-3': 'With respect the status of the pentaquark search performed, these states cannot be ruled out and, since they cannot decay via strong process, their total widths should be narrow.', '1212.5079-1-26-0': '# Summary and Discussions', '1212.5079-1-27-0': 'We studied the chiral partner structure of heavy baryons in the bound state approach including the vector meson exchanging effects through the hidden local symmetry.', '1212.5079-1-27-1': 'We showed that, in the large [MATH] limit and the heavy quark limit, the ground state heavy baryon made of the ground state heavy-light meson and the nucleon has the chiral partner made of the excited heavy-light meson and the nucleon.', '1212.5079-1-27-2': 'Our explicit calculation shows that the chiral partner of [MATH] is a heavy quark doublet of [MATH] and [MATH].', '1212.5079-1-27-3': 'This contrasts to the perdition made in the pioneering work in Ref. [CITATION], where the chiral partner is the singlet under the heavy quark spin transformation.', '1212.5079-1-27-4': 'Our prediction of the mass is about [MATH]GeV, which indicates that the [MATH] and [MATH] listed in the PDG table [CITATION] should be interpreted as the [MATH] excitation of [MATH].', '1212.5079-1-28-0': 'We also studied the bound states in the pentaquark channel.', '1212.5079-1-28-1': 'We found that the [MATH] channel forms bound states for anti-[MATH] doublet ([MATH], [MATH]), while the [MATH] channel forms bound states for anti-[MATH] doublet ([MATH]).', '1212.5079-1-28-2': 'It is found that the predicted masses of the pentaquark states made of both anti-[MATH] doublet and anti-[MATH] doublet are below the [MATH] threshold which cannot be ruled out by the present data [CITATION].', '1212.5079-1-29-0': 'In the present analysis, we take the infinite heavy soliton and heavy quark limits, so that both the soliton and heavy-light meson are sitting at the origin.', '1212.5079-1-29-1': 'This picture cannot be applied to the bound states with non-zero [MATH].', '1212.5079-1-29-2': 'Since in the present analysis, the chiral partner of heavy [MATH] has the mass of about [MATH]GeV which is a bound state of soliton and heavy-light mesons in [MATH] doublet, one can expect that it has a broad width due to the broad width of constituent [MATH]-wave mesons in [MATH] doublet.', '1212.5079-1-29-3': 'From the numerical results in Table [REF] we conclude that the spectrum of the heavy baryons with bottom quark is consistent with PDG [CITATION] for [MATH] and [MATH].', '1212.5079-1-30-0': 'It should be noted that, in the present analysis, we consider that the chiral partner to the nucleon is itself: The left-handed nucleon is the chiral partner of the right-handed nucleon, and vice versa.', '1212.5079-1-30-1': 'So that the chiral partner to the heavy baryon as the bound state of the [MATH] doublet and the nucleon is the one made of [MATH] doublet and the nucleon.', '1212.5079-1-30-2': 'This implies that the chiral partner structure of the heavy baryons in our approach arises from the chiral partner structure of the constituent heavy-light mesons.', '1212.5079-1-30-3': 'On the other hand, in the mirror scenario for the light baryon [CITATION], the chiral partner to the nucleon is considered as [MATH].', '1212.5079-1-30-4': 'In such a case, the full picture of the chiral partner structure of heavy baryons becomes complicated.', '1212.5079-1-30-5': 'We will not consider this scenario in the present work.', '1212.5079-1-31-0': '# Matrix element of heavy-light meson isospin operator', '1212.5079-1-32-0': 'Using the Wigner-Eckart theorem, we can express the matrix element of heavy-light meson isospin operator [MATH] in terms of the matrix element of the operator [MATH], i.e., [EQUATION]', '1212.5079-1-33-0': 'We are grateful to Y. Oh, B.-Y. Park and M. Rho for valuable comments and critical reading of the manuscript.', '1212.5079-1-33-1': 'This work is supported in part by Grant-in-Aid for Scientific Research on Innovative Areas (No. 2104) "Quest on New Hadrons with Variety of Flavors" from MEXT.', '1212.5079-1-33-2': 'The work of M.H. is supported in part by the Grant-in-Aid for Nagoya University Global COE Program "Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos" from MEXT, the JSPS Grant-in-Aid for Scientific Research (S) [MATH] 22224003.', '1212.5079-1-33-3': 'The work of Y.M. is supported in part by the National Science Foundation of China (NSFC) under grant No. 10905060.'}

{'1212.5079-2-0-0': 'The chiral partner structure of heavy baryons is studied using the bound state approach by binding the heavy-light mesons to the nucleon as the soliton in an effective Lagrangian for the pseudoscalar and vector mesons based on hidden local symmetry.', '1212.5079-2-0-1': 'In the heavy-light meson sector, we regard the [MATH] doublet and [MATH] doublet as chiral partners and couple them to light mesons with a minimal derivative.', '1212.5079-2-0-2': 'We find that the chiral partner of [MATH] is the [MATH] heavy quark doublet with a mass of about [MATH]GeV but not ([MATH], [MATH]), which might be interpreted as an orbital excitation of [MATH].', '1212.5079-2-0-3': 'The same model is applied to the bottom sector, and the chiral partner of [MATH] is shown to have a mass of about [MATH]GeV.', '1212.5079-2-0-4': 'We also discuss the chiral partner structures for the isospin vector heavy baryons.', '1212.5079-2-0-5': 'For the pentaquark states, we find that the masses of the pentaquark state made of a ground state heavy-light meson and its chiral partner are similar, and both of them are below the [MATH] threshold, which therefore cannot be ruled out by the present data.', '1212.5079-2-1-0': '# Introduction', '1212.5079-2-2-0': 'Chiral symmetry plays an important role in hadron physics.', '1212.5079-2-2-1': 'When we set [MATH] flavor light quarks to be massless, quantum chromodynamics (QCD) has an [MATH] chiral symmetry at the Lagrangian level.', '1212.5079-2-2-2': 'The chiral symmetry is not preserved by QCD vacuum but broken dynamically to its vector component, i.e., [MATH].', '1212.5079-2-2-3': 'This dynamical chiral symmetry breaking splits the degeneracy of the chiral parters, which are supposed to be degenerate when the chiral symmetry is restored so that the study of the chiral partner structure of hadrons can help us reveal the magnitude of the chiral symmetry breaking, i.e., the order parameter.', '1212.5079-2-3-0': 'In the light meson sector, the chiral partner structure might be complicated since two light quarks are involved (see, e.g., Refs. [CITATION]).', '1212.5079-2-3-1': 'On the other hand, studying the chiral partner structure in the heavy-light meson sector would be easier since they include only one light quark, the dynamics of which is controlled by the chiral symmetry.', '1212.5079-2-3-2': 'In addition to the chiral symmetry, the dynamics of the heavy-light mesons is also controlled by spin-flavor symmetry due to their heavy quark constituent [CITATION].', '1212.5079-2-3-3': 'Based on this heavy quark symmetry, the ground states form a doublet with spin-parity quantum numbers [MATH], and the first excited states belong to the [MATH] doublet.', '1212.5079-2-3-4': 'In Ref. [CITATION], it was proposed that these two doublets were chiral partners to each other in the QCD-like models.', '1212.5079-2-3-5': 'As a signature of the chiral partner structure, the mass splitting of them is induced by the dynamical breaking of the chiral symmetry so that the mass difference is about the constituent quark mass.', '1212.5079-2-3-6': 'This was confirmed by the spectrum of the relevant particles; [MATH]MeV is at the same order of [MATH]MeV (see, e.g., Refs. [CITATION]).', '1212.5079-2-4-0': 'In Ref. [CITATION], the analysis on the chiral partner structure of heavy baryons was made based on the bound state picture [CITATION] together with the heavy quark symmetry in which the heavy baryon is introduced [CITATION] as the heavy mesons bound with the nucleon as the soliton.', '1212.5079-2-4-1': 'In the analysis, the excited heavy baryon [MATH] with [MATH] is regarded as the chiral partner to the ground state baryon [MATH]).', '1212.5079-2-5-0': 'In this paper, we revisit the chiral partner structure of the heavy baryons in the bound state approach based on our recent progress in the soliton property [CITATION] and the effective Lagrangian for the heavy-light meson chiral partner structure [CITATION].', '1212.5079-2-5-1': 'In the light meson sector, we considered all the [MATH] terms of hidden local symmetry (HLS), all the [MATH], [MATH], and homogeneous Wess-Zumino (hWZ) terms [CITATION].', '1212.5079-2-5-2': 'In the heavy and light meson interaction sector, we start with the interaction Lagrangian for the heavy-light meson and light mesons analyzed in Refs. [CITATION], where the chiral partner is introduced in the framework of a linear sigma model.', '1212.5079-2-5-3': 'We integrate out the scalar mesons and integrate in the vector mesons to construct an effective Lagrangian for heavy mesons interacting with the pseudoscalar mesons and vector mesons based on the HLS [CITATION] (see, e.g., Refs. [CITATION] for alternative approaches), and the heavy quark symmetry.', '1212.5079-2-5-4': 'Then, we consider that the static soliton couples to the heavy-light meson and study their spectrum.', '1212.5079-2-5-5': 'After the derivation of the heavy baryon spectrum in the static case, we consider the collective coordinate quantization to make states definite quantum numbers.', '1212.5079-2-5-6': 'Our explicit calculation shows that, in the heavy quark limit and large [MATH] limit, up to the [MATH] terms of HLS, the chiral partner of [MATH] is predicted to be the [MATH] heavy quark doublet with a mass of about [MATH]GeV but not ([MATH], [MATH]) listed in the PDG table [CITATION], which might be interpreted as an orbital excitation of [MATH].', '1212.5079-2-5-7': 'Extending our approach to the bottom sector we predicted the chiral partner structure of bottom baryons.', '1212.5079-2-5-8': 'We finally studied the pentaquark spectrum using our framework and found that the masses of the pentaquark states made of a ground state heavy-light meson and its chiral partner are similar and both of them are below the [MATH] threshold, which therefore cannot be ruled out by the present data [CITATION].', '1212.5079-2-6-0': 'This paper is organized as follows: In Sec. [REF], the chiral partner structure of heavy baryons is studied.', '1212.5079-2-6-1': "We derive the analytic forms of the heavy baryons' masses.", '1212.5079-2-6-2': "The heavy baryon spectrum with chiral partner structure is estimated in Sec. [REF], and the pentaquark state's spectrum is estimated in Sec. [REF].", '1212.5079-2-6-3': 'The last section is for a summary and discussions.', '1212.5079-2-6-4': 'Some useful explicit derivations are given in the Appendix.', '1212.5079-2-7-0': '# Heavy Baryons in the Effective Lagrangian for Heavy-light Mesons with Chiral Doubling', '1212.5079-2-8-0': '## Effective Lagrangian for heavy-light mesons with chiral doubling', '1212.5079-2-9-0': 'Here, we construct the effective Lagrangian describing the interaction between the heavy-light mesons and the light mesons.', '1212.5079-2-9-1': 'With respect to the chiral transformation property of the light quarks in the heavy-light mesons, the heavy-light meson field can be decomposed into a right-handed component [MATH] and left-handed one [MATH] [CITATION].', '1212.5079-2-9-2': 'Under chiral [MATH] symmetry, they transform as [EQUATION]', '1212.5079-2-9-3': 'In Refs. [CITATION] these fields are used to construct a Lagrangian where the chiral symmetry are realized linearly.', '1212.5079-2-9-4': 'In the present paper, for the study of the heavy baryon spectrum in the bound state approach, we adopt the nonlinear realization of the chiral symmetry.', '1212.5079-2-9-5': 'Then, by replacing [MATH] in Eq. (29) of Refs. [CITATION] with [MATH] where [MATH], we obtain [EQUATION] where [MATH] and [MATH] are parameters.', '1212.5079-2-9-6': 'In the present work, we include the vector mesons using the HLS [CITATION] by introducing the matrix valued variables [MATH] and [MATH] as [MATH].', '1212.5079-2-9-7': 'Then, similarly to Ref. [CITATION] we convert the heavy meson fields as [EQUATION] which under the full symmetry transformation [MATH] transform as [EQUATION]', '1212.5079-2-9-8': 'Associated with the field redefinitions in Eq. ([REF]), it is convenient to use the following quantities for the [MATH] fields: [EQUATION] where the covariant derivative [MATH] is given by [MATH] with [MATH] being the gauge field of the HLS.', '1212.5079-2-9-9': 'By using these quantities, the above Lagrangian is extended to include the vector mesons as [EQUATION] where [MATH] is defined as [MATH] with [MATH] being a real parameter measuring the magnitude of the violation of the vector meson dominance.', '1212.5079-2-10-0': 'To study the chiral partner structure of the heavy baryons, we rewrite the Lagrangian ([REF]) in terms of the heavy-light meson doublets [MATH] and [MATH] with quantum numbers [MATH] and [MATH]; specifically, we make the substitution [EQUATION]', '1212.5079-2-10-1': 'In terms of the physical states, the [MATH] and [MATH] doublets can be explicitly expressed as [EQUATION]', '1212.5079-2-10-2': 'Substituting Eq. ([REF]) into Eq. ([REF]), we obtain [EQUATION]', '1212.5079-2-10-3': 'This expression explicitly shows that the [MATH] term splits the spectrum of [MATH] and [MATH] doublets while the [MATH] term shifts the masses of these two doublets toward the same direction.', '1212.5079-2-10-4': 'In the present paper, we use the physical masses of heavy mesons as inputs to calculate the heavy baryon masses so that we drop the [MATH] term and the [MATH] term in the following calculation of the masses of heavy baryons.', '1212.5079-2-10-5': 'Note that due to the chiral partner structure adopted here, the magnitudes of the coupling constants in the last two terms of Eq. ([REF]) are the same; therefore, the chiral partner spectrum is predictable.', '1212.5079-2-11-0': '## Heavy baryon masses from the bound state approach', '1212.5079-2-12-0': 'In this subsection we derive the heavy baryon masses based on the bound state approach [CITATION].', '1212.5079-2-13-0': 'To make the mesonic theory a baryonic one, we follow the standard procedure to take the Hedgehog ansatze for a classical soliton [CITATION] [EQUATION] with [MATH] as the Pauli matrices and the subscript [MATH] standing for the classical solution.', '1212.5079-2-13-1': 'From the Hedgehog ansatze ([REF]), one can easily see that [MATH] transforms under separate spatial rotation and isospin rotation but is invariant under the combined rotation; i.e., the hedgehog profile correlates the angular momentum and the isospin.', '1212.5079-2-13-2': 'For the vector mesons, their profile functions can be parametrized as [CITATION] [EQUATION]', '1212.5079-2-13-3': 'From the hedgehog ansatze ([REF]) and profile functions ([REF]) we express the quantities [MATH] and [MATH] as [EQUATION] where [EQUATION]', '1212.5079-2-13-4': 'In the rest frame of the heavy-light meson, i.e., [MATH], the [MATH] doublet has nonvanishing elements only in the upper-right [MATH] sub-block while the [MATH] doublet has nonvanishing elements only in the upper-left [MATH] sub-block.', '1212.5079-2-13-5': 'The matrix forms of [MATH] and [MATH] doublets become [EQUATION]', '1212.5079-2-13-6': 'Then, the Lagrangian ([REF]) is reduced to [EQUATION]', '1212.5079-2-13-7': 'Since the hedgehog ansatz for the Skyrme soliton correlates the angular momentum and isospin, the bound states should be invariant under the "grand spin" rotation with the operator defined as [EQUATION] where [MATH], and [MATH] are the ordinary orbital angular momentum between the soliton and heavy-light meson, heavy meson spin, and the heavy meson isospin operators.', '1212.5079-2-13-8': 'Taking into account that the heavy quark spin is conserved in the heavy quark limit, one simply defines the "light quark grand spin" operator [EQUATION] with [MATH] as the spin operator of the light degree of freedom of the heavy-light meson, and in both [MATH] and [MATH] doublets, the eigenvalue of the operator [MATH] is [MATH] so that the eigenmodes of the heavy baryons can be classified by the third component of heavy quark spin [MATH] and the light quark grand spin [MATH] and the parity [MATH].', '1212.5079-2-14-0': 'Taking into account the isospin, light quark spin and heavy quark spin indices that the heavy-light meson has, one can write the static wave functions of the heavy-light mesons as [CITATION] [EQUATION] where [MATH], and [MATH] denote the indices for the isospin of the heavy-light meson, the spin of the light degree of freedom, and the heavy quark spin, respectively.', '1212.5079-2-14-1': '[MATH] is the eigenvalue of the operator [EQUATION] with [MATH] as its third component.', '1212.5079-2-14-2': '[MATH] is factorized out due to the conservation of the heavy quark spin.', '1212.5079-2-14-3': '[MATH] is a radial function which is strongly peaked at the origin and normalized as [MATH] [CITATION].', '1212.5079-2-14-4': 'This implies that the relevant matrix elements are independent of the quantum number [MATH] [CITATION].', '1212.5079-2-14-5': 'The generalized "angular" wave function [MATH] can be expanded as [CITATION] [EQUATION] where [MATH] stands for the standard spherical harmonic representing orbital angular momentum [MATH] while [MATH] denotes the ordinary Clebsch-Gordan coefficients.', '1212.5079-2-14-6': '[MATH] represents a wave function in which the "light spin" and "light isospin" referring to the "light cloud" component of the heavy meson are added vectorially to give [MATH] with eigenvalues [MATH].', '1212.5079-2-14-7': 'Note that in the present analysis, [MATH] is a good quantum number since the relevant matrix elements are independent of the quantum number [MATH].', '1212.5079-2-14-8': 'Furthermore, both the quantum numbers for the "light spin" and "light isospin" are given by [MATH] so that the possible values of [MATH] are either [MATH] or [MATH].', '1212.5079-2-14-9': 'The normalization of the eigenstate [MATH] is [EQUATION] where [MATH] is the solid angle integration.', '1212.5079-2-15-0': 'From Lagrangian ([REF]), we parametrize the potential as [EQUATION]', '1212.5079-2-15-1': 'By substituting the ansatz ([REF]), [MATH] and [MATH] are obtained as [EQUATION]', '1212.5079-2-15-2': 'Next, we make a quantization by a time dependent [MATH] rotation of the fields in the HLS Lagrangian in the light-quark sector as [EQUATION] where [MATH] is a time dependent unitary matrix satisfying [MATH].', '1212.5079-2-15-3': 'Accordingly, the heavy-light meson fields are rotated as [EQUATION]', '1212.5079-2-15-4': 'This collective rotation gives an additional contribution to the Lagrangian [EQUATION] where the angular velocity [MATH] corresponding to the collective coordinate rotation is defined as [EQUATION] [MATH] is the moment of the inertia of the soliton configuration.', '1212.5079-2-15-5': 'By using [MATH], the light baryon masses are expressed as [EQUATION] where [MATH] is the soliton mass and [MATH] is the spin of light baryon.', '1212.5079-2-15-6': 'Using the nucleon mass [MATH] and the delta mass [MATH] as inputs, [MATH] and [MATH] are given as [EQUATION]', '1212.5079-2-15-7': 'From Eq. ([REF]) one obtains [CITATION] [EQUATION] where the coefficient [MATH] is calculated as [EQUATION] with [MATH] being the spin quantum number of the light degree of freedom of tne heavy-light meson.', '1212.5079-2-15-8': 'For convenience we show the derivation in the Appendix.', '1212.5079-2-15-9': 'The Hamiltonian of the collective rotated system is obtained by the standard Legendre transformation as [EQUATION] where [MATH] is the canonical momentum conjugating to the collective variable [MATH]: [EQUATION]', '1212.5079-2-15-10': 'The first term [MATH] is the isospin operator for the light baryon sector while the second term [MATH] is the isospin operator for the heavy-light mesons interacting with the light baryon so that [MATH] is identical to the isospin operator for the heavy baryon [MATH]: [EQUATION]', '1212.5079-2-15-11': 'After the collective coordinate rotation, the total spin of the light degrees of freedom in the heavy baryon is defined as [EQUATION]', '1212.5079-2-15-12': 'By including the heavy quark spin, the spin operator for the heavy baryon is expressed as [MATH] with eigenvalues [MATH] (in the case of [MATH], only [MATH] exists).', '1212.5079-2-15-13': 'Then, we can express the collectively rotated Hamiltonian as [EQUATION]', '1212.5079-2-15-14': 'Gathering all the above contributions, we finally obtain the heavy baryon mass as [EQUATION] where [MATH]) is the binding energy corresponding to the heavy meson [MATH]).', '1212.5079-2-15-15': '[MATH] are the weight-averaged heavy meson masses with [MATH] and [MATH].', '1212.5079-2-15-16': 'Note that each combination of [MATH] generates a pair of degenerate states with [MATH].', '1212.5079-2-16-0': '# Chiral Doubling of Heavy Baryons', '1212.5079-2-17-0': 'In this section, we study the chiral doubling structure using the formulas obtained in the previous section.', '1212.5079-2-17-1': 'In the following analysis, we restrict ourselves to the case with [MATH] and use the following values of heavy meson masses as inputs: [EQUATION] which lead to [EQUATION]', '1212.5079-2-17-2': 'Furthermore, to simulate the profile functions [MATH] and [MATH], which are necessary to evaluate the binding energy [MATH] and [MATH] expressed in Eq. ([REF]), we use the following inputs: [EQUATION] which yield soliton mass [MATH] GeV and the inverse of the moment of inertia [MATH] GeV.', '1212.5079-2-17-3': 'Then, using the relevant expressions from HLS up to [MATH] including the hWZ terms given in Refs. [CITATION], by taking [MATH] and [MATH] as free parameters to fit the inputs ([REF]), we obtain [MATH]MeV and [MATH]MeV and the values of the profile functions at origin as [EQUATION]', '1212.5079-2-17-4': 'Moreover, we take the parameter [MATH] [CITATION] and fix the universal coupling constant [MATH] in HLS through [EQUATION]', '1212.5079-2-17-5': 'Let us first consider the binding energy in order to determine which channel can form the bound state and calculate the spectra of the chiral partners.', '1212.5079-2-17-6': 'Using Eq. ([REF]), we obtain the binding energy between the heavy-light mesons in the [MATH] doublet and soliton as [EQUATION]', '1212.5079-2-17-7': 'The value of [MATH] is determined through the [MATH] decay as [MATH] [CITATION].', '1212.5079-2-17-8': 'It does not seem possible to determine the [MATH] coupling constant [MATH] from the available experimental data for the heavy meson decay.', '1212.5079-2-17-9': 'In the case of the vector meson dominance we have [MATH]; therefore, it is reasonable to regard [MATH].', '1212.5079-2-17-10': 'Then we conclude that the [MATH] channel gives a bound state when [MATH].', '1212.5079-2-17-11': 'This bound state can be naturally identified with [MATH] from the quantum number so that we assume [MATH] in the following analysis.', '1212.5079-2-17-12': 'Since the collective energy is zero for the [MATH] state, we use the experimental value of the mass of [MATH] as an input to determine the value of [MATH].', '1212.5079-2-17-13': 'Using [MATH]GeV, we obtain [MATH].', '1212.5079-2-18-0': 'Next, we consider the bound states made from the [MATH] doublet.', '1212.5079-2-18-1': 'The binding energy is expressed as [EQUATION]', '1212.5079-2-18-2': 'As we can see easily, [MATH] for [MATH]) so that there is no bound state in the [MATH] channel.', '1212.5079-2-18-3': 'For the [MATH] channel, using [MATH] determined above, we obtain [MATH]GeV, which implies that the [MATH] channel is actually bound.', '1212.5079-2-19-0': 'From Eq. ([REF]) the total spin of the light degrees of freedom becomes [MATH] so that the resultant heavy baryons form a heavy-quark doublet consisting of [MATH] and [MATH].', '1212.5079-2-19-1': 'Combined with the collective energy, the mass of the bound state is expressed as [EQUATION] which leads to [EQUATION]', '1212.5079-2-19-2': 'This value is much larger than the experimental values for the masses of negative parity baryons: [MATH]GeV and [MATH]GeV.', '1212.5079-2-19-3': 'So, we conclude that the negative parity baryons found by experiments [MATH] and [MATH] are not the chiral partner to the ground state baryon [MATH].', '1212.5079-2-19-4': 'In the present bound state approach, they should be regarded as the [MATH] state made from the [MATH] doublet and nucleon.', '1212.5079-2-19-5': 'Then, we expect to have a doublet for the chiral partner around the [MATH]GeV region.', '1212.5079-2-20-0': 'We next study the [MATH] baryons.', '1212.5079-2-20-1': 'In the positive parity baryon sector, the [MATH] channel is bound so that the total spin of the light degrees of freedom in the heavy baryon becomes [MATH].', '1212.5079-2-20-2': 'As a result, the spin of [MATH] baryons with positive parity is either [MATH] or [MATH].', '1212.5079-2-20-3': 'In the mass formula in Eq. ([REF]), only [MATH] changes its value depending on the isospin of baryons.', '1212.5079-2-20-4': 'Since [MATH] for [MATH], the mass difference between the [MATH] and the [MATH] in the positive negative parity is obtained as [EQUATION]', '1212.5079-2-20-5': 'In the negative parity baryon sector, the [MATH] channel is bound, then the eigenvalue for [MATH] is either [MATH], [MATH], or [MATH].', '1212.5079-2-20-6': 'We summarize our predicted results for the charm baryon spectrum in Tables [REF] and [REF].', '1212.5079-2-21-0': 'We next study the mass spectrum of the bottom baryon by substituting the bottom meson masses into the charm meson masses in Eq. ([REF]).', '1212.5079-2-21-1': 'In the bottom meson spectrum, the masses of the ground states [MATH] and [MATH] are well measured but masses of the mesons in the [MATH] doublets are not well established.', '1212.5079-2-21-2': 'Here, we naively estimate them using [MATH]MeV and [MATH]MeV which lead to [MATH]MeV and [MATH]MeV.', '1212.5079-2-21-3': 'Our numerical results for the masses of the heavy baryons including the bottom quark with the corresponding quantum numbers are given in Tables [REF] and [REF].', '1212.5079-2-22-0': '# Pentaquarks with heavy quark', '1212.5079-2-23-0': 'We next consider the pentaquark channel.', '1212.5079-2-23-1': 'Although the existence of these kinds of states still needs experimental confirmation, theoretical study of them is meaningful.', '1212.5079-2-23-2': 'For a pentaquark state, the large component of the antiheavy quark can be projected out with the projection operator [MATH] so that in case the heavy meson is at rest, the [MATH] doublet has nonvanishing elements only in the lower-left [MATH] sub-block while the [MATH] doublet has nonvanishing elements only in the lower-right [MATH] sub-block, i.e., [EQUATION]', '1212.5079-2-23-3': 'Then, substituting [MATH] with [MATH], following the above derivation, one can see that both the binding energies given by Eq. ([REF]) change a sign.', '1212.5079-2-23-4': 'As a result, the binding energies for the pentaquark states made from the anti-[MATH] doublet and the pentaquark states made from the anti-[MATH] doublet are expressed as [EQUATION]', '1212.5079-2-23-5': 'Therefore, for the anti-[MATH] doublet, the [MATH] channel gives the bound states with binding energy [MATH] MeV, while for anti-[MATH] doublet, the [MATH] channel gives the bound states with binding energy [MATH] MeV.', '1212.5079-2-24-0': 'Substituting relevant numerical results, we obtain the spectrum of the pentaquark states.', '1212.5079-2-24-1': 'We list our results in Tables [REF] and [REF] for pentaquark states consisting anti-charm quark and Tables [REF] and [REF] for pentaquark states consisting anti-bottom quark.', '1212.5079-2-25-0': 'The results in Tables [REF] and [REF] show that the lightest charmed pentaquark states made of soliton and heavy-light mesons in the anti-[MATH] doublet have masses of about [MATH]GeV, and their chiral partner made of the anti-[MATH] doublet has a mass of about [MATH]GeV.', '1212.5079-2-25-1': 'Both of them are below the [MATH] threshold.', '1212.5079-2-25-2': 'The reason that the pentaquark states from the anti-[MATH] doublet and anti-[MATH] doublet have similar masses is because the binding energy of the anti-[MATH] doublet is about [MATH] MeV smaller than that of the anti-[MATH] doublet, and the collective rotation energy, which is about [MATH] MeV, does not contribute to the latter.', '1212.5079-2-25-3': 'With respect to the status of the pentaquark search performed, these states cannot be ruled out, and since they cannot decay via a strong process, their total widths should be narrow.', '1212.5079-2-26-0': '# Summary and Discussions', '1212.5079-2-27-0': 'We studied the chiral partner structure of heavy baryons in the bound state approach including the vector meson exchanging effects through the hidden local symmetry.', '1212.5079-2-27-1': 'We showed that in the large [MATH] limit and the heavy quark limit, the ground state heavy baryon made of the ground state heavy-light meson and the nucleon has a chiral partner made of an excited heavy-light meson and nucleon.', '1212.5079-2-27-2': 'Our explicit calculation showed that the chiral partner of [MATH] is a heavy quark doublet of [MATH] and [MATH].', '1212.5079-2-27-3': 'This contrasted to the perdition made in the pioneering work in Ref. [CITATION], where the chiral partner was the singlet under the heavy quark spin transformation.', '1212.5079-2-27-4': 'Our prediction of the mass was about [MATH]GeV, which indicated that the [MATH] and [MATH] listed in the PDG table [CITATION] should be interpreted as the [MATH] excitation of [MATH].', '1212.5079-2-27-5': 'To calculate the spectrum of the [MATH] states, one should consider the relative motion of the soliton with respect to the heavy mesons [CITATION].', '1212.5079-2-27-6': 'This is beyond the scope of the present paper.', '1212.5079-2-28-0': 'We also studied the bound states in the pentaquark channel.', '1212.5079-2-28-1': 'We found that the [MATH] channel forms bound states for the anti-[MATH] doublet ([MATH], [MATH]), while the [MATH] channel forms bound states for the anti-[MATH] doublet ([MATH]).', '1212.5079-2-28-2': 'It was found that the predicted masses of the pentaquark states made of the anti-[MATH] doublet and anti-[MATH] doublet were below the [MATH] threshold, which cannot be ruled out by the present data [CITATION].', '1212.5079-2-29-0': 'In the present analysis, we took the infinite heavy soliton and heavy quark limits so that both the soliton and heavy-light meson were sitting at the origin.', '1212.5079-2-29-1': 'This picture could not be applied to the bound states with nonzero [MATH].', '1212.5079-2-29-2': 'Since in the present analysis, the chiral partner of heavy [MATH] had a mass of about [MATH]GeV, which was a bound state of soliton and heavy-light mesons in the [MATH] doublet, one could expect that it had broad width due to the broad width of the constituent [MATH]-wave mesons in the [MATH] doublet.', '1212.5079-2-29-3': 'From the numerical results in Tables [REF] and [REF] we concluded that the spectrum of the heavy baryons with bottom quark was consistent with PDG [CITATION] for [MATH] and [MATH].', '1212.5079-2-30-0': 'It should be noted that in the present analysis, we considered that the chiral partner to the nucleon was itself: The left-handed nucleon was the chiral partner of the right-handed nucleon, and vice versa so that the chiral partner to the heavy baryon as the bound state of the [MATH] doublet and the nucleon was the one made of the [MATH] doublet and the nucleon.', '1212.5079-2-30-1': 'This implied that the chiral partner structure of the heavy baryons in our approach arose from the chiral partner structure of the constituent heavy-light mesons.', '1212.5079-2-30-2': 'On the other hand, in the mirror scenario for the light baryon [CITATION], the chiral partner to the nucleon was considered as [MATH].', '1212.5079-2-30-3': 'In such a case, the full picture of the chiral partner structure of heavy baryons became complicated, and we did not consider this scenario in the present work.', '1212.5079-2-31-0': '# Matrix element of heavy-light meson isospin operator', '1212.5079-2-32-0': 'Using the Wigner-Eckart theorem, we can express the matrix element of heavy-light meson isospin operator [MATH] in terms of the matrix element of the operator [MATH], i.e., [EQUATION]', '1212.5079-2-33-0': 'We are grateful to Y. Oh, B.-Y. Park, and M. Rho for valuable comments and critical reading of the manuscript.', '1212.5079-2-33-1': 'This work is supported in part by Grant-in-Aid for Scientific Research on Innovative Areas Grant No. 2104, "Quest on New Hadrons with Variety of Flavors" from MEXT.', '1212.5079-2-33-2': 'The work of M.H. is supported in part by the Grant-in-Aid for Nagoya University Global COE Program "Quest for Fundamental Principles in the Universe: From Particles to the Solar System and the Cosmos" from MEXT, the JSPS Grant-in-Aid for Scientic Research Grabts No. (C) 24540266 and No. (S) 22224003.', '1212.5079-2-33-3': 'The work of Y.M. is supported in part by the National Science Foundation of China (NSFC) under Grant No. 10905060.'}

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['1212.5079-1-14-8', '1212.5079-2-14-8'], ['1212.5079-1-21-0', '1212.5079-2-21-0'], ['1212.5079-1-21-3', '1212.5079-2-21-3'], ['1212.5079-1-30-0', '1212.5079-2-30-0'], ['1212.5079-1-30-1', '1212.5079-2-30-0'], ['1212.5079-1-30-4', '1212.5079-2-30-3'], ['1212.5079-1-30-5', '1212.5079-2-30-3'], ['1212.5079-1-29-1', '1212.5079-2-29-1'], ['1212.5079-1-0-4', '1212.5079-2-0-4'], ['1212.5079-1-23-2', '1212.5079-2-23-2'], ['1212.5079-1-23-3', '1212.5079-2-23-2'], ['1212.5079-1-2-3', '1212.5079-2-2-3'], ['1212.5079-1-2-4', '1212.5079-2-2-3']]

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[]

[['1212.5079-1-30-0', '1212.5079-2-30-0'], ['1212.5079-1-30-1', '1212.5079-2-30-0'], ['1212.5079-1-30-4', '1212.5079-2-30-3'], ['1212.5079-1-30-5', '1212.5079-2-30-3'], ['1212.5079-1-29-1', '1212.5079-2-29-1'], ['1212.5079-1-0-4', '1212.5079-2-0-4'], ['1212.5079-1-23-2', '1212.5079-2-23-2'], ['1212.5079-1-23-3', '1212.5079-2-23-2'], ['1212.5079-1-2-3', '1212.5079-2-2-3'], ['1212.5079-1-2-4', '1212.5079-2-2-3']]

[]

['1212.5079-1-5-3', '1212.5079-1-5-4', '1212.5079-1-15-9', '1212.5079-1-33-0', '1212.5079-2-33-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1212.5079

astro-ph-0402009

{'astro-ph-0402009-1-0-0': "We present simulations of interferometric Sunyaev-Zel'dovich effect (SZE) and optical weak lenisng observations for the forthcoming AMiBA experiment, aiming at searching for high-redshift clusters of galaxies.", 'astro-ph-0402009-1-0-1': 'On the basis of simulated sky maps, we have derived theoretical halo number counts and redshift distributions of selected halo samples for an AMiBA SZE survey and a weak lensing follow-up survey.', 'astro-ph-0402009-1-0-2': 'By utilizing the conditional number counts of weak lensing halos with the faint SZE detection, we show that a combined SZE and weak lensing survey can gain an additional fainter halo sample at a given false positive rate, which cannot be obtained from either survey alone.', 'astro-ph-0402009-1-1-0': 'PACS Nos.: include PACS Nos.', 'astro-ph-0402009-1-2-0': '# Introduction', 'astro-ph-0402009-1-3-0': "The thermal Sunyaev-Zel'dovich effect (SZE[CITATION]) is a spectral distortion of the Cosmic Microwave Background (CMB) radiation due to the inverse-Compton scattering of CMB photons by high energy electrons in the intracluster medium (ICM).", 'astro-ph-0402009-1-3-1': 'The most remarkable properties of the SZE is that its surface brightness is redshift independent, which make it as an ideal probe of the high-redshift universe.', 'astro-ph-0402009-1-3-2': 'Further, since the SZE is proportional to the thermal energy content of the ICM, SZE imaging surveys allow us to select clusters over a wide range of the redshift with physically meaningful selection criteria.', 'astro-ph-0402009-1-3-3': 'Weak gravitational lensing, on the other hand, probes the total mass projected along the line-of-sight, and hence provides complementary information on the mass of galaxy clusters.', 'astro-ph-0402009-1-3-4': '[CITATION] Array for Microwave Background Anisotropy (AMiBA[CITATION]) is a 19-element interferometric array with full polarization capabilities operating at [MATH]GHz, specifically designed for the CMB observations.', 'astro-ph-0402009-1-3-5': 'One of the main scientific goals of AMiBA is to conduct blind SZE surveys to search for high-redshift clusters.', 'astro-ph-0402009-1-3-6': 'AMiBA will also conduct follow-up optical imaging observations with wide-field camera, MegaCam, at Canada France Hawaii Telescope (CFHT).', 'astro-ph-0402009-1-3-7': 'In this paper, we simulate the forthcoming AMiBA SZE experiment combined with the planned follow-up weak lensing observations to examine the expected cluster number counts for individual surveys, and explore the potential of a combined AMiBA SZE and weak lensing cluser survey.', 'astro-ph-0402009-1-4-0': '# Simulation Data and Mock Observations', 'astro-ph-0402009-1-5-0': 'To make sky maps with realistic SZE and weak lensing signals, we use results from preheating cosmological simulations of a [MATH]CDM model [MATH] in a [MATH]Mpc co-moving box which reproduce the observed cluster [MATH]-[MATH] and [MATH]-[MATH] relations at [MATH].', 'astro-ph-0402009-1-5-1': '[CITATION] We construct 36 SZE sky maps each with [MATH]deg[MATH] on a [MATH] grid, by projecting the electron pressure through the randomly displaced and oriented simulation boxes, separated by [MATH] Mpc, along a viewing cone out to the redshift of [MATH].', 'astro-ph-0402009-1-5-2': 'Similarly, weak lensing convergence ([MATH]) maps are constructed by projecting the distance-weighted mass over-density [MATH] out to a source plane at [MATH].', 'astro-ph-0402009-1-6-0': 'As an AMiBA specification, we adopt a close-packed hexagonal configuration of [MATH]m dishes on a single platform.', 'astro-ph-0402009-1-6-1': 'This array configuration yields a synthesized beam of [MATH]Mpc at [MATH]), which is optimized to detect high-redshift clusters.', 'astro-ph-0402009-1-6-2': 'The field-of-view of the primary beam is about [MATH].', 'astro-ph-0402009-1-6-3': 'To generate the mock AMiBA visibility data, we perform a mosaic survey of [MATH] pointings with each exposure of [MATH] hours covering [MATH]deg[MATH] with spacing of [MATH].', 'astro-ph-0402009-1-6-4': 'The AMiBA sensitivity is then given as [MATH] mJy/beam, where [MATH] depends on the mosaicking strategy.', 'astro-ph-0402009-1-6-5': 'The non-linear maximum entropy method is applied to reconstruct an AMiBA SZE map from the mock visibility data.', 'astro-ph-0402009-1-6-6': 'For a weak lensing survey, we assume moderately deep optical observations with a mean source number density of [MATH] and [MATH].', 'astro-ph-0402009-1-6-7': 'We take into account noise in observable image ellipticities due to the random-phase intrinsic source ellipticities.', 'astro-ph-0402009-1-6-8': 'We choose the intrinsic dispersion of [MATH].', 'astro-ph-0402009-1-6-9': 'We then apply a pixelization on the mock weak lensing data using a Gaussian filter with [MATH], and perform a linear mass inversion to reconstruct the [MATH]-map.', 'astro-ph-0402009-1-6-10': 'The sensitivity in reconstructed [MATH]-maps is given as [MATH].', 'astro-ph-0402009-1-6-11': 'We carry out the statistical analysis on reconstructed sky maps by using the inner [MATH] subfield in order to avoid the noisier boundaries.', 'astro-ph-0402009-1-6-12': 'Thus the effective survey area used for our statistical analysis is [MATH].', 'astro-ph-0402009-1-7-0': '# Results', 'astro-ph-0402009-1-8-0': 'Figure 1 shows the theoretical halo number counts [MATH] and false positive rate as a function of peak threshold [MATH] obtained from mock AMiBA SZE and weak lensing surveys.', 'astro-ph-0402009-1-8-1': 'To quantify the spurious peak detections due to experimental noise, we followed the prescription given by Ref. Pen.', 'astro-ph-0402009-1-8-2': 'The limiting halo masses [MATH] for both surveys are shown in Figure 2.', 'astro-ph-0402009-1-8-3': 'For both surveys, halo samples are defined at a false positive rate of [MATH], and the redshift distributions of the halo samples are derived (see Figure 2).', 'astro-ph-0402009-1-8-4': 'Further, we utilize the conditional number counts of weak lensing halos with faint SZE detection, [MATH], to explore the potential of a combined AMiBA SZE and weak lensing survey.', 'astro-ph-0402009-1-8-5': 'Here we choose the SZE peak threshold of [MATH] (see Figure 1).', 'astro-ph-0402009-1-8-6': 'Table 1 summarizes the main properties of halos samples obtained from mock AMiBA SZE and weak lensing observations.', 'astro-ph-0402009-1-9-0': '# Conclusions', 'astro-ph-0402009-1-10-0': 'We have examined the expected cluster number counts and redshift distributions of cluster samples for the forthcoming AMiBA SZE/weak lensing cluster survey on the basis of [MATH]CDM cosmological simulations.', 'astro-ph-0402009-1-10-1': 'By utilizing the conditional halo number counts [MATH], we have demonstrated that a combined SZE and weak lensing survey can gain an additional fainter halo sample at a given false positive rate, which cannot be obtained from either survey alone.'}

{'astro-ph-0402009-2-0-0': "We present simulations of interferometric Sunyaev-Zel'dovich effect (SZE) and optical weak lenisng observations for the forthcoming AMiBA experiment, aiming at searching for high-redshift clusters of galaxies.", 'astro-ph-0402009-2-0-1': 'On the basis of simulated sky maps, we have derived theoretical halo number counts and redshift distributions of selected halo samples for an AMiBA SZE survey and a weak lensing follow-up survey.', 'astro-ph-0402009-2-0-2': 'By utilizing the conditional number counts of weak lensing halos with the faint SZE detection, we show that a combined SZE and weak lensing survey can gain an additional fainter halo sample at a given false positive rate, which cannot be obtained from either survey alone.', 'astro-ph-0402009-2-1-0': 'PACS Nos.: include PACS Nos.', 'astro-ph-0402009-2-2-0': '# Introduction', 'astro-ph-0402009-2-3-0': "The thermal Sunyaev-Zel'dovich effect (SZE[CITATION]) is a spectral distortion of the Cosmic Microwave Background (CMB) radiation due to the inverse-Compton scattering of CMB photons by high energy electrons in the intracluster medium (ICM).", 'astro-ph-0402009-2-3-1': 'The most remarkable properties of the SZE is that its surface brightness is redshift independent, which make it as an ideal probe of the high-redshift universe.', 'astro-ph-0402009-2-3-2': 'Further, since the SZE is proportional to the thermal energy content of the ICM, SZE imaging surveys allow us to select clusters over a wide range of the redshift with physically meaningful selection criteria.', 'astro-ph-0402009-2-3-3': 'Weak gravitational lensing, on the other hand, probes the total mass projected along the line-of-sight, and hence provides complementary information on the mass of galaxy clusters.', 'astro-ph-0402009-2-3-4': '[CITATION] Array for Microwave Background Anisotropy (AMiBA[CITATION]) is a 19-element interferometric array with full polarization capabilities operating at [MATH]GHz, specifically designed for the CMB observations.', 'astro-ph-0402009-2-3-5': 'One of the main scientific goals of AMiBA is to conduct blind SZE surveys to search for high-redshift clusters.', 'astro-ph-0402009-2-3-6': 'AMiBA will also conduct follow-up optical imaging observations with wide-field camera, MegaCam, at Canada France Hawaii Telescope (CFHT).', 'astro-ph-0402009-2-3-7': 'In this paper, we simulate the forthcoming AMiBA SZE experiment combined with the planned follow-up weak lensing observations to examine the expected cluster number counts for individual surveys, and explore the potential of a combined AMiBA SZE and weak lensing cluser survey.', 'astro-ph-0402009-2-4-0': '# Simulation Data and Mock Observations', 'astro-ph-0402009-2-5-0': 'To make sky maps with realistic SZE and weak lensing signals, we use results from preheating cosmological simulations of a [MATH]CDM model [MATH] in a [MATH]Mpc co-moving box which reproduce the observed cluster [MATH]-[MATH] and [MATH]-[MATH] relations at [MATH].', 'astro-ph-0402009-2-5-1': '[CITATION] We construct 36 SZE sky maps each with [MATH]deg[MATH] on a [MATH] grid, by projecting the electron pressure through the randomly displaced and oriented simulation boxes, separated by [MATH] Mpc, along a viewing cone out to the redshift of [MATH].', 'astro-ph-0402009-2-5-2': 'Similarly, weak lensing convergence ([MATH]) maps are constructed by projecting the distance-weighted mass over-density [MATH] out to a source plane at [MATH].', 'astro-ph-0402009-2-6-0': 'As an AMiBA specification, we adopt a close-packed hexagonal configuration of [MATH]m dishes on a single platform.', 'astro-ph-0402009-2-6-1': 'This array configuration yields a synthesized beam of [MATH]Mpc at [MATH]), which is optimized to detect high-redshift clusters.', 'astro-ph-0402009-2-6-2': 'The field-of-view of the primary beam is about [MATH].', 'astro-ph-0402009-2-6-3': 'To generate the mock AMiBA visibility data, we perform a mosaic survey of [MATH] pointings with each exposure of [MATH] hours covering [MATH]deg[MATH] with spacing of [MATH].', 'astro-ph-0402009-2-6-4': 'The AMiBA sensitivity is then given as [MATH] mJy/beam, where [MATH] depends on the mosaicking strategy.', 'astro-ph-0402009-2-6-5': 'The non-linear maximum entropy method is applied to reconstruct an AMiBA SZE map from the mock visibility data.', 'astro-ph-0402009-2-6-6': 'For a weak lensing survey, we assume moderately deep optical observations with a mean source number density of [MATH] and [MATH].', 'astro-ph-0402009-2-6-7': 'We take into account noise in observable image ellipticities due to the random-phase intrinsic source ellipticities.', 'astro-ph-0402009-2-6-8': 'We choose the intrinsic dispersion of [MATH].', 'astro-ph-0402009-2-6-9': 'We then apply a pixelization on the mock weak lensing data using a Gaussian filter with [MATH], and perform a linear mass inversion to reconstruct the [MATH]-map.', 'astro-ph-0402009-2-6-10': 'The sensitivity in reconstructed [MATH]-maps is given as [MATH].', 'astro-ph-0402009-2-6-11': 'We carry out the statistical analysis on reconstructed sky maps by using the inner [MATH] subfield in order to avoid the noisier boundaries.', 'astro-ph-0402009-2-6-12': 'Thus the effective survey area used for our statistical analysis is [MATH].', 'astro-ph-0402009-2-7-0': '# Results', 'astro-ph-0402009-2-8-0': 'Figure 1 shows the theoretical halo number counts [MATH] and false positive rate as a function of peak threshold [MATH] obtained from mock AMiBA SZE and weak lensing surveys.', 'astro-ph-0402009-2-8-1': 'To quantify the spurious peak detections due to experimental noise, we followed the prescription given by Ref. Pen.', 'astro-ph-0402009-2-8-2': 'The limiting halo masses [MATH] for both surveys are shown in Figure 2.', 'astro-ph-0402009-2-8-3': 'For both surveys, halo samples are defined at a false positive rate of [MATH], and the redshift distributions of the halo samples are derived (see Figure 2).', 'astro-ph-0402009-2-8-4': 'Further, we utilize the conditional number counts of weak lensing halos with faint SZE detection, [MATH], to explore the potential of a combined AMiBA SZE and weak lensing survey.', 'astro-ph-0402009-2-8-5': 'Here we choose the SZE peak threshold of [MATH] (see Figure 1).', 'astro-ph-0402009-2-8-6': 'Table 1 summarizes the main properties of halos samples obtained from mock AMiBA SZE and weak lensing observations.', 'astro-ph-0402009-2-9-0': '# Conclusions', 'astro-ph-0402009-2-10-0': 'We have examined the expected cluster number counts and redshift distributions of cluster samples for the forthcoming AMiBA SZE/weak lensing cluster survey on the basis of [MATH]CDM cosmological simulations.', 'astro-ph-0402009-2-10-1': 'By utilizing the conditional halo number counts [MATH], we have demonstrated that a combined SZE and weak lensing survey can gain an additional fainter halo sample at a given false positive rate, which cannot be obtained from either survey alone.'}

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[]

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['astro-ph-0402009-1-1-0', 'astro-ph-0402009-2-1-0']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0402009

astro-ph-0210471

{'astro-ph-0210471-1-0-0': 'We investigate the physical significance of a new spectral parameter, [MATH].', 'astro-ph-0210471-1-0-1': 'This parameter was defined based on a Principal Component Analysis of the 2dF Galaxy Redshift Survey, to retain astrophysical information while minimising the effect of measurement uncertainties.', 'astro-ph-0210471-1-0-2': 'We find that while [MATH] is correlated with morphological type, there is a large scatter in this relationship.', 'astro-ph-0210471-1-0-3': 'A tighter empirical relationship is found between [MATH] and the equivalent width of the H[MATH] line, suggesting a connection with the relative star formation rate.', 'astro-ph-0210471-1-0-4': 'We pursue this connection using spectral synthesis models.', 'astro-ph-0210471-1-0-5': 'Using models in which the star formation history is parameterised in terms of an exponentially decreasing function of time, we find that there is a tight correlation between [MATH] and the ratio of the present to the past-averaged rate of star-formation, often known as the "birthrate" parameter [MATH].', 'astro-ph-0210471-1-0-6': 'This correlation also holds in models with much more complicated star formation histories, generated by a semi-analytic model of galaxy formation based upon the hierarchical formation scenario.', 'astro-ph-0210471-1-0-7': 'There are two possible causes for the tight correlations we find, between [MATH] and [MATH], in those galaxies with the most complex star formation histories; Firstly, the spectra themselves may be degenerate to the actual long-term star formation history of each galaxy in the optical wavelength range probed by the 2dFGRS.', 'astro-ph-0210471-1-0-8': 'Secondly, the birthrate parameter [MATH] may represent a physically fundamental quality of galaxy halos - their over-density relative to the background density - such that small-[MATH] galaxies form in high peaks (which collapse early) while large-[MATH] galaxies represent lower peaks, which collapse later.', 'astro-ph-0210471-1-0-9': 'We conclude that the tight connection with [MATH] makes [MATH] a physically meaningful, as well as convenient and robust, statistic for galaxy parameterisation and classification.', 'astro-ph-0210471-1-1-0': '# Introduction', 'astro-ph-0210471-1-2-0': 'Galaxy redshift surveys are now probing the galaxy distribution of the local Universe more accurately than ever before, and in so doing they are establishing many fundamental properties of the galaxy population and its large-scale structure.', 'astro-ph-0210471-1-2-1': 'The 2dF Galaxy Redshift Survey (2dFGRS) is one such ambitious project, conceived with the aim of mapping the galaxy distribution to an extinction corrected [MATH] magnitude limit.', 'astro-ph-0210471-1-2-2': 'This particular survey is now essentially complete with approximately 230,000 galaxy spectra having already been acquired, and has already started to yield significant results (e.g. Percival et al. 2001; Madgwick et al. 2002 and others).', 'astro-ph-0210471-1-3-0': 'Apart from the main science goals of quantifying the large-scale structure of the Universe, one of the most significant contributions of galaxy redshift surveys is to our understanding of the galaxy population itself, through the information about galaxy properties contained in the observed spectra.', 'astro-ph-0210471-1-3-1': 'Having a data set of 230,000 galaxy spectra - as in the case of the 2dFGRS - allows us to test the validity of galaxy formation and evolution scenarios with unprecedented accuracy.', 'astro-ph-0210471-1-3-2': 'However, the sheer size of the data set presents its own unique problems.', 'astro-ph-0210471-1-3-3': "Clearly, in order to make the spectral data set more 'digestible', some form of data compression is necessary.", 'astro-ph-0210471-1-3-4': 'Familiar statistics such as equivalent width measurements, morphological types, and broad-band colours are really just compression techniques in some sense.', 'astro-ph-0210471-1-3-5': 'These quantities can be compared with theoretical predictions and simulations, and hence can set constraints on scenarios for galaxy formation and biasing.', 'astro-ph-0210471-1-4-0': 'The approach that has been adopted in Madgwick et al. (2002) is to define a spectral indicator, [MATH], based on Principal Component Analysis (PCA).', 'astro-ph-0210471-1-4-1': 'Because of instrumental limitations, the flux calibration of the 2dF spectra is unreliable, and hence robust measurements of the shape of the spectral continua are not possible.', 'astro-ph-0210471-1-4-2': 'For this reason, the [MATH] statistic was designed to (in a loose sense) preserve the most discriminatory information in the spectra, while being robust to the instrumental uncertainties.', 'astro-ph-0210471-1-4-3': 'Effectively, the parameter [MATH] measures the strength of absorption and emission lines, while remaining insensitive to the slope of the continuum (or broad band colour).', 'astro-ph-0210471-1-5-0': 'The motivation of the definition of the [MATH] statistic was in part pragmatic.', 'astro-ph-0210471-1-5-1': 'The goal of this paper is to address the fundamental question of the physical significance of this parameter, in terms of how it relates to the physical properties of galaxies.', 'astro-ph-0210471-1-5-2': 'A substantial amount of work has been carried out in the past on relating the observed spectra of galaxies to the physical processes occurring in them (see e.g Kennicutt, Tamblyn Congdon 1994; Ronen, Aragon-Salamanca Lahav 1999; Carter, Fabricant, Geller Kurtz 2001 and references therein), and the advent of such a large and uniform survey such as the 2dFGRS will result in many more advances in this field.', 'astro-ph-0210471-1-5-3': 'However, for the present analysis we restrict ourselves to relating only the 2dFGRS spectral classification parameter, [MATH], to these processes and leave a fuller development of this subject for future works.', 'astro-ph-0210471-1-5-4': 'The investigation presented here is particularly timely as many new results from the 2dFGRS are being presented which make exclusive use of this parameter to characterise and partition the galaxy population (e.g. Madgwick et al. 2002; Norberg et al. 2002; Martinez et al. 2002).', 'astro-ph-0210471-1-6-0': 'We make use of two kinds of models to pursue this question.', 'astro-ph-0210471-1-6-1': 'In the first approach, we adopt a simple parameterisation to describe the star formation history of each galaxy in terms of an exponentially decreasing function of time.', 'astro-ph-0210471-1-6-2': 'For a given redshift of observation, the star formation history is then characterised by a single parameter: the timescale of the exponential decline.', 'astro-ph-0210471-1-6-3': "These star formation histories are then convolved with spectral synthesis models, using the PEGASE (Projet d'Etude des GAlaxies par Synthese Evolutive) code, developed by Fioc Rocca-Volmerange (1997).", 'astro-ph-0210471-1-6-4': 'A particular advantage of this package compared to others available in the literature (e.g. Bruzual Charlot 1993) is that it is possible to include modelling of nebular emission from star forming regions in each galaxy - an important ingredient in the calculation of the [MATH] parameter.', 'astro-ph-0210471-1-6-5': 'We refer to these models as the "simple" models, because of the rather simplified nature of the parametrised star formation histories.', 'astro-ph-0210471-1-7-0': 'The limitation of this simple method is that we must explicitly assume some family of star-formation histories for our galaxy population.', 'astro-ph-0210471-1-7-1': 'We therefore extend this approach to incorporate the cosmological framework of the Cold Dark Matter (CDM) scenario, by using semi-analytic techniques (see e.g. Kauffmann, White Guiderdoni 1993; Cole et al. 1994; Somerville Primack 1999).', 'astro-ph-0210471-1-7-2': 'We make use of a mock catalogue of synthetic spectra which was designed to match the selection criteria of the 2dFGRS, analysed previously in Slonim et al. (2001), and created using an updated version of the code described in Somerville Primack (1999).', 'astro-ph-0210471-1-7-3': 'There are several advantages of using a semi-analytic model to create spectra for this exercise.', 'astro-ph-0210471-1-7-4': 'The star formation history of each galaxy will reflect at least some of the complexity of the interconnected processes of dark matter clustering, gas cooling, galaxy merging, supernova feedback, etc., modelled self-consistently within a specific cosmological framework.', 'astro-ph-0210471-1-7-5': 'In addition, as we can select galaxies so as to reproduce the same redshift and luminosity distributions displayed by the observed 2dF galaxies, the resulting ensemble should contain a mix of galaxies with different sorts of star formation histories that is similar to the actual observed sample.', 'astro-ph-0210471-1-7-6': 'Finally, the semi-analytic model also yields many other physical parameters for each galaxy, giving us the potential to further probe the physical processes that may be related to [MATH].', 'astro-ph-0210471-1-8-0': 'We find that in the simple models, there is essentially a one-to-one relationship between [MATH] and the ratio of the present rate of star formation to its past time-averaged value.', 'astro-ph-0210471-1-8-1': 'This quantity is sometimes known as the "birthrate" parameter, [MATH] (Scalo 1986).', 'astro-ph-0210471-1-8-2': 'Even more surprisingly, we find that there is a strong correlation between [MATH] and [MATH] even in the spectra produced by the semi-analytic models, despite the complex and diverse nature of the underlying star formation histories.', 'astro-ph-0210471-1-8-3': 'Empirically, the [MATH] parameter is known to be tightly correlated with morphological type and colour.', 'astro-ph-0210471-1-8-4': 'We therefore argue that the [MATH] parameter presents a practical and robust means of extracting a fundamental measure of galaxy type from the relatively low-quality spectra typical of present-day large redshift surveys.', 'astro-ph-0210471-1-9-0': 'Section 2 of this paper briefly reviews the operational definition of and motivation behind the spectral classification parameter [MATH], used in the 2dFGRS.', 'astro-ph-0210471-1-9-1': 'In Section 3 we make use of the "simple" models based on exponentially declining star formation histories to show how [MATH] is related to the star-formation history of a galaxy.', 'astro-ph-0210471-1-9-2': 'Section 4 then generalises these results using formation histories generated with a semi-analytic model.', 'astro-ph-0210471-1-9-3': 'We conclude with a discussion of our results and ways to build upon the results presented here.', 'astro-ph-0210471-1-10-0': '# Spectral Classification in the 2dFGRS', 'astro-ph-0210471-1-11-0': '## Principal Component Analysis', 'astro-ph-0210471-1-12-0': 'Principal component analysis (PCA) is a well established statistical technique which has proved very useful in dealing with high dimensional data sets (see e.g. Murtagh Heck 1987; Connolly et al. 1995).', 'astro-ph-0210471-1-12-1': 'In the particular case of galaxy spectra we are typically presented with approximately 1000 spectral channels per galaxy, however when used in applications this is usually compressed down to just a few numbers, either by integrating over small line features - yielding equivalent widths - or over broad band filters.', 'astro-ph-0210471-1-12-2': 'The key advantage of using PCA in our data compression is that it allows us to make use of all the information contained in the spectrum in a statistically unbiased way, i.e. without the use of such ad hoc filters.', 'astro-ph-0210471-1-13-0': 'In order to perform the PCA on our galaxy spectra we first construct a representative volume limited sample of the galaxies.', 'astro-ph-0210471-1-13-1': 'We compute the "mean spectrum" from this ensemble, and subtract this mean from all of the galaxy spectra.', 'astro-ph-0210471-1-13-2': 'When we apply the PCA to this sample an orthogonal set of components (eigenspectra) is constructed, which span the wavelength space occupied by the galaxy spectra.', 'astro-ph-0210471-1-13-3': 'These components have been specifically chosen by the PCA in such a way that as much information (variance) is contained in the first eigenspectrum as possible, and that the amount of the remaining information in all the subsequent eigenspectra is likewise maximised.', 'astro-ph-0210471-1-13-4': 'Therefore, if the information contained in the first [MATH] eigenspectra is found to be significantly greater than that in the remaining eigenspectra we can significantly compress the data set by swapping each galaxy spectrum (described by 1000 channels) with its projections onto just those [MATH] eigenspectra.', 'astro-ph-0210471-1-14-0': 'In the case of the 2dF galaxy spectra we find that approximately two thirds of the total variance (including the noise) in the spectra can be represented in terms of only the first two projections ([MATH], [MATH]).', 'astro-ph-0210471-1-14-1': 'So, at least to a first approximation, galaxy spectra can be thought of as a two dimensional sequence in terms of these two projections.', 'astro-ph-0210471-1-15-0': 'In Fig. [REF] we show these first two eigenspectra.', 'astro-ph-0210471-1-15-1': 'It can be seen from this figure that whilst the first eigenspectrum contains information from both lines and the continuum, the second is dominated by absorption and emission lines.', 'astro-ph-0210471-1-15-2': 'Because of this it is possible to take two simple linear combinations which isolate either the continuum or the emission/absorption line features.', 'astro-ph-0210471-1-15-3': 'In effect what we are doing when we utilise these linear combinations is rotating the axes defined by the PCA to make the interpretation of the components more straightforward.', 'astro-ph-0210471-1-15-4': 'In so doing we can see that a parameterisation in terms of [MATH] and [MATH] is essentially equivalent to a two dimensional sequence in colour (continuum slope) and the average emission/absorption line strength.', 'astro-ph-0210471-1-15-5': 'This ability to isolate the continuum and line components of the galaxy spectra turns out to be very useful, as will be discussed in Section 2.2.', 'astro-ph-0210471-1-16-0': '## The [MATH] Parameter', 'astro-ph-0210471-1-17-0': 'The 2dF instrument makes use of up to 400 optical fibres with a diameter of 140[MATH]m (corresponding to [MATH] on the sky, depending on plate position.', 'astro-ph-0210471-1-17-1': 'See Lewis et al. 2002).', 'astro-ph-0210471-1-17-2': 'The instrument itself was designed to measure large numbers of redshifts in as short an observing time as possible.', 'astro-ph-0210471-1-17-3': 'However, in order to optimise the number of redshifts that can be measured in a given period of time, compromises had to be made with respect to the spectral quality of the observations.', 'astro-ph-0210471-1-17-4': 'Therefore if one wishes to characterise the observed galaxy population in terms of their spectral properties, care must be taken in order to ensure that these properties are robust to the instrumental uncertainties.', 'astro-ph-0210471-1-18-0': 'The quality and representativeness of the observed spectra can be compromised in several ways and a more detailed discussion of these issues is presented in previous work (see e.g. Madgwick et al. 2002).', 'astro-ph-0210471-1-18-1': 'The net effect is that the uncertainties introduced into the fibre-spectra predominantly effect the calibration of the continuum slope and have relatively little impact on the emission/absorption line strengths.', 'astro-ph-0210471-1-18-2': 'For this reason any given galaxy spectrum which is projected onto the plane defined by ([MATH],[MATH]) will not be uniquely defined in the direction of varying continuum but will be robust in the orthogonal direction (which measures the average line strength).', 'astro-ph-0210471-1-19-0': 'The linear combination of the first two eigenspectra which is robust to these uncertainties is shown in Fig. [REF] (b) and denoted by [MATH] (ETATYPE in the 2dFGRS catalogue).', 'astro-ph-0210471-1-19-1': 'It is simply [EQUATION] where [MATH] is a constant which we find empirically to be [MATH].', 'astro-ph-0210471-1-20-0': 'We have now identified the single statistically dominant component of the galaxy spectra which is robust to the known instrumental uncertainties.', 'astro-ph-0210471-1-20-1': 'We have therefore chosen to adopt this (continuous) variable as our measure of spectral type.', 'astro-ph-0210471-1-20-2': 'Having defined [MATH] in this formal and pragmatic manner, however, we are left with the question of whether it is physically meaningful, and how it is to be interpreted.', 'astro-ph-0210471-1-20-3': 'This is the issue that we now address.', 'astro-ph-0210471-1-21-0': '## Correlation of [MATH] with traditional galaxy classifiers', 'astro-ph-0210471-1-22-0': 'We show the observed distribution of [MATH] for the 2dF galaxies in Fig. [REF].', 'astro-ph-0210471-1-22-1': 'It is rather intriguing that this distribution is strongly bimodal.', 'astro-ph-0210471-1-22-2': 'Fig. [REF] also shows the [MATH] projections of spectra from galaxies in the Kennicutt Atlas (Kennicutt 1992), which have known morphologies.', 'astro-ph-0210471-1-22-3': 'There is a correspondence between morphology and [MATH] in the expected sense: the value of [MATH] increases as one moves towards later type objects in the Hubble sequence.', 'astro-ph-0210471-1-22-4': 'Larger values of [MATH] indicate more dominant emission lines, and it is well-known that later type galaxies have stronger emission lines.', 'astro-ph-0210471-1-22-5': 'From this diagram we see that [MATH] seems to be correlated with morphological type, at least based on this small sample of galaxies.', 'astro-ph-0210471-1-22-6': 'Norberg et al. (2002) showed the distribution of [MATH] for a larger sample of galaxies that had been classified into four morphological types (E, S0, Sp and Irr), and showed that while there is a correlation between morphological type and [MATH], there is a large scatter.', 'astro-ph-0210471-1-22-7': 'The correspondence between [MATH] and morphological type is investigated in more detail in Madgwick (2002).', 'astro-ph-0210471-1-23-0': 'We can also compare the value of [MATH] derived for each galaxy to the equivalent width of the H[MATH] emission line, for a sample of high signal-to-noise ratio emission line spectra.', 'astro-ph-0210471-1-23-1': 'This is shown in Fig. [REF], from which it can be seen that there is a strong correlation between these quantities.', 'astro-ph-0210471-1-23-2': 'It is well-known that the equivalent width of H[MATH] is a measure of relative star formation; indeed it has been used to estimate the birthrate parameter [MATH] (the ratio of the present rate of star formation to the past averaged value - see e.g. Figure 3 of Kennicutt, Tamblyn Congdon 1994).', 'astro-ph-0210471-1-23-3': 'We return to this point in Section [REF], where we use galaxy formation models to investigate how [MATH] relates to the star-formation history.', 'astro-ph-0210471-1-24-0': '# Comparison with Simple Models', 'astro-ph-0210471-1-25-0': 'The results shown above suggest that [MATH] is related both to galaxy morphology and to star formation history.', 'astro-ph-0210471-1-25-1': 'In this section we further explore and test this hypothesis with models produced by combining simple parameterised star formation histories with spectral synthesis models.', 'astro-ph-0210471-1-25-2': "In this way, we can make a direct connection between a galaxy's star formation history and the value of our spectral classification statistic, [MATH].", 'astro-ph-0210471-1-26-0': '## Model Ingredients', 'astro-ph-0210471-1-27-0': 'Stellar population models combine theoretical temperature-luminosity tracks for stars of various masses with model stellar atmospheres and an assumed Initial Mass Function (IMF) in order to produce synthetic spectra for a "single burst" population of a uniform age and metalicity.', 'astro-ph-0210471-1-27-1': 'In order to use these models to obtain predictions for a composite stellar population (i.e. a galaxy), the "single burst" models must be combined with an assumed star formation history, which provides the distribution of stellar ages and metalicities within the galaxy.', 'astro-ph-0210471-1-27-2': 'Since the classical work on spectral synthesis by Tinsley (1980), many authors have adopted a simple parametrisation of the star formation history in terms of an exponential function of time (e.g. Bruzual 1983): [EQUATION]', 'astro-ph-0210471-1-27-3': 'Here, [MATH] is the time at which star formation first commences, and [MATH] is the characteristic timescale of star formation.', 'astro-ph-0210471-1-27-4': 'Most observed spectra can be modelled rather well using this simple function with the appropriate choice of [MATH].', 'astro-ph-0210471-1-27-5': 'A typical spiral galaxy spectrum might be well-fit with [MATH] Gyr, while later type galaxies require larger values and early type galaxies require smaller values of [MATH].', 'astro-ph-0210471-1-28-0': 'Nebular emission lines from ionised H[MATH] regions and extinction due to dust also contribute to the appearance of a galaxy spectrum.', 'astro-ph-0210471-1-28-1': 'Modelling these processes is more complicated, and requires assumptions about the ionisation state, geometry, and metalicity of H[MATH] regions, and the dust mass, composition and geometry.', 'astro-ph-0210471-1-29-0': 'The models presented in this Section are based on this simple but well-defined picture, and are realized using the PEGASE software package.', 'astro-ph-0210471-1-29-1': 'We have used version 2.0 of the PEGASE code, which is based on the original package described in Fioc Rocca-Volmerange (1997), but includes several new features.', 'astro-ph-0210471-1-29-2': 'Stellar evolutionary tracks with non-solar metalicities are available, the library of stellar spectra of Lejeune et al. (1997, 1998) replaces that of Gunn Stryker (1983) and the models include a simple treatment of internal extinction due to dust.', 'astro-ph-0210471-1-29-3': 'Also the metalicity of the galaxy is traced in a self-consistent manner using the prescription of Woosley Weaver (1995) to model the enrichment of the inter-stellar medium.', 'astro-ph-0210471-1-29-4': 'In all of the models shown we have adopted a Kennicutt (1983) initial-mass function with stellar masses in the range [MATH].', 'astro-ph-0210471-1-30-0': 'Nebular emission lines result from light re-emitted by the ionised gas in star-forming regions.', 'astro-ph-0210471-1-30-1': 'A prescription for determining the strengths of these lines is implemented in the PEGASE code.', 'astro-ph-0210471-1-30-2': 'This process involves the absorption of Lyman continuum photons (below 912) by the nebular gas, which gets ionised, and reaches recombination equilibrium (Osterbrock 1989).', 'astro-ph-0210471-1-30-3': 'It is assumed that 70% of these photons are absorbed by the gas at solar metalicity.', 'astro-ph-0210471-1-30-4': 'In this approach, the strength of the nebular emission lines is a function of the age of the stellar population only, and metalicity and geometric effects are neglected.', 'astro-ph-0210471-1-31-0': 'The PEGASE code provides a simple way to model the effects of dust extinction on the synthetic spectra, in which the optical depth is estimated from the mass of gas and the metalicity.', 'astro-ph-0210471-1-31-1': 'The absorption is then estimated using observational data for a mixture of graphites and silicates as in the Milky Way and the Magellanic Clouds (Draine Lee 1984 and Pei 1992).', 'astro-ph-0210471-1-31-2': 'In making this calculation it is also necessary to make an assumption about the geometry of the galaxy.', 'astro-ph-0210471-1-31-3': 'We investigated models in which we assumed a spheroidal geometry and also an inclination-averaged disk geometry.', 'astro-ph-0210471-1-32-0': 'The spectra generated by the PEGASE code over the optical wavelength range are generally given in 10 bins, and the emission line strengths are specified separately in terms of their peak fluxes over the continuum.', 'astro-ph-0210471-1-32-1': 'In order to make comparisons between these spectra and the 2dFGRS we therefore interpolate the synthetic spectra onto the 4 binned wavelength range of the 2dFGRS.', 'astro-ph-0210471-1-32-2': 'Because the given emission lines are not resolved in this binning we create lines by superposing Gaussian profiles with the specified peak flux and FWHM corresponding to that calculated for the 2dF spectrograph using arc line measurements ([MATH] pixels).', 'astro-ph-0210471-1-32-3': 'We convert the given synthetic spectra to units of counts/bin by multiplying the flux (erg/s/) by the wavelength.', 'astro-ph-0210471-1-32-4': 'We then normalise each spectrum to have mean counts of [MATH] over our entire wavelength range.', 'astro-ph-0210471-1-32-5': 'The spectra processed in this way are directly comparable to those of the 2dFGRS.', 'astro-ph-0210471-1-33-0': 'Spectral histories of a given galaxy are compiled for a range of formation times ([MATH] Gyr), and a grid of values of [MATH]: 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, and 2.0 Gyr.', 'astro-ph-0210471-1-33-1': 'We also calculate spectral histories for an instantaneous burst of star-formation at [MATH].', 'astro-ph-0210471-1-33-2': 'Because this grid does not comprise a well-defined sample of galaxies, to compare with the results from the 2dFGRS, we first subtract the mean spectrum from the 2dFGRS volume limited sample described in the previous Section from each spectrum.', 'astro-ph-0210471-1-33-3': 'We then compute the projection of each spectrum on the eigenspectra derived from the 2dFGRS volume limited sample.', 'astro-ph-0210471-1-33-4': 'The first two projections ([MATH], [MATH]) derived in this manner are shown for our grid of models in Fig. [REF], along with the corresponding projections for a random subsample of galaxies in the [MATH] volume-limited 2dFGRS sample.', 'astro-ph-0210471-1-33-5': 'Model grids are shown both with and without including the effects of dust extinction.', 'astro-ph-0210471-1-33-6': 'It can be seen that dust does not have a dramatic impact on the results, but does seem to improve the agreement between the model tracks and the locus of the observed galaxies.', 'astro-ph-0210471-1-34-0': 'Already it is interesting that these simple models seem to cover the same range of the ([MATH], [MATH]) parameter space occupied by the observed galaxy population.', 'astro-ph-0210471-1-34-1': 'Evidently, at least in this simple scheme, galaxies are "born" in the lower right-hand corner of the plot, and progress towards the upper left-hand corner as they age.', 'astro-ph-0210471-1-34-2': 'As well, galaxies in the upper right part of the diagram tend to have smaller values of the characteristic timescale for star formation, [MATH], associated with early type galaxies, while as one moves diagonally towards the lower left corner, galaxies have the more extended timescales for star formation associated with late type galaxies.', 'astro-ph-0210471-1-34-3': 'Perhaps unsurprisingly, the vast majority of the 2dF galaxies (which are fairly luminous) are consistent with times since formation [MATH] between 2-10 Gyr.', 'astro-ph-0210471-1-34-4': 'It should be kept in mind, however, that the instrumental effects discussed in Section [REF] can introduce random scatter in this diagram.', 'astro-ph-0210471-1-34-5': 'We therefore turn now to the more robust [MATH] parameter.', 'astro-ph-0210471-1-35-0': '## The Physical Significance of [MATH]', 'astro-ph-0210471-1-36-0': 'We argued based on a visual inspection of the eigenspectrum used to define the [MATH] projection that [MATH] essentially measures the strength of stellar and interstellar absorption line features and the strength of nebular emission line features.', 'astro-ph-0210471-1-36-1': 'The strength of the absorption features mainly indicates the age and metalicity of the stellar population that dominates the optical luminosity of the galaxy, and the emission lines are strongly correlated with star formation activity.', 'astro-ph-0210471-1-36-2': 'The comparison with classical galaxy classification methods such as visual morphology or equivalent width, shown in Section [REF], suggested that [MATH] may be connected with a measure of the star formation relative to the existing older stellar population, such as the birthrate parameter, [MATH] (Scalo 1986).', 'astro-ph-0210471-1-36-3': 'The birthrate parameter is defined as the ratio between the current and past-averaged star-formation rate, which in the case of the simple exponential star formation law is given analytically by [EQUATION]', 'astro-ph-0210471-1-36-4': 'In this section we investigate our hypothesis that [MATH] might be correlated with the birthrate parameter, [MATH].', 'astro-ph-0210471-1-37-0': 'The relationship between [MATH] and [MATH] for the grid of models with simple star formation histories (as before) is shown in Fig. [REF].', 'astro-ph-0210471-1-37-1': 'For a fixed formation time, [MATH], there is a well-defined relationship between the star formation history, as characterised by the birthrate parameter [MATH], and the [MATH] spectral parameter.', 'astro-ph-0210471-1-37-2': 'This helps in the interpretation of [MATH] for the observed population of galaxies.', 'astro-ph-0210471-1-37-3': 'At least in this simple scheme, galaxies with small values of [MATH] formed most of their stars in the past, and are currently evolving rather passively.', 'astro-ph-0210471-1-37-4': 'This fits in well with the classical picture of early type galaxy formation.', 'astro-ph-0210471-1-37-5': 'Galaxies with larger values of [MATH] have had a significant amount of recent star formation, characteristic of spiral and irregular galaxies.', 'astro-ph-0210471-1-37-6': 'Recalling from Fig. [REF] that the 2dFGRS sample spans a range of [MATH] values of about -5 to 10, this suggests again that most of the observed galaxies in the 2dF require times since formation greater than about 1 Gyr.', 'astro-ph-0210471-1-37-7': 'From this diagram, we arrive at a prediction that the distribution of [MATH] for galaxies observed at larger look-back times should exhibit a shift towards higher values.', 'astro-ph-0210471-1-37-8': 'We find that the effect of extinction corrections is to slightly lower the calculated [MATH] for a given star formation scenario, however, this is only a small effect and does not alter the qualitative interpretation that we have suggested.', 'astro-ph-0210471-1-38-0': 'A limitation of the simple characterisation of the star formation histories that we have adopted here is manifested by the asymptote of the [MATH] values towards unity: because the star formation rate was assumed to be monotonically decreasing, [MATH] is always less than one.', 'astro-ph-0210471-1-38-1': 'It is clear that we must be cautious in over-interpreting these results, as real galaxies presumably may have much more complex star formation histories than we have assumed here.', 'astro-ph-0210471-1-38-2': 'We now pursue a similar investigation for models with much more complex star formation histories, generated using a cosmological semi-analytic model of galaxy formation.', 'astro-ph-0210471-1-39-0': '# Comparison with Semi-Analytic Models', 'astro-ph-0210471-1-40-0': 'In this section, we again extract the [MATH] parameter from model galaxy spectra created using stellar population synthesis models.', 'astro-ph-0210471-1-40-1': 'The difference is that instead of using a simple parameterisation for the star formation history, the formation history of each galaxy is now modelled based on physical recipes set within the framework of hierarchical structure formation.', 'astro-ph-0210471-1-40-2': 'The ensemble of model spectra that we use here was created using the Somerville et al. semi-analytic models (e.g. Somerville Primack 1999; Somerville, Primack Faber 2001) and has been described previously in Slonim et al. (2001).', 'astro-ph-0210471-1-40-3': 'We refer the reader to those works for details, and here sketch the ingredients of the models very briefly.', 'astro-ph-0210471-1-41-0': 'In the hierarchical picture, present-day galaxies such as the ones observed by the 2dFGRS formed by the merging and accretion of smaller objects over time.', 'astro-ph-0210471-1-41-1': 'In this picture, the star formation history of a galaxy is determined by the mass accretion/merging history of its host dark matter halo, and the efficiency of gas cooling within those halos.', 'astro-ph-0210471-1-41-2': 'As the hot gas is enriched with heavy elements by metal-rich winds from massive stars and supernovae, the cooling efficiency is increased.', 'astro-ph-0210471-1-41-3': 'Galaxy mergers may trigger powerful bursts of star formation.', 'astro-ph-0210471-1-41-4': 'In turn, violent star formation events may inhibit future star formation by heating the interstellar medium or driving winds that blow it out of the galaxy.', 'astro-ph-0210471-1-41-5': 'Morphology and star formation history are connected in these models, in that major mergers are assumed both to destroy disks, leaving any pre-existing stars in a spheroidal configuration, and to trigger bursts of star formation.', 'astro-ph-0210471-1-41-6': 'A spheroid that does not happen to accrete any new gas will be classified as an elliptical galaxy, while in other cases a spheroid created in this way may grow a new disk and be classified as a spiral with a bulge.', 'astro-ph-0210471-1-41-7': 'Therefore, although the star formation histories generated by the semi-analytic models are highly diverse, complex and non-monotonic, we still expect there to be a connection between star formation history, morphology, and spectral properties.', 'astro-ph-0210471-1-42-0': 'These star formation histories are convolved with stellar population models to produce model spectra in much the same way as in the "simple" models discussed in the previous section.', 'astro-ph-0210471-1-42-1': 'In the models used here, we have used the multi-metalicity GISSEL models (Bruzual Charlot, in preparation) with a Salpeter IMF to calculate the stellar part of the spectra.', 'astro-ph-0210471-1-42-2': 'These models are very similar to the PEGASE stellar population models used in the previous section, and for the purposes of this investigation, this difference should not significantly affect our results.', 'astro-ph-0210471-1-42-3': 'As well, the Salpeter IMF differs from the Kennicutt IMF mainly in the zero-point of the mass-to-light ratio, which again does not affect the present calculation.', 'astro-ph-0210471-1-42-4': 'Nebular emission lines are added to the spectra in the same way as before, using the empirical library from PEGASE.', 'astro-ph-0210471-1-43-0': 'Dust extinction is included using an approach similar to that of Guiderdoni Rocca-Volmerange (1987), which is also very similar to the approach implemented in the PEGASE package.', 'astro-ph-0210471-1-43-1': 'Here, the mass of dust is assumed to be proportional to the gas fraction times the metalicity of the cold gas.', 'astro-ph-0210471-1-43-2': 'We then use a standard Galactic extinction curve and a "slab" model to compute the extinction as a function of wavelength and inclination.', 'astro-ph-0210471-1-43-3': 'The extinction correction is applied indiscriminantly to the stellar and line emission.', 'astro-ph-0210471-1-43-4': 'This is probably unphysical as it is likely that the star-forming regions that produce the emission lines are more heavily extinguished than the underlying old stellar population, but at our current level of modelling, we ignore this effect.', 'astro-ph-0210471-1-44-0': 'The recipes for star formation, supernova feedback, and chemical evolution involve free parameters, which we set by requiring a halo with a virial velocity of 220 km/s to host a galaxy with an average I-band luminosity of about [MATH], and with an average gas fraction of [MATH] to [MATH], consistent with observations of local spiral galaxies.', 'astro-ph-0210471-1-44-1': 'If we assume that mergers with mass ratios greater than [MATH] 1:3 form spheroids, we find that the models produce the correct morphological mix of spirals, S0s and ellipticals at the present day.', 'astro-ph-0210471-1-44-2': 'It has previously been shown in numerous works (e.g. Somerville Primack 1999) that this approach leads to fairly good agreement with numerous key galaxy observations, such as the local luminosity function, colours, and clustering properties.', 'astro-ph-0210471-1-45-0': 'We construct a "mock 2dF catalogue" of [MATH] model galaxies with the same magnitude limit, wavelength coverage and spectral resolution, and redshift range as the 2dF survey.', 'astro-ph-0210471-1-45-1': 'The synthetic spectra are expressed in terms of photon counts and the total number of counts in each spectrum is normalised to unity, as in the prepared observed spectra.', 'astro-ph-0210471-1-46-0': 'The calculation of the strength of the nebular emission features is one of the less precise aspects of the spectral synthesis packages.', 'astro-ph-0210471-1-46-1': 'For this reason in Fig. [REF] we show the first two principal components ([MATH],[MATH]) for the SAM ensemble of synthetic spectra.', 'astro-ph-0210471-1-46-2': 'It can be seen that the distribution of the mock catalogue principal components is quite similar to that of the observed 2dFGRS galaxies.', 'astro-ph-0210471-1-46-3': 'To guide the eye, the same set of evolutionary tracks shown in Fig. [REF], derived from the "simple" models, is also shown.', 'astro-ph-0210471-1-46-4': 'The most noticeable difference between the observed distribution and that of the mock, is that the former is substantially broader (most likely due to observational effects such as noise, reddening and evolution combined with the fact that the observed galaxies form a much larger sample).', 'astro-ph-0210471-1-46-5': 'It is also interesting that the simple model tracks span the same locus of the ([MATH], [MATH]) parameter space as the SAM galaxies, despite the fact that as we have emphasised the star formation histories are very different in the two kinds of models.', 'astro-ph-0210471-1-47-0': 'We can now discover whether the strong connection between [MATH] and the birthrate parameter [MATH] that we demonstrated for the simple models hold true for galaxies with more arbitrary star formation histories.', 'astro-ph-0210471-1-47-1': 'In Fig. [REF], we show [MATH] vs. [MATH] for the mock-2dF ensemble of SAM galaxy spectra.', 'astro-ph-0210471-1-47-2': 'Again we see a strong relationship between [MATH] and the [MATH] parameter.', 'astro-ph-0210471-1-47-3': 'Also, it is encouraging to note that the range of [MATH] spanned by the SAM galaxies (-5 to 15) is similar to that spanned by the 2dF galaxies.', 'astro-ph-0210471-1-47-4': 'The fact that the spectral properties of the galaxies in the SAM ensemble appear similar to those of the real 2dF galaxies enables us to use the results of the SAMs to draw a correspondence between the actual numerical value of [MATH] and [MATH].', 'astro-ph-0210471-1-47-5': 'This leads to the interesting conclusion that the dividing line between the two "bumps" in the bimodal distribution of [MATH] seen in Fig. [REF], [MATH], corresponds to galaxies that are forming stars at about 1/10th of their past-averaged rate.', 'astro-ph-0210471-1-47-6': 'Similarly, the other two [MATH] thresholds adopted by Madgwick et al. (2002) to calculate luminosity functions per spectral type may be matched up with [MATH] values using Fig. [REF].', 'astro-ph-0210471-1-48-0': 'We have also investigated correlations between [MATH] and other physical properties of galaxies in the SAMs, such as bulge-to-total ratio or mean stellar age.', 'astro-ph-0210471-1-48-1': 'While other correlations exist, none are as tight as the correlation shown in Fig. [REF] between [MATH] and the birthrate parameter [MATH].', 'astro-ph-0210471-1-48-2': 'Of all the correlations we investigated, the connection between [MATH] and star formation history seems to present the most straightforward interpretation of the [MATH] parameter.', 'astro-ph-0210471-1-48-3': 'We defer a more detailed investigation of the other physical correlations in the SAMs and comparison with the 2dF and other data sets to a future work.', 'astro-ph-0210471-1-49-0': '# Discussion', 'astro-ph-0210471-1-50-0': 'In this paper we have studied the physical interpretation of a PCA-based spectral parameter, [MATH], defined for the analysis of the 2dFGRS spectra (Madgwick et al. 2002).', 'astro-ph-0210471-1-50-1': "Although the definition of [MATH] was motivated by the need to compromise between the desire to extract the maximum amount of statistical information from the spectra (in the sense of PCA's maximum variance) and the limitations imposed by the instrumental uncertainties of the 2dF, we have argued here that this parameter has a straightforward and physically meaningful interpretation.", 'astro-ph-0210471-1-50-2': 'We find that [MATH] is an indicator of the star formation history of the galaxy, and is tightly correlated with the birthrate parameter [MATH], which characterises the ratio of present to past-averaged star formation.', 'astro-ph-0210471-1-51-0': 'A first indication of this correspondence is the correlation we find between [MATH] and and the equivalent width of the H[MATH] emission line, EW(H[MATH]), which has been used as a direct measure of the birthrate parameter in previous studies (e.g. Kennicutt, Tamblyn Congdon 1994).', 'astro-ph-0210471-1-51-1': 'We show that in models with simple star formation histories, parameterised in terms of an exponential function of time, there is a one-to-one correspondence between [MATH] and the birthrate parameter [MATH] (for a fixed formation time).', 'astro-ph-0210471-1-51-2': 'Perhaps this is not too surprising in the context of classical spectral synthesis work - it has long been known (e.g. Tinsley 1970) that a simple SF history can be used to interpret spectral appearance, i.e. [MATH] in this context.', 'astro-ph-0210471-1-51-3': 'More surprisingly, we show also that a strong relationship between [MATH] and [MATH] is also exhibited by model galaxies with much more complex star formation histories, created within a hierarchical cosmological framework using a semi-analytic model.', 'astro-ph-0210471-1-52-0': 'The explanation of this surprisingly simple result must lie in the physical processes that determine the appearance of a galaxy spectrum.', 'astro-ph-0210471-1-52-1': 'First, it is important to remember that at optical wavelengths, where all of this analysis has been carried out, the spectrum is dominated by the most luminous stars, and is therefore biased towards the most recent significant star formation activity.', 'astro-ph-0210471-1-52-2': 'The slope of the optical continuum and the strength of stellar absorption lines evolve on the timescale of the lifetimes of intermediate type main sequence stars, roughly several Gigayears.', 'astro-ph-0210471-1-52-3': 'The strengths of nebular emission features, however, evolve on much shorter timescales, as they require the presence of very hot, short lived O and B type stars.', 'astro-ph-0210471-1-52-4': 'These features therefore depend on the star formation history on the timescale of 5-10 Megayears.', 'astro-ph-0210471-1-52-5': 'The [MATH] parameter has been defined so as to be insensitive to the continuum slope (for practical reasons), and hence represents a sequence from strong absorption lines to strong emission lines.', 'astro-ph-0210471-1-52-6': 'It is well-known that the strengths of stellar absorption lines (especially Hydrogen recombination lines such as the Balmer series) are good indicators of stellar age, while the strength of the nebular emission lines are indicators of present star formation rate.', 'astro-ph-0210471-1-52-7': 'With all this in mind, we can deduce that the rapid evolution in the value of [MATH] over the first several hundred Myr of a galaxies life (see Fig. [REF]) reflects the fading of the emission lines as the star formation rate declines and the young stars burn out.', 'astro-ph-0210471-1-52-8': 'Galaxies then tend to "pile up" at low values of [MATH] as the age of their dominant stellar population exceeds a few Gyr.', 'astro-ph-0210471-1-53-0': 'In this sense it is straight-forward to understand the relationship between [MATH] and [MATH] for our "simple" models, of monotonically decreasing star formation activity.', 'astro-ph-0210471-1-53-1': 'However, the star formation histories of galaxies created by the semi-analytic models can be quite complex.', 'astro-ph-0210471-1-53-2': 'The star formation rate fluctuates dramatically and non-monotonically over time as the galaxy exhausts its gas and then accretes a new gas supply, or experiences bursts of star formation triggered by mergers.', 'astro-ph-0210471-1-53-3': 'The issue we must now address is how an optical diagnostic such as [MATH], which is dominated by only the most recent events of star-formation, can be so closely related to the birth-rate parameter, [MATH], which incorporates the entire formation history of a galaxy.', 'astro-ph-0210471-1-54-0': 'The basis of the hierarchical picture of galaxy formation - from which our mock catalogue of SAM galaxies has been derived - is that galaxy properties are determined by the merger histories of their host dark matter halos.', 'astro-ph-0210471-1-54-1': 'In essence, the history of a specific halo (or galaxy) can be understood in terms of how the density on the scale of that structure compares to the background density.', 'astro-ph-0210471-1-54-2': 'A region with density much above the average (a "many sigma" peak, in the language of Gaussian random fields) collapses early, while lower density peaks collapse later.', 'astro-ph-0210471-1-54-3': 'It can therefore be conjectured that the early collapse of a dense dark matter halo can be associated with early star formation, early consumption of all available gas, and low present-day star formation rates.', 'astro-ph-0210471-1-54-4': 'Conversely, late-forming objects will have a sustained gas supply and ongoing star formation.', 'astro-ph-0210471-1-54-5': 'This suggests that the birthrate parameter [MATH] reflects perhaps the most theoretically fundamental property of a galaxy (or of its host halo): small-[MATH] galaxies represent rare, many-sigma peaks in the primordial density field, while large-[MATH] galaxies are formed in more common, lower density peaks.', 'astro-ph-0210471-1-54-6': 'This trend is manifested in the numerous empirical correlations between galaxy "type" (as characterised by morphology, colour, or spectral type), luminosity, and environment.', 'astro-ph-0210471-1-54-7': 'We expect this simple picture to be complicated by the details of gas cooling, star formation, feedback, etc., but the results from the semi-analytic models (which include all of these effects at some level, although of course the real Universe is likely to be even more complicated) suggest that this only introduces a moderate amount of scatter on top of the general trend.', 'astro-ph-0210471-1-55-0': 'We therefore conclude that the [MATH] parameter represents a promising candidate for galaxy classification in large modern redshift surveys.', 'astro-ph-0210471-1-55-1': 'It was adopted in previous analyses of the 2dFGRS because of its practical advantages: it is straightforward and efficient to compute in an automated fashion, and it is robust to the instrumental uncertainties commonly associated with fibre-based multi-object spectrographs.', 'astro-ph-0210471-1-55-2': 'Here we have shown that [MATH] also has a straightforward physical interpretation which can be intuitively connected both with traditional classifiers such as morphology, and that it seems to be connected with theoretically fundamental properties of galaxies within the modern hierarchical structure formation paradigm.'}

{'astro-ph-0210471-2-0-0': 'We investigate the physical significance of a new spectral parameter, [MATH].', 'astro-ph-0210471-2-0-1': 'This parameter was defined based on a Principal Component Analysis of the 2dF Galaxy Redshift Survey, to retain astrophysical information while minimising the effect of measurement uncertainties.', 'astro-ph-0210471-2-0-2': 'We find that while [MATH] is correlated with morphological type, there is a large scatter in this relationship.', 'astro-ph-0210471-2-0-3': 'A tighter empirical relationship is found between [MATH] and the equivalent width of the H[MATH] line, suggesting a connection with the star formation rate.', 'astro-ph-0210471-2-0-4': 'We pursue this connection using spectral synthesis models.', 'astro-ph-0210471-2-0-5': 'Using models in which the star formation history is parameterised in terms of an exponentially decreasing function of time, we find that there is a tight correlation between [MATH] and the ratio of the present to the past-averaged rate of star-formation, often known as the "birthrate" parameter [MATH].', 'astro-ph-0210471-2-0-6': 'This correlation also holds in models with much more complicated star formation histories, generated by a semi-analytic model of galaxy formation based upon the hierarchical formation scenario.', 'astro-ph-0210471-2-0-7': 'There are two possible causes for the tight correlations we find, between [MATH] and [MATH], in those galaxies with the most complex star formation histories; Firstly, the spectra themselves may be degenerate to the actual long-term star formation history of each galaxy in the optical wavelength range probed by the 2dFGRS.', 'astro-ph-0210471-2-0-8': 'Secondly, the birthrate parameter [MATH] may represent a physically fundamental quality of galaxy halos - their over-density relative to the background density - such that small-[MATH] galaxies form in high peaks (which collapse early) while large-[MATH] galaxies represent lower peaks, which collapse later.', 'astro-ph-0210471-2-0-9': 'We conclude that the tight connection with [MATH] makes [MATH] a physically meaningful, as well as convenient and robust, statistic for galaxy parameterisation and classification.', 'astro-ph-0210471-2-1-0': '# Introduction', 'astro-ph-0210471-2-2-0': 'Galaxy redshift surveys are now probing the galaxy distribution of the local Universe more accurately than ever before, and in so doing they are establishing many fundamental properties of the galaxy population and its large-scale structure.', 'astro-ph-0210471-2-2-1': 'The 2dF Galaxy Redshift Survey (2dFGRS) is one such ambitious project, conceived with the aim of mapping the galaxy distribution to an extinction corrected [MATH] magnitude limit.', 'astro-ph-0210471-2-2-2': 'This particular survey is now essentially complete with approximately 230,000 galaxy spectra having already been acquired, and has already started to yield significant results (e.g. Percival et al. 2001; Madgwick et al. 2002 and others).', 'astro-ph-0210471-2-2-3': 'In addition to this survey, the Sloan Digital Sky Survey (Strauss et al., 2002) is also underway, and once complete will obtain up to 1,000,000 individual galaxy spectra.', 'astro-ph-0210471-2-3-0': 'Apart from the main science goals of quantifying the large-scale structure of the Universe, one of the most significant contributions of galaxy redshift surveys is to our understanding of the galaxy population itself, through the information about galaxy properties contained in the observed spectra.', 'astro-ph-0210471-2-3-1': 'Having a data set of 230,000 galaxy spectra - as in the case of the 2dFGRS - allows us to test the validity of galaxy formation and evolution scenarios with unprecedented accuracy.', 'astro-ph-0210471-2-3-2': 'However, the sheer size of the data set presents its own unique problems.', 'astro-ph-0210471-2-3-3': "Clearly, in order to make the spectral data set more 'digestible', some form of data compression is necessary.", 'astro-ph-0210471-2-3-4': 'Familiar statistics such as equivalent width measurements, morphological types, and broad-band colours are really just compression techniques in some sense.', 'astro-ph-0210471-2-3-5': 'These quantities can be compared with theoretical predictions and simulations, and hence can set constraints on scenarios for galaxy formation and biasing.', 'astro-ph-0210471-2-4-0': 'The approach that has been adopted in Madgwick et al. (2002) is to define a spectral indicator, [MATH], based on Principal Component Analysis (PCA).', 'astro-ph-0210471-2-4-1': 'Because of instrumental limitations, the flux calibration of the 2dF spectra is unreliable, and hence robust measurements of the shape of the spectral continua are not possible.', 'astro-ph-0210471-2-4-2': 'For this reason, the [MATH] statistic was designed to (in a loose sense) preserve the most discriminatory information in the spectra, while being robust to the instrumental uncertainties.', 'astro-ph-0210471-2-4-3': 'Effectively, the parameter [MATH] measures the strength of absorption and emission lines, while remaining insensitive to the slope of the continuum (or broad band colour).', 'astro-ph-0210471-2-5-0': 'The motivation of the definition of the [MATH] statistic was in part pragmatic.', 'astro-ph-0210471-2-5-1': 'The goal of this paper is to address the fundamental question of the physical significance of this parameter, in terms of how it relates to the physical properties of galaxies.', 'astro-ph-0210471-2-5-2': 'A substantial amount of work has been carried out in the past on relating the observed spectra of galaxies to the physical processes occurring in them (see e.g Kennicutt, Tamblyn Congdon 1994; Ronen, Aragon-Salamanca Lahav 1999; Carter, Fabricant, Geller Kurtz 2001 and references therein), and the advent of a large and uniform survey such as the 2dFGRS will result in many more advances in this field.', 'astro-ph-0210471-2-5-3': 'However, for the present analysis we restrict ourselves to relating only the 2dFGRS spectral classification parameter, [MATH], to these processes and leave a fuller development of this subject for future works.', 'astro-ph-0210471-2-5-4': 'The investigation presented here is particularly timely as many new results from the 2dFGRS are being presented which make exclusive use of this parameter to characterise and partition the galaxy population (e.g. Madgwick et al. 2002; Norberg et al. 2002; Martinez et al. 2002).', 'astro-ph-0210471-2-6-0': 'We make use of two kinds of models to pursue this question.', 'astro-ph-0210471-2-6-1': 'In the first approach, we adopt a simple parameterisation to describe the star formation history of each galaxy in terms of an exponentially decreasing function of time.', 'astro-ph-0210471-2-6-2': 'For a given redshift of observation, the star formation history is then characterised by a single parameter: the timescale of the exponential decline.', 'astro-ph-0210471-2-6-3': "These star formation histories are then convolved with spectral synthesis models, using the PEGASE (Projet d'Etude des GAlaxies par Synthese Evolutive) code, developed by Fioc Rocca-Volmerange (1997).", 'astro-ph-0210471-2-6-4': 'A particular advantage of this package compared to others available in the literature (e.g. Bruzual Charlot 1993) is that it is possible to include modelling of nebular emission from star forming regions in each galaxy - an important ingredient in the calculation of the [MATH] parameter.', 'astro-ph-0210471-2-6-5': 'We refer to these models as the "simple" models, because of the rather simplified nature of the parametrised star formation histories.', 'astro-ph-0210471-2-7-0': 'The limitation of this simple method is that we must explicitly assume some family of star-formation histories for our galaxy population.', 'astro-ph-0210471-2-7-1': 'We therefore extend this approach to incorporate the cosmological framework of the Cold Dark Matter (CDM) scenario, by using semi-analytic techniques (see e.g. Kauffmann, White Guiderdoni 1993; Cole et al. 1994; Somerville Primack 1999).', 'astro-ph-0210471-2-7-2': 'We make use of a mock catalogue of synthetic spectra which was designed to match the selection criteria of the 2dFGRS, analysed previously in Slonim et al. (2001), and created using an updated version of the code described in Somerville Primack (1999).', 'astro-ph-0210471-2-7-3': 'There are several advantages of using a semi-analytic model to create spectra for this exercise.', 'astro-ph-0210471-2-7-4': 'The star formation history of each galaxy will reflect at least some of the complexity of the interconnected processes of dark matter clustering, gas cooling, galaxy merging, supernova feedback, etc., modelled self-consistently within a specific cosmological framework.', 'astro-ph-0210471-2-7-5': 'In addition, as we can select galaxies so as to reproduce the same redshift and luminosity distributions displayed by the observed 2dF galaxies, the resulting ensemble should contain a mix of galaxies with different sorts of star formation histories that is similar to the actual observed sample.', 'astro-ph-0210471-2-7-6': 'Finally, the semi-analytic model also yields many other physical parameters for each galaxy, giving us the potential to further probe the physical processes that may be related to [MATH].', 'astro-ph-0210471-2-8-0': 'We find that in the simple models, there is essentially a one-to-one relationship between [MATH] and the ratio of the present rate of star formation to its past time-averaged value.', 'astro-ph-0210471-2-8-1': 'This quantity is sometimes known as the "birthrate" parameter, [MATH] (Scalo 1986).', 'astro-ph-0210471-2-8-2': 'Even more surprisingly, we find that there is a strong correlation between [MATH] and [MATH] even in the spectra produced by the semi-analytic models, despite the complex and diverse nature of the underlying star formation histories.', 'astro-ph-0210471-2-8-3': 'Empirically, the [MATH] parameter is known to be tightly correlated with morphological type and colour.', 'astro-ph-0210471-2-8-4': 'We therefore argue that the [MATH] parameter presents a practical and robust means of extracting a fundamental measure of galaxy type from the relatively low-quality spectra typical of present-day large redshift surveys.', 'astro-ph-0210471-2-9-0': 'Section 2 of this paper briefly reviews the operational definition of and motivation behind the spectral classification parameter [MATH], used in the 2dFGRS.', 'astro-ph-0210471-2-9-1': 'In Section 3 we make use of the "simple" models based on exponentially declining star formation histories to show how [MATH] is related to the star-formation history of a galaxy.', 'astro-ph-0210471-2-9-2': 'Section 4 then generalises these results using formation histories generated with a semi-analytic model.', 'astro-ph-0210471-2-9-3': 'We conclude with a discussion of our results and ways to build upon the results presented here.', 'astro-ph-0210471-2-10-0': '# Spectral Classification in the 2dFGRS', 'astro-ph-0210471-2-11-0': '## Principal Component Analysis', 'astro-ph-0210471-2-12-0': 'Principal component analysis (PCA) is a well established statistical technique which has proved very useful in dealing with high dimensional data sets (see e.g. Murtagh Heck 1987; Connolly et al. 1995).', 'astro-ph-0210471-2-12-1': 'In the particular case of galaxy spectra we are typically presented with approximately 1000 spectral channels per galaxy, however when used in applications this is usually compressed down to just a few numbers, either by integrating over small line features - yielding equivalent widths - or over broad band filters.', 'astro-ph-0210471-2-12-2': 'The key advantage of using PCA in our data compression is that it allows us to make use of all the information contained in the spectrum in a statistically unbiased way, i.e. without the use of such ad hoc filters.', 'astro-ph-0210471-2-13-0': 'In order to perform the PCA on our galaxy spectra we first construct a representative volume limited sample of the galaxies.', 'astro-ph-0210471-2-13-1': 'We compute the "mean spectrum" from this ensemble, and subtract this mean from all of the galaxy spectra.', 'astro-ph-0210471-2-13-2': 'When we apply the PCA to this sample an orthogonal set of components (eigenspectra) is constructed, which span the wavelength space occupied by the galaxy spectra.', 'astro-ph-0210471-2-13-3': 'These components have been specifically chosen by the PCA in such a way that as much information (variance) is contained in the first eigenspectrum as possible, and that the amount of the remaining information in all the subsequent eigenspectra is likewise maximised.', 'astro-ph-0210471-2-13-4': 'Therefore, if the information contained in the first [MATH] eigenspectra is found to be significantly greater than that in the remaining eigenspectra we can significantly compress the data set by swapping each galaxy spectrum (described by 1000 channels) with its projections onto just those [MATH] eigenspectra.', 'astro-ph-0210471-2-14-0': 'In the case of the 2dF galaxy spectra we find that approximately two thirds of the total variance (including the noise) in the spectra can be represented in terms of only the first two projections ([MATH], [MATH]).', 'astro-ph-0210471-2-14-1': 'So, at least to a first approximation, galaxy spectra can be thought of as a two dimensional sequence in terms of these two projections.', 'astro-ph-0210471-2-15-0': 'In Fig. [REF] we show these first two eigenspectra.', 'astro-ph-0210471-2-15-1': 'It can be seen from this figure that whilst the first eigenspectrum contains information from both lines and the continuum, the second is dominated by absorption and emission lines.', 'astro-ph-0210471-2-15-2': 'Because of this it is possible to take two simple linear combinations which isolate either the continuum or the emission/absorption line features.', 'astro-ph-0210471-2-15-3': 'In effect what we are doing when we utilise these linear combinations is rotating the axes defined by the PCA to make the interpretation of the components more straightforward.', 'astro-ph-0210471-2-15-4': 'In so doing we can see that a parameterisation in terms of [MATH] and [MATH] is essentially equivalent to a two dimensional sequence in colour (continuum slope) and the average emission/absorption line strength.', 'astro-ph-0210471-2-15-5': 'This ability to isolate the continuum and line components of the galaxy spectra turns out to be very useful, as will be discussed in Section 2.2.', 'astro-ph-0210471-2-16-0': '## The [MATH] Parameter', 'astro-ph-0210471-2-17-0': 'The 2dF instrument makes use of up to 400 optical fibres with a diameter of 140[MATH]m (corresponding to [MATH] on the sky, depending on plate position.', 'astro-ph-0210471-2-17-1': 'See Lewis et al. 2002).', 'astro-ph-0210471-2-17-2': 'The instrument itself was designed to measure large numbers of redshifts in as short an observing time as possible.', 'astro-ph-0210471-2-17-3': 'However, in order to optimise the number of redshifts that can be measured in a given period of time, compromises had to be made with respect to the spectral quality of the observations.', 'astro-ph-0210471-2-17-4': 'Therefore if one wishes to characterise the observed galaxy population in terms of their spectral properties, care must be taken in order to ensure that these properties are robust to the instrumental uncertainties.', 'astro-ph-0210471-2-18-0': 'The quality and representativeness of the observed spectra can be compromised in several ways and a more detailed discussion of these issues is presented in previous work (see e.g. Madgwick et al. 2002).', 'astro-ph-0210471-2-18-1': 'The net effect is that the uncertainties introduced into the fibre-spectra predominantly affect the calibration of the continuum slope and have relatively little impact on the emission/absorption line strengths.', 'astro-ph-0210471-2-18-2': 'For this reason any given galaxy spectrum which is projected onto the plane defined by ([MATH],[MATH]) will not be uniquely defined in the direction of varying continuum but will be robust in the orthogonal direction (which measures the average line strength).', 'astro-ph-0210471-2-19-0': 'The linear combination of the first two eigenspectra which is robust to these uncertainties is shown in Fig. [REF] (b) and denoted by [MATH] (ETATYPE in the 2dFGRS catalogue).', 'astro-ph-0210471-2-19-1': 'It is simply [EQUATION] where [MATH] is a constant which we find empirically to be [MATH].', 'astro-ph-0210471-2-20-0': 'We have now identified the single statistically dominant component of the galaxy spectra which is robust to the known instrumental uncertainties.', 'astro-ph-0210471-2-20-1': 'We have therefore chosen to adopt this (continuous) variable as our measure of spectral type.', 'astro-ph-0210471-2-20-2': 'Having defined [MATH] in this formal and pragmatic manner, however, we are left with the question of whether it is physically meaningful, and how it is to be interpreted.', 'astro-ph-0210471-2-20-3': 'This is the issue that we now address.', 'astro-ph-0210471-2-21-0': '## Correlation of [MATH] with traditional galaxy classifiers', 'astro-ph-0210471-2-22-0': 'We show the observed distribution of [MATH] for the 2dF galaxies in Fig. [REF].', 'astro-ph-0210471-2-22-1': 'It is rather intriguing that this distribution is strongly bimodal.', 'astro-ph-0210471-2-22-2': 'Fig. [REF] also shows the [MATH] projections of spectra from galaxies in the Kennicutt Atlas (Kennicutt 1992), which have known morphologies.', 'astro-ph-0210471-2-22-3': 'There is a correspondence between morphology and [MATH] in the expected sense: the value of [MATH] increases as one moves towards later type objects in the Hubble sequence.', 'astro-ph-0210471-2-22-4': 'Larger values of [MATH] indicate more dominant emission lines, and it is well-known that later type galaxies have stronger emission lines.', 'astro-ph-0210471-2-22-5': 'From this diagram we see that [MATH] seems to be correlated with morphological type, at least based on this small sample of galaxies.', 'astro-ph-0210471-2-22-6': 'Norberg et al. (2002) showed the distribution of [MATH] for a larger sample of galaxies that had been classified into four morphological types (E, S0, Sp and Irr), and showed that while there is a correlation between morphological type and [MATH], there is a large scatter.', 'astro-ph-0210471-2-22-7': 'The correspondence between [MATH] and morphological type is investigated in more detail in Madgwick (2002).', 'astro-ph-0210471-2-23-0': 'We can also compare the value of [MATH] derived for each galaxy to the equivalent width of the H[MATH] emission line, for a sample of high signal-to-noise ratio emission line spectra.', 'astro-ph-0210471-2-23-1': 'This is shown in Fig. [REF], from which it can be seen that there is a strong correlation between these quantities.', 'astro-ph-0210471-2-23-2': 'It is well-known that the equivalent width of H[MATH] is a measure of star formation; indeed it has been used to estimate the birthrate parameter [MATH] (the ratio of the present rate of star formation to the past averaged value - see e.g. Figure 3 of Kennicutt, Tamblyn Congdon 1994).', 'astro-ph-0210471-2-23-3': 'We return to this point in Section [REF], where we use galaxy formation models to investigate how [MATH] relates to the star-formation history.', 'astro-ph-0210471-2-24-0': '# Comparison with Simple Models', 'astro-ph-0210471-2-25-0': 'The results shown above suggest that [MATH] is related both to galaxy morphology and to star formation history.', 'astro-ph-0210471-2-25-1': 'In this section we further explore and test this hypothesis with models produced by combining simple parameterised star formation histories with spectral synthesis models.', 'astro-ph-0210471-2-25-2': "In this way, we can make a direct connection between a galaxy's star formation history and the value of our spectral classification statistic, [MATH].", 'astro-ph-0210471-2-26-0': '## Model Ingredients', 'astro-ph-0210471-2-27-0': 'Stellar population models combine theoretical temperature-luminosity tracks for stars of various masses with model stellar atmospheres and an assumed Initial Mass Function (IMF) in order to produce synthetic spectra for a "single burst" population of a uniform age and metalicity.', 'astro-ph-0210471-2-27-1': 'In order to use these models to obtain predictions for a composite stellar population (i.e. a galaxy), the "single burst" models must be combined with an assumed star formation history, which provides the distribution of stellar ages and metalicities within the galaxy.', 'astro-ph-0210471-2-27-2': 'Since the classical work on spectral synthesis by Tinsley (1980), many authors have adopted a simple parametrisation of the star formation history in terms of an exponential function of time (e.g. Bruzual 1983): [EQUATION]', 'astro-ph-0210471-2-27-3': 'Here, [MATH] is the time at which star formation first commences, and [MATH] is the characteristic timescale of star formation.', 'astro-ph-0210471-2-27-4': 'Most observed spectra can be modelled rather well using this simple function with the appropriate choice of [MATH].', 'astro-ph-0210471-2-27-5': 'A typical spiral galaxy spectrum might be well-fit with [MATH] Gyr, while later type galaxies require larger values and early type galaxies require smaller values of [MATH].', 'astro-ph-0210471-2-28-0': 'Nebular emission lines from ionised H[MATH] regions and extinction due to dust also contribute to the appearance of a galaxy spectrum.', 'astro-ph-0210471-2-28-1': 'Modelling these processes is more complicated, and requires assumptions about the ionisation state, geometry, and metalicity of H[MATH] regions, and the dust mass, composition and geometry.', 'astro-ph-0210471-2-29-0': 'The models presented in this Section are based on this simple but well-defined picture, and are realized using version 2.0 of the PEGASE software package (Fioc Rocca-Volmerange) In all of the models shown we have adopted a Kennicutt (1983) initial-mass function with stellar masses in the range [MATH].', 'astro-ph-0210471-2-29-1': 'In addition, the metalicity of each galaxy is traced in a self-consistent manner using the prescription of Woosley Weaver (1995), to model the enrichment of the inter-stellar medium.', 'astro-ph-0210471-2-30-0': 'Nebular emission lines result from light re-emitted by the ionised gas in star-forming regions.', 'astro-ph-0210471-2-30-1': 'A prescription for determining the strengths of these lines is implemented in the PEGASE code.', 'astro-ph-0210471-2-30-2': 'This process involves the absorption of Lyman continuum photons (below 912) by the nebular gas, which gets ionised, and reaches recombination equilibrium (Osterbrock 1989).', 'astro-ph-0210471-2-30-3': 'It is assumed that 70% of these photons are absorbed by the gas at solar metalicity.', 'astro-ph-0210471-2-30-4': 'In this approach, the strength of the nebular emission lines is a function of the age of the stellar population only, and metalicity and geometric effects are neglected.', 'astro-ph-0210471-2-31-0': 'The PEGASE code provides a simple way to model the effects of dust extinction on the synthetic spectra, in which the optical depth is estimated from the mass of gas and the metalicity.', 'astro-ph-0210471-2-31-1': 'The absorption is then estimated using observational data for a mixture of graphites and silicates as in the Milky Way and the Magellanic Clouds (Draine Lee 1984 and Pei 1992).', 'astro-ph-0210471-2-31-2': 'In making this calculation it is also necessary to make an assumption about the geometry of the galaxy.', 'astro-ph-0210471-2-31-3': 'We investigated models in which we assumed a spheroidal geometry and also an inclination-averaged disk geometry.', 'astro-ph-0210471-2-32-0': 'The spectra generated by the PEGASE code over the optical wavelength range are generally given in 10 bins, and the emission line strengths are specified separately in terms of their peak fluxes over the continuum.', 'astro-ph-0210471-2-32-1': 'In order to make comparisons between these spectra and the 2dFGRS we therefore interpolate the synthetic spectra onto the 4 binned wavelength range of the 2dFGRS.', 'astro-ph-0210471-2-32-2': 'Because the given emission lines are not resolved in this binning we create lines by superposing Gaussian profiles with the specified peak flux and FWHM corresponding to that calculated for the 2dF spectrograph using arc line measurements ([MATH] pixels).', 'astro-ph-0210471-2-32-3': 'We convert the given synthetic spectra to units of counts/bin by multiplying the flux (erg/s/) by the wavelength.', 'astro-ph-0210471-2-32-4': 'We then normalise each spectrum to have mean counts of [MATH] over our entire wavelength range.', 'astro-ph-0210471-2-32-5': 'The spectra processed in this way are directly comparable to those of the 2dFGRS.', 'astro-ph-0210471-2-33-0': 'Spectral histories of a given galaxy are compiled for a range of formation times ([MATH] Gyr), and a grid of values of [MATH]: 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, and 2.0 Gyr.', 'astro-ph-0210471-2-33-1': 'We also calculate spectral histories for an instantaneous burst of star-formation at [MATH].', 'astro-ph-0210471-2-33-2': 'Because this grid does not comprise a well-defined sample of galaxies, to compare with the results from the 2dFGRS, we first subtract the mean spectrum from the 2dFGRS volume limited sample described in the previous Section from each spectrum.', 'astro-ph-0210471-2-33-3': 'We then compute the projection of each spectrum on the eigenspectra derived from the 2dFGRS volume limited sample.', 'astro-ph-0210471-2-33-4': 'The first two projections ([MATH], [MATH]) derived in this manner are shown for our grid of models in Fig. [REF], along with the corresponding projections for a random subsample of galaxies in the [MATH] volume-limited 2dFGRS sample.', 'astro-ph-0210471-2-33-5': 'Model grids are shown both with and without including the effects of dust extinction.', 'astro-ph-0210471-2-33-6': 'It can be seen that dust does not have a dramatic impact on the results, but does seem to improve the agreement between the model tracks and the locus of the observed galaxies.', 'astro-ph-0210471-2-34-0': 'Already it is interesting that these simple models seem to cover the same range of the ([MATH], [MATH]) parameter space occupied by the observed galaxy population.', 'astro-ph-0210471-2-34-1': 'Evidently, at least in this simple scheme, galaxies are "born" in the lower right-hand corner of the plot, and progress towards the upper left-hand corner as they age.', 'astro-ph-0210471-2-34-2': 'As well, galaxies in the upper right part of the diagram tend to have smaller values of the characteristic timescale for star formation, [MATH], associated with early type galaxies, while as one moves diagonally towards the lower left corner, galaxies have the more extended timescales for star formation associated with late type galaxies.', 'astro-ph-0210471-2-34-3': 'Perhaps unsurprisingly, the vast majority of the 2dF galaxies (which are fairly luminous) are consistent with times since formation [MATH] between 2-10 Gyr.', 'astro-ph-0210471-2-34-4': 'It should be kept in mind, however, that the instrumental effects discussed in Section [REF] can introduce random scatter in this diagram.', 'astro-ph-0210471-2-34-5': 'We therefore turn now to the more robust [MATH] parameter.', 'astro-ph-0210471-2-35-0': '## The Physical Significance of [MATH]', 'astro-ph-0210471-2-36-0': 'We argued based on a visual inspection of the eigenspectrum used to define the [MATH] projection that [MATH] essentially measures the strength of stellar and interstellar absorption line features and the strength of nebular emission line features.', 'astro-ph-0210471-2-36-1': 'The strength of the absorption features mainly indicates the age and metalicity of the stellar population that dominates the optical luminosity of the galaxy, and the emission lines are strongly correlated with star formation activity.', 'astro-ph-0210471-2-36-2': 'The comparison with classical galaxy classification methods such as visual morphology or equivalent width, shown in Section [REF], suggested that [MATH] may be connected with a measure of the star formation relative to the existing older stellar population, such as the birthrate parameter, [MATH] (Scalo 1986).', 'astro-ph-0210471-2-36-3': 'The birthrate parameter is defined as the ratio between the current and past-averaged star-formation rate, which in the case of the simple exponential star formation law is given analytically by [EQUATION]', 'astro-ph-0210471-2-36-4': 'In this section we investigate our hypothesis that [MATH] might be correlated with the birthrate parameter, [MATH].', 'astro-ph-0210471-2-37-0': 'The relationship between [MATH] and [MATH] for the grid of models with simple star formation histories (as before) is shown in Fig. [REF].', 'astro-ph-0210471-2-37-1': 'For a fixed formation time, [MATH], there is a well-defined relationship between the star formation history, as characterised by the birthrate parameter [MATH], and the [MATH] spectral parameter.', 'astro-ph-0210471-2-37-2': 'This helps in the interpretation of [MATH] for the observed population of galaxies.', 'astro-ph-0210471-2-37-3': 'At least in this simple scheme, galaxies with small values of [MATH] formed most of their stars in the past, and are currently evolving rather passively.', 'astro-ph-0210471-2-37-4': 'This fits in well with the classical picture of early type galaxy formation.', 'astro-ph-0210471-2-37-5': 'Galaxies with larger values of [MATH] have had a significant amount of recent star formation, characteristic of spiral and irregular galaxies.', 'astro-ph-0210471-2-37-6': 'Recalling from Fig. [REF] that the 2dFGRS sample spans a range of [MATH] values of about -5 to 10, this suggests again that most of the observed galaxies in the 2dF require times since formation greater than about 1 Gyr.', 'astro-ph-0210471-2-37-7': 'From this diagram, we arrive at a prediction that the distribution of [MATH] for galaxies observed at larger look-back times should exhibit a shift towards higher values.', 'astro-ph-0210471-2-37-8': 'We find that the effect of extinction corrections is to slightly lower the calculated [MATH] for a given star formation scenario, however, this is only a small effect and does not alter the qualitative interpretation that we have suggested.', 'astro-ph-0210471-2-38-0': 'A limitation of the simple characterisation of the star formation histories that we have adopted here is manifested by the asymptote of the [MATH] values towards unity: because the star formation rate was assumed to be monotonically decreasing, [MATH] is always less than one.', 'astro-ph-0210471-2-38-1': 'It is clear that we must be cautious in over-interpreting these results, as real galaxies presumably may have much more complex star formation histories than we have assumed here.', 'astro-ph-0210471-2-38-2': 'We now pursue a similar investigation for models with much more complex star formation histories, generated using a cosmological semi-analytic model of galaxy formation.', 'astro-ph-0210471-2-39-0': '# Comparison with Semi-Analytic Models', 'astro-ph-0210471-2-40-0': 'In this section, we again extract the [MATH] parameter from model galaxy spectra created using stellar population synthesis models.', 'astro-ph-0210471-2-40-1': 'The difference is that instead of using a simple parameterisation for the star formation history, the formation history of each galaxy is now modelled based on physical recipes set within the framework of hierarchical structure formation.', 'astro-ph-0210471-2-40-2': 'The ensemble of model spectra that we use here was created using the Somerville et al. semi-analytic models (e.g. Somerville Primack 1999; Somerville, Primack Faber 2001) and has been described previously in Slonim et al. (2001).', 'astro-ph-0210471-2-40-3': 'We refer the reader to those works for details, and here sketch the ingredients of the models very briefly.', 'astro-ph-0210471-2-41-0': 'In the hierarchical picture, present-day galaxies such as the ones observed by the 2dFGRS formed by the merging and accretion of smaller objects over time.', 'astro-ph-0210471-2-41-1': 'In this picture, the star formation history of a galaxy is determined by the mass accretion/merging history of its host dark matter halo, and the efficiency of gas cooling within those halos.', 'astro-ph-0210471-2-41-2': 'As the hot gas is enriched with heavy elements by metal-rich winds from massive stars and supernovae, the cooling efficiency is increased.', 'astro-ph-0210471-2-41-3': 'Galaxy mergers may trigger powerful bursts of star formation.', 'astro-ph-0210471-2-41-4': 'In turn, violent star formation events may inhibit future star formation by heating the interstellar medium or driving winds that blow it out of the galaxy.', 'astro-ph-0210471-2-42-0': 'These star formation histories are convolved with stellar population models to produce model spectra in much the same way as in the "simple" models discussed in the previous section.', 'astro-ph-0210471-2-42-1': 'In the models used here, we have used the multi-metalicity GISSEL models (Bruzual Charlot, in preparation) with a Salpeter IMF to calculate the stellar part of the spectra.', 'astro-ph-0210471-2-42-2': 'These models are very similar to the PEGASE stellar population models used in the previous section, and for the purposes of this investigation, this difference should not significantly affect our results.', 'astro-ph-0210471-2-42-3': 'Nebular emission lines are added to the spectra in the same way as before, using the empirical library from PEGASE.', 'astro-ph-0210471-2-43-0': 'Dust extinction is included using an approach similar to that of Guiderdoni Rocca-Volmerange (1987), which is also very similar to the approach implemented in the PEGASE package.', 'astro-ph-0210471-2-43-1': 'Here, the mass of dust is assumed to be proportional to the gas fraction times the metalicity of the cold gas.', 'astro-ph-0210471-2-43-2': 'We then use a standard Galactic extinction curve and a "slab" model to compute the extinction as a function of wavelength and inclination.', 'astro-ph-0210471-2-44-0': 'The recipes for star formation, supernova feedback, and chemical evolution involve free parameters, which we set by requiring a halo with a virial velocity of 220 km/s to host a galaxy with an average I-band luminosity of about [MATH], and with an average gas fraction of [MATH] to [MATH], consistent with observations of local spiral galaxies.', 'astro-ph-0210471-2-44-1': 'If we assume that mergers with mass ratios greater than [MATH] 1:3 form spheroids, we find that the models produce the correct morphological mix of spirals, S0s and ellipticals at the present day.', 'astro-ph-0210471-2-44-2': 'It has previously been shown in numerous works (e.g. Somerville Primack 1999) that this approach leads to fairly good agreement with numerous key galaxy observations, such as the local luminosity function, colours, and clustering properties.', 'astro-ph-0210471-2-45-0': 'We construct a "mock 2dF catalogue" of [MATH] model galaxies with the same magnitude limit, wavelength coverage and spectral resolution, and redshift range as the 2dF survey.', 'astro-ph-0210471-2-45-1': 'The synthetic spectra are expressed in terms of photon counts and the total number of counts in each spectrum is normalised to unity, as in the prepared observed spectra.', 'astro-ph-0210471-2-46-0': 'The calculation of the strength of the nebular emission features is one of the less precise aspects of the spectral synthesis packages.', 'astro-ph-0210471-2-46-1': 'For this reason in Fig. [REF] we show the first two principal components ([MATH],[MATH]) for the SAM ensemble of synthetic spectra.', 'astro-ph-0210471-2-46-2': 'It can be seen that the distribution of the mock catalogue principal components is quite similar to that of the observed 2dFGRS galaxies.', 'astro-ph-0210471-2-46-3': 'To guide the eye, the same set of evolutionary tracks shown in Fig. [REF], derived from the "simple" models, is also shown.', 'astro-ph-0210471-2-46-4': 'The most noticeable difference between the observed distribution and that of the mock, is that the former is substantially broader (most likely due to observational effects such as noise, reddening and evolution combined with the fact that the observed galaxies form a much larger sample).', 'astro-ph-0210471-2-46-5': 'It is also interesting that the simple model tracks span the same locus of the ([MATH], [MATH]) parameter space as the SAM galaxies, despite the fact that as we have emphasised the star formation histories are very different in the two kinds of models.', 'astro-ph-0210471-2-47-0': 'We can now discover whether the strong connection between [MATH] and the birthrate parameter [MATH] that we demonstrated for the simple models hold true for galaxies with more arbitrary star formation histories.', 'astro-ph-0210471-2-47-1': 'In Fig. [REF], we show [MATH] vs. [MATH] for the mock-2dF ensemble of SAM galaxy spectra.', 'astro-ph-0210471-2-47-2': 'Again we see a strong relationship between [MATH] and the [MATH] parameter.', 'astro-ph-0210471-2-47-3': 'Also, it is encouraging to note that the range of [MATH] spanned by the SAM galaxies (-5 to 15) is similar to that spanned by the 2dF galaxies.', 'astro-ph-0210471-2-47-4': 'The fact that the spectral properties of the galaxies in the SAM ensemble appear similar to those of the real 2dF galaxies enables us to use the results of the SAMs to draw a correspondence between the actual numerical value of [MATH] and [MATH].', 'astro-ph-0210471-2-47-5': 'This leads to the interesting conclusion that the dividing line between the two "bumps" in the bimodal distribution of [MATH] seen in Fig. [REF], [MATH], corresponds to galaxies that are forming stars at about 1/10th of their past-averaged rate.', 'astro-ph-0210471-2-47-6': 'Similarly, the other two [MATH] thresholds adopted by Madgwick et al. (2002) to calculate luminosity functions per spectral type may be matched up with [MATH] values using Fig. [REF].', 'astro-ph-0210471-2-48-0': 'We have also investigated correlations between [MATH] and other physical properties of galaxies in the SAMs, such as bulge-to-total ratio or mean stellar age.', 'astro-ph-0210471-2-48-1': 'While other correlations exist, none are as tight as the correlation shown in Fig. [REF] between [MATH] and the birthrate parameter [MATH].', 'astro-ph-0210471-2-48-2': 'Of all the correlations we investigated, the connection between [MATH] and star formation history seems to present the most straightforward interpretation of the [MATH] parameter.', 'astro-ph-0210471-2-48-3': 'We defer a more detailed investigation of the other physical correlations in the SAMs and comparison with the 2dF and other data sets to a future work.', 'astro-ph-0210471-2-49-0': '# Discussion', 'astro-ph-0210471-2-50-0': 'In this paper we have studied the physical interpretation of a PCA-based spectral parameter, [MATH], defined for the analysis of the 2dFGRS spectra (Madgwick et al. 2002).', 'astro-ph-0210471-2-50-1': "Although the definition of [MATH] was motivated by the need to compromise between the desire to extract the maximum amount of statistical information from the spectra (in the sense of PCA's maximum variance) and the limitations imposed by the instrumental uncertainties of the 2dF, we have argued here that this parameter has a straightforward and physically meaningful interpretation.", 'astro-ph-0210471-2-50-2': 'We find that [MATH] is an indicator of the star formation history of the galaxy, and is tightly correlated with the birthrate parameter [MATH], which characterises the ratio of present to past-averaged star formation.', 'astro-ph-0210471-2-51-0': 'A first indication of this correspondence is the correlation we find between [MATH] and and the equivalent width of the H[MATH] emission line, EW(H[MATH]), which has been used as a direct measure of the birthrate parameter in previous studies (e.g. Kennicutt, Tamblyn Congdon 1994).', 'astro-ph-0210471-2-51-1': 'We show that in models with simple star formation histories, parameterised in terms of an exponential function of time, there is a one-to-one correspondence between [MATH] and the birthrate parameter [MATH] (for a fixed formation time).', 'astro-ph-0210471-2-51-2': 'Perhaps this is not too surprising in the context of classical spectral synthesis work - it has long been known (e.g. Tinsley 1970) that a simple SF history can be used to interpret spectral appearance, i.e. [MATH] in this context.', 'astro-ph-0210471-2-51-3': 'More surprisingly, we show also that a strong relationship between [MATH] and [MATH] is also exhibited by model galaxies with much more complex star formation histories, created within a hierarchical cosmological framework using a semi-analytic model.', 'astro-ph-0210471-2-52-0': 'The explanation of this surprisingly simple result must lie in the physical processes that determine the appearance of a galaxy spectrum.', 'astro-ph-0210471-2-52-1': 'First, it is important to remember that at optical wavelengths, where all of this analysis has been carried out, the spectrum is dominated by the most luminous stars, and is therefore biased towards the most recent significant star formation activity.', 'astro-ph-0210471-2-52-2': 'The slope of the optical continuum and the strength of stellar absorption lines evolve on the timescale of the lifetimes of intermediate type main sequence stars, roughly several Gigayears.', 'astro-ph-0210471-2-52-3': 'The strengths of nebular emission features, however, evolve on much shorter timescales, as they require the presence of very hot, short lived O and B type stars.', 'astro-ph-0210471-2-52-4': 'These features therefore depend on the star formation history on the timescale of 5-10 Megayears.', 'astro-ph-0210471-2-52-5': 'The [MATH] parameter has been defined so as to be insensitive to the continuum slope (for practical reasons), and hence represents a sequence from strong absorption lines to strong emission lines.', 'astro-ph-0210471-2-52-6': 'It is well-known that the strengths of stellar absorption lines (especially Hydrogen recombination lines such as the Balmer series) are good indicators of stellar age, while the strength of the nebular emission lines are indicators of present star formation rate.', 'astro-ph-0210471-2-52-7': "With all this in mind, we can deduce that the rapid evolution in the value of [MATH] over the first several hundred Myr of a galaxy's life (see Fig. [REF]) reflects the fading of the emission lines as the star formation rate declines and the young stars burn out.", 'astro-ph-0210471-2-52-8': 'Galaxies then tend to "pile up" at low values of [MATH] as the age of their dominant stellar population exceeds a few Gyr.', 'astro-ph-0210471-2-53-0': 'In this sense it is straightforward to understand the relationship between [MATH] and [MATH] for our "simple" models, of monotonically decreasing star formation activity.', 'astro-ph-0210471-2-53-1': 'However, the star formation histories of galaxies created by the semi-analytic models can be quite complex.', 'astro-ph-0210471-2-53-2': 'The star formation rate fluctuates dramatically and non-monotonically over time as the galaxy exhausts its gas and then accretes a new gas supply, or experiences bursts of star formation triggered by mergers.', 'astro-ph-0210471-2-53-3': 'The issue we must now address is how an optical diagnostic such as [MATH], which is dominated by only the most recent events of star-formation, can be so closely related to the birth-rate parameter, [MATH], which incorporates the entire formation history of a galaxy.', 'astro-ph-0210471-2-54-0': 'The basis of the hierarchical picture of galaxy formation - from which our mock catalogue of SAM galaxies has been derived - is that galaxy properties are determined by the merger histories of their host dark matter halos.', 'astro-ph-0210471-2-54-1': 'In essence, the history of a specific halo (or galaxy) can be understood in terms of how the density on the scale of that structure compares to the background density.', 'astro-ph-0210471-2-54-2': 'A region with density much above the average (a "many sigma" peak, in the language of Gaussian random fields) collapses early, while lower density peaks collapse later.', 'astro-ph-0210471-2-54-3': 'It can therefore be conjectured that the early collapse of a dense dark matter halo can be associated with early star formation, early consumption of all available gas, and low present-day star formation rates.', 'astro-ph-0210471-2-54-4': 'Conversely, late-forming objects will have a sustained gas supply and ongoing star formation.', 'astro-ph-0210471-2-54-5': 'This suggests that the birthrate parameter [MATH] reflects perhaps the most theoretically fundamental property of a galaxy (or of its host halo): small-[MATH] galaxies represent rare, many-sigma peaks in the primordial density field, while large-[MATH] galaxies are formed in more common, lower density peaks.', 'astro-ph-0210471-2-54-6': 'This trend is manifested in the numerous empirical correlations between galaxy "type" (as characterised by morphology, colour, or spectral type), luminosity, and environment.', 'astro-ph-0210471-2-54-7': 'We expect this simple picture to be complicated by the details of gas cooling, star formation, feedback, etc., but the results from the semi-analytic models (which include all of these effects at some level, although of course the real Universe is likely to be even more complicated) suggest that this only introduces a moderate amount of scatter on top of the general trend.', 'astro-ph-0210471-2-55-0': 'We therefore conclude that the [MATH] parameter represents a promising candidate for galaxy classification in large modern redshift surveys.', 'astro-ph-0210471-2-55-1': 'It was adopted in previous analyses of the 2dFGRS because of its practical advantages: it is straightforward and efficient to compute in an automated fashion, and it is robust to the instrumental uncertainties commonly associated with fibre-based multi-object spectrographs.', 'astro-ph-0210471-2-55-2': 'Here we have shown that [MATH] also has a straightforward physical interpretation which can be intuitively connected both with traditional classifiers such as morphology, and that it seems to be connected with theoretically fundamental properties of galaxies within the modern hierarchical structure formation paradigm.'}

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[]

[['astro-ph-0210471-1-29-0', 'astro-ph-0210471-2-29-0'], ['astro-ph-0210471-1-29-1', 'astro-ph-0210471-2-29-0'], ['astro-ph-0210471-1-29-4', 'astro-ph-0210471-2-29-0'], ['astro-ph-0210471-2-28-1', 'astro-ph-0210471-3-28-1']]

[]

['astro-ph-0210471-1-17-1', 'astro-ph-0210471-2-17-1', 'astro-ph-0210471-3-17-1']

{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}

https://arxiv.org/abs/astro-ph/0210471

{'astro-ph-0210471-3-0-0': 'We investigate the physical significance of a new spectral parameter, [MATH].', 'astro-ph-0210471-3-0-1': 'This parameter was defined based on a Principal Component Analysis of the 2dF Galaxy Redshift Survey, to retain astrophysical information while minimising the effect of measurement uncertainties.', 'astro-ph-0210471-3-0-2': 'We find that while [MATH] is correlated with morphological type, there is a large scatter in this relationship.', 'astro-ph-0210471-3-0-3': 'A tighter empirical relationship is found between [MATH] and the equivalent width of the H[MATH] line, suggesting a connection with the star formation rate.', 'astro-ph-0210471-3-0-4': 'We pursue this connection using spectral synthesis models.', 'astro-ph-0210471-3-0-5': 'Using models in which the star formation history is parameterised in terms of an exponentially decreasing function of time, we find that there is a tight correlation between [MATH] and the ratio of the present to the past-averaged rate of star-formation, often known as the "birthrate" parameter [MATH].', 'astro-ph-0210471-3-0-6': 'This correlation also holds in models with much more complicated star formation histories, generated by a semi-analytic model of galaxy formation based upon the hierarchical formation scenario.', 'astro-ph-0210471-3-0-7': 'There are two possible causes for the tight correlations we find, between [MATH] and [MATH], in those galaxies with the most complex star formation histories; Firstly, the spectra themselves may be degenerate to the actual long-term star formation history of each galaxy in the optical wavelength range probed by the 2dFGRS.', 'astro-ph-0210471-3-0-8': 'Secondly, the birthrate parameter [MATH] may represent a physically fundamental quality of galaxy halos - their over-density relative to the background density - such that small-[MATH] galaxies form in high peaks (which collapse early) while large-[MATH] galaxies represent lower peaks, which collapse later.', 'astro-ph-0210471-3-0-9': 'We conclude that the tight connection with [MATH] makes [MATH] a physically meaningful, as well as convenient and robust, statistic for galaxy parameterisation and classification.', 'astro-ph-0210471-3-1-0': '# Introduction', 'astro-ph-0210471-3-2-0': 'Galaxy redshift surveys are now probing the galaxy distribution of the local Universe more accurately than ever before, and in so doing they are establishing many fundamental properties of the galaxy population and its large-scale structure.', 'astro-ph-0210471-3-2-1': 'The 2dF Galaxy Redshift Survey (2dFGRS) is one such ambitious project, conceived with the aim of mapping the galaxy distribution to an extinction corrected magnitude limit of [MATH].', 'astro-ph-0210471-3-2-2': 'This particular survey is now essentially complete with approximately 230,000 galaxy spectra having already been acquired, and has already started to yield significant results (e.g. Percival et al. 2001; Madgwick et al. 2002 and others).', 'astro-ph-0210471-3-2-3': 'In addition to this survey, the Sloan Digital Sky Survey (Strauss et al., 2002) is also underway, and once complete will obtain up to 1,000,000 individual galaxy spectra.', 'astro-ph-0210471-3-3-0': 'Apart from the main scientific goals of quantifying the large-scale structure of the Universe, one of the most significant contributions of galaxy redshift surveys is to our understanding of the galaxy population itself, through the information about galaxy properties contained in the observed spectra.', 'astro-ph-0210471-3-3-1': 'Having a data set of 230,000 galaxy spectra - as in the case of the 2dFGRS - allows us to test the validity of galaxy formation and evolution scenarios with unprecedented accuracy.', 'astro-ph-0210471-3-3-2': 'However, the sheer size of the data set presents its own unique problems.', 'astro-ph-0210471-3-3-3': "Clearly, in order to make the spectral data set more 'digestible', some form of data compression is necessary.", 'astro-ph-0210471-3-3-4': 'Familiar statistics such as equivalent width measurements, morphological types, and broad-band colours are really just compression techniques in some sense.', 'astro-ph-0210471-3-3-5': 'These quantities can be compared with theoretical predictions and simulations, and hence can set constraints on scenarios for galaxy formation and biasing.', 'astro-ph-0210471-3-4-0': 'The approach that has been adopted in Madgwick et al. (2002) is to define a spectral indicator, [MATH], based on Principal Component Analysis (PCA).', 'astro-ph-0210471-3-4-1': 'Because of instrumental limitations, the flux calibration of the 2dF spectra is unreliable, and hence robust measurements of the shape of the spectral continua are not possible.', 'astro-ph-0210471-3-4-2': 'For this reason, the [MATH] statistic was designed to (in a loose sense) preserve the most discriminatory information in the spectra, while being robust to the instrumental uncertainties.', 'astro-ph-0210471-3-4-3': 'Effectively, the parameter [MATH] measures the strength of absorption and emission lines, while remaining insensitive to the slope of the continuum (or broad band colour).', 'astro-ph-0210471-3-5-0': 'The motivation of the definition of the [MATH] statistic was in part pragmatic.', 'astro-ph-0210471-3-5-1': 'The goal of this paper is to address the fundamental question of the physical significance of this parameter, in terms of how it relates to the physical properties of galaxies.', 'astro-ph-0210471-3-5-2': 'A substantial amount of work has been carried out in the past on relating the observed spectra of galaxies to the physical processes occurring in them (see e.g Kennicutt, Tamblyn Congdon 1994; Ronen, Aragon-Salamanca Lahav 1999; Carter, Fabricant, Geller Kurtz 2001 and references therein), and the advent of a large and uniform survey such as the 2dFGRS will result in many more advances in this field.', 'astro-ph-0210471-3-5-3': 'However, for the present analysis we restrict ourselves to relating only the 2dFGRS spectral classification parameter, [MATH], to these processes and leave a fuller development of this subject for future works.', 'astro-ph-0210471-3-5-4': 'The investigation presented here is particularly timely as many new results from the 2dFGRS are being presented which make exclusive use of this parameter to characterise and partition the galaxy population (e.g. Madgwick et al. 2002; Norberg et al. 2002; Martinez et al. 2002).', 'astro-ph-0210471-3-6-0': 'We make use of two kinds of models to pursue this question.', 'astro-ph-0210471-3-6-1': 'In the first approach, we adopt a simple parameterisation to describe the star formation history of each galaxy in terms of an exponentially decreasing function of time.', 'astro-ph-0210471-3-6-2': 'For a given redshift of observation, the star formation history is then characterised by a single parameter: the timescale of the exponential decline.', 'astro-ph-0210471-3-6-3': "These star formation histories are then convolved with spectral synthesis models, using the PEGASE (Projet d'Etude des GAlaxies par Synthese Evolutive) code, developed by Fioc Rocca-Volmerange (1997).", 'astro-ph-0210471-3-6-4': 'A particular advantage of this package compared to others available in the literature (e.g. Bruzual Charlot 1993) is that it is possible to include modelling of nebular emission from star forming regions in each galaxy - an important ingredient in the calculation of the [MATH] parameter.', 'astro-ph-0210471-3-6-5': 'We refer to these models as the "simple" models, because of the rather simplified nature of the parametrised star formation histories.', 'astro-ph-0210471-3-7-0': 'The limitation of this simple method is that we must explicitly assume some family of star-formation histories for our galaxy population.', 'astro-ph-0210471-3-7-1': 'We therefore extend this approach to incorporate the cosmological framework of the Cold Dark Matter (CDM) scenario, by using semi-analytic techniques (see e.g. Kauffmann, White Guiderdoni 1993; Cole et al. 1994; Somerville Primack 1999).', 'astro-ph-0210471-3-7-2': 'We make use of a mock catalogue of synthetic spectra which was designed to match the selection criteria of the 2dFGRS, analysed previously in Slonim et al. (2001), and created using an updated version of the code described in Somerville Primack (1999).', 'astro-ph-0210471-3-7-3': 'There are several advantages of using a semi-analytic model to create spectra for this exercise.', 'astro-ph-0210471-3-7-4': 'The star formation history of each galaxy will reflect at least some of the complexity of the interconnected processes of dark matter clustering, gas cooling, galaxy merging, supernova feedback, etc., modelled self-consistently within a specific cosmological framework.', 'astro-ph-0210471-3-7-5': 'In addition, as we can select galaxies so as to reproduce the same redshift and luminosity distributions displayed by the observed 2dF galaxies, the resulting ensemble should contain a mix of galaxies with different sorts of star formation histories that is similar to the actual observed sample.', 'astro-ph-0210471-3-7-6': 'Finally, the semi-analytic model also yields many other physical parameters for each galaxy, giving us the potential to further probe the physical processes that may be related to [MATH].', 'astro-ph-0210471-3-8-0': 'We find that in the simple models, there is essentially a one-to-one relationship between [MATH] and the ratio of the present rate of star formation to its past time-averaged value.', 'astro-ph-0210471-3-8-1': 'This quantity is sometimes known as the "birthrate" parameter, [MATH] (Scalo 1986).', 'astro-ph-0210471-3-8-2': 'Even more surprisingly, we find that there is a strong correlation between [MATH] and [MATH] even in the spectra produced by the semi-analytic models, despite the complex and diverse nature of the underlying star formation histories.', 'astro-ph-0210471-3-8-3': 'Empirically, the [MATH] parameter is known to be tightly correlated with morphological type and colour.', 'astro-ph-0210471-3-8-4': 'We therefore argue that the [MATH] parameter presents a practical and robust means of extracting a fundamental measure of galaxy type from the relatively low-quality spectra typical of present-day large redshift surveys.', 'astro-ph-0210471-3-9-0': 'Section 2 of this paper briefly reviews the operational definition of and motivation behind the spectral classification parameter [MATH], used in the 2dFGRS.', 'astro-ph-0210471-3-9-1': 'In Section 3 we make use of the "simple" models based on exponentially declining star formation histories to show how [MATH] is related to the star-formation history of a galaxy.', 'astro-ph-0210471-3-9-2': 'Section 4 then generalises these results using formation histories generated with a semi-analytic model.', 'astro-ph-0210471-3-9-3': 'We conclude with a discussion of our results and ways to build upon the results presented here.', 'astro-ph-0210471-3-10-0': '# Spectral Classification in the 2dFGRS', 'astro-ph-0210471-3-11-0': '## Principal Component Analysis', 'astro-ph-0210471-3-12-0': 'Principal component analysis (PCA) is a well established statistical technique which has proved very useful in dealing with high dimensional data sets (see e.g. Murtagh Heck 1987; Connolly et al. 1995).', 'astro-ph-0210471-3-12-1': 'In the particular case of galaxy spectra we are typically presented with approximately 1000 spectral channels per galaxy, however when used in applications this is usually compressed down to just a few numbers, either by integrating over small line features - yielding equivalent widths - or over broad band filters.', 'astro-ph-0210471-3-12-2': 'The key advantage of using PCA in our data compression is that it allows us to make use of all the information contained in the spectrum in a statistically unbiased way, i.e. without the use of such ad hoc filters.', 'astro-ph-0210471-3-13-0': 'In order to perform the PCA on our galaxy spectra we first construct a representative volume limited sample of the galaxies.', 'astro-ph-0210471-3-13-1': 'We compute the "mean spectrum" from this ensemble, and subtract this mean from all of the galaxy spectra.', 'astro-ph-0210471-3-13-2': 'When we apply the PCA to this sample an orthogonal set of components (eigenspectra) is constructed, which span the wavelength space occupied by the galaxy spectra.', 'astro-ph-0210471-3-13-3': 'These components have been specifically chosen by the PCA in such a way that as much information (variance) is contained in the first eigenspectrum as possible, and that the amount of the remaining information in all the subsequent eigenspectra is likewise maximised.', 'astro-ph-0210471-3-13-4': 'Therefore, if the information contained in the first [MATH] eigenspectra is found to be significantly greater than that in the remaining eigenspectra we can significantly compress the data set by swapping each galaxy spectrum (described by 1000 channels) with its projections onto just those [MATH] eigenspectra.', 'astro-ph-0210471-3-14-0': 'In the case of the 2dF galaxy spectra we find that approximately two thirds of the total variance (including the noise) in the spectra can be represented in terms of only the first two projections ([MATH], [MATH]).', 'astro-ph-0210471-3-14-1': 'So, at least to a first approximation, galaxy spectra can be thought of as a two dimensional sequence in terms of these two projections.', 'astro-ph-0210471-3-15-0': 'In Fig. [REF] we show these first two eigenspectra.', 'astro-ph-0210471-3-15-1': 'It can be seen from this figure that whilst the first eigenspectrum contains information from both lines and the continuum, the second is dominated by absorption and emission lines.', 'astro-ph-0210471-3-15-2': 'Because of this it is possible to take two simple linear combinations which isolate either the continuum or the emission/absorption line features.', 'astro-ph-0210471-3-15-3': 'In effect what we are doing when we utilise these linear combinations is rotating the axes defined by the PCA to make the interpretation of the components more straightforward.', 'astro-ph-0210471-3-15-4': 'In so doing we can see that a parameterisation in terms of [MATH] and [MATH] is essentially equivalent to a two dimensional sequence in colour (continuum slope) and the average emission/absorption line strength.', 'astro-ph-0210471-3-15-5': 'This ability to isolate the continuum and line components of the galaxy spectra turns out to be very useful, as will be discussed in Section 2.2.', 'astro-ph-0210471-3-16-0': '## The [MATH] Parameter', 'astro-ph-0210471-3-17-0': 'The 2dF instrument makes use of up to 400 optical fibres with a diameter of 140[MATH]m (corresponding to [MATH] on the sky, depending on plate position.', 'astro-ph-0210471-3-17-1': 'See Lewis et al. 2002).', 'astro-ph-0210471-3-17-2': 'The instrument itself was designed to measure large numbers of redshifts in as short an observing time as possible.', 'astro-ph-0210471-3-17-3': 'However, in order to optimise the number of redshifts that can be measured in a given period of time, compromises had to be made with respect to the spectral quality of the observations.', 'astro-ph-0210471-3-17-4': 'Therefore if one wishes to characterise the observed galaxy population in terms of their spectral properties, care must be taken in order to ensure that these properties are robust to the instrumental uncertainties.', 'astro-ph-0210471-3-18-0': 'The quality and representativeness of the observed spectra can be compromised in several ways and a more detailed discussion of these issues is presented in previous work (see e.g. Madgwick et al. 2002).', 'astro-ph-0210471-3-18-1': 'The net effect is that the uncertainties introduced into the fibre-spectra predominantly affect the calibration of the continuum slope and have relatively little impact on the emission/absorption line strengths.', 'astro-ph-0210471-3-18-2': 'For this reason any given galaxy spectrum which is projected onto the plane defined by ([MATH],[MATH]) will not be uniquely defined in the direction of varying continuum but will be robust in the orthogonal direction (which measures the average line strength).', 'astro-ph-0210471-3-19-0': 'The linear combination of the first two eigenspectra which is robust to these uncertainties is shown in Fig. [REF] (b) and denoted by [MATH] (ETATYPE in the 2dFGRS catalogue).', 'astro-ph-0210471-3-19-1': 'It is simply [EQUATION] where [MATH] is a constant which we find empirically to be [MATH].', 'astro-ph-0210471-3-20-0': 'We have now identified the single statistically dominant component of the galaxy spectra which is robust to the known instrumental uncertainties.', 'astro-ph-0210471-3-20-1': 'We have therefore chosen to adopt this (continuous) variable as our measure of spectral type.', 'astro-ph-0210471-3-20-2': 'Having defined [MATH] in this formal and pragmatic manner, however, we are left with the question of whether it is physically meaningful, and how it is to be interpreted.', 'astro-ph-0210471-3-20-3': 'This is the issue that we now address.', 'astro-ph-0210471-3-21-0': '## Correlation of [MATH] with traditional galaxy classifiers', 'astro-ph-0210471-3-22-0': 'We show the observed distribution of [MATH] for the 2dF galaxies in Fig. [REF].', 'astro-ph-0210471-3-22-1': 'It is rather intriguing that this distribution is strongly bimodal.', 'astro-ph-0210471-3-22-2': 'Fig. [REF] also shows the [MATH] projections of spectra from galaxies in the Kennicutt Atlas (Kennicutt 1992), which have known morphologies.', 'astro-ph-0210471-3-22-3': 'There is a correspondence between morphology and [MATH] in the expected sense: the value of [MATH] increases as one moves towards later type objects in the Hubble sequence.', 'astro-ph-0210471-3-22-4': 'Larger values of [MATH] indicate more dominant emission lines, and it is well-known that later type galaxies have stronger emission lines.', 'astro-ph-0210471-3-22-5': 'From this diagram we see that [MATH] seems to be correlated with morphological type, at least based on this small sample of galaxies.', 'astro-ph-0210471-3-22-6': 'Norberg et al. (2002) showed the distribution of [MATH] for a larger sample of galaxies that had been classified into four morphological types (E, S0, Sp and Irr), and showed that while there is a correlation between morphological type and [MATH], there is a large scatter.', 'astro-ph-0210471-3-22-7': 'The correspondence between [MATH] and morphological type is investigated in more detail in Madgwick (2002).', 'astro-ph-0210471-3-23-0': 'We can also compare the value of [MATH] derived for each galaxy to the equivalent width of the H[MATH] emission line, for a sample of high signal-to-noise ratio emission line spectra.', 'astro-ph-0210471-3-23-1': 'This is shown in Fig. [REF], from which it can be seen that there is a strong correlation between these quantities.', 'astro-ph-0210471-3-23-2': 'It is well-known that the equivalent width of H[MATH] is a measure of star formation; indeed it has been used to estimate the birthrate parameter [MATH] (the ratio of the present rate of star formation to the past averaged value - see e.g. Figure 3 of Kennicutt, Tamblyn Congdon 1994).', 'astro-ph-0210471-3-23-3': 'We return to this point in Section [REF], where we use galaxy formation models to investigate how [MATH] relates to the star-formation history.', 'astro-ph-0210471-3-24-0': '# Comparison with Simple Models', 'astro-ph-0210471-3-25-0': 'The results shown above suggest that [MATH] is related both to galaxy morphology and to star formation history.', 'astro-ph-0210471-3-25-1': 'In this section we further explore and test this hypothesis with models produced by combining simple parameterised star formation histories with spectral synthesis models.', 'astro-ph-0210471-3-25-2': "In this way, we can make a direct connection between a galaxy's star formation history and the value of our spectral classification statistic, [MATH].", 'astro-ph-0210471-3-26-0': '## Model Ingredients', 'astro-ph-0210471-3-27-0': 'Stellar population models combine theoretical temperature-luminosity tracks for stars of various masses with model stellar atmospheres and an assumed Initial Mass Function (IMF) in order to produce synthetic spectra for a "single burst" population of a uniform age and metalicity.', 'astro-ph-0210471-3-27-1': 'In order to use these models to obtain predictions for a composite stellar population (i.e. a galaxy), the "single burst" models must be combined with an assumed star formation history, which provides the distribution of stellar ages and metalicities within the galaxy.', 'astro-ph-0210471-3-27-2': 'Since the classical work on spectral synthesis by Tinsley (1980), many authors have adopted a simple parametrisation of the star formation history in terms of an exponential function of time (e.g. Bruzual 1983): [EQUATION]', 'astro-ph-0210471-3-27-3': 'Here, [MATH] is the time at which star formation first commences, and [MATH] is the characteristic timescale of star formation.', 'astro-ph-0210471-3-27-4': 'Most observed spectra can be modelled rather well using this simple function with the appropriate choice of [MATH].', 'astro-ph-0210471-3-27-5': 'A typical spiral galaxy spectrum might be well-fit with [MATH] Gyr, while later type galaxies require larger values and early type galaxies require smaller values of [MATH].', 'astro-ph-0210471-3-28-0': 'Nebular emission lines from ionised H[MATH] regions and extinction due to dust also contribute to the appearance of a galaxy spectrum.', 'astro-ph-0210471-3-28-1': 'Modelling these processes is complicated, requiring assumptions about the ionization state, the geometry, the metalicity, the dust mass and the composition of H[MATH] regions.', 'astro-ph-0210471-3-29-0': 'The models presented in this Section are based on this simple but well-defined picture, and are realized using version 2.0 of the PEGASE software package (Fioc Rocca-Volmerange) In all of the models shown we have adopted a Kennicutt (1983) initial-mass function with stellar masses in the range [MATH].', 'astro-ph-0210471-3-29-1': 'In addition, the metalicity of each galaxy is traced in a self-consistent manner using the prescription of Woosley Weaver (1995), to model the enrichment of the inter-stellar medium.', 'astro-ph-0210471-3-30-0': 'Nebular emission lines result from light re-emitted by the ionised gas in star-forming regions.', 'astro-ph-0210471-3-30-1': 'A prescription for determining the strengths of these lines is implemented in the PEGASE code.', 'astro-ph-0210471-3-30-2': 'This process involves the absorption of Lyman continuum photons (below 912) by the nebular gas, which gets ionised, and reaches recombination equilibrium (Osterbrock 1989).', 'astro-ph-0210471-3-30-3': 'It is assumed that 70% of these photons are absorbed by the gas at solar metalicity.', 'astro-ph-0210471-3-30-4': 'In this approach, the strength of the nebular emission lines is a function of the age of the stellar population only, and metalicity and geometric effects are neglected.', 'astro-ph-0210471-3-31-0': 'The PEGASE code provides a simple way to model the effects of dust extinction on the synthetic spectra, in which the optical depth is estimated from the mass of gas and the metalicity.', 'astro-ph-0210471-3-31-1': 'The absorption is then estimated using observational data for a mixture of graphites and silicates as in the Milky Way and the Magellanic Clouds (Draine Lee 1984 and Pei 1992).', 'astro-ph-0210471-3-31-2': 'In making this calculation it is also necessary to make an assumption about the geometry of the galaxy.', 'astro-ph-0210471-3-31-3': 'We investigated models in which we assumed a spheroidal geometry and also an inclination-averaged disk geometry.', 'astro-ph-0210471-3-32-0': 'The spectra generated by the PEGASE code over the optical wavelength range are generally given in 10 bins, and the emission line strengths are specified separately in terms of their peak fluxes over the continuum.', 'astro-ph-0210471-3-32-1': 'In order to make comparisons between these spectra and the 2dFGRS we therefore interpolate the synthetic spectra onto the 4 binned wavelength range of the 2dFGRS.', 'astro-ph-0210471-3-32-2': 'Because the given emission lines are not resolved in this binning we create lines by superposing Gaussian profiles with the specified peak flux and FWHM corresponding to that calculated for the 2dF spectrograph using arc line measurements ([MATH] pixels).', 'astro-ph-0210471-3-32-3': 'We convert the given synthetic spectra to units of counts/bin by multiplying the flux (erg/s/) by the wavelength.', 'astro-ph-0210471-3-32-4': 'We then normalise each spectrum to have mean counts of [MATH] over our entire wavelength range.', 'astro-ph-0210471-3-32-5': 'The spectra processed in this way are directly comparable to those of the 2dFGRS.', 'astro-ph-0210471-3-33-0': 'Spectral histories of a given galaxy are compiled for a range of formation times ([MATH] Gyr), and a grid of values of [MATH]: 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, and 2.0 Gyr.', 'astro-ph-0210471-3-33-1': 'We also calculate spectral histories for an instantaneous burst of star-formation at [MATH].', 'astro-ph-0210471-3-33-2': 'Because this grid does not comprise a well-defined sample of galaxies, to compare with the results from the 2dFGRS, we first subtract the mean spectrum from the 2dFGRS volume limited sample (described in the previous section) from each spectrum.', 'astro-ph-0210471-3-33-3': 'We then compute the projection of each spectrum on the eigenspectra derived from the 2dFGRS volume limited sample.', 'astro-ph-0210471-3-33-4': 'The first two projections ([MATH], [MATH]) derived in this manner are shown for our grid of models in Fig. [REF], along with the corresponding projections for a random subsample of galaxies in the [MATH] volume-limited 2dFGRS sample.', 'astro-ph-0210471-3-33-5': 'Model grids are shown both with and without including the effects of dust extinction.', 'astro-ph-0210471-3-33-6': 'It can be seen that dust does not have a dramatic impact on the results, but does seem to improve the agreement between the model tracks and the locus of the observed galaxies.', 'astro-ph-0210471-3-34-0': 'Already it is interesting that these simple models seem to cover the same range of the ([MATH], [MATH]) parameter space occupied by the observed galaxy population.', 'astro-ph-0210471-3-34-1': 'Evidently, at least in this simple scheme, galaxies are "born" in the lower right-hand corner of the plot, and progress towards the upper left-hand corner as they age.', 'astro-ph-0210471-3-34-2': 'As well, galaxies in the upper right part of the diagram tend to have smaller values of the characteristic timescale for star formation, [MATH], associated with early type galaxies, while as one moves diagonally towards the lower left corner, galaxies have the more extended timescales for star formation associated with late type galaxies.', 'astro-ph-0210471-3-34-3': 'Perhaps unsurprisingly, the vast majority of the 2dF galaxies (which are fairly luminous) are consistent with times since formation [MATH] between 2-10 Gyr.', 'astro-ph-0210471-3-34-4': 'It should be kept in mind, however, that the instrumental effects discussed in Section [REF] can introduce random scatter in this diagram.', 'astro-ph-0210471-3-34-5': 'We therefore turn now to the more robust [MATH] parameter.', 'astro-ph-0210471-3-35-0': '## The Physical Significance of [MATH]', 'astro-ph-0210471-3-36-0': 'We argued based on a visual inspection of the eigenspectrum used to define the [MATH] projection that [MATH] essentially measures the strength of stellar and interstellar absorption line features and the strength of nebular emission line features.', 'astro-ph-0210471-3-36-1': 'The strength of the absorption features mainly indicates the age and metalicity of the stellar population that dominates the optical luminosity of the galaxy, and the emission lines are strongly correlated with star formation activity.', 'astro-ph-0210471-3-36-2': 'The comparison with classical galaxy classification methods such as visual morphology or equivalent width, shown in Section [REF], suggested that [MATH] may be connected with a measure of the star formation relative to the existing older stellar population, such as the birthrate parameter, [MATH] (Scalo 1986).', 'astro-ph-0210471-3-36-3': 'The birthrate parameter is defined as the ratio between the current and past-averaged star-formation rate, which in the case of the simple exponential star formation law is given analytically by [EQUATION]', 'astro-ph-0210471-3-36-4': 'In this section we investigate our hypothesis that [MATH] might be correlated with the birthrate parameter, [MATH].', 'astro-ph-0210471-3-37-0': 'The relationship between [MATH] and [MATH] for the grid of models with simple star formation histories (as before) is shown in Fig. [REF].', 'astro-ph-0210471-3-37-1': 'For a fixed formation time, [MATH], there is a well-defined relationship between the star formation history, as characterised by the birthrate parameter [MATH], and the [MATH] spectral parameter.', 'astro-ph-0210471-3-37-2': 'This helps in the interpretation of [MATH] for the observed population of galaxies.', 'astro-ph-0210471-3-37-3': 'At least in this simple scheme, galaxies with small values of [MATH] formed most of their stars in the past, and are currently evolving rather passively.', 'astro-ph-0210471-3-37-4': 'This fits in well with the classical picture of early type galaxy formation.', 'astro-ph-0210471-3-37-5': 'Galaxies with larger values of [MATH] have had a significant amount of recent star formation, characteristic of spiral and irregular galaxies.', 'astro-ph-0210471-3-37-6': 'Recalling from Fig. [REF] that the 2dFGRS sample spans a range of [MATH] values of about -5 to 10, this suggests again that most of the observed galaxies in the 2dF require times since formation greater than about 1 Gyr.', 'astro-ph-0210471-3-37-7': 'From this diagram, we arrive at a prediction that the distribution of [MATH] for galaxies observed at larger look-back times should exhibit a shift towards higher values.', 'astro-ph-0210471-3-37-8': 'We find that the effect of extinction corrections is to slightly lower the calculated [MATH] for a given star formation scenario, however, this is only a small effect and does not alter the qualitative interpretation that we have suggested.', 'astro-ph-0210471-3-38-0': 'A limitation of the simple characterisation of the star formation histories that we have adopted here is manifested by the asymptote of the [MATH] values towards unity: because the star formation rate was assumed to be monotonically decreasing, [MATH] is always less than one.', 'astro-ph-0210471-3-38-1': 'It is clear that we must be cautious in over-interpreting these results, as real galaxies presumably may have much more complex star formation histories than we have assumed here.', 'astro-ph-0210471-3-38-2': 'We now pursue a similar investigation for models with much more complex star formation histories, generated using a cosmological semi-analytic model of galaxy formation.', 'astro-ph-0210471-3-39-0': '# Comparison with Semi-Analytic Models', 'astro-ph-0210471-3-40-0': 'In this section, we again extract the [MATH] parameter from model galaxy spectra created using stellar population synthesis models.', 'astro-ph-0210471-3-40-1': 'The difference is that instead of using a simple parameterisation for the star formation history, the formation history of each galaxy is now modelled based on physical recipes set within the framework of hierarchical structure formation.', 'astro-ph-0210471-3-40-2': 'The ensemble of model spectra that we use here was created using the Somerville et al. semi-analytic models (e.g. Somerville Primack 1999; Somerville, Primack Faber 2001) and has been described previously in Slonim et al. (2001).', 'astro-ph-0210471-3-40-3': 'We refer the reader to those works for details, and here sketch the ingredients of the models very briefly.', 'astro-ph-0210471-3-41-0': 'In the hierarchical picture, present-day galaxies such as the ones observed by the 2dFGRS formed by the merging and accretion of smaller objects over time.', 'astro-ph-0210471-3-41-1': 'In this picture, the star formation history of a galaxy is determined by the mass accretion and merging histories of its host dark matter halo, and the efficiency of gas cooling within those halos.', 'astro-ph-0210471-3-41-2': 'As the hot gas is enriched with heavy elements by metal-rich winds from massive stars and supernovae, the cooling efficiency is increased.', 'astro-ph-0210471-3-41-3': 'Galaxy mergers may trigger powerful bursts of star formation.', 'astro-ph-0210471-3-41-4': 'In turn, violent star formation events may inhibit future star formation by heating the interstellar medium or driving winds that blow it out of the galaxy.', 'astro-ph-0210471-3-42-0': 'These star formation histories are convolved with stellar population models to produce model spectra in much the same way as in the "simple" models discussed in the previous section.', 'astro-ph-0210471-3-42-1': 'In the models used here, we have used the multi-metalicity GISSEL models (Bruzual Charlot, 1983) with a Salpeter IMF to calculate the stellar part of the spectra.', 'astro-ph-0210471-3-42-2': 'These models are very similar to the PEGASE stellar population models used in the previous section, and for the purposes of this investigation, this difference should not significantly affect our results.', 'astro-ph-0210471-3-42-3': 'Nebular emission lines are added to the spectra in the same way as before, using the empirical library from PEGASE.', 'astro-ph-0210471-3-43-0': 'Dust extinction is included using an approach similar to that of Guiderdoni Rocca-Volmerange (1987), which is also very similar to the approach implemented in the PEGASE package.', 'astro-ph-0210471-3-43-1': 'Here, the mass of dust is assumed to be proportional to the gas fraction times the metalicity of the cold gas.', 'astro-ph-0210471-3-43-2': 'We then use a standard Galactic extinction curve and a "slab" model to compute the extinction as a function of wavelength and inclination.', 'astro-ph-0210471-3-44-0': 'The recipes for star formation, supernova feedback, and chemical evolution involve free parameters, which we set by requiring a halo with a virial velocity of 220 km/s to host a galaxy with an average I-band luminosity of about [MATH], and with an average gas fraction of [MATH] to [MATH], consistent with observations of local spiral galaxies.', 'astro-ph-0210471-3-44-1': 'If we assume that mergers with mass ratios greater than [MATH] 1:3 form spheroids, we find that the models produce the correct morphological mix of spirals, S0s and ellipticals at the present day.', 'astro-ph-0210471-3-44-2': 'It has previously been shown in numerous works (e.g. Somerville Primack 1999) that this approach leads to fairly good agreement with numerous key galaxy observations, such as the local luminosity function, colours, and clustering properties.', 'astro-ph-0210471-3-45-0': 'We construct a "mock 2dF catalogue" of [MATH] model galaxies with the same magnitude limit, wavelength coverage and spectral resolution, and redshift range as the 2dF survey.', 'astro-ph-0210471-3-45-1': 'The synthetic spectra are expressed in terms of photon counts and the total number of counts in each spectrum is normalised to unity, as in the prepared observed spectra.', 'astro-ph-0210471-3-46-0': 'The calculation of the strength of the nebular emission features is one of the less precise aspects of the spectral synthesis packages.', 'astro-ph-0210471-3-46-1': 'For this reason in Fig. [REF] we show the first two principal components ([MATH],[MATH]) for the semi-analytic model (SAM) ensemble of synthetic spectra.', 'astro-ph-0210471-3-46-2': 'It can be seen that the distribution of the mock catalogue principal components is quite similar to that of the observed 2dFGRS galaxies.', 'astro-ph-0210471-3-46-3': 'To guide the eye, the same set of evolutionary tracks shown in Fig. [REF], derived from the "simple" models, is also shown.', 'astro-ph-0210471-3-46-4': 'The most noticeable difference between the observed distribution and that of the mock, is that the former is substantially broader (most likely due to observational effects such as noise, reddening and evolution combined with the fact that the observed galaxies form a much larger sample).', 'astro-ph-0210471-3-46-5': 'It is also interesting that the simple model tracks span the same locus of the ([MATH], [MATH]) parameter space as the SAM galaxies, despite the fact that as we have emphasised the star formation histories are very different in the two kinds of models.', 'astro-ph-0210471-3-47-0': 'We can now discover whether the strong connection between [MATH] and the birthrate parameter [MATH] that we demonstrated for the simple models hold true for galaxies with more arbitrary star formation histories.', 'astro-ph-0210471-3-47-1': 'In Fig. [REF], we show [MATH] vs. [MATH] for the mock-2dF ensemble of SAM galaxy spectra.', 'astro-ph-0210471-3-47-2': 'Again we see a strong relationship between [MATH] and the [MATH] parameter.', 'astro-ph-0210471-3-47-3': 'Also, it is encouraging to note that the range of [MATH] spanned by the SAM galaxies (-5 to 15) is similar to that spanned by the 2dF galaxies.', 'astro-ph-0210471-3-47-4': 'The fact that the spectral properties of the galaxies in the SAM ensemble appear similar to those of the real 2dF galaxies enables us to use the results of the SAMs to draw a correspondence between the actual numerical value of [MATH] and [MATH].', 'astro-ph-0210471-3-47-5': 'This leads to the interesting conclusion that the dividing line between the two "bumps" in the bimodal distribution of [MATH] seen in Fig. [REF], [MATH], corresponds to galaxies that are forming stars at about 1/10th of their past-averaged rate.', 'astro-ph-0210471-3-47-6': 'Similarly, the other two [MATH] thresholds adopted by Madgwick et al. (2002) to calculate luminosity functions per spectral type may be matched up with [MATH] values using Fig. [REF].', 'astro-ph-0210471-3-48-0': 'We have also investigated correlations between [MATH] and other physical properties of galaxies in the SAMs, such as bulge-to-total ratio or mean stellar age.', 'astro-ph-0210471-3-48-1': 'While other correlations exist, none are as tight as the correlation shown in Fig. [REF] between [MATH] and the birthrate parameter [MATH].', 'astro-ph-0210471-3-48-2': 'Of all the correlations we investigated, the connection between [MATH] and star formation history seems to present the most straightforward interpretation of the [MATH] parameter.', 'astro-ph-0210471-3-48-3': 'We defer a more detailed investigation of the other physical correlations in the SAMs and comparison with the 2dF and other data sets to a future work.', 'astro-ph-0210471-3-49-0': '# Discussion', 'astro-ph-0210471-3-50-0': 'In this paper we have studied the physical interpretation of a PCA-based spectral parameter, [MATH], defined for the analysis of the 2dFGRS spectra (Madgwick et al. 2002).', 'astro-ph-0210471-3-50-1': "Although the definition of [MATH] was motivated by the need to compromise between the desire to extract the maximum amount of statistical information from the spectra (in the sense of PCA's maximum variance) and the limitations imposed by the instrumental uncertainties of the 2dF, we have argued here that this parameter has a straightforward and physically meaningful interpretation.", 'astro-ph-0210471-3-50-2': 'We find that [MATH] is an indicator of the star formation history of the galaxy, and is tightly correlated with the birthrate parameter [MATH], which characterises the ratio of present to past-averaged star formation.', 'astro-ph-0210471-3-51-0': 'A first indication of this correspondence is the correlation we find between [MATH] and and the equivalent width of the H[MATH] emission line, EW(H[MATH]), which has been used as a direct measure of the birthrate parameter in previous studies (e.g. Kennicutt, Tamblyn Congdon 1994).', 'astro-ph-0210471-3-51-1': 'We show that in models with simple star formation histories, parameterised in terms of an exponential function of time, there is a one-to-one correspondence between [MATH] and the birthrate parameter [MATH] (for a fixed formation time).', 'astro-ph-0210471-3-51-2': 'Perhaps this is not too surprising in the context of classical spectral synthesis work - it has long been known (e.g. Tinsley 1980) that a simple SF history can be used to interpret spectral appearance, i.e. [MATH] in this context.', 'astro-ph-0210471-3-51-3': 'More surprisingly, we show also that a strong relationship between [MATH] and [MATH] is also exhibited by model galaxies with much more complex star formation histories, created within a hierarchical cosmological framework using a semi-analytic model.', 'astro-ph-0210471-3-52-0': 'The explanation of this surprisingly simple result must lie in the physical processes that determine the appearance of a galaxy spectrum.', 'astro-ph-0210471-3-52-1': 'First, it is important to remember that at optical wavelengths, where all of this analysis has been carried out, the spectrum is dominated by the most luminous stars, and is therefore biased towards the most recent significant star formation activity.', 'astro-ph-0210471-3-52-2': 'The slope of the optical continuum and the strength of stellar absorption lines evolve on the timescale of the lifetimes of intermediate type main sequence stars, roughly several Gigayears.', 'astro-ph-0210471-3-52-3': 'The strengths of nebular emission features, however, evolve on much shorter timescales, as they require the presence of very hot, short lived O and B type stars.', 'astro-ph-0210471-3-52-4': 'These features therefore depend on the star formation history on the timescale of 5-10 Megayears.', 'astro-ph-0210471-3-52-5': 'The [MATH] parameter has been defined so as to be insensitive to the continuum slope (for practical reasons), and hence represents a sequence from strong absorption lines to strong emission lines.', 'astro-ph-0210471-3-52-6': 'It is well-known that the strengths of stellar absorption lines (especially Hydrogen recombination lines such as the Balmer series) are good indicators of stellar age, while the strength of the nebular emission lines are indicators of present star formation rate.', 'astro-ph-0210471-3-52-7': "With all this in mind, we can deduce that the rapid evolution in the value of [MATH] over the first several hundred Myr of a galaxy's life (see Fig. [REF]) reflects the fading of the emission lines as the star formation rate declines and the young stars burn out.", 'astro-ph-0210471-3-52-8': 'Galaxies then tend to "pile up" at low values of [MATH] as the age of their dominant stellar population exceeds a few Gyr.', 'astro-ph-0210471-3-53-0': 'In this sense it is straightforward to understand the relationship between [MATH] and [MATH] for our "simple" models, of monotonically decreasing star formation activity.', 'astro-ph-0210471-3-53-1': 'However, the star formation histories of galaxies created by the semi-analytic models can be quite complex.', 'astro-ph-0210471-3-53-2': 'The star formation rate fluctuates dramatically and non-monotonically over time as the galaxy exhausts its gas and then accretes a new gas supply, or experiences bursts of star formation triggered by mergers.', 'astro-ph-0210471-3-53-3': 'The issue we must now address is how an optical diagnostic such as [MATH], which is dominated by only the most recent events of star-formation, can be so closely related to the birth-rate parameter, [MATH], which incorporates the entire formation history of a galaxy.', 'astro-ph-0210471-3-54-0': 'The basis of the hierarchical picture of galaxy formation - from which our mock catalogue of SAM galaxies has been derived - is that galaxy properties are determined by the merger histories of their host dark matter halos.', 'astro-ph-0210471-3-54-1': 'In essence, the history of a specific halo (or galaxy) can be understood in terms of how the density on the scale of that structure compares to the background density.', 'astro-ph-0210471-3-54-2': 'A region with density much above the average (a "many sigma" peak, in the language of Gaussian random fields) collapses early, while lower density peaks collapse later.', 'astro-ph-0210471-3-54-3': 'It can therefore be conjectured that the early collapse of a dense dark matter halo can be associated with early star formation, early consumption of all available gas, and low present-day star formation rates.', 'astro-ph-0210471-3-54-4': 'Conversely, late-forming objects will have a sustained gas supply and ongoing star formation.', 'astro-ph-0210471-3-54-5': 'This suggests that the birthrate parameter [MATH] reflects perhaps the most theoretically fundamental property of a galaxy (or of its host halo): small-[MATH] galaxies represent rare, many-sigma peaks in the primordial density field, while large-[MATH] galaxies are formed in more common, lower density peaks.', 'astro-ph-0210471-3-54-6': 'This trend is manifested in the numerous empirical correlations between galaxy "type" (as characterised by morphology, colour, or spectral type), luminosity, and environment.', 'astro-ph-0210471-3-54-7': 'We expect this simple picture to be complicated by the details of gas cooling, star formation, feedback, etc., but the results from the semi-analytic models (which include all of these effects at some level, although of course the real Universe is likely to be even more complicated) suggest that this only introduces a moderate amount of scatter on top of the general trend.', 'astro-ph-0210471-3-55-0': 'We therefore conclude that the [MATH] parameter represents a promising candidate for galaxy classification in large modern redshift surveys.', 'astro-ph-0210471-3-55-1': 'It was adopted in previous analyses of the 2dFGRS because of its practical advantages: it is straightforward and efficient to compute in an automated fashion, and it is robust to the instrumental uncertainties commonly associated with fibre-based multi-object spectrographs.', 'astro-ph-0210471-3-55-2': 'Here we have shown that [MATH] also has a straightforward physical interpretation which can be intuitively connected both with traditional classifiers such as morphology, and that it seems to be connected with theoretically fundamental properties of galaxies within the modern hierarchical structure formation paradigm.'}

1810.04247

{'1810.04247-1-0-0': 'In this study, we propose a novel non-parametric embedded feature selection method based on minimizing the [MATH] norm of the vector of an indicator variable, whose point-wise product of an input selects a subset of features.', '1810.04247-1-0-1': 'Our approach relies on the continuous relaxation of Bernoulli distributions, which allows our model to learn the parameters of the approximate Bernoulli distributions via tractable methods.', '1810.04247-1-0-2': 'Using these tools we present a general neural network that simultaneously minimizes a loss function while selecting relevant features.', '1810.04247-1-0-3': 'We also provide an information-theoretic justification of incorporating Bernoulli distribution into our approach.', '1810.04247-1-0-4': 'Finally, we demonstrate the potential of the approach on synthetic and real-life applications.', '1810.04247-1-1-0': '# Introduction', '1810.04247-1-2-0': 'Technological advances are leading to the generation of large data sets both in sample size and dimensionality.', '1810.04247-1-2-1': 'Scientists are collecting complex high dimensional measurements.', '1810.04247-1-2-2': 'The collected data sets encapsulate both opportunities and challenges.', '1810.04247-1-2-3': 'For instance, in biology, we have access to tremendous amounts of biological markers on patients and wish to model complex interactions for prediction purposes.', '1810.04247-1-2-4': 'Unfortunately, that requires far more data than is generally available from clinical trials or in testing settings where experiments can be expensive.', '1810.04247-1-2-5': 'A method to mitigate this challenge is to identify the key set of features that influence prediction.', '1810.04247-1-2-6': 'Finding a subset of meaningful features, might not only improve the analytic task but also provide new scientific findings and improve the interpretability of machine-based models [CITATION].', '1810.04247-1-2-7': 'In bio-medical research, identifying a small set of relevant predictive bio-markers is essential [CITATION].', '1810.04247-1-2-8': 'There has been numerous works on non-parametric feature selection in high-dimensional settings [CITATION] and [CITATION].', '1810.04247-1-2-9': 'However, the scalability of these methods on huge datasets is limited in practice.', '1810.04247-1-3-0': 'Feature selection involves finding a subset of features that are sufficient for building a model.', '1810.04247-1-3-1': 'In the supervised setting, selected features might improve classification or regression performance.', '1810.04247-1-3-2': 'Furthermore, reducing the number of features has computational advantages and has been shown to improve model generalization on unseen data [CITATION].', '1810.04247-1-4-0': 'Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods.', '1810.04247-1-4-1': 'Filter methods are task-independent; they attempt to filter out the nonrelevant features prior to classification.', '1810.04247-1-4-2': 'Typically, a relevance score is created for each feature using some statistical comparison between each feature and the response (or class label).', '1810.04247-1-4-3': 'Filter methods have been demonstrated to be useful for various applications in [CITATION].', '1810.04247-1-4-4': 'More recent filter methods, such as [CITATION] use kernels or the Hilbert-Schmidt Independence Criterion (HSIC) [CITATION] to extract the most relevant features.', '1810.04247-1-4-5': 'Wrapper methods use the classifiers outcome to determine the relevance of each feature.', '1810.04247-1-4-6': 'As it is usually unfeasible to check all [MATH] subset of features, various works have been proposed to select subsets of features.', '1810.04247-1-4-7': 'Tree-based feature selection methods such as [CITATION], sequential wrapper methods such as [CITATION] an an iterative kernel based wrapper method was proposed in [CITATION].', '1810.04247-1-4-8': 'The main disadvantage of all mentioned wrapper methods is that they require recomputing the classifier for each subset of features.', '1810.04247-1-5-0': 'Embedded methods aim at relieving the computational burden.', '1810.04247-1-5-1': 'This is done by simultaneously learning the classifier and the subset of most relevant features.', '1810.04247-1-5-2': 'The Mutual Information (MI) is used by [CITATION], to select the important features by incorporating it in the classification procedure.', '1810.04247-1-5-3': 'Decision trees and random forest are naturally used for feature selection.', '1810.04247-1-5-4': 'A feature may be ranked based on the number of improvements it achieves as a splitter node in the tree.', '1810.04247-1-5-5': 'Perhaps the most common embedded method is the least absolute shrinkage and selection operator (LASSO) [CITATION].', '1810.04247-1-5-6': 'LASSO minimizes an objective function while enforcing an [MATH] constraint on the weights of the features.', '1810.04247-1-5-7': 'This method results in very efficient and scalable feature selection procedure.', '1810.04247-1-5-8': 'Although LASSO has been extended in various works [CITATION], it remains a linear method, which limits the classification capabilities.', '1810.04247-1-5-9': '[CITATION] extends to nonlinear functions by developing a feature wise kernelized LASSO.', '1810.04247-1-6-0': 'There have been a few attempts to use a neural network for feature selection.', '1810.04247-1-6-1': '[CITATION] trains different networks on subsets of features then selects them based on the performance of the network.', '1810.04247-1-6-2': '[CITATION] propose a heuristical procedure for adding features based on their contribution to a partially trained network.', '1810.04247-1-6-3': '[CITATION] use the activation values of the first layer to rank the importance of the features.', '1810.04247-1-6-4': 'All these methods are wrapper methods and do not provide an approach to simultaneously train a network and select the features.', '1810.04247-1-7-0': 'To the best of our knowledge, this study provides the first practical embedded feature selection method based on a neural network.', '1810.04247-1-7-1': 'The proposed learning procedure aims at simultaneously minimizing the [MATH] norm of a randomly selected subset of features along with a general loss function.', '1810.04247-1-7-2': 'Our method relies on the recent efforts of developing a continuous differentiable approximation to categorical distribution called Concrete Relaxation or Gumbel-softmax trick [CITATION], [CITATION].', '1810.04247-1-7-3': 'This enables us to introduce stochastic gates on the inputs, whose probability being active will be jointly learned with the model parameters via gradient descent.', '1810.04247-1-7-4': 'Our formulation naturally extends from linear models to neural networks by introducing indicator variables in the input layer of a network.', '1810.04247-1-7-5': 'We also provide an information-theoretic interpretation on the Bernoulli relaxation of the best subset selection, which justifies the introduction of randomness for feature selection in our risk minimization.', '1810.04247-1-7-6': 'Finally, we apply our feature selection method to various artificial and real data sets to demonstrate its effectiveness.', '1810.04247-1-8-0': '# Background', '1810.04247-1-9-0': 'Feature selection can be considered as finding the subset of features that leads to the largest possible generalization or equivalently to minimal risk.', '1810.04247-1-9-1': 'Every subset of features is modeled by a vector of indicator variables [MATH]z[MATH], where the point-wise product with an input [MATH]x[MATH] provides a subset of features.', '1810.04247-1-10-0': 'Given a parameterized family of regression functions, [MATH][MATH] we try to find a vector of indicator variables [MATH]z[MATH] and a vector of parameters [MATH][MATH][MATH] that minimizes the expected risk: [EQUATION] where [MATH] denotes the point-wise product, [MATH] is a loss function and [MATH] is a probability measure on the domain of the training data [MATH]X,Y[MATH].', '1810.04247-1-10-1': 'Where [MATH]X[MATH]x[MATH], is a realization of the input random variable and [MATH], is a realization of the response random variable or class label.', '1810.04247-1-10-2': 'In some cases we will also have the additional constraint [MATH]z[MATH], where [MATH] measures the sparsity of a given indicator variable [MATH]z[MATH].', '1810.04247-1-11-0': '## Feature Selection as an Optimization Problem', '1810.04247-1-12-0': 'Most penalized linear models expressed are in the terms of the following minimization: [EQUATION] where [MATH]x[MATH] measures the loss on the training point [MATH]x[MATH] and [MATH][MATH] is a penalizing term.', '1810.04247-1-12-1': 'The hyper-parameter [MATH] is a trade-off coefficient balancing the empirical error with this penalizing term.', '1810.04247-1-12-2': 'Examples of empirical errors are the [MATH] hinge loss, the [MATH] loss, and the logistic loss.', '1810.04247-1-12-3': 'Common penalizing terms include the [MATH] norm and the [MATH] norm, both have been shown to enforce sparsity.', '1810.04247-1-13-0': 'The minimization in Eq. ([REF]) inspired the least absolute shrinkage and selection operator (LASSO).', '1810.04247-1-13-1': 'LASSO enables a computationally efficient feature selection procedure.', '1810.04247-1-13-2': 'LASSO may be described as follows; let [MATH] be the sample size, [MATH] be the number of features, [MATH] be the response variable, [MATH]x[MATH], and [MATH][MATH] be the vector of weight parameters.', '1810.04247-1-13-3': 'The objective function of LASSO can be written as follows: [EQUATION]', '1810.04247-1-13-4': 'The above minimization problem has been applicable in numerous scientific fields and in other domains.', '1810.04247-1-14-0': '# Proposed Method', '1810.04247-1-15-0': 'The rise of [MATH] norm constraints in sparse estimation was partially due to its computational efficiency - the [MATH] norm is the closest convex function to the [MATH] norm [CITATION].', '1810.04247-1-15-1': "By adding the [MATH] norm as a regularization term LASSO's minimization procedure selects the most relevant features.", '1810.04247-1-15-2': 'We take a different approach to approximate the [MATH] norm, which can extend to non-linear models while remaining computationally efficient.', '1810.04247-1-15-3': 'In this section, we describe how to incorporate a continuous approximation of the Bernoulli distribution (presented in [CITATION], [CITATION]) into a general differentiable loss function.', '1810.04247-1-15-4': 'Applying backpropagation on the regularized loss function provides a natural network-based feature selection procedure.', '1810.04247-1-15-5': 'We denote this method as Deep Neural Network Feature Selection (DNN-FS).', '1810.04247-1-16-0': 'Penalizing the [MATH] norm in the empirical risk minimization can be understood as the best subset selection.', '1810.04247-1-16-1': 'In logistic regression, for example, the objective can be written as follows: [EQUATION] where [MATH] is a tuning parameter and [MATH][MATH].', '1810.04247-1-17-0': 'By introducing a vector [MATH]z[MATH] of indicator variables, where [MATH] indicates that [MATH]-th feature is included and [MATH] indicates that [MATH]-th feature is absent.', '1810.04247-1-17-1': 'Then, by taking the pointwise product of [MATH]z[MATH] and [MATH][MATH], we can represent any subset of features.', '1810.04247-1-17-2': 'To view the above optimization from a probabilistic perspective one can introduce a Bernoulli distribution [MATH] over each indicator variable [MATH].', '1810.04247-1-17-3': 'Then, using a logistic regression loss the expected risk from Eq. ([REF]) becomes [EQUATION] where [MATH]z[MATH][MATH] is a Bernoulli distribution, and [MATH]x[MATH]X[MATH] are the realization of input and response variables from the data distribution.', '1810.04247-1-17-4': "The goal of empirical risk minimization is to find [MATH][MATH] and [MATH][MATH] that minimizes [MATH][MATH][MATH]; However, due to the discrete nature of [MATH]z[MATH], optimizing the above objective with respect to the distribution parameters [MATH][MATH] doesn't admit tractable methods such as gradient descent.", '1810.04247-1-18-0': 'Following the recent work [CITATION], we review how to take gradients of the objective with respect to [MATH][MATH] by relaxing the discrete nature of [MATH]z[MATH].', '1810.04247-1-18-1': 'This allows us to compute Monte Carlo estimates to the expected risk and enable gradient descent to minimize the loss function.', '1810.04247-1-19-0': 'Our goal is to introduce a distribution that is continuous and differentiable with respect to its parameter as a replacement of Bernoulli distribution.', '1810.04247-1-19-1': 'To do this, we approximate a Bernoulli random variable represented as one-hot 2-dimensional vector with parameters [MATH] and [MATH] such that [MATH].', '1810.04247-1-19-2': 'This approximation is called Binary Concrete [CITATION], [CITATION]; here we briefly describe its derivation.', '1810.04247-1-20-0': 'First, a random perturbation is independently drawn for each category from a Gumbel(0,1) distribution: [EQUATION]', '1810.04247-1-20-1': 'Then we apply a temperature-dependent sigmoid function as a surrogate to finding a one-hot vector through an argmax operator: [EQUATION]', '1810.04247-1-20-2': 'The resulting random 2-dimensional vector is called Binary Concrete vector, which we denote as [EQUATION]', '1810.04247-1-20-3': 'Since [MATH], we set [MATH] to be the first coordinate of [MATH]s[MATH].', '1810.04247-1-20-4': 'We will introduce this variable [MATH] for each feature, forming the vector of approximate indicator variables, whose point-wise product [MATH]s[MATH]x[MATH] provides a subset of features.', '1810.04247-1-21-0': 'The recent paper [CITATION] proposed a hard-concrete distribution, which extends the above Binary Concrete distribution so that more probability mass can concentrate around [MATH], this increases sparsity in the final solution.', '1810.04247-1-21-1': 'The derivation of the hard-concrete distribution appears in the Appendix of this manuscript.', '1810.04247-1-21-2': 'By replacing the Bernoulli distribution with the hard-concrete distribution, we can now compute the derivative of the objective [MATH][MATH][MATH] with respect to the distribution parameters [MATH][MATH]: [EQUATION] where [MATH][MATH]s[MATH][MATH]s[MATH] can be computed via back-propagation since [MATH]s[MATH] is a deterministic function of [MATH][MATH], and the randomness only comes from [MATH] (defined in Eq. [REF]).', '1810.04247-1-21-3': 'The expectation over [MATH]x[MATH]X[MATH] is approximated based on the empirical per batch realizations.', '1810.04247-1-22-0': '# Related Work', '1810.04247-1-23-0': 'The two most related works to this study are [CITATION] and [CITATION].', '1810.04247-1-23-1': '[CITATION] introduces the hard concrete distribution.', '1810.04247-1-23-2': 'They show that incorporating it into a deep neural network sparsifies the learned weights.', '1810.04247-1-23-3': 'The authors demonstrate how this sparsification leads to fast convergence and improved generalization.', '1810.04247-1-23-4': 'In this study, we focus on a different goal.', '1810.04247-1-23-5': 'By applying the stochastic gates only to the input layer, we demonstrate how the hard concrete distribution is useful for parametric feature selection.', '1810.04247-1-23-6': 'This allows us to increase the feature size up to a number of thousands of features (as shown in the Section [REF]); this high dimensional regime is common in a field such as bioinformatics.', '1810.04247-1-23-7': 'In [CITATION], the Gumbel-softmax trick is used to develop a framework for interpreting a pre-trained model [CITATION].', '1810.04247-1-23-8': 'The main difference between our work and [CITATION] is that their method is focused on finding a subset of features given a particular instance, whereas our method aims to construct a subset of features based on all the training examples.', '1810.04247-1-23-9': 'This approach is more appropriate when we want to apply feature selection methods to data that are known to have consistent important features such as gene data.', '1810.04247-1-24-0': 'Furthermore, their method aims to optimize over a family of distributions, whose cardinality is [MATH].', '1810.04247-1-24-1': 'In order to reduce the number of possible candidates, they employ a subset sampling procedure based on the approximate categorical distribution for the purpose of instance-wise feature selection.', '1810.04247-1-24-2': 'This is not suitable for feature selection in general since when the dimension of the feature space is large, the variance of the samples from the categorical distribution becomes large, whereas our method employs an approximate Bernoulli for each dimension of the features, whose variance of the samples is independent of the the dimension of the feature space.', '1810.04247-1-25-0': 'Some authors tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models.', '1810.04247-1-25-1': 'Although [CITATION] and [CITATION] have a similar goal as ours, their empirical result does not achieve enough sparsity as practitioners would like.', '1810.04247-1-25-2': 'Our approach, which utilizes stochastic gates instead of relying on regularizing [MATH] norms, has an advantage in terms of achieving high sparsity level while maintaining good performance.', '1810.04247-1-26-0': '# Connection to Mutual Information', '1810.04247-1-27-0': 'In this section, we demonstrate that replacing the original subset selection problem with the Bernoulli probabilistic setting can be justified from a mutual information perspective in the feature selection setting.', '1810.04247-1-27-1': 'This is motivated by, but different than the work done by [CITATION].', '1810.04247-1-27-2': 'Recall that the mutual information between two random variables can be defined as [EQUATION] where [MATH]Y[MATH]Y[MATH]X[MATH] are the entropy of [MATH]y[MATH] and the conditional entropy of [MATH]Y[MATH]X[MATH]y[MATH]x[MATH], respectively [CITATION].', '1810.04247-1-27-3': 'Next, we present our two assumptions for this section:', '1810.04247-1-28-0': 'The first assumption means that if we do not include an element from [MATH], then we can improve our prediction accuracy by adding it.', '1810.04247-1-28-1': 'The second assumption means that we only need the variables in [MATH] to predict [MATH].', '1810.04247-1-28-2': 'Any additional variables are superfluous.', '1810.04247-1-28-3': 'The assumptions are quite benign.', '1810.04247-1-28-4': 'For instance they are satisfied if [MATH]X[MATH] is drawn Gaussian with a non-degenerate covariance matrix and [MATH]X_S^*[MATH], where [MATH] is noise independent of [MATH]X[MATH] and [MATH] is not degenerate.', '1810.04247-1-28-5': 'With this in place we can present our results.', '1810.04247-1-29-0': 'Suppose that the above assumptions hold for the model.', '1810.04247-1-29-1': 'Then, solving the optimization [EQUATION] is equivalent to solving the optimization [EQUATION] where the coordinates of [MATH] are drawn according to a Bernoulli distribution with parameter [MATH].', '1810.04247-1-30-0': 'Due to length constraints, we leave the proof of this proposition to the appendix.', '1810.04247-1-31-0': '# Experiments', '1810.04247-1-32-0': 'In this section, we perform a variety of experiments to evaluate the potential of using the proposed method (DNN-FS).', '1810.04247-1-32-1': 'In the first two experiments, we generate artificial samples by randomly sampling points based on some parametric distribution; then we assign a label based on a non-linear function.', '1810.04247-1-32-2': 'Each data set is concatenated with nuisance noisy coordinates; these coordinates do not hold information regarding the class identity.', '1810.04247-1-32-3': 'We compute both classification accuracy and feature weights depicted by our proposed approach.', '1810.04247-1-32-4': 'To evaluate the strength of DNN-FS, we compare it to recursive feature elimination (RFE) [CITATION], feature ranking using support vectors classification (SVC) [CITATION], LASSO [CITATION], and two tree-based methods [CITATION] and [CITATION] which we denote Tree and RF respectively.', '1810.04247-1-32-5': 'We also compare our method to a neural network with the same architecture without the feature selection layer, we denote this method as DNN.', '1810.04247-1-32-6': 'Each method extracts a weight for the relevance of each feature, we denote this weight by [MATH].', '1810.04247-1-32-7': 'The wrapper methods, such as RFE, SVC, Tree, and RF are retrained based on the features with highest extracted weights.', '1810.04247-1-32-8': 'In the following experiments, we compare classification accuracy as well as a metric which we denote as FEAT.', '1810.04247-1-32-9': 'FEAT is defined as the sum of weights [MATH] over the informative features divided by the sum over all weights.', '1810.04247-1-33-0': 'The architecture of the deep neural network is similar for all experiments.', '1810.04247-1-33-1': 'We use 3 layers with a hyperbolic tangent (Tanh) activation function.', '1810.04247-1-33-2': 'For the classification experiments, the final layer performs logistic regression.', '1810.04247-1-33-3': 'Whereas for the Cox hazard model (Section [REF]) we use the scaled exponential linear units (Selu) and partial likelihood as the loss.', '1810.04247-1-34-0': 'Setting the hyperparameter [MATH] effects the portion of selected features.', '1810.04247-1-34-1': 'In the artificial experiments (Sections [REF], [REF] and [REF]) we evaluate the effect of [MATH] on the convergence of the network.', '1810.04247-1-34-2': 'For the biological experiments (Sections [REF], [REF]) we vary lambda until the networks selects roughly [MATH] of original features and evaluate the networks generalization on the test set.', '1810.04247-1-35-0': '## Two Moons classification with nuisance features', '1810.04247-1-36-0': 'In the first experiment, we construct a dataset [MATH]X[MATH] based on "two moons" shape classes, concatenated with noisy features.', '1810.04247-1-36-1': 'The first two coordinates [MATH] are generated by adding Gaussian noise with zero mean to two nested half circles.', '1810.04247-1-36-2': 'An example of one realization of the first two coordinates is presented in Fig. [REF].', '1810.04247-1-36-3': 'Each addition nuisance features [MATH], is drawn from a Gaussian distribution with zero mean and variance of [MATH].', '1810.04247-1-36-4': 'Classification accuracy and the portion of weights assigned to the real signal (FEAT) are presented in Fig. [REF] and [REF].', '1810.04247-1-36-5': 'Based on the classification accuracies, it is evident that all methods correctly identify the most relevant features.', '1810.04247-1-36-6': 'As expected, DNN-FS and Random Forest are the only methods that achieve near perfect classification.', '1810.04247-1-36-7': 'DNN-FS and LASSO are the only methods which naturally sparsify the feature space.', '1810.04247-1-36-8': 'The parameter was set to [MATH]; this value seems to hold for a wide range of dimensions.', '1810.04247-1-36-9': "Moreover, the performance seemed stable for a wide range of [MATH]'s.", '1810.04247-1-37-0': '## Noisy binary XOR classification', '1810.04247-1-38-0': 'The binary XOR is a challenging classification problem.', '1810.04247-1-38-1': 'The first two coordinates [MATH] are drawn from a binary "fair" Bernoulli distribution.', '1810.04247-1-38-2': 'The response variable is set as an XOR of the first coordinates, such that [MATH].', '1810.04247-1-38-3': 'As in the previous experiment [MATH] are noisy features drawn from a Gaussian with zero mean and variance of [MATH].', '1810.04247-1-38-4': 'The number of points is [MATH], and [MATH] are used as training of all methods.', '1810.04247-1-38-5': 'The experiment is repeated 50 times for each value of [MATH], the average test classification accuracies are presented in Fig. [REF].', '1810.04247-1-38-6': 'In Fig. [REF] the average FEAT metric is presented.', '1810.04247-1-38-7': 'It seems that although every single feature is statistically independent of [MATH], the proposed method manages to select the relevant feature even for high [MATH] and reasonably small [MATH].', '1810.04247-1-39-0': '## Sparse Handwritten digits classification', '1810.04247-1-40-0': "In the following toy example, we attempt to classify between images of handwritten digits of 3's and 8's.", '1810.04247-1-40-1': 'Image features represent spatial information, therefore for this binary classification, we expect that some of the left side pixels would be sufficient for the separation.', '1810.04247-1-40-2': 'The experiment is performed as followed.', '1810.04247-1-40-3': 'We omit [MATH] of the data as the test set, and train on the remaining [MATH].', '1810.04247-1-40-4': 'We then apply DNN-FS and evaluate the classification accuracy and selected features.', '1810.04247-1-40-5': 'The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials.', '1810.04247-1-40-6': 'We noticed a relatively small number of selected features which are positioned Southwest and close to the center of the images achieve very high classification accuracy.', '1810.04247-1-40-7': 'An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. [REF].', '1810.04247-1-40-8': 'In this experiment, we further evaluate the effect of [MATH] on the sparsification and accuracy of the method.', '1810.04247-1-40-9': 'We apply the approach to a randomly sampled training set of size [MATH] and change [MATH] in the range of [MATH].', '1810.04247-1-40-10': "In Fig. [REF] we compare the accuracy and sparsity level for a wide range of [MATH]'s.", '1810.04247-1-40-11': 'It seems that within this wide range DNN-FS converges to a sparse solution with reasonable classification accuracy.', '1810.04247-1-41-0': '## Purified populations of peripheral blood monocytes (PBMCs)', '1810.04247-1-42-0': 'Single-cell RNA sequencing (scRNA-seq) is a novel technology that measures gene expression levels of hundreds of thousands of individual cells, simultaneously.', '1810.04247-1-42-1': 'This new tool is revolutionizing our understanding of cellular biology as it enables, among other things, the discovery of new cell types as well as the detection of subtle differences between similar but distinct cells.', '1810.04247-1-42-2': '[CITATION] have subjected more than 90,000 purified populations of peripheral blood monocytes (PBMCs) to scRNA-seq analysis.', '1810.04247-1-42-3': 'Such blood cells have been thoroughly characterized and studied, some of the cells are well separated based on their functionality.', '1810.04247-1-42-4': 'Here we focus on B-cells and cytotoxic T-cells.', '1810.04247-1-42-5': 'There are various marker genes that are known as unique to these cell types.', '1810.04247-1-42-6': 'In the following experiment, we attempt to extract these known markers by applying DNN-FS to a set of labeled PBMCs.', '1810.04247-1-42-7': 'We first filter out the genes that are lowly expressed in the cells, this leaves us with [MATH] genes (features).', '1810.04247-1-42-8': 'The number of cells in these two classes is [MATH], of which we only use [MATH] of the data.', '1810.04247-1-42-9': 'This again is a challenging regime for a neural network.', '1810.04247-1-42-10': 'DNN-FS achieves classification accuracy of [MATH] using [MATH] genes only.', '1810.04247-1-42-11': 'Intriguingly, when inspecting these selected genes there are various know markers of B and T cells or genes related to proteins which bind with them.', '1810.04247-1-42-12': 'Among some of these selected genes are CD3E, which is unique to T cells, CD79A and CD79B which are unique to B cells, and CD37 that is more abundant in B cells than in cytotoxic T-cells.', '1810.04247-1-43-0': '## Cox Proportional Hazard Deep Network Models for Survival Analysis', '1810.04247-1-44-0': 'In survival analysis, we are interested in building a predictive model for the survival time [MATH] of an individual [MATH] based on the covariates [MATH]x[MATH].', '1810.04247-1-44-1': 'Survival times are assumed to follow a distribution, which is characterized by survival function [MATH].', '1810.04247-1-44-2': 'A hazard function, which measures the instantaneous rate of death, is defined by [EQUATION]', '1810.04247-1-44-3': 'We can relate the two functions in the following way: [MATH].', '1810.04247-1-45-0': 'Proportional hazard models assume a multiplicative effect of the covariates [MATH] on the hazard function such that [EQUATION] where [MATH] is a baseline hazard function, which is often the exponential or Weibull distribution, and [MATH] is the parameter of interests.', '1810.04247-1-46-0': 'One of the difficulties in estimating [MATH][MATH] in survival analysis is that a large portion of the available data is censored.', '1810.04247-1-46-1': 'However, in order to obtain estimates, Cox observed that it is sufficient to maximize the partial-likelihood, which is defined as follows: [EQUATION]', '1810.04247-1-46-2': 'The previous work proposed DeepSurv [CITATION], which uses a deep neural network model to replace the linear relations between the covariate [MATH] and [MATH], demonstrating improvements of survival time prediction over existing models such as CPH and the random survival forest [CITATION], [CITATION].', '1810.04247-1-47-0': 'We apply our feature selection method in DeepSurv to see how our feature selection procedure improves the performance on the breast cancer dataset called METABRIC.', '1810.04247-1-47-1': 'The Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) uses gene and protein expression profiles to determine new breast cancer subgroups in order to help physicians provide better treatment recommendations.', '1810.04247-1-47-2': 'The METABRIC dataset consists of gene expression data and clinical features for [MATH] patients, and [MATH] have an observed death due to breast cancer with a median survival time of 116 months.', '1810.04247-1-48-0': 'We have picked 16 genes that were used in the Oncotype DX test, a genomic test that analyzes the activity of genes that could affect breast cancer behavior.', '1810.04247-1-48-1': 'We joined these genes with the patient’s clinical features (hormone treatment indicator, radiotherapy indicator, chemotherapy indicator, ER-positive indicator, age at diagnosis).', '1810.04247-1-48-2': 'We then reserved [MATH] of the patients as the test set.', '1810.04247-1-49-0': 'We first used DeepSurv on this dataset without feature selection.', '1810.04247-1-49-1': 'We tested 4 different feed-forward neural network architectures and repeated the same experiment 5 times.', '1810.04247-1-49-2': 'Then, we evaluated the predictive ability of the learned models based on the concordance index (CI), which measures the quality of rankings.', '1810.04247-1-49-3': 'The concordance index is a standard performance measure for model assessment in survival analysis.', '1810.04247-1-49-4': 'The best average CI is 0.639.', '1810.04247-1-50-0': 'To see how our method performs, we randomly sample genes to construct nuisance features.', '1810.04247-1-50-1': 'In total, we obtain 221 features, where 200 features are randomly sampled.', '1810.04247-1-50-2': 'We have applied DNN-FS to this data.', '1810.04247-1-50-3': 'With [MATH], the model picks 3 features, which is roughly [MATH] of the feature size.', '1810.04247-1-50-4': 'It achieves the average CI of 0.649 over 5 runs.', '1810.04247-1-50-5': 'This demonstrates that with just three features (GRB7, age at diagnosis, chemotherapy indicator), our model is able to perform better than the original DeepSurv model.', '1810.04247-1-51-0': '# Conclusion', '1810.04247-1-52-0': 'In this paper, we propose a novel embedded feature selection method based on stochastic gates.', '1810.04247-1-52-1': 'It has an advantage over previous LASSO based methods in terms of achieving a high level of sparsity in non-linear models such as neural networks.', '1810.04247-1-52-2': 'We relate the [MATH] norm minimization of the input weight to Mutual Information Maximization under the specific choice of variational family.', '1810.04247-1-52-3': 'In experiments, we observe that our method consistently outperforms existing embedded feature selection methods in both synthetic datasets and real biological datasets.'}

{'1810.04247-2-0-0': 'Feature selection problems have been extensively studied for linear estimation, for instance, Lasso, but less emphasis has been placed on feature selection for non-linear functions.', '1810.04247-2-0-1': 'In this study, we propose a method for feature selection in high-dimensional non-linear function estimation problems.', '1810.04247-2-0-2': 'The new procedure is based on minimizing the [MATH] norm of the vector of indicator variables that represent if a feature is selected or not.', '1810.04247-2-0-3': 'Our approach relies on the continuous relaxation of Bernoulli distributions, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent.', '1810.04247-2-0-4': 'This general framework simultaneously minimizes a loss function while selecting relevant features.', '1810.04247-2-0-5': 'Furthermore, we provide an information-theoretic justification of incorporating Bernoulli distribution into our approach and demonstrate the potential of the approach on synthetic and real-life applications.', '1810.04247-2-1-0': '# Introduction', '1810.04247-2-2-0': 'Technological advances are leading to the generation of large complex data sets both in sample size and dimensionality.', '1810.04247-2-2-1': 'The collected data sets encapsulate both opportunities and challenges.', '1810.04247-2-2-2': 'For instance, in biology, we have access to tremendous amounts of biological markers and wish to model their interactions for prediction purposes.', '1810.04247-2-2-3': 'Unfortunately, that requires far more data than is generally available from clinical trials.', '1810.04247-2-2-4': 'A method to mitigate this challenge is to identify the key set of features that influence prediction.', '1810.04247-2-2-5': 'Finding a subset of meaningful features might not only help the analytic task but also provide new scientific findings and improve the interpretability of models [CITATION].', '1810.04247-2-2-6': 'Furthermore, reducing the number of features has computational advantages and has been shown to improve model generalization on unseen data [CITATION].', '1810.04247-2-2-7': 'In high-dimensional settings, there has been numerous works on non-parametric feature selection [CITATION].', '1810.04247-2-3-0': 'Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods.', '1810.04247-2-3-1': 'Filter methods attempt to remove irrelevant features prior to classification.', '1810.04247-2-3-2': 'Typically, a relevance score is created for each feature based on some statistical measure.', '1810.04247-2-3-3': 'Filter methods have been demonstrated to be useful for various applications in [CITATION].', '1810.04247-2-3-4': 'More recent filter methods, such as [CITATION] use kernels to represent nonlinear interactions between features.', '1810.04247-2-3-5': 'Wrapper methods use the classifiers outcome to determine the relevance of each feature.', '1810.04247-2-3-6': 'Among some of the wrapper approaches are Tree-based feature selection methods such as [CITATION], sequential wrapper methods such as [CITATION] and an iterative kernel based wrapper method [CITATION].', '1810.04247-2-3-7': 'A few neural network based wrapper methods include [CITATION].', '1810.04247-2-3-8': 'The main disadvantage of all mentioned wrapper methods is that they require recomputing the classifier for each subset of features.', '1810.04247-2-4-0': 'Embedded methods aim at relieving the computational burden.', '1810.04247-2-4-1': 'This is done by simultaneously learning the model and the subset of most relevant features.', '1810.04247-2-4-2': 'The Mutual Information (MI) based embedded methods include [CITATION].', '1810.04247-2-4-3': 'Perhaps the most common embedded method is the Least Absolute Shrinkage and Selection Operator (LASSO) [CITATION].', '1810.04247-2-4-4': 'LASSO minimizes an objective function while enforcing an [MATH] constraint on the weights of the features.', '1810.04247-2-4-5': 'Although LASSO has been extended in various works [CITATION], it remains a linear method, which limits the classification and regression capabilities.', '1810.04247-2-4-6': 'LASSO is extended to nonlinear functions in [CITATION] by applying feature wise kernels.', '1810.04247-2-4-7': 'A few attempts have been made to generalize the objective used in LASSO to a neural network.', '1810.04247-2-4-8': 'In [CITATION] a combination of [MATH] and [MATH] applied to the weights from the first layer are added as a regularization term to the training loss.', '1810.04247-2-4-9': 'This idea is extended in [CITATION] by imposing the regularization simultaneously on groups of weights outgoing from the same neuron.', '1810.04247-2-4-10': 'In practice these methods require an additional thresholding of the weights and their performance deteriorates for a large number of layers.', '1810.04247-2-5-0': 'To overcome these limitations, we develop a practical fully embedded feature selection method for nonlinear functions.', '1810.04247-2-5-1': 'The proposed learning procedure aims at simultaneously minimizing the [MATH] norm of a randomly selected subset of features along with a general loss function.', '1810.04247-2-5-2': 'Inspired by the recent efforts of developing a continuous differentiable approximation to discrete distributions [CITATION], [CITATION], we present a simple relaxation of Bernoulli distributions, which enables us to introduce stochastic gates on the inputs, whose probability being active will be jointly learned with the model parameters via gradient descent.', '1810.04247-2-5-3': 'Our formulation naturally extends from linear models to neural networks by introducing indicator variables in the input layer of a network.', '1810.04247-2-5-4': 'We also provide an information-theoretic interpretation on the Bernoulli relaxation of the best subset selection, which justifies the introduction of randomness for feature selection in our risk minimization.', '1810.04247-2-5-5': 'We apply our feature selection method to various artificial and real datasets to demonstrate its effectiveness.', '1810.04247-2-5-6': 'Finally, we demonstrate that for feature selection tasks, our method has an advantage over the previously proposed relaxation.', '1810.04247-2-6-0': '# Background', '1810.04247-2-7-0': 'Feature selection can be considered as finding the subset of features that lead to the largest possible generalization or equivalently to minimal risk.', '1810.04247-2-7-1': 'Every subset of features is modeled by a vector of indicator variables [MATH]z[MATH], where the point-wise product with an input [MATH]x[MATH] provides a subset of features.', '1810.04247-2-8-0': 'Given a parameterized family of regression functions [MATH][MATH], we try to find a vector of indicator variables [MATH]z[MATH] and a vector of parameters [MATH][MATH][MATH] that minimize the expected risk: [EQUATION] where [MATH] denotes the point-wise product, [MATH] is a loss function and [MATH]X,Y[MATH] come from some data distribution [MATH].', '1810.04247-2-8-1': 'The empirical risk minimizer is calculated using the observations [MATH]x[MATH]y[MATH], where [MATH]x[MATH] is a feature vector and [MATH] is a response variable or class label.', '1810.04247-2-8-2': 'In some cases, we will also have the additional constraint [MATH]z[MATH], where [MATH] measures the sparsity of a given indicator variable [MATH]z[MATH].', '1810.04247-2-9-0': '## Feature Selection as an Optimization Problem', '1810.04247-2-10-0': 'Most penalized linear models are expressed in terms of the following minimization: [EQUATION] where [MATH][MATH] is a penalizing term.', '1810.04247-2-10-1': 'The hyper-parameter [MATH] balances the empirical error with the penalizing term.', '1810.04247-2-10-2': 'Examples of losses are the [MATH] hinge loss, the [MATH] loss, and the logistic loss.', '1810.04247-2-10-3': 'Common penalizing terms include the [MATH] norm and the [MATH] norm.', '1810.04247-2-11-0': 'The minimization in Eq. ([REF]) inspired the least absolute shrinkage and selection operator (LASSO), which enables a computationally efficient feature selection procedure.', '1810.04247-2-11-1': 'The objective function of LASSO can be written as follows: [EQUATION]', '1810.04247-2-11-2': 'The above minimization problem has been applicable in numerous scientific fields and in other domains.', '1810.04247-2-12-0': '# Proposed Method', '1810.04247-2-13-0': 'The rise of [MATH] norm constraints in sparse estimation was partially due to its computational efficiency - the [MATH] norm is the closest convex function to the [MATH] norm [CITATION].', '1810.04247-2-13-1': "By adding the [MATH] norm as a regularization term, LASSO's minimization procedure is prone to select more relavent features.", '1810.04247-2-13-2': 'We take a probabilistic approach to approximate the [MATH] norm, which can extend to non-linear models while remaining computationally efficient.', '1810.04247-2-14-0': 'To view the above optimization from a probabilistic perspective, one can introduce a Bernoulli distribution [MATH], where [MATH] is the parameter of indicator variable [MATH].', '1810.04247-2-15-0': 'Then, Eq. ([REF]) becomes [EQUATION] where [MATH]z[MATH][MATH] is the product of [MATH] independent Bernoulli distributions, and [MATH]x[MATH]y[MATH]X[MATH]Y[MATH] are the input and response variables from the data distribution.', '1810.04247-2-16-0': 'Our goal is to find [MATH][MATH] and [MATH][MATH] that minimize [MATH][MATH][MATH] via tractable methods such as gradient descent.', '1810.04247-2-17-0': '## Issues in Gradient Estimation', '1810.04247-2-18-0': 'The first term of the empirical risk [MATH][MATH][MATH] is expressed as [EQUATION]', '1810.04247-2-18-1': 'In practice, we have to replace the outer sum, which enumerates [MATH] possibilities of the indicator variables, with Monte Carlo samples from the product of Bernoulli distributions [MATH]z[MATH][MATH].', '1810.04247-2-18-2': 'However, a Monte Carlo estimate of [MATH][MATH][MATH] suffers from high variance.', '1810.04247-2-19-0': 'The exact gradient of the empirical risk with respect to [MATH] is [EQUATION] where [MATH] and we absorb the model [MATH] and the data into [MATH].', '1810.04247-2-19-1': 'We can see that even the sign of the gradient estimate becomes inaccurate if we can only access a small number of Monte Carlo samples, which creates difficulties in optimization.', '1810.04247-2-20-0': '## Continuous Relaxation', '1810.04247-2-21-0': 'As detailed in Subsection [REF], the optimization of a loss which consists of a discrete random variable suffers from high variance.', '1810.04247-2-21-1': 'To address this limitation, we introduce a continuous relaxation of Bernoulli distributions, which allows the model to utilize the gradient information of the loss evaluated at approximated Bernoulli samples, and thus reduce the variance of the gradient estimation in practice.', '1810.04247-2-21-2': 'For each feature [MATH], we first sample a random variable [MATH] from the standard Gaussian distribution [MATH], then we shift and scale it using parameters [MATH] and [MATH].', '1810.04247-2-21-3': 'We pass the rescaled variable to the hard-sigmoid function [MATH], where [MATH] is the slope parameter.', '1810.04247-2-21-4': 'The procedure is summarized as follows: [EQUATION]', '1810.04247-2-21-5': 'The resulting random variable [MATH] acts as a continuous approximation to a Bernoulli random variable, whose gradient with respect to its parameters can be computed via backpropagation; [MATH] is a deterministic function of [MATH], and the randomness only comes from [MATH].', '1810.04247-2-22-0': 'The new objective is now [EQUATION]', '1810.04247-2-22-1': 'The resulting gradient estimator becomes [EQUATION] where [MATH] is the number of Monte Carlo samples.', '1810.04247-2-22-2': 'We note that if we replace [MATH] with 1, the above gradient estimator is reduced to the Straight-Through estimator [CITATION].', '1810.04247-2-22-3': "In feature selection problems, [MATH] carries the critical information regarding feature's relevance.", '1810.04247-2-23-0': '## Implementation of the Regularization Term', '1810.04247-2-24-0': 'Under the continuous relaxation we have employed, the expected regularization term in the objective [MATH][MATH][MATH] is calculated by the sum of the probability of the gate [MATH] being active for [MATH].', '1810.04247-2-24-1': 'Using the the cumulative distribution function of the Gaussian distribution, it is expressed as [MATH], which can be implemented using the Gauss error function available in a standard machine learning library such as TensorFlow.', '1810.04247-2-24-2': 'However, the direct application of the Gaussian CDF does not optimize the loss since the gradient vanishes too quickly as [MATH] moves away from the origin.', '1810.04247-2-24-3': 'To address this issue in our implementation, we employ a function such as [MATH] that decays slower than the Gauss error function.', '1810.04247-2-25-0': '# Proposed Method', '1810.04247-2-26-0': 'The rise of [MATH] norm constraints in sparse estimation was partially due to its computational efficiency - the [MATH] norm is the closest convex function to the [MATH] norm [CITATION].', '1810.04247-2-26-1': "By adding the [MATH] norm as a regularization term LASSO's minimization procedure selects the most relevant features.", '1810.04247-2-26-2': 'We take a different approach to approximate the [MATH] norm, which can extend to non-linear models while remaining computationally efficient.', '1810.04247-2-26-3': 'In this section, we describe how to incorporate a continuous approximation of the Bernoulli distribution (presented in [CITATION], [CITATION]) into a general differentiable loss function.', '1810.04247-2-26-4': 'Applying backpropagation on the regularized loss function provides a natural network-based feature selection procedure.', '1810.04247-2-26-5': 'We denote this method as Deep Neural Network Feature Selection (DNN-FS).', '1810.04247-2-27-0': 'Penalizing the [MATH] norm in the empirical risk minimization can be understood as the best subset selection.', '1810.04247-2-27-1': 'In logistic regression, for example, the objective can be written as follows: [EQUATION] where [MATH] is a tuning parameter and [MATH][MATH].', '1810.04247-2-28-0': 'By introducing a vector [MATH]z[MATH] of indicator variables, where [MATH] indicates that [MATH]-th feature is included and [MATH] indicates that [MATH]-th feature is absent.', '1810.04247-2-28-1': 'Then, by taking the pointwise product of [MATH]z[MATH] and [MATH][MATH], we can represent any subset of features.', '1810.04247-2-29-0': 'To view the above optimization from a probabilistic perspective one can introduce a Bernoulli distribution [MATH] over each indicator variable [MATH].', '1810.04247-2-30-0': 'Then, using a logistic regression loss the expected risk from Eq. ([REF]) becomes [EQUATION] where [MATH]z[MATH][MATH] is a Bernoulli distribution, and [MATH]x[MATH]X[MATH] are the realization of input and response variables from the data distribution.', '1810.04247-2-31-0': "The goal of empirical risk minimization is to find [MATH][MATH] and [MATH][MATH] that minimizes [MATH][MATH][MATH]; However, due to the discrete nature of [MATH]z[MATH], optimizing the above objective with respect to the distribution parameters [MATH][MATH] doesn't admit tractable methods such as gradient descent.", '1810.04247-2-32-0': 'Following the recent work [CITATION], we review how to take gradients of the objective with respect to [MATH][MATH] by relaxing the discrete nature of [MATH]z[MATH].', '1810.04247-2-32-1': 'This allows us to compute Monte Carlo estimates to the expected risk and enable gradient descent to minimize the loss function.', '1810.04247-2-33-0': 'Our goal is to introduce a distribution that is continuous and differentiable with respect to its parameter as a replacement of Bernoulli distribution.', '1810.04247-2-33-1': 'To do this, we approximate a Bernoulli random variable represented as one-hot 2-dimensional vector with parameters [MATH] and [MATH] such that [MATH].', '1810.04247-2-33-2': 'This approximation is called Binary Concrete [CITATION], [CITATION]; here we briefly describe its derivation.', '1810.04247-2-34-0': 'First, a random perturbation is independently drawn for each category from a Gumbel(0,1) distribution: [EQUATION]', '1810.04247-2-34-1': 'Then we apply a temperature-dependent sigmoid function as a surrogate to finding a one-hot vector through an argmax operator: [EQUATION]', '1810.04247-2-34-2': 'The resulting random 2-dimensional vector is called Binary Concrete vector, which we denote as [EQUATION]', '1810.04247-2-34-3': 'Since [MATH], we set [MATH] to be the first coordinate of [MATH]s[MATH].', '1810.04247-2-34-4': 'We will introduce this variable [MATH] for each feature, forming the vector of approximate indicator variables, whose point-wise product [MATH]s[MATH]x[MATH] provides a subset of features.', '1810.04247-2-35-0': 'The recent paper [CITATION] proposed a hard-concrete distribution, which extends the above Binary Concrete distribution so that more probability mass can concentrate around [MATH], this increases sparsity in the final solution.', '1810.04247-2-35-1': 'The derivation of the hard-concrete distribution appears in the Appendix of this manuscript.', '1810.04247-2-36-0': 'By replacing the Bernoulli distribution with the hard-concrete distribution, we can now compute the derivative of the objective [MATH][MATH][MATH] with respect to the distribution parameters [MATH][MATH]: [EQUATION] where [MATH][MATH]s[MATH][MATH]s[MATH] can be computed via back-propagation since [MATH]s[MATH] is a deterministic function of [MATH][MATH], and the randomness only comes from [MATH] (defined in Eq. [REF]).', '1810.04247-2-36-1': 'The expectation over [MATH]x[MATH]X[MATH] is approximated based on the empirical per batch realizations.', '1810.04247-2-37-0': '# Connection to Mutual Information', '1810.04247-2-38-0': 'In this section, we demonstrate that replacing the original subset selection problem (Eq. [REF]) with the Bernoulli probabilistic setting can be justified from a mutual information perspective in the feature selection setting.', '1810.04247-2-38-1': 'This is motivated by, but different than the work done by [CITATION].', '1810.04247-2-38-2': 'Recall that the mutual information between two random variables can be defined as [EQUATION] where [MATH]Y[MATH]Y[MATH]X[MATH] are the entropy of [MATH]y[MATH] and the conditional entropy of [MATH]Y[MATH]X[MATH]y[MATH]x[MATH], respectively [CITATION].', '1810.04247-2-38-3': 'Next, we present our two assumptions for this section:', '1810.04247-2-39-0': 'The first assumption means that if we do not include an element from [MATH], then we can improve our prediction accuracy by adding it.', '1810.04247-2-39-1': 'The second assumption means that we only need the variables in [MATH] to predict [MATH].', '1810.04247-2-39-2': 'Any additional variables are superfluous.', '1810.04247-2-39-3': 'The assumptions are quite benign.', '1810.04247-2-39-4': 'For instance they are satisfied if [MATH]X[MATH] is drawn Gaussian with a non-degenerate covariance matrix and [MATH]X[MATH], where [MATH] is noise independent of [MATH]X[MATH] and [MATH] is not degenerate.', '1810.04247-2-39-5': 'With this in place we can present our results.', '1810.04247-2-40-0': 'Due to length constraints, we leave the proof of this proposition to the appendix.', '1810.04247-2-41-0': '# Related Work', '1810.04247-2-42-0': 'The two most related works to this study are [CITATION] and [CITATION].', '1810.04247-2-42-1': 'In [CITATION], they introduce the Hard-Concrete distribution as a continuous surrogate for Bernoulli distributions in the context of model compression.', '1810.04247-2-42-2': 'The authors demonstrate how their method leads to fast convergence and improved generalization in deep neural networks due to the sparsification effect.', '1810.04247-2-42-3': 'In this study, we focus on a different goal, and demonstrate that a simple relaxation of Bernoulli distributions is sufficient and works better than the Hard-Concrete distribution for feature selection tasks.', '1810.04247-2-42-4': 'Unlike the full sparsification framework, our method enables us to increase the feature size up to a number of thousands of features (as shown in the Section [REF]); this high dimensional regime is common in a field such as bioinformatics.', '1810.04247-2-42-5': 'In [CITATION], the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained models.', '1810.04247-2-42-6': 'Their method is focused on finding a subset of features given a particular instance, and therefore is not appropriate for general feature selection.', '1810.04247-2-42-7': 'In the case of gene data, for instance, we are often interested in finding consistent important features.', '1810.04247-2-43-0': 'Some authors tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models.', '1810.04247-2-43-1': 'Although [CITATION] and [CITATION] have a similar goal as ours, their empirical result does not achieve sufficient sparsity as practitioners would like.', '1810.04247-2-43-2': 'Our approach, which utilizes stochastic gates along with the [MATH] norm instead of relying on regularizing [MATH] norms, has an advantage in terms of achieving high sparsity level while maintaining good performance.', '1810.04247-2-44-0': '# Experiments', '1810.04247-2-45-0': 'In this section, we perform a variety of experiments to evaluate the potential of using the proposed Stochastic Gates (STG) for feature selection.', '1810.04247-2-45-1': 'In the first two experiments, we generate artificial samples by randomly sampling points based on some parametric distribution; then we assign a label based on a non-linear function.', '1810.04247-2-45-2': 'Each data set is concatenated with nuisance noisy coordinates; these coordinates do not hold information regarding the class identity.', '1810.04247-2-45-3': 'We compute both classification accuracy and feature weights depicted by our proposed approach.', '1810.04247-2-45-4': 'To evaluate the strength of STG, we compare it to feature ranking using support vectors classification (SVC) [CITATION], LASSO [CITATION], and two tree-based methods [CITATION] and [CITATION] which we denote Tree and Random Forests (RF) respectively.', '1810.04247-2-45-5': 'We also compare to network based feature selection methods, deep feature selection (DFS), group sparse regularized NN (SG-L1-NN) and to a neural network without any feature selection layer (DNN).', '1810.04247-2-45-6': 'In each experiment, all neural network based methods use the same architecture.', '1810.04247-2-45-7': 'Each method extracts a weight for the relevance of each feature, we denote this weight by [MATH].', '1810.04247-2-45-8': 'The wrapper methods, SVC, Tree, and RF are retrained based on the features with highest extracted weights.', '1810.04247-2-45-9': 'In the following experiments, we compare classification accuracy as well as the Informative Features Weight Ratio (IFWR).', '1810.04247-2-45-10': 'IFWR is defined as the sum of weights [MATH] over the informative features divided by the sum over all weights.', '1810.04247-2-46-0': 'The architecture of the neural networks is similar for all experiments.', '1810.04247-2-46-1': 'We use between [MATH] to [MATH] layers with widely used activation functions such as Linear, Relu or Tanh.', '1810.04247-2-46-2': 'For the artificial datasets, the exact architecture is optimized such that the network reaches near-optimal performance when no nuisance variables are added.', '1810.04247-2-46-3': 'For the classification experiments, the final layer performs logistic regression.', '1810.04247-2-46-4': 'Whereas for the Cox hazard model (Section [REF]) we use the scaled exponential linear units (Selu) and partial likelihood as the loss.', '1810.04247-2-47-0': 'Setting the hyperparameter [MATH] effects the portion of selected features.', '1810.04247-2-47-1': 'In the handwritten digits and biological experiments (Sections [REF] and [REF]) we evaluate the effect of [MATH] on the amount of features selected by the network.', '1810.04247-2-48-0': '## Two Moons classification with nuisance features', '1810.04247-2-49-0': 'In the first experiment, we construct a dataset based on "two moons" shape classes, concatenated with noisy features.', '1810.04247-2-49-1': 'The first two coordinates [MATH] are generated by adding a Gaussian noise with zero mean and the variance of [MATH] onto two nested half circles, as presented in Fig. [REF].', '1810.04247-2-49-2': 'Nuisance features [MATH], are drawn from a Gaussian distribution with zero mean and variance of [MATH].', '1810.04247-2-49-3': 'The classification accuracy and the portion of weights assigned to the informative feature (IFWR) are presented in Fig. [REF] and [REF].', '1810.04247-2-49-4': 'Based on the classification accuracies and assigned weights, it is evident that for a small number of nuisance dimensions all methods correctly identify the most relevant features.', '1810.04247-2-49-5': 'The proposed method (STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy for a wide range of nuisance dimensions.', '1810.04247-2-49-6': 'The other NN based methods (DFS and SG-L1-NN) seem to converge to sub-optimal solutions.', '1810.04247-2-49-7': 'STG and LASSO are the only methods which naturally sparsity the feature space.', '1810.04247-2-49-8': 'The parameter was set to [MATH]; this value appears to perform well for a wide range of nuisance dimensions.', '1810.04247-2-49-9': "Moreover, the performance seemed stable for a wide range of [MATH]'s.", '1810.04247-2-50-0': '## Noisy binary XOR classification', '1810.04247-2-51-0': 'The Junta problem was originally suggested by Godel, the goal is to learn a Boolean function which depends on [MATH] variables out of a set of size [MATH] [CITATION].', '1810.04247-2-51-1': 'Here, we consider this problem using a binary XOR for classification task.', '1810.04247-2-51-2': 'The first two coordinates [MATH] are drawn from a binary "fair" Bernoulli distribution.', '1810.04247-2-51-3': 'The response variable is set as an XOR of the first coordinates, such that [MATH].', '1810.04247-2-51-4': 'The coordinates [MATH] are nuisance features also drawn from a binary "fair" Bernoulli distribution.', '1810.04247-2-51-5': 'The number of points is [MATH], and [MATH] are used as a training set while the rest are used for test.', '1810.04247-2-51-6': 'The experiment is repeated 50 times for different values of [MATH], and the average test classification accuracy and standard deviation are presented in Fig. [REF].', '1810.04247-2-51-7': 'In Fig. [REF] the average IFWR metric is presented.', '1810.04247-2-51-8': 'Although every single feature is statistically independent of [MATH], the proposed method manages to reach perfect accuracy for large [MATH] and reasonably small [MATH].', '1810.04247-2-52-0': '## Regression using synthetic and real datasets', '1810.04247-2-53-0': 'In this section, we evaluate our method for regression tasks against two other embedded feature selection methods: LASSO and Sparse Random Fourier Features [CITATION].', '1810.04247-2-53-1': 'Following the same format as [CITATION], the following functions are used to generate synthetic data: (SE1: 100/5) [MATH].', '1810.04247-2-53-2': '(SE2: 18/5) [MATH].', '1810.04247-2-53-3': '(SE3: 1000/10) [MATH].', '1810.04247-2-53-4': 'The numbers next to the experiment code indicate total dimensions/irrelevant dimensions in the feature space.', '1810.04247-2-53-5': 'We also evaluate our method using a real dataset (RCP: 21/-), which measures computer systems activity, taken from the LIACC repository .', '1810.04247-2-53-6': 'For each dataset, we generate 30 different replications and randomly split the data into train, validation, and test set.', '1810.04247-2-53-7': 'For data preparation, we use the code publicly made available by [CITATION].', '1810.04247-2-53-8': 'The root mean squared error on the test set averaged over 30 random replicated datasets are reported in Table 1.', '1810.04247-2-53-9': 'Our method outperforms the other two methods for most cases.', '1810.04247-2-53-10': 'We note that (SE1) is generated using the sine function, which is in favor of SRFF, the random Fourier feature based method.', '1810.04247-2-54-0': '## Comparison to Hard-Concrete distribution', '1810.04247-2-55-0': 'To evaluate the strength of the proposed continuous relaxation of the Bernoulli distribution, described in Subsection [REF], we compare it with the Hard-Concrete distribution [CITATION], another continuous surrogate for Bernoulli distributions, which was originally developed for neural network model compression.', '1810.04247-2-55-1': 'The details of the Hard-Concrete distribution is deferred to the appendix.', '1810.04247-2-56-0': 'The main difference is that their distribution is based on the logistic distribution, which has a heavier tail than the Gaussian distribution we have employed.', '1810.04247-2-56-1': 'The heavy-tailness encourages "exploration" in terms of feature selection during training, but results in the instability as shown in Fig. 3; Our method converges much faster and more reliably than the feature selection method using the Hard-Concrete distribution on a the two-moons dataset (Subsection [REF]).', '1810.04247-2-57-0': '## Sparse Handwritten digits classification', '1810.04247-2-58-0': "In the following toy example, we attempt to distinguish between images of handwritten digits of 3's and 8's using samples from MNIST [CITATION].", '1810.04247-2-58-1': "The orientation and location of the digits is more or less the same throughout this dataset, therefore for these two classes (3's and 8's), we expect that some of the left side features (pixels) would be sufficient for the separation.", '1810.04247-2-58-2': 'The experiment is performed as followed.', '1810.04247-2-58-3': 'We omit [MATH] of the data as the test set, and train on the remaining [MATH].', '1810.04247-2-58-4': 'We then apply STG and evaluate the classification accuracy and selected features.', '1810.04247-2-58-5': 'The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials.', '1810.04247-2-58-6': 'We noticed a relatively small number of selected features which are positioned southwest and close to the center of the images achieve very high classification accuracy.', '1810.04247-2-58-7': 'An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. [REF].', '1810.04247-2-58-8': 'In this experiment, we further evaluate the effect of [MATH] on the sparsification and accuracy of the method.', '1810.04247-2-58-9': 'We apply the approach to a randomly sampled training set of size [MATH] and change [MATH] in the range of [MATH].', '1810.04247-2-58-10': 'In Fig. [REF] we present the accuracy and sparsity level vs. the [MATH] parameter.', '1810.04247-2-58-11': 'This expirement demonstrates the improved performence of the proposed distribution compared to the Hard-Concrete (HC [CITATION]).', '1810.04247-2-58-12': 'Not only that the overall accuracy is superior, but it seems that the transition as a function of [MATH] is smoother, this suggests that the method is less sensitive to the choice of [MATH].', '1810.04247-2-59-0': '## Purified populations of peripheral blood monocytes (PBMCs)', '1810.04247-2-60-0': 'Single-cell RNA sequencing (scRNA-seq) is a novel technology that measures gene expression levels of hundreds of thousands of individual cells, simultaneously.', '1810.04247-2-60-1': 'This new tool is revolutionizing our understanding of cellular biology as it enables, among other things, the discovery of new cell types as well as the detection of subtle differences between similar but distinct cells.', '1810.04247-2-60-2': 'The auturs in [CITATION], have subjected more than 90,000 purified populations of peripheral blood monocytes (PBMCs) to scRNA-seq analysis.', '1810.04247-2-60-3': 'Such blood cells have been thoroughly characterized and studied.', '1810.04247-2-60-4': 'Many of these cells fall into well separated clusters (subpopulations).', '1810.04247-2-60-5': 'Here we focus on two subpopulations of T-cells, namely the Naive and regulatory T-cells.', '1810.04247-2-60-6': 'The Naive CD4+ T-cells are responsible for activating the immune system against antigens, and the regulatory T-cells prevent activation against self antigens (autoimmune diseases).', '1810.04247-2-60-7': 'In the following experiment, we use the proposed method to select a subset of genes for which the network discriminates between Naive and regulatory T-cells.', '1810.04247-2-60-8': 'We first filter out the genes that are lowly expressed in the cells, this leaves us with [MATH] genes (features).', '1810.04247-2-60-9': 'The total number of cells in these two classes is [MATH], of which we only use [MATH] of the data for training.', '1810.04247-2-60-10': 'This again is a challenging regime for a classification task.', '1810.04247-2-60-11': 'We repeat the experiment for different values of [MATH] and report the number of selected features and classification accuracy on the test set.', '1810.04247-2-60-12': 'Here we compare our performance to Random Forests and LASSO.', '1810.04247-2-60-13': 'A scatter plot of the accuracy vs. number of selected features is presented in Fig. 5.', '1810.04247-2-60-14': 'For visibility convenience, we have added a least squares polynomial fit to all scatter plots in Fig. 5.', '1810.04247-2-60-15': 'We have also evaluated the performance of the Hard-Concrete applied to all layers (HC-Full), following the procedure in [CITATION].', '1810.04247-2-60-16': 'It seems that using this type of regularization provides inferior capabilities in terms of feature selection.', '1810.04247-2-60-17': 'Moreover, when the method converges to a larger subset of features [MATH], it does not generalize at all and the test accuracy is around [MATH].', '1810.04247-2-61-0': 'DNN-FS achieves classification accuracy of [MATH] using [MATH] genes only.', '1810.04247-2-61-1': 'Intriguingly, when inspecting these selected genes there are various known markers of B and T cells or genes related to proteins which bind with them.', '1810.04247-2-61-2': 'Among some of these selected genes are CD3E, which is unique to T cells, CD79A and CD79B which are unique to B cells, and CD37 that is more abundant in B cells than in cytotoxic T-cells.', '1810.04247-2-62-0': '## Cox Proportional Hazard Models for Survival Analysis', '1810.04247-2-63-0': 'In survival analysis, we are interested in building a predictive model for the survival time [MATH] of an individual [MATH] based on the covariates [MATH]x[MATH].', '1810.04247-2-63-1': 'Survival times are assumed to follow a distribution, which is characterized by the survival function [MATH].', '1810.04247-2-63-2': 'A hazard function, which measures the instantaneous rate of death, is defined by [EQUATION]', '1810.04247-2-63-3': 'We can relate the two functions in the following way: [MATH].', '1810.04247-2-64-0': 'Proportional hazard models assume a multiplicative effect of the covariates [MATH] on the hazard function such that [EQUATION] where [MATH] is a baseline hazard function, which is often the exponential or Weibull distribution, and [MATH] is the parameter of interests.', '1810.04247-2-65-0': 'One of the difficulties in estimating [MATH][MATH] in survival analysis is that a large portion of the available data is censored.', '1810.04247-2-65-1': 'However, in order to obtain estimates, Cox observed that it is sufficient to maximize the partial-likelihood, which is defined as follows: [EQUATION].', '1810.04247-2-66-0': 'In [CITATION], the authors propose DeepSurv, which uses a deep neural network model to replace the linear relations between the covariate [MATH]x[MATH] and [MATH][MATH], demonstrating improvements of survival time prediction over existing models such as CPH and the random survival forest [CITATION], [CITATION].', '1810.04247-2-67-0': 'We apply our feature selection method in DeepSurv to see how our procedure improves the performance on the breast cancer dataset called METABRIC [CITATION].', '1810.04247-2-68-0': 'The Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) uses gene and protein expression profiles to determine new breast cancer subgroups in order to help physicians provide better treatment recommendations.', '1810.04247-2-68-1': 'The METABRIC dataset consists of gene expression data and clinical features for [MATH] patients, and [MATH] have an observed death due to breast cancer with a median survival time of 116 months.', '1810.04247-2-69-0': 'We pick 16 genes used in the Oncotype DX test, a genomic test that analyzes the activity of genes that could affect breast cancer behavior.', '1810.04247-2-69-1': 'We join these genes with the patient’s clinical features (hormone treatment indicator, radiotherapy indicator, chemotherapy indicator, ER-positive indicator, age at diagnosis).', '1810.04247-2-69-2': 'To evaluate our feature selection method, we randomly sample genes to construct nuisance features.', '1810.04247-2-69-3': 'In total, we obtain 221 features, where 200 features are randomly sampled.', '1810.04247-2-69-4': 'We then hold out [MATH] of the patients as the test set.', '1810.04247-2-70-0': 'We evaluate the predictive ability of the learned models based on the concordance index (CI), which measures the agreement between the rankings of the predicted and observed survival times.', '1810.04247-2-70-1': 'The concordance index is a standard performance measure for model assessment in survival analysis.', '1810.04247-2-71-0': 'We compared our method (Cox-STG) against three other methods: Cox model with [MATH] regularization (Cox-LASSO; available as a python glmnet package), Random Survival Forest (RSF; available as an R package), and the original DeepSurv (DSurv).', '1810.04247-2-72-0': 'The performance in terms of the c-index is reported in Table [REF].', '1810.04247-2-72-1': 'We see that Cox-STG outperforms all the other methods.', '1810.04247-2-72-2': 'We note that the original Cox model using neural nets (DSurv) overfits to the training set, resulting in the poor generalization, whereas Cox-STG successfully shrinks the feature size and achieves good performance on the test set.', '1810.04247-2-73-0': '# Conclusion', '1810.04247-2-74-0': 'In this paper, we propose a novel embedded feature selection method based on stochastic gates.', '1810.04247-2-74-1': 'It has an advantage over previous [MATH] regularization based methods in terms of achieving a high level of sparsity in non-linear models such as neural networks, without hurting the performance.', '1810.04247-2-74-2': 'We justify our probabilistic feature selection framework from the information theoretic perspective.', '1810.04247-2-74-3': 'In experiments, we demonstrate that our method consistently outperforms existing embedded feature selection methods in both synthetic datasets and real datasets.', '1810.04247-2-75-0': '# Conclusion', '1810.04247-2-76-0': 'In this paper, we propose a novel embedded feature selection method based on stochastic gates.', '1810.04247-2-76-1': 'It has an advantage over previous [MATH] regularization based methods in terms of achieving a high level of sparsity in non-linear models such as neural networks, without hurting the performance.', '1810.04247-2-76-2': 'We justify our probabilistic feature selection framework from the information theoretic perspective.', '1810.04247-2-76-3': 'In experiments, we demonstrate that our method consistently outperforms existing embedded feature selection methods in both synthetic datasets and real datasets.'}

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['1810.04247-3-52-13', '1810.04247-4-54-13'], ['1810.04247-3-44-0', '1810.04247-4-46-0'], ['1810.04247-3-44-1', '1810.04247-4-46-1'], ['1810.04247-3-44-2', '1810.04247-4-46-2'], ['1810.04247-3-44-3', '1810.04247-4-46-3'], ['1810.04247-3-44-4', '1810.04247-4-46-4'], ['1810.04247-3-44-5', '1810.04247-4-46-5'], ['1810.04247-3-44-6', '1810.04247-4-46-6'], ['1810.04247-3-44-7', '1810.04247-4-46-7'], ['1810.04247-3-44-8', '1810.04247-4-46-8'], ['1810.04247-3-44-9', '1810.04247-4-46-9'], ['1810.04247-3-44-10', '1810.04247-4-46-10'], ['1810.04247-3-59-0', '1810.04247-4-60-0'], ['1810.04247-3-59-3', '1810.04247-4-60-3'], ['1810.04247-3-30-0', '1810.04247-4-31-0'], ['1810.04247-3-82-0', '1810.04247-4-83-0'], ['1810.04247-3-82-1', '1810.04247-4-83-1'], ['1810.04247-3-82-2', '1810.04247-4-83-2'], ['1810.04247-3-80-0', '1810.04247-4-81-0'], ['1810.04247-3-80-1', '1810.04247-4-81-1'], ['1810.04247-3-80-2', '1810.04247-4-81-2'], ['1810.04247-3-80-3', '1810.04247-4-81-3'], ['1810.04247-3-80-4', '1810.04247-4-81-4'], ['1810.04247-3-80-5', '1810.04247-4-81-5'], ['1810.04247-3-80-6', '1810.04247-4-81-6'], ['1810.04247-3-80-7', '1810.04247-4-81-7'], ['1810.04247-3-80-8', '1810.04247-4-81-8'], ['1810.04247-3-80-9', '1810.04247-4-81-9'], ['1810.04247-3-2-0', '1810.04247-4-2-0'], ['1810.04247-3-2-1', '1810.04247-4-2-1'], ['1810.04247-3-2-2', '1810.04247-4-2-2'], ['1810.04247-3-2-3', '1810.04247-4-2-3'], ['1810.04247-3-2-4', '1810.04247-4-2-4'], ['1810.04247-3-2-5', '1810.04247-4-2-5'], ['1810.04247-3-2-6', '1810.04247-4-2-6'], ['1810.04247-3-2-7', '1810.04247-4-2-7'], ['1810.04247-3-35-0', '1810.04247-4-36-0'], ['1810.04247-3-35-1', '1810.04247-4-36-1'], ['1810.04247-3-35-2', '1810.04247-4-36-2'], ['1810.04247-3-35-3', '1810.04247-4-36-3'], ['1810.04247-3-15-0', '1810.04247-4-16-0'], ['1810.04247-3-15-1', '1810.04247-4-16-1'], ['1810.04247-3-15-2', '1810.04247-4-16-2'], ['1810.04247-3-15-3', '1810.04247-4-16-3'], ['1810.04247-3-15-4', '1810.04247-4-16-4'], ['1810.04247-3-75-0', 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[]

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{'1': 'http://creativecommons.org/publicdomain/zero/1.0/', '2': 'http://creativecommons.org/publicdomain/zero/1.0/', '3': 'http://creativecommons.org/publicdomain/zero/1.0/', '4': 'http://creativecommons.org/publicdomain/zero/1.0/', '5': 'http://creativecommons.org/publicdomain/zero/1.0/'}

https://arxiv.org/abs/1810.04247

{'1810.04247-3-0-0': 'Feature selection problems have been extensively studied for linear estimation, for instance, Lasso, but less emphasis has been placed on feature selection for non-linear functions.', '1810.04247-3-0-1': 'In this study, we propose a method for feature selection in high-dimensional non-linear function estimation problems.', '1810.04247-3-0-2': 'The new procedure is based on minimizing the [MATH] norm of the vector of indicator variables that represent if a feature is selected or not.', '1810.04247-3-0-3': 'Our approach relies on the continuous relaxation of Bernoulli distributions, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent.', '1810.04247-3-0-4': 'This general framework simultaneously minimizes a loss function while selecting relevant features.', '1810.04247-3-0-5': 'Furthermore, we provide an information-theoretic justification of incorporating Bernoulli distribution into our approach and demonstrate the potential of the approach on synthetic and real-life applications.', '1810.04247-3-1-0': '# Introduction', '1810.04247-3-2-0': 'Technological advances have led to the generation of large complex data sets both in sample size and dimensionality.', '1810.04247-3-2-1': 'The collected data sets encapsulate both opportunities and challenges.', '1810.04247-3-2-2': 'For instance, in biology, we have access to tremendous amounts of biological markers and wish to model their interactions for prediction purposes.', '1810.04247-3-2-3': 'Unfortunately, in many cases, the number of features exceeds the number of samples.', '1810.04247-3-2-4': 'A method to mitigate this challenge is to identify the key set of features that influence prediction.', '1810.04247-3-2-5': 'Finding a subset of meaningful features not only helps the analytic task but also provides new scientific findings and improves the interpretability of models [CITATION].', '1810.04247-3-2-6': 'Furthermore, reducing the number of features has computational advantages and has been shown to improve model generalization on unseen data [CITATION].', '1810.04247-3-2-7': 'In high-dimensional settings, there has been numerous works studying the theoretical and empirical properties of linear [CITATION] and non-linear feature selection methods [CITATION].', '1810.04247-3-3-0': 'Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods.', '1810.04247-3-3-1': 'Filter methods attempt to remove irrelevant features prior to learning a model.', '1810.04247-3-3-2': 'These methods filter features based on a per-feature relevance score that is created based on some statistical measure [CITATION].', '1810.04247-3-3-3': 'Wrapper methods, such as [CITATION], use the outcome of a classifier to determine the relevance of each feature, which requires recomputing the classifier for each subset of features.', '1810.04247-3-3-4': 'This becomes computationally expensive for neural network based wrapper methods [CITATION].', '1810.04247-3-3-5': 'Embedded methods aim to remove this burden by learning the model while simultaneously selecting the subset of relevant features.', '1810.04247-3-3-6': 'The Least Absolute Shrinkage and Selection Operator (LASSO) [CITATION] is a well-known embedded method, whose objective is to minimize the loss while enforcing an [MATH] constraint on the weights of the features.', '1810.04247-3-3-7': 'Although LASSO is scalable and widely used [CITATION], it is restricted to the domain of linear functions.', '1810.04247-3-3-8': 'Therefore, it is appealing to consider the nonlinear extension of the LASSO formulation.', '1810.04247-3-4-0': 'We develop a fully embedded feature selection method based directly on approximating the [MATH] penalty.', '1810.04247-3-4-1': 'This method is applicable to learning non-linear functions modeled by neural networks.', '1810.04247-3-4-2': 'Thus, it can also be used as an alternative to LASSO for linear problems.', '1810.04247-3-4-3': 'We demonstrate a number of comparisons in Section [REF].', '1810.04247-3-4-4': 'Our contributions are as follows:', '1810.04247-3-5-0': 'Inspired by the recent efforts to develop a continuous and differentiable approximation to discrete distributions [CITATION], [CITATION], we present a simple relaxation of Bernoulli distributions, which enables us to introduce a stochastic selection gate for each feature.', '1810.04247-3-5-1': 'We apply the proposed gates to an input layer of a neural network, which allows us to approximate the [MATH] penalty.', '1810.04247-3-5-2': 'We then learn the parameters of the model and gates simultaneously via gradient descent.', '1810.04247-3-6-0': 'We justify our probabilistic approach by analyzing the constrained Mutual Information maximization objective.', '1810.04247-3-6-1': 'By implementing our procedure using a neural network, we demonstrate its capabilities for regression and classification on artificial and real data sets.', '1810.04247-3-7-0': 'Notation: We refer to vectors as bold lowercase [MATH]x[MATH] and random vectors as bold uppercase letters [MATH]X[MATH].', '1810.04247-3-7-1': 'Scalars are non-bold case [MATH], while random variables are capital case [MATH].', '1810.04247-3-7-2': 'A set is represented by script fonts [MATH], [MATH], [MATH].', '1810.04247-3-7-3': 'For example the [MATH] vector-valued observation is denoted as [MATH]x[MATH] whereas [MATH] represents the [MATH] feature of the vector-valued random variable [MATH]X[MATH].', '1810.04247-3-7-4': 'Let [MATH].', '1810.04247-3-7-5': 'For a set [MATH] let the vector [MATH] be the characteristic function for the set.', '1810.04247-3-7-6': 'That is [MATH] if [MATH] and [MATH] otherwise.', '1810.04247-3-7-7': 'For two vectors [MATH]x[MATH] and [MATH]z[MATH] we denote [MATH] to be the element-wise product between [MATH] and [MATH].', '1810.04247-3-7-8': 'Thus, if we let [MATH] be the characteristic vector of [MATH], then we may define [MATH].', '1810.04247-3-7-9': 'The [MATH] norm of a vector is denoted as [MATH].', '1810.04247-3-7-10': 'Finally, the [MATH] norm of a vector is denoted as [MATH] and counts the total number of non-zero entries in the vector [MATH].', '1810.04247-3-8-0': '# Problem Setup and Background', '1810.04247-3-9-0': 'Let [MATH] be the input domain with corresponding response domain [MATH].', '1810.04247-3-9-1': 'Given realizations from some unknown data distribution [MATH] the goal of embedded feature selection methods is to simultaneously find a subset of indices [MATH] and construct a model that predicts [MATH] based on the selected features [MATH]X[MATH].', '1810.04247-3-10-0': '## Risk minimization objective', '1810.04247-3-11-0': 'We assume that we are given a family of functions [MATH] such that any function [MATH] is indexed by a set of parameters [MATH].', '1810.04247-3-11-1': 'Given some loss [MATH], a selection of features [MATH], and a choice of parameters [MATH], we denote the risk of our model as [EQUATION] where we recall that [MATH]s[MATH] is a vector of indicator variables for the set [MATH] and [MATH] denotes the point-wise product.', '1810.04247-3-11-2': 'Thus, the goal of the feature selection problem is to find the parameters [MATH] and [MATH] that minimize [MATH] such that [MATH] is small compared to [MATH].', '1810.04247-3-12-0': '## Feature Selection for Linear Models', '1810.04247-3-13-0': 'Before proceeding with our proposed method, we review the feature selection problem in the linear regression setting with the least squares loss.', '1810.04247-3-13-1': 'Thus, we restrict [MATH] to be the space of linear functions and the loss function to be the quadratic loss.', '1810.04247-3-14-0': 'Given observations [MATH]x[MATH] we may consider the constrained empirical risk minimization problem [EQUATION]', '1810.04247-3-14-1': 'Since the above problem is intractable, a number of authors replace the [MATH] constraint with a surrogate function [MATH][MATH] designed to penalize the number of selected features in [MATH].', '1810.04247-3-14-2': 'A natural choice for [MATH] is the [MATH] norm, which yields a convex problem and more precisely the LASSO optimization problem [CITATION].', '1810.04247-3-14-3': 'Subsequently, a number of authors have developed computationally efficient algorithms for solving the problem [CITATION].', '1810.04247-3-14-4': 'While the original Lasso problem focuses on the constrained optimization problem, the regularized least squares problem, which is often used in practice, yields the following minimization objective: [EQUATION]', '1810.04247-3-14-5': 'The hyperparameter [MATH] trades off the amount of regularization versus the fit of the objective.', '1810.04247-3-14-6': 'The [MATH] regularized method is very effective for feature selection and prediction; however, it achieves this through shrinkage of the coefficients.', '1810.04247-3-14-7': 'As a result, authors have considered non-convex choices for [MATH] as well [CITATION] that perform well both theoretically and empirically for prediction and feature selection.', '1810.04247-3-15-0': 'Our goal is to apply such regularization techniques to perform feature selection while learning a non-linear function.', '1810.04247-3-15-1': 'Kernel methods have been considered [CITATION], but scale quadratically in the number of observations.', '1810.04247-3-15-2': 'An alternative approach is to model [MATH] using a neural network with [MATH] regularization on the input weights [CITATION].', '1810.04247-3-15-3': 'However, in practice, introducing an [MATH] penalty into gradient descent does not provide sufficient specification.', '1810.04247-3-15-4': 'Below, we discuss our method that works to directly use an [MATH] penalty.', '1810.04247-3-16-0': '# Proposed Method', '1810.04247-3-17-0': 'We take a probabilistic approach to approximate the [MATH] norm, which can extend to non-linear models while remaining computationally efficient.', '1810.04247-3-17-1': 'To motivate our approach, we provide theoretical support (see Section [REF]) based on a Mutual Information perspective for the feature selection problem.', '1810.04247-3-18-0': 'To view the [MATH] regularized version of the risk (Eq. [REF]) from a probabilistic perspective, one can introduce a Bernoulli random vector [MATH]S[MATH] whose entries are independent and the [MATH] entry satisfies [MATH] for [MATH].', '1810.04247-3-18-1': 'If we denote the empirical expectation over our observations as [MATH], then, the empirical regularized risk (Eq. [REF]) becomes [EQUATION] where we have [MATH]S[MATH] and we constrain [MATH].', '1810.04247-3-18-2': 'Clearly, this formulation is equivalent to equation [REF], with a regularized penalty on cardinality rather than an explicit constraint.', '1810.04247-3-18-3': 'We may then relax the discrete constraint on [MATH] to be [MATH].', '1810.04247-3-19-0': 'Now, our goal is to find the model parameters [MATH][MATH] and Bernoulli parameters [MATH][MATH] that minimize the empirical risk [MATH][MATH][MATH] via gradient descent.', '1810.04247-3-19-1': 'However, an optimization of a loss function which includes discrete random variables suffers from high variance (See supplementary for more details).', '1810.04247-3-19-2': 'Therefore, inspired by a recently developed continuous approximation for discrete random variables, suggested by [CITATION], we develop and use a novel and simple continuous distribution that is fully differentiable and suited to the task of feature selection.', '1810.04247-3-20-0': '## Continuous Relaxation', '1810.04247-3-21-0': 'Our continuous relaxation for the Bernoulli variables [MATH] for [MATH] relies on the reparametrization trick, which is widely used for reducing the variance of gradient estimators [CITATION].', '1810.04247-3-21-1': 'To construct a continuous approximation to Bernoulli random variable via the reparametrization trick, we define [MATH] where [MATH] is drawn from a Gaussian distribution [MATH], where [MATH] is fixed throughout training.', '1810.04247-3-21-2': 'This approximation can be viewed as a clipped, mean-shifted, Gaussian random vector.', '1810.04247-3-21-3': 'Furthermore, the gradient of the objective with respect to [MATH] can be computed via the chain rule.', '1810.04247-3-21-4': 'We can now rewrite the objective in Eq. [REF] as [EQUATION] where [MATH]Z[MATH] is a random vector with [MATH] independent variables [MATH] for [MATH].', '1810.04247-3-21-5': 'To optimize the empirical surrogate of the objective [REF], we first differentiate it with respect to [MATH][MATH].', '1810.04247-3-21-6': 'Then, Monte Carlo sampling leads us to the following gradient estimator [EQUATION] where [MATH] is the number of Monte Carlo samples.', '1810.04247-3-21-7': 'Thus, we can update the parameters [MATH] for [MATH] via gradient descent.', '1810.04247-3-21-8': 'We note that if we replace [MATH] with 1, the above gradient estimator for [MATH] is reduced to the Straight-Through estimator [CITATION].', '1810.04247-3-22-0': 'Under the continuous relaxation, the expected regularization term in the objective [MATH][MATH][MATH] is simply the sum of the probability that the gates [MATH] are active, which is equal to [MATH], where [MATH] is the standard Gaussian CDF.', '1810.04247-3-22-1': "To conclude, we can now optimize the objective [REF] using gradient descent over the model parameters [MATH][MATH] and the parameters [MATH][MATH] representing the Gaussian's mean (instead of the Bernoulli parameters [MATH][MATH]).", '1810.04247-3-23-0': 'After training, to remove the stochasticity from the learned gates, we set [MATH], which informs what features are selected.', '1810.04247-3-23-1': 'Note that when [MATH] is less than [MATH], [MATH] returns the value between [MATH].', '1810.04247-3-23-2': 'In such a case, we can treat the value of [MATH] as feature importance or employ an additional thresholding (i.e. 1 if [MATH] and [MATH] otherwise) depending on application-specific needs.', '1810.04247-3-23-3': 'In the supplementary material, we provide the pseudo-code of our algorithm as well as the discussion of the choice of [MATH].', '1810.04247-3-24-0': '# Connection to Mutual Information', '1810.04247-3-25-0': 'In this section we show an equivalence between the Bernoulli formulation of the feature selection problem and the [MATH] regularized approach.', '1810.04247-3-26-0': '## Mutual Information based objective', '1810.04247-3-27-0': 'From an information theoretic perspective, the goal of feature selection is to find a subset of features [MATH] that has the highest Mutual Information (MI) with the target variable [MATH].', '1810.04247-3-27-1': 'Recall that the MI between two random variables can be defined as [MATH]X[MATH]X[MATH] where [MATH]X[MATH] are the entropy of [MATH] and the conditional entropy of [MATH]X[MATH], respectively [CITATION].', '1810.04247-3-27-2': 'Then we can formulate the task as selecting [MATH] such that the mutual information between [MATH]X[MATH] and [MATH] are maximized: [EQUATION] where [MATH] is the hypothesised number of relevant features.', '1810.04247-3-28-0': '## Introducing randomness', '1810.04247-3-29-0': 'We first demonstrate that under mild assumptions we can replace the deterministic search over the set [MATH] (or corresponding indicator vector [MATH]s[MATH]), by a search over the parameters of the distributions that model [MATH]s[MATH].', '1810.04247-3-29-1': 'Our proposition is based on the following two assumptions: Assumption 1: There exists a subset of indices [MATH] with cardinality equal to [MATH] such that for any [MATH] we have [MATH]X[MATH].', '1810.04247-3-29-2': 'Assumption 2: [MATH]X[MATH]X[MATH].', '1810.04247-3-29-3': 'Discussion of assumptions: Assumption 1 that including an element from [MATH] improves prediction accuracy.', '1810.04247-3-29-4': 'This assumption is equivalent to stating that feature [MATH] is strongly relevant [CITATION].', '1810.04247-3-29-5': 'Assumption 2 simply states that [MATH] is a superset of the Markov Blanket of the variable [MATH] [CITATION].', '1810.04247-3-29-6': 'The assumptions are quite benign.', '1810.04247-3-29-7': 'For instance they are satisfied if [MATH]X[MATH] is drawn from a Gaussian with a non-degenerate covariance matrix and [MATH]X[MATH], where [MATH] is noise independent of [MATH]X[MATH] and [MATH] is not degenerate.', '1810.04247-3-29-8': 'With these assumptions in hand, we may present our result.', '1810.04247-3-30-0': 'Due to length constraints, we leave the proof of this proposition and how it bridges the MI maximization [REF] and risk minimization [REF] in the supplementary material.', '1810.04247-3-31-0': '# Related Work', '1810.04247-3-32-0': 'The two most related works to this study are [CITATION] and [CITATION].', '1810.04247-3-32-1': 'In [CITATION], they introduce the Hard-Concrete distribution as a continuous surrogate for Bernoulli distributions in the context of model compression.', '1810.04247-3-32-2': 'The authors demonstrate how their method leads to fast convergence and improved generalization in deep neural networks due to the sparsification effect.', '1810.04247-3-32-3': 'In this study, we present a sparsification method that is aimed specificaly at feature selection that can then be extended to general non-linear function estimation.', '1810.04247-3-32-4': 'We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and works better than the Hard-Concrete distribution for feature selection tasks (see the supplementary material).', '1810.04247-3-32-5': 'Unlike the full sparsification framework, our method enables us to increase the feature size to a number of thousands of features (as shown in the Section [REF]); this high dimensional regime is common in a field such as bioinformatics.', '1810.04247-3-32-6': 'In [CITATION], the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained models.', '1810.04247-3-32-7': 'Their method is focused on finding a subset of features given a particular instance, and therefore is not appropriate for general feature selection.', '1810.04247-3-33-0': 'Some authors tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models.', '1810.04247-3-33-1': 'Other authors [CITATION] and [CITATION] have a similar goal as ours in performing feature selection, but instead rely on the [MATH] relaxation to the [MATH].', '1810.04247-3-33-2': 'Our approach, which utilizes stochastic gates coupled with the [MATH] norm achieves better empirical performance when compared against other baselines including the [MATH] relaxation.', '1810.04247-3-33-3': 'This point is explored in the next section.', '1810.04247-3-34-0': '# Experiments', '1810.04247-3-35-0': 'Here we provide empirical evaluation of our method in a wide range of settings.', '1810.04247-3-35-1': 'We start from the simple setting of linear regression, and move to nonlinear function estimation.', '1810.04247-3-35-2': 'The hyperparamters of all the methods are optimized using validation sets via Optuna [CITATION].', '1810.04247-3-35-3': 'In the supplemental material, we provide all details of the parameter tuning procedure as well as additional experiments, including an evaluation of our regularization parameter [MATH] and comparison to the Hard Concrete [CITATION].', '1810.04247-3-36-0': '## Phase Transition in the Linear Setting', '1810.04247-3-37-0': 'The problem of signal support recovery is tied to the problem of feature selection.', '1810.04247-3-37-1': 'The ability to recover the support of a vector based on its noisy linear observation is useful for various tasks, among which are compressed sensing, image denoising, and channel estimation.', '1810.04247-3-37-2': 'The authors in [CITATION] provide an analysis of the probability of successful support recovery of LASSO in the following experimental setup.', '1810.04247-3-37-3': 'Let [MATH][MATH] be a fixed sparse vector, such that [MATH] (with equal probability) if [MATH], and [MATH] otherwise.', '1810.04247-3-37-4': 'Suppose the cardinally of the support [MATH] is known.', '1810.04247-3-37-5': 'Given a matrix of measurements [MATH]X[MATH] with values drawn independently from [MATH], the response [MATH]y[MATH] is defined as [EQUATION] where the values of the noise [MATH]w[MATH] are drawn independently from [MATH].', '1810.04247-3-38-0': 'Here, we reproduce this setting to evaluate the probability of prefect support recovery of [MATH][MATH], based on our method.', '1810.04247-3-38-1': 'As in [CITATION] we use a sparsity that scales with [MATH] such that [MATH].', '1810.04247-3-38-2': 'For each number of samples [MATH] in the range [MATH], we run [MATH] simulations and count the portion of correctly recovered supports.', '1810.04247-3-38-3': 'We repeat this process for [MATH] different values of [MATH] and compare our performance to LASSO.', '1810.04247-3-38-4': 'For LASSO, the regularization parameter was set to its optimal value [MATH] [CITATION].', '1810.04247-3-38-5': 'For STG, we set [MATH], such that [MATH] is a constant, which is selected using a grid search in the range [0.1,10].', '1810.04247-3-38-6': 'As evident from Fig. [REF], even when restricting to linear functions our method has a clear advantage over LASSO.', '1810.04247-3-38-7': 'This implies that the [MATH] based penalty, even though not convex in nature, allows us to recover the support of [MATH] using less samples.', '1810.04247-3-39-0': '## Noisy Binary XOR Classification', '1810.04247-3-40-0': 'To evaluate the capability of our method to capture nonlinear relationships between [MATH]x[MATH] and [MATH], we consider the problem of learning a binary XOR function for classification task.', '1810.04247-3-40-1': 'The first two coordinates [MATH] are drawn from a binary "fair" Bernoulli distribution.', '1810.04247-3-40-2': 'The response variable is set as an XOR of the first coordinates, such that [MATH].', '1810.04247-3-40-3': 'The coordinates [MATH] are nuisance features, also drawn from a binary "fair" Bernoulli distribution.', '1810.04247-3-40-4': 'The number of points we generate is [MATH], of which [MATH] are reserved for test and [MATH] of the remaining training set was reserved for validation.', '1810.04247-3-40-5': 'We compare the proposed method to four embedded feature selection methods (LASSO [CITATION], C-support vectors (SVC) [CITATION], deep feature selection (DFS) [CITATION], sparse group regularized NN (SG-L1-NN) [CITATION]).', '1810.04247-3-40-6': 'To provide more benchmarks, we also compare our embedded method against three wrapper methods (Extremely Randomized Trees (Tree) [CITATION], Random Forests (RF) [CITATION]) and Extreme Gradient Boosting (XGBOOST) [CITATION].', '1810.04247-3-41-0': 'To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio (IFWR).', '1810.04247-3-41-1': 'IFWR is defined as the sum of weights [MATH] over the informative features divided by the sum over all weights.', '1810.04247-3-41-2': '(See the supplementary material for more details.)', '1810.04247-3-42-0': 'The experiment is repeated [MATH] times for different values of [MATH], and the average test classification accuracy and standard deviation are presented in Fig. [REF], followed by the IFWR in Fig. [REF].', '1810.04247-3-42-1': 'The number of selected features affects the accuracy.', '1810.04247-3-42-2': 'Therefore, to treat all the methods in a fair manner, we tune the hyperparameter that controls the sparsity level using Optuna [CITATION] based on a cross validation procedure to optimize the overall accuracy across different [MATH]s. For instance, the wrapper methods (Tree, RF and XGBOOST) has a threshold value to retain features.', '1810.04247-3-42-3': 'We retrain them using only such features whose weight is higher than the threshold, which we can optimize to control the sparsity.', '1810.04247-3-42-4': 'In terms of feature ranking, only the tree based methods and the proposed STG provide the optimal median rank (which is [MATH]) for the two relevant features.', '1810.04247-3-43-0': '## MADELON dataset', '1810.04247-3-44-0': 'The MADELON dataset, first suggested for the NIPS 2003 feature selection problem, is a multivariate highly nonlinear binary classification problem.', '1810.04247-3-44-1': 'The MADELON dataset is generated using 32 groups of data points placed on a [MATH] dimensional hyper-cube and randomly labeling them by one of the two class labels.', '1810.04247-3-44-2': 'The first [MATH] informative features are then used to create [MATH] additional coordinates that are formed based on a random linear transformation of the first [MATH].', '1810.04247-3-44-3': 'A Gaussian noise [MATH] is added to each feature.', '1810.04247-3-44-4': 'Next, additional [MATH] nuisance coordinates are added in the same manner.', '1810.04247-3-44-5': 'These features have no effect on the class label.', '1810.04247-3-44-6': 'Finally, [MATH] of the labels are flipped.', '1810.04247-3-44-7': 'We use [MATH] points from this dataset, and evaluate our proposed method in terms of its predictive power and ability to detect the informative features.', '1810.04247-3-44-8': 'We vary the regularizaton parameter [MATH] in the range [MATH] and evaluate the classification accuracy using [MATH] folds cross validation.', '1810.04247-3-44-9': 'In this example, we restrict our comparison to Random Forest and LASSO.', '1810.04247-3-44-10': 'We focus on Random Forest as it was the strongest competitor to our method in all of our experiments, while LASSO is evaluated because it is a widely used embedded feature selection method.', '1810.04247-3-45-0': 'As evident from Fig. [REF], our method achieves the highest accuracy while using less features.', '1810.04247-3-45-1': 'Moreover, as depicted from this figure, peak performence occurs when selecting [MATH] features, thus, our method provides a clear indication to the true number of informative features.', '1810.04247-3-45-2': 'Both LASSO and RF on the other hand, do not provide a clear indication of the true number of relevant features.', '1810.04247-3-45-3': 'In Fig. [REF], we evaluate the effect of [MATH] on the number and quality of selected features.', '1810.04247-3-45-4': "As shown in this plot, there is a wide range of [MATH]'s such that our method only selects relevant features.", '1810.04247-3-45-5': "Finally, as evident from the plato on the right hand side of the red plot, there is a range of [MATH]'s such that exactly [MATH] features are selected.", '1810.04247-3-46-0': '## Two Moons classification with nuisance features', '1810.04247-3-47-0': 'In this experiment, we construct a dataset based on "two moons" shape classes, concatenated with noisy features.', '1810.04247-3-47-1': 'The first two coordinates [MATH] are generated by adding a Gaussian noise with zero mean and the variance of [MATH] onto two nested half circles, as presented in Fig. [REF].', '1810.04247-3-47-2': 'Nuisance features [MATH], are drawn from a Gaussian distribution with zero mean and variance of [MATH].', '1810.04247-3-47-3': 'We reserve the [MATH] as a test set, and use [MATH] of the remaining training set as a validation set.', '1810.04247-3-47-4': 'We follow the same hyperparameter tuning procedure as in the XOR experiment.', '1810.04247-3-47-5': 'The classification accuracy is in Fig. [REF].', '1810.04247-3-47-6': 'Based on the classification accuracies, it is evident that for a small number of nuisance dimensions all methods correctly identify the most relevant features.', '1810.04247-3-47-7': 'The proposed method (STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy for a wide range of nuisance dimensions.', '1810.04247-3-47-8': 'The other NN based method (DFS) seem to converge to sub-optimal solutions.', '1810.04247-3-47-9': 'We note that the median rank for all the methods is 1.5.', '1810.04247-3-48-0': '## Comparison to Hard-Concrete distribution', '1810.04247-3-49-0': 'To evaluate the strength of the proposed continuous relaxation of the Bernoulli distribution, described in Subsection 3.1, we compare it with the Hard-Concrete distribution [CITATION], another continuous surrogate for Bernoulli distributions, which was originally developed for neural network model compression.', '1810.04247-3-49-1': 'The details of the Hard-Concrete distribution is described in the Appendix.', '1810.04247-3-50-0': 'The main difference between our proposed distribution and the Hard-Concrete [CITATION] distribution is that the latter is based on the logistic distribution, which has a heavier tail than the Gaussian distribution we have employed.', '1810.04247-3-50-1': 'As shown in Fig [REF], the heavy-tailness results in instability during training.', '1810.04247-3-50-2': 'Furthermore, our method converges much faster and more reliably than the feature selection method using the Hard-Concrete distribution on a the two-moons dataset (Subsection [REF]).', '1810.04247-3-51-0': '## Sparse Handwritten digits classification', '1810.04247-3-52-0': "In the following toy example, we attempt to distinguish between images of handwritten digits of 3's and 8's using samples from MNIST [CITATION].", '1810.04247-3-52-1': "The orientation and location of the digits is more or less the same throughout this dataset, therefore for these two classes (3's and 8's), we expect that some of the left side features (pixels) would be sufficient for the separation.", '1810.04247-3-52-2': 'The experiment is performed as followed.', '1810.04247-3-52-3': 'We reserve [MATH] of the data as the test set, and train on the remaining [MATH].', '1810.04247-3-52-4': 'We then apply STG and evaluate the classification accuracy and the number of selected features.', '1810.04247-3-52-5': 'We use the architecture [200, 50, 10] with tanh activations.', '1810.04247-3-52-6': 'The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials.', '1810.04247-3-52-7': 'We noticed a relatively small number of selected features, which are positioned southwest and close to the center of the images, achieve very high classification accuracy.', '1810.04247-3-52-8': 'An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. [REF].', '1810.04247-3-52-9': 'Furthermore, we also evaluate the effect of [MATH] on the the number of selected features and accuracy of the method.', '1810.04247-3-52-10': 'We apply our method (STG) and its variant using the Hard-Concrete distribution to a randomly sampled training set of size [MATH] and vary [MATH] in the range of [MATH].', '1810.04247-3-52-11': 'In Fig. [REF] we present the accuracy and sparsity level vs. the [MATH] parameter.', '1810.04247-3-52-12': 'This experiment demonstrates the improved performance of the proposed distribution compared to the Hard-Concrete (HC [CITATION]), which was designed for neural net model compression.', '1810.04247-3-52-13': 'Not only that the overall accuracy is superior, but also it seems that the transition as a function of [MATH] is smoother, which suggests that the method is less sensitive to the choice of [MATH].', '1810.04247-3-53-0': '## Regression using synthetic and real datasets', '1810.04247-3-54-0': 'In this section, we evaluate our method for regression tasks against two other embedded feature selection methods: LASSO and Sparse Random Fourier Features [CITATION].', '1810.04247-3-54-1': 'Following the same format as [CITATION], the following functions are used to generate synthetic data: (SE1: 100/5) [MATH].', '1810.04247-3-54-2': '(SE2: 18/5) [MATH].', '1810.04247-3-54-3': '(SE3: 1000/10) [MATH], where each [MATH] is drawn from the standard Gaussian [MATH].', '1810.04247-3-54-4': 'For (SE3), the five consecutive coordinates are generated by [MATH], where [MATH] for [MATH] and [MATH].', '1810.04247-3-54-5': 'The numbers next to the experiment code indicate total dimensions/relevant dimensions in the feature space.', '1810.04247-3-54-6': 'We also evaluate our method using a real dataset (RCP: 21/-), which measures computer systems activity, taken from the LIACC repository .', '1810.04247-3-54-7': 'For each dataset, we generate 30 different replications and randomly split the data into train, validation, and test set.', '1810.04247-3-54-8': 'For data preparation, we use the code publicly made available by [CITATION].', '1810.04247-3-54-9': 'The root mean squared error on the test set averaged over 30 random replicated datasets are reported in Table 1.', '1810.04247-3-54-10': 'Our method outperforms the other two methods for most cases.', '1810.04247-3-54-11': 'We note that (SE1) is generated using the sine function, which is in favor of the random Fourier feature based method (SRFF).', '1810.04247-3-55-0': '## Purified populations of peripheral blood monocytes (PBMCs)', '1810.04247-3-56-0': 'Single-cell RNA sequencing (scRNA-seq) is a novel technology that measures gene expression levels of hundreds of thousands of individual cells, simultaneously.', '1810.04247-3-56-1': 'This new tool is revolutionizing our understanding of cellular biology as it enables, among other things, the discovery of new cell types as well as the detection of subtle differences between similar but distinct cells.', '1810.04247-3-56-2': 'The authors in [CITATION], have subjected more than [MATH] purified populations of peripheral blood monocytes (PBMCs) to scRNA-seq analysis.', '1810.04247-3-56-3': 'Here we focus on classifying two subpopulations of T-cells, namely the Naive and regulatory T-cells.', '1810.04247-3-56-4': 'The Naive CD4+ T-cells are responsible for activating the immune system against antigens, and the regulatory T-cells prevent activation against self antigens (autoimmune diseases).', '1810.04247-3-56-5': 'In the following experiment, we use the proposed method to select a subset of genes for which the network discriminates between Naive and regulatory T-cells.', '1810.04247-3-56-6': 'We first filter out the genes that are lowly expressed in the cells, which leaves us with [MATH] genes (features).', '1810.04247-3-56-7': 'The total number of cells in these two classes is [MATH], of which we only use [MATH] of the data for training.', '1810.04247-3-56-8': 'We apply the proposed method for different values of [MATH] and report the number of selected features and classification accuracy on the test set.', '1810.04247-3-56-9': 'Here we compare our performance to RF and LASSO.', '1810.04247-3-56-10': 'A scatter plot of the accuracy vs. number of selected features is presented in Fig. [REF].', '1810.04247-3-56-11': 'For visibility convenience, we have added a least squares polynomial fit to all scatter plots in Fig. [REF].', '1810.04247-3-56-12': 'We have also evaluated the performance of the Hard-Concrete applied to all layers (HC-Full), following the procedure in [CITATION].', '1810.04247-3-56-13': 'It seems that using this type of regularization provides inferior capabilities in terms of feature selection.', '1810.04247-3-56-14': 'Moreover, when the method converges to a larger subset of features [MATH], it does not generalize at all and the test accuracy is around [MATH].', '1810.04247-3-57-0': 'DNN-FS achieves classification accuracy of [MATH] using [MATH] genes only.', '1810.04247-3-57-1': 'Intriguingly, when inspecting these selected genes there are various known markers of B and T cells or genes related to proteins which bind with them.', '1810.04247-3-57-2': 'Among some of these selected genes are CD3E, which is unique to T cells, CD79A and CD79B which are unique to B cells, and CD37 that is more abundant in B cells than in cytotoxic T-cells.', '1810.04247-3-58-0': '## Cox Proportional Hazard Models for Survival Analysis', '1810.04247-3-59-0': 'A standard model for survival analysis is Cox Proportional Hazard Model.', '1810.04247-3-59-1': 'DeepSurv was proposed to extend the model to neural networks [CITATION].', '1810.04247-3-59-2': 'We incorporate our feature selection method into DeepSurv to see how our procedure improves survival analysis based on gene expression profiles from the breast cancer dataset called METABRIC [CITATION] and additional commonly used clinical variables.', '1810.04247-3-59-3': 'See Supplimentary for more details about the dataset and experimental setup.', '1810.04247-3-60-0': 'We compared our method (Cox-STG) against two other methods: Cox model with [MATH] regularization (Cox-LASSO; available as a python glmnet package) and the original DeepSurv.', '1810.04247-3-60-1': 'We evaluate the predictive ability of the learned models based on the concordance index (CI), a standard performance metric for model assessment in survival analysis, which measures the agreement between the rankings of the predicted and observed survival times.', '1810.04247-3-60-2': 'The performance in terms of the CI is reported in Table [REF].', '1810.04247-3-60-3': 'We see that Cox-STG outperforms all the other methods, indicating that Cox-STG successfully shrinks the feature size and achieves good performance on the test set.', '1810.04247-3-61-0': '# Conclusion', '1810.04247-3-62-0': 'In this paper, we propose a novel embedded feature selection method based on stochastic gates.', '1810.04247-3-62-1': 'It has an advantage over previous [MATH] regularization based methods in terms of achieving a high level of sparsity in non-linear models such as neural networks, without hurting the performance.', '1810.04247-3-62-2': 'We justify our probabilistic feature selection framework from the information theoretic perspective.', '1810.04247-3-62-3': 'In experiments, we demonstrate that our method consistently outperforms existing embedded feature selection methods in both synthetic datasets and real datasets.', '1810.04247-3-63-0': '# Algorithm', '1810.04247-3-64-0': 'Here we present the pseudo code of our method, presented in Algorithm [REF].', '1810.04247-3-64-1': 'The loss [MATH] is typically negative-log likelihood for classification and squared loss for regression.', '1810.04247-3-64-2': '[MATH] is the sample size, [MATH] is the number of features, [MATH] is the number of Monte Carlo samples.', '1810.04247-3-65-0': '[!', '1810.04247-3-65-1': 'htb] STG: Feature selection using stochastic gates Input: [MATH]X[MATH], target variables [MATH]y[MATH], regularization parameter [MATH], number of epochs [MATH], learning rate [MATH].', '1810.04247-3-66-0': 'Output: Trained model [MATH][MATH] and parameter [MATH][MATH].', '1810.04247-3-67-0': '[1] Initialize the model parameter [MATH][MATH].', '1810.04247-3-67-1': 'Set [MATH][MATH] Sample [MATH] Compute the gate [MATH] Set [MATH]', '1810.04247-3-68-0': 'Set [MATH]z[MATH] Compute the loss [MATH]x[MATH]z[MATH] Compute the regularization term [MATH] Update [MATH][MATH][MATH][MATH] and [MATH][MATH][MATH][MATH]', '1810.04247-3-69-0': 'Note that we empirically observe that setting the number of Monte Carlo samples [MATH] and the standard deviation of the Gaussian distribution [MATH] for [MATH] suffices for feature selection in our experiments.', '1810.04247-3-69-1': 'See Section 4 in the supplementary material for more details about [MATH].', '1810.04247-3-69-2': 'After training, the set of indices for selected features is: [MATH].', '1810.04247-3-70-0': '# Proof of Proposition [REF]', '1810.04247-3-71-0': 'We now give the proof of the proposition showing the equivalence between the stochastic optimization [REF] and the deterministic one [REF].', '1810.04247-3-71-1': 'Let [MATH] be a subset such that [MATH].', '1810.04247-3-71-2': 'That is there exists some element in [MATH] that is not in [MATH].', '1810.04247-3-71-3': 'For any such set [MATH] we have that [MATH]X[MATH]X[MATH].', '1810.04247-3-71-4': 'Indeed, if we let [MATH] then we have [EQUATION] where the final inequality follows by Assumption 1.', '1810.04247-3-71-5': 'Assumption 2 also yields that for any set [MATH] such that [MATH], we have [MATH]X[MATH]X[MATH].', '1810.04247-3-71-6': 'Now, when we consider the Bernoulli optimization problem we have [EQUATION]', '1810.04247-3-71-7': 'The mutual information can be expanded as [EQUATION] where we have used the fact that [MATH]S[MATH] is independent of everything else.', '1810.04247-3-71-8': 'Recall that in optimization [REF] the coordinates of [MATH]S[MATH] are sampled at random.', '1810.04247-3-71-9': 'Therefore, the distribution that is being optimized over [MATH][MATH] is a product distribution.', '1810.04247-3-71-10': 'Our goal is to understand the form of this distribution.', '1810.04247-3-71-11': 'To that end, we will consider a problem dropping the independence constraint.', '1810.04247-3-71-12': 'If we can show that the distribution found by solving this new optimization problem with less constraints is still a product distribution, then we obtain a solution to the original optimization [REF].', '1810.04247-3-72-0': 'Now, from above we know that the optimal value of the optimization is [MATH]X[MATH]S[MATH] for any set [MATH].', '1810.04247-3-72-1': 'Hence, any unconstrained distribution should place all of its mass on such subsets in order to maximize the mutual information.', '1810.04247-3-72-2': 'As a result [MATH]S[MATH].', '1810.04247-3-72-3': 'However, there is an optimization constraint that [MATH]S[MATH].', '1810.04247-3-72-4': 'Therefore, [MATH] for any [MATH].', '1810.04247-3-72-5': 'Hence, the optimal solution is to select the distribution so that all of the mass is placed on the subset [MATH] and no mass elsewhere.', '1810.04247-3-72-6': 'As this is also a product distribution, this complete the proof of the claim.', '1810.04247-3-73-0': '# Bridging the Two Perspectives', '1810.04247-3-74-0': 'To motivate the introduction of randomness into the risk, we have looked at the feature selection problem from a MI perspective.', '1810.04247-3-74-1': 'Based on the MI objective, we have observed that introducing randomness into the constrained maximization procedure, does not change the objective (Proposition 1).', '1810.04247-3-74-2': 'Here we provide a relation between the MI objective [REF] to the empirical risk [REF], which supports our proposed procedure.', '1810.04247-3-75-0': 'We first note that the MI maximization over the set [MATH] can be reformulated as the minimization of the conditional entropy [MATH]X[MATH] since [MATH] does not depend on [MATH]: [EQUATION]', '1810.04247-3-75-1': 'Recall that [MATH]X[MATH]X[MATH]S[MATH].', '1810.04247-3-75-2': 'By Proposition 1, we rewrite the the deterministic search over the set [MATH] by a search over the Bernoulli parameters [MATH][MATH]: [EQUATION] where the expectation is over [MATH]X[MATH][MATH] which is the true data distribution, and [MATH][MATH].', '1810.04247-3-75-3': 'Put our model distribution as [MATH].', '1810.04247-3-75-4': 'Then we rewrite the right hand side as: [EQUATION]', '1810.04247-3-75-5': 'Since [MATH]X[MATH]S[MATH]X[MATH]S[MATH] is non-negative, [MATH]X[MATH]S[MATH]X[MATH]S[MATH] is also non-negative because it is a weighted sum of non-negative terms.', '1810.04247-3-75-6': 'Noting that [EQUATION] we can conclude that [EQUATION]', '1810.04247-3-75-7': 'If we consider the negative log likelihood of the target given the observations (i.e. [MATH]X[MATH]S[MATH]) as a loss function [MATH] (which encodes the classification or regression function [MATH][MATH]), then we see that the minimizing the risk approximately maximizes the MI objective in [REF].', '1810.04247-3-76-0': 'We now turn our focus on relating the MI objective (Eq. [REF]) to the empirical risk [REF].', '1810.04247-3-76-1': 'To do this, we find a lower bound for the MI [REF] that enables a tractable maximization search for the subset [MATH].', '1810.04247-3-76-2': 'This setup allows us to ignore the constant entropy of [MATH], [MATH] during the opimization.', '1810.04247-3-76-3': 'Next, by introducing a variational distribution [MATH] and by noting that KL divergence is non-negative, we can achieve the following lower bound (cite Barber, Agakov (2003)): [EQUATION] where an equality holds when [MATH] and [MATH] are equal.', '1810.04247-3-76-4': 'Using this lower bound, we can relax the maximization problem described in Eq. [REF] as follows: [EQUATION] where [MATH] is considered as a model distribution approximating the true conditional distribution [MATH].', '1810.04247-3-76-5': 'If we consider the negative log likelihood of the target given the observations (i.e. [MATH]) as a loss function [MATH] (which encodes the classification or regression function [MATH]), then the maximization in [REF] is equivalent to the following minimization: [EQUATION] where the [MATH] norm counts the number of non-zero components.', '1810.04247-3-76-6': 'However, the above optimization is intractable as it requires to enumerate all [MATH] possibilities to find the minimum of the objective.', '1810.04247-3-76-7': 'In the following section, we show an introduction of randomness into [MATH]S[MATH] allows us to develop a computationally tractable optimization procedure.', '1810.04247-3-77-0': '# Details of Regularization Term', '1810.04247-3-78-0': 'Here we provide the detail description of the regularization term.', '1810.04247-3-78-1': 'For the vector of stochastic gates [MATH]z[MATH], the regularization term is expressed as follows: [EQUATION]', '1810.04247-3-78-2': 'The derivative of the regularization term with respect to the distribution parameter [MATH] is simply the Gaussian PDF: [EQUATION]', '1810.04247-3-78-3': 'The effect of [MATH] can be understood by looking at the value of [MATH]Z[MATH].', '1810.04247-3-78-4': 'In the first iteration during the training, [MATH] is 0.', '1810.04247-3-78-5': 'Therefore, during the initial phase of training, it is close to [MATH].', '1810.04247-3-78-6': 'In order to remove irrelevant features, this term has to be greater than the derivative of the loss with respect to [MATH] because otherwise [MATH] is updated in the incorrect direction.', '1810.04247-3-78-7': 'To encourage such behavior, we set [MATH], which is around the maximum of the gradient during the initial phase as shown in Fig. [REF].', '1810.04247-3-78-8': 'Although the point that attains the maximum moves as [MATH] changes, we empirically observe that setting [MATH] performs well in our experiments when the regularization parameter [MATH] is appropriately set.', '1810.04247-3-79-0': '# Issues in Gradient Estimation of Discrete Random Variables', '1810.04247-3-80-0': 'In Section [REF], we have introduced Bernoulli random [MATH] variables with corresponding parameters [MATH] into the risk objective [REF].', '1810.04247-3-80-1': 'Taking the expectation over the [MATH] norm of [MATH]S[MATH] boils down to the sum of the Bernoulli parameters [MATH].', '1810.04247-3-80-2': 'However, the optimization of the resulting objective suffers from high variance due to the discrete nature of [MATH]S[MATH].', '1810.04247-3-80-3': 'Here, we attempt to convey this problem by analyzing the risk term in the objective [REF].', '1810.04247-3-80-4': 'Using the Bernoulli paramterization the empirical risk [MATH][MATH][MATH] is expressed as [EQUATION]', '1810.04247-3-80-5': 'In practice, as the outer sum involves enumerating [MATH] possibilities of the indicator variables, one can replace the outer sum with Monte Carlo samples from the product of Bernoulli distributions [MATH]z[MATH][MATH].', '1810.04247-3-80-6': 'However, a Monte Carlo estimate of [MATH][MATH][MATH] suffers from high variance.', '1810.04247-3-80-7': 'To see this, consider the following exact gradient of the empirical risk with respect to [MATH], which is [EQUATION] where [MATH], by absorbing the model [MATH] and the data into [MATH].', '1810.04247-3-80-8': 'Due to the discrete nature of [MATH]z[MATH], we see that even the sign of the gradient estimate becomes inaccurate if we can only access a small number of Monte Carlo samples.', '1810.04247-3-80-9': 'While a score-function estimator such as REINFORCE [CITATION] can be used, it is known that the reparametrization trick reduces the variance more in practice.', '1810.04247-3-81-0': '# Hard-Concrete distribution', '1810.04247-3-82-0': 'The authors in [CITATION] introduce a modification of Binary Concrete, whose sampling procedure is as follows: [EQUATION] where [MATH] is an interval, with [MATH] and [MATH].', '1810.04247-3-82-1': 'This induces a new distribution, whose support is [MATH] instead of [MATH].', '1810.04247-3-82-2': 'With [MATH], the probability density concentrates its mass near the end points, since values larger than [MATH] are rounded to one, whereas values smaller than [MATH] are rounded to zero.', '1810.04247-3-83-0': 'The CDF of [MATH] is [EQUATION] and so the CDF of [MATH] is [EQUATION] where [MATH]', '1810.04247-3-84-0': 'Now, the probability of being the gate [MATH] being active is [MATH] and can be written as [EQUATION]', '1810.04247-3-85-0': '# Additional Experimental Details', '1810.04247-3-86-0': 'Here we provide a full description of the procedures we have performed in the experimental parts of the paper.', '1810.04247-3-87-0': 'For synthetic datasets are first split into train, validation and test.', '1810.04247-3-87-1': 'Validation is always [MATH] of the train, while the exact ratios between train and test is detailed for each experiment separately.', '1810.04247-3-87-2': 'All the neural network weights are initialized by drawing from [MATH] and bias terms are set to 0.', '1810.04247-3-87-3': 'All the batch sizes are equal to the number of training samples.', '1810.04247-3-87-4': 'In table [REF], we detail the the search range of hyperparamters as well as the exact values used in our experiments.', '1810.04247-3-87-5': 'We set n-trials = 1000 for Optuna.', '1810.04247-3-87-6': 'We use SGD for all the experiments, except for the Cox model where we use Adam.', '1810.04247-3-87-7': 'All the experiments are conducted using Intel(R) Xeon(R) CPU E5-2620 v3 @2.4Ghz x2 (12 cores total).', '1810.04247-3-88-0': 'For the Phase Transition experiment, we use 0.1 as a learning rate.', '1810.04247-3-88-1': 'For all the experiments when we use Tree and RF, we use the default value for max-depth, so that nodes are expanded based on the purity.', '1810.04247-3-88-2': 'For the XOR problem, the exact architectures used for the NN based methods are: (STG): [MATH] with Tanh, (DFS): [MATH] with Tanh, (SG-L1-NN): [MATH] with Tanh.', '1810.04247-3-88-3': 'For the two moons we use (STG): [MATH] with Tanh, (DFS): [MATH] with Tanh, (SG-L1-NN): [MATH] with Tanh.', '1810.04247-3-88-4': 'For the XOR problem, we attempted to use Optuna to optimize parameters of DFS and SG-L1-NN, but we ended up using the architecture suggested by the authors [CITATION] as they outperform the values suggested by Optuna.', '1810.04247-3-88-5': 'The number of epochs used for the XOR problem is [MATH] for STG, DFS and SG-L1-NN respectively.', '1810.04247-3-88-6': 'Regularization parameters are [MATH] and [MATH] respectively.', '1810.04247-3-88-7': 'The number of epochs used for the two-moons problem is [MATH] for STG, DFS and SG-L1-NN respectively.', '1810.04247-3-88-8': 'Regularization parameters are [MATH] and [MATH] respectively.', '1810.04247-3-88-9': 'We note that the regularization parameters and learning procedure is different in nature, as we use an [MATH] type penalty.', '1810.04247-3-88-10': 'For the PBMC experiment, the architecture was hand-tuned using the validation set and set as [MATH] with Tanh.', '1810.04247-3-88-11': 'Learning rate was [MATH] and the number of epochs [MATH].', '1810.04247-3-88-12': 'The hyperparameter [MATH] varies in the range [MATH] to achieve different levels of sparsity.', '1810.04247-3-88-13': 'For MADELON, we use the architecture optimized for the binary XOR classification.', '1810.04247-3-88-14': 'The number of epochs used is [MATH], the learning rate is [MATH] and [MATH] varies in the range [MATH].', '1810.04247-3-88-15': 'For regression, we manually select the architecture and set the number of epochs using the validation set.', '1810.04247-3-88-16': 'The learning rate and the [MATH] are optimized via Optuna using the search range [MATH] and [MATH] based on validation.', '1810.04247-3-88-17': 'The parameters used are the following: (SE1) architecture [600, 200, 100, 50, 1] with ReLu activations, num epochs: 5, [MATH], [MATH] (SE2) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs=2000, [MATH], [MATH] (SE3) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs : 1000, [MATH], [MATH].', '1810.04247-3-88-18': '(RCP) architecture [1000, 300, 150, 60, 20] with ReLu activation, num-epochs: 2000, [MATH], [MATH].', '1810.04247-3-88-19': 'The ratio of train/test/valid split is 1:1:1 for synthetic data.', '1810.04247-3-88-20': 'For the real data (RCP), the train size is 6000, the test and valid size is 1000 samples.', '1810.04247-3-89-0': 'In order to define the IFWR, for STG, the [MATH] feature weight is set to [MATH].', '1810.04247-3-89-1': 'For other neural net based methods, it is given by [MATH], where [MATH] is the weight matrix of the first layer.', '1810.04247-3-89-2': 'For other methods we just used the feature relevance returned by the trained model.', '1810.04247-3-89-3': "Finally, the LASSO's IFWR in the XOR experiment was omitted from the manuscript as it suffered from high variance.", '1810.04247-3-90-0': 'Regarding the comparison performed in the two-moons and XOR problem, we believe that adding IFWR along with classification accuracy versus number of feature selected provides a complementary perspective in demonstrating the efficacy of feature selection techniques.', '1810.04247-3-90-1': 'We emphasize that our goal is not to just rank features but select features by assigning the weight of [MATH] to irrelevant features while simultaneously obtaining good predictive accuracy.'}

{'1810.04247-4-0-0': 'Feature selection problems have been extensively studied for linear estimation, for instance, Lasso, but less emphasis has been placed on feature selection for non-linear functions.', '1810.04247-4-0-1': 'In this study, we propose a method for feature selection in high-dimensional non-linear function estimation problems.', '1810.04247-4-0-2': 'The new procedure is based on minimizing the [MATH] norm of the vector of indicator variables that represent if a feature is selected or not.', '1810.04247-4-0-3': 'Our approach relies on the continuous relaxation of Bernoulli distributions, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent.', '1810.04247-4-0-4': 'This general framework simultaneously minimizes a loss function while selecting relevant features.', '1810.04247-4-0-5': 'Furthermore, we provide an information-theoretic justification of incorporating Bernoulli distribution into our approach and demonstrate the potential of the approach on synthetic and real-life applications.', '1810.04247-4-1-0': '# Introduction', '1810.04247-4-2-0': 'Technological advances have led to the generation of large complex data sets both in sample size and dimensionality.', '1810.04247-4-2-1': 'The collected data sets encapsulate both opportunities and challenges.', '1810.04247-4-2-2': 'For instance, in biology, we have access to tremendous amounts of biological markers and wish to model their interactions for prediction purposes.', '1810.04247-4-2-3': 'Unfortunately, in many cases, the number of features exceeds the number of samples.', '1810.04247-4-2-4': 'A method to mitigate this challenge is to identify the key set of features that influence prediction.', '1810.04247-4-2-5': 'Finding a subset of meaningful features not only helps the analytic task but also provides new scientific findings and improves the interpretability of models [CITATION].', '1810.04247-4-2-6': 'Furthermore, reducing the number of features has computational advantages and has been shown to improve model generalization on unseen data [CITATION].', '1810.04247-4-2-7': 'In high-dimensional settings, there has been numerous works studying the theoretical and empirical properties of linear [CITATION] and non-linear feature selection methods [CITATION].', '1810.04247-4-3-0': 'Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods.', '1810.04247-4-3-1': 'Filter methods attempt to remove irrelevant features prior to learning a model.', '1810.04247-4-3-2': 'These methods filter features based on a per-feature relevance score that is created based on some statistical measure [CITATION].', '1810.04247-4-3-3': 'Wrapper methods, such as [CITATION], use the outcome of a classifier to determine the relevance of each feature, which requires recomputing the classifier for each subset of features.', '1810.04247-4-3-4': 'This becomes computationally expensive for neural network based wrapper methods [CITATION].', '1810.04247-4-3-5': 'Embedded methods aim to remove this burden by learning the model while simultaneously selecting the subset of relevant features.', '1810.04247-4-3-6': 'The Least Absolute Shrinkage and Selection Operator (LASSO) [CITATION] is a well-known embedded method, whose objective is to minimize the loss while enforcing an [MATH] constraint on the weights of the features.', '1810.04247-4-3-7': 'Although LASSO is scalable and widely used [CITATION], it is restricted to the domain of linear functions.', '1810.04247-4-3-8': 'Therefore, it is appealing to consider the nonlinear extension of the LASSO formulation.', '1810.04247-4-4-0': 'We develop a fully embedded feature selection method based directly on approximating the [MATH] penalty.', '1810.04247-4-4-1': 'This method is applicable to learning non-linear functions modeled by neural networks.', '1810.04247-4-4-2': 'Thus, it can also be used as an alternative to LASSO for linear and non linear problems.', '1810.04247-4-4-3': 'We demonstrate using numerous examples (see Section [REF] and supplementary) the applicability if our method.', '1810.04247-4-4-4': 'Our contributions are as follows:', '1810.04247-4-5-0': 'By utilizing a recent development of continuous and differentiable approximation to discrete distributions ([CITATION], [CITATION], [CITATION]), we introduce a solution to the long standing problem of feature selection with an [MATH] regularization.', '1810.04247-4-5-1': 'We present a novel simple relaxation of Bernoulli distribution to sparsify the input layer (the feature space) which we call stochastic gate (STG) and show its advantage over the distribution presented by [CITATION] both in performance and convergence time.', '1810.04247-4-6-0': 'By applying these two relaxations to an input layer of a neural network, we perform embedded feature selection in classification, regression or survival analysis tasks and demonstrate its capabilities on artificial and real data sets.', '1810.04247-4-7-0': 'We justify our probabilistic approach by analyzing the constrained Mutual Information maximization objective of feature selection.', '1810.04247-4-8-0': 'Notation: We refer to vectors as bold lowercase [MATH]x[MATH] and random vectors as bold uppercase letters [MATH]X[MATH].', '1810.04247-4-8-1': 'Scalars are non-bold case [MATH], while random variables are capital case [MATH].', '1810.04247-4-8-2': 'A set is represented by script fonts [MATH], [MATH], [MATH].', '1810.04247-4-8-3': 'For example the [MATH] vector-valued observation is denoted as [MATH]x[MATH] whereas [MATH] represents the [MATH] feature of the vector-valued random variable [MATH]X[MATH].', '1810.04247-4-8-4': 'Let [MATH].', '1810.04247-4-8-5': 'For a set [MATH] let the vector [MATH] be the characteristic function for the set.', '1810.04247-4-8-6': 'That is [MATH] if [MATH] and [MATH] otherwise.', '1810.04247-4-8-7': 'For two vectors [MATH]x[MATH] and [MATH]z[MATH] we denote [MATH] to be the element-wise product between [MATH] and [MATH].', '1810.04247-4-8-8': 'Thus, if we let [MATH] be the characteristic vector of [MATH], then we may define [MATH].', '1810.04247-4-8-9': 'The [MATH] norm of a vector is denoted as [MATH].', '1810.04247-4-8-10': 'Finally, the [MATH] norm of a vector is denoted as [MATH] and counts the total number of non-zero entries in the vector [MATH].', '1810.04247-4-9-0': '# Problem Setup and Background', '1810.04247-4-10-0': 'Let [MATH] be the input domain with corresponding response domain [MATH].', '1810.04247-4-10-1': 'Given realizations from some unknown data distribution [MATH] the goal of embedded feature selection methods is to simultaneously find a subset of indices [MATH] and construct a model that predicts [MATH] based on the selected features [MATH]X[MATH].', '1810.04247-4-11-0': '## Risk minimization objective', '1810.04247-4-12-0': 'We assume that we are given a family of functions [MATH] such that any function [MATH] is indexed by a set of parameters [MATH].', '1810.04247-4-12-1': 'Given some loss [MATH], a selection of features [MATH], and a choice of parameters [MATH], we denote the risk of our model as [EQUATION] where we recall that [MATH]s[MATH] is a vector of indicator variables for the set [MATH] and [MATH] denotes the point-wise product.', '1810.04247-4-12-2': 'Thus, the goal of the feature selection problem is to find the parameters [MATH] and [MATH] that minimize [MATH] such that [MATH] is small compared to [MATH].', '1810.04247-4-13-0': '## Feature Selection for Linear Models', '1810.04247-4-14-0': 'Before proceeding with our proposed method, we review the feature selection problem in the linear regression setting with the least squares loss.', '1810.04247-4-14-1': 'Thus, we restrict [MATH] to be the space of linear functions and the loss function to be the quadratic loss.', '1810.04247-4-15-0': 'Given observations [MATH]x[MATH] we may consider the constrained empirical risk minimization problem [EQUATION]', '1810.04247-4-15-1': 'Since the above problem is intractable, a number of authors replace the [MATH] constraint with a surrogate function [MATH][MATH] designed to penalize the number of selected features in [MATH].', '1810.04247-4-15-2': 'A natural choice for [MATH] is the [MATH] norm, which yields a convex problem and more precisely the LASSO optimization problem [CITATION].', '1810.04247-4-15-3': 'Subsequently, a number of authors have developed computationally efficient algorithms for solving the problem [CITATION].', '1810.04247-4-15-4': 'While the original Lasso problem focuses on the constrained optimization problem, the regularized least squares problem, which is often used in practice, yields the following minimization objective: [EQUATION]', '1810.04247-4-15-5': 'The hyperparameter [MATH] trades off the amount of regularization versus the fit of the objective.', '1810.04247-4-15-6': 'The [MATH] regularized method is very effective for feature selection and prediction; however, it achieves this through shrinkage of the coefficients.', '1810.04247-4-15-7': 'As a result, authors have considered non-convex choices for [MATH] as well [CITATION] that perform well both theoretically and empirically for prediction and feature selection.', '1810.04247-4-16-0': 'Our goal is to apply such regularization techniques to perform feature selection while learning a non-linear function.', '1810.04247-4-16-1': 'Kernel methods have been considered [CITATION], but scale quadratically in the number of observations.', '1810.04247-4-16-2': 'An alternative approach is to model [MATH] using a neural network with [MATH] regularization on the input weights [CITATION].', '1810.04247-4-16-3': 'However, in practice, introducing an [MATH] penalty into gradient descent does not provide sufficient specification.', '1810.04247-4-16-4': 'Below, we discuss our method that works to directly use an [MATH] penalty.', '1810.04247-4-17-0': '# Proposed Method', '1810.04247-4-18-0': 'We take a probabilistic approach to approximate the [MATH] norm, which can extend to non-linear models while remaining computationally efficient.', '1810.04247-4-18-1': 'To motivate our approach, we provide theoretical support (see Section [REF]) based on a Mutual Information perspective for the feature selection problem.', '1810.04247-4-19-0': 'To view the [MATH] regularized version of the risk (Eq. [REF]) from a probabilistic perspective, one can introduce a Bernoulli random vector [MATH]S[MATH] whose entries are independent and the [MATH] entry satisfies [MATH] for [MATH].', '1810.04247-4-19-1': 'If we denote the empirical expectation over our observations as [MATH], then, the empirical regularized risk (Eq. [REF]) becomes [EQUATION] where we have [MATH]S[MATH] and we constrain [MATH].', '1810.04247-4-19-2': 'Clearly, this formulation is equivalent to equation [REF], with a regularized penalty on cardinality rather than an explicit constraint.', '1810.04247-4-19-3': 'We may then relax the discrete constraint on [MATH] to be [MATH].', '1810.04247-4-20-0': 'Now, our goal is to find the model parameters [MATH][MATH] and Bernoulli parameters [MATH][MATH] that minimize the empirical risk [MATH][MATH][MATH] via gradient descent.', '1810.04247-4-20-1': 'However, an optimization of a loss function which includes discrete random variables suffers from high variance (See supplementary for more details).', '1810.04247-4-20-2': 'Therefore, inspired by a recently developed continuous approximation for discrete random variables, suggested by [CITATION], we develop and use a novel and simple continuous distribution that is fully differentiable and suited to the task of feature selection.', '1810.04247-4-21-0': '## Continuous Relaxation', '1810.04247-4-22-0': 'Our continuous relaxation for the Bernoulli variables [MATH] for [MATH] relies on the reparametrization trick, which is widely used for reducing the variance of gradient estimators [CITATION].', '1810.04247-4-22-1': 'To construct a continuous approximation to Bernoulli random variable via the reparametrization trick, we define [MATH] where [MATH] is drawn from a Gaussian distribution [MATH], where [MATH] is fixed throughout training.', '1810.04247-4-22-2': 'This approximation can be viewed as a clipped, mean-shifted, Gaussian random vector.', '1810.04247-4-22-3': 'Furthermore, the gradient of the objective with respect to [MATH] can be computed via the chain rule.', '1810.04247-4-22-4': 'We can now rewrite the objective in Eq. [REF] as [EQUATION] where [MATH]Z[MATH] is a random vector with [MATH] independent variables [MATH] for [MATH].', '1810.04247-4-22-5': 'To optimize the empirical surrogate of the objective [REF], we first differentiate it with respect to [MATH][MATH].', '1810.04247-4-22-6': 'Then, Monte Carlo sampling leads us to the following gradient estimator [EQUATION] where [MATH] is the number of Monte Carlo samples.', '1810.04247-4-22-7': 'Thus, we can update the parameters [MATH] for [MATH] via gradient descent.', '1810.04247-4-22-8': 'We note that if we replace [MATH] with 1, the above gradient estimator for [MATH] is reduced to the Straight-Through estimator [CITATION].', '1810.04247-4-23-0': 'Under the continuous relaxation, the expected regularization term in the objective [MATH][MATH][MATH] is simply the sum of the probability that the gates [MATH] are active, which is equal to [MATH], where [MATH] is the standard Gaussian CDF.', '1810.04247-4-23-1': "To conclude, we can now optimize the objective [REF] using gradient descent over the model parameters [MATH][MATH] and the parameters [MATH][MATH] representing the Gaussian's mean (instead of the Bernoulli parameters [MATH][MATH]).", '1810.04247-4-24-0': 'After training, to remove the stochasticity from the learned gates, we set [MATH], which informs what features are selected.', '1810.04247-4-24-1': 'Note that when [MATH] is less than [MATH], [MATH] returns the value between [MATH].', '1810.04247-4-24-2': 'In such a case, we can treat the value of [MATH] as feature importance or employ an additional thresholding (i.e. 1 if [MATH] and [MATH] otherwise) depending on application-specific needs.', '1810.04247-4-24-3': 'In the supplementary material, we provide the pseudo-code of our algorithm as well as the discussion of the choice of [MATH].', '1810.04247-4-25-0': '# Connection to Mutual Information', '1810.04247-4-26-0': 'In this section we show an equivalence between the Bernoulli formulation of the feature selection problem and the [MATH] regularized approach.', '1810.04247-4-27-0': '## Mutual Information based objective', '1810.04247-4-28-0': 'From an information theoretic perspective, the goal of feature selection is to find a subset of features [MATH] that has the highest Mutual Information (MI) with the target variable [MATH].', '1810.04247-4-28-1': 'Recall that the MI between two random variables can be defined as [MATH]X[MATH]X[MATH] where [MATH]X[MATH] are the entropy of [MATH] and the conditional entropy of [MATH]X[MATH], respectively [CITATION].', '1810.04247-4-28-2': 'Then we can formulate the task as selecting [MATH] such that the mutual information between [MATH]X[MATH] and [MATH] are maximized: [EQUATION] where [MATH] is the hypothesised number of relevant features.', '1810.04247-4-29-0': '## Introducing randomness', '1810.04247-4-30-0': 'We first demonstrate that under mild assumptions we can replace the deterministic search over the set [MATH] (or corresponding indicator vector [MATH]s[MATH]), by a search over the parameters of the distributions that model [MATH]s[MATH].', '1810.04247-4-30-1': 'Our proposition is based on the following two assumptions: Assumption 1: There exists a subset of indices [MATH] with cardinality equal to [MATH] such that for any [MATH] we have [MATH]X[MATH].', '1810.04247-4-30-2': 'Assumption 2: [MATH]X[MATH]X[MATH].', '1810.04247-4-30-3': 'Discussion of assumptions: Assumption 1 that including an element from [MATH] improves prediction accuracy.', '1810.04247-4-30-4': 'This assumption is equivalent to stating that feature [MATH] is strongly relevant [CITATION].', '1810.04247-4-30-5': 'Assumption 2 simply states that [MATH] is a superset of the Markov Blanket of the variable [MATH] [CITATION].', '1810.04247-4-30-6': 'The assumptions are quite benign.', '1810.04247-4-30-7': 'For instance they are satisfied if [MATH]X[MATH] is drawn from a Gaussian with a non-degenerate covariance matrix and [MATH]X[MATH], where [MATH] is noise independent of [MATH]X[MATH] and [MATH] is not degenerate.', '1810.04247-4-30-8': 'With these assumptions in hand, we may present our result.', '1810.04247-4-31-0': 'Due to length constraints, we leave the proof of this proposition and how it bridges the MI maximization [REF] and risk minimization [REF] in the supplementary material.', '1810.04247-4-32-0': '# Related Work', '1810.04247-4-33-0': 'The two most related works to this study are [CITATION] and [CITATION].', '1810.04247-4-33-1': 'In [CITATION], they introduce the Hard-Concrete distribution as a continuous surrogate for Bernoulli distributions in the context of model compression.', '1810.04247-4-33-2': 'The authors demonstrate how their method leads to fast convergence and improved generalization in deep neural networks due to the sparsification effect.', '1810.04247-4-33-3': 'In this study, we present a sparsification method that is aimed specificaly at feature selection that can then be extended to general non-linear function estimation.', '1810.04247-4-33-4': 'We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and works better than the Hard-Concrete distribution for feature selection tasks (see the supplementary material).', '1810.04247-4-33-5': 'Unlike the full sparsification framework, our method enables us to increase the feature size to a number of thousands of features (as shown in the Section [REF]); this high dimensional regime is common in a field such as bioinformatics.', '1810.04247-4-33-6': 'In [CITATION], the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained models.', '1810.04247-4-33-7': 'Their method is focused on finding a subset of features given a particular instance, and therefore is not appropriate for general feature selection.', '1810.04247-4-34-0': 'Some authors tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models.', '1810.04247-4-34-1': 'Other authors [CITATION] and [CITATION] have a similar goal as ours in performing feature selection, but instead rely on the [MATH] relaxation to the [MATH].', '1810.04247-4-34-2': 'Our approach, which utilizes stochastic gates coupled with the [MATH] norm achieves better empirical performance when compared against other baselines including the [MATH] relaxation.', '1810.04247-4-34-3': 'This point is explored in the next section.', '1810.04247-4-35-0': '# Experiments', '1810.04247-4-36-0': 'Here we provide empirical evaluation of our method in a wide range of settings.', '1810.04247-4-36-1': 'We start from the simple setting of linear regression, and move to nonlinear function estimation.', '1810.04247-4-36-2': 'The hyperparamters of all the methods are optimized using validation sets via Optuna [CITATION].', '1810.04247-4-36-3': 'In the supplemental material, we provide all details of the parameter tuning procedure as well as additional experiments, including an evaluation of our regularization parameter [MATH] and comparison to the Hard Concrete [CITATION].', '1810.04247-4-37-0': '## Phase Transition in the Linear Setting', '1810.04247-4-38-0': 'The problem of signal support recovery is tied to the problem of feature selection.', '1810.04247-4-38-1': 'The ability to recover the support of a vector based on its noisy linear observation is useful for various tasks, among which are compressed sensing, image denoising, and channel estimation.', '1810.04247-4-38-2': '[CITATION] provide an analysis of the probability of successful support recovery of LASSO in the following experimental setup.', '1810.04247-4-38-3': 'Let [MATH][MATH] be a fixed sparse vector, such that [MATH] (with equal probability) if [MATH], and [MATH] otherwise.', '1810.04247-4-38-4': 'Suppose the cardinality of the support [MATH] is known.', '1810.04247-4-38-5': 'Given a matrix of measurements [MATH]X[MATH] with values drawn independently from [MATH], the response [MATH]y[MATH] is defined as [EQUATION] where the values of the noise [MATH]w[MATH] are drawn independently from [MATH].', '1810.04247-4-39-0': 'Here, we reproduce this setting to evaluate the probability of perfect support recovery of [MATH][MATH], based on our method.', '1810.04247-4-39-1': 'As suggested by [CITATION], we use a sparsity that scales with [MATH] such that [MATH].', '1810.04247-4-39-2': 'For each number of samples [MATH] in the range [MATH], we run [MATH] simulations and count the portion of correctly recovered supports.', '1810.04247-4-39-3': 'We repeat this process for [MATH] different values of [MATH] and compare our performance to LASSO.', '1810.04247-4-39-4': 'For LASSO, the regularization parameter was set to its optimal value [MATH] ([CITATION]).', '1810.04247-4-39-5': 'For STG and HC we set [MATH], such that [MATH] is a constant, which is selected using a grid search in the range [0.1,10].', '1810.04247-4-39-6': 'As evident from Fig. [REF], even when restricting to linear functions our method has a clear advantage over LASSO.', '1810.04247-4-39-7': 'This implies that the [MATH] based penalty, even though not convex in nature, allows us to recover the support of [MATH] using less samples.', '1810.04247-4-39-8': 'Furthermore, the proposed STG requires less samples than the HC distribution for perfect support recovery.', '1810.04247-4-40-0': '## Noisy Binary XOR Classification', '1810.04247-4-41-0': 'To evaluate the capability of our method to capture nonlinear relationships between [MATH]x[MATH] and [MATH], we consider the problem of learning a binary XOR function for classification task.', '1810.04247-4-41-1': 'The first two coordinates [MATH] are drawn from a binary "fair" Bernoulli distribution.', '1810.04247-4-41-2': 'The response variable is set as an XOR of the first coordinates, such that [MATH].', '1810.04247-4-41-3': 'The coordinates [MATH] are nuisance features, also drawn from a binary "fair" Bernoulli distribution.', '1810.04247-4-41-4': 'The number of points we generate is [MATH], of which [MATH] are reserved for test and [MATH] of the remaining training set was reserved for validation.', '1810.04247-4-41-5': 'We compare the proposed method to four embedded feature selection methods (LASSO [CITATION], C-support vectors (SVC) [CITATION], deep feature selection (DFS) [CITATION], sparse group regularized NN (SG-L1-NN) [CITATION]).', '1810.04247-4-41-6': 'To provide more benchmarks, we also compare our embedded method against three wrapper methods (Extremely Randomized Trees (Tree) [CITATION], Random Forests (RF) [CITATION]) and Extreme Gradient Boosting (XGBOOST) [CITATION].', '1810.04247-4-42-0': 'To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio (IFWR).', '1810.04247-4-42-1': 'IFWR is defined as the sum of weights [MATH] over the informative features divided by the sum over all weights.', '1810.04247-4-42-2': 'In the case of binary weights the IFWR is in fact a recall measure for the relevant features (See the supplementary material for more details.)', '1810.04247-4-43-0': 'The experiment is repeated [MATH] times for different values of [MATH], and the average test classification accuracy and standard deviation are presented in Fig. [REF], followed by the IFWR in Fig. [REF].', '1810.04247-4-43-1': 'The number of selected features affects the accuracy.', '1810.04247-4-43-2': 'Therefore, to treat all the methods in a fair manner, we tune the hyperparameter that controls the sparsity level using Optuna ([CITATION]) based on a cross validation procedure to optimize the overall accuracy across different [MATH]s. For instance, the wrapper methods (Tree, RF and XGBOOST) has a threshold value to retain features.', '1810.04247-4-43-3': 'We retrain them using only such features whose weight is higher than the threshold, which we can optimize to control the sparsity.', '1810.04247-4-44-0': 'In terms of feature ranking, only the tree based methods and the proposed STG provide the optimal median rank (which is [MATH]) for the two relevant features.', '1810.04247-4-45-0': '## MADELON dataset', '1810.04247-4-46-0': 'The MADELON dataset, first suggested for the NIPS 2003 feature selection problem, is a multivariate highly nonlinear binary classification problem.', '1810.04247-4-46-1': 'The MADELON dataset is generated using 32 groups of data points placed on a [MATH] dimensional hyper-cube and randomly labeling them by one of the two class labels.', '1810.04247-4-46-2': 'The first [MATH] informative features are then used to create [MATH] additional coordinates that are formed based on a random linear transformation of the first [MATH].', '1810.04247-4-46-3': 'A Gaussian noise [MATH] is added to each feature.', '1810.04247-4-46-4': 'Next, additional [MATH] nuisance coordinates are added in the same manner.', '1810.04247-4-46-5': 'These features have no effect on the class label.', '1810.04247-4-46-6': 'Finally, [MATH] of the labels are flipped.', '1810.04247-4-46-7': 'We use [MATH] points from this dataset, and evaluate our proposed method in terms of its predictive power and ability to detect the informative features.', '1810.04247-4-46-8': 'We vary the regularizaton parameter [MATH] in the range [MATH] and evaluate the classification accuracy using [MATH] folds cross validation.', '1810.04247-4-46-9': 'In this example, we restrict our comparison to Random Forest and LASSO.', '1810.04247-4-46-10': 'We focus on Random Forest as it was the strongest competitor to our method in all of our experiments, while LASSO is evaluated because it is a widely used embedded feature selection method.', '1810.04247-4-47-0': 'As evident from Fig. [REF], our method achieves the highest accuracy while using less features.', '1810.04247-4-47-1': 'Moreover, as depicted from this figure, peak performence occurs when selecting [MATH] features, thus, our method provides a clear indication to the true number of informative features.', '1810.04247-4-47-2': 'Both LASSO and RF on the other hand, do not provide a clear indication of the true number of relevant features.', '1810.04247-4-47-3': 'In Fig. [REF], we evaluate the effect of [MATH] on the number and quality of selected features.', '1810.04247-4-47-4': "As shown in this plot, there is a wide range of [MATH]'s such that our method only selects relevant features.", '1810.04247-4-47-5': "Finally, as evident from the plato on the right hand side of the red plot, there is a range of [MATH]'s such that exactly [MATH] features are selected.", '1810.04247-4-48-0': '## Two Moons classification with nuisance features', '1810.04247-4-49-0': 'In this experiment, we construct a dataset based on "two moons" shape classes, concatenated with noisy features.', '1810.04247-4-49-1': 'The first two coordinates [MATH] are generated by adding a Gaussian noise with zero mean and the variance of [MATH] onto two nested half circles, as presented in Fig. [REF].', '1810.04247-4-49-2': 'Nuisance features [MATH], are drawn from a Gaussian distribution with zero mean and variance of [MATH].', '1810.04247-4-49-3': 'We reserve the [MATH] as a test set, and use [MATH] of the remaining training set as a validation set.', '1810.04247-4-49-4': 'We follow the same hyperparameter tuning procedure as in the XOR experiment.', '1810.04247-4-49-5': 'The classification accuracy is in Fig. [REF].', '1810.04247-4-49-6': 'Based on the classification accuracies, it is evident that for a small number of nuisance dimensions all methods correctly identify the most relevant features.', '1810.04247-4-49-7': 'The proposed method (STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy for a wide range of nuisance dimensions.', '1810.04247-4-49-8': 'The other NN based method (DFS) seem to converge to sub-optimal solutions.', '1810.04247-4-49-9': 'We note that the median rank for all the methods is 1.5.', '1810.04247-4-50-0': '## Comparison to Hard-Concrete distribution', '1810.04247-4-51-0': 'To evaluate the strength of the proposed continuous relaxation of the Bernoulli distribution, described in Subsection 3.1, we compare it with the Hard-Concrete distribution [CITATION], another continuous surrogate for Bernoulli distributions, which was originally developed for neural network model compression.', '1810.04247-4-51-1': 'The details of the Hard-Concrete distribution is described in the Appendix.', '1810.04247-4-52-0': 'The main difference between our proposed distribution (STG) and the Hard-Concrete [CITATION] distribution is that the latter is based on the logistic distribution, which has a heavier tail than the Gaussian distribution we have employed.', '1810.04247-4-52-1': 'As shown in Fig [REF], the heavy-tailness results in instability during training.', '1810.04247-4-52-2': 'Furthermore, our method converges much faster and more reliably than the feature selection method using the Hard-Concrete distribution on a the two-moons, XOR and MADELON datasets (see Subsection [REF] and [REF]).', '1810.04247-4-52-3': 'A similar phenomenon is also demonstrated in the MNIST experiment (see Subsection [REF]).', '1810.04247-4-53-0': '## Sparse Handwritten digits classification', '1810.04247-4-54-0': "In the following toy example, we attempt to distinguish between images of handwritten digits of 3's and 8's using samples from MNIST [CITATION].", '1810.04247-4-54-1': "The orientation and location of the digits is more or less the same throughout this dataset, therefore for these two classes (3's and 8's), we expect that some of the left side features (pixels) would be sufficient for the separation.", '1810.04247-4-54-2': 'The experiment is performed as followed.', '1810.04247-4-54-3': 'We reserve [MATH] of the data as the test set, and train on the remaining [MATH].', '1810.04247-4-54-4': 'We then apply STG and evaluate the classification accuracy and the number of selected features.', '1810.04247-4-54-5': 'We use the architecture [200, 50, 10] with tanh activations.', '1810.04247-4-54-6': 'The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials.', '1810.04247-4-54-7': 'We noticed a relatively small number of selected features, which are positioned southwest and close to the center of the images, achieve very high classification accuracy.', '1810.04247-4-54-8': 'An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. [REF].', '1810.04247-4-54-9': 'Furthermore, we also evaluate the effect of [MATH] on the the number of selected features and accuracy of the method.', '1810.04247-4-54-10': 'We apply our method (STG) and its variant using the Hard-Concrete distribution to a randomly sampled training set of size [MATH] and vary [MATH] in the range of [MATH].', '1810.04247-4-54-11': 'In Fig. [REF] we present the accuracy and sparsity level vs. the [MATH] parameter.', '1810.04247-4-54-12': 'This experiment demonstrates the improved performance of the proposed distribution compared to the Hard-Concrete (HC [CITATION]), which was designed for neural net model compression.', '1810.04247-4-54-13': 'Not only that the overall accuracy is superior, but also it seems that the transition as a function of [MATH] is smoother, which suggests that the method is less sensitive to the choice of [MATH].', '1810.04247-4-55-0': '## Regression using synthetic and real datasets', '1810.04247-4-56-0': 'In this section, we evaluate our method for regression tasks against two other embedded feature selection methods: LASSO and Sparse Random Fourier Features [CITATION].', '1810.04247-4-56-1': 'Following the same format as [CITATION], the following functions are used to generate synthetic data: (SE1: 100/5) [MATH].', '1810.04247-4-56-2': '(SE2: 18/5) [MATH].', '1810.04247-4-56-3': '(SE3: 1000/10) [MATH], where each [MATH] is drawn from the standard Gaussian [MATH].', '1810.04247-4-56-4': 'For (SE3), the five consecutive coordinates are generated by [MATH], where [MATH] for [MATH] and [MATH].', '1810.04247-4-56-5': 'The numbers next to the experiment code indicate total dimensions/relevant dimensions in the feature space.', '1810.04247-4-56-6': 'We also evaluate our method using three real datasets: (RCP: 21/-), of computer systems activity, (REL 17/-) of F16 elevators and (RAI 39/-) of F16 ailernos all taken from the LIACC repository .', '1810.04247-4-56-7': 'For each dataset, we generate 30 different replications and randomly split the data into train, validation, and test set (see supplementary for more details).', '1810.04247-4-56-8': 'For data preparation, we use the code publicly made available by [CITATION].', '1810.04247-4-56-9': 'The root mean squared error on the test set averaged over 30 random replicated datasets are reported in Table 1.', '1810.04247-4-56-10': 'Our method outperforms the other two methods for most cases.', '1810.04247-4-56-11': 'We note that (SE1) is generated using the sine function, which is in favor of the random Fourier feature based method (SRFF).', '1810.04247-4-57-0': '## Purified populations of peripheral blood monocytes (PBMCs)', '1810.04247-4-58-0': 'Single-cell RNA sequencing (scRNA-seq) is a novel technology that measures gene expression levels of hundreds of thousands of individual cells.', '1810.04247-4-58-1': 'This new tool is revolutionizing our understanding of cellular biology as it enables, among other things, the discovery of new cell types as well as the detection of subtle differences between similar but distinct cells.', '1810.04247-4-58-2': '[CITATION], have subjected more than [MATH] purified populations of peripheral blood monocytes (PBMCs) to scRNA-seq analysis.', '1810.04247-4-58-3': 'Here we focus on classifying two subpopulations of T-cells, namely the Naive and regulatory T-cells.', '1810.04247-4-58-4': 'The Naive CD4+ T-cells are responsible for activating the immune system against antigens, and the regulatory T-cells prevent activation against self antigens (autoimmune diseases).', '1810.04247-4-58-5': 'We use the proposed method to select a subset of genes for which the network discriminates between Naive and regulatory T-cells.', '1810.04247-4-58-6': 'We first filter out the genes that are lowly expressed in the cells, which leaves us with [MATH] genes (features).', '1810.04247-4-58-7': 'The total number of cells in these two classes is [MATH], of which we only use [MATH] of the data for training.', '1810.04247-4-58-8': 'We apply the proposed method for different values of [MATH] and report the number of selected features and classification accuracy on the test set.', '1810.04247-4-58-9': 'Here we compare our performance (STG and HC) to RF and LASSO.', '1810.04247-4-58-10': 'A least squares polynomial fit plot of the accuracy vs. number of selected features is presented in Fig. [REF].', '1810.04247-4-58-11': 'We have also evaluated the performance of the Hard-Concrete applied to all layers (HC-Full), following the procedure in [CITATION].', '1810.04247-4-58-12': 'Empirically we observed that using this type of regularization across all layers provides inferior capabilities in terms of feature selection.', '1810.04247-4-58-13': 'Moreover, when the method converges to a larger subset of features [MATH], it does not generalize at all and the test accuracy is around [MATH].', '1810.04247-4-59-0': '## Cox Proportional Hazard Models for Survival Analysis', '1810.04247-4-60-0': 'A standard model for survival analysis is Cox Proportional Hazard Model.', '1810.04247-4-60-1': '[CITATION] proposed DeepSurv that extends the COX model to neural networks .', '1810.04247-4-60-2': 'We incorporate our method into DeepSurv to see how our procedure improves survival analysis based on gene expression profiles from the breast cancer dataset called METABRIC ([CITATION]) along with additional commonly used clinical variables.', '1810.04247-4-60-3': 'See Supplimentary for more details about the dataset and experimental setup.', '1810.04247-4-61-0': 'We compared our method (Cox-STG and COX-HC) against three other methods: Cox model with [MATH] regularization (Cox-LASSO), Random Survival Forest (RSF) ([CITATION]) and the original DeepSurv.', '1810.04247-4-61-1': 'We evaluate the predictive ability of the learned models based on the concordance index (CI), a standard performance metric for model assessment in survival analysis, which measures the agreement between the rankings of the predicted and observed survival times.', '1810.04247-4-61-2': 'The performance in terms of the CI is reported in Table [REF].', '1810.04247-4-61-3': 'We see that Cox-HC and Cox-STG outperform the other methods, indicating that our method shrinks the feature size while maintaining high performance.', '1810.04247-4-62-0': '# Conclusion', '1810.04247-4-63-0': 'In this paper, we propose a novel embedded feature selection method based on stochastic gates.', '1810.04247-4-63-1': 'It has an advantage over previous [MATH] regularization based methods in terms of achieving a high level of sparsity in non-linear models such as neural networks, without hurting the performance.', '1810.04247-4-63-2': 'We justify our probabilistic feature selection framework from the information theoretic perspective.', '1810.04247-4-63-3': 'In experiments, we demonstrate that our method consistently outperforms existing embedded feature selection methods in both synthetic datasets and real datasets.', '1810.04247-4-64-0': '# Algorithm', '1810.04247-4-65-0': 'Here we present the pseudo code of our method, presented in Algorithm [REF].', '1810.04247-4-65-1': 'The loss [MATH] is typically negative-log likelihood for classification and squared loss for regression.', '1810.04247-4-65-2': '[MATH] is the sample size, [MATH] is the number of features, [MATH] is the number of Monte Carlo samples.', '1810.04247-4-66-0': '[!', '1810.04247-4-66-1': 'htb] STG: Feature selection using stochastic gates Input: [MATH]X[MATH], target variables [MATH]y[MATH], regularization parameter [MATH], number of epochs [MATH], learning rate [MATH].', '1810.04247-4-67-0': 'Output: Trained model [MATH][MATH] and parameter [MATH][MATH].', '1810.04247-4-68-0': '[1] Initialize the model parameter [MATH][MATH].', '1810.04247-4-68-1': 'Set [MATH][MATH] Sample [MATH] Compute the gate [MATH] Set [MATH]', '1810.04247-4-69-0': 'Set [MATH]z[MATH] Compute the loss [MATH]x[MATH]z[MATH] Compute the regularization term [MATH] Update [MATH][MATH][MATH][MATH] and [MATH][MATH][MATH][MATH]', '1810.04247-4-70-0': 'Note that we empirically observe that setting the number of Monte Carlo samples [MATH] and the standard deviation of the Gaussian distribution [MATH] for [MATH] suffices for feature selection in our experiments.', '1810.04247-4-70-1': 'See Section 4 in the supplementary material for more details about [MATH].', '1810.04247-4-70-2': 'After training, the set of indices for selected features is: [MATH].', '1810.04247-4-71-0': '# Proof of Proposition [REF]', '1810.04247-4-72-0': 'We now give the proof of the proposition showing the equivalence between the stochastic optimization [REF] and the deterministic one [REF].', '1810.04247-4-72-1': 'Let [MATH] be a subset such that [MATH].', '1810.04247-4-72-2': 'That is there exists some element in [MATH] that is not in [MATH].', '1810.04247-4-72-3': 'For any such set [MATH] we have that [MATH]X[MATH]X[MATH].', '1810.04247-4-72-4': 'Indeed, if we let [MATH] then we have [EQUATION] where the final inequality follows by Assumption 1.', '1810.04247-4-72-5': 'Assumption 2 also yields that for any set [MATH] such that [MATH], we have [MATH]X[MATH]X[MATH].', '1810.04247-4-72-6': 'Now, when we consider the Bernoulli optimization problem we have [EQUATION]', '1810.04247-4-72-7': 'The mutual information can be expanded as [EQUATION] where we have used the fact that [MATH]S[MATH] is independent of everything else.', '1810.04247-4-72-8': 'Recall that in optimization [REF] the coordinates of [MATH]S[MATH] are sampled at random.', '1810.04247-4-72-9': 'Therefore, the distribution that is being optimized over [MATH][MATH] is a product distribution.', '1810.04247-4-72-10': 'Our goal is to understand the form of this distribution.', '1810.04247-4-72-11': 'To that end, we will consider a problem dropping the independence constraint.', '1810.04247-4-72-12': 'If we can show that the distribution found by solving this new optimization problem with less constraints is still a product distribution, then we obtain a solution to the original optimization [REF].', '1810.04247-4-73-0': 'Now, from above we know that the optimal value of the optimization is [MATH]X[MATH]S[MATH] for any set [MATH].', '1810.04247-4-73-1': 'Hence, any unconstrained distribution should place all of its mass on such subsets in order to maximize the mutual information.', '1810.04247-4-73-2': 'As a result [MATH]S[MATH].', '1810.04247-4-73-3': 'However, there is an optimization constraint that [MATH]S[MATH].', '1810.04247-4-73-4': 'Therefore, [MATH] for any [MATH].', '1810.04247-4-73-5': 'Hence, the optimal solution is to select the distribution so that all of the mass is placed on the subset [MATH] and no mass elsewhere.', '1810.04247-4-73-6': 'As this is also a product distribution, this complete the proof of the claim.', '1810.04247-4-74-0': '# Bridging the Two Perspectives', '1810.04247-4-75-0': 'To motivate the introduction of randomness into the risk, we have looked at the feature selection problem from a MI perspective.', '1810.04247-4-75-1': 'Based on the MI objective, we have observed that introducing randomness into the constrained maximization procedure, does not change the objective (Proposition 1).', '1810.04247-4-75-2': 'Here we provide a relation between the MI objective [REF] to the empirical risk [REF], which supports our proposed procedure.', '1810.04247-4-76-0': 'We first note that the MI maximization over the set [MATH] can be reformulated as the minimization of the conditional entropy [MATH]X[MATH] since [MATH] does not depend on [MATH]: [EQUATION]', '1810.04247-4-76-1': 'Recall that [MATH]X[MATH]X[MATH]S[MATH].', '1810.04247-4-76-2': 'By Proposition 1, we rewrite the the deterministic search over the set [MATH] by a search over the Bernoulli parameters [MATH][MATH]: [EQUATION] where the expectation is over [MATH]X[MATH][MATH] which is the true data distribution, and [MATH][MATH].', '1810.04247-4-76-3': 'Put our model distribution as [MATH].', '1810.04247-4-76-4': 'Then we rewrite the right hand side as: [EQUATION]', '1810.04247-4-76-5': 'Since [MATH]X[MATH]S[MATH]X[MATH]S[MATH] is non-negative, [MATH]X[MATH]S[MATH]X[MATH]S[MATH] is also non-negative because it is a weighted sum of non-negative terms.', '1810.04247-4-76-6': 'Noting that [EQUATION] we can conclude that [EQUATION]', '1810.04247-4-76-7': 'If we consider the negative log likelihood of the target given the observations (i.e. [MATH]X[MATH]S[MATH]) as a loss function [MATH] (which encodes the classification or regression function [MATH][MATH]), then we see that the minimizing the risk approximately maximizes the MI objective in [REF].', '1810.04247-4-77-0': 'We now turn our focus on relating the MI objective (Eq. [REF]) to the empirical risk [REF].', '1810.04247-4-77-1': 'To do this, we find a lower bound for the MI [REF] that enables a tractable maximization search for the subset [MATH].', '1810.04247-4-77-2': 'This setup allows us to ignore the constant entropy of [MATH], [MATH] during the opimization.', '1810.04247-4-77-3': 'Next, by introducing a variational distribution [MATH] and by noting that KL divergence is non-negative, we can achieve the following lower bound (cite Barber, Agakov (2003)): [EQUATION] where an equality holds when [MATH] and [MATH] are equal.', '1810.04247-4-77-4': 'Using this lower bound, we can relax the maximization problem described in Eq. [REF] as follows: [EQUATION] where [MATH] is considered as a model distribution approximating the true conditional distribution [MATH].', '1810.04247-4-77-5': 'If we consider the negative log likelihood of the target given the observations (i.e. [MATH]) as a loss function [MATH] (which encodes the classification or regression function [MATH]), then the maximization in [REF] is equivalent to the following minimization: [EQUATION] where the [MATH] norm counts the number of non-zero components.', '1810.04247-4-77-6': 'However, the above optimization is intractable as it requires to enumerate all [MATH] possibilities to find the minimum of the objective.', '1810.04247-4-77-7': 'In the following section, we show an introduction of randomness into [MATH]S[MATH] allows us to develop a computationally tractable optimization procedure.', '1810.04247-4-78-0': '# Details of Regularization Term', '1810.04247-4-79-0': 'Here we provide the detail description of the regularization term.', '1810.04247-4-79-1': 'For the vector of stochastic gates [MATH]z[MATH], the regularization term is expressed as follows: [EQUATION]', '1810.04247-4-79-2': 'The derivative of the regularization term with respect to the distribution parameter [MATH] is simply the Gaussian PDF: [EQUATION]', '1810.04247-4-79-3': 'The effect of [MATH] can be understood by looking at the value of [MATH]Z[MATH].', '1810.04247-4-79-4': 'In the first iteration during the training, [MATH] is 0.', '1810.04247-4-79-5': 'Therefore, during the initial phase of training, it is close to [MATH].', '1810.04247-4-79-6': 'In order to remove irrelevant features, this term has to be greater than the derivative of the loss with respect to [MATH] because otherwise [MATH] is updated in the incorrect direction.', '1810.04247-4-79-7': 'To encourage such behavior, we set [MATH], which is around the maximum of the gradient during the initial phase as shown in Fig. [REF].', '1810.04247-4-79-8': 'Although the point that attains the maximum moves as [MATH] changes, we empirically observe that setting [MATH] performs well in our experiments when the regularization parameter [MATH] is appropriately set.', '1810.04247-4-80-0': '# Issues in Gradient Estimation of Discrete Random Variables', '1810.04247-4-81-0': 'In Section [REF], we have introduced Bernoulli random [MATH] variables with corresponding parameters [MATH] into the risk objective [REF].', '1810.04247-4-81-1': 'Taking the expectation over the [MATH] norm of [MATH]S[MATH] boils down to the sum of the Bernoulli parameters [MATH].', '1810.04247-4-81-2': 'However, the optimization of the resulting objective suffers from high variance due to the discrete nature of [MATH]S[MATH].', '1810.04247-4-81-3': 'Here, we attempt to convey this problem by analyzing the risk term in the objective [REF].', '1810.04247-4-81-4': 'Using the Bernoulli paramterization the empirical risk [MATH][MATH][MATH] is expressed as [EQUATION]', '1810.04247-4-81-5': 'In practice, as the outer sum involves enumerating [MATH] possibilities of the indicator variables, one can replace the outer sum with Monte Carlo samples from the product of Bernoulli distributions [MATH]z[MATH][MATH].', '1810.04247-4-81-6': 'However, a Monte Carlo estimate of [MATH][MATH][MATH] suffers from high variance.', '1810.04247-4-81-7': 'To see this, consider the following exact gradient of the empirical risk with respect to [MATH], which is [EQUATION] where [MATH], by absorbing the model [MATH] and the data into [MATH].', '1810.04247-4-81-8': 'Due to the discrete nature of [MATH]z[MATH], we see that even the sign of the gradient estimate becomes inaccurate if we can only access a small number of Monte Carlo samples.', '1810.04247-4-81-9': 'While a score-function estimator such as REINFORCE [CITATION] can be used, it is known that the reparametrization trick reduces the variance more in practice.', '1810.04247-4-82-0': '# Hard-Concrete distribution', '1810.04247-4-83-0': 'The authors in [CITATION] introduce a modification of Binary Concrete, whose sampling procedure is as follows: [EQUATION] where [MATH] is an interval, with [MATH] and [MATH].', '1810.04247-4-83-1': 'This induces a new distribution, whose support is [MATH] instead of [MATH].', '1810.04247-4-83-2': 'With [MATH], the probability density concentrates its mass near the end points, since values larger than [MATH] are rounded to one, whereas values smaller than [MATH] are rounded to zero.', '1810.04247-4-84-0': 'The CDF of [MATH] is [EQUATION] and so the CDF of [MATH] is [EQUATION] where [MATH]', '1810.04247-4-85-0': 'Now, the probability of being the gate [MATH] being active is [MATH] and can be written as [EQUATION]', '1810.04247-4-86-0': '# Additional Experimental Details', '1810.04247-4-87-0': 'Here we provide a full description of the procedures we have performed in the experimental parts of the paper.', '1810.04247-4-88-0': 'For synthetic datasets are first split into train, validation and test.', '1810.04247-4-88-1': 'Validation is always [MATH] of the train, while the exact ratios between train and test is detailed for each experiment separately.', '1810.04247-4-88-2': 'All the neural network weights are initialized by drawing from [MATH] and bias terms are set to 0.', '1810.04247-4-88-3': 'All the batch sizes are equal to the number of training samples.', '1810.04247-4-88-4': 'In table [REF], we detail the the search range of hyperparamters as well as the exact values used in our experiments.', '1810.04247-4-88-5': 'We set n-trials = 1000 for Optuna.', '1810.04247-4-88-6': 'We use SGD for all the experiments, except for the Cox model where we use Adam.', '1810.04247-4-88-7': 'All the experiments are conducted using Intel(R) Xeon(R) CPU E5-2620 v3 @2.4Ghz x2 (12 cores total).', '1810.04247-4-89-0': 'For the Phase Transition experiment, we use 0.1 as a learning rate.', '1810.04247-4-89-1': 'For all the experiments when we use Tree and RF, we use the default value for max-depth, so that nodes are expanded based on the purity.', '1810.04247-4-89-2': 'For the XOR problem, the exact architectures used for the NN based methods are: (STG/HC): [MATH] with Tanh, (DFS): [MATH] with Tanh, (SG-L1-NN): [MATH] with Tanh.', '1810.04247-4-89-3': 'For the two moons we use (STG): [MATH] with Tanh, (DFS): [MATH] with Tanh, (SG-L1-NN): [MATH] with Tanh.', '1810.04247-4-89-4': 'For the XOR problem, we attempted to use Optuna to optimize parameters of DFS and SG-L1-NN, but we ended up using the architecture suggested by the authors [CITATION] as they outperform the values suggested by Optuna.', '1810.04247-4-89-5': 'The number of epochs used for the XOR problem is [MATH] for STG/HC, DFS and SG-L1-NN respectively.', '1810.04247-4-89-6': 'Regularization parameters are [MATH] and [MATH] respectively.', '1810.04247-4-89-7': 'The number of epochs used for the two-moons problem is [MATH] for STG/HC, DFS and SG-L1-NN respectively.', '1810.04247-4-89-8': 'Regularization parameters are [MATH] and [MATH] respectively.', '1810.04247-4-89-9': 'We note that the regularization parameters and learning procedure is different in nature, as we use an [MATH] type penalty.', '1810.04247-4-89-10': 'For the PBMC experiment, the architecture was hand-tuned using the validation set and set as [MATH] with Tanh.', '1810.04247-4-89-11': 'Learning rate was [MATH] and the number of epochs [MATH].', '1810.04247-4-89-12': 'The hyperparameter [MATH] varies in the range [MATH] to achieve different levels of sparsity.', '1810.04247-4-89-13': 'For MADELON, we use the architecture optimized for the binary XOR classification.', '1810.04247-4-89-14': 'The number of epochs used is [MATH], the learning rate is [MATH] and [MATH] varies in the range [MATH].', '1810.04247-4-89-15': 'For regression, the learning rate and the [MATH] are optimized via Optuna using the search range [MATH] and [MATH] based on validation.', '1810.04247-4-89-16': 'The parameters used are the following: (SE1) architecture [600, 200, 100, 50, 1] with ReLu activations, num epochs: 5, [MATH], [MATH] (SE2) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs=2000, [MATH], [MATH] (SE3) architecture [600, 300, 150, 60, 20] with ReLu activations, num epochs : 1000, [MATH], [MATH].', '1810.04247-4-89-17': '(RCP) architecture [1000, 300, 150, 60, 20] with ReLu activation, num-epochs: 2000, [MATH], [MATH].', '1810.04247-4-89-18': '(REL) architecture [26,91,63] with ReLu activation, num-epochs: 1600, [MATH], [MATH].', '1810.04247-4-89-19': '(RAI) architecture [10,177] with ReLu activation, num-epochs: 1800, [MATH], [MATH].', '1810.04247-4-89-20': 'Architectures for SE1-SE3 and RCP where tuned manually.', '1810.04247-4-90-0': 'The ratio of train/test/valid split is 1:1:1 for synthetic data.', '1810.04247-4-90-1': 'For the real data (RCP and REL), the train size is 6000, the test and valid size is 1000 samples.', '1810.04247-4-90-2': 'For RAI the train size is 5000, the test and valid size is 1000 samples.', '1810.04247-4-91-0': 'In order to define the IFWR, for STG, the [MATH] feature weight is set to [MATH].', '1810.04247-4-91-1': 'For other neural net based methods, it is given by [MATH], where [MATH] is the weight matrix of the first layer.', '1810.04247-4-91-2': 'For other methods we just used the feature relevance returned by the trained model.', '1810.04247-4-91-3': "Finally, the LASSO's IFWR in the XOR experiment was omitted from the manuscript as it suffered from high variance.", '1810.04247-4-92-0': 'Regarding the comparison performed in the two-moons and XOR problem, we believe that adding IFWR along with classification accuracy versus number of feature selected provides a complementary perspective in demonstrating the efficacy of feature selection techniques.', '1810.04247-4-92-1': 'We emphasize that our goal is not to just rank features but select features by assigning the weight of [MATH] to irrelevant features while simultaneously obtaining good predictive accuracy.'}

{'1810.04247-5-0-0': 'Feature selection problems have been extensively studied in the setting of linear estimation, for instance LASSO, but less emphasis has been placed on feature selection for non-linear functions.', '1810.04247-5-0-1': 'In this study, we propose a method for feature selection in non-linear function estimation problems.', '1810.04247-5-0-2': 'The new procedure is based on directly penalizing the [MATH] norm of features, or the count of the number of selected features.', '1810.04247-5-0-3': 'Our [MATH] based regularization relies on a continuous relaxation of the Bernoulli distribution, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent.', '1810.04247-5-0-4': 'The proposed framework simultaneously learns a non-linear regression or classification function while selecting a small subset of features.', '1810.04247-5-0-5': 'We provide an information-theoretic justification for incorporating Bernoulli distribution into our approach.', '1810.04247-5-0-6': 'Furthermore, we evaluate our method using synthetic and real-life data and demonstrate that our approach outperforms other embedded methods in terms of predictive performance and feature selection.', '1810.04247-5-1-0': '# Introduction', '1810.04247-5-2-0': 'Feature selection is a fundamental task in machine learning and statistics.', '1810.04247-5-2-1': 'Feature selection leads to a number of potential benefits: reducing experimental costs [CITATION], enhancing interpretability [CITATION], computational speed up and even improving model generalization on unseen data [CITATION].', '1810.04247-5-2-2': 'In biomedicine, scientists collect multitude datasets comprising of many biomarkers (e.g., genes or proteins) that require development of effective diagnostics or prognostics models.', '1810.04247-5-2-3': 'For instance, in Genome wide association studies (GWAS), feature selection can help identify such models and lead to improved risk assessment and reduced cost.', '1810.04247-5-3-0': 'Feature selection methods may be classified into three major categories: filter methods, wrapper methods, and embedded methods.', '1810.04247-5-3-1': 'Filter methods attempt to remove irrelevant features prior to learning a model.', '1810.04247-5-3-2': 'These methods filter features using a per-feature relevance score that is created based on some statistical measure [CITATION].', '1810.04247-5-3-3': 'Wrapper methods [CITATION] use the outcome of a classifier to determine the relevance of each feature, which requires recomputing the classifier for each subset of features.', '1810.04247-5-3-4': 'This becomes computationally expensive for neural network based wrapper methods [CITATION].', '1810.04247-5-4-0': 'Embedded methods aim to remove this burden by learning the model while simultaneously selecting the subset of relevant features.', '1810.04247-5-4-1': 'The Least Absolute Shrinkage and Selection Operator (LASSO) [CITATION] is a well-known embedded method, whose objective is to minimize the loss while enforcing an [MATH] constraint on the weights of the features.', '1810.04247-5-4-2': 'Although LASSO is scalable and widely used [CITATION], it is restricted to the domain of linear functions.', '1810.04247-5-4-3': 'To allow the model to capture nonlinear interaction, it is appealing to consider the non-convex extension of the LASSO formulation using neural networks.', '1810.04247-5-5-0': 'We develop a fully embedded feature selection method for nonlinear models.', '1810.04247-5-5-1': 'To the best of our knowledge it is the first [MATH] embedded feature selection method.', '1810.04247-5-5-2': 'Our method improves upon the LASSO formulation in two aspects: a) it captures nonlinear interactions between the features via neural network modeling and b) it employs an [MATH]-like regularization using gates whose weights are parametrized by a smooth variant of a Bernoulli distribution.', '1810.04247-5-5-3': 'Altogether these twofold improvements are formulated as a fully differentiable neural network.', '1810.04247-5-5-4': 'Specifically, our contributions are as follows:', '1810.04247-5-6-0': 'By utilizing a recent development of continuous and differentiable approximation to discrete distributions [CITATION], [CITATION], [CITATION], we introduce a solution to the long standing problem of feature selection with an [MATH] regularization.', '1810.04247-5-6-1': 'We present a novel simple relaxation of Bernoulli distribution to sparsify the input layer (the feature space) which we call stochastic gate (STG) and show its advantage over the distribution presented by [CITATION] both in performance and convergence time.', '1810.04247-5-7-0': 'By applying these two relaxations to an input layer of a neural network, we perform embedded feature selection in classification, regression or survival analysis tasks and demonstrate its capabilities on artificial and real data sets.', '1810.04247-5-8-0': 'We justify our probabilistic approach by analyzing the constrained Mutual Information maximization objective of feature selection.', '1810.04247-5-8-1': 'We demonstrate the applicability of our method using numerous examples (see Section [REF] and Appendix).', '1810.04247-5-9-0': 'Notation: We refer to vectors as bold lowercase [MATH]x[MATH] and random vectors as bold uppercase letters [MATH]X[MATH].', '1810.04247-5-9-1': 'Scalars are non-bold case [MATH], while random variables are capital case [MATH].', '1810.04247-5-9-2': 'A set is represented by script fonts [MATH], [MATH], [MATH].', '1810.04247-5-9-3': 'For example the [MATH] vector-valued observation is denoted as [MATH]x[MATH] whereas [MATH] represents the [MATH] feature of the vector-valued random variable [MATH]X[MATH].', '1810.04247-5-9-4': 'Let [MATH].', '1810.04247-5-9-5': 'For a set [MATH] let the vector [MATH] be the characteristic function for the set.', '1810.04247-5-9-6': 'That is [MATH] if [MATH] and [MATH] otherwise.', '1810.04247-5-9-7': 'For two vectors [MATH]x[MATH] and [MATH]z[MATH] we denote [MATH] to be the element-wise product between [MATH] and [MATH].', '1810.04247-5-9-8': 'Thus, if we let [MATH] be the characteristic vector of [MATH], then we may define [MATH].', '1810.04247-5-9-9': 'The [MATH] norm of a vector is denoted as [MATH].', '1810.04247-5-9-10': 'Finally, the [MATH] norm of a vector is denoted as [MATH] and counts the total number of non-zero entries in the vector [MATH].', '1810.04247-5-10-0': '# Problem Setup and Background', '1810.04247-5-11-0': 'Let [MATH] be the input domain with corresponding response domain [MATH].', '1810.04247-5-11-1': 'Given realizations from some unknown data distribution [MATH] the goal of embedded feature selection methods is to simultaneously find a subset of indices [MATH] and construct a model that predicts [MATH] based on the selected features [MATH]X[MATH].', '1810.04247-5-12-0': '## Risk minimization objective', '1810.04247-5-13-0': 'We assume that we are given a family of functions [MATH] such that any function [MATH] is indexed by a set of parameters [MATH].', '1810.04247-5-13-1': 'Given some loss [MATH], a selection of features [MATH], and a choice of parameters [MATH], we denote the risk of our model as [EQUATION] where we recall that [MATH]s[MATH] is a vector of indicator variables for the set [MATH] and [MATH] denotes the point-wise product.', '1810.04247-5-13-2': 'Thus, the goal of the feature selection problem is to find the parameters [MATH] and [MATH] that minimize [MATH] such that [MATH] is small compared to [MATH].', '1810.04247-5-14-0': '## Feature Selection for Linear Models', '1810.04247-5-15-0': 'Before proceeding with our proposed method, we review the feature selection problem in the linear regression setting with the least squares loss.', '1810.04247-5-15-1': 'Thus, we restrict [MATH] to be the space of linear functions and the loss function to be the quadratic loss.', '1810.04247-5-16-0': 'Given observations [MATH]x[MATH] we may consider the constrained empirical risk minimization problem [EQUATION]', '1810.04247-5-16-1': 'Since the above problem is intractable, a number of authors replace the [MATH] constraint with a surrogate function [MATH][MATH] designed to penalize the number of selected features in [MATH].', '1810.04247-5-16-2': 'A natural choice for [MATH] is the [MATH] norm, which yields a convex problem and more precisely the LASSO optimization problem [CITATION].', '1810.04247-5-16-3': 'Subsequently, a number of authors have developed computationally efficient algorithms for solving the problem [CITATION].', '1810.04247-5-16-4': 'While the original Lasso problem focuses on the constrained optimization problem, the regularized least squares problem, which is often used in practice, yields the following minimization objective: [EQUATION]', '1810.04247-5-16-5': 'The hyperparameter [MATH] trades off the amount of regularization versus the fit of the objective.', '1810.04247-5-16-6': 'The [MATH] regularized method is very effective for feature selection and prediction; however, it achieves this through shrinkage of the coefficients.', '1810.04247-5-16-7': 'As a result, authors have considered non-convex choices for [MATH] as well [CITATION] that perform well both theoretically and empirically for prediction and feature selection.', '1810.04247-5-17-0': 'Our goal is to apply such regularization techniques to perform feature selection while learning a non-linear function.', '1810.04247-5-17-1': 'Kernel methods have been considered [CITATION], but scale quadratically in the number of observations.', '1810.04247-5-17-2': 'An alternative approach is to model [MATH] using a neural network with [MATH] regularization on the input weights [CITATION].', '1810.04247-5-17-3': 'However, in practice, introducing an [MATH] penalty into gradient descent does not provide sufficient specification.', '1810.04247-5-17-4': 'Below, we discuss our method that works to directly use an [MATH] penalty.', '1810.04247-5-18-0': '# Proposed Method', '1810.04247-5-19-0': 'We take a probabilistic approach to approximate the [MATH] norm, which can extend to non-linear models while remaining computationally efficient.', '1810.04247-5-19-1': 'To motivate our approach, we provide theoretical support (see Section [REF]) based on a Mutual Information perspective for the feature selection problem.', '1810.04247-5-20-0': 'To view the [MATH] regularized version of the risk (Eq. [REF]) from a probabilistic perspective, one can introduce a Bernoulli random vector [MATH]S[MATH] whose entries are independent and the [MATH] entry satisfies [MATH] for [MATH].', '1810.04247-5-20-1': 'If we denote the empirical expectation over our observations as [MATH], then, the empirical regularized risk (Eq. [REF]) becomes [EQUATION] where we have [MATH]S[MATH] and we constrain [MATH].', '1810.04247-5-20-2': 'Clearly, this formulation is equivalent to equation [REF], with a regularized penalty on cardinality rather than an explicit constraint.', '1810.04247-5-20-3': 'We may then relax the discrete constraint on [MATH] to be [MATH].', '1810.04247-5-21-0': 'Now, our goal is to find the model parameters [MATH][MATH] and Bernoulli parameters [MATH][MATH] that minimize the empirical risk [MATH][MATH][MATH] via gradient descent.', '1810.04247-5-21-1': 'However, an optimization of a loss function which includes discrete random variables suffers from high variance (See supplementary for more details).', '1810.04247-5-21-2': 'Therefore, inspired by a recently developed continuous approximation for discrete random variables, suggested by [CITATION], we develop and use a novel and simple continuous distribution that is fully differentiable and suited to the task of feature selection.', '1810.04247-5-22-0': '## Continuous Relaxation', '1810.04247-5-23-0': 'Our continuous relaxation for the Bernoulli variables [MATH] for [MATH] is termed stochastic gate (STG).', '1810.04247-5-23-1': 'The STG relies on the reparametrization trick, which is widely used for reducing the variance of gradient estimators [CITATION].', '1810.04247-5-23-2': 'To construct a continuous approximation to Bernoulli random variable via the reparametrization trick, we define [MATH] where [MATH] is drawn from a Gaussian distribution [MATH], where [MATH] is fixed throughout training.', '1810.04247-5-23-3': 'This approximation can be viewed as a clipped, mean-shifted, Gaussian random vector.', '1810.04247-5-23-4': 'Furthermore, the gradient of the objective with respect to [MATH] can be computed via the chain rule.', '1810.04247-5-24-0': 'We can now rewrite the objective in Eq. [REF] as [EQUATION] where [MATH]Z[MATH] is a random vector with [MATH] independent variables [MATH] for [MATH].', '1810.04247-5-24-1': 'To optimize the empirical surrogate of the objective (Eq. [REF]), we first differentiate it with respect to [MATH][MATH].', '1810.04247-5-24-2': 'Then, Monte Carlo sampling leads us to the following gradient estimator [EQUATION] where [MATH] is the number of Monte Carlo samples.', '1810.04247-5-24-3': 'Thus, we can update the parameters [MATH] for [MATH] via gradient descent.', '1810.04247-5-24-4': 'We note that if we replace [MATH] with 1, the above gradient estimator for [MATH] is reduced to the Straight-Through estimator [CITATION].', '1810.04247-5-25-0': 'Under the continuous relaxation, the expected regularization term in the objective [MATH][MATH][MATH] is simply the sum of the probability that the gates [MATH] are active, which is equal to [MATH], where [MATH] is the standard Gaussian CDF.', '1810.04247-5-25-1': "To conclude, we can now optimize the objective in Eq. [REF] using gradient descent over the model parameters [MATH][MATH] and the parameters [MATH][MATH] representing the Gaussian's mean (instead of the Bernoulli parameters [MATH][MATH]).", '1810.04247-5-26-0': 'After training, to remove the stochasticity from the learned gates, we set [MATH], which informs what features are selected.', '1810.04247-5-26-1': 'Note that when [MATH] is less than [MATH], [MATH] returns the value between [MATH].', '1810.04247-5-26-2': 'In such a case, we can treat the value of [MATH] as feature importance or employ an additional thresholding (i.e. 1 if [MATH] and [MATH] otherwise) depending on application-specific needs.', '1810.04247-5-26-3': 'In the Appendix, we provide the pseudo-code of our algorithm as well as the discussion of the choice of [MATH].', '1810.04247-5-27-0': '# Connection to Mutual Information', '1810.04247-5-28-0': 'In this section we show an equivalence between the Bernoulli formulation of the feature selection problem and the [MATH] regularized approach.', '1810.04247-5-29-0': '## Mutual Information based objective', '1810.04247-5-30-0': 'From an information theoretic perspective, the goal of feature selection is to find a subset of features [MATH] that has the highest Mutual Information (MI) with the target variable [MATH].', '1810.04247-5-30-1': 'Recall that the MI between two random variables can be defined as [MATH]X[MATH]X[MATH] where [MATH]X[MATH] are the entropy of [MATH] and the conditional entropy of [MATH]X[MATH], respectively [CITATION].', '1810.04247-5-30-2': 'Then we can formulate the task as selecting [MATH] such that the mutual information between [MATH]X[MATH] and [MATH] are maximized: [EQUATION] where [MATH] is the hypothesised number of relevant features.', '1810.04247-5-31-0': '## Introducing randomness', '1810.04247-5-32-0': 'We first demonstrate that under mild assumptions we can replace the deterministic search over the set [MATH] (or corresponding indicator vector [MATH]s[MATH]), by a search over the parameters of the distributions that model [MATH]s[MATH].', '1810.04247-5-32-1': 'Our proposition is based on the following two assumptions: Assumption 1: There exists a subset of indices [MATH] with cardinality equal to [MATH] such that for any [MATH] we have [MATH]X[MATH].', '1810.04247-5-32-2': 'Assumption 2: [MATH]X[MATH]X[MATH].', '1810.04247-5-32-3': 'Discussion of assumptions: Assumption 1 that including an element from [MATH] improves prediction accuracy.', '1810.04247-5-32-4': 'This assumption is equivalent to stating that feature [MATH] is strongly relevant [CITATION].', '1810.04247-5-32-5': 'Assumption 2 simply states that [MATH] is a superset of the Markov Blanket of the variable [MATH] [CITATION].', '1810.04247-5-32-6': 'The assumptions are quite benign.', '1810.04247-5-32-7': 'For instance they are satisfied if [MATH]X[MATH] is drawn from a Gaussian with a non-degenerate covariance matrix and [MATH]X[MATH], where [MATH] is noise independent of [MATH]X[MATH] and [MATH] is not degenerate.', '1810.04247-5-32-8': 'With these assumptions in hand, we may present our result.', '1810.04247-5-33-0': 'Due to length constraints, we leave the proof of this proposition and how it bridges the MI maximization [REF] and risk minimization [REF] in the supplementary material.', '1810.04247-5-34-0': '# Related Work', '1810.04247-5-35-0': 'The two most related works to this study are [CITATION] and [CITATION].', '1810.04247-5-35-1': 'In [CITATION], they introduce the Hard-Concrete distribution as a continuous surrogate for Bernoulli distributions in the context of model compression.', '1810.04247-5-35-2': 'The authors demonstrate how their method leads to fast convergence and improved generalization in deep neural networks due to the sparsification effect.', '1810.04247-5-35-3': 'In this study, we present a sparsification method that is aimed specificaly at feature selection that can then be extended to general non-linear function estimation.', '1810.04247-5-35-4': 'We demonstrate that a simple relaxation of Bernoulli distributions is sufficient and works better than the Hard-Concrete distribution for feature selection tasks (see the supplementary material).', '1810.04247-5-35-5': 'Unlike the full sparsification framework, our method enables us to increase the feature size to a number of thousands of features (as shown in the Section [REF]); this high dimensional regime is common in a field such as bioinformatics.', '1810.04247-5-35-6': 'In [CITATION], the Gumbel-softmax trick is used to develop a framework for interpreting pre-trained models.', '1810.04247-5-35-7': 'Their method is focused on finding a subset of features given a particular instance, and therefore is not appropriate for general feature selection.', '1810.04247-5-36-0': 'Some authors tackle embedded feature selection problems by extending LASSO and group LASSO to neural network models.', '1810.04247-5-36-1': 'Other authors [CITATION] and [CITATION] have a similar goal as ours in performing feature selection, but instead rely on the [MATH] relaxation to the [MATH].', '1810.04247-5-36-2': 'Our approach, which utilizes stochastic gates coupled with the [MATH] norm achieves better empirical performance when compared against other baselines including the [MATH] relaxation.', '1810.04247-5-36-3': 'This point is explored in the next section.', '1810.04247-5-37-0': '# Experiments', '1810.04247-5-38-0': 'Here we provide empirical evaluation of our method in a wide range of settings.', '1810.04247-5-38-1': 'We start from the simple setting of linear regression, and move to nonlinear function estimation.', '1810.04247-5-38-2': 'The hyperparamters of all the methods are optimized using validation sets via Optuna [CITATION].', '1810.04247-5-38-3': 'In the supplemental material, we provide all details of the parameter tuning procedure as well as additional experiments, including an evaluation of our regularization parameter [MATH] and comparison to the Hard Concrete [CITATION].', '1810.04247-5-39-0': '## Phase Transition in the Linear Setting', '1810.04247-5-40-0': 'The problem of signal support recovery is tied to the problem of feature selection.', '1810.04247-5-40-1': 'The ability to recover the support of a vector based on its noisy linear observation is useful for various tasks, among which are compressed sensing, image denoising, and channel estimation.', '1810.04247-5-40-2': '[CITATION] provide an analysis of the probability of successful support recovery of LASSO in the following experimental setup.', '1810.04247-5-40-3': 'Let [MATH][MATH] be a fixed sparse vector, such that [MATH] (with equal probability) if [MATH], and [MATH] otherwise.', '1810.04247-5-40-4': 'Suppose the cardinality of the support [MATH] is known.', '1810.04247-5-40-5': 'Given a matrix of measurements [MATH]X[MATH] with values drawn independently from [MATH], the response [MATH]y[MATH] is defined as [EQUATION] where the values of the noise [MATH]w[MATH] are drawn independently from [MATH].', '1810.04247-5-41-0': 'Here, we reproduce this setting to evaluate the probability of perfect support recovery of [MATH][MATH], based on our method.', '1810.04247-5-41-1': 'As suggested by [CITATION], we use a sparsity that scales with [MATH] such that [MATH].', '1810.04247-5-41-2': 'For each number of samples [MATH] in the range [MATH], we run [MATH] simulations and count the portion of correctly recovered supports.', '1810.04247-5-41-3': 'We repeat this process for [MATH] different values of [MATH] and compare our performance to LASSO.', '1810.04247-5-41-4': 'For LASSO, the regularization parameter was set to its optimal value [MATH] ([CITATION]).', '1810.04247-5-41-5': 'For STG and HC we set [MATH], such that [MATH] is a constant, which is selected using a grid search in the range [0.1,10].', '1810.04247-5-41-6': 'As evident from Fig. [REF], even when restricting to linear functions our method has a clear advantage over LASSO.', '1810.04247-5-41-7': 'This implies that the [MATH] based penalty, even though not convex in nature, allows us to recover the support of [MATH] using less samples.', '1810.04247-5-41-8': 'Furthermore, the proposed STG requires less samples than the HC distribution for perfect support recovery.', '1810.04247-5-42-0': '## Noisy Binary XOR Classification', '1810.04247-5-43-0': 'In the following evaluation, we consider the problem of learning a binary XOR function for classification task.', '1810.04247-5-43-1': 'The first two coordinates [MATH] are drawn from a binary "fair" Bernoulli distribution.', '1810.04247-5-43-2': 'The response variable is set as an XOR of the first coordinates, such that [MATH].', '1810.04247-5-43-3': 'The coordinates [MATH] are nuisance features, also drawn from a binary "fair" Bernoulli distribution.', '1810.04247-5-43-4': 'The number of points we generate is [MATH], of which [MATH] are reserved for test and [MATH] of the remaining training set was reserved for validation.', '1810.04247-5-43-5': 'We compare the proposed method to four embedded feature selection methods (LASSO [CITATION], C-support vectors (SVC) [CITATION], deep feature selection (DFS) [CITATION], sparse group regularized NN (SG-L1-NN) [CITATION]).', '1810.04247-5-43-6': 'To provide more benchmarks, we also compare our embedded method against three wrapper methods (Extremely Randomized Trees (Tree) [CITATION], Random Forests (RF) [CITATION]) and Extreme Gradient Boosting (XGBOOST) [CITATION].', '1810.04247-5-44-0': 'To evaluate the feature selection performance, we calculate the Informative Features Weight Ratio (IFWR).', '1810.04247-5-44-1': 'IFWR is defined as the sum of weights [MATH] over the informative features divided by the sum over all weights.', '1810.04247-5-44-2': 'In the case of binary weights the IFWR is in fact a recall measure for the relevant features (See the Appendix for more details.)', '1810.04247-5-45-0': 'The experiment is repeated [MATH] times for different values of [MATH], and the average test classification accuracy and standard deviation are presented in Fig. [REF], followed by the IFWR in Fig. [REF].', '1810.04247-5-45-1': 'The number of selected features affects the accuracy.', '1810.04247-5-45-2': 'Therefore, to treat all the methods in a fair manner, we tune the hyperparameter that controls the sparsity level using Optuna [CITATION] which optimizes the overall accuracy across different [MATH]s. For instance, the wrapper methods (Tree, RF and XGBOOST) has a threshold value to retain features.', '1810.04247-5-45-3': 'We retrain them using only such features whose weight is higher than the threshold.', '1810.04247-5-45-4': 'In terms of feature ranking (see Fig. [REF], only the tree based methods and the proposed (STG and HC based) provide the optimal median rank (which is [MATH]) for the two relevant features.', '1810.04247-5-45-5': 'Nonetheless, the ranking provided by STG is the most stable comparing to all the alternative methods.', '1810.04247-5-46-0': '## MADELON dataset', '1810.04247-5-47-0': 'The MADELON dataset, first suggested for the NIPS 2003 feature selection problem, is a multivariate highly nonlinear binary classification problem.', '1810.04247-5-47-1': 'The MADELON dataset is generated using 32 groups of data points placed on a [MATH] dimensional hyper-cube and randomly labeling them by one of the two class labels.', '1810.04247-5-47-2': 'The first [MATH] informative features are then used to create [MATH] additional coordinates that are formed based on a random linear transformation of the first [MATH].', '1810.04247-5-47-3': 'A Gaussian noise [MATH] is added to each feature.', '1810.04247-5-47-4': 'Next, additional [MATH] nuisance coordinates are added in the same manner.', '1810.04247-5-47-5': 'These features have no effect on the class label.', '1810.04247-5-47-6': 'Finally, [MATH] of the labels are flipped.', '1810.04247-5-47-7': 'To be consistent with the XOR and two moons experiments, we use [MATH] points from this dataset, and evaluate our proposed method in terms of its predictive power and ability to detect the informative features.', '1810.04247-5-47-8': 'We vary the regularizaton parameter [MATH] in the range [MATH] and evaluate the classification accuracy using [MATH] folds cross validation.', '1810.04247-5-47-9': 'In this example, we restrict our comparison to Random Forest and LASSO.', '1810.04247-5-47-10': 'We focus on Random Forest as it was the strongest competitor to our method in all of our experiments, while LASSO is evaluated because it is a widely used embedded feature selection method.', '1810.04247-5-48-0': 'As evident from Fig. [REF], our method achieves the highest accuracy while using less features.', '1810.04247-5-48-1': 'Moreover, as depicted from this figure, peak performence occurs when selecting [MATH] features, thus, our method provides a clear indication to the true number of informative features.', '1810.04247-5-48-2': 'Both LASSO and RF on the other hand, do not provide a clear indication of the true number of relevant features.', '1810.04247-5-48-3': 'In Fig. [REF], we evaluate the effect of [MATH] on the number and quality of selected features.', '1810.04247-5-48-4': "As shown in this plot, there is a wide range of [MATH]'s such that our method only selects relevant features.", '1810.04247-5-48-5': "Finally, as evident from the plato on the right hand side of the red plot, there is a range of [MATH]'s such that exactly [MATH] features are selected.", '1810.04247-5-49-0': '## Two Moons classification with nuisance features', '1810.04247-5-50-0': 'In this experiment, we construct a dataset based on "two moons" shape classes, concatenated with noisy features.', '1810.04247-5-50-1': 'The first two coordinates [MATH] are generated by adding a Gaussian noise with zero mean and the variance of [MATH] onto two nested half circles, as presented in Fig. [REF].', '1810.04247-5-50-2': 'Nuisance features [MATH], are drawn from a Gaussian distribution with zero mean and variance of [MATH].', '1810.04247-5-50-3': 'We reserve the [MATH] as a test set, and use [MATH] of the remaining training set as a validation set.', '1810.04247-5-50-4': 'We follow the same hyperparameter tuning procedure as in the XOR experiment.', '1810.04247-5-50-5': 'The classification accuracy is in Fig. [REF].', '1810.04247-5-50-6': 'Based on the classification accuracies, it is evident that for a small number of nuisance dimensions all methods correctly identify the most relevant features.', '1810.04247-5-50-7': 'The proposed method (STG) and Random Forest (RF) are the only methods that achieve near perfect classification accuracy for a wide range of nuisance dimensions.', '1810.04247-5-50-8': 'The other NN based method (DFS) seem to converge to sub-optimal solutions.', '1810.04247-5-50-9': 'We note that the median rank for all the methods is 1.5.', '1810.04247-5-51-0': '## Comparison to Hard-Concrete distribution', '1810.04247-5-52-0': 'To evaluate the strength of the proposed continuous relaxation of the Bernoulli distribution, described in Subsection 3.1, we compare it with the Hard-Concrete distribution [CITATION], another continuous surrogate for Bernoulli distributions, which was originally developed for neural network model compression.', '1810.04247-5-52-1': 'The details of the Hard-Concrete distribution is described in the Appendix.', '1810.04247-5-53-0': 'The main difference between our proposed distribution (STG) and the Hard-Concrete [CITATION] distribution is that the latter is based on the logistic distribution, which has a heavier tail than the Gaussian distribution we have employed.', '1810.04247-5-53-1': 'As shown in Fig [REF], the heavy-tailness results in instability during training.', '1810.04247-5-53-2': 'Furthermore, our method converges much faster and more reliably than the feature selection method using the Hard-Concrete distribution on a the two-moons, XOR and MADELON datasets (see Subsection [REF] and [REF]).', '1810.04247-5-53-3': 'A similar phenomenon is also demonstrated in the MNIST experiment (see Subsection [REF]).', '1810.04247-5-54-0': '## Sparse Handwritten digits classification', '1810.04247-5-55-0': "In the following toy example, we attempt to distinguish between images of handwritten digits of 3's and 8's using samples from MNIST [CITATION].", '1810.04247-5-55-1': "The orientation and location of the digits is more or less the same throughout this dataset, therefore for these two classes (3's and 8's), we expect that some of the left side features (pixels) would be sufficient for the separation.", '1810.04247-5-55-2': 'The experiment is performed as followed.', '1810.04247-5-55-3': 'We reserve [MATH] of the data as the test set, and train on the remaining [MATH].', '1810.04247-5-55-4': 'We then apply STG and evaluate the classification accuracy and the number of selected features.', '1810.04247-5-55-5': 'We use the architecture [200, 50, 10] with tanh activations.', '1810.04247-5-55-6': 'The experiment was repeated 10 times, the extracted features and accuracies were consistent over 20 trials.', '1810.04247-5-55-7': 'We noticed a relatively small number of selected features, which are positioned southwest and close to the center of the images, achieve very high classification accuracy.', '1810.04247-5-55-8': 'An example of 9 randomly selected samples overlaid with the weights of the selected features is presented in Fig. [REF].', '1810.04247-5-55-9': 'Furthermore, we also evaluate the effect of [MATH] on the the number of selected features and accuracy of the method.', '1810.04247-5-55-10': 'We apply our method (STG) and its variant using the Hard-Concrete distribution to a randomly sampled training set of size [MATH] and vary [MATH] in the range of [MATH].', '1810.04247-5-55-11': 'In Fig. [REF] we present the accuracy and sparsity level vs. the [MATH] parameter.', '1810.04247-5-55-12': 'This experiment demonstrates the improved performance of the proposed distribution compared to the Hard-Concrete (HC [CITATION]), which was designed for neural net model compression.', '1810.04247-5-55-13': 'Not only that the overall accuracy is superior, but also it seems that the transition as a function of [MATH] is smoother, which suggests that the method is less sensitive to the choice of [MATH].', '1810.04247-5-56-0': '## Regression using synthetic and real datasets', '1810.04247-5-57-0': 'In this section, we evaluate our method for regression tasks against two other embedded feature selection methods: LASSO and Sparse Random Fourier Features [CITATION].', '1810.04247-5-57-1': 'Following the same format as [CITATION], the following functions are used to generate synthetic data: (SE1: 100/5) [MATH].', '1810.04247-5-57-2': '(SE2: 18/5) [MATH].', '1810.04247-5-57-3': '(SE3: 1000/10) [MATH], where each [MATH] is drawn from the standard Gaussian [MATH].', '1810.04247-5-57-4': 'For (SE3), the five consecutive coordinates are generated by [MATH], where [MATH] for [MATH] and [MATH].', '1810.04247-5-57-5': 'The numbers next to the experiment code indicate total dimensions/relevant dimensions in the feature space.', '1810.04247-5-57-6': 'We also evaluate our method using three real datasets: (RCP: 21/-), of computer systems activity, (REL 17/-) of F16 elevators and (RAI 39/-) of F16 ailernos all taken from the LIACC repository .', '1810.04247-5-57-7': 'For each dataset, we generate 30 different replications and randomly split the data into train, validation, and test set (see Appendix for more details).', '1810.04247-5-57-8': 'The root mean squared error on the test set averaged over 30 random replicated datasets are reported in Table 1.', '1810.04247-5-57-9': 'Our method outperforms all alternative methods for most cases.', '1810.04247-5-57-10': 'Note that (SE1) is generated using a sine function, which is in favor of the random Fourier feature based method (SRFF).', '1810.04247-5-58-0': '# Application', '1810.04247-5-59-0': '## Gisette dataset', '1810.04247-5-60-0': 'The Gisette is a handwritten dataset also appears in the NIPS 2003 feature selection challenge.', '1810.04247-5-60-1': "The data consists of [MATH] handwritten digits of '4' and '9'.", '1810.04247-5-60-2': 'The original digits have [MATH] pixels, which were modified for the feature selection challenge.', '1810.04247-5-60-3': 'Specifically, a random subset of the features are embedded in a [MATH] dimensional space and additional [MATH] irrelevant probes are added.', '1810.04247-5-60-4': 'The train, validation and test dimensions are [MATH] respectively.', '1810.04247-5-60-5': 'We apply the proposed STG based feature selection method and compare its performance to Random Forests (RF) and Extremely Randomized Trees (Tree).', '1810.04247-5-60-6': 'As evident from Fig. [REF], we obtain a high accuracy even for a dramatic reduction in the feature size.', '1810.04247-5-60-7': 'LASSO is not presented in this experiment as its performance is dramatically inferior to the alternatives (accuracy [MATH]).', '1810.04247-5-61-0': '## Reuters Corpus Volume I', '1810.04247-5-62-0': 'The Reuters Corpus Volume I (RCV1) consists of [MATH] newswire stories manually labeled by [MATH] categories.', '1810.04247-5-62-1': 'This is a multilable regime, i.e. each story is assigned to multiple categories.', '1810.04247-5-62-2': 'Here we focus on a binary subset of this corpus, with [MATH] stories.', '1810.04247-5-62-3': 'The total number of feature is [MATH] and the train, validation and test portions are [MATH] respectively.', '1810.04247-5-62-4': 'We evaluate the performance of our method using a [MATH]-fold cross validation.', '1810.04247-5-62-5': 'A comparison the LASSO and RF appears in Fig. [REF].', '1810.04247-5-62-6': 'This example demonstrates that our method is also effective in an extremely high dimensional regime of non linear function estimation.', '1810.04247-5-63-0': '## Purified populations of peripheral blood monocytes (PBMCs)', '1810.04247-5-64-0': 'Single-cell RNA sequencing (scRNA-seq) is a novel technology that measures gene expression levels of hundreds of thousands of individual cells.', '1810.04247-5-64-1': 'This new tool is revolutionizing our understanding of cellular biology as it enables, among other things, the discovery of new cell types as well as the detection of subtle differences between similar but distinct cells.', '1810.04247-5-64-2': '[CITATION], have subjected more than [MATH] purified populations of peripheral blood monocytes (PBMCs) to scRNA-seq analysis.', '1810.04247-5-64-3': 'Here we focus on classifying two subpopulations of T-cells, namely the Naive and regulatory T-cells.', '1810.04247-5-64-4': 'The Naive CD4+ T-cells are responsible for activating the immune system against antigens, and the regulatory T-cells prevent activation against self antigens (autoimmune diseases).', '1810.04247-5-64-5': 'We use the proposed method to select a subset of genes for which the network discriminates between Naive and regulatory T-cells.', '1810.04247-5-64-6': 'We first filter out the genes that are lowly expressed in the cells, which leaves us with [MATH] genes (features).', '1810.04247-5-64-7': 'The total number of cells in these two classes is [MATH], of which we only use [MATH] of the data for training.', '1810.04247-5-64-8': 'We apply the proposed method for different values of [MATH] and report the number of selected features and classification accuracy on the test set.', '1810.04247-5-64-9': 'Here we compare our performance (STG and HC) to RF and LASSO.', '1810.04247-5-64-10': 'A least squares polynomial fit plot of the accuracy vs. number of selected features is presented in Fig. [REF].', '1810.04247-5-64-11': 'We have also evaluated the performance of the Hard-Concrete applied to all layers (HC-Full), following the procedure in [CITATION].', '1810.04247-5-64-12': 'Empirically we observed that using this type of regularization across all layers provides inferior capabilities in terms of feature selection.', '1810.04247-5-64-13': 'Moreover, when the method converges to a larger subset of features [MATH], it does not generalize at all and the test accuracy is around [MATH].', '1810.04247-5-65-0': '## Cox Proportional Hazard Models for Survival Analysis', '1810.04247-5-66-0': 'A standard model for survival analysis is Cox Proportional Hazard Model.', '1810.04247-5-66-1': '[CITATION] proposed DeepSurv that extends the COX model to neural networks .', '1810.04247-5-66-2': 'We incorporate our method into DeepSurv to see how our procedure improves survival analysis based on gene expression profiles from the breast cancer dataset called METABRIC ([CITATION]) along with additional commonly used clinical variables.', '1810.04247-5-66-3': 'See Supplimentary for more details about the dataset and experimental setup.', '1810.04247-5-67-0': 'We compared our method (Cox-STG and COX-HC) against three other methods: Cox model with [MATH] regularization (Cox-LASSO), Random Survival Forest (RSF) ([CITATION]) and the original DeepSurv.', '1810.04247-5-67-1': 'We evaluate the predictive ability of the learned models based on the concordance index (CI), a standard performance metric for model assessment in survival analysis, which measures the agreement between the rankings of the predicted and observed survival times.', '1810.04247-5-67-2': 'The performance in terms of the CI is reported in Table [REF].', '1810.04247-5-67-3': 'We see that Cox-HC and Cox-STG outperform the other methods, indicating that our method shrinks the feature size while maintaining high performance.', '1810.04247-5-68-0': '# Conclusion', '1810.04247-5-69-0': 'In this paper, we propose a novel embedded feature selection method based on stochastic gates.', '1810.04247-5-69-1': 'It has an advantage over previous [MATH] regularization based methods in terms of achieving a high level of sparsity in non-linear models such as neural networks, without hurting the performance.', '1810.04247-5-69-2': 'We justify our probabilistic feature selection framework from the information theoretic perspective.', '1810.04247-5-69-3': 'In experiments, we demonstrate that our method consistently outperforms existing embedded feature selection methods in both synthetic datasets and real datasets.'}

1803.11326

{'1803.11326-1-0-0': 'Slot filling is a critical task in natural language understanding (NLU) for dialog systems.', '1803.11326-1-0-1': 'State-of-the-art solutions regard it as a sequence labeling task and adopt BiLSTM-CRF models.', '1803.11326-1-0-2': 'While BiLSTM-CRF models works relatively well on standard datasets it faces challenges in Chinese E-commerce slot filling due to more informative slot labels and richer expressions.', '1803.11326-1-0-3': 'In this paper, we propose a deep multi-task learning model with cascade and residual connections.', '1803.11326-1-0-4': 'Experimental results show that our framework not only achieves competitive performance with state-of-the-arts on a standard dataset, but also significantly outperforms strong baselines by a substantial gain of 14.6% on a Chinese E-commerce dataset.', '1803.11326-1-1-0': 'UTF8gbsn', '1803.11326-1-2-0': '# Introduction', '1803.11326-1-3-0': 'An intelligent E-commerce online shopping guide assistant is a comprehensive human-like system providing various services such as pre-sale and after-sale inquiries, product recommendations, and user complaints processing, all of which seek to give the customers better shopping experience.', '1803.11326-1-3-1': 'The core of such assistant is a dialog system which has the ability to understand natural language utterances from a user and then give natural language responses.', '1803.11326-1-3-2': 'The architecture of a task-oriented dialog system for online shopping guide assistant is illustrated in Figure [REF].', '1803.11326-1-3-3': 'Natural Language Understanding (NLU), which aims to interpret the semantic meanings conveyed by input utterances is a main component in task-oriented dialog systems.', '1803.11326-1-3-4': 'Slot filling is a subproblem in NLU, which identifies the properties and their values about the task to be performed in the dialog.', '1803.11326-1-4-0': 'Slot filling extracts semantic constituents by using the words of input text to fill in pre-defined slots in a semantic frame [CITATION].', '1803.11326-1-4-1': 'It can be regarded as sequence labeling task, which assigns an appropriate semantic label to each word in the given input utterance.', '1803.11326-1-4-2': 'In the case of E-commerce shopping, there are three named entity types including Category, Property Key and Property Value.', '1803.11326-1-4-3': 'We show a real example in Table [REF] with In/Out/Begin(IOB) scheme.', '1803.11326-1-4-4': 'In the named entity level, "连衣裙"(dress) is a Category (B-CG/I-CG), while "品牌"(brand) is labeled as Property Key (B-PK/I-PK), which is the name of one product property.', '1803.11326-1-4-5': '"耐克"(Nike) and "黑色"(black) are labeled as Property Value (B-PV/I-PV) since they are concrete property values.', '1803.11326-1-4-6': 'However, labeling as Property Value is not good enough for NLU.', '1803.11326-1-4-7': 'Thus, in Slot Filling level, we further label "耐克"(Nike) as Brand Property (B-Brand/I-Brand), and "黑色"(black) as Color Property (B-Color/I-Color).', '1803.11326-1-4-8': 'In the meantime, other words in the example utterance that carry no semantic meaning are assigned O label.', '1803.11326-1-5-0': 'State-of-the-art sequence labeling models are typically based on BiLSTM and CRF [CITATION] and evaluated on a commonly used standard dataset ATIS [CITATION] in slot filling area.', '1803.11326-1-5-1': 'This dataset is in the domain of airline travel in America and Table [REF] shows an example utterance.', '1803.11326-1-5-2': 'However, the vocabulary size of ATIS is too small (only 572) and slot labels are not diverse enough since airline travel is a relatively small and specific domain, such that recent deep learning models can achieved very high F1 scores (nearly 0.96).', '1803.11326-1-6-0': 'Compared to ATIS, our E-commerce shopping guide assistant dataset is more complex.', '1803.11326-1-6-1': 'This dataset comes from a real world application and the semantic slots are more diverse and informal than ATIS, which increases the difficulty for the task.', '1803.11326-1-6-2': 'For example, to describe different properties for a product for purpose of utterance understanding and query rewrite, we should define large amount of informative slot labels such as color, brand, style, season, gender and so on.', '1803.11326-1-6-3': 'While most semantic labels of ATIS are related to only time and location.', '1803.11326-1-6-4': 'On the other hand, the spoken E-commerce Chinese language is more complex and enriched expression makes it harder to understand.', '1803.11326-1-6-5': 'For example, "红色" and "红" both mean red, "品牌" and "牌子" both mean brand, "耐克" and "Nike" and "Niky" all mean Nike.', '1803.11326-1-6-6': 'While in ATIS, expression can be simpler, and most expressions are standard locations or time.', '1803.11326-1-7-0': 'Besides, Chinese language, like many other Asian languages, are not word segmented by nature, and word segmentation is a difficult first step in many NLP tasks.', '1803.11326-1-7-1': 'Without proper word segmentation, sequence labeling becomes very challenging as the errors from segmentation will propagate.', '1803.11326-1-7-2': 'On the other hand, more than 97% of the chunks in ATIS data have only one or two words, in which segment (or chunking) may not be a serious problem.', '1803.11326-1-8-0': 'In this paper, we are the first to employ multi-task sequence labeling model to tackle slot filling in a novel Chinese E-commerce dialog system.', '1803.11326-1-8-1': 'We divide the slot filling task into two lower-level tasks: named entity tagging and segment tagging.', '1803.11326-1-8-2': 'Example labels of these two tasks are shown in the bottom two rows of Table [REF].', '1803.11326-1-8-3': 'Segment tagging and named entity tagging can be regarded as syntactic labeling, while slot filling is more like semantic labeling.', '1803.11326-1-8-4': 'Once we know the syntactic structure of an input sentence, filling the semantic labels becomes easier.', '1803.11326-1-8-5': 'Compared to directly attacking slot filling, these two low-level tasks are much easier to solve due to fewer labels.', '1803.11326-1-8-6': 'To this end, we propose a Deep Cascade Multi-task Learning model, and co-train three tasks in the same framework with a goal of optimizing the target slot filling task.', '1803.11326-1-9-0': 'The contributions of this paper are summarized below:', '1803.11326-1-10-0': '# Problem', '1803.11326-1-11-0': 'Given an utterance containing a sequence of words [MATH], the goal of our problem is to find a sequence of slot labels [MATH], one for each word in the utterance, such that: [EQUATION]', '1803.11326-1-11-1': 'In this paper we only define a slot filling problem in Dress category domain for simplification.', '1803.11326-1-11-2': 'That means we know in advance the defined category classification (intent).', '1803.11326-1-11-3': 'There are thousands of category classifications in E-commerce domain and in each category, there can be dozens of properties that are totally different.', '1803.11326-1-11-4': 'Performing slot filling with tens of thousands slot labels with cross-domains is impractical at this point.', '1803.11326-1-11-5': 'A joint model for category classification and slot filling is our future research.', '1803.11326-1-12-0': 'Along with Property Key(PK), Category(CG) and O, there are altogether 29 (57 in the IOB scheme) slot labels in our problem.', '1803.11326-1-12-1': 'Examples are listed in Table [REF].', '1803.11326-1-12-2': 'Notice that terms such as "brand", "color" appearing in an utterance will be labeled as PK, while Color and Brand can be pre-defined slot labels and will be assigned to terms like "black" and "Nike".', '1803.11326-1-13-0': '# Approach', '1803.11326-1-14-0': 'In this section we describe our approach in detail.', '1803.11326-1-14-1': 'Figure [REF] gives an overview of the proposed architectures.', '1803.11326-1-14-2': 'First we introduce the most common and popular BiLSTM-CRF model (Figure [REF](a)) for sequence labeling tasks.', '1803.11326-1-14-3': 'Then we move on to multi-task learning perspective (Figure [REF](b) and (c)).', '1803.11326-1-14-4': 'Finally we propose our new method, which is called Deep Cascade Multi-task Learning in Figure [REF](d).', '1803.11326-1-15-0': '## RNN Sequence Labeling', '1803.11326-1-16-0': 'Figure [REF](a) shows the principle architecture of a BiLSTM-CRF model, which is the state-of-the-art model for various sequence labeling tasks [CITATION].', '1803.11326-1-16-1': 'BiLSTM-CRF model consists of a BiLSTM layer and a CRF layer.', '1803.11326-1-17-0': 'Bidirectional LSTMs enable the hidden states to capture both historical and future context information of the words.', '1803.11326-1-17-1': 'Mathematically, the input of this BiLSTM layer is a sequence of input vectors, denoted as [MATH].', '1803.11326-1-17-2': 'The output of BiLSTM layer is a sequence of the hidden states for each input word, denoted as [MATH].', '1803.11326-1-17-3': 'Each final hidden state is the concatenation of the forward [MATH] and backward [MATH] hidden states.', '1803.11326-1-17-4': 'We view BiLSTM as a function [MATH]: [EQUATION]', '1803.11326-1-17-5': 'Most of time we stack multiple BiLSTMs to make the model deeper, in which the output [MATH] of layer [MATH] becomes the input of layer [MATH], e.g. [MATH].', '1803.11326-1-18-0': 'It is always beneficial to consider the correlations between the current label and neighboring labels, since there are many syntactical constraints in natural language sentences.', '1803.11326-1-18-1': 'If we simply feed the above mentioned hidden states independently to a softmax layer to predict the labels [CITATION], such constraints are more likely to be violated.', '1803.11326-1-18-2': 'Linear-chain Conditional Random Field (CRF) [CITATION] is the most popular way to control the structure prediction and its basic idea is to use a series of potential functions to approximate the conditional probability of the output label sequence given the input word sequence.', '1803.11326-1-19-0': 'Formally, we take the above sequence of hidden states [MATH] as input to a CRF layer, and the output of the CRF is the final prediction label sequence [MATH], where [MATH] is in the set of pre-defined target labels.', '1803.11326-1-19-1': 'We denote [MATH] as the set of all possible label sequences.', '1803.11326-1-19-2': 'Then we derive the conditional probability of the output sequence, given the input hidden state sequence is: [EQUATION] where [MATH] are potential functions and [MATH] and [MATH] are weight vector and bias of label pair [MATH].', '1803.11326-1-19-3': 'To train the CRF layer, we use the classic maximum conditional likelihood estimate and gradient ascent.', '1803.11326-1-19-4': 'For a training dataset [MATH], the final log-likelihood is: [EQUATION].', '1803.11326-1-19-5': 'Finally, the Viterbi algorithm is adopted to decode the optimal output sequence [MATH]: [EQUATION]', '1803.11326-1-20-0': '## Multi-task Learning', '1803.11326-1-21-0': 'While directly attacking the slot filling task is hard, low-level tasks with fewer labels are much easier to solve.', '1803.11326-1-21-1': 'Once we know the syntactic structure of a sentence, filling in semantic labels will become easier accordingly.', '1803.11326-1-21-2': 'Thus, it is reasonable to solve the problem in a multi-task learning framework.', '1803.11326-1-21-3': 'In our problem, we can devise three individual tasks: slot filling, named entity tagging and segment tagging.', '1803.11326-1-21-4': 'Slot filling is our target task; named entity tagging is to classify which named entity type (PV/PK/CG) a word is; and segment tagging is to judge whether a word is begin (B), in (I) or out (O) of a trunking.', '1803.11326-1-22-0': 'In a multi-task learning (MTL) setting, we have several prediction tasks over the same input sequence, where each task has its own output vocabulary (a set of task specified labels).', '1803.11326-1-22-1': 'Intuitively, the three tasks do share a lot of information.', '1803.11326-1-22-2': 'Consider the example in Table [REF] again.', '1803.11326-1-22-3': 'Knowing the named entity type of "黑色"(black) being B-PV can definitely help determine its slot label, which is B-Color.', '1803.11326-1-22-4': 'Similarly, knowing its segment type (B) also helps with both named entity tagging and slot filling.', '1803.11326-1-22-5': 'Thus it is reasonable for these tasks to share parameters and learn in the same framework cooperatively.', '1803.11326-1-23-0': '### Vanilla Multi-task Learning', '1803.11326-1-24-0': 'The general idea of multi-task learning is to share parameters of encoding part of the network.', '1803.11326-1-24-1': 'As Figure [REF](b) shows, this is naturally achieved by sharing the [MATH]-layers BiLSTM part of the network across three tasks.', '1803.11326-1-24-2': 'Based on that, we use a separate CRF decoder for each task [MATH]: [MATH], where [MATH] and [MATH] are task-specific parameters.', '1803.11326-1-24-3': 'This encourages the deep BiLSTM network to learn a hidden representation [MATH] which benefits all three different tasks.', '1803.11326-1-25-0': '### Hierarchy Multi-task Learning', '1803.11326-1-26-0': 'Previous discussion indicates that there is a natural order among the different tasks: slot filling may benefit more from named entity tagging, than the other way around.', '1803.11326-1-26-1': 'This motivates us to employ low-level tasks at lower BiLSTM layers, while high level tasks are trained at higher layers.', '1803.11326-1-26-2': 'This idea was first proposed by Anders and Yoav [CITATION].', '1803.11326-1-26-3': 'As shown in Figure [REF](c), instead of decoding all tasks separately at the outermost BiLSTM layer, we associate each BiLSTM layer [MATH] with one task [MATH].', '1803.11326-1-26-4': 'Then the conditional probabilities of the output sequence for each task are: [EQUATION]', '1803.11326-1-26-5': 'Here [MATH], [MATH] and [MATH] represent the tasks of segment tagging, named entity tagging and slot filling, respectively.', '1803.11326-1-26-6': '[MATH] is the word embeddings of input sequence [MATH] and [MATH].', '1803.11326-1-26-7': 'We call this model hierarchy multi-task learning, since some layers are shared by all tasks while the others are only related to specific tasks.', '1803.11326-1-27-0': '## Deep Cascade Multi-task Learning', '1803.11326-1-28-0': 'Hierarchy multi-task learning share parameters among different tasks, and allow low-level tasks help adjust the result of high-level target task.', '1803.11326-1-28-1': 'It is effective for those tasks which are weakly correlated, such as POS tagging, syntactic chunking and CCG supertagging [CITATION].', '1803.11326-1-28-2': 'However, when it comes to problems where different tasks maintain a strict order, in another word, the performance of high-level task dramatically depends on low-level tasks, the hierarchy structure is not compact and effective enough.', '1803.11326-1-28-3': 'Therefore, we propose cascade and residual connections to enable high-level tasks taking the tagging results and hidden states of low-level tasks as additional input.', '1803.11326-1-28-4': 'These connections serves as "shortcuts" that create a more closely coupled and efficient model.', '1803.11326-1-28-5': 'We call it deep cascade multi-task learning, and the framework is shown in Figure [REF](d).', '1803.11326-1-29-0': '### Cascade Connection', '1803.11326-1-30-0': 'Here we feed the tagging output of the task at lower layer e.g. [MATH] or [MATH] to the upper BiLSTM layer as its additional input.', '1803.11326-1-30-1': 'Now the hidden states of each task layer become: [EQUATION] where [MATH] is the weight parameter for cascade connection.', '1803.11326-1-31-0': 'At training time, [MATH] and [MATH] can be the true tagging outputs.', '1803.11326-1-31-1': 'At inference time, we simply take the greedy path of our cascade model without doing search, where the model emits the best [MATH] and [MATH] by Viterbi inference algorithm.', '1803.11326-1-31-2': 'Alternatively, one can do beam search [CITATION] by maintaining a set of [MATH] best partial hypotheses at each cascade layer.', '1803.11326-1-31-3': 'However, unlike traditional seq2seq models e.g., in machine translation, where each inference step is just based on probability of a discrete variable (by softmax function), our inference for tagging output is a structured probability distribution defined by the CRF output.', '1803.11326-1-31-4': 'Efficient beam search method for this structured cascade model is left to our future work.', '1803.11326-1-32-0': '### Residual Connection', '1803.11326-1-33-0': 'To encourage the information sharing among different tasks, we also introduce the residual connection, where we add the input of a previous layer to the current input: [EQUATION]', '1803.11326-1-33-1': 'Deep residual learning [CITATION] is first introduced to ease the gradient vanish problem for training very deep neural networks.', '1803.11326-1-33-2': 'Here we propose that residual connection between different layers can help for multi-task learning.', '1803.11326-1-34-0': '### Training', '1803.11326-1-35-0': 'For our multi-task setting, we define three loss functions (refer to Section [REF]): [MATH], [MATH] and [MATH] for tasks of segment tagging, named entity tagging and slot filling respectively.', '1803.11326-1-35-1': 'We construct three training set, [MATH], [MATH] and [MATH], where each of them (called [MATH] generically) contains a set of input-output sequence pair [MATH].', '1803.11326-1-35-2': 'The input utterance [MATH] is shared across tasks, but the output [MATH] is task dependent.', '1803.11326-1-36-0': 'For vanilla multi-task learning, we define a loss function [MATH], where [MATH] and [MATH] are hyper-parameters.', '1803.11326-1-37-0': 'As for hierarchy multi-task learning and cascade multi-task learning, we choose a random task [MATH] at each training step, followed by a random training batch [MATH].', '1803.11326-1-37-1': 'Then we update the model parameters by back-propagating the corresponding loss [MATH].', '1803.11326-1-38-0': '# Experiments', '1803.11326-1-39-0': 'In this section we first introduce the popular ATIS dataset, and describe how we collect our E-commerce Shopping Guide Assistant (ECSGA) dataset.', '1803.11326-1-39-1': 'Then we show the implementation details for our model.', '1803.11326-1-39-2': 'Finally we demonstrate the evaluation results on both ATIS and ECSGA dataset and give some discussions.', '1803.11326-1-39-3': 'In the following experiments, we call our proposed Deep Cascade Multi-task Learning method as DCMTL for short.', '1803.11326-1-40-0': '## Dataset', '1803.11326-1-41-0': 'ATIS Dataset: The ATIS corpus is the most commonly used dataset for slot filling research, which consists of reservation requests from the air travel domain.', '1803.11326-1-41-1': 'There are 84 different slot labels (127 with IOB prefix).', '1803.11326-1-41-2': 'We randomly selected 80% of the training data for model training and the remaining 20% as the validation set [CITATION].', '1803.11326-1-41-3': 'Apart from the ground-truth slot labels, we also generate its corresponding segment labels for our multi-task model setting.', '1803.11326-1-42-0': 'ECSGA Dataset: To create large amounts of gold standard data for the training of our model, we adopt an unsupervised method to automatically tag the input utterances.', '1803.11326-1-42-1': 'All the utterances are extracted from the user input logs (either from text or voice) on an online shopping guide assistant system.', '1803.11326-1-42-2': 'Our E-commerce knowledge base is a dictionary consisting of pairs of word terms and their ground-truth slot labels such as "红色-颜色"(red-color).', '1803.11326-1-42-3': 'We use a dynamic programming algorithm to match terms in the utterances and then assign each word with its slot label in IOB scheme.', '1803.11326-1-42-4': 'We filter utterances whose matching result is ambiguous and only reserve those that can be perfectly matched (all words can be tagged by only one unique label) as our training and testing data.', '1803.11326-1-42-5': 'With the slot labels of each word, we can induce the named entity labels and segment labels straightforwardly.', '1803.11326-1-42-6': 'Our goal is to develop a sequence labeling algorithm with the ability to generalize to vocabulary outside of the dictionary.', '1803.11326-1-43-0': "To evaluate model's ability to generalize, we randomly split the dictionary into three parts.", '1803.11326-1-43-1': 'One part is used to generate testing data and the other two to generate training data.', '1803.11326-1-43-2': "If we don't split the dictionary and use the whole to generate both training and testing data, then the trained model may remember the whole dictionary and the results will not reflect the true performance of the models.", '1803.11326-1-44-0': 'The following experiments use a dataset of 24,892 training pairs and 2,723 testing pairs.', '1803.11326-1-44-1': 'Each pair contains an input utterance [MATH], its corresponding gold sequence of slot labels [MATH], named entity labels [MATH] and segment labels [MATH].', '1803.11326-1-45-0': '## Implementation Details', '1803.11326-1-46-0': 'For the RNN component in our system, we use a 3-layers LSTM network for ECSGA and 2-layers LSTM network for ATIS, they are all with unit size 100.', '1803.11326-1-46-1': 'All input sentences are padded to a maximum sequence length of 21 in ECSGA dataset and 46 in ATIS dataset.', '1803.11326-1-46-2': 'The input in ECSGA is a sequence of Chinese characters rather that words since there is no segmentation.', '1803.11326-1-46-3': 'The dimension of embedding layer [MATH] and BiLSTM network output hidden state are set to 200.', '1803.11326-1-47-0': 'For experiments on ECSGA dataset, the size of the labels for slot filling, named entity tagging and segment tagging are 57, 7 and 3 respectively.', '1803.11326-1-47-1': 'For experiments on ATIS dataset, the size of the labels for slot filling and segment tagging are 127 and 3 (no named entity tagging in this case).', '1803.11326-1-48-0': 'We perform a mini-batch log-likelihood loss training with a batch size of 32 sentences for 10 training epochs.', '1803.11326-1-48-1': 'We use Adam optimizer, and the learning rate is initialized to 0.001.', '1803.11326-1-49-0': '## Results and Discussions', '1803.11326-1-50-0': 'Eval on ATIS: We compare the ATIS results of our DCMTL model with current published results in Table [REF].', '1803.11326-1-50-1': 'Almost all the methods (including ours) reach very high F1 score of around 0.96.', '1803.11326-1-50-2': 'This makes us wonder whether it is meaningful enough to continue evaluating on this dataset, for minor changes between different results may be given rise by model or data variance.', '1803.11326-1-51-0': 'Eval on ECSGA: On ECSGA dataset, we evaluate different models including Basic BiLSTM-CRF, Vanilla Multi-task, Hierarchy Multi-task and Deep Cascade Multi-task on testing data regarding slot filling as the target task.', '1803.11326-1-51-1': 'We report Precision, Recall and F1 in Table [REF].', '1803.11326-1-52-0': 'The Basic BiLSTM-CRF model achieves an F1 score of 0.43.', '1803.11326-1-52-1': 'To show the impact of the lower tasks to slot filling, we "cheated" by using the ground-truth segment type (cond.', '1803.11326-1-52-2': 'SEG) or named entity type (cond.', '1803.11326-1-52-3': 'NE) as the extra features for each word in the Basic BiLSTM-CRF model.', '1803.11326-1-52-4': 'Row 3 and 4 (with *) in Table [REF] show that the slot filling performance can be improved by 85% and 109% if the correct segment type or named entity type is pre-known.', '1803.11326-1-52-5': "Of course in practice, the model doesn't know the true values of these types.", '1803.11326-1-53-0': 'Our further experiments show that DCMTL outperforms the baselines on both precision and recall.', '1803.11326-1-53-1': 'DCMTL achieves the best F1 score of 0.5105, which improves by a relative margin of 14.6% against the strong baseline method (see Table [REF]).', '1803.11326-1-53-2': 'Multi-task models generally perform better than the Basic BiLSTM with single-task target.', '1803.11326-1-53-3': 'The exception is the vanilla multi-task setting.', '1803.11326-1-53-4': 'This is mainly because vanilla multi-task shares parameters across all the layers which are likely to be disturbed by the interaction of three tasks.', '1803.11326-1-53-5': 'It is preferable to let the target task dominate the weights at high-level layers.', '1803.11326-1-54-0': 'We further investigate the learning trend of our proposed approach against baseline methods.', '1803.11326-1-54-1': 'Figure [REF](a)(b)(c) shows the typical learning curves of performance measured by Precision, Recall and F1.', '1803.11326-1-54-2': 'We can observe that our method DCMTL performs worse than other baseline methods for the first [MATH] batch steps.', '1803.11326-1-54-3': 'After that, other methods converge quickly and DCMTL perform much better after [MATH] batch steps and finally converge to the best F1 score.', '1803.11326-1-54-4': 'We believe that in the beginning, high-level task in DCMTL is affected more by the noise of low-level tasks comparing to others, but as the training goes on, the high-level slot filling task slowly reaps the benefits from low-level tasks.', '1803.11326-1-55-0': 'Ablation Test: We also investigate how our model DCMTL performs with or without cascade and residual connections.', '1803.11326-1-55-1': 'As shown in Table [REF], F1 score increases from 0.4840 to 0.5105 when residual connection is applied, which verifies its benefit.', '1803.11326-1-55-2': 'If we remove cascade connection from DCMTL, the model actually degenerates into hierarchy multi-task model with residual connection.', '1803.11326-1-55-3': 'From the table we find that it performs better than basic hierarchy multi-task model.', '1803.11326-1-55-4': 'Meanwhile, we can conclude that cascade connection plays a more important role than residual connection in our DCMTL model.', '1803.11326-1-56-0': 'Furthermore, we explore how DCMTL performs with different cascade connection methods.', '1803.11326-1-56-1': 'We compare three different types of cascade connection illustrated in Figure [REF](a): [1.]', '1803.11326-1-56-2': 'Segment labeling skipped to slot filling (SLOT+SEG).', '1803.11326-1-56-3': 'Named entity labeling directly connected to slot filling (SLOT+NE).', '1803.11326-1-56-4': 'Segment labeling, named entity labeling and slot filling in sequence (SLOT+NE+SEG).', '1803.11326-1-57-0': 'From Figure [REF](d), we find that cascade connection with type 3 performs the best and then with type 2, while cascade method with skipped connection (type 1) performs the worst.', '1803.11326-1-57-1': 'Therefore, we design the networks with a cascade connection in a hierarchical fashion and do not apply skipped connection for the cascade inputs (Figure [REF](b)).', '1803.11326-1-57-2': 'This phenomenon here may also be proved by our case study above.', '1803.11326-1-57-3': 'Slot filling performance with pre-known named entity type is much better than with pre-known segment type (rows with * in Table [REF]).', '1803.11326-1-58-0': '# Related Work', '1803.11326-1-59-0': 'Slot filling is considered a sequence labeling problem that is traditionally solved by generative models such as Hidden Markov Models (HMMs) [CITATION], hidden vector state model [CITATION], and discriminative models such as conditional random fields (CRFs) [CITATION] and Support Vector Machine (SVMs) [CITATION].', '1803.11326-1-59-1': 'In recent years, deep learning approaches have been explored due to its successful application in many NLP tasks.', '1803.11326-1-59-2': 'Many neural network architectures have been used such as simple RNNs [CITATION], convolutional neural networks (CNNs) [CITATION], LSTMs [CITATION] and variations like encoder-decoder [CITATION] and external memory [CITATION].', '1803.11326-1-59-3': 'In general, these works adopt a BiLSTM as the major labeling architecture to extract various features, then use a CRF layer [CITATION] to model the label dependency.', '1803.11326-1-59-4': 'We also adopt a BiLSTM-CRF model as baseline and claim that a multi-task learning framework is working better than directly applying it on Chinese E-commerce dataset.', '1803.11326-1-59-5': 'Previous works only apply joint model of slot filling and intent detection [CITATION].', '1803.11326-1-59-6': 'Our work is the first to propose a multi-task sequence labeling model with deep neural networks on slot filling problem.', '1803.11326-1-60-0': 'Multi-task learning (MTL) has attracted increasing attention in both academia and industry recently.', '1803.11326-1-60-1': 'By jointly learning across multiple tasks [CITATION], we can improve performance on each task and reduce the need for labeled data.', '1803.11326-1-60-2': 'There has been several attempts of using multi-task learning on sequence labeling task [CITATION], where most of these works learn all tasks at the out-most layer.', '1803.11326-1-60-3': 'Søgaard and Goldberg [CITATION] is the first to assume the existence of a hierarchy between the different tasks in a stacking BiRNN model.', '1803.11326-1-60-4': 'Compared to these works, our DCMTL model further improves this idea even thorough with cascade and residual connection.', '1803.11326-1-61-0': '# Conclusion', '1803.11326-1-62-0': 'In this paper, we attempt to solve the real-world slot filling task in a novel Chinese E-commerce shopping guide assistant.', '1803.11326-1-62-1': 'We proposed a deep multi-task sequence learning framework with cascade and residual connection.', '1803.11326-1-62-2': 'Our model achieves comparable results with several state-of-the-art models on the common slot filling dataset ATIS.', '1803.11326-1-62-3': 'On our released real-world Chinese E-commerce dataset ECSGA, our proposed model DCMTL also achieves best F1 score comparing to several strong baselines.'}

{'1803.11326-2-0-0': 'Slot filling is a critical task in natural language understanding (NLU) for dialog systems.', '1803.11326-2-0-1': 'State-of-the-art solutions regard it as a sequence labeling task and adopt BiLSTM-CRF models.', '1803.11326-2-0-2': 'While BiLSTM-CRF models works relatively well on standard datasets it faces challenges in Chinese E-commerce slot filling due to more informative slot labels and richer expressions.', '1803.11326-2-0-3': 'In this paper, we propose a deep multi-task learning model with cascade and residual connections.', '1803.11326-2-0-4': 'Experimental results show that our framework not only achieves competitive performance with state-of-the-arts on a standard dataset, but also significantly outperforms strong baselines by a substantial gain of 14.6% on a Chinese E-commerce dataset.', '1803.11326-2-1-0': 'UTF8gbsn', '1803.11326-2-2-0': '# Introduction', '1803.11326-2-3-0': 'An intelligent E-commerce online shopping guide assistant is a comprehensive human-like system providing various services such as pre-sale and after-sale inquiries, product recommendations, and user complaints processing, all of which seek to give the customers better shopping experience.', '1803.11326-2-3-1': 'The core of such assistant is a dialog system which has the ability to understand natural language utterances from a user and then give natural language responses.', '1803.11326-2-3-2': 'The architecture of a task-oriented dialog system for online shopping guide assistant is illustrated in Figure [REF].', '1803.11326-2-3-3': 'Natural Language Understanding (NLU), which aims to interpret the semantic meanings conveyed by input utterances is a main component in task-oriented dialog systems.', '1803.11326-2-3-4': 'Slot filling is a subproblem in NLU, which identifies the properties and their values about the task to be performed in the dialog.', '1803.11326-2-4-0': 'Slot filling extracts semantic constituents by using the words of input text to fill in pre-defined slots in a semantic frame [CITATION].', '1803.11326-2-4-1': 'It can be regarded as sequence labeling task, which assigns an appropriate semantic label to each word in the given input utterance.', '1803.11326-2-4-2': 'In the case of E-commerce shopping, there are three named entity types including Category, Property Key and Property Value.', '1803.11326-2-4-3': 'We show a real example in Table [REF] with In/Out/Begin(IOB) scheme.', '1803.11326-2-4-4': 'In the named entity level, "连衣裙"(dress) is a Category (B-CG/I-CG), while "品牌"(brand) is labeled as Property Key (B-PK/I-PK), which is the name of one product property.', '1803.11326-2-4-5': '"耐克"(Nike) and "黑色"(black) are labeled as Property Value (B-PV/I-PV) since they are concrete property values.', '1803.11326-2-4-6': 'However, labeling as Property Value is not good enough for NLU.', '1803.11326-2-4-7': 'Thus, in Slot Filling level, we further label "耐克"(Nike) as Brand Property (B-Brand/I-Brand), and "黑色"(black) as Color Property (B-Color/I-Color).', '1803.11326-2-4-8': 'In the meantime, other words in the example utterance that carry no semantic meaning are assigned O label.', '1803.11326-2-5-0': 'State-of-the-art sequence labeling models are typically based on BiLSTM and CRF [CITATION] and evaluated on a commonly used standard dataset ATIS [CITATION] in slot filling area.', '1803.11326-2-5-1': 'This dataset is in the domain of airline travel in America and Table [REF] shows an example utterance.', '1803.11326-2-5-2': 'However, the vocabulary size of ATIS is too small (only 572) and slot labels are not diverse enough since airline travel is a relatively small and specific domain, such that recent deep learning models can achieved very high F1 scores (nearly 0.96).', '1803.11326-2-6-0': 'Compared to ATIS, our E-commerce shopping guide assistant dataset is more complex.', '1803.11326-2-6-1': 'This dataset comes from a real world application and the semantic slots are more diverse and informal than ATIS, which increases the difficulty for the task.', '1803.11326-2-6-2': 'For example, to describe different properties for a product for purpose of utterance understanding and query rewrite, we should define large amount of informative slot labels such as color, brand, style, season, gender and so on.', '1803.11326-2-6-3': 'While most semantic labels of ATIS are related to only time and location.', '1803.11326-2-6-4': 'On the other hand, the spoken E-commerce Chinese language is more complex and enriched expression makes it harder to understand.', '1803.11326-2-6-5': 'For example, "红色" and "红" both mean red, "品牌" and "牌子" both mean brand, "耐克" and "Nike" and "Niky" all mean Nike.', '1803.11326-2-6-6': 'While in ATIS, expression can be simpler, and most expressions are standard locations or time.', '1803.11326-2-7-0': 'Besides, Chinese language, like many other Asian languages, are not word segmented by nature, and word segmentation is a difficult first step in many NLP tasks.', '1803.11326-2-7-1': 'Without proper word segmentation, sequence labeling becomes very challenging as the errors from segmentation will propagate.', '1803.11326-2-7-2': 'On the other hand, more than 97% of the chunks in ATIS data have only one or two words, in which segment (or chunking) may not be a serious problem.', '1803.11326-2-8-0': 'In this paper, we are the first to employ multi-task sequence labeling model to tackle slot filling in a novel Chinese E-commerce dialog system.', '1803.11326-2-8-1': 'We divide the slot filling task into two lower-level tasks: named entity tagging and segment tagging.', '1803.11326-2-8-2': 'Example labels of these two tasks are shown in the bottom two rows of Table [REF].', '1803.11326-2-8-3': 'Segment tagging and named entity tagging can be regarded as syntactic labeling, while slot filling is more like semantic labeling.', '1803.11326-2-8-4': 'Once we know the syntactic structure of an input sentence, filling the semantic labels becomes easier.', '1803.11326-2-8-5': 'Compared to directly attacking slot filling, these two low-level tasks are much easier to solve due to fewer labels.', '1803.11326-2-8-6': 'To this end, we propose a Deep Cascade Multi-task Learning model, and co-train three tasks in the same framework with a goal of optimizing the target slot filling task.', '1803.11326-2-9-0': 'The contributions of this paper are summarized below:', '1803.11326-2-10-0': '# Problem', '1803.11326-2-11-0': 'Given an utterance containing a sequence of words [MATH], the goal of our problem is to find a sequence of slot labels [MATH], one for each word in the utterance, such that: [EQUATION]', '1803.11326-2-11-1': 'In this paper we only define a slot filling problem in Dress category domain for simplification.', '1803.11326-2-11-2': 'That means we know in advance the defined category classification (intent).', '1803.11326-2-11-3': 'There are thousands of category classifications in E-commerce domain and in each category, there can be dozens of properties that are totally different.', '1803.11326-2-11-4': 'Performing slot filling with tens of thousands slot labels with cross-domains is impractical at this point.', '1803.11326-2-11-5': 'A joint model for category classification and slot filling is our future research.', '1803.11326-2-12-0': 'Along with Property Key(PK), Category(CG) and O, there are altogether 29 (57 in the IOB scheme) slot labels in our problem.', '1803.11326-2-12-1': 'Examples are listed in Table [REF].', '1803.11326-2-12-2': 'Notice that terms such as "brand", "color" appearing in an utterance will be labeled as PK, while Color and Brand can be pre-defined slot labels and will be assigned to terms like "black" and "Nike".', '1803.11326-2-13-0': '# Approach', '1803.11326-2-14-0': 'In this section we describe our approach in detail.', '1803.11326-2-14-1': 'Figure [REF] gives an overview of the proposed architectures.', '1803.11326-2-14-2': 'First we introduce the most common and popular BiLSTM-CRF model (Figure [REF](a)) for sequence labeling tasks.', '1803.11326-2-14-3': 'Then we move on to multi-task learning perspective (Figure [REF](b) and (c)).', '1803.11326-2-14-4': 'Finally we propose our new method, which is called Deep Cascade Multi-task Learning in Figure [REF](d).', '1803.11326-2-15-0': '## RNN Sequence Labeling', '1803.11326-2-16-0': 'Figure [REF](a) shows the principle architecture of a BiLSTM-CRF model, which is the state-of-the-art model for various sequence labeling tasks [CITATION].', '1803.11326-2-16-1': 'BiLSTM-CRF model consists of a BiLSTM layer and a CRF layer.', '1803.11326-2-17-0': 'Bidirectional LSTMs enable the hidden states to capture both historical and future context information of the words.', '1803.11326-2-17-1': 'Mathematically, the input of this BiLSTM layer is a sequence of input vectors, denoted as [MATH].', '1803.11326-2-17-2': 'The output of BiLSTM layer is a sequence of the hidden states for each input word, denoted as [MATH].', '1803.11326-2-17-3': 'Each final hidden state is the concatenation of the forward [MATH] and backward [MATH] hidden states.', '1803.11326-2-17-4': 'We view BiLSTM as a function [MATH]: [EQUATION]', '1803.11326-2-17-5': 'Most of time we stack multiple BiLSTMs to make the model deeper, in which the output [MATH] of layer [MATH] becomes the input of layer [MATH], e.g. [MATH].', '1803.11326-2-18-0': 'It is always beneficial to consider the correlations between the current label and neighboring labels, since there are many syntactical constraints in natural language sentences.', '1803.11326-2-18-1': 'If we simply feed the above mentioned hidden states independently to a softmax layer to predict the labels [CITATION], such constraints are more likely to be violated.', '1803.11326-2-18-2': 'Linear-chain Conditional Random Field (CRF) [CITATION] is the most popular way to control the structure prediction and its basic idea is to use a series of potential functions to approximate the conditional probability of the output label sequence given the input word sequence.', '1803.11326-2-19-0': 'Formally, we take the above sequence of hidden states [MATH] as input to a CRF layer, and the output of the CRF is the final prediction label sequence [MATH], where [MATH] is in the set of pre-defined target labels.', '1803.11326-2-19-1': 'We denote [MATH] as the set of all possible label sequences.', '1803.11326-2-19-2': 'Then we derive the conditional probability of the output sequence, given the input hidden state sequence is: [EQUATION] where [MATH] are potential functions and [MATH] and [MATH] are weight vector and bias of label pair [MATH].', '1803.11326-2-19-3': 'To train the CRF layer, we use the classic maximum conditional likelihood estimate and gradient ascent.', '1803.11326-2-19-4': 'For a training dataset [MATH], the final log-likelihood is: [EQUATION].', '1803.11326-2-19-5': 'Finally, the Viterbi algorithm is adopted to decode the optimal output sequence [MATH]: [EQUATION]', '1803.11326-2-20-0': '## Multi-task Learning', '1803.11326-2-21-0': 'While directly attacking the slot filling task is hard, low-level tasks with fewer labels are much easier to solve.', '1803.11326-2-21-1': 'Once we know the syntactic structure of a sentence, filling in semantic labels will become easier accordingly.', '1803.11326-2-21-2': 'Thus, it is reasonable to solve the problem in a multi-task learning framework.', '1803.11326-2-21-3': 'In our problem, we can devise three individual tasks: slot filling, named entity tagging and segment tagging.', '1803.11326-2-21-4': 'Slot filling is our target task; named entity tagging is to classify which named entity type (PV/PK/CG) a word is; and segment tagging is to judge whether a word is begin (B), in (I) or out (O) of a trunking.', '1803.11326-2-22-0': 'In a multi-task learning (MTL) setting, we have several prediction tasks over the same input sequence, where each task has its own output vocabulary (a set of task specified labels).', '1803.11326-2-22-1': 'Intuitively, the three tasks do share a lot of information.', '1803.11326-2-22-2': 'Consider the example in Table [REF] again.', '1803.11326-2-22-3': 'Knowing the named entity type of "黑色"(black) being B-PV can definitely help determine its slot label, which is B-Color.', '1803.11326-2-22-4': 'Similarly, knowing its segment type (B) also helps with both named entity tagging and slot filling.', '1803.11326-2-22-5': 'Thus it is reasonable for these tasks to share parameters and learn in the same framework cooperatively.', '1803.11326-2-23-0': '### Vanilla Multi-task Learning', '1803.11326-2-24-0': 'The general idea of multi-task learning is to share parameters of encoding part of the network.', '1803.11326-2-24-1': 'As Figure [REF](b) shows, this is naturally achieved by sharing the [MATH]-layers BiLSTM part of the network across three tasks.', '1803.11326-2-24-2': 'Based on that, we use a separate CRF decoder for each task [MATH]: [MATH], where [MATH] and [MATH] are task-specific parameters.', '1803.11326-2-24-3': 'This encourages the deep BiLSTM network to learn a hidden representation [MATH] which benefits all three different tasks.', '1803.11326-2-25-0': '### Hierarchy Multi-task Learning', '1803.11326-2-26-0': 'Previous discussion indicates that there is a natural order among the different tasks: slot filling may benefit more from named entity tagging, than the other way around.', '1803.11326-2-26-1': 'This motivates us to employ low-level tasks at lower BiLSTM layers, while high level tasks are trained at higher layers.', '1803.11326-2-26-2': 'This idea was first proposed by Anders and Yoav [CITATION].', '1803.11326-2-26-3': 'As shown in Figure [REF](c), instead of decoding all tasks separately at the outermost BiLSTM layer, we associate each BiLSTM layer [MATH] with one task [MATH].', '1803.11326-2-26-4': 'Then the conditional probabilities of the output sequence for each task are: [EQUATION]', '1803.11326-2-26-5': 'Here [MATH], [MATH] and [MATH] represent the tasks of segment tagging, named entity tagging and slot filling, respectively.', '1803.11326-2-26-6': '[MATH] is the word embeddings of input sequence [MATH] and [MATH].', '1803.11326-2-26-7': 'We call this model hierarchy multi-task learning, since some layers are shared by all tasks while the others are only related to specific tasks.', '1803.11326-2-27-0': '## Deep Cascade Multi-task Learning', '1803.11326-2-28-0': 'Hierarchy multi-task learning share parameters among different tasks, and allow low-level tasks help adjust the result of high-level target task.', '1803.11326-2-28-1': 'It is effective for those tasks which are weakly correlated, such as POS tagging, syntactic chunking and CCG supertagging [CITATION].', '1803.11326-2-28-2': 'However, when it comes to problems where different tasks maintain a strict order, in another word, the performance of high-level task dramatically depends on low-level tasks, the hierarchy structure is not compact and effective enough.', '1803.11326-2-28-3': 'Therefore, we propose cascade and residual connections to enable high-level tasks taking the tagging results and hidden states of low-level tasks as additional input.', '1803.11326-2-28-4': 'These connections serves as "shortcuts" that create a more closely coupled and efficient model.', '1803.11326-2-28-5': 'We call it deep cascade multi-task learning, and the framework is shown in Figure [REF](d).', '1803.11326-2-29-0': '### Cascade Connection', '1803.11326-2-30-0': 'Here we feed the tagging output of the task at lower layer e.g. [MATH] or [MATH] to the upper BiLSTM layer as its additional input.', '1803.11326-2-30-1': 'Now the hidden states of each task layer become: [EQUATION] where [MATH] is the weight parameter for cascade connection.', '1803.11326-2-31-0': 'At training time, [MATH] and [MATH] can be the true tagging outputs.', '1803.11326-2-31-1': 'At inference time, we simply take the greedy path of our cascade model without doing search, where the model emits the best [MATH] and [MATH] by Viterbi inference algorithm.', '1803.11326-2-31-2': 'Alternatively, one can do beam search [CITATION] by maintaining a set of [MATH] best partial hypotheses at each cascade layer.', '1803.11326-2-31-3': 'However, unlike traditional seq2seq models e.g., in machine translation, where each inference step is just based on probability of a discrete variable (by softmax function), our inference for tagging output is a structured probability distribution defined by the CRF output.', '1803.11326-2-31-4': 'Efficient beam search method for this structured cascade model is left to our future work.', '1803.11326-2-32-0': '### Residual Connection', '1803.11326-2-33-0': 'To encourage the information sharing among different tasks, we also introduce the residual connection, where we add the input of a previous layer to the current input: [EQUATION]', '1803.11326-2-33-1': 'Deep residual learning [CITATION] is first introduced to ease the gradient vanish problem for training very deep neural networks.', '1803.11326-2-33-2': 'Here we propose that residual connection between different layers can help for multi-task learning.', '1803.11326-2-34-0': '### Training', '1803.11326-2-35-0': 'For our multi-task setting, we define three loss functions (refer to Section [REF]): [MATH], [MATH] and [MATH] for tasks of segment tagging, named entity tagging and slot filling respectively.', '1803.11326-2-35-1': 'We construct three training set, [MATH], [MATH] and [MATH], where each of them (called [MATH] generically) contains a set of input-output sequence pair [MATH].', '1803.11326-2-35-2': 'The input utterance [MATH] is shared across tasks, but the output [MATH] is task dependent.', '1803.11326-2-36-0': 'For vanilla multi-task learning, we define a loss function [MATH], where [MATH] and [MATH] are hyper-parameters.', '1803.11326-2-37-0': 'As for hierarchy multi-task learning and cascade multi-task learning, we choose a random task [MATH] at each training step, followed by a random training batch [MATH].', '1803.11326-2-37-1': 'Then we update the model parameters by back-propagating the corresponding loss [MATH].', '1803.11326-2-38-0': '# Experiments', '1803.11326-2-39-0': 'In this section we first introduce the popular ATIS dataset, and describe how we collect our E-commerce Shopping Guide Assistant (ECSGA) dataset.', '1803.11326-2-39-1': 'Then we show the implementation details for our model.', '1803.11326-2-39-2': 'Finally we demonstrate the evaluation results on both ATIS and ECSGA dataset and give some discussions.', '1803.11326-2-39-3': 'In the following experiments, we call our proposed Deep Cascade Multi-task Learning method as DCMTL for short.', '1803.11326-2-40-0': '## Dataset', '1803.11326-2-41-0': 'ATIS Dataset: The ATIS corpus is the most commonly used dataset for slot filling research, which consists of reservation requests from the air travel domain.', '1803.11326-2-41-1': 'There are 84 different slot labels (127 with IOB prefix).', '1803.11326-2-41-2': 'We randomly selected 80% of the training data for model training and the remaining 20% as the validation set [CITATION].', '1803.11326-2-41-3': 'Apart from the ground-truth slot labels, we also generate its corresponding segment labels for our multi-task model setting.', '1803.11326-2-42-0': 'ECSGA Dataset: To create large amounts of gold standard data for the training of our model, we adopt an unsupervised method to automatically tag the input utterances.', '1803.11326-2-42-1': 'All the utterances are extracted from the user input logs (either from text or voice) on an online shopping guide assistant system.', '1803.11326-2-42-2': 'Our E-commerce knowledge base is a dictionary consisting of pairs of word terms and their ground-truth slot labels such as "红色-颜色"(red-color).', '1803.11326-2-42-3': 'We use a dynamic programming algorithm to match terms in the utterances and then assign each word with its slot label in IOB scheme.', '1803.11326-2-42-4': 'We filter utterances whose matching result is ambiguous and only reserve those that can be perfectly matched (all words can be tagged by only one unique label) as our training and testing data.', '1803.11326-2-42-5': 'With the slot labels of each word, we can induce the named entity labels and segment labels straightforwardly.', '1803.11326-2-42-6': 'Our goal is to develop a sequence labeling algorithm with the ability to generalize to vocabulary outside of the dictionary.', '1803.11326-2-43-0': "To evaluate model's ability to generalize, we randomly split the dictionary into three parts.", '1803.11326-2-43-1': 'One part is used to generate testing data and the other two to generate training data.', '1803.11326-2-43-2': "If we don't split the dictionary and use the whole to generate both training and testing data, then the trained model may remember the whole dictionary and the results will not reflect the true performance of the models.", '1803.11326-2-44-0': 'The following experiments use a dataset of 24,892 training pairs and 2,723 testing pairs.', '1803.11326-2-44-1': 'Each pair contains an input utterance [MATH], its corresponding gold sequence of slot labels [MATH], named entity labels [MATH] and segment labels [MATH].', '1803.11326-2-45-0': '## Implementation Details', '1803.11326-2-46-0': 'For the RNN component in our system, we use a 3-layers LSTM network for ECSGA and 2-layers LSTM network for ATIS, they are all with unit size 100.', '1803.11326-2-46-1': 'All input sentences are padded to a maximum sequence length of 21 in ECSGA dataset and 46 in ATIS dataset.', '1803.11326-2-46-2': 'The input in ECSGA is a sequence of Chinese characters rather that words since there is no segmentation.', '1803.11326-2-46-3': 'The dimension of embedding layer [MATH] and BiLSTM network output hidden state are set to 200.', '1803.11326-2-47-0': 'For experiments on ECSGA dataset, the size of the labels for slot filling, named entity tagging and segment tagging are 57, 7 and 3 respectively.', '1803.11326-2-47-1': 'For experiments on ATIS dataset, the size of the labels for slot filling and segment tagging are 127 and 3 (no named entity tagging in this case).', '1803.11326-2-48-0': 'We perform a mini-batch log-likelihood loss training with a batch size of 32 sentences for 10 training epochs.', '1803.11326-2-48-1': 'We use Adam optimizer, and the learning rate is initialized to 0.001.', '1803.11326-2-49-0': '## Results and Discussions', '1803.11326-2-50-0': 'Eval on ATIS: We compare the ATIS results of our DCMTL model with current published results in Table [REF].', '1803.11326-2-50-1': 'Almost all the methods (including ours) reach very high F1 score of around 0.96.', '1803.11326-2-50-2': 'This makes us wonder whether it is meaningful enough to continue evaluating on this dataset, for minor changes between different results may be given rise by model or data variance.', '1803.11326-2-51-0': 'Eval on ECSGA: On ECSGA dataset, we evaluate different models including Basic BiLSTM-CRF, Vanilla Multi-task, Hierarchy Multi-task and Deep Cascade Multi-task on testing data regarding slot filling as the target task.', '1803.11326-2-51-1': 'We report Precision, Recall and F1 in Table [REF].', '1803.11326-2-52-0': 'The Basic BiLSTM-CRF model achieves an F1 score of 0.43.', '1803.11326-2-52-1': 'To show the impact of the lower tasks to slot filling, we "cheated" by using the ground-truth segment type (cond.', '1803.11326-2-52-2': 'SEG) or named entity type (cond.', '1803.11326-2-52-3': 'NE) as the extra features for each word in the Basic BiLSTM-CRF model.', '1803.11326-2-52-4': 'Row 3 and 4 (with *) in Table [REF] show that the slot filling performance can be improved by 85% and 109% if the correct segment type or named entity type is pre-known.', '1803.11326-2-52-5': "Of course in practice, the model doesn't know the true values of these types.", '1803.11326-2-53-0': 'Our further experiments show that DCMTL outperforms the baselines on both precision and recall.', '1803.11326-2-53-1': 'DCMTL achieves the best F1 score of 0.5105, which improves by a relative margin of 14.6% against the strong baseline method (see Table [REF]).', '1803.11326-2-53-2': 'Multi-task models generally perform better than the Basic BiLSTM with single-task target.', '1803.11326-2-53-3': 'The exception is the vanilla multi-task setting.', '1803.11326-2-53-4': 'This is mainly because vanilla multi-task shares parameters across all the layers which are likely to be disturbed by the interaction of three tasks.', '1803.11326-2-53-5': 'It is preferable to let the target task dominate the weights at high-level layers.', '1803.11326-2-54-0': 'We further investigate the learning trend of our proposed approach against baseline methods.', '1803.11326-2-54-1': 'Figure [REF](a)(b)(c) shows the typical learning curves of performance measured by Precision, Recall and F1.', '1803.11326-2-54-2': 'We can observe that our method DCMTL performs worse than other baseline methods for the first [MATH] batch steps.', '1803.11326-2-54-3': 'After that, other methods converge quickly and DCMTL perform much better after [MATH] batch steps and finally converge to the best F1 score.', '1803.11326-2-54-4': 'We believe that in the beginning, high-level task in DCMTL is affected more by the noise of low-level tasks comparing to others, but as the training goes on, the high-level slot filling task slowly reaps the benefits from low-level tasks.', '1803.11326-2-55-0': 'Ablation Test: We also investigate how our model DCMTL performs with or without cascade and residual connections.', '1803.11326-2-55-1': 'As shown in Table [REF], F1 score increases from 0.4840 to 0.5105 when residual connection is applied, which verifies its benefit.', '1803.11326-2-55-2': 'If we remove cascade connection from DCMTL, the model actually degenerates into hierarchy multi-task model with residual connection.', '1803.11326-2-55-3': 'From the table we find that it performs better than basic hierarchy multi-task model.', '1803.11326-2-55-4': 'Meanwhile, we can conclude that cascade connection plays a more important role than residual connection in our DCMTL model.', '1803.11326-2-56-0': 'Furthermore, we explore how DCMTL performs with different cascade connection methods.', '1803.11326-2-56-1': 'We compare three different types of cascade connection illustrated in Figure [REF](a): [1.]', '1803.11326-2-56-2': 'Segment labeling skipped to slot filling (SLOT+SEG).', '1803.11326-2-56-3': 'Named entity labeling directly connected to slot filling (SLOT+NE).', '1803.11326-2-56-4': 'Segment labeling, named entity labeling and slot filling in sequence (SLOT+NE+SEG).', '1803.11326-2-57-0': 'From Figure [REF](d), we find that cascade connection with type 3 performs the best and then with type 2, while cascade method with skipped connection (type 1) performs the worst.', '1803.11326-2-57-1': 'Therefore, we design the networks with a cascade connection in a hierarchical fashion and do not apply skipped connection for the cascade inputs (Figure [REF](b)).', '1803.11326-2-57-2': 'This phenomenon here may also be proved by our case study above.', '1803.11326-2-57-3': 'Slot filling performance with pre-known named entity type is much better than with pre-known segment type (rows with * in Table [REF]).', '1803.11326-2-58-0': '# Related Work', '1803.11326-2-59-0': 'Slot filling is considered a sequence labeling problem that is traditionally solved by generative models such as Hidden Markov Models (HMMs) [CITATION], hidden vector state model [CITATION], and discriminative models such as conditional random fields (CRFs) [CITATION] and Support Vector Machine (SVMs) [CITATION].', '1803.11326-2-59-1': 'In recent years, deep learning approaches have been explored due to its successful application in many NLP tasks.', '1803.11326-2-59-2': 'Many neural network architectures have been used such as simple RNNs [CITATION], convolutional neural networks (CNNs) [CITATION], LSTMs [CITATION] and variations like encoder-decoder [CITATION] and external memory [CITATION].', '1803.11326-2-59-3': 'In general, these works adopt a BiLSTM as the major labeling architecture to extract various features, then use a CRF layer [CITATION] to model the label dependency.', '1803.11326-2-59-4': 'We also adopt a BiLSTM-CRF model as baseline and claim that a multi-task learning framework is working better than directly applying it on Chinese E-commerce dataset.', '1803.11326-2-59-5': 'Previous works only apply joint model of slot filling and intent detection [CITATION].', '1803.11326-2-59-6': 'Our work is the first to propose a multi-task sequence labeling model with deep neural networks on slot filling problem.', '1803.11326-2-60-0': 'Multi-task learning (MTL) has attracted increasing attention in both academia and industry recently.', '1803.11326-2-60-1': 'By jointly learning across multiple tasks [CITATION], we can improve performance on each task and reduce the need for labeled data.', '1803.11326-2-60-2': 'There has been several attempts of using multi-task learning on sequence labeling task [CITATION], where most of these works learn all tasks at the out-most layer.', '1803.11326-2-60-3': 'Søgaard and Goldberg [CITATION] is the first to assume the existence of a hierarchy between the different tasks in a stacking BiRNN model.', '1803.11326-2-60-4': 'Compared to these works, our DCMTL model further improves this idea even thorough with cascade and residual connection.', '1803.11326-2-61-0': '# Conclusion', '1803.11326-2-62-0': 'In this paper, we attempt to solve the real-world slot filling task in a novel Chinese E-commerce shopping guide assistant.', '1803.11326-2-62-1': 'We proposed a deep multi-task sequence learning framework with cascade and residual connection.', '1803.11326-2-62-2': 'Our model achieves comparable results with several state-of-the-art models on the common slot filling dataset ATIS.', '1803.11326-2-62-3': 'On our released real-world Chinese E-commerce dataset ECSGA, our proposed model DCMTL also achieves best F1 score comparing to several strong baselines.'}

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['1803.11326-3-67-0', '1803.11326-4-67-0'], ['1803.11326-3-67-1', '1803.11326-4-67-1'], ['1803.11326-3-67-2', '1803.11326-4-67-2'], ['1803.11326-3-67-3', '1803.11326-4-67-3'], ['1803.11326-3-67-5', '1803.11326-4-67-5'], ['1803.11326-3-67-6', '1803.11326-4-67-6'], ['1803.11326-3-67-7', '1803.11326-4-67-7'], ['1803.11326-3-13-0', '1803.11326-4-13-0'], ['1803.11326-3-13-1', '1803.11326-4-13-1'], ['1803.11326-3-6-0', '1803.11326-4-6-0'], ['1803.11326-3-6-1', '1803.11326-4-6-1'], ['1803.11326-3-6-2', '1803.11326-4-6-2'], ['1803.11326-3-6-3', '1803.11326-4-6-3'], ['1803.11326-3-6-4', '1803.11326-4-6-4'], ['1803.11326-3-3-0', '1803.11326-4-3-0'], ['1803.11326-3-3-1', '1803.11326-4-3-1'], ['1803.11326-3-3-2', '1803.11326-4-3-2'], ['1803.11326-3-3-3', '1803.11326-4-3-3'], ['1803.11326-3-3-4', '1803.11326-4-3-4'], ['1803.11326-3-3-5', '1803.11326-4-3-5'], ['1803.11326-3-3-6', '1803.11326-4-3-6'], ['1803.11326-3-3-7', '1803.11326-4-3-7'], ['1803.11326-3-35-0', '1803.11326-4-35-0'], ['1803.11326-3-35-1', '1803.11326-4-35-1'], ['1803.11326-3-34-0', '1803.11326-4-34-0'], ['1803.11326-3-34-1', '1803.11326-4-34-1'], ['1803.11326-3-55-0', '1803.11326-4-55-0'], ['1803.11326-3-55-1', '1803.11326-4-55-1'], ['1803.11326-3-55-2', '1803.11326-4-55-2'], ['1803.11326-3-55-3', '1803.11326-4-55-3'], ['1803.11326-3-55-4', '1803.11326-4-55-4'], ['1803.11326-3-40-0', '1803.11326-4-40-0'], ['1803.11326-3-40-1', '1803.11326-4-40-1'], ['1803.11326-3-40-2', '1803.11326-4-40-2'], ['1803.11326-3-29-0', '1803.11326-4-29-0'], ['1803.11326-3-29-1', '1803.11326-4-29-1'], ['1803.11326-3-29-2', '1803.11326-4-29-2'], ['1803.11326-3-29-3', '1803.11326-4-29-3'], ['1803.11326-3-29-4', '1803.11326-4-29-4'], ['1803.11326-3-70-0', '1803.11326-4-70-0'], ['1803.11326-3-70-1', '1803.11326-4-70-1'], ['1803.11326-3-70-2', '1803.11326-4-70-2'], ['1803.11326-3-70-3', '1803.11326-4-70-3'], ['1803.11326-3-70-4', '1803.11326-4-70-4'], ['1803.11326-3-70-5', '1803.11326-4-70-5'], ['1803.11326-3-70-6', '1803.11326-4-70-6'], ['1803.11326-3-70-7', '1803.11326-4-70-7'], ['1803.11326-3-44-0', '1803.11326-4-44-0'], ['1803.11326-3-44-1', '1803.11326-4-44-1'], ['1803.11326-3-44-2', '1803.11326-4-44-2'], ['1803.11326-3-44-3', '1803.11326-4-44-3'], ['1803.11326-3-44-4', '1803.11326-4-44-4'], ['1803.11326-3-44-5', '1803.11326-4-44-5'], ['1803.11326-2-56-0', '1803.11326-3-59-0'], ['1803.11326-2-56-2', '1803.11326-3-60-1'], ['1803.11326-2-56-3', '1803.11326-3-60-3'], ['1803.11326-2-56-4', '1803.11326-3-60-5'], ['1803.11326-2-48-0', '1803.11326-3-46-3'], ['1803.11326-2-48-1', '1803.11326-3-46-4']]

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[]

[['1803.11326-2-59-0', '1803.11326-3-67-0'], ['1803.11326-2-59-0', '1803.11326-3-67-1'], ['1803.11326-2-59-6', '1803.11326-3-67-7'], ['1803.11326-2-41-0', '1803.11326-3-40-0'], ['1803.11326-2-33-2', '1803.11326-3-31-2'], ['1803.11326-2-5-1', '1803.11326-3-4-2'], ['1803.11326-2-4-2', '1803.11326-3-3-0'], ['1803.11326-2-4-4', '1803.11326-3-3-2'], ['1803.11326-2-4-5', '1803.11326-3-3-3'], ['1803.11326-2-62-0', '1803.11326-3-70-0'], ['1803.11326-2-42-2', '1803.11326-3-42-2'], ['1803.11326-2-42-5', '1803.11326-3-42-6'], ['1803.11326-2-50-2', '1803.11326-3-50-1'], ['1803.11326-2-54-1', '1803.11326-3-55-1'], ['1803.11326-2-55-0', '1803.11326-3-58-3'], ['1803.11326-2-55-4', '1803.11326-3-58-6'], ['1803.11326-2-21-3', '1803.11326-3-19-3'], ['1803.11326-2-6-4', '1803.11326-3-5-4'], ['1803.11326-2-8-1', '1803.11326-3-7-1'], ['1803.11326-2-8-4', '1803.11326-3-7-4'], ['1803.11326-2-0-1', '1803.11326-3-0-1'], ['1803.11326-2-0-2', '1803.11326-3-0-2'], ['1803.11326-2-0-3', '1803.11326-3-0-3'], ['1803.11326-2-46-0', '1803.11326-3-46-0']]

[]

['1803.11326-1-1-0', '1803.11326-1-9-0', '1803.11326-2-1-0', '1803.11326-2-9-0', '1803.11326-3-8-0', '1803.11326-3-59-1', '1803.11326-3-60-0', '1803.11326-3-60-2', '1803.11326-3-60-4', '1803.11326-4-8-0', '1803.11326-4-59-1', '1803.11326-4-60-0', '1803.11326-4-60-2', '1803.11326-4-60-4']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1803.11326

{'1803.11326-3-0-0': 'Slot filling is a critical task in natural language understanding (NLU) for dialog systems.', '1803.11326-3-0-1': 'State-of-the-art approaches treat it as a sequence labeling problem and adopt such models as BiLSTM-CRF.', '1803.11326-3-0-2': 'While these models work relatively well on standard benchmark datasets, they face challenges in the context of E-commerce where the slot labels are more informative and carry richer expressions.', '1803.11326-3-0-3': 'In this work, inspired by the unique structure of E-commerce knowledge base, we propose a novel multi-task model with cascade and residual connections, which jointly learns segment tagging, named entity tagging and slot filling.', '1803.11326-3-0-4': 'Experiments show the effectiveness of the proposed cascade and residual structures.', '1803.11326-3-0-5': 'Our model has a 14.6 advantage in F1 score over the strong baseline methods on a new Chinese E-commerce shopping assistant dataset, while achieving competitive accuracies on a standard dataset.', '1803.11326-3-0-6': 'Furthermore, online test deployed on such dominant E-commerce platform shows 130 improvement on accuracy of understanding user utterances.', '1803.11326-3-0-7': 'Our model has already gone into production in the E-commerce platform.', '1803.11326-3-1-0': '# Introduction', '1803.11326-3-2-0': 'An intelligent online shopping assistant offers services such as pre-sale and after-sale inquiries, product recommendations, and user complaints processing, all of which seek to give the customers better shopping experience.', '1803.11326-3-2-1': 'The core of such assistant is a task-oriented dialog system which has the ability to understand natural language utterances from a user and then give natural language responses [CITATION].', '1803.11326-3-2-2': 'Natural Language Understanding (NLU), which aims to interpret the semantic meanings conveyed by input utterances, is a main component in task-oriented dialog systems.', '1803.11326-3-2-3': 'Slot filling, a sub-problem of NLU, extracts semantic constituents by using the words of input utterance to fill in pre-defined slots in a semantic frame [CITATION].', '1803.11326-3-3-0': 'In the case of E-commerce shopping, there are three named entity types: Category, Property Key and Property Value, according to typical E-commerce knowledge base such as the one in Figure [REF].', '1803.11326-3-3-1': 'We show a real example in Table [REF] with In/Out/Begin (I/O/B) scheme.', '1803.11326-3-3-2': 'In the named entity level, "dress" is a Category (CG), while "brand" is labeled as Property Key (PK), which is the name of one product property.', '1803.11326-3-3-3': '"Nike" and "black" are labeled as Property Value (PV) since they are concrete property values.', '1803.11326-3-3-4': 'However, merely labeling as Property Value is not sufficient as the shopping assistant needs more fine-grained semantics.', '1803.11326-3-3-5': 'Therefore, in the Slot Filling level, we further label "Nike" as Brand Property (Brand), and "black" as Color Property (Color).', '1803.11326-3-3-6': 'In Table [REF], B-CG refers to Begin-Category (the meaning of other labels can also be inferred).', '1803.11326-3-3-7': 'In the meantime, other words in the example utterance that carry no semantic meaning are assigned O label.', '1803.11326-3-4-0': 'Traditionally, slot filling problem can be regarded as a sequence labeling task, which assigns an appropriate semantic label to each word in the given input utterance.', '1803.11326-3-4-1': 'State-of-the-art sequence labeling models are typically based on BiLSTM-CRF [CITATION] and evaluated on a commonly used standard dataset ATIS [CITATION] in the slot filling area.', '1803.11326-3-4-2': 'This dataset is about airline travel in the United States.', '1803.11326-3-4-3': 'However, the vocabulary size of ATIS is small (only 572) and slot labels are not diverse enough (mostly related to only time and location) since airline travel is a relatively small and specific domain, such that recent deep learning models can achieve very high F1 scores (nearly 0.96).', '1803.11326-3-4-4': 'Recently, a detailed quantitative and qualitative study of this dataset comes to the same conclusion that slot filling models should be tested on a much more real and complex dataset [CITATION].', '1803.11326-3-5-0': 'In this paper, we try to tackle a real-world slot filling problem for one of the largest E-commerce platform in China.', '1803.11326-3-5-1': 'The semantic slots are much more diverse and informative than ATIS.', '1803.11326-3-5-2': 'For example, to describe different properties of a product for the purpose of utterance understanding, we define large amount of informative slot labels such as color, brand, style, season, gender and so on.', '1803.11326-3-5-3': 'In contrast, most semantic labels of ATIS are related to only time and location.', '1803.11326-3-5-4': 'Furthermore, the Chinese language used for e-commerce is more complex and the semantically rich expressions make it harder to understand.', '1803.11326-3-5-5': 'Whereas in ATIS, expression can be simpler, and most expressions are standard locations or time.', '1803.11326-3-5-6': 'Thus, large scale semantic slots and more complex expressions bring problem such as data sparsity.', '1803.11326-3-5-7': 'Traditional end-to-end sequence labeling model may not be able to handle it.', '1803.11326-3-6-0': 'Besides, Chinese language, like many other Asian languages, is not word segmented by nature, and word segmentation is a difficult first step in many NLP tasks.', '1803.11326-3-6-1': 'Without proper word segmentation, sequence labeling becomes very challenging as the errors from segmentation will propagate.', '1803.11326-3-6-2': 'On the other hand, more than 97% of the chunks in ATIS data have only one or two words, in which segmentation (or chunking) is not a serious problem.', '1803.11326-3-6-3': 'Due to these reasons, if we simply apply basic sequence labeling models, which can be regarded as an end-to-end method, the sentences may not be segmented correctly in the first place.', '1803.11326-3-6-4': 'Then the errors will propagate and the resulting slot labels will be incorrect.', '1803.11326-3-7-0': 'In this paper, we propose to employ multi-task sequence labeling model to tackle slot filling in a novel Chinese E-commerce dialog system.', '1803.11326-3-7-1': 'Inspired by the natural structure of E-commerce knowledge base shown in Figure [REF], we extract two additional lower-level tasks from the slot filling task: named entity tagging and segment tagging.', '1803.11326-3-7-2': 'Example labels of these two tasks are shown in the bottom two rows of Table [REF].', '1803.11326-3-7-3': 'Segment tagging and named entity tagging can be regarded as syntactic labeling, while slot filling is more like semantic labeling.', '1803.11326-3-7-4': 'With the help of information sharing ability of multi-task learning, once we learn the information of syntactic structure of an input sentence, filling the semantic labels becomes much easier.', '1803.11326-3-7-5': 'Compared to directly attacking slot filling, these two low-level tasks are much easier to solve due to fewer labels.', '1803.11326-3-7-6': 'To this end, we propose a Deep Cascade Multi-task Learning model, and co-train three tasks in the same framework with a goal of optimizing the target slot filling task.', '1803.11326-3-8-0': 'The contributions of this paper are summarized below:', '1803.11326-3-9-0': '# Approach', '1803.11326-3-10-0': 'In this section we describe our approach in detail.', '1803.11326-3-10-1': 'Figure [REF] gives an overview of the proposed architectures.', '1803.11326-3-10-2': 'First we introduce the most common and popular BiLSTM-CRF model (Figure [REF](a)) for sequence labeling tasks.', '1803.11326-3-10-3': 'Then we move on to multi-task learning perspective (Figure [REF](b) and (c)).', '1803.11326-3-10-4': 'Finally we propose our new method, which is called Deep Cascade Multi-task Learning in Figure [REF](d).', '1803.11326-3-11-0': 'Given an utterance containing a sequence of words [MATH], the goal of our problem is to find a sequence of slot labels [MATH], one for each word in the utterance, such that: [EQUATION]', '1803.11326-3-11-1': 'We use "word" in problem and model description, but "word" actually means Chinese char in our problem.', '1803.11326-3-11-2': 'And a "term" consists of one or several words.', '1803.11326-3-12-0': '## RNN Sequence Labeling', '1803.11326-3-13-0': 'Figure [REF](a) shows the principle architecture of a BiLSTM-CRF model, which is the state-of-the-art model for various sequence labeling tasks [CITATION].', '1803.11326-3-13-1': 'BiLSTM-CRF model consists of a BiLSTM layer and a CRF layer.', '1803.11326-3-14-0': 'BiLSTM (Bidirectional-LSTM) enables the hidden states to capture both historical and future context information of the words.', '1803.11326-3-14-1': 'Mathematically, the input of this BiLSTM layer is a sequence of input vectors, denoted as [MATH].', '1803.11326-3-14-2': 'The output of BiLSTM layer is a sequence of the hidden states for each input word, denoted as [MATH].', '1803.11326-3-14-3': 'Each final hidden state is the concatenation of the forward [MATH] and backward [MATH] hidden states.', '1803.11326-3-14-4': 'We view BiLSTM as a function [MATH]: [EQUATION]', '1803.11326-3-14-5': 'Most of time we stack multiple BiLSTMs to make the model deeper, in which the output [MATH] of layer [MATH] becomes the input of layer [MATH], e.g. [MATH].', '1803.11326-3-15-0': 'It is always beneficial to consider the correlations between the current label and neighboring labels, since there are many syntactical constraints in natural language sentences.', '1803.11326-3-15-1': 'For example, I-Brand is never followed by a B-Color.', '1803.11326-3-15-2': 'If we simply feed the above mentioned hidden states independently to a softmax layer to predict the labels [CITATION], such constraints are more likely to be violated.', '1803.11326-3-15-3': 'Linear-chain Conditional Random Field (CRF) [CITATION] is the most popular way to control the structure prediction and its basic idea is to use a series of potential functions to approximate the conditional probability of the output label sequence given the input word sequence.', '1803.11326-3-16-0': 'Formally, we take the above sequence of hidden states [MATH] as input to a CRF layer, and the output of the CRF is the final prediction label sequence [MATH], where [MATH] is in the set of pre-defined target labels.', '1803.11326-3-16-1': 'We denote [MATH] as the set of all possible label sequences.', '1803.11326-3-16-2': 'Then we derive the conditional probability of the output sequence, given the input hidden state sequence is: [EQUATION] where [MATH] are potential functions and [MATH] and [MATH] are weight vector and bias of label pair [MATH].', '1803.11326-3-16-3': 'To train the CRF layer, we use the classic maximum conditional likelihood estimate and gradient ascent.', '1803.11326-3-16-4': 'For a training dataset [MATH], the final log-likelihood is: [EQUATION]', '1803.11326-3-16-5': 'Finally, the Viterbi algorithm is adopted to decode the optimal output sequence [MATH]: [EQUATION]', '1803.11326-3-17-0': '## Multi-task Learning', '1803.11326-3-18-0': 'The slot labels are large-scaled, informative and diverse in the case of E-commerce, and the syntactic structure of input Chinese utterance are complicated, so that the slot filling problem becomes hard to solve.', '1803.11326-3-18-1': 'If we directly train an end-to-end sequential model, the tagging performance will suffer from data sparsity severely.', '1803.11326-3-18-2': 'When we try to handle slot filling (can be seen as semantic labeling task), some low-level tasks such as named entity tagging or segment tagging (can be seen as syntactic labeling task) may first make mistakes.', '1803.11326-3-18-3': 'If the low-level tasks get wrong, so as to the target slot filling task.', '1803.11326-3-18-4': 'That is to say it is easy to make wrong decisions in the low-level tasks, if we try to fill in all the labels at once.', '1803.11326-3-18-5': 'Then the error will propagate and lead to a bad performance of slot filling, which is our high-level target.', '1803.11326-3-19-0': 'While directly attacking the slot filling task is hard, low-level tasks with fewer labels are much easier to solve.', '1803.11326-3-19-1': 'Once we know the syntactic structure of a sentence, filling in semantic labels will become easier accordingly.', '1803.11326-3-19-2': 'Thus, it is reasonable to solve the problem in a multi-task learning framework.', '1803.11326-3-19-3': 'In our problem, following the special structure of E-commerce knowledge base (Figure [REF]), we can devise three individual tasks: slot filling, named entity tagging and segment tagging.', '1803.11326-3-19-4': 'Slot filling is our target task; named entity tagging is to classify which named entity type (PV/PK/CG) a word is; and segment tagging is to judge whether a word is begin (B), in (I) or out (O) of a trunking.', '1803.11326-3-20-0': 'In a multi-task learning (MTL) setting, we have several prediction tasks over the same input sequence, where each task has its own output vocabulary (a set of task specified labels).', '1803.11326-3-20-1': 'Intuitively, the three tasks do share a lot of information.', '1803.11326-3-20-2': 'Consider the example in Table [REF] again.', '1803.11326-3-20-3': 'Knowing the named entity type of B-PV[MATH]I-PV can definitely help determine its slot label, which is B-Color[MATH]I-Color.', '1803.11326-3-20-4': 'Similarly, knowing its segment type (B[MATH]I) also helps with both named entity tagging and slot filling.', '1803.11326-3-20-5': 'Thus it is reasonable for these tasks to share parameters and learn in the same framework cooperatively.', '1803.11326-3-21-0': '### Vanilla Multi-task Learning', '1803.11326-3-22-0': 'The general idea of multi-task learning is to share parameters of encoding part of the network.', '1803.11326-3-22-1': 'As Figure [REF](b) shows, this is naturally achieved by sharing the [MATH]-layers BiLSTM part of the network across three tasks.', '1803.11326-3-22-2': 'Based on that, we use a separate CRF decoder for each task [MATH]: [MATH], where [MATH] and [MATH] are task-specific parameters.', '1803.11326-3-22-3': 'This encourages the deep BiLSTM network to learn a hidden representation [MATH] which benefits all three different tasks.', '1803.11326-3-23-0': '### Hierarchy Multi-task Learning', '1803.11326-3-24-0': 'Previous discussion indicates that there is a natural order among the different tasks: slot filling may benefit more from named entity tagging, than the other way around.', '1803.11326-3-24-1': 'This motivates us to employ low-level tasks at lower BiLSTM layers, while high level tasks are trained at higher layers.', '1803.11326-3-24-2': 'We borrow the idea of involving a hierarchical neural networks structure [CITATION].', '1803.11326-3-24-3': 'As shown in Figure [REF](c), instead of decoding all tasks separately at the outermost BiLSTM layer, we associate each BiLSTM layer [MATH] with one task [MATH].', '1803.11326-3-24-4': 'Then the conditional probabilities of the output sequence for each task are: [EQUATION]', '1803.11326-3-24-5': 'Here [MATH], [MATH] and [MATH] represent the tasks of segment tagging, named entity tagging and slot filling, respectively.', '1803.11326-3-24-6': '[MATH] is the word embeddings of input sequence [MATH] and [MATH].', '1803.11326-3-24-7': 'We call this model hierarchy multi-task learning, since some layers are shared by all tasks while the others are only related to specific tasks.', '1803.11326-3-25-0': '## Deep Cascade Multi-task Learning', '1803.11326-3-26-0': 'Hierarchy multi-task learning share parameters among different tasks, and allow low-level tasks help adjust the result of high-level target task.', '1803.11326-3-26-1': 'It is effective for those tasks which are weakly correlated, such as POS tagging, syntactic chunking and CCG supertagging [CITATION].', '1803.11326-3-26-2': 'However, when it comes to problems where different tasks maintain a strict order, in another word, the performance of high-level task dramatically depends on low-level tasks, the hierarchy structure is not compact and effective enough.', '1803.11326-3-26-3': 'Therefore, we propose cascade and residual connections to allow high-level tasks to take the tagging results and hidden states from low-level tasks as additional input.', '1803.11326-3-26-4': 'These connections serves as "shortcuts" that create a more closely coupled and efficient model.', '1803.11326-3-26-5': 'We call it deep cascade multi-task learning, and the framework is shown in Figure [REF](d).', '1803.11326-3-27-0': '### Cascade Connection', '1803.11326-3-28-0': 'Here we feed the tagging output of the task at lower layer e.g. [MATH] or [MATH] to the upper BiLSTM layer as its additional input.', '1803.11326-3-28-1': 'Now the hidden states of each task layer become: [EQUATION] where [MATH] and [MATH] are the weight parameters for cascade connection.', '1803.11326-3-29-0': 'At training time, [MATH] and [MATH] can be the true tagging outputs.', '1803.11326-3-29-1': 'At inference time, we simply take the greedy path of our cascade model without doing search, where the model emits the best [MATH] and [MATH] by Viterbi inference algorithm.', '1803.11326-3-29-2': 'Alternatively, one can do beam search [CITATION] by maintaining a set of [MATH] best partial hypotheses at each cascade layer.', '1803.11326-3-29-3': 'However, unlike traditional seq2seq models e.g., in machine translation, where each inference step is just based on probability of a discrete variable (by softmax function), our inference for tagging output is a structured probability distribution defined by the CRF output.', '1803.11326-3-29-4': 'Efficient beam search method for this structured cascade model is left to our future work.', '1803.11326-3-30-0': '### Residual Connection', '1803.11326-3-31-0': 'To encourage the information sharing among different tasks, we also introduce the residual connection, where we add the input of a previous layer to the current input: [EQUATION]', '1803.11326-3-31-1': 'Deep residual learning [CITATION] is introduced to ease the gradient vanish problem for training very deep neural networks.', '1803.11326-3-31-2': 'Here we borrow the idea of cross residual learning method for multi-task visual recognition [CITATION] and believe the residual connection between different layers can benefit our multi-task sequence learning.', '1803.11326-3-31-3': 'We propose cascade residual connection instead of cross residual connection because different tasks are connected via cascading in our problem, while they are organized via branching in visual recognition.', '1803.11326-3-32-0': '## Training', '1803.11326-3-33-0': 'For our multi-task setting, we define three loss functions (refer to Section [REF]): [MATH], [MATH] and [MATH] for tasks of segment tagging, named entity tagging and slot filling respectively.', '1803.11326-3-33-1': 'We construct three training set, [MATH], [MATH] and [MATH], where each of them (called [MATH] generically) contains a set of input-output sequence pair [MATH].', '1803.11326-3-33-2': 'The input utterance [MATH] is shared across tasks, but the output [MATH] is task dependent.', '1803.11326-3-34-0': 'For vanilla multi-task learning, we define a unified loss function [MATH], where [MATH] and [MATH] are hyper-parameters.', '1803.11326-3-34-1': 'And we update the model parameters by loss [MATH].', '1803.11326-3-35-0': 'As for hierarchy multi-task learning and cascade multi-task learning, we choose a random task [MATH] at each training step, followed by a random training batch [MATH].', '1803.11326-3-35-1': 'Then we update the model parameters by back-propagating the corresponding loss [MATH].', '1803.11326-3-36-0': '# Experiments', '1803.11326-3-37-0': 'In this section we first introduce the popular ATIS dataset, then describe how we collect our E-commerce Shopping Assistant (ECSA) dataset.', '1803.11326-3-37-1': 'Then we show the implementation details for our model.', '1803.11326-3-37-2': 'Finally we demonstrate the evaluation results on both ATIS and ECSA dataset and give some discussions.', '1803.11326-3-37-3': 'In the following experiments, we call our proposed Deep Cascade Multi-Task Learning method as DCMTL for short.', '1803.11326-3-38-0': '## Dataset', '1803.11326-3-39-0': '### ATIS Dataset', '1803.11326-3-40-0': 'The ATIS corpus, the most commonly used dataset for slot filling research, contains reservation requests for air travel.', '1803.11326-3-40-1': 'It contains 4,978 training and 893 testing sentences in total, with a vocabulary size of 572 [CITATION].', '1803.11326-3-40-2': 'Apart from the ground-truth slot labels, we also generate its corresponding segment labels for our multi-task model setting.', '1803.11326-3-41-0': '### ECSA Dataset', '1803.11326-3-42-0': 'To create large amounts of gold standard data to train our model, we adopt an unsupervised method to automatically tag the input utterances.', '1803.11326-3-42-1': 'All the utterances are extracted from the user input logs (either from text or voice) on our online shopping assistant system.', '1803.11326-3-42-2': 'Besides our E-commerce knowledge-base is a dictionary consisting of pairs of word terms and their ground-truth slot labels such as "red-Color" or "Nike-Brand".', '1803.11326-3-42-3': 'Since this resource is created by human beings, we will use it to create gold standard.', '1803.11326-3-42-4': 'We use a dynamic programming algorithm of max-matching to match words in the utterances and then assign each word with its slot label in IOB scheme.', '1803.11326-3-42-5': 'We filter utterances whose matching result is ambiguous and only reserve those that can be perfectly matched (all words can be tagged by only one unique label) as our training and testing data.', '1803.11326-3-42-6': 'With the slot labels of each word, we can induce the named entity labels and segment labels straightforwardly via the E-commerce knowledge-base.', '1803.11326-3-42-7': 'For we only extract the perfectly matched sentences, the quality of our ECSA dataset can be guaranteed.', '1803.11326-3-42-8': 'It can be considered as a long-distance supervision method [CITATION].', '1803.11326-3-43-0': "To evaluate model's ability to generalize, we randomly split the dictionary into three parts.", '1803.11326-3-43-1': 'One part is used to generate testing data and the other two to generate training data.', '1803.11326-3-43-2': "If we don't split the dictionary and use the whole to generate both training and testing data, then the trained model may remember the whole dictionary and the results will not reflect the true performance of the models.", '1803.11326-3-44-0': 'This unsupervised approach alleviates human annotations, and we can produce a large volume of labeled data automatically.', '1803.11326-3-44-1': 'The following experiments use a dataset of 24,892 training pairs and 2,723 testing pairs.', '1803.11326-3-44-2': 'Each pair contains an input utterance [MATH], its corresponding gold sequence of slot labels [MATH], named entity labels [MATH] and segment labels [MATH].', '1803.11326-3-44-3': 'The vocabulary size of ECSA is 1265 (Chinese characters), and the amount of segmented terms can be much larger.', '1803.11326-3-44-4': 'The Out-of-Vocabulary (OOV) rate of ESCA dataset is 85.3% (Meaning 85.3% of terms in testing data never appear in training data) while the OOV rate of ATIS is lower than 1%.', '1803.11326-3-44-5': 'Apparently slot filling task on ESCA dataset is more challenging.', '1803.11326-3-45-0': '## Implementation Details', '1803.11326-3-46-0': 'For the RNN component in our system, we use a 3-layers BiLSTM networks for ECSA and 2-layers BiLSTM networks for ATIS (no named entity tagging in this case), and all LSTM networks come with hidden state size 100.', '1803.11326-3-46-1': 'The input in ECSA is a sequence of Chinese characters rather that words since there is no segmentation.', '1803.11326-3-46-2': 'The dimension of embedding layer [MATH] and BiLSTM network output state (concatenation of the forward and backward LSTM) are set to 200.', '1803.11326-3-46-3': 'We perform a mini-batch log-likelihood loss training with a batch size of 32 sentences for 10 training epochs.', '1803.11326-3-46-4': 'We use Adam optimizer, and the learning rate is initialized to 0.001.', '1803.11326-3-46-5': 'To prevent the gradient explosion problem for training LSTM networks, we set gradient clip-norm as 5.', '1803.11326-3-47-0': '## Results and Discussions', '1803.11326-3-48-0': '### Evaluation on ATIS', '1803.11326-3-49-0': 'We compare the ATIS results of our DCMTL model with current published results in Table [REF].', '1803.11326-3-49-1': 'We split the methods into two categories: one is Sequence Labeling based method, and the other is Encoder-Decoder based method.', '1803.11326-3-49-2': 'Sequence Labeling based method generally adopts a sequential network (RNN [CITATION] or CNN [CITATION]) and calculate a loss function (such as CRF loss [CITATION], cross entropy loss [CITATION] or ranking loss [CITATION]) on top of the network output.', '1803.11326-3-49-3': 'Encoder-Decoder based method, on the other hand, usually employs a RNN to encode the whole sentence and another RNN to decode the labels [CITATION].', '1803.11326-3-49-4': 'The decoder will attend to the whole encoding sequence with attention mechanism [CITATION].', '1803.11326-3-49-5': 'Our method follows the Sequence Labeling framework and we design a novel multi-task sequence labeling model which achieve the best performance against the published Sequence Labeling based method (F1+0.22) and compatible result against the best Encoder-Decoder based method (F1-0.03).', '1803.11326-3-49-6': 'As we claim in Section [REF], more than 97 of chunks in ATIS dataset have only one or two words and there are no named entity labels at all.', '1803.11326-3-49-7': 'These two reasons prevent our proposed DCMTL model from further improving the performance on ATIS dataset.', '1803.11326-3-49-8': 'Thus, we will mainly focus on ECSA dataset, which is much larger and more sophisticated, to prove the effectiveness of our proposed model.', '1803.11326-3-50-0': 'Besides, almost all the methods (including ours) reach very high F1 score of around 0.96.', '1803.11326-3-50-1': 'This also makes us wonder whether it is meaningful enough to continue evaluating on this dataset, for minor differences in the results may be attributed to data variance more than the models.', '1803.11326-3-50-2': 'Apparently high performance on ATIS does not mean working on real-world application which contains more informative semantic slot labels and more complicated expressions as in the case of online shopping assistant.', '1803.11326-3-51-0': '### Evaluation on ECSA', '1803.11326-3-52-0': 'On ECSA dataset, we evaluate different models including Basic BiLSTM-CRF, Vanilla Multi-task, Hierarchy Multi-task and Deep Cascade Multi-task on testing data regarding slot filling as the target task.', '1803.11326-3-52-1': 'We report Precision, Recall and F1 in Table [REF].', '1803.11326-3-53-0': 'The Basic BiLSTM-CRF model achieves an F1 score of 0.43.', '1803.11326-3-53-1': 'To show the usefulness of the lower tasks to slot filling, we "cheated" by using the ground-truth segment type (cond.', '1803.11326-3-53-2': 'SEG) or named entity type (cond.', '1803.11326-3-53-3': 'NE) as the extra features for each word in the Basic BiLSTM-CRF model.', '1803.11326-3-53-4': 'Row 3 and 4 (with *) in Table [REF] show that the slot filling performance can be improved by 85% and 109% if the correct segment type or named entity type is pre-known.', '1803.11326-3-53-5': 'It can perfectly verify our claim that low-level syntactic tasks can significantly affect to the slot filling performance.', '1803.11326-3-53-6': "Of course in practice, the model doesn't know the true values of these types during prediction.", '1803.11326-3-54-0': 'Our further experiments show that DCMTL outperforms the baselines on both precision and recall.', '1803.11326-3-54-1': 'DCMTL achieves the best F1 score of 0.5105, which improves by a relative margin of 14.6% against the strong baseline method (see Table [REF]).', '1803.11326-3-54-2': 'Multi-task models generally perform better than the Basic BiLSTM with single-task target.', '1803.11326-3-54-3': 'The exception is the vanilla multi-task setting.', '1803.11326-3-54-4': 'This is mainly because vanilla multi-task shares parameters across all the layers, and these parameters are likely to be disturbed by the interaction of three tasks.', '1803.11326-3-54-5': 'It is more desirable to let the target task dominate the weights at high-level layers.', '1803.11326-3-55-0': 'We further investigate the learning trend of our proposed approach against baseline methods.', '1803.11326-3-55-1': 'Figure [REF](a) shows the typical learning curves of performance measured by F1.', '1803.11326-3-55-2': 'We can observe that our method DCMTL performs worse than other baseline methods for the first [MATH] batch steps.', '1803.11326-3-55-3': 'After that, other methods converge quickly and DCMTL perform much better after [MATH] batch steps and finally converge to the best F1 score.', '1803.11326-3-55-4': 'We believe that in the beginning, high-level task in DCMTL is affected more by the noise of low-level tasks comparing to others, but as the training goes on, the high-level slot filling task slowly reaps the benefits from low-level tasks.', '1803.11326-3-56-0': 'To make our experiments more solid, we implemented two previous best performing models on ATIS dataset: Sequential CNN [CITATION] (Sequence Labeling based) and Neural Sequence Chunking [CITATION] (Encoder-Decoder based).', '1803.11326-3-56-1': 'They achieved 0.2877 and 0.4355 F1 scores respectively, while our DCMTL model scores 0.5105 F1 and outperforms both of them (by 77% and 17% improvements).', '1803.11326-3-57-0': '### Ablation Test', '1803.11326-3-58-0': 'Our "shortcuts" connections come in two flavors: cascade connection and residual connection.', '1803.11326-3-58-1': 'Multi-task outputs and "shortcuts" connections are highly related since without the multi-task framework, there will be no cascade connections.', '1803.11326-3-58-2': 'We go on to show that both multi-task setting and the "shortcuts" connections are effective and useful in Table [REF], where F1 score improves from 0.4302 to 0.4455 and 0.5105 respectively.', '1803.11326-3-58-3': 'We also investigate how our model DCMTL performs with or without cascade and residual connections (rows with ** prefix in Table [REF]).', '1803.11326-3-58-4': 'F1 score increases from 0.4840 to 0.5105 when residual connection is applied, which verifies its benefit.', '1803.11326-3-58-5': 'If we remove cascade connection from DCMTL, the model actually degenerates into hierarchy multi-task model with residual connection and performs 0.4633 F1 score.', '1803.11326-3-58-6': 'Thus we can conclude that both connections are helpful for our DCMTL model.', '1803.11326-3-58-7': 'However, the cascade connection, which relies on the multi-task, is more effective than the residual connection.', '1803.11326-3-58-8': 'We can verify it from the fact that DCMTL model without cascade connection performs much worse than without residual connection (0.4633 vs. 0.4840 F1 scores).', '1803.11326-3-59-0': 'Furthermore, we explore how DCMTL performs with different cascade connection methods.', '1803.11326-3-59-1': 'We compare three different types of cascade connection illustrated in Figure [REF](a):', '1803.11326-3-60-0': '[1.]', '1803.11326-3-60-1': 'Segment labeling skipped to slot filling (SLOT+SEG).', '1803.11326-3-60-2': '[2.]', '1803.11326-3-60-3': 'Named entity labeling directly connected to slot filling (SLOT+NE).', '1803.11326-3-60-4': '[3.]', '1803.11326-3-60-5': 'Segment labeling, named entity labeling and slot filling in sequence (SLOT+NE+SEG).', '1803.11326-3-61-0': 'From Figure [REF](b), we find that cascade connection with type 3 performs the best and then with type 2, while cascade method with skipped connection (type 1) performs the worst.', '1803.11326-3-61-1': 'Therefore, we design the networks with a cascade connection in a hierarchical fashion and do not apply skipped connection for the cascade inputs (Figure [REF](b)).', '1803.11326-3-61-2': 'This phenomenon here may also be proved by our "cheated" case study above.', '1803.11326-3-61-3': 'Slot filling performance with pre-known named entity type is much better than with pre-known segment type (rows with * in Table [REF]).', '1803.11326-3-62-0': '## Online Testing', '1803.11326-3-63-0': 'Previous experimental results have proven the advantages of our proposed DCMTL approach, so we deploy it in a real world online environment to test its practical performance.', '1803.11326-3-64-0': 'For online A/B testing, we extracted users query log with the slot filling results for one day.', '1803.11326-3-64-1': 'There are in total 251,409 unique queries.', '1803.11326-3-64-2': 'We let three persons to manually evaluate whether a query is slotted perfectly with the strategy where the minority obeys the majority.', '1803.11326-3-64-3': 'A query is slotted perfectly means all terms in query are assigned with the correct slot labels.', '1803.11326-3-64-4': 'Our DCMTL model results in 152,178 perfectly slotted queries which is 60.53 accuracy.', '1803.11326-3-64-5': 'While the original online max-matching algorithm with E-commerce knowledge base (more details in Section [REF]) only covers 66,302 perfectly slotted queries with 26.37 accuracy.', '1803.11326-3-64-6': 'Thus, the accuracy of query slot filling in such online shopping assistant system is improved by 130 after deploying DCMTL model.', '1803.11326-3-64-7': 'This demonstrates that our model can effectively extract the semantic attributes of users query which is extremely helpful E-commerce Shopping Assistant system.', '1803.11326-3-65-0': '# Related Work', '1803.11326-3-66-0': 'There are mainly two lines of research that are related to our work: slot filling for dialog system and multi-task learning in natural language processing.', '1803.11326-3-67-0': 'Slot Filling is considered a sequence labeling problem that is traditionally solved by generative models.', '1803.11326-3-67-1': 'such as Hidden Markov Models (HMMs) [CITATION], hidden vector state model [CITATION], and discriminative models such as conditional random fields (CRFs) [CITATION] and Support Vector Machine (SVMs) [CITATION].', '1803.11326-3-67-2': 'In recent years, deep learning approaches have been explored due to its successful application in many NLP tasks.', '1803.11326-3-67-3': 'Many neural network architectures have been used such as simple RNNs [CITATION], convolutional neural networks (CNNs) [CITATION], LSTMs [CITATION] and variations like encoder-decoder [CITATION] and external memory [CITATION].', '1803.11326-3-67-4': 'In general, these works adopt a BiLSTM as the major labeling architecture to extract various features, then use a CRF layer [CITATION] to model the label dependency.', '1803.11326-3-67-5': 'We also adopt a BiLSTM-CRF model as baseline and claim that a multi-task learning framework is working better than directly applying it on Chinese E-commerce dataset.', '1803.11326-3-67-6': 'Previous works only apply joint model of slot filling and intent detection [CITATION].', '1803.11326-3-67-7': 'Our work is the first to propose a multi-task sequence labeling model with novel cascade and residual connections based on deep neural networks to tackle real-world slot filling problem.', '1803.11326-3-68-0': 'Multi-task Learning (MTL) has attracted increasing attention in both academia and industry recently.', '1803.11326-3-68-1': 'By jointly learning across multiple tasks [CITATION], we can improve performance on each task and reduce the need for labeled data.', '1803.11326-3-68-2': 'There has been several attempts of using multi-task learning on sequence labeling task [CITATION], where most of these works learn all tasks at the out-most layer.', '1803.11326-3-68-3': 'Søgaard and Goldberg [CITATION] is the first to assume the existence of a hierarchy between the different tasks in a stacking BiRNN model.', '1803.11326-3-68-4': 'Compared to these works, our DCMTL model further improves this idea even thorough with cascade and residual connection.', '1803.11326-3-69-0': '# Conclusion', '1803.11326-3-70-0': 'In this paper, we tackle the real-world slot filling task in a novel Chinese online shopping assistant system.', '1803.11326-3-70-1': 'We proposed a deep multi-task sequence learning framework with cascade and residual connection.', '1803.11326-3-70-2': 'Our model achieves comparable results with several state-of-the-art models on the common slot filling dataset ATIS.', '1803.11326-3-70-3': 'On our real-world Chinese E-commerce dataset ECSA, our proposed model DCMTL also achieves best F1 score comparing to several strong baselines.', '1803.11326-3-70-4': 'DCMTL has been deployed on the online shopping assistant of a dominant Chinese E-commerce platform.', '1803.11326-3-70-5': 'Online testing results show that our model meets better understanding of users utterances and improves customer’s shopping experience.', '1803.11326-3-70-6': 'Our future research may include a joint model for category classification and slot filling.', '1803.11326-3-70-7': 'Active learning for slot filling can also be investigated by involving human-beings interaction with our system.'}

{'1803.11326-4-0-0': 'Slot filling is a critical task in natural language understanding (NLU) for dialog systems.', '1803.11326-4-0-1': 'State-of-the-art approaches treat it as a sequence labeling problem and adopt such models as BiLSTM-CRF.', '1803.11326-4-0-2': 'While these models work relatively well on standard benchmark datasets, they face challenges in the context of E-commerce where the slot labels are more informative and carry richer expressions.', '1803.11326-4-0-3': 'In this work, inspired by the unique structure of E-commerce knowledge base, we propose a novel multi-task model with cascade and residual connections, which jointly learns segment tagging, named entity tagging and slot filling.', '1803.11326-4-0-4': 'Experiments show the effectiveness of the proposed cascade and residual structures.', '1803.11326-4-0-5': 'Our model has a 14.6 advantage in F1 score over the strong baseline methods on a new Chinese E-commerce shopping assistant dataset, while achieving competitive accuracies on a standard dataset.', '1803.11326-4-0-6': 'Furthermore, online test deployed on such dominant E-commerce platform shows 130 improvement on accuracy of understanding user utterances.', '1803.11326-4-0-7': 'Our model has already gone into production in the E-commerce platform.', '1803.11326-4-1-0': '# Introduction', '1803.11326-4-2-0': 'An intelligent online shopping assistant offers services such as pre-sale and after-sale inquiries, product recommendations, and user complaints processing, all of which seek to give the customers better shopping experience.', '1803.11326-4-2-1': 'The core of such assistant is a task-oriented dialog system which has the ability to understand natural language utterances from a user and then give natural language responses [CITATION].', '1803.11326-4-2-2': 'Natural Language Understanding (NLU), which aims to interpret the semantic meanings conveyed by input utterances, is a main component in task-oriented dialog systems.', '1803.11326-4-2-3': 'Slot filling, a sub-problem of NLU, extracts semantic constituents by using the words of input utterance to fill in pre-defined slots in a semantic frame [CITATION].', '1803.11326-4-3-0': 'In the case of E-commerce shopping, there are three named entity types: Category, Property Key and Property Value, according to typical E-commerce knowledge base such as the one in Figure [REF].', '1803.11326-4-3-1': 'We show a real example in Table [REF] with In/Out/Begin (I/O/B) scheme.', '1803.11326-4-3-2': 'In the named entity level, "dress" is a Category (CG), while "brand" is labeled as Property Key (PK), which is the name of one product property.', '1803.11326-4-3-3': '"Nike" and "black" are labeled as Property Value (PV) since they are concrete property values.', '1803.11326-4-3-4': 'However, merely labeling as Property Value is not sufficient as the shopping assistant needs more fine-grained semantics.', '1803.11326-4-3-5': 'Therefore, in the Slot Filling level, we further label "Nike" as Brand Property (Brand), and "black" as Color Property (Color).', '1803.11326-4-3-6': 'In Table [REF], B-CG refers to Begin-Category (the meaning of other labels can also be inferred).', '1803.11326-4-3-7': 'In the meantime, other words in the example utterance that carry no semantic meaning are assigned O label.', '1803.11326-4-4-0': 'Traditionally, slot filling problem can be regarded as a sequence labeling task, which assigns an appropriate semantic label to each word in the given input utterance.', '1803.11326-4-4-1': 'State-of-the-art sequence labeling models are typically based on BiLSTM-CRF [CITATION] and evaluated on a commonly used standard dataset ATIS [CITATION] in the slot filling area.', '1803.11326-4-4-2': 'This dataset is about airline travel in the United States.', '1803.11326-4-4-3': 'However, the vocabulary size of ATIS is small (only 572) and slot labels are not diverse enough (mostly related to only time and location) since airline travel is a relatively small and specific domain, such that recent deep learning models can achieve very high F1 scores (nearly 0.96).', '1803.11326-4-4-4': 'Recently, a detailed quantitative and qualitative study of this dataset comes to the same conclusion that slot filling models should be tested on a much more real and complex dataset [CITATION].', '1803.11326-4-5-0': 'In this paper, we try to tackle a real-world slot filling problem for one of the largest E-commerce platform in China.', '1803.11326-4-5-1': 'The semantic slots are much more diverse and informative than ATIS.', '1803.11326-4-5-2': 'For example, to describe different properties of a product for the purpose of utterance understanding, we define large amount of informative slot labels such as color, brand, style, season, gender and so on.', '1803.11326-4-5-3': 'In contrast, most semantic labels of ATIS are related to only time and location.', '1803.11326-4-5-4': 'Furthermore, the Chinese language used for e-commerce is more complex and the semantically rich expressions make it harder to understand.', '1803.11326-4-5-5': 'Whereas in ATIS, expression can be simpler, and most expressions are standard locations or time.', '1803.11326-4-5-6': 'Thus, large scale semantic slots and more complex expressions bring problem such as data sparsity.', '1803.11326-4-5-7': 'Traditional end-to-end sequence labeling model may not be able to handle it.', '1803.11326-4-6-0': 'Besides, Chinese language, like many other Asian languages, is not word segmented by nature, and word segmentation is a difficult first step in many NLP tasks.', '1803.11326-4-6-1': 'Without proper word segmentation, sequence labeling becomes very challenging as the errors from segmentation will propagate.', '1803.11326-4-6-2': 'On the other hand, more than 97% of the chunks in ATIS data have only one or two words, in which segmentation (or chunking) is not a serious problem.', '1803.11326-4-6-3': 'Due to these reasons, if we simply apply basic sequence labeling models, which can be regarded as an end-to-end method, the sentences may not be segmented correctly in the first place.', '1803.11326-4-6-4': 'Then the errors will propagate and the resulting slot labels will be incorrect.', '1803.11326-4-7-0': 'In this paper, we propose to employ multi-task sequence labeling model to tackle slot filling in a novel Chinese E-commerce dialog system.', '1803.11326-4-7-1': 'Inspired by the natural structure of E-commerce knowledge base shown in Figure [REF], we extract two additional lower-level tasks from the slot filling task: named entity tagging and segment tagging.', '1803.11326-4-7-2': 'Example labels of these two tasks are shown in the bottom two rows of Table [REF].', '1803.11326-4-7-3': 'Segment tagging and named entity tagging can be regarded as syntactic labeling, while slot filling is more like semantic labeling.', '1803.11326-4-7-4': 'With the help of information sharing ability of multi-task learning, once we learn the information of syntactic structure of an input sentence, filling the semantic labels becomes much easier.', '1803.11326-4-7-5': 'Compared to directly attacking slot filling, these two low-level tasks are much easier to solve due to fewer labels.', '1803.11326-4-7-6': 'To this end, we propose a Deep Cascade Multi-task Learning model, and co-train three tasks in the same framework with a goal of optimizing the target slot filling task.', '1803.11326-4-8-0': 'The contributions of this paper are summarized below:', '1803.11326-4-9-0': '# Approach', '1803.11326-4-10-0': 'In this section we describe our approach in detail.', '1803.11326-4-10-1': 'Figure [REF] gives an overview of the proposed architectures.', '1803.11326-4-10-2': 'First we introduce the most common and popular BiLSTM-CRF model (Figure [REF](a)) for sequence labeling tasks.', '1803.11326-4-10-3': 'Then we move on to multi-task learning perspective (Figure [REF](b) and (c)).', '1803.11326-4-10-4': 'Finally we propose our new method, which is called Deep Cascade Multi-task Learning in Figure [REF](d).', '1803.11326-4-11-0': 'Given an utterance containing a sequence of words [MATH], the goal of our problem is to find a sequence of slot labels [MATH], one for each word in the utterance, such that: [EQUATION]', '1803.11326-4-11-1': 'We use "word" in problem and model description, but "word" actually means Chinese char in our problem.', '1803.11326-4-11-2': 'And a "term" consists of one or several words.', '1803.11326-4-12-0': '## RNN Sequence Labeling', '1803.11326-4-13-0': 'Figure [REF](a) shows the principle architecture of a BiLSTM-CRF model, which is the state-of-the-art model for various sequence labeling tasks [CITATION].', '1803.11326-4-13-1': 'BiLSTM-CRF model consists of a BiLSTM layer and a CRF layer.', '1803.11326-4-14-0': 'BiLSTM (Bidirectional-LSTM) enables the hidden states to capture both historical and future context information of the words.', '1803.11326-4-14-1': 'Mathematically, the input of this BiLSTM layer is a sequence of input vectors, denoted as [MATH].', '1803.11326-4-14-2': 'The output of BiLSTM layer is a sequence of the hidden states for each input word, denoted as [MATH].', '1803.11326-4-14-3': 'Each final hidden state is the concatenation of the forward [MATH] and backward [MATH] hidden states.', '1803.11326-4-14-4': 'We view BiLSTM as a function [MATH]: [EQUATION]', '1803.11326-4-14-5': 'Most of time we stack multiple BiLSTMs to make the model deeper, in which the output [MATH] of layer [MATH] becomes the input of layer [MATH], e.g. [MATH].', '1803.11326-4-15-0': 'It is always beneficial to consider the correlations between the current label and neighboring labels, since there are many syntactical constraints in natural language sentences.', '1803.11326-4-15-1': 'For example, I-Brand is never followed by a B-Color.', '1803.11326-4-15-2': 'If we simply feed the above mentioned hidden states independently to a softmax layer to predict the labels [CITATION], such constraints are more likely to be violated.', '1803.11326-4-15-3': 'Linear-chain Conditional Random Field (CRF) [CITATION] is the most popular way to control the structure prediction and its basic idea is to use a series of potential functions to approximate the conditional probability of the output label sequence given the input word sequence.', '1803.11326-4-16-0': 'Formally, we take the above sequence of hidden states [MATH] as input to a CRF layer, and the output of the CRF is the final prediction label sequence [MATH], where [MATH] is in the set of pre-defined target labels.', '1803.11326-4-16-1': 'We denote [MATH] as the set of all possible label sequences.', '1803.11326-4-16-2': 'Then we derive the conditional probability of the output sequence, given the input hidden state sequence is: [EQUATION] where [MATH] are potential functions and [MATH] and [MATH] are weight vector and bias of label pair [MATH].', '1803.11326-4-16-3': 'To train the CRF layer, we use the classic maximum conditional likelihood estimate and gradient ascent.', '1803.11326-4-16-4': 'For a training dataset [MATH], the final log-likelihood is: [EQUATION]', '1803.11326-4-16-5': 'Finally, the Viterbi algorithm is adopted to decode the optimal output sequence [MATH]: [EQUATION]', '1803.11326-4-17-0': '## Multi-task Learning', '1803.11326-4-18-0': 'The slot labels are large-scaled, informative and diverse in the case of E-commerce, and the syntactic structure of input Chinese utterance are complicated, so that the slot filling problem becomes hard to solve.', '1803.11326-4-18-1': 'If we directly train an end-to-end sequential model, the tagging performance will suffer from data sparsity severely.', '1803.11326-4-18-2': 'When we try to handle slot filling (can be seen as semantic labeling task), some low-level tasks such as named entity tagging or segment tagging (can be seen as syntactic labeling task) may first make mistakes.', '1803.11326-4-18-3': 'If the low-level tasks get wrong, so as to the target slot filling task.', '1803.11326-4-18-4': 'That is to say it is easy to make wrong decisions in the low-level tasks, if we try to fill in all the labels at once.', '1803.11326-4-18-5': 'Then the error will propagate and lead to a bad performance of slot filling, which is our high-level target.', '1803.11326-4-19-0': 'While directly attacking the slot filling task is hard, low-level tasks with fewer labels are much easier to solve.', '1803.11326-4-19-1': 'Once we know the syntactic structure of a sentence, filling in semantic labels will become easier accordingly.', '1803.11326-4-19-2': 'Thus, it is reasonable to solve the problem in a multi-task learning framework.', '1803.11326-4-19-3': 'In our problem, following the special structure of E-commerce knowledge base (Figure [REF]), we can devise three individual tasks: slot filling, named entity tagging and segment tagging.', '1803.11326-4-19-4': 'Slot filling is our target task; named entity tagging is to classify which named entity type (PV/PK/CG) a word is; and segment tagging is to judge whether a word is begin (B), in (I) or out (O) of a trunking.', '1803.11326-4-20-0': 'In a multi-task learning (MTL) setting, we have several prediction tasks over the same input sequence, where each task has its own output vocabulary (a set of task specified labels).', '1803.11326-4-20-1': 'Intuitively, the three tasks do share a lot of information.', '1803.11326-4-20-2': 'Consider the example in Table [REF] again.', '1803.11326-4-20-3': 'Knowing the named entity type of B-PV[MATH]I-PV can definitely help determine its slot label, which is B-Color[MATH]I-Color.', '1803.11326-4-20-4': 'Similarly, knowing its segment type (B[MATH]I) also helps with both named entity tagging and slot filling.', '1803.11326-4-20-5': 'Thus it is reasonable for these tasks to share parameters and learn in the same framework cooperatively.', '1803.11326-4-21-0': '### Vanilla Multi-task Learning', '1803.11326-4-22-0': 'The general idea of multi-task learning is to share parameters of encoding part of the network.', '1803.11326-4-22-1': 'As Figure [REF](b) shows, this is naturally achieved by sharing the [MATH]-layers BiLSTM part of the network across three tasks.', '1803.11326-4-22-2': 'Based on that, we use a separate CRF decoder for each task [MATH]: [MATH], where [MATH] and [MATH] are task-specific parameters.', '1803.11326-4-22-3': 'This encourages the deep BiLSTM network to learn a hidden representation [MATH] which benefits all three different tasks.', '1803.11326-4-23-0': '### Hierarchy Multi-task Learning', '1803.11326-4-24-0': 'Previous discussion indicates that there is a natural order among the different tasks: slot filling may benefit more from named entity tagging, than the other way around.', '1803.11326-4-24-1': 'This motivates us to employ low-level tasks at lower BiLSTM layers, while high level tasks are trained at higher layers.', '1803.11326-4-24-2': 'We borrow the idea of involving a hierarchical neural networks structure [CITATION].', '1803.11326-4-24-3': 'As shown in Figure [REF](c), instead of decoding all tasks separately at the outermost BiLSTM layer, we associate each BiLSTM layer [MATH] with one task [MATH].', '1803.11326-4-24-4': 'Then the conditional probabilities of the output sequence for each task are: [EQUATION]', '1803.11326-4-24-5': 'Here [MATH], [MATH] and [MATH] represent the tasks of segment tagging, named entity tagging and slot filling, respectively.', '1803.11326-4-24-6': '[MATH] is the word embeddings of input sequence [MATH] and [MATH].', '1803.11326-4-24-7': 'We call this model hierarchy multi-task learning, since some layers are shared by all tasks while the others are only related to specific tasks.', '1803.11326-4-25-0': '## Deep Cascade Multi-task Learning', '1803.11326-4-26-0': 'Hierarchy multi-task learning share parameters among different tasks, and allow low-level tasks help adjust the result of high-level target task.', '1803.11326-4-26-1': 'It is effective for those tasks which are weakly correlated, such as POS tagging, syntactic chunking and CCG supertagging [CITATION].', '1803.11326-4-26-2': 'However, when it comes to problems where different tasks maintain a strict order, in another word, the performance of high-level task dramatically depends on low-level tasks, the hierarchy structure is not compact and effective enough.', '1803.11326-4-26-3': 'Therefore, we propose cascade and residual connections to allow high-level tasks to take the tagging results and hidden states from low-level tasks as additional input.', '1803.11326-4-26-4': 'These connections serves as "shortcuts" that create a more closely coupled and efficient model.', '1803.11326-4-26-5': 'We call it deep cascade multi-task learning, and the framework is shown in Figure [REF](d).', '1803.11326-4-27-0': '### Cascade Connection', '1803.11326-4-28-0': 'Here we feed the tagging output of the task at lower layer e.g. [MATH] or [MATH] to the upper BiLSTM layer as its additional input.', '1803.11326-4-28-1': 'Now the hidden states of each task layer become: [EQUATION] where [MATH] and [MATH] are the weight parameters for cascade connection.', '1803.11326-4-29-0': 'At training time, [MATH] and [MATH] can be the true tagging outputs.', '1803.11326-4-29-1': 'At inference time, we simply take the greedy path of our cascade model without doing search, where the model emits the best [MATH] and [MATH] by Viterbi inference algorithm.', '1803.11326-4-29-2': 'Alternatively, one can do beam search [CITATION] by maintaining a set of [MATH] best partial hypotheses at each cascade layer.', '1803.11326-4-29-3': 'However, unlike traditional seq2seq models e.g., in machine translation, where each inference step is just based on probability of a discrete variable (by softmax function), our inference for tagging output is a structured probability distribution defined by the CRF output.', '1803.11326-4-29-4': 'Efficient beam search method for this structured cascade model is left to our future work.', '1803.11326-4-30-0': '### Residual Connection', '1803.11326-4-31-0': 'To encourage the information sharing among different tasks, we also introduce the residual connection, where we add the input of a previous layer to the current input: [EQUATION]', '1803.11326-4-31-1': 'Deep residual learning [CITATION] is introduced to ease the gradient vanish problem for training very deep neural networks.', '1803.11326-4-31-2': 'Here we borrow the idea of cross residual learning method for multi-task visual recognition [CITATION] and believe the residual connection between different layers can benefit our multi-task sequence learning.', '1803.11326-4-31-3': 'We propose cascade residual connection instead of cross residual connection because different tasks are connected via cascading in our problem, while they are organized via branching in visual recognition.', '1803.11326-4-32-0': '## Training', '1803.11326-4-33-0': 'For our multi-task setting, we define three loss functions (refer to Section [REF]): [MATH], [MATH] and [MATH] for tasks of segment tagging, named entity tagging and slot filling respectively.', '1803.11326-4-33-1': 'We construct three training set, [MATH], [MATH] and [MATH], where each of them (called [MATH] generically) contains a set of input-output sequence pair [MATH].', '1803.11326-4-33-2': 'The input utterance [MATH] is shared across tasks, but the output [MATH] is task dependent.', '1803.11326-4-34-0': 'For vanilla multi-task learning, we define a unified loss function [MATH], where [MATH] and [MATH] are hyper-parameters.', '1803.11326-4-34-1': 'And we update the model parameters by loss [MATH].', '1803.11326-4-35-0': 'As for hierarchy multi-task learning and cascade multi-task learning, we choose a random task [MATH] at each training step, followed by a random training batch [MATH].', '1803.11326-4-35-1': 'Then we update the model parameters by back-propagating the corresponding loss [MATH].', '1803.11326-4-36-0': '# Experiments', '1803.11326-4-37-0': 'In this section we first introduce the popular ATIS dataset, then describe how we collect our E-commerce Shopping Assistant (ECSA) dataset.', '1803.11326-4-37-1': 'Then we show the implementation details for our model.', '1803.11326-4-37-2': 'Finally we demonstrate the evaluation results on both ATIS and ECSA dataset and give some discussions.', '1803.11326-4-37-3': 'In the following experiments, we call our proposed Deep Cascade Multi-Task Learning method as DCMTL for short.', '1803.11326-4-38-0': '## Dataset', '1803.11326-4-39-0': '### ATIS Dataset', '1803.11326-4-40-0': 'The ATIS corpus, the most commonly used dataset for slot filling research, contains reservation requests for air travel.', '1803.11326-4-40-1': 'It contains 4,978 training and 893 testing sentences in total, with a vocabulary size of 572 [CITATION].', '1803.11326-4-40-2': 'Apart from the ground-truth slot labels, we also generate its corresponding segment labels for our multi-task model setting.', '1803.11326-4-41-0': '### ECSA Dataset', '1803.11326-4-42-0': 'To create large amounts of gold standard data to train our model, we adopt an unsupervised method to automatically tag the input utterances.', '1803.11326-4-42-1': 'All the utterances are extracted from the user input logs (either from text or voice) on our online shopping assistant system.', '1803.11326-4-42-2': 'Besides our E-commerce knowledge-base is a dictionary consisting of pairs of word terms and their ground-truth slot labels such as "red-Color" or "Nike-Brand".', '1803.11326-4-42-3': 'Since this resource is created by human beings, we will use it to create gold standard.', '1803.11326-4-42-4': 'We use a dynamic programming algorithm of max-matching to match words in the utterances and then assign each word with its slot label in IOB scheme.', '1803.11326-4-42-5': 'We filter utterances whose matching result is ambiguous and only reserve those that can be perfectly matched (all words can be tagged by only one unique label) as our training and testing data.', '1803.11326-4-42-6': 'With the slot labels of each word, we can induce the named entity labels and segment labels straightforwardly via the E-commerce knowledge-base.', '1803.11326-4-42-7': 'For we only extract the perfectly matched sentences, the quality of our ECSA dataset can be guaranteed.', '1803.11326-4-42-8': 'It can be considered as a long-distance supervision method [CITATION].', '1803.11326-4-43-0': "To evaluate model's ability to generalize, we randomly split the dictionary into three parts.", '1803.11326-4-43-1': 'One part is used to generate testing data and the other two to generate training data.', '1803.11326-4-43-2': "If we don't split the dictionary and use the whole to generate both training and testing data, then the trained model may remember the whole dictionary and the results will not reflect the true performance of the models.", '1803.11326-4-44-0': 'This unsupervised approach alleviates human annotations, and we can produce a large volume of labeled data automatically.', '1803.11326-4-44-1': 'The following experiments use a dataset of 24,892 training pairs and 2,723 testing pairs.', '1803.11326-4-44-2': 'Each pair contains an input utterance [MATH], its corresponding gold sequence of slot labels [MATH], named entity labels [MATH] and segment labels [MATH].', '1803.11326-4-44-3': 'The vocabulary size of ECSA is 1265 (Chinese characters), and the amount of segmented terms can be much larger.', '1803.11326-4-44-4': 'The Out-of-Vocabulary (OOV) rate of ESCA dataset is 85.3% (Meaning 85.3% of terms in testing data never appear in training data) while the OOV rate of ATIS is lower than 1%.', '1803.11326-4-44-5': 'Apparently slot filling task on ESCA dataset is more challenging.', '1803.11326-4-45-0': '## Implementation Details', '1803.11326-4-46-0': 'For the RNN component in our system, we use a 3-layers BiLSTM networks for ECSA and 2-layers BiLSTM networks for ATIS (no named entity tagging in this case), and all LSTM networks come with hidden state size 100.', '1803.11326-4-46-1': 'The input in ECSA is a sequence of Chinese characters rather that words since there is no segmentation.', '1803.11326-4-46-2': 'The dimension of embedding layer [MATH] and BiLSTM network output state (concatenation of the forward and backward LSTM) are set to 200.', '1803.11326-4-46-3': 'We perform a mini-batch log-likelihood loss training with a batch size of 32 sentences for 10 training epochs.', '1803.11326-4-46-4': 'We use Adam optimizer, and the learning rate is initialized to 0.001.', '1803.11326-4-46-5': 'To prevent the gradient explosion problem for training LSTM networks, we set gradient clip-norm as 5.', '1803.11326-4-47-0': '## Results and Discussions', '1803.11326-4-48-0': '### Evaluation on ATIS', '1803.11326-4-49-0': 'We compare the ATIS results of our DCMTL model with current published results in Table [REF].', '1803.11326-4-49-1': 'We split the methods into two categories: one is Sequence Labeling based method, and the other is Encoder-Decoder based method.', '1803.11326-4-49-2': 'Sequence Labeling based method generally adopts a sequential network (RNN [CITATION] or CNN [CITATION]) and calculate a loss function (such as CRF loss [CITATION], cross entropy loss [CITATION] or ranking loss [CITATION]) on top of the network output.', '1803.11326-4-49-3': 'Encoder-Decoder based method, on the other hand, usually employs a RNN to encode the whole sentence and another RNN to decode the labels [CITATION].', '1803.11326-4-49-4': 'The decoder will attend to the whole encoding sequence with attention mechanism [CITATION].', '1803.11326-4-49-5': 'Our method follows the Sequence Labeling framework and we design a novel multi-task sequence labeling model which achieve the best performance against the published Sequence Labeling based method (F1+0.22) and compatible result against the best Encoder-Decoder based method (F1-0.03).', '1803.11326-4-49-6': 'As we claim in Section [REF], more than 97 of chunks in ATIS dataset have only one or two words and there are no named entity labels at all.', '1803.11326-4-49-7': 'These two reasons prevent our proposed DCMTL model from further improving the performance on ATIS dataset.', '1803.11326-4-49-8': 'Thus, we will mainly focus on ECSA dataset, which is much larger and more sophisticated, to prove the effectiveness of our proposed model.', '1803.11326-4-50-0': 'Besides, almost all the methods (including ours) reach very high F1 score of around 0.96.', '1803.11326-4-50-1': 'This also makes us wonder whether it is meaningful enough to continue evaluating on this dataset, for minor differences in the results may be attributed to data variance more than the models.', '1803.11326-4-50-2': 'Apparently high performance on ATIS does not mean working on real-world application which contains more informative semantic slot labels and more complicated expressions as in the case of online shopping assistant.', '1803.11326-4-51-0': '### Evaluation on ECSA', '1803.11326-4-52-0': 'On ECSA dataset, we evaluate different models including Basic BiLSTM-CRF, Vanilla Multi-task, Hierarchy Multi-task and Deep Cascade Multi-task on testing data regarding slot filling as the target task.', '1803.11326-4-52-1': 'We report Precision, Recall and F1 in Table [REF].', '1803.11326-4-53-0': 'The Basic BiLSTM-CRF model achieves an F1 score of 0.43.', '1803.11326-4-53-1': 'To show the usefulness of the lower tasks to slot filling, we "cheated" by using the ground-truth segment type (cond.', '1803.11326-4-53-2': 'SEG) or named entity type (cond.', '1803.11326-4-53-3': 'NE) as the extra features for each word in the Basic BiLSTM-CRF model.', '1803.11326-4-53-4': 'Row 3 and 4 (with *) in Table [REF] show that the slot filling performance can be improved by 85% and 109% if the correct segment type or named entity type is pre-known.', '1803.11326-4-53-5': 'It can perfectly verify our claim that low-level syntactic tasks can significantly affect to the slot filling performance.', '1803.11326-4-53-6': "Of course in practice, the model doesn't know the true values of these types during prediction.", '1803.11326-4-54-0': 'Our further experiments show that DCMTL outperforms the baselines on both precision and recall.', '1803.11326-4-54-1': 'DCMTL achieves the best F1 score of 0.5105, which improves by a relative margin of 14.6% against the strong baseline method (see Table [REF]).', '1803.11326-4-54-2': 'Multi-task models generally perform better than the Basic BiLSTM with single-task target.', '1803.11326-4-54-3': 'The exception is the vanilla multi-task setting.', '1803.11326-4-54-4': 'This is mainly because vanilla multi-task shares parameters across all the layers, and these parameters are likely to be disturbed by the interaction of three tasks.', '1803.11326-4-54-5': 'It is more desirable to let the target task dominate the weights at high-level layers.', '1803.11326-4-55-0': 'We further investigate the learning trend of our proposed approach against baseline methods.', '1803.11326-4-55-1': 'Figure [REF](a) shows the typical learning curves of performance measured by F1.', '1803.11326-4-55-2': 'We can observe that our method DCMTL performs worse than other baseline methods for the first [MATH] batch steps.', '1803.11326-4-55-3': 'After that, other methods converge quickly and DCMTL perform much better after [MATH] batch steps and finally converge to the best F1 score.', '1803.11326-4-55-4': 'We believe that in the beginning, high-level task in DCMTL is affected more by the noise of low-level tasks comparing to others, but as the training goes on, the high-level slot filling task slowly reaps the benefits from low-level tasks.', '1803.11326-4-56-0': 'To make our experiments more solid, we implemented two previous best performing models on ATIS dataset: Sequential CNN [CITATION] (Sequence Labeling based) and Neural Sequence Chunking [CITATION] (Encoder-Decoder based).', '1803.11326-4-56-1': 'They achieved 0.2877 and 0.4355 F1 scores respectively, while our DCMTL model scores 0.5105 F1 and outperforms both of them (by 77% and 17% improvements).', '1803.11326-4-57-0': '### Ablation Test', '1803.11326-4-58-0': 'Our "shortcuts" connections come in two flavors: cascade connection and residual connection.', '1803.11326-4-58-1': 'Multi-task outputs and "shortcuts" connections are highly related since without the multi-task framework, there will be no cascade connections.', '1803.11326-4-58-2': 'We go on to show that both multi-task setting and the "shortcuts" connections are effective and useful in Table [REF], where F1 score improves from 0.4302 to 0.4455 and 0.5105 respectively.', '1803.11326-4-58-3': 'We also investigate how our model DCMTL performs with or without cascade and residual connections (rows with ** prefix in Table [REF]).', '1803.11326-4-58-4': 'F1 score increases from 0.4840 to 0.5105 when residual connection is applied, which verifies its benefit.', '1803.11326-4-58-5': 'If we remove cascade connection from DCMTL, the model actually degenerates into hierarchy multi-task model with residual connection and performs 0.4633 F1 score.', '1803.11326-4-58-6': 'Thus we can conclude that both connections are helpful for our DCMTL model.', '1803.11326-4-58-7': 'However, the cascade connection, which relies on the multi-task, is more effective than the residual connection.', '1803.11326-4-58-8': 'We can verify it from the fact that DCMTL model without cascade connection performs much worse than without residual connection (0.4633 vs. 0.4840 F1 scores).', '1803.11326-4-59-0': 'Furthermore, we explore how DCMTL performs with different cascade connection methods.', '1803.11326-4-59-1': 'We compare three different types of cascade connection illustrated in Figure [REF](a):', '1803.11326-4-60-0': '[1.]', '1803.11326-4-60-1': 'Segment labeling skipped to slot filling (SLOT+SEG).', '1803.11326-4-60-2': '[2.]', '1803.11326-4-60-3': 'Named entity labeling directly connected to slot filling (SLOT+NE).', '1803.11326-4-60-4': '[3.]', '1803.11326-4-60-5': 'Segment labeling, named entity labeling and slot filling in sequence (SLOT+NE+SEG).', '1803.11326-4-61-0': 'From Figure [REF](b), we find that cascade connection with type 3 performs the best and then with type 2, while cascade method with skipped connection (type 1) performs the worst.', '1803.11326-4-61-1': 'Therefore, we design the networks with a cascade connection in a hierarchical fashion and do not apply skipped connection for the cascade inputs (Figure [REF](b)).', '1803.11326-4-61-2': 'This phenomenon here may also be proved by our "cheated" case study above.', '1803.11326-4-61-3': 'Slot filling performance with pre-known named entity type is much better than with pre-known segment type (rows with * in Table [REF]).', '1803.11326-4-62-0': '## Online Testing', '1803.11326-4-63-0': 'Previous experimental results have proven the advantages of our proposed DCMTL approach, so we deploy it in a real world online environment to test its practical performance.', '1803.11326-4-64-0': 'For online A/B testing, we extracted users query log with the slot filling results for one day.', '1803.11326-4-64-1': 'There are in total 251,409 unique queries.', '1803.11326-4-64-2': 'We let three persons to manually evaluate whether a query is slotted perfectly with the strategy where the minority obeys the majority.', '1803.11326-4-64-3': 'A query is slotted perfectly means all terms in query are assigned with the correct slot labels.', '1803.11326-4-64-4': 'Our DCMTL model results in 152,178 perfectly slotted queries which is 60.53 accuracy.', '1803.11326-4-64-5': 'While the original online max-matching algorithm with E-commerce knowledge base (more details in Section [REF]) only covers 66,302 perfectly slotted queries with 26.37 accuracy.', '1803.11326-4-64-6': 'Thus, the accuracy of query slot filling in such online shopping assistant system is improved by 130 after deploying DCMTL model.', '1803.11326-4-64-7': 'This demonstrates that our model can effectively extract the semantic attributes of users query which is extremely helpful E-commerce Shopping Assistant system.', '1803.11326-4-65-0': '# Related Work', '1803.11326-4-66-0': 'There are mainly two lines of research that are related to our work: slot filling for dialog system and multi-task learning in natural language processing.', '1803.11326-4-67-0': 'Slot Filling is considered a sequence labeling problem that is traditionally solved by generative models.', '1803.11326-4-67-1': 'such as Hidden Markov Models (HMMs) [CITATION], hidden vector state model [CITATION], and discriminative models such as conditional random fields (CRFs) [CITATION] and Support Vector Machine (SVMs) [CITATION].', '1803.11326-4-67-2': 'In recent years, deep learning approaches have been explored due to its successful application in many NLP tasks.', '1803.11326-4-67-3': 'Many neural network architectures have been used such as simple RNNs [CITATION], convolutional neural networks (CNNs) [CITATION], LSTMs [CITATION] and variations like encoder-decoder [CITATION] and external memory [CITATION].', '1803.11326-4-67-4': 'In general, these works adopt a BiLSTM [CITATION] as the major labeling architecture to extract various features, then use a CRF layer [CITATION] to model the label dependency.', '1803.11326-4-67-5': 'We also adopt a BiLSTM-CRF model as baseline and claim that a multi-task learning framework is working better than directly applying it on Chinese E-commerce dataset.', '1803.11326-4-67-6': 'Previous works only apply joint model of slot filling and intent detection [CITATION].', '1803.11326-4-67-7': 'Our work is the first to propose a multi-task sequence labeling model with novel cascade and residual connections based on deep neural networks to tackle real-world slot filling problem.', '1803.11326-4-68-0': 'Multi-task Learning (MTL) has attracted increasing attention in both academia and industry recently.', '1803.11326-4-68-1': 'By jointly learning across multiple tasks [CITATION], we can improve performance on each task and reduce the need for labeled data.', '1803.11326-4-68-2': 'There has been several attempts of using multi-task learning on sequence labeling task [CITATION], where most of these works learn all tasks at the out-most layer.', '1803.11326-4-68-3': 'Søgaard and Goldberg [CITATION] is the first to assume the existence of a hierarchy between the different tasks in a stacking BiRNN model.', '1803.11326-4-68-4': 'Compared to these works, our DCMTL model further improves this idea even thorough with cascade and residual connection.', '1803.11326-4-69-0': '# Conclusion', '1803.11326-4-70-0': 'In this paper, we tackle the real-world slot filling task in a novel Chinese online shopping assistant system.', '1803.11326-4-70-1': 'We proposed a deep multi-task sequence learning framework with cascade and residual connection.', '1803.11326-4-70-2': 'Our model achieves comparable results with several state-of-the-art models on the common slot filling dataset ATIS.', '1803.11326-4-70-3': 'On our real-world Chinese E-commerce dataset ECSA, our proposed model DCMTL also achieves best F1 score comparing to several strong baselines.', '1803.11326-4-70-4': 'DCMTL has been deployed on the online shopping assistant of a dominant Chinese E-commerce platform.', '1803.11326-4-70-5': 'Online testing results show that our model meets better understanding of users utterances and improves customer’s shopping experience.', '1803.11326-4-70-6': 'Our future research may include a joint model for category classification and slot filling.', '1803.11326-4-70-7': 'Active learning for slot filling can also be investigated by involving human-beings interaction with our system.'}

1603.08866

{'1603.08866-1-0-0': 'We give a tight scheme for teleporting a quantum state between two parties, whose reference frames are misaligned by an action of a finite symmetry group.', '1603.08866-1-0-1': 'Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size.', '1603.08866-1-0-2': 'In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.', '1603.08866-1-1-0': '# Introduction', '1603.08866-1-2-0': '## The problem and our result', '1603.08866-1-3-0': "Recently many authors have recognised the importance of developing a theory of quantum information which takes account of the reference frames by which a system's state is defined [CITATION].", '1603.08866-1-3-1': 'Quantum teleportation is a foundational quantum protocol with important applications [CITATION]; it was recognised some time ago [CITATION] that a shared reference frame is a hidden additional resource assumed in classical teleportation protocols.', '1603.08866-1-3-2': 'Indeed, it is highly likely that two parties using a teleportation protocol to transmit a quantum state over long distances will be uncertain about the alignment of their respective reference frames.', '1603.08866-1-3-3': 'The problem of teleporting a quantum state between two parties who do not share a reference frame, to be treated in this paper, is therefore natural and important.', '1603.08866-1-4-0': 'In this paper we exhibit a new teleportation protocol, requiring only local operations and classical communication, which in many cases allows perfect teleportation between two parties even when their reference frames are not aligned.', '1603.08866-1-4-1': 'No additional resources or prior communication are required; we require only that the group [MATH] of reference frame transformations is finite.', '1603.08866-1-4-2': 'We demonstrate that these tight reference frame-independent (RFI) teleportation protocols correspond exactly to [MATH]-equivariant unitary error bases, a structure we define; we show these bases do not exist for every representation, and develop methods for constructing them when they do.', '1603.08866-1-4-3': 'In particular, we provide a simple sufficient condition for the existence of RFI teleportation protocols for systems of dimension less than 5.', '1603.08866-1-5-0': "The nature of the classical channel through which the result of Alice's measurement is communicated is crucial to our new protocol.", '1603.08866-1-5-1': 'In the terminology of Peres and Scudo [CITATION], we use a classical channel capable of transmitting unspeakable information, rather than a generic channel, which can transmit only speakable information.', '1603.08866-1-5-2': 'An example of unspeakable information is the choice of a direction in space, to be agreed by two parties who do not share a common directional reference; no amount of communication through a generic shared classical channel can decide the matter, but the fidelitous transfer from one party to the other of a single oriented physical system, such as an arrow, is sufficient.', '1603.08866-1-6-0': 'Importantly, the fundamental concept of a [MATH]-equivariant error basis was developed from investigations in categorical quantum mechanics.', '1603.08866-1-6-1': 'Indeed, we characterised teleportation schemes as structures internal to the category of finite dimensional Hilbert spaces, and investigated corresponding structures in the category of unitary representations of a finite group; these corresponding structures were exactly the [MATH]-equivariant unitary error bases.', '1603.08866-1-6-2': 'Following their discovery, further investigation demonstrated their relevance to the problem of RFI teleportation.', '1603.08866-1-6-3': 'This exemplifies the utility of categorical quantum mechanics as a toolkit for developing new and interesting concepts in quantum information.', '1603.08866-1-7-0': 'In Section [REF] we give an informal worked example of our procedure, and in Section [REF] we carefully analyse its mathematical basis.', '1603.08866-1-7-1': 'In Section [REF] we show how reference frame independent teleportation schemes are related to ideas in categorical quantum mechanics.', '1603.08866-1-7-2': 'In Section [REF] we prove a variety of existence, nonexistence and construction results for [MATH]-equivariant unitary error bases.', '1603.08866-1-8-0': '## Previous results', '1603.08866-1-9-0': "The problem of RFI teleportation was discussed at length in [CITATION], where the authors distinguished teleportation of speakable information, where the transferred state need only have the same coordinates in Bob's reference frame as it had in Alice's; and teleportation of unspeakable information, where an external observer would see that the state of the system itself is transferred.", '1603.08866-1-9-1': "Here we will be concerned with teleportation of unspeakable information; from this point forward we will use the word 'teleportation' to mean only this.", '1603.08866-1-9-2': 'We are only interested in teleportation protocols with zero probability of failure.', '1603.08866-1-10-0': 'It was demonstrated in [CITATION] that teleportation is impossible when the group of reference frame transformations is a nontrivial compact connected Lie group and the representation on the system to be teleported does not factor through a representation of a finite group.', '1603.08866-1-10-1': 'As observed in that paper, this leaves open the question of whether tight quantum teleportation is possible in the case of a finite group of reference frame transformations.', '1603.08866-1-10-2': 'The protocol we are to exhibit resolves this question.', '1603.08866-1-11-0': 'A number of other resolutions for the finite case have previously been proposed, but all are in some way deficient.', '1603.08866-1-11-1': 'In [CITATION], it was suggested that Alice could transmit half of a maximally entangled token state, in the regular representation, in advance of performing the protocol; the two parties could then use this to synchronise their operations.', '1603.08866-1-11-2': 'This method, however, requires Alice to be able to initialise an entangled state on a pair of systems each carrying the regular representation of the transformation group, a procedure which may be experimentally difficult or impossible.', '1603.08866-1-12-0': 'Another relevant result can be found in [CITATION].', '1603.08866-1-12-1': 'In that work, it was shown that it is possible to perform quantum protocols using a second shared system as a quantum reference frame.', '1603.08866-1-12-2': 'This general result, intended for application to quantum cryptography, may be used to create reference frame-independent teleportation protocols.', '1603.08866-1-12-3': 'These protocols are formally identical to the token state method, but operationally more practicable; Alice and Bob simply initialise an additional shared entangled state at the same time as they create the first, take half each and use it to synchronise their operations.', '1603.08866-1-12-4': 'The problems of the token state method therefore persist in this case, although without the additional difficulty of communicating half of the second entangled state from one party to the other.', '1603.08866-1-13-0': "Finally, various solutions have also been proposed which use prior communication to align both parties' reference frames in advance of performing a normal teleportation protocol; methods are described in [CITATION].", '1603.08866-1-13-1': 'This increases the amount of classical information that must be communicated for successful teleportation.', '1603.08866-1-13-2': "Moreover, this procedure is not robust against changes in the alignment of Alice and Bob's reference frames, which must stay constant if the protocol is to succeed.", '1603.08866-1-13-3': 'Our protocol, in contrast, is robust against reference frame changes even while the classical message from Alice to Bob is in transit.', '1603.08866-1-13-4': "We only require that the alignment of Bob's reference frame is constant for the short time between his receiving the classical information and his applying the appropriate unitary correction.", '1603.08866-1-14-0': 'However, we note that our solution is limited in its applicability to the situation where the representation of the group of reference frame transformations on the system being teleported admits a [MATH]-equivariant unitary error basis; when this is not the case, one of the approaches described above must be used.', '1603.08866-1-15-0': '# Example of the procedure', '1603.08866-1-16-0': 'In this section we give an informal account of the problem of reference frame-independent quantum teleportation, and of our new solution.', '1603.08866-1-16-1': 'This is followed by a more mathematically precise description in the next section.', '1603.08866-1-17-0': "Alice and Bob's laboratories are separated by a considerable distance; they share a generic classical channel through which they can communicate by sending classical bits.", '1603.08866-1-17-1': 'In the past, they met in person and created an entangled pair of spin 1/2 particles in the Bell state [MATH] .', '1603.08866-1-17-2': '(Here the degree of freedom is, as usual, the spin component in the [MATH]-direction).', '1603.08866-1-17-3': 'They each took one half of this pair and went back to their respective laboratories.', '1603.08866-1-17-4': 'Alice now has another spin 1/2 particle in a certain state, and wants to use a teleportation protocol to communicate it to Bob.', '1603.08866-1-17-5': 'There is a serious problem, however; they both forgot which direction they were using as the [MATH]-direction when they created the entangled state.', '1603.08866-1-18-0': 'Before they begin the procedure, then, they must attempt to align their reference frames in order to coordinate their operations.', '1603.08866-1-18-1': 'They use the following method.', '1603.08866-1-18-2': 'Alice sends a beam of photons, polarized vertically with respect to her local reference frame, to Bob; Bob then rotates his laboratory until his local vertical coordinate is aligned with the polarization direction.', '1603.08866-1-19-0': "This method has an obvious flaw: one party's axis might be upside down relative to the other, since polarization has a [MATH] symmetry.", '1603.08866-1-19-1': 'Nonetheless, Alice and Bob continue with the traditional teleportation procedure.', '1603.08866-1-19-2': 'Using their classical communication channel, they decide to follow a procedure whereby Alice will measure her initial system together with her part of the entangled state in the basis [MATH] and communicate the result to Bob, who will apply the corresponding correction [MATH].', '1603.08866-1-19-3': 'We define [MATH], where [MATH] denotes the transpose of the matrix [MATH], the symbol [MATH] denotes the 2-by-2 identity matrix, and [MATH] is the Bell state given above.', '1603.08866-1-19-4': 'The [MATH] are defined as follows: [EQUATION]', '1603.08866-1-19-5': 'It can readily be checked that this data forms an unitary error basis, and so by the results of Werner [CITATION] gives correct data for the execution of an ordinary quantum teleportation procedure for a single qubit, when the shared state is the Bell state [MATH].', '1603.08866-1-20-0': "If Bob was fortunate and managed to align his reference frame correctly with Alice's, then the parties are carrying out standard quantum teleportation, and the procedure will be successful.", '1603.08866-1-20-1': "However, if his reference frame is upside-down with respect to Alice's, then the difference between his and Alice's representation of the state of his qubit will correspond to a unitary action of the group [MATH] on its Hilbert space.", '1603.08866-1-20-2': 'We suppose that this action is as follows, where [MATH] is the non-identity element of [MATH]: [EQUATION]', '1603.08866-1-20-3': 'In general, the details of this unitary operator will depend on the actual direction of the spin axis on which Alice and Bob have agreed, relative to the spin axis they used when they created the Bell state.', '1603.08866-1-20-4': "From Alice's perspective frame, Bob's correction is not the unitary [MATH] corresponding to her measurement result; rather, it is the unitary [MATH].", '1603.08866-1-20-5': 'A straightforward calculation then shows that Bob will receive a mixed state; quantum information has been irrevocably lost, and the correctness of the traditional teleportation scheme has been destroyed by the misalignment in their reference frames.', '1603.08866-1-20-6': "From Bob's perspective, he correctly applied the unitary [MATH], but the teleportation failed because the measurement result Alice communicated to him did not correspond to the measurement operation she actually performed.", '1603.08866-1-21-0': 'We now provide a resolution.', '1603.08866-1-21-1': 'In our new procedure, rather than communicate two bits to Bob using their shared classical channel, she sends two physical objects: arrows, of the sort a medieval archer might use.', '1603.08866-1-21-2': 'She orients these arrows according to the measurement result that she obtained, using the following encoding: [EQUATION]', '1603.08866-1-21-3': 'Each arrow is pointing either up or down with respect to the agreed spin axis.', '1603.08866-1-21-4': "One at a time, she physically sends these arrows through space to Bob's laboratory, who observes their local orientations and infers the measurement result 0, 1, 2 or 3 that Alice obtained.", '1603.08866-1-21-5': "Suppose Bob's laboratory is correctly aligned with Alice's; then he will correctly infer Alice's measurement result, and he will apply the corresponding unitary correction.", '1603.08866-1-21-6': 'In this case, the two parties have executed a traditional quantum teleportation protocol, albeit one where the two classical bits of information were transferred from Alice to Bob in an unusual way.', '1603.08866-1-22-0': "Now we suppose that Bob's laboratory is aligned upside-down with respect to Alice's.", '1603.08866-1-22-1': 'Then, if Alice attempts to send the message 0, 1, 2 or 3, Bob will receive it as 1, 0, 3 or 2 respectively, since the arrows will appear to him with the opposite orientations.', '1603.08866-1-22-2': "Furthermore, just as before, when Bob applies a unitary [MATH], its action is seen in Alice's reference frame as [MATH].", '1603.08866-1-22-3': 'We now see the point of the entire construction: the unitary error basis is carefully chosen so that these effects cancel out.', '1603.08866-1-22-4': 'Indeed, the following equations can be easily verified: [EQUATION]', '1603.08866-1-22-5': "As a result, the quantum teleporation will still conclude successfully, even though Bob's reference frame was misaligned.", '1603.08866-1-23-0': 'In summary, by a careful choice of the unitary error basis, and by transferring the measurement result as unspeakable rather than speakable information, the quantum teleportation procedure can be carried out in a way which is robust against this restricted sort of reference frame error.', '1603.08866-1-23-1': 'Note in particular that only 2 bits of classical information were transferred from Alice to Bob, just as with the traditional teleportation procedure, and the Hilbert space of the entangled resource is of minimal dimension, so this procedure is tight in the sense of Werner [CITATION].', '1603.08866-1-23-2': "Also note that the unspeakable information Bob receives from Alice is uniformly random, since Alice's measurement results are; in particular, Bob receives no information during the protocol about whether his axis is actually upside down with respect to Alice's.", '1603.08866-1-23-3': "Finally, it is clear that the procedure would succeed even if Bob's reference frame were constantly changing between the two alignments, as long as the alignment stays constant between Bob's receiving the arrows and applying the unitary correction.", '1603.08866-1-24-0': '# Mathematical description of the proposal', '1603.08866-1-25-0': '## Traditional teleportation', '1603.08866-1-26-0': 'Teleportation is a well-understood procedure.', '1603.08866-1-26-1': "In its traditional form [CITATION], where all parties' reference frames are perfectly aligned, it works as follows.", '1603.08866-1-26-2': 'Note that we only consider tight quantum teleportation in this paper; that is, where the state spaces of the initial system and the entangled systems all have the same dimension, which is equal to the number of classical bits transferred, and the procedure succeeds with probability 1.', '1603.08866-1-27-0': '[Traditional quantum teleportation]', '1603.08866-1-28-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally entangled bipartite state [MATH], and Bob the other.', '1603.08866-1-28-1': 'She performs a measurement with respect to an orthonormal basis of effects [MATH] on the bipartite system built from her initial system and her half of the entangled state.', '1603.08866-1-28-2': 'She sends the measurement result [MATH] to Bob through a perfect classical channel.', '1603.08866-1-28-3': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-1-28-4': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-1-29-0': 'A complete description of correct data [MATH] was given by Werner [CITATION], as follows.', '1603.08866-1-30-0': 'For a Hilbert space [MATH], a unitary error basis is a basis of unitary operators [MATH], which are orthonormal under the Hilbert-Schmidt inner product: [EQUATION] [Werner]', '1603.08866-1-31-0': 'Up to equivalence, teleportation schemes for systems with Hilbert space [MATH] are in one to one correspondence with unitary error bases on [MATH].', '1603.08866-1-32-0': 'Under this correspondence, the shared entangled state [MATH] is the Bell state [MATH] for any orthonormal basis [MATH].', '1603.08866-1-32-1': 'Alice measures in the maximally entangled basis [MATH], where [MATH] is defined as [MATH].', '1603.08866-1-32-2': "Bob's correction for the measurement outcome [MATH] is [MATH].", '1603.08866-1-33-0': '## Reference frame-independent quantum teleportation', '1603.08866-1-34-0': 'We now fully describe the problem we solve.', '1603.08866-1-35-0': '[Reference frame-independent quantum teleportation] Alice and Bob are spatially-separated quantum information theorists capable of performing local operations and classical communication.', '1603.08866-1-35-1': 'Each party has one half of an entangled state which they created at some point in the past, when they were together and their reference frames were aligned; Alice wants to use this entanglement to communicate a quantum state to Bob by teleportation.', '1603.08866-1-35-2': 'However, since then, their reference frames have shifted.', '1603.08866-1-35-3': 'This shift in reference frames is described by the action of unknown elements [MATH] of the group [MATH] of reference frame transformations.', '1603.08866-1-36-0': 'This problem is a useful foundation for our analysis, but hardly captures the full range of situations where it will be useful.', '1603.08866-1-36-1': "Our analysis can be applied almost verbatim, for instance, to any situation where some active group of transformations acts on Alice and Bob's systems, provided that the same group element acts on both of Alice's systems.", '1603.08866-1-36-2': 'Even if this latter condition does not hold, many elements of the analysis will still be relevant.', '1603.08866-1-37-0': 'We now demonstrate how the shift in reference frames causes a serious problem for Procedure [REF].', '1603.08866-1-37-1': 'First we note the following lemma.', '1603.08866-1-38-0': 'Under the conditions of Problem [REF], Procedure [REF] will work for all reference frame alignments if and only if the operations [MATH] and [MATH] are all intertwiners for the group action.', '1603.08866-1-39-0': 'We express the operations with reference to the original shared reference frame.', '1603.08866-1-39-1': "Let Alice and Bob's frame shifts be described by group elements [MATH] and [MATH] respectively.", '1603.08866-1-39-2': 'Alice measures [MATH] relative to her reference frame; in the original frame the operation she has performed is [MATH].', '1603.08866-1-39-3': 'She then sends the result [MATH] to Bob, who performs the operation [MATH] relative to his frame; in the original frame the operation he has performed will be [MATH].', '1603.08866-1-39-4': 'In general, the channel will therefore only work for all reference frame configurations when, for all [MATH], [MATH] and [MATH], for some action of [MATH] on the set of measurement outcomes.', '1603.08866-1-39-5': 'Since [MATH] for the identity [MATH], this clearly implies that [MATH] for all [MATH].', '1603.08866-1-39-6': 'The result follows.', '1603.08866-1-40-0': 'We now demonstrate that Procedure [REF] works only for a trivial [MATH]-action.', '1603.08866-1-41-0': 'Procedure [REF] will only work for all reference frame alignments when [MATH] is a direct sum of identical one-dimensional representations of [MATH]; that is, when [MATH] acts by a global phase.', '1603.08866-1-42-0': 'By Theorem [REF] and Proposition [REF], Procedure [REF] will work only if all projections [MATH] and corrections [MATH] are intertwiners.', '1603.08866-1-42-1': 'By the definition of [MATH] in Theorem [REF], it is sufficient that all [MATH] be intertwiners.', '1603.08866-1-42-2': 'Let us assume that this is the case.', '1603.08866-1-42-3': 'Since the [MATH]-action is trivial on a basis of [MATH], it must be completely trivial on [MATH].', '1603.08866-1-42-4': 'Therefore we have [MATH].', '1603.08866-1-42-5': 'By straightforward character theory, there can only be one copy of [MATH] in the product of an irreducible representation with its dual.', '1603.08866-1-42-6': 'Breaking [MATH] up into simple factors, it follows by counting dimensions that they must all be identical and one dimensional.', '1603.08866-1-43-0': '## Our new scheme', '1603.08866-1-44-0': 'We now give a precise statement of our new procedure.', '1603.08866-1-44-1': '[Reference frame-independent quantum teleportation]', '1603.08866-1-45-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-1-45-1': 'She forms the bipartite system given by her initial system together with her half of the entangled state, and sends it to Bob through a classical channel which is decoherent in the basis [MATH].', '1603.08866-1-45-2': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-1-45-3': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-1-46-0': 'In the language of Peres and Scudo, Proposition [REF] states that tight teleportation is impossible when only speakable classical information is communicated from Alice to Bob as in Procedure [REF].', '1603.08866-1-46-1': 'In Procedure [REF], Alice communicates unspeakable information to Bob.', '1603.08866-1-46-2': 'Indeed, misalignment of reference frames will not affect the way Bob perceives the data arriving through a generic classical channel, but it will affect his perception of the decohered bipartite system.', '1603.08866-1-46-3': 'This means that the information Bob receives from Alice will depend in a nontrivial way on the alignment of his reference frame.', '1603.08866-1-47-0': 'In fact, it might not even be necessary for Alice to send the decohered system itself, should the parties have access to some other means of classically communicating unspeakable information.', '1603.08866-1-47-1': 'The important thing is that the classical data should itself carry a [MATH]-action, allowing us to encode the measurement results in a way compatible with the [MATH]-equivariant UEB we are using (we will introduce [MATH]-equivariant UEBs shortly).', '1603.08866-1-47-2': "An example was given in Section [REF], in which Alice's measurement result was encoded in the spatial orientations of some physical objects.", '1603.08866-1-47-3': 'We leave further examples to the ingenuity of Alice and Bob.', '1603.08866-1-48-0': 'The basic data of Procedure [REF] is the same as for Procedure [REF], and so Theorem [REF] still applies.', '1603.08866-1-48-1': 'However, not all unitary error bases give rise to successful teleporation schemes under this procedure.', '1603.08866-1-48-2': 'We now investigate which of them do.', '1603.08866-1-49-0': '[[MATH]-equivariant unitary error basis] For a Hilbert space [MATH] equipped with a unitary representation of [MATH], a unitary error basis is [MATH]-equivariant when the elements are permuted by the natural action [MATH] of [MATH] on [MATH].', '1603.08866-1-49-1': 'Explicitly, [MATH] for some permutation [MATH] of the set [MATH].', '1603.08866-1-50-0': 'Procedure [REF] will succeed for any reference frame misalignment [MATH] just when the unitary error basis [MATH] is [MATH]-equivariant.', '1603.08866-1-51-0': "We again work in Alice and Bob's original lab frame.", '1603.08866-1-51-1': 'Alice decoheres in the orthonormal basis [MATH].', '1603.08866-1-51-2': 'Bob then measures in the orthonormal basis [MATH], and, depending on his measurement outcome [MATH], performs the corresponding correction [MATH].', '1603.08866-1-52-0': "We first note that, putting Alice's decoherence and Bob's measurement together as one operation, we get a teleportation scheme under Definition [REF].", '1603.08866-1-52-1': "Therefore, by Theorem [REF], Alice's decoherence operation followed by Bob's measurement must be a measurement in some orthonormal basis of maximally entangled states; clearly that must be the basis that Bob measures in.", '1603.08866-1-52-2': "Letting Bob's measurement channel be [MATH] and Alice's decohering channel be [MATH], it follows that [MATH]; this can clearly only be true if the projection basis for [MATH] is the same as the projection basis for [MATH].", '1603.08866-1-52-3': 'We therefore see that the basis [MATH] must be some reordering of the basis [MATH], for all [MATH].', '1603.08866-1-52-4': 'This is exactly [MATH]-equivariance of the UEB [MATH].', '1603.08866-1-53-0': 'We now demonstrate that this condition is sufficient to guarantee success for Procedure [REF].', '1603.08866-1-53-1': 'Suppose [MATH] is [MATH]-equivariant.', '1603.08866-1-53-2': "Then Alice's decohering operation is exactly the same as it would have been if her reference frame had not shifted at all.", '1603.08866-1-53-3': 'Bob measures and performs the correction; the correction therefore corresponds to the measurement and the result follows.', '1603.08866-1-54-0': '# Classical structures in [MATH]', '1603.08866-1-55-0': 'Teleportation in the context of a finite group [MATH] can be described elegantly in the framework of categorical quantum mechanics [CITATION].', '1603.08866-1-55-1': 'One key strategy in this research programme is to understand features of quantum information in terms of the category [MATH] of finite-dimensional Hilbert spaces and linear maps, and then to generalize them by applying them in different categories.', '1603.08866-1-55-2': 'The concept of [MATH]-equivariant quantum teleportation arises by understanding the categorical structure of the traditional quantum teleportation procedure, and then applying it in [MATH], as we now explore.', '1603.08866-1-55-3': 'This technical section of the paper will make use of well-known ideas from categorical quantum mechanics, of which full details are available in the provided references; for reasons of space, proofs in this section are omitted.', '1603.08866-1-56-0': 'The following definition gives our abstract categorical description of quantum teleportation, in terms of classical structures in a symmetric monoidal category [CITATION].', '1603.08866-1-57-0': 'In a dagger-compact category, a quantum teleportation procedure on an object [MATH] with a right dual is a classical structure on the object [MATH], satisfying the following condition, where [MATH] is some scalar: [EQUATION]', '1603.08866-1-57-1': "This definition is motivated by the following theorem; recall Werner's Theorem [REF].", '1603.08866-1-58-0': 'Quantum teleportation procedures in [MATH] correspond precisely to unitary error bases.', '1603.08866-1-59-0': 'We now summarize the application of these ideas in a group representation category.', '1603.08866-1-60-0': 'For a group [MATH], the dagger-compact category [MATH] has objects given by unitary representations of [MATH], morphisms given by intertwiners, and a dagger-compact structure inherited from the underlying Hilbert spaces.', '1603.08866-1-61-0': 'Quantum teleportation procedures in [MATH] correspond precisely to [MATH]-equivariant unitary error bases.', '1603.08866-1-62-0': 'Finally, we observe that the constructions of unitary error bases in Theorem [REF] and Remark [REF] carry over straightforwardly to the [MATH]-equivariant setting because they are essentially categorical constructions; the Hadamard construction, for instance, is defined in terms of two special commutative dagger Frobenius algebras and an isomorphism between them.', '1603.08866-1-62-1': 'In Rep([MATH]), this reduces exactly to the intertwining Hadamard matrix and [MATH]-equivariant orthonormal basis of Theorem [REF].', '1603.08866-1-62-2': 'In this sense, these constructions are much more natural than, for instance, the construction of unitary error bases using projective group representations [CITATION]; indeed, it is difficult to see how the latter construction could be brought into the [MATH]-equivariant framework.', '1603.08866-1-63-0': '# Existence and construction of RFI teleportation protocols', '1603.08866-1-64-0': 'We have demonstrated that [MATH]-equivariant UEBs are exactly the structures we need to perform reference frame-independent teleportation protocols, but it is still unclear how to construct them for a given representation [MATH], if they exist at all.', '1603.08866-1-64-1': 'We cannot hope for a general classification of [MATH]-equivariant UEBs, since there is not even a classification in the case where the [MATH]-action is trivial, although many construction methods exist [CITATION].', '1603.08866-1-64-2': 'In this section we will demonstrate that [MATH]-equivariant unitary error bases need not exist on every representation, meaning that RFI teleportation is not always possible.', '1603.08866-1-64-3': 'We will then demonstrate that several UEB constructions carry over naturally to the [MATH]-equivariant setting, allowing us to construct RFI teleportation protcols for a wide variety of systems.', '1603.08866-1-65-0': 'We begin with a definition.', '1603.08866-1-66-0': 'A [MATH]-equivariant orthonormal basis for some representation [MATH] is an orthonormal basis of [MATH] whose elements are permuted by the action of [MATH].', '1603.08866-1-67-0': '[MATH]-equivariant unitary error bases are [MATH]-equivariant orthonormal bases of [MATH], all of whose elements are unitary maps.', '1603.08866-1-68-0': 'It will transpire that we can use [MATH]-equivariant orthonormal bases on [MATH] to construct [MATH]-equivariant UEBs for [MATH].', '1603.08866-1-68-1': 'Moreover, if we prove that there are no [MATH]-equivariant orthonormal bases on [MATH], it follows by Remark [REF] that there will be no [MATH]-equivariant UEBs for [MATH]; we will use this fact to demonstrate that RFI teleportation protocols need not always exist.', '1603.08866-1-68-2': 'Our first step is therefore a classification of [MATH]-equivariant orthonormal bases.', '1603.08866-1-69-0': '## A classification of [MATH]-equivariant orthonormal bases', '1603.08866-1-70-0': 'We begin with a simple lemma.', '1603.08866-1-70-1': 'Let [MATH]-Set be the category whose objects are sets carrying an action of [MATH], and whose morphisms are [MATH]-equivariant functions between them.', '1603.08866-1-70-2': 'Then there exists a functor [MATH]G[MATH]G[MATH], which, given a [MATH]-set, constructs the free Hilbert space on its elements, and extends the [MATH]-action and morphisms linearly.', '1603.08866-1-71-0': '[MATH]-equivariant orthonormal bases exist only on representations isomorphic to those in the image of [MATH].', '1603.08866-1-72-0': 'Immediate, since a [MATH]-equivariant orthonormal basis has an underlying Hilbert space isomorphic to the free Hilbert space on the elements of the chosen basis, which [MATH] acts on by permutations.', '1603.08866-1-73-0': 'We begin by presenting a simple classification of [MATH]-sets due to Burnside [CITATION].', '1603.08866-1-74-0': 'Given two [MATH]-sets [MATH] and [MATH], their disjoint union [MATH] is the disjoint union of [MATH] and [MATH] as sets with the natural induced action.', '1603.08866-1-75-0': 'Given a subgroup [MATH] of [MATH], the coset space [MATH] is the [MATH]-set whose elements are the cosets of [MATH] in [MATH], and whose [MATH]-action [MATH] is the natural action of [MATH] by left multiplication on those cosets.', '1603.08866-1-76-0': 'Any [MATH]-set is isomorphic to a disjoint union of coset spaces.', '1603.08866-1-76-1': 'Two coset spaces are isomorphic as [MATH]-sets if and only they correspond to conjugate subgroups.', '1603.08866-1-77-0': 'See [CITATION].', '1603.08866-1-78-0': 'In modern language, Lemma [REF] states that [MATH]-Set is a semisimple fusion category whose simple objects correspond to conjugacy classes of subgroups in [MATH].', '1603.08866-1-78-1': 'It is easy to see that the functor [MATH] is additive; the disjoint union of two [MATH]-sets will be sent under [MATH] to the direct sum of their corresponding representations.', '1603.08866-1-78-2': 'In order to classify all objects in the image of [MATH], therefore, it is sufficient to find the image of the coset spaces under [MATH].', '1603.08866-1-78-3': 'We will call those representations the basic permutation representations.', '1603.08866-1-79-0': 'In order to identify the basic permutation representations, we now state an obvious but critical lemma regarding the character of the permutation representation induced by [MATH] on a [MATH]-set.', '1603.08866-1-80-0': 'Given a [MATH]-set [MATH], let [MATH] be the character of [MATH].', '1603.08866-1-80-1': 'Then the following holds: [EQUATION]', '1603.08866-1-80-2': 'The character [MATH] is exactly the trace of the matrix representing [MATH]; the result follows trivially from the definition of [MATH].', '1603.08866-1-81-0': 'We may therefore identify the basic permutation representations by taking a representative of every conjugacy class of subgroups of [MATH], finding the number of fixed points of the action of each element of [MATH] on the corresponding coset spaces, then decomposing the resulting characters using the character table to find the corresponding representations.', '1603.08866-1-82-0': '## Existence of RFI teleportation protocols', '1603.08866-1-83-0': 'Using the results of Subsection [REF], we now exhibit a representation for which no [MATH]-equivariant UEBs exist, and on which quantum teleportation is therefore impossible.', '1603.08866-1-84-0': 'There is no RFI protocol to teleport the state of the 2-dimensional irreducible representation [MATH] of [MATH].', '1603.08866-1-85-0': 'Using the method outlined in Subsection [REF], we find that the characters of the basic permutation representations are as follows:', '1603.08866-1-86-0': 'The character of [MATH] is [MATH], which clearly cannot be composed as a sum of characters of basic permutation representations.', '1603.08866-1-86-1': 'By Remark [REF], the result follows.', '1603.08866-1-87-0': 'This argument does not extend to all irreducible representations.', '1603.08866-1-87-1': 'The endomorphism space of the 2-dimensional irreducible representation of [MATH], for instance, is a sum of basic permutation representations.', '1603.08866-1-88-0': '## Construction of RFI teleportation protocols', '1603.08866-1-89-0': 'Although RFI teleportation protocols need not always exist, they can often be constructed.', '1603.08866-1-89-1': 'We now demonstrate that, if we can find a [MATH]-equivariant orthonormal basis on [MATH], and a Hadamard matrix which commutes with all [MATH] in that basis, we can perform RFI teleportation on [MATH].', '1603.08866-1-90-0': 'Let [MATH] be a [MATH]-equivariant orthonormal basis on [MATH].', '1603.08866-1-90-1': 'In this basis all [MATH] will be permutation matrices.', '1603.08866-1-90-2': 'Let [MATH] be a Hadamard matrix that commutes with all [MATH] in this basis.', '1603.08866-1-90-3': 'Then the following family is a [MATH]-equivariant UEB: [EQUATION]', '1603.08866-1-90-4': 'It was already proved in [CITATION] that this is a UEB; we therefore need only show that it is [MATH]-equivariant.', '1603.08866-1-90-5': 'Since [MATH] we have that [MATH].', '1603.08866-1-90-6': 'We see easily that [MATH].', '1603.08866-1-90-7': 'Now note that the fact that [MATH] commutes with all elements of [MATH] means that permuting the columns of [MATH] is exactly the same as permuting the rows, since [MATH] for all [MATH].', '1603.08866-1-90-8': 'So [MATH].', '1603.08866-1-90-9': 'A similar argument works for [MATH].', '1603.08866-1-91-0': 'If the assumptions of Theorem [REF] are satisfied, it is possible to construct many more [MATH]-equivariant UEBs using quantum Latin squares (QLSs) [CITATION]; this construction will give [MATH]-equivariant UEBs provided the linear map defining the QLS is an intertwiner.', '1603.08866-1-92-0': 'We finish this section with a simple sufficient condition for the existence of tight RFI protocols on systems of dimension less than 5.', '1603.08866-1-92-1': 'Firstly we prove a lemma.', '1603.08866-1-93-0': 'Let [MATH] be a matrix of dimension [MATH] defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-1-93-1': 'Let [MATH] where [MATH] and [MATH].', '1603.08866-1-93-2': 'Then [MATH] is unitary precisely when the following conditions are satisfied:', '1603.08866-1-94-0': '0.3 [EQUATION] 0.33 [EQUATION] 0.36 [EQUATION] 0pt', '1603.08866-1-95-0': 'For unitarity it is sufficient that the rows form an orthonormal basis.', '1603.08866-1-95-1': 'It is clear from the symmetry of [MATH] that it is sufficient for one row vector to be normal, and one pair of row vectors to be orthogonal.', '1603.08866-1-95-2': 'This gives us two equations in [MATH] and [MATH]: [EQUATION]', '1603.08866-1-95-3': 'We will demonstrate that ([REF]) is necessary and sufficient for us to find [MATH] satisfying these equations.', '1603.08866-1-95-4': 'It is obvious that ([REF]) is satisfiable if and only if [MATH].', '1603.08866-1-95-5': 'Letting [MATH], Equation ([REF]) becomes [EQUATION].', '1603.08866-1-95-6': 'Since [MATH] and [MATH] can be freely adjusted to give [MATH] any value in that range, we see that the following is necessary and sufficient for ([REF]) to be soluble: [EQUATION]', '1603.08866-1-95-7': 'Use of the identity ([MATH]) and a short calculation demonstrates that this is equivalent to the lower bound in the inequality ([REF]).', '1603.08866-1-96-0': 'Suppose [MATH] admits a [MATH]-equivariant orthonormal basis, and is of dimension less than 5.', '1603.08866-1-96-1': 'Then there exists a RFI teleportation protocol for [MATH].', '1603.08866-1-97-0': 'We construct a [MATH]-equivariant UEB for [MATH].', '1603.08866-1-97-1': 'Expressed in the [MATH]-equivariant orthonormal basis, [MATH] will be some subgroup of the permutation matrices [MATH].', '1603.08866-1-97-2': 'To use Theorem [REF], we must find a Hadamard matrix commuting with [MATH].', '1603.08866-1-97-3': 'In the worst case, [MATH] will be the whole group [MATH] of permutation matrices.', '1603.08866-1-97-4': '(This situation is realised for the representation [MATH] of [MATH], where [MATH] is the fundamental [MATH]-dimensional representation of [MATH]).', '1603.08866-1-98-0': 'We will demonstrate that, when [MATH] is of dimension less than 5, we can find a Hadamard matrix which commutes with all the permutation matrices.', '1603.08866-1-98-1': 'First we eliminate the degenerate cases [MATH] and [MATH].', '1603.08866-1-98-2': 'Clearly for [MATH] we can perform RFI teleportation by Proposition [REF], For [MATH] the following family of Hadamard matrices commutes with [MATH], where [MATH] and [MATH]: [EQUATION]', '1603.08866-1-98-3': 'From now on we may therefore assume [MATH].', '1603.08866-1-99-0': 'It is easy to see that the centraliser [MATH] is the set of matrices defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-1-99-1': 'The conditions necessary for such a matrix to be unitary were given in Lemma [REF].', '1603.08866-1-99-2': 'Setting [MATH] in ([REF]), it follows that [MATH].', '1603.08866-1-99-3': 'This is compatible with ([REF]) only for [MATH].'}

{'1603.08866-2-0-0': 'We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group.', '1603.08866-2-0-1': 'Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size.', '1603.08866-2-0-2': 'In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.', '1603.08866-2-1-0': '# Introduction', '1603.08866-2-2-0': '## The problem and our result', '1603.08866-2-3-0': "Recently many authors have recognised the importance of developing a theory of quantum information which takes account of the reference frames by which a system's state is defined [CITATION].", '1603.08866-2-3-1': 'Quantum teleportation is a foundational quantum protocol with important applications [CITATION]; it was recognised some time ago [CITATION] that a shared reference frame is a hidden additional resource assumed in classical teleportation protocols.', '1603.08866-2-3-2': 'Indeed, it is highly likely that two parties using a teleportation protocol to transmit a quantum state over long distances will be uncertain about the alignment of their respective reference frames.', '1603.08866-2-3-3': 'The problem of teleporting a quantum state between two parties who do not share a reference frame, to be treated in this paper, is therefore natural and important.', '1603.08866-2-4-0': 'In this paper we exhibit a new teleportation protocol, requiring only local operations and classical communication, which in many cases allows perfect teleportation between two parties even when their reference frames are not aligned.', '1603.08866-2-4-1': 'No additional resources or prior communication are required; we require only that the group [MATH] of reference frame transformations is finite.', '1603.08866-2-4-2': 'We demonstrate that these tight reference frame-independent (RFI) teleportation protocols correspond exactly to [MATH]-equivariant unitary error bases, a structure we define; we show these bases do not exist for every representation, and develop methods for constructing them when they do.', '1603.08866-2-4-3': 'In particular, we provide a simple sufficient condition for the existence of RFI teleportation protocols for systems of dimension less than 5.', '1603.08866-2-5-0': "The nature of the classical channel through which the result of Alice's measurement is communicated is crucial to our new protocol.", '1603.08866-2-5-1': 'In the terminology of Peres and Scudo [CITATION], we use a classical channel capable of transmitting unspeakable information, rather than a generic channel, which can transmit only speakable information.', '1603.08866-2-5-2': 'An example of unspeakable information is the choice of a direction in space, to be agreed by two parties who do not share a common directional reference; no amount of communication through a generic shared classical channel can decide the matter, but the fidelitous transfer from one party to the other of a single oriented physical system, such as an arrow, is sufficient.', '1603.08866-2-6-0': 'The fundamental concept of a [MATH]-equivariant error basis was developed from investigations in categorical quantum mechanics.', '1603.08866-2-6-1': 'We characterised teleportation schemes as structures internal to the category of finite dimensional Hilbert spaces, and investigated corresponding structures in the category of unitary representations of a finite group; these corresponding structures were exactly the [MATH]-equivariant unitary error bases.', '1603.08866-2-6-2': 'Following their discovery, further investigation demonstrated their relevance to the problem of RFI teleportation.', '1603.08866-2-6-3': 'This exemplifies the utility of categorical quantum mechanics as a toolkit for developing new and interesting concepts in quantum information.', '1603.08866-2-7-0': 'In Section [REF] we give an informal worked example of our procedure, and in Section [REF] we carefully analyse its mathematical basis.', '1603.08866-2-7-1': 'In Section [REF] we show how reference frame-independent teleportation schemes are related to ideas in categorical quantum mechanics.', '1603.08866-2-7-2': 'In Section [REF] we prove a variety of existence, nonexistence and construction results for [MATH]-equivariant unitary error bases.', '1603.08866-2-8-0': '## Previous results', '1603.08866-2-9-0': "The problem of RFI teleportation was discussed at length in [CITATION], where the authors distinguished teleportation of speakable information, where the transferred state need only have the same coordinates in Bob's reference frame as it had in Alice's; and teleportation of unspeakable information, where an external observer would see that the state of the system itself is transferred.", '1603.08866-2-9-1': "Here we will be concerned with teleportation of unspeakable information; from this point forward we will use the word 'teleportation' to mean only this.", '1603.08866-2-9-2': 'We are only interested in teleportation protocols with zero probability of failure.', '1603.08866-2-10-0': 'It was demonstrated in [CITATION] that teleportation is impossible when the group of reference frame transformations is a nontrivial compact connected Lie group and the representation on the system to be teleported does not factor through a representation of a finite group.', '1603.08866-2-10-1': 'As observed in that paper, this leaves open the question of whether tight quantum teleportation is possible in the case of a finite group of reference frame transformations.', '1603.08866-2-10-2': 'The protocol we are to exhibit resolves this question.', '1603.08866-2-11-0': 'A number of other resolutions for the finite case have previously been proposed.', '1603.08866-2-11-1': 'In [CITATION], it was suggested that Alice could transmit half of a maximally-entangled token state, in the regular representation, in advance of performing the protocol; the two parties could then use this to synchronise their operations.', '1603.08866-2-11-2': 'This method, however, requires Alice to be able to initialise an entangled state on a pair of systems each carrying the regular representation of the transformation group, a procedure which may be experimentally difficult or impossible.', '1603.08866-2-12-0': 'Another relevant result can be found in [CITATION].', '1603.08866-2-12-1': 'In that work, it was shown that it is possible to perform quantum protocols using a second shared system as a quantum reference frame.', '1603.08866-2-12-2': 'This general result, intended for application to quantum cryptography, may be used to create reference frame-independent teleportation protocols.', '1603.08866-2-12-3': 'These protocols are formally identical to the token state method, but operationally more practicable; Alice and Bob simply initialise an additional shared entangled state at the same time as they create the first, take half each and use it to synchronise their operations.', '1603.08866-2-12-4': 'The problems of the token state method therefore persist in this case, although without the additional difficulty of communicating half of the second entangled state from one party to the other.', '1603.08866-2-13-0': "Finally, various solutions have also been proposed which use prior communication to align both parties' reference frames in advance of performing a normal teleportation protocol; methods are described in [CITATION].", '1603.08866-2-13-1': 'This increases the amount of classical information that must be communicated for successful teleportation.', '1603.08866-2-13-2': "Moreover, this procedure is not robust against changes in the alignment of Alice and Bob's reference frames, which must stay constant if the protocol is to succeed.", '1603.08866-2-13-3': 'Our protocol, in contrast, is robust against reference frame changes even while the classical message from Alice to Bob is in transit.', '1603.08866-2-13-4': "We only require that the alignment of Bob's reference frame is constant for the short time between his receipt of the classical information and his application of the appropriate unitary correction.", '1603.08866-2-14-0': 'However, we note that our solution is limited in its applicability to the situation where the representation of the group of reference frame transformations on the system being teleported admits a [MATH]-equivariant unitary error basis; when this is not the case, one of the approaches described above must be used.', '1603.08866-2-15-0': '# Example of the procedure', '1603.08866-2-16-0': 'In this section we give an informal account of the problem of reference frame-independent quantum teleportation, and of our new solution.', '1603.08866-2-16-1': 'This is followed by a more mathematically precise description in the next section.', '1603.08866-2-17-0': "Alice and Bob's laboratories are separated by a considerable distance; they share a generic classical channel through which they can communicate by sending classical bits.", '1603.08866-2-17-1': 'They work with quantum systems whose state is defined with reference to a direction in space.', '1603.08866-2-17-2': 'In the past, they met and created an entangled pair of systems in the Bell state [MATH].', '1603.08866-2-17-3': 'They each took one half of this pair and went back to their respective laboratories.', '1603.08866-2-17-4': 'Alice now has another system in a certain state, and wants to use a teleportation protocol to communicate it to Bob.', '1603.08866-2-17-5': 'There is a serious problem, however; Bob forgot which direction they were using as a reference when they created the entangled state.', '1603.08866-2-18-0': 'Before they begin the procedure, Alice communicates the direction to Bob in order to coordinate their operations.', '1603.08866-2-18-1': 'They lack the necessary equipment to agree on a direction using some absolute method, such as reference to the stars, so they use the following approach.', '1603.08866-2-18-2': 'Alice sends a beam of light to Bob, polarized vertically with respect to her local reference frame; he then uses the polarisation direction as his reference.', '1603.08866-2-19-0': "This method has an obvious flaw: Bob's direction might be upside down relative to Alice's, since polarization has a [MATH] symmetry.", '1603.08866-2-19-1': 'Nonetheless, Alice and Bob continue with the traditional teleportation procedure.', '1603.08866-2-19-2': 'Using their classical communication channel, they decide to follow a procedure whereby Alice will measure her initial system together with her part of the entangled state in the basis [MATH] and communicate the result to Bob, who will apply the corresponding correction [MATH].', '1603.08866-2-19-3': 'We define [MATH], where [MATH] denotes the transpose of the matrix [MATH], the symbol [MATH] denotes the 2-by-2 identity matrix, and [MATH] is the Bell state given above.', '1603.08866-2-19-4': 'The [MATH] are defined as follows: [EQUATION]', '1603.08866-2-19-5': 'It can readily be checked that this data forms an unitary error basis, and so by the results of Werner [CITATION] gives correct data for the execution of an ordinary quantum teleportation procedure for a single qubit, when the shared state is the Bell state [MATH].', '1603.08866-2-20-0': "If Bob was fortunate and aligned his reference direction correctly with Alice's, then the parties are carrying out standard quantum teleportation, and the procedure will be successful.", '1603.08866-2-20-1': "However, if his reference direction is upside-down with respect to Alice's, then the difference between his and Alice's representation of the state of his qubit will correspond to a unitary action of the group [MATH] on its Hilbert space.", '1603.08866-2-20-2': 'We suppose that this action is as follows, where [MATH] is the non-identity element of [MATH]: [EQUATION]', '1603.08866-2-20-3': 'The fact that this matrix is nontrivial corresponds to the fact that the wavefunction of the system is not rotationally symmetric.', '1603.08866-2-20-4': 'In this case, teleportation does not proceed successfully.', '1603.08866-2-20-5': "From Alice's perspective, Bob's correction is not in fact the unitary [MATH] corresponding to her measurement result, but rather the unitary [MATH]; a straightforward calculation then shows that Bob will receive a mixed state, and quantum information has been irrevocably lost.", '1603.08866-2-20-6': "From Bob's perspective, he correctly applied the unitary [MATH], but the teleportation failed because the measurement result [MATH] that Alice communicated to him did not correspond to the state she actually measured, which was [MATH].", '1603.08866-2-21-0': 'We now provide a resolution.', '1603.08866-2-21-1': 'In our new procedure, rather than communicate two bits to Bob using their shared classical channel, Alice sends two physical objects: arrows, of the sort a medieval archer might use.', '1603.08866-2-21-2': 'She orients these arrows according to the measurement result that she obtained, using the following encoding: [EQUATION]', '1603.08866-2-21-3': "Each arrow is oriented with respect to Alice's reference direction.", '1603.08866-2-21-4': "One at a time, she physically sends these arrows through space to Bob's laboratory.", '1603.08866-2-21-5': 'Bob observes their local orientations and infers the measurement result 0, 1, 2 or 3 that Alice obtained.', '1603.08866-2-21-6': "Suppose Bob's laboratory is correctly aligned with Alice's; then he will correctly infer Alice's measurement result, and he will apply the corresponding unitary correction.", '1603.08866-2-21-7': 'In this case, the two parties have executed a traditional quantum teleportation protocol, albeit one where the two classical bits of information were transferred from Alice to Bob in an unusual way.', '1603.08866-2-22-0': "Now we suppose that Bob's laboratory is aligned upside-down with respect to Alice's.", '1603.08866-2-22-1': 'Then, if Alice attempts to send the message 0, 1, 2 or 3, Bob will receive it as 1, 0, 3 or 2 respectively, since the arrows will appear to him with the opposite orientations.', '1603.08866-2-22-2': "Furthermore, just as before, when Bob applies a unitary [MATH], its action is seen in Alice's reference frame as [MATH].", '1603.08866-2-22-3': 'We now see the point of the entire construction: the unitary error basis is carefully chosen so that these effects cancel out.', '1603.08866-2-22-4': 'Indeed, the following equations can be easily verified: [EQUATION]', '1603.08866-2-22-5': "As a result, the quantum teleporation will still conclude successfully, even though Bob's reference direction was wrong.", '1603.08866-2-23-0': 'In summary, by a careful choice of the unitary error basis, and by transferring the measurement result as unspeakable rather than speakable information, the quantum teleportation procedure can be carried out in a way which is robust against this restricted sort of reference frame error.', '1603.08866-2-23-1': 'Note in particular that only 2 bits of classical information were transferred from Alice to Bob, just as with the traditional teleportation procedure, and the Hilbert space of the entangled resource is of minimal dimension, so this procedure is tight in the sense of Werner [CITATION].', '1603.08866-2-23-2': "Also note that the unspeakable information Bob receives from Alice is uniformly random, since Alice's measurement results are; in particular, Bob receives no information during the protocol about whether his reference direction is actually upside down with respect to Alice's.", '1603.08866-2-23-3': "Finally, it is clear that the procedure would succeed even if Bob's reference frame were constantly changing between the two alignments, as long as the alignment stays constant between his receipt of the arrows and his application of the unitary correction.", '1603.08866-2-24-0': '# Mathematical description of the proposal', '1603.08866-2-25-0': '## Traditional teleportation', '1603.08866-2-26-0': 'Teleportation is a well-understood procedure.', '1603.08866-2-26-1': "In its traditional form [CITATION], where all parties' reference frames are perfectly aligned, it works as follows.", '1603.08866-2-26-2': 'Note that we only consider tight quantum teleportation in this paper; that is, where the state spaces of the initial system and the entangled systems all have the same dimension, which is equal to the number of classical bits transferred, and the procedure succeeds with probability 1.', '1603.08866-2-27-0': '[Traditional quantum teleportation]', '1603.08866-2-28-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-2-28-1': 'She performs a measurement with respect to an orthonormal basis of effects [MATH] on the bipartite system built from her initial system and her half of the entangled state.', '1603.08866-2-28-2': 'She sends the measurement result [MATH] to Bob through a perfect classical channel.', '1603.08866-2-28-3': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-2-28-4': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-2-29-0': 'A complete description of correct data [MATH] was given by Werner [CITATION], as follows.', '1603.08866-2-30-0': 'For a Hilbert space [MATH], a unitary error basis is a basis of unitary operators [MATH], which are orthonormal under the Hilbert-Schmidt inner product: [EQUATION] [Werner]', '1603.08866-2-31-0': 'Up to equivalence, teleportation schemes for systems with Hilbert space [MATH] are in one to one correspondence with unitary error bases on [MATH].', '1603.08866-2-32-0': 'Under this correspondence, the shared entangled state [MATH] is the Bell state [MATH] for any orthonormal basis [MATH].', '1603.08866-2-32-1': 'Alice measures in the maximally-entangled basis [MATH], where [MATH] is defined as [MATH].', '1603.08866-2-32-2': "Bob's correction for the measurement outcome [MATH] is [MATH].", '1603.08866-2-33-0': '## Reference frame-independent quantum teleportation', '1603.08866-2-34-0': 'We now fully describe the problem we solve.', '1603.08866-2-35-0': '[Reference frame-independent quantum teleportation] Alice and Bob are spatially-separated quantum information theorists capable of performing local operations and classical communication.', '1603.08866-2-35-1': 'Each party has one half of an entangled state which they created at some point in the past, when they were together and their reference frames were aligned; Alice wants to use this entanglement to communicate a quantum state to Bob by teleportation.', '1603.08866-2-35-2': 'However, since then, their reference frames have shifted.', '1603.08866-2-35-3': 'This shift in reference frames is described by the action of unknown elements [MATH] of the group [MATH] of reference frame transformations.', '1603.08866-2-36-0': 'This problem is a useful foundation for our analysis, but hardly captures the full range of situations where it will be useful.', '1603.08866-2-36-1': "Our analysis can be applied almost verbatim, for instance, to any situation where some active group of transformations acts on Alice and Bob's systems, provided that the same group element acts on both of Alice's systems.", '1603.08866-2-36-2': 'Even if this latter condition does not hold, many elements of the analysis will still be relevant.', '1603.08866-2-37-0': 'We now demonstrate how the shift in reference frames causes a serious problem for Procedure [REF].', '1603.08866-2-37-1': 'First we note the following lemma.', '1603.08866-2-38-0': 'Under the conditions of Problem [REF], Procedure [REF] will work for all reference frame alignments if and only if the operations [MATH] and [MATH] are all intertwiners for the group action.', '1603.08866-2-39-0': 'We express the operations with reference to the original shared reference frame.', '1603.08866-2-39-1': "Let Alice and Bob's frame shifts be described by group elements [MATH] and [MATH] respectively.", '1603.08866-2-39-2': 'Alice measures [MATH] relative to her reference frame; in the original frame the operation she has performed is [MATH].', '1603.08866-2-39-3': 'She then sends the result [MATH] to Bob, who performs the operation [MATH] relative to his frame; in the original frame the operation he has performed will be [MATH].', '1603.08866-2-39-4': 'In general, the channel will therefore only work for all reference frame configurations when, for all [MATH], [MATH] and [MATH], for some action of [MATH] on the set of measurement outcomes.', '1603.08866-2-39-5': 'Since [MATH] for the identity [MATH], this clearly implies that [MATH] for all [MATH].', '1603.08866-2-39-6': 'The result follows.', '1603.08866-2-40-0': 'We now demonstrate that Procedure [REF] works only for a trivial [MATH]-action.', '1603.08866-2-41-0': 'Procedure [REF] will only work for all reference frame alignments when [MATH] is a direct sum of identical one-dimensional representations of [MATH]; that is, when [MATH] acts by a global phase.', '1603.08866-2-42-0': 'By Theorem [REF] and Proposition [REF], Procedure [REF] will work only if all projections [MATH] and corrections [MATH] are intertwiners.', '1603.08866-2-42-1': 'By the definition of [MATH] in Theorem [REF], it is sufficient that all [MATH] be intertwiners.', '1603.08866-2-42-2': 'Let us assume that this is the case.', '1603.08866-2-42-3': 'Since the [MATH]-action is trivial on a basis of [MATH], it must be completely trivial on [MATH].', '1603.08866-2-42-4': 'Therefore we have [MATH].', '1603.08866-2-42-5': 'By straightforward character theory, there can only be one copy of [MATH] in the product of an irreducible representation with its dual.', '1603.08866-2-42-6': 'Breaking [MATH] up into simple factors, it follows by counting dimensions that they must all be identical and one dimensional.', '1603.08866-2-43-0': '## Our new scheme', '1603.08866-2-44-0': 'We now give a precise statement of our new procedure.', '1603.08866-2-44-1': '[Reference frame-independent quantum teleportation]', '1603.08866-2-45-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-2-45-1': 'She forms the bipartite system given by her initial system together with her half of the entangled state, and sends it to Bob through a classical channel which is decoherent in the basis [MATH].', '1603.08866-2-45-2': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-2-45-3': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-2-46-0': 'In the language of Peres and Scudo, Proposition [REF] states that tight teleportation is impossible when only speakable classical information is communicated from Alice to Bob as in Procedure [REF].', '1603.08866-2-46-1': 'In Procedure [REF], Alice communicates unspeakable information to Bob.', '1603.08866-2-46-2': 'Indeed, misalignment of reference frames will not affect the way Bob perceives the data arriving through a generic classical channel, but it will affect his perception of the decohered bipartite system.', '1603.08866-2-46-3': 'This means that the information Bob receives from Alice will depend in a nontrivial way on the alignment of his reference frame.', '1603.08866-2-47-0': 'In fact, it might not even be necessary for Alice to send the decohered system itself, should the parties have access to some other means of classically communicating unspeakable information.', '1603.08866-2-47-1': 'The important thing is that the classical data should itself carry a [MATH]-action, allowing us to encode the measurement results in a way compatible with the [MATH]-equivariant UEB we are using (we will introduce [MATH]-equivariant UEBs shortly).', '1603.08866-2-47-2': "An example was given in Section [REF], in which Alice's measurement result was encoded in the spatial orientations of some physical objects.", '1603.08866-2-47-3': 'We leave further examples to the ingenuity of Alice and Bob.', '1603.08866-2-48-0': 'The basic data of Procedure [REF] is the same as for Procedure [REF], and so Theorem [REF] still applies.', '1603.08866-2-48-1': 'However, not all unitary error bases give rise to successful teleporation schemes under this procedure.', '1603.08866-2-48-2': 'We now investigate which of them do.', '1603.08866-2-49-0': '[[MATH]-equivariant unitary error basis] For a Hilbert space [MATH] equipped with a unitary representation of [MATH], a unitary error basis is [MATH]-equivariant when the elements are permuted by the natural action [MATH] of [MATH] on [MATH].', '1603.08866-2-49-1': 'Explicitly, [MATH] for some permutation [MATH] of the set [MATH].', '1603.08866-2-50-0': 'Procedure [REF] will succeed for any reference frame misalignment [MATH] just when the unitary error basis [MATH] is [MATH]-equivariant.', '1603.08866-2-51-0': "We again work in Alice and Bob's original lab frame.", '1603.08866-2-51-1': 'Alice decoheres in the orthonormal basis [MATH].', '1603.08866-2-51-2': 'Bob then measures in the orthonormal basis [MATH], and, depending on his measurement outcome [MATH], performs the corresponding correction [MATH].', '1603.08866-2-52-0': "We first note that, putting Alice's decoherence and Bob's measurement together as one operation, we get a teleportation scheme under Definition [REF].", '1603.08866-2-52-1': "Therefore, by Theorem [REF], Alice's decoherence operation followed by Bob's measurement must be a measurement in some orthonormal basis of maximally-entangled states; clearly that must be the basis that Bob measures in.", '1603.08866-2-52-2': "Letting Bob's measurement channel be [MATH] and Alice's decohering channel be [MATH], it follows that [MATH]; this can clearly only be true if the projection basis for [MATH] is the same as the projection basis for [MATH].", '1603.08866-2-52-3': 'We therefore see that the basis [MATH] must be some reordering of the basis [MATH], for all [MATH].', '1603.08866-2-52-4': 'This is exactly [MATH]-equivariance of the UEB [MATH].', '1603.08866-2-53-0': 'We now demonstrate that this condition is sufficient to guarantee success for Procedure [REF].', '1603.08866-2-53-1': 'Suppose [MATH] is [MATH]-equivariant.', '1603.08866-2-53-2': "Then Alice's decohering operation is exactly the same as it would have been if her reference frame had not shifted at all.", '1603.08866-2-53-3': 'Bob measures and performs the correction; the correction therefore corresponds to the measurement and the result follows.', '1603.08866-2-54-0': '# Classical structures in [MATH]', '1603.08866-2-55-0': 'Teleportation in the context of a finite group [MATH] can be described elegantly in the framework of categorical quantum mechanics [CITATION].', '1603.08866-2-55-1': 'One key strategy in this research programme is to understand features of quantum information in terms of the category [MATH] of finite-dimensional Hilbert spaces and linear maps, and then to generalize them by applying them in different categories.', '1603.08866-2-55-2': 'The concept of [MATH]-equivariant quantum teleportation arises by understanding the categorical structure of the traditional quantum teleportation procedure, and then applying it in [MATH], as we now explore.', '1603.08866-2-55-3': 'This technical section of the paper will make use of well-known ideas from categorical quantum mechanics, of which full details are available in the provided references.', '1603.08866-2-56-0': 'The following definition gives our abstract categorical description of quantum teleportation, in terms of classical structures in a symmetric monoidal category [CITATION].', '1603.08866-2-57-0': 'In a dagger-compact category, a quantum teleportation procedure on an object [MATH] with a right dual is a classical structure on the object [MATH], satisfying the following condition, where [MATH] is some scalar: [EQUATION]', '1603.08866-2-57-1': "This definition is motivated by the following theorem; recall Werner's Theorem [REF].", '1603.08866-2-58-0': 'Quantum teleportation procedures in [MATH] correspond precisely to unitary error bases.', '1603.08866-2-59-0': 'We now summarize the application of these ideas in a group representation category.', '1603.08866-2-60-0': 'For a group [MATH], the dagger-compact category [MATH] has objects given by unitary representations of [MATH], morphisms given by intertwiners, and a dagger-compact structure inherited from the underlying Hilbert spaces.', '1603.08866-2-61-0': 'Quantum teleportation procedures in [MATH] correspond precisely to [MATH]-equivariant unitary error bases.', '1603.08866-2-62-0': 'Finally, we observe that the constructions of unitary error bases in Theorem [REF] and Remark [REF] carry over straightforwardly to the [MATH]-equivariant setting because they are essentially categorical constructions; the Hadamard construction, for instance, is defined in terms of two special commutative dagger Frobenius algebras and an isomorphism between them.', '1603.08866-2-62-1': 'In Rep([MATH]), this reduces exactly to the intertwining Hadamard matrix and [MATH]-equivariant orthonormal basis of Theorem [REF].', '1603.08866-2-62-2': 'In this sense, these constructions are much more natural than, for instance, the construction of unitary error bases using projective group representations [CITATION]; indeed, it is difficult to see how the latter construction could be brought into the [MATH]-equivariant framework.', '1603.08866-2-63-0': '# Existence and construction of RFI teleportation protocols', '1603.08866-2-64-0': 'We have demonstrated that [MATH]-equivariant UEBs are exactly the structures we need to perform reference frame-independent teleportation protocols, but it is still unclear how to construct them for a given representation [MATH], if they exist at all.', '1603.08866-2-64-1': 'We cannot hope for a general classification of [MATH]-equivariant UEBs, since there is not even a classification in the case where the [MATH]-action is trivial, although many construction methods exist [CITATION].', '1603.08866-2-64-2': 'In this section we will demonstrate that [MATH]-equivariant unitary error bases need not exist on every representation, meaning that RFI teleportation is not always possible.', '1603.08866-2-64-3': 'We will then demonstrate that several UEB constructions carry over naturally to the [MATH]-equivariant setting, allowing us to construct RFI teleportation protcols for a wide variety of systems.', '1603.08866-2-65-0': 'We begin with a definition.', '1603.08866-2-66-0': 'A [MATH]-equivariant orthonormal basis for some representation [MATH] is an orthonormal basis of [MATH] whose elements are permuted by the action of [MATH].', '1603.08866-2-67-0': '[MATH]-equivariant unitary error bases are [MATH]-equivariant orthonormal bases of [MATH], all of whose elements are unitary maps.', '1603.08866-2-68-0': 'It will transpire that we can use [MATH]-equivariant orthonormal bases on [MATH] to construct [MATH]-equivariant UEBs for [MATH].', '1603.08866-2-68-1': 'Moreover, if we prove that there are no [MATH]-equivariant orthonormal bases on [MATH], it follows by Remark [REF] that there will be no [MATH]-equivariant UEBs for [MATH]; we will use this fact to demonstrate that RFI teleportation protocols need not always exist.', '1603.08866-2-68-2': 'Our first step is therefore a classification of [MATH]-equivariant orthonormal bases.', '1603.08866-2-69-0': '## A classification of [MATH]-equivariant orthonormal bases', '1603.08866-2-70-0': 'We begin with a simple lemma.', '1603.08866-2-70-1': 'Let [MATH]-Set be the category whose objects are sets carrying an action of [MATH], and whose morphisms are [MATH]-equivariant functions between them.', '1603.08866-2-70-2': 'Then there exists a functor [MATH]G[MATH]G[MATH], which, given a [MATH]-set, constructs the free Hilbert space on its elements, and extends the [MATH]-action and morphisms linearly.', '1603.08866-2-71-0': '[MATH]-equivariant orthonormal bases exist only on representations isomorphic to those in the image of [MATH].', '1603.08866-2-72-0': 'Immediate, since a [MATH]-equivariant orthonormal basis has an underlying Hilbert space isomorphic to the free Hilbert space on the elements of the chosen basis, which [MATH] acts on by permutations.', '1603.08866-2-73-0': 'We begin by presenting a simple classification of [MATH]-sets due to Burnside [CITATION].', '1603.08866-2-74-0': 'Given two [MATH]-sets [MATH] and [MATH], their disjoint union [MATH] is the disjoint union of [MATH] and [MATH] as sets with the natural induced action.', '1603.08866-2-75-0': 'Given a subgroup [MATH] of [MATH], the coset space [MATH] is the [MATH]-set whose elements are the cosets of [MATH] in [MATH], and whose [MATH]-action [MATH] is the natural action of [MATH] by left multiplication on those cosets.', '1603.08866-2-76-0': 'Any [MATH]-set is isomorphic to a disjoint union of coset spaces.', '1603.08866-2-76-1': 'Two coset spaces are isomorphic as [MATH]-sets if and only they correspond to conjugate subgroups.', '1603.08866-2-77-0': 'See [CITATION].', '1603.08866-2-78-0': 'In modern language, Lemma [REF] states that [MATH]-Set is a semisimple fusion category whose simple objects correspond to conjugacy classes of subgroups in [MATH].', '1603.08866-2-78-1': 'It is easy to see that the functor [MATH] is additive; the disjoint union of two [MATH]-sets will be sent under [MATH] to the direct sum of their corresponding representations.', '1603.08866-2-78-2': 'In order to classify all objects in the image of [MATH], therefore, it is sufficient to find the image of the coset spaces under [MATH].', '1603.08866-2-78-3': 'We will call those representations the basic permutation representations.', '1603.08866-2-79-0': 'In order to identify the basic permutation representations, we now state an obvious but critical lemma regarding the character of the permutation representation induced by [MATH] on a [MATH]-set.', '1603.08866-2-80-0': 'Given a [MATH]-set [MATH], let [MATH] be the character of [MATH].', '1603.08866-2-80-1': 'Then the following holds: [EQUATION]', '1603.08866-2-80-2': 'The character [MATH] is exactly the trace of the matrix representing [MATH]; the result follows trivially from the definition of [MATH].', '1603.08866-2-81-0': 'We may therefore identify the basic permutation representations by taking a representative of every conjugacy class of subgroups of [MATH], finding the number of fixed points of the action of each element of [MATH] on the corresponding coset spaces, then decomposing the resulting characters using the character table to find the corresponding representations.', '1603.08866-2-82-0': '## Existence of RFI teleportation protocols', '1603.08866-2-83-0': 'Using the results of Subsection [REF], we now exhibit a representation for which no [MATH]-equivariant UEBs exist, and on which quantum teleportation is therefore impossible.', '1603.08866-2-84-0': 'There is no RFI protocol to teleport the state of the 2-dimensional irreducible representation [MATH] of [MATH].', '1603.08866-2-85-0': 'Using the method outlined in Subsection [REF], we find that the characters of the basic permutation representations are as follows:', '1603.08866-2-86-0': 'The character of [MATH] is [MATH], which clearly cannot be composed as a sum of characters of basic permutation representations.', '1603.08866-2-86-1': 'By Remark [REF], the result follows.', '1603.08866-2-87-0': 'This argument does not extend to all irreducible representations.', '1603.08866-2-87-1': 'The endomorphism space of the 2-dimensional irreducible representation of [MATH], for instance, is a sum of basic permutation representations.', '1603.08866-2-88-0': '## Construction of RFI teleportation protocols', '1603.08866-2-89-0': 'Although RFI teleportation protocols need not always exist, they can often be constructed.', '1603.08866-2-89-1': 'We now demonstrate that, if we can find a [MATH]-equivariant orthonormal basis on [MATH], and a Hadamard matrix which commutes with all [MATH] in that basis, we can perform RFI teleportation on [MATH].', '1603.08866-2-90-0': 'Let [MATH] be a [MATH]-equivariant orthonormal basis on [MATH].', '1603.08866-2-90-1': 'In this basis all [MATH] will be permutation matrices.', '1603.08866-2-90-2': 'Let [MATH] be a Hadamard matrix that commutes with all [MATH] in this basis.', '1603.08866-2-90-3': 'Then the following family is a [MATH]-equivariant UEB: [EQUATION]', '1603.08866-2-90-4': 'It was already proved in [CITATION] that this is a UEB; we therefore need only show that it is [MATH]-equivariant.', '1603.08866-2-90-5': 'Since [MATH] we have that [MATH].', '1603.08866-2-90-6': 'We see easily that [MATH].', '1603.08866-2-90-7': 'Now note that the fact that [MATH] commutes with all elements of [MATH] means that permuting the columns of [MATH] is exactly the same as permuting the rows, since [MATH] for all [MATH].', '1603.08866-2-90-8': 'So [MATH].', '1603.08866-2-90-9': 'A similar argument works for [MATH].', '1603.08866-2-91-0': 'If the assumptions of Theorem [REF] are satisfied, it is possible to construct many more [MATH]-equivariant UEBs using quantum Latin squares (QLSs) [CITATION]; this construction will give [MATH]-equivariant UEBs provided the linear map defining the QLS is an intertwiner.', '1603.08866-2-92-0': 'We finish this section with a simple sufficient condition for the existence of tight RFI protocols on systems of dimension less than 5.', '1603.08866-2-92-1': 'Firstly we prove a lemma.', '1603.08866-2-93-0': 'Let [MATH] be a matrix of dimension [MATH] defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-2-93-1': 'Let [MATH] where [MATH] and [MATH].', '1603.08866-2-93-2': 'Then [MATH] is unitary precisely when the following conditions are satisfied:', '1603.08866-2-94-0': '0.3 [EQUATION] 0.33 [EQUATION] 0.36 [EQUATION] 0pt', '1603.08866-2-95-0': 'For unitarity it is sufficient that the rows form an orthonormal basis.', '1603.08866-2-95-1': 'It is clear from the symmetry of [MATH] that it is sufficient for one row vector to be normal, and one pair of row vectors to be orthogonal.', '1603.08866-2-95-2': 'This gives us two equations in [MATH] and [MATH]: [EQUATION]', '1603.08866-2-95-3': 'We will demonstrate that ([REF]) is necessary and sufficient for us to find [MATH] satisfying these equations.', '1603.08866-2-95-4': 'It is obvious that ([REF]) is satisfiable if and only if [MATH].', '1603.08866-2-95-5': 'Letting [MATH], Equation ([REF]) becomes [EQUATION].', '1603.08866-2-95-6': 'Since [MATH] and [MATH] can be freely adjusted to give [MATH] any value in that range, we see that the following is necessary and sufficient for ([REF]) to be soluble: [EQUATION]', '1603.08866-2-95-7': 'Use of the identity ([MATH]) and a short calculation demonstrates that this is equivalent to the lower bound in the inequality ([REF]).', '1603.08866-2-96-0': 'Suppose [MATH] admits a [MATH]-equivariant orthonormal basis, and is of dimension less than 5.', '1603.08866-2-96-1': 'Then there exists a RFI teleportation protocol for [MATH].', '1603.08866-2-97-0': 'We construct a [MATH]-equivariant UEB for [MATH].', '1603.08866-2-97-1': 'Expressed in the [MATH]-equivariant orthonormal basis, [MATH] will be some subgroup of the permutation matrices [MATH].', '1603.08866-2-97-2': 'To use Theorem [REF], we must find a Hadamard matrix commuting with [MATH].', '1603.08866-2-97-3': 'In the worst case, [MATH] will be the whole group [MATH] of permutation matrices.', '1603.08866-2-97-4': '(This situation is realised for the representation [MATH] of [MATH], where [MATH] is the fundamental [MATH]-dimensional representation of [MATH]).', '1603.08866-2-98-0': 'We will demonstrate that, when [MATH] is of dimension less than 5, we can find a Hadamard matrix which commutes with all the permutation matrices.', '1603.08866-2-98-1': 'First we eliminate the degenerate cases [MATH] and [MATH].', '1603.08866-2-98-2': 'Clearly for [MATH] we can perform RFI teleportation by Proposition [REF], For [MATH] the following family of Hadamard matrices commutes with [MATH], where [MATH] and [MATH]: [EQUATION]', '1603.08866-2-98-3': 'From now on we may therefore assume [MATH].', '1603.08866-2-99-0': 'It is easy to see that the centraliser [MATH] is the set of matrices defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-2-99-1': 'The conditions necessary for such a matrix to be unitary were given in Lemma [REF].', '1603.08866-2-99-2': 'Setting [MATH] in ([REF]), it follows that [MATH].', '1603.08866-2-99-3': 'This is compatible with ([REF]) only for [MATH].'}

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[]

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[]

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{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1603.08866

{'1603.08866-3-0-0': 'We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group.', '1603.08866-3-0-1': 'Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size.', '1603.08866-3-0-2': 'In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.', '1603.08866-3-1-0': '# Introduction', '1603.08866-3-2-0': '## The problem and our result', '1603.08866-3-3-0': "Recently many authors have recognised the importance of developing a theory of quantum information which takes account of the reference frames by which a system's state is defined [CITATION].", '1603.08866-3-3-1': 'It was recognised some time ago [CITATION] that a shared reference frame is a hidden implicit assumption in the conventional description of quantum teleportation, an assumption which it is reasonable to challenge, since it is clearly possible that two parties attempting teleportation across a large distance will be uncertain about their relative reference frames.', '1603.08866-3-3-2': 'Quantum teleportation is a foundational quantum protocol with important applications [CITATION]; the problem of teleporting a quantum state between two parties who do not share a reference frame is therefore natural and important.', '1603.08866-3-4-0': "A successful reference frame-independent teleportation protocol is one where any observer with their own fixed reference frame will agree that Alice's quantum state has been successfully transferred to Bob, regardless of the orientation of Alice and Bob's frames.", '1603.08866-3-4-1': "Here we consider perfect teleportation, where Alice's state is transferred to Bob with certainty.", '1603.08866-3-5-0': 'The effect of a change in reference frame alignment on the perceived state of a system may be encoded formally as the unitary action of a group of reference frame transformations on the Hilbert space of the system under consideration.', '1603.08866-3-5-1': 'It has been shown [CITATION] that if the group of reference frame transformations contains [MATH] and the action is nontrivial, then reference frame-independent teleportation is impossible.', '1603.08866-3-5-2': 'In this paper we show that when the group of reference frame transformations is finite, reference frame-independent teleportation is possible in some nontrivial cases.', '1603.08866-3-6-0': 'We will demonstrate that reference frame-independent (RFI) teleportation protocols correspond exactly to [MATH]-equivariant unitary error bases, a structure we define, and develop methods of constructing these bases when they exist.', '1603.08866-3-6-1': 'In particular, we provide a simple sufficient condition for the existence of RFI teleportation protocols for systems of dimension less than 5.', '1603.08866-3-7-0': "The nature of the classical channel through which the result of Alice's measurement is communicated is crucial to our new protocol.", '1603.08866-3-7-1': 'In the terminology of Peres and Scudo [CITATION], our protocol requires classical transmission of unspeakable information, rather than of speakable information.', '1603.08866-3-7-2': 'An example of unspeakable information is the choice of a direction in space, to be agreed by two parties who do not share a common directional reference; no amount of communication through a generic shared classical channel can decide the matter, but the fidelitous transfer from one party to the other of a single oriented physical system, such as an arrow, is sufficient.', '1603.08866-3-8-0': 'We foresee two further applications of this result, both of which merit further investigation:', '1603.08866-3-9-0': 'The fundamental concept of a [MATH]-equivariant unitary error basis was developed from investigations in categorical quantum mechanics [CITATION].', '1603.08866-3-9-1': 'We characterised teleportation schemes as structures internal to the category of finite dimensional Hilbert spaces, and investigated corresponding structures in the category of unitary representations of a finite group; these corresponding structures were exactly the [MATH]-equivariant unitary error bases.', '1603.08866-3-9-2': 'Following their discovery, further investigation demonstrated their relevance to the problem of RFI teleportation.', '1603.08866-3-9-3': 'This exemplifies the utility of categorical quantum mechanics as a toolkit for developing new and interesting concepts in quantum information.', '1603.08866-3-10-0': 'The outline of the paper is as follows.', '1603.08866-3-10-1': 'In Section [REF] we give an informal worked example of our procedure, and in Section [REF] we provide a more formal presentation, along with further examples.', '1603.08866-3-10-2': 'In Section [REF] we show how reference frame-independent teleportation schemes are related to ideas in categorical quantum mechanics.', '1603.08866-3-10-3': 'In Section [REF] we prove a variety of existence, nonexistence and construction results for [MATH]-equivariant unitary error bases.', '1603.08866-3-11-0': '## Previous results', '1603.08866-3-12-0': 'It was demonstrated in [CITATION] that teleportation is impossible when the group of reference frame transformations is a infinite compact connected Lie group and the representation on the system to be teleported does not factor through a representation of a finite group.', '1603.08866-3-12-1': 'As observed in that paper, this leaves open the question of whether quantum teleportation is possible in the case of a finite group of reference frame transformations, without additional resources or prior communication.', '1603.08866-3-12-2': 'The protocol we exhibit provides an affirmative answer to this question.', '1603.08866-3-13-0': 'A number of other solutions for the finite case have been proposed.', '1603.08866-3-13-1': 'In [CITATION], it was suggested that Alice could transmit half of a maximally-entangled token state, in the regular representation, in advance of performing the protocol; the two parties could then use this to synchronise their operations.', '1603.08866-3-13-2': 'This method, however, requires Alice to be able to initialise an entangled state on a pair of systems each carrying the regular representation of the transformation group, a procedure which may be experimentally difficult or impossible.', '1603.08866-3-14-0': 'Another relevant result can be found in [CITATION], where it was shown that it is possible to perform quantum protocols using a second shared system as a quantum reference frame.', '1603.08866-3-14-1': 'This general result, intended for application to quantum cryptography, may be used to create reference frame-independent teleportation protocols.', '1603.08866-3-14-2': 'These protocols are formally identical to the token state method, but operationally more practicable; Alice and Bob simply initialise an additional shared entangled state at the same time as they create the first, take half each and use it to synchronise their operations.', '1603.08866-3-14-3': 'The problems of the token state method therefore persist in this case, although without the additional difficulty of communicating half of the second entangled state from one party to the other.', '1603.08866-3-15-0': "Finally, various solutions have also been proposed which use prior communication to align both parties' reference frames in advance of performing a normal teleportation protocol; see [CITATION].", '1603.08866-3-15-1': 'This increases the amount of classical information that must be communicated for successful teleportation.', '1603.08866-3-15-2': "Moreover, this procedure is not robust against changes in the alignment of Alice and Bob's reference frames, which must stay constant if the protocol is to succeed.", '1603.08866-3-15-3': 'Our protocol, in contrast, is robust against reference frame changes even while the classical message from Alice to Bob is in transit.', '1603.08866-3-15-4': "We only require that the alignment of Bob's reference frame is constant for the short time between his receipt of the classical information and his application of the unitary correction.", '1603.08866-3-16-0': 'However, we emphasize that our solution applies only when the system to be teleported carries a representation of the group of reference frame transformations [MATH] which admits a [MATH]-equivariant unitary error basis.', '1603.08866-3-16-1': 'The other approaches described above do not have this limitation.', '1603.08866-3-17-0': '# Example of the procedure', '1603.08866-3-18-0': 'In this section we give an informal account of the problem of reference frame-independent quantum teleportation, in the specific case where the systems are two-dimensional and the reference frame corresponds to a choice of spatial direction.', '1603.08866-3-18-1': 'This is followed by a more general and mathematically precise treatment in the next section.', '1603.08866-3-19-0': 'Alice and Bob are quantum information theorists in separate laboratories, which do not necessarily have the same orientation in space.', '1603.08866-3-19-1': "However, their relative orientations are not completely unknown: we are given some finite subgroup [MATH], the group of rigid spatial rotations, with the promise that there is some element [MATH] which relates Alice's and Bob's frames.", '1603.08866-3-19-2': 'The group [MATH] is common knowledge to both parties.', '1603.08866-3-20-0': 'The task is to perform teleportation of a quantum state from Alice to Bob, without revealing their spatial orientations, either to each other or to a potential eavesdropper.', '1603.08866-3-20-1': 'There are a variety of reasons why this may be advantageous: this information may be strategically or cryptographically valuable, and hence they may prefer not to divulge it for reasons of privacy; they may prefer to conserve limited bandwidth, and hence to not communicate redundant reference frame alignment information if it can be avoided; or they may simply be disoriented, and not aware of their own orientations.', '1603.08866-3-21-0': 'In this example, we consider the case that [MATH], meaning that their laboratories may be in one of two possible orientations which are related by a 180[MATH] rotation about some given axis.', '1603.08866-3-21-1': 'We suppose that the nontrivial element [MATH] acts on the qubit to be teleported as follows: [EQUATION].', '1603.08866-3-21-2': 'The fact that this matrix is nontrivial corresponds to the fact that the quantum system is not symmetric under the rotation operation.', '1603.08866-3-22-0': 'Alice and Bob agree in advance to perform quantum teleportation as follows.', '1603.08866-3-22-1': 'Alice will measure her initial system together with her part of the entangled state in the basis [MATH] and communicate the result to Bob, who will apply the corresponding correction [MATH].', '1603.08866-3-22-2': 'We define [MATH], where [MATH] denotes the transpose of the matrix [MATH], the symbol [MATH] denotes the 2-by-2 identity matrix, and [MATH] is the Bell state given above.', '1603.08866-3-22-3': 'The [MATH] are defined as follows: [EQUATION]', '1603.08866-3-22-4': 'It can readily be checked that this data forms an unitary error basis, and so by the results of Werner [CITATION] gives correct data for the execution of an ordinary quantum teleportation procedure for a single qubit, when the shared state is the Bell state [MATH].', '1603.08866-3-23-0': "If Bob's reference frame is correctly aligned with Alice's, then they are carrying out ordinary quantum teleportation, and the procedure will be successful.", '1603.08866-3-23-1': "However, if his reference direction is upside-down with respect to Alice's, then teleportation does not proceed successfully.", '1603.08866-3-23-2': "From Alice's perspective, Bob's correction is not in fact the unitary [MATH] corresponding to her measurement result, but rather the unitary [MATH]; a straightforward calculation then shows that Bob will receive a mixed state, and quantum information has been irrevocably lost.", '1603.08866-3-23-3': "From Bob's perspective, he correctly applied the unitary [MATH], but the teleportation failed because the measurement result [MATH] that Alice communicated to him did not correspond to the state she actually measured, which was [MATH].", '1603.08866-3-24-0': 'We now provide a resolution.', '1603.08866-3-24-1': 'In our new procedure, rather than communicating the two bits encoding the measurement result to Bob using their shared classical channel, Alice sends two physical objects to Bob: arrows, of the sort a medieval archer might use.', '1603.08866-3-24-2': 'She orients these arrows according to the measurement result that she obtained, using the following encoding, where her reference direction is written as [MATH]: [EQUATION]', '1603.08866-3-24-3': "One at a time, she physically sends these arrows through space to Bob's laboratory.", '1603.08866-3-25-0': 'Bob observes their local orientations and infers the measurement result 0, 1, 2 or 3 that Alice obtained.', '1603.08866-3-25-1': "Suppose Bob's laboratory is correctly aligned with Alice's; then he will correctly infer Alice's measurement result, and he will apply the corresponding unitary correction.", '1603.08866-3-25-2': 'In this case, the two parties have executed a traditional quantum teleportation protocol, albeit one where the two classical bits of information were transferred from Alice to Bob in an unusual way.', '1603.08866-3-26-0': "Now we suppose that Bob's laboratory is aligned upside-down with respect to Alice's.", '1603.08866-3-26-1': 'If Alice attempts to send the message 0, 1, 2 or 3, Bob will receive it as 1, 0, 3 or 2 respectively, since the arrows will appear to him with the opposite orientations.', '1603.08866-3-26-2': "Furthermore, just as before, when Bob applies a unitary [MATH], its action is seen in Alice's reference frame as [MATH].", '1603.08866-3-26-3': 'We now see the point of the entire construction: the unitary error basis is carefully chosen so that these effects cancel out.', '1603.08866-3-26-4': 'Indeed, the following equations can be easily verified: [EQUATION]', '1603.08866-3-26-5': "As a result, the quantum teleporation will conclude successfully, even though Alice and Bob's reference directions were misaligned.", '1603.08866-3-27-0': 'In summary, by a careful choice of the unitary error basis, and by transferring the measurement result as unspeakable rather than speakable information, the quantum teleportation procedure can be carried out in a way which is robust against this restricted sort of reference frame error.', '1603.08866-3-27-1': 'Note in particular that only 2 bits of classical information were transferred from Alice to Bob, just as with the traditional teleportation procedure, and the Hilbert space of the entangled resource is of minimal dimension, so this procedure is tight in the sense of Werner [CITATION].', '1603.08866-3-27-2': "Also note that the unspeakable information Bob receives from Alice is uniformly random, since Alice's measurement results are; in particular, Bob receives no information during the protocol about the relative alignment between the two reference frames.", '1603.08866-3-27-3': "Finally, it is clear that the procedure would succeed even if Bob's reference frame were constantly changing between the two alignments, as long as the alignment stays constant between Bob's receipt of the arrows and his application of the unitary correction.", '1603.08866-3-28-0': '# Mathematical description of the proposal', '1603.08866-3-29-0': '## Traditional teleportation', '1603.08866-3-30-0': 'Teleportation is a well-understood procedure.', '1603.08866-3-30-1': 'It is traditionally formulated under the assumption that both parties have aligned reference frames [CITATION].', '1603.08866-3-30-2': "(Note that we only consider tight quantum teleportation in this paper, where the state spaces of the initial system and of the entangled systems all have the same dimension, which is equal to the number of classical bits transferred, and the procedure succeeds with probability 1; this corresponds to the informal restriction of 'no additional resources or prior communication'.)", '1603.08866-3-30-3': 'The traditional formulation is as follows.', '1603.08866-3-31-0': '[Teleportation without communication of unspeakable information]', '1603.08866-3-32-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-3-32-1': 'She performs a measurement with respect to an orthonormal basis of effects [MATH] on the bipartite system built from her initial system and her half of the entangled state.', '1603.08866-3-32-2': 'She sends the measurement result [MATH] to Bob through a generic classical channel.', '1603.08866-3-32-3': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-3-32-4': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-3-33-0': 'A complete description of correct data [MATH] was given by Werner [CITATION], as follows.', '1603.08866-3-34-0': 'For a Hilbert space [MATH], a unitary error basis is a basis of unitary operators [MATH], which are orthonormal under the Hilbert-Schmidt inner product: [EQUATION] [Werner]', '1603.08866-3-35-0': 'Up to equivalence, teleportation schemes for systems with Hilbert space [MATH] are in one to one correspondence with unitary error bases on [MATH].', '1603.08866-3-36-0': 'Under this correspondence, the shared entangled state [MATH] is the Bell state [MATH] for any orthonormal basis [MATH].', '1603.08866-3-36-1': 'Alice measures in the maximally-entangled basis [MATH], where [MATH] is defined as [MATH].', '1603.08866-3-36-2': "Bob's correction for the measurement outcome [MATH] is [MATH].", '1603.08866-3-37-0': '## Reference frame-independent quantum teleportation', '1603.08866-3-38-0': 'We now fully describe the problem we solve.', '1603.08866-3-39-0': '[Reference frame-independent quantum teleportation] Alice and Bob are spatially-separated quantum information theorists capable of performing local operations and classical communication.', '1603.08866-3-39-1': 'Each party has one half of an entangled state which they created at some point in the past using a shared reference frame.', '1603.08866-3-39-2': 'Alice wants to use this entanglement to communicate a quantum state to Bob by teleportation.', '1603.08866-3-39-3': "However, their reference frames are now misaligned; in some observer's fixed reference frame, their frame alignments will be described by the action of unknown elements [MATH] of the group [MATH] of reference frame transformations.", '1603.08866-3-40-0': 'We will now show that the answer to (i) is almost always negative.', '1603.08866-3-40-1': 'First we note the following lemma.', '1603.08866-3-41-0': 'Under the conditions of Problem [REF], Procedure [REF] will work for all reference frame alignments if and only if the operations [MATH] and [MATH] are intertwiners for the group action.', '1603.08866-3-42-0': 'We express the operations with reference to the original shared reference frame.', '1603.08866-3-42-1': "Let Alice and Bob's frame shifts be described by group elements [MATH] and [MATH] respectively.", '1603.08866-3-42-2': 'Alice measures [MATH] relative to her reference frame; in the original frame the operation she has performed is [MATH].', '1603.08866-3-42-3': 'She then sends the result [MATH] to Bob, who performs the operation [MATH] relative to his frame; in the original frame the operation he has performed will be [MATH].', '1603.08866-3-42-4': 'In general, the channel will therefore only work for all reference frame configurations when, for all [MATH], [MATH] and [MATH], for some action of [MATH] on the set of measurement outcomes.', '1603.08866-3-42-5': 'Since [MATH] for the identity [MATH], this clearly implies that [MATH] for all [MATH].', '1603.08866-3-42-6': 'The result follows.', '1603.08866-3-43-0': 'We now demonstrate that Procedure [REF] works only for a trivial [MATH]-action, rendering it inadequate for reference frame-independent teleportation in any nontrivial case.', '1603.08866-3-44-0': 'Procedure [REF] will only work for all reference frame alignments when [MATH] acts by a global phase.', '1603.08866-3-45-0': 'By Theorem [REF] and Proposition [REF], Procedure [REF] will work only if all projections [MATH] and corrections [MATH] are intertwiners.', '1603.08866-3-45-1': 'By the definition of [MATH] in Theorem [REF], it is sufficient that all [MATH] be intertwiners.', '1603.08866-3-45-2': 'Let us assume that this is the case.', '1603.08866-3-45-3': 'Since the [MATH]-action is trivial on a basis of [MATH], it must be completely trivial on [MATH].', '1603.08866-3-45-4': 'Therefore we have [MATH].', '1603.08866-3-45-5': 'By straightforward character theory, there can only be one copy of [MATH] in the product of an irreducible representation with its dual.', '1603.08866-3-45-6': 'Breaking [MATH] up into simple factors, it follows by counting dimensions that they must all be identical and one dimensional.', '1603.08866-3-46-0': '## Our new scheme', '1603.08866-3-47-0': 'In answer to (ii), we will now present our new scheme for teleportation using unspeakable information transfer.', '1603.08866-3-47-1': '[Teleportation with communication of unspeakable information]', '1603.08866-3-48-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-3-48-1': 'She forms the bipartite system given by her initial system together with her half of the entangled state, and sends it to Bob through a classical channel which is decoherent in the basis [MATH].', '1603.08866-3-48-2': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-3-48-3': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-3-49-0': 'The key aspect of Procedure [REF] is that misalignment of reference frames will not affect the way Bob perceives the data arriving through a generic classical channel, but it will affect his perception of the decohered bipartite system.', '1603.08866-3-49-1': 'In other words, the information Bob receives from Alice will depend in a nontrivial way on the alignment of his reference frame.', '1603.08866-3-50-0': 'The basic data of Procedure [REF] is the same as for Procedure [REF], and so Theorem [REF] still applies.', '1603.08866-3-50-1': 'However, not all unitary error bases give rise to successful teleporation schemes under this procedure.', '1603.08866-3-50-2': 'We now investigate which of them do.', '1603.08866-3-51-0': '[[MATH]-equivariant unitary error basis] For a Hilbert space [MATH] equipped with a unitary representation of [MATH], a unitary error basis is [MATH]-equivariant when the elements are permuted by the natural action [MATH] of [MATH] on [MATH].', '1603.08866-3-51-1': 'Explicitly, [MATH] for some permutation [MATH] of the set [MATH].', '1603.08866-3-52-0': 'Procedure [REF] will succeed for any reference frame misalignment [MATH] just when the unitary error basis [MATH] is [MATH]-equivariant.', '1603.08866-3-53-0': "We again work in Alice and Bob's original lab frame.", '1603.08866-3-53-1': 'Alice decoheres in the orthonormal basis [MATH].', '1603.08866-3-53-2': 'Bob then measures in the orthonormal basis [MATH], and, depending on his measurement outcome [MATH], performs the corresponding correction [MATH].', '1603.08866-3-54-0': "We first note that, putting Alice's decoherence and Bob's measurement together as one operation, we get a teleportation scheme under Definition [REF].", '1603.08866-3-54-1': "Therefore, by Theorem [REF], Alice's decoherence operation followed by Bob's measurement must be a measurement in some orthonormal basis of maximally-entangled states; clearly that must be the basis that Bob measures in.", '1603.08866-3-54-2': "Letting Bob's measurement channel be [MATH] and Alice's decohering channel be [MATH], it follows that [MATH]; this can clearly only be true if the projection basis for [MATH] is the same as the projection basis for [MATH].", '1603.08866-3-54-3': 'We therefore see that the basis [MATH] must be some reordering of the basis [MATH], for all [MATH].', '1603.08866-3-54-4': 'This is exactly [MATH]-equivariance of the UEB [MATH].', '1603.08866-3-55-0': 'We now demonstrate that this condition is sufficient to guarantee success for Procedure [REF].', '1603.08866-3-55-1': 'Suppose [MATH] is [MATH]-equivariant.', '1603.08866-3-55-2': "Then Alice's decohering operation is exactly the same as it would have been if her reference frame had not shifted at all.", '1603.08866-3-55-3': 'Bob measures and performs the correction; the correction therefore corresponds to the measurement and the result follows.', '1603.08866-3-56-0': 'In Procedure [REF], we have specified that Alice send the decohered bipartite system itself, since this is always theoretically possible.', '1603.08866-3-56-1': 'However, in practise it may be experimentally more practicable to use some other means of classically communicating unspeakable information; the important thing is that the classical data should itself carry the same [MATH]-action as the corresponding [MATH]-equivariant measurement basis.', '1603.08866-3-56-2': "An example was given in Section [REF], where Alice's measurement result was encoded in the spatial orientations of some physical objects.", '1603.08866-3-56-3': 'In order to demonstrate that this approach is applicable to other types of reference frame uncertainty, we provide two further examples of unspeakable encodings of classical information.', '1603.08866-3-57-0': '[(i)] Time.', '1603.08866-3-57-1': "Suppose the computational basis states of Alice and Bob's systems are nondegenerate energy eigenstates (for example, eigenstates of the photon number operator).", '1603.08866-3-57-2': 'Here they will need to share a time reference.', '1603.08866-3-57-3': 'Let the time translation operator [MATH] have periodicity [MATH] for some [MATH].', '1603.08866-3-57-4': 'Suppose that the group [MATH] of time translations has been discretised to some cyclic subgroup [MATH] of translations by [MATH].', '1603.08866-3-57-5': "Alice and Bob's reference frame configurations will correspond to their zeroes of time.", '1603.08866-3-57-6': 'Signals sent by Alice to to Bob which arrive, according to her reference frame, at time [MATH], arrive for Bob at a different time [MATH], depending on the difference between their reference frames.', '1603.08866-3-57-7': 'By encoding her measurement result in the time of arrival of signals, Alice may construct [MATH]-equivariant teleportation protocols.', '1603.08866-3-58-0': '[(ii)] Circular polarisation.', '1603.08866-3-58-1': 'Suppose Alice and Bob are working with photonic qubits whose computational basis states are left and right circular polarisation.', '1603.08866-3-58-2': 'In this case, the group of reference frame transformations will correspond to planar rotations of the axes perpendicular to the propagation direction.', '1603.08866-3-58-3': 'Suppose that the group [MATH] of planar rotations has been discretised to some cyclic subgroup [MATH] of rigid rotations by multiples of [MATH], and that Alice can classically communicate linearly polarised light to Bob.', '1603.08866-3-58-4': "By communicating frame configurations using beams of linearly polarised light, Alice may encode measurement results in the angle difference between Bob's frame and the commmunicated frame, allowing her to construct [MATH]-equivariant teleportation protocols.", '1603.08866-3-59-0': '# Classical structures in [MATH]', '1603.08866-3-60-0': 'Teleportation in the context of a finite group [MATH] can be described elegantly in the framework of categorical quantum mechanics [CITATION].', '1603.08866-3-60-1': 'One key strategy in this research programme is to understand features of quantum information in terms of the category [MATH] of finite-dimensional Hilbert spaces and linear maps, and then to generalize them by applying them in different categories.', '1603.08866-3-60-2': 'The concept of [MATH]-equivariant quantum teleportation arises by understanding the categorical structure of the traditional quantum teleportation procedure, and then applying it in [MATH], as we now explore.', '1603.08866-3-60-3': 'This technical section of the paper will make use of well-known ideas from categorical quantum mechanics, of which full details are available in the provided references.', '1603.08866-3-61-0': 'The following definition gives our abstract categorical description of quantum teleportation, in terms of classical structures in a symmetric monoidal category [CITATION].', '1603.08866-3-62-0': 'In a dagger-compact category, a quantum teleportation procedure on an object [MATH] with a right dual is a classical structure on the object [MATH], satisfying the following condition, where [MATH] is some scalar: [EQUATION]', '1603.08866-3-62-1': "This definition is motivated by the following theorem; recall Werner's Theorem [REF].", '1603.08866-3-63-0': 'Quantum teleportation procedures in [MATH] correspond precisely to unitary error bases.', '1603.08866-3-64-0': 'We now summarize the application of these ideas in a group representation category.', '1603.08866-3-65-0': 'For a group [MATH], the dagger-compact category [MATH] has objects given by unitary representations of [MATH], morphisms given by intertwiners, and a dagger-compact structure inherited from the underlying Hilbert spaces.', '1603.08866-3-66-0': 'Quantum teleportation procedures in [MATH] correspond precisely to [MATH]-equivariant unitary error bases.', '1603.08866-3-67-0': 'Finally, we observe that the constructions of unitary error bases in Theorem [REF] and Remark [REF] carry over straightforwardly to the [MATH]-equivariant setting because they are essentially categorical constructions; the Hadamard construction, for instance, is defined in terms of two special commutative dagger Frobenius algebras and an isomorphism between them.', '1603.08866-3-67-1': 'In Rep([MATH]), this reduces exactly to the intertwining Hadamard matrix and [MATH]-equivariant orthonormal basis of Theorem [REF].', '1603.08866-3-67-2': 'In this sense, these constructions are much more natural than, for instance, the construction of unitary error bases using projective group representations [CITATION]; indeed, it is difficult to see how the latter construction could be brought into the [MATH]-equivariant framework.', '1603.08866-3-68-0': '# Existence and construction of RFI teleportation protocols', '1603.08866-3-69-0': 'We have demonstrated that [MATH]-equivariant UEBs are exactly the structures we need to perform reference frame-independent teleportation protocols, but it is still unclear how to construct them for a given representation [MATH], if they exist at all.', '1603.08866-3-69-1': 'We cannot hope for a general classification of [MATH]-equivariant UEBs, since there is not even a classification in the case where the [MATH]-action is trivial, although many construction methods exist [CITATION].', '1603.08866-3-69-2': 'In this section we will demonstrate that [MATH]-equivariant unitary error bases need not exist on every representation, meaning that RFI teleportation is not always possible.', '1603.08866-3-69-3': 'We will then demonstrate that several UEB constructions carry over naturally to the [MATH]-equivariant setting, allowing us to construct RFI teleportation protcols for a wide variety of systems.', '1603.08866-3-70-0': 'We begin with a definition.', '1603.08866-3-71-0': 'A [MATH]-equivariant orthonormal basis for some representation [MATH] is an orthonormal basis of [MATH] whose elements are permuted by the action of [MATH].', '1603.08866-3-72-0': '[MATH]-equivariant unitary error bases are [MATH]-equivariant orthonormal bases of [MATH], all of whose elements are unitary maps.', '1603.08866-3-73-0': 'It will transpire that we can use [MATH]-equivariant orthonormal bases on [MATH] to construct [MATH]-equivariant UEBs for [MATH].', '1603.08866-3-73-1': 'Moreover, if we prove that there are no [MATH]-equivariant orthonormal bases on [MATH], it follows by Remark [REF] that there will be no [MATH]-equivariant UEBs for [MATH]; we will use this fact to demonstrate that RFI teleportation protocols need not always exist.', '1603.08866-3-73-2': 'Our first step is therefore a classification of [MATH]-equivariant orthonormal bases.', '1603.08866-3-74-0': '## A classification of [MATH]-equivariant orthonormal bases', '1603.08866-3-75-0': 'We begin with a simple lemma.', '1603.08866-3-75-1': 'Let [MATH]-Set be the category whose objects are sets carrying an action of [MATH], and whose morphisms are [MATH]-equivariant functions between them.', '1603.08866-3-75-2': 'Then there exists a functor [MATH]G[MATH]G[MATH], which, given a [MATH]-set, constructs the free Hilbert space on its elements, and extends the [MATH]-action and morphisms linearly.', '1603.08866-3-76-0': '[MATH]-equivariant orthonormal bases exist only on representations isomorphic to those in the image of [MATH].', '1603.08866-3-77-0': 'Immediate, since a [MATH]-equivariant orthonormal basis has an underlying Hilbert space isomorphic to the free Hilbert space on the elements of the chosen basis, which [MATH] acts on by permutations.', '1603.08866-3-78-0': 'We begin by presenting a simple classification of [MATH]-sets due to Burnside [CITATION].', '1603.08866-3-79-0': 'Given two [MATH]-sets [MATH] and [MATH], their disjoint union [MATH] is the disjoint union of [MATH] and [MATH] as sets with the natural induced action.', '1603.08866-3-80-0': 'Given a subgroup [MATH] of [MATH], the coset space [MATH] is the [MATH]-set whose elements are the cosets of [MATH] in [MATH], and whose [MATH]-action [MATH] is the natural action of [MATH] by left multiplication on those cosets.', '1603.08866-3-81-0': 'Any [MATH]-set is isomorphic to a disjoint union of coset spaces.', '1603.08866-3-81-1': 'Two coset spaces are isomorphic as [MATH]-sets if and only they correspond to conjugate subgroups.', '1603.08866-3-82-0': 'See [CITATION].', '1603.08866-3-83-0': 'In modern language, Lemma [REF] states that [MATH]-Set is a semisimple fusion category whose simple objects correspond to conjugacy classes of subgroups in [MATH].', '1603.08866-3-83-1': 'It is easy to see that the functor [MATH] is additive; the disjoint union of two [MATH]-sets will be sent under [MATH] to the direct sum of their corresponding representations.', '1603.08866-3-83-2': 'In order to classify all objects in the image of [MATH], therefore, it is sufficient to find the image of the coset spaces under [MATH].', '1603.08866-3-83-3': 'We will call those representations the basic permutation representations.', '1603.08866-3-84-0': 'In order to identify the basic permutation representations, we now state an obvious but critical lemma regarding the character of the permutation representation induced by [MATH] on a [MATH]-set.', '1603.08866-3-85-0': 'Given a [MATH]-set [MATH], let [MATH] be the character of [MATH].', '1603.08866-3-85-1': 'Then the following holds: [EQUATION]', '1603.08866-3-85-2': 'The character [MATH] is exactly the trace of the matrix representing [MATH]; the result follows trivially from the definition of [MATH].', '1603.08866-3-86-0': 'We may therefore identify the basic permutation representations by taking a representative of every conjugacy class of subgroups of [MATH], finding the number of fixed points of the action of each element of [MATH] on the corresponding coset spaces, then decomposing the resulting characters using the character table to find the corresponding representations.', '1603.08866-3-87-0': '## Existence of RFI teleportation protocols', '1603.08866-3-88-0': 'Using the results of Subsection [REF], we now exhibit a representation for which no [MATH]-equivariant UEBs exist, and on which quantum teleportation is therefore impossible.', '1603.08866-3-89-0': 'There is no RFI protocol to teleport the state of the 2-dimensional irreducible representation [MATH] of [MATH].', '1603.08866-3-90-0': 'Using the method outlined in Subsection [REF], we find that the characters of the basic permutation representations are as follows:', '1603.08866-3-91-0': 'The character of [MATH] is [MATH], which clearly cannot be composed as a sum of characters of basic permutation representations.', '1603.08866-3-91-1': 'By Remark [REF], the result follows.', '1603.08866-3-92-0': 'This argument does not extend to all irreducible representations.', '1603.08866-3-92-1': 'The endomorphism space of the 2-dimensional irreducible representation of [MATH], for instance, is a sum of basic permutation representations.', '1603.08866-3-93-0': '## Construction of RFI teleportation protocols', '1603.08866-3-94-0': 'Although RFI teleportation protocols need not always exist, they can often be constructed.', '1603.08866-3-94-1': 'We now demonstrate that, if we can find a [MATH]-equivariant orthonormal basis on [MATH], and a Hadamard matrix which commutes with all [MATH] in that basis, we can perform RFI teleportation on [MATH].', '1603.08866-3-95-0': 'Let [MATH] be a [MATH]-equivariant orthonormal basis on [MATH].', '1603.08866-3-95-1': 'In this basis all [MATH] will be permutation matrices.', '1603.08866-3-95-2': 'Let [MATH] be a Hadamard matrix that commutes with all [MATH] in this basis.', '1603.08866-3-95-3': 'Then the following family is a [MATH]-equivariant UEB: [EQUATION]', '1603.08866-3-95-4': 'It was already proved in [CITATION] that this is a UEB; we therefore need only show that it is [MATH]-equivariant.', '1603.08866-3-95-5': 'Since [MATH] we have that [MATH].', '1603.08866-3-95-6': 'We see easily that [MATH].', '1603.08866-3-95-7': 'Now note that the fact that [MATH] commutes with all elements of [MATH] means that permuting the columns of [MATH] is exactly the same as permuting the rows, since [MATH] for all [MATH].', '1603.08866-3-95-8': 'So [MATH].', '1603.08866-3-95-9': 'A similar argument works for [MATH].', '1603.08866-3-96-0': 'If the assumptions of Theorem [REF] are satisfied, it is possible to construct many more [MATH]-equivariant UEBs using quantum Latin squares (QLSs) [CITATION]; this construction will give [MATH]-equivariant UEBs provided the linear map defining the QLS is an intertwiner.', '1603.08866-3-97-0': 'We finish this section with a simple sufficient condition for the existence of tight RFI protocols on systems of dimension less than 5.', '1603.08866-3-97-1': 'Firstly we prove a lemma.', '1603.08866-3-98-0': 'Let [MATH] be a matrix of dimension [MATH] defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-3-98-1': 'Let [MATH] where [MATH] and [MATH].', '1603.08866-3-98-2': 'Then [MATH] is unitary precisely when the following conditions are satisfied:', '1603.08866-3-99-0': '0.3 [EQUATION] 0.33 [EQUATION] 0.36 [EQUATION] 0pt', '1603.08866-3-100-0': 'For unitarity it is sufficient that the rows form an orthonormal basis.', '1603.08866-3-100-1': 'It is clear from the symmetry of [MATH] that it is sufficient for one row vector to be normal, and one pair of row vectors to be orthogonal.', '1603.08866-3-100-2': 'This gives us two equations in [MATH] and [MATH]: [EQUATION]', '1603.08866-3-100-3': 'We will demonstrate that ([REF]) is necessary and sufficient for us to find [MATH] satisfying these equations.', '1603.08866-3-100-4': 'It is obvious that ([REF]) is satisfiable if and only if [MATH].', '1603.08866-3-100-5': 'Letting [MATH], Equation ([REF]) becomes [EQUATION].', '1603.08866-3-100-6': 'Since [MATH] and [MATH] can be freely adjusted to give [MATH] any value in that range, we see that the following is necessary and sufficient for ([REF]) to be soluble: [EQUATION]', '1603.08866-3-100-7': 'Use of the identity ([MATH]) and a short calculation demonstrates that this is equivalent to the lower bound in the inequality ([REF]).', '1603.08866-3-101-0': 'Suppose [MATH] admits a [MATH]-equivariant orthonormal basis, and is of dimension less than 5.', '1603.08866-3-101-1': 'Then there exists a RFI teleportation protocol for [MATH].', '1603.08866-3-102-0': 'We construct a [MATH]-equivariant UEB for [MATH].', '1603.08866-3-102-1': 'Expressed in the [MATH]-equivariant orthonormal basis, [MATH] will be some subgroup of the permutation matrices [MATH].', '1603.08866-3-102-2': 'To use Theorem [REF], we must find a Hadamard matrix commuting with [MATH].', '1603.08866-3-102-3': 'In the worst case, [MATH] will be the whole group [MATH] of permutation matrices.', '1603.08866-3-102-4': '(This situation is realised for the representation [MATH] of [MATH], where [MATH] is the fundamental [MATH]-dimensional representation of [MATH]).', '1603.08866-3-103-0': 'We will demonstrate that, when [MATH] is of dimension less than 5, we can find a Hadamard matrix which commutes with all the permutation matrices.', '1603.08866-3-103-1': 'First we eliminate the degenerate cases [MATH] and [MATH].', '1603.08866-3-103-2': 'Clearly for [MATH] we can perform RFI teleportation by Proposition [REF], For [MATH] the following family of Hadamard matrices commutes with [MATH], where [MATH] and [MATH]: [EQUATION]', '1603.08866-3-103-3': 'From now on we may therefore assume [MATH].', '1603.08866-3-104-0': 'It is easy to see that the centraliser [MATH] is the set of matrices defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-3-104-1': 'The conditions necessary for such a matrix to be unitary were given in Lemma [REF].', '1603.08866-3-104-2': 'Setting [MATH] in ([REF]), it follows that [MATH].', '1603.08866-3-104-3': 'This is compatible with ([REF]) only for [MATH].'}

{'1603.08866-4-0-0': 'We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group.', '1603.08866-4-0-1': 'Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size.', '1603.08866-4-0-2': 'In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.', '1603.08866-4-1-0': '# Introduction', '1603.08866-4-2-0': '## The problem and our result', '1603.08866-4-3-0': "Recently many authors have recognised the importance of developing a theory of quantum information which takes account of the reference frames by which a system's state is defined [CITATION].", '1603.08866-4-3-1': 'It was recognised some time ago [CITATION] that a shared reference frame is a hidden implicit assumption in the conventional description of quantum teleportation, an assumption which it is reasonable to challenge, since it is clearly possible that two parties attempting teleportation across a large distance will be uncertain about their relative reference frames.', '1603.08866-4-3-2': 'Quantum teleportation is a foundational quantum protocol with important applications [CITATION]; the problem of teleporting a quantum state between two parties who do not share a reference frame is therefore natural and important.', '1603.08866-4-4-0': "A successful reference frame-independent teleportation protocol is one where any observer with their own fixed reference frame will agree that Alice's quantum state has been successfully transferred to Bob, regardless of the orientation of Alice and Bob's frames; this was referred to in [CITATION] as 'teleportation of unspeakable information'.", '1603.08866-4-4-1': "Here we consider perfect teleportation, where Alice's state is transferred to Bob with certainty.", '1603.08866-4-5-0': 'The effect of a change in reference frame alignment on the perceived state of a system may be encoded formally as the unitary action of a group of reference frame transformations on the Hilbert space of the system under consideration.', '1603.08866-4-5-1': 'It has been shown [CITATION] that if the group of reference frame transformations contains [MATH] and the action is nontrivial, then reference frame-independent teleportation is impossible.', '1603.08866-4-5-2': 'In this paper we show that when the group of reference frame transformations is finite, reference frame-independent teleportation is possible in some nontrivial cases.', '1603.08866-4-6-0': 'We will demonstrate that reference frame-independent (RFI) teleportation protocols correspond exactly to [MATH]-equivariant unitary error bases, a structure we define, and develop methods of constructing these bases when they exist.', '1603.08866-4-6-1': 'In particular, we provide a simple sufficient condition for the existence of RFI teleportation protocols for systems of dimension less than 5.', '1603.08866-4-7-0': "The nature of the classical channel through which the result of Alice's measurement is communicated is crucial to our new protocol.", '1603.08866-4-7-1': 'In the terminology of Peres and Scudo [CITATION], our protocol requires classical transmission of unspeakable information, rather than of speakable information.', '1603.08866-4-7-2': 'An example of unspeakable information is the choice of a direction in space, to be agreed by two parties who do not share a common directional reference; no amount of communication through a generic shared classical channel can decide the matter, but the fidelitous transfer from one party to the other of a single oriented physical system, such as an arrow, is sufficient.', '1603.08866-4-8-0': 'We foresee two further applications of this result, both of which merit further investigation:', '1603.08866-4-9-0': 'The fundamental concept of a [MATH]-equivariant unitary error basis was developed from investigations in categorical quantum mechanics [CITATION].', '1603.08866-4-9-1': 'We characterised teleportation schemes as structures internal to the category of finite dimensional Hilbert spaces, and investigated corresponding structures in the category of unitary representations of a finite group; these corresponding structures were exactly the [MATH]-equivariant unitary error bases.', '1603.08866-4-9-2': 'Following their discovery, further investigation demonstrated their relevance to the problem of RFI teleportation.', '1603.08866-4-9-3': 'This exemplifies the utility of categorical quantum mechanics as a toolkit for developing new and interesting concepts in quantum information.', '1603.08866-4-10-0': 'The outline of the paper is as follows.', '1603.08866-4-10-1': 'In Section [REF] we give an informal worked example of our procedure, and in Section [REF] we provide a more formal presentation, along with further examples.', '1603.08866-4-10-2': 'In Section [REF] we show how reference frame-independent teleportation schemes are related to ideas in categorical quantum mechanics.', '1603.08866-4-10-3': 'In Section [REF] we prove a variety of existence, nonexistence and construction results for [MATH]-equivariant unitary error bases.', '1603.08866-4-11-0': '## Previous results', '1603.08866-4-12-0': 'It was demonstrated in [CITATION] that teleportation is impossible when the group of reference frame transformations is a infinite compact connected Lie group and the representation on the system to be teleported does not factor through a representation of a finite group.', '1603.08866-4-12-1': 'As observed in that paper, this leaves open the question of whether quantum teleportation is possible in the case of a finite group of reference frame transformations, without additional resources or prior communication.', '1603.08866-4-12-2': 'The protocol we exhibit provides an affirmative answer to this question.', '1603.08866-4-13-0': 'A number of other solutions for the finite case have been proposed.', '1603.08866-4-13-1': 'In [CITATION], it was suggested that Alice could transmit half of a maximally-entangled token state, in the regular representation, in advance of performing the protocol; the two parties could then use this to synchronise their operations.', '1603.08866-4-13-2': 'This method, however, requires Alice to be able to initialise an entangled state on a pair of systems each carrying the regular representation of the transformation group, a procedure which may be experimentally difficult or impossible.', '1603.08866-4-14-0': 'Another relevant result can be found in [CITATION], where it was shown that it is possible to perform quantum protocols using a second shared system as a quantum reference frame.', '1603.08866-4-14-1': 'This general result, intended for application to quantum cryptography, may be used to create reference frame-independent teleportation protocols.', '1603.08866-4-14-2': 'These protocols are formally identical to the token state method, but operationally more practicable; Alice and Bob simply initialise an additional shared entangled state at the same time as they create the first, take half each and use it to synchronise their operations.', '1603.08866-4-14-3': 'The problems of the token state method therefore persist in this case, although without the additional difficulty of communicating half of the second entangled state from one party to the other.', '1603.08866-4-15-0': "Finally, various solutions have also been proposed which use prior communication to align both parties' reference frames in advance of performing a normal teleportation protocol; see [CITATION].", '1603.08866-4-15-1': 'This increases the amount of classical information that must be communicated for successful teleportation.', '1603.08866-4-15-2': "Moreover, this procedure is not robust against changes in the alignment of Alice and Bob's reference frames, which must stay constant if the protocol is to succeed.", '1603.08866-4-15-3': 'Our protocol, in contrast, is robust against reference frame changes even while the classical message from Alice to Bob is in transit.', '1603.08866-4-15-4': "We only require that the alignment of Bob's reference frame is constant for the short time between his receipt of the classical information and his application of the unitary correction.", '1603.08866-4-16-0': 'However, we emphasize that our solution applies only when the system to be teleported carries a representation of the group of reference frame transformations [MATH] which admits a [MATH]-equivariant unitary error basis.', '1603.08866-4-16-1': 'The other approaches described above do not have this limitation.', '1603.08866-4-17-0': '# Example of the procedure', '1603.08866-4-18-0': 'In this section we give an informal account of the problem of reference frame-independent quantum teleportation, in the specific case where the systems are two-dimensional and the reference frame corresponds to a choice of spatial direction.', '1603.08866-4-18-1': 'This is followed by a more general and mathematically precise treatment in the next section.', '1603.08866-4-19-0': 'Alice and Bob are quantum information theorists in separate laboratories, which do not necessarily have the same orientation in space.', '1603.08866-4-19-1': "However, their relative orientations are not completely unknown: we are given some finite subgroup [MATH], the group of rigid spatial rotations, with the promise that there is some element [MATH] which relates Alice's and Bob's frames.", '1603.08866-4-19-2': 'The group [MATH] is common knowledge to both parties.', '1603.08866-4-20-0': 'The task is to perform teleportation of a quantum state from Alice to Bob, without revealing their spatial orientations, either to each other or to a potential eavesdropper.', '1603.08866-4-20-1': 'There are a variety of reasons why this may be advantageous: this information may be strategically or cryptographically valuable, and hence they may prefer not to divulge it for reasons of privacy; they may prefer to conserve limited bandwidth, and hence to not communicate redundant reference frame alignment information if it can be avoided; or they may simply be disoriented, and not aware of their own orientations.', '1603.08866-4-21-0': 'In this example, we consider the case that [MATH], meaning that their laboratories may be in one of two possible orientations which are related by a 180[MATH] rotation about some given axis.', '1603.08866-4-21-1': 'We suppose that the nontrivial element [MATH] acts on the qubit to be teleported as follows: [EQUATION].', '1603.08866-4-21-2': 'The fact that this matrix is nontrivial corresponds to the fact that the quantum system is not symmetric under the rotation operation.', '1603.08866-4-22-0': 'Alice and Bob agree in advance to perform quantum teleportation as follows.', '1603.08866-4-22-1': 'Alice will measure her initial system together with her part of the entangled state in the basis [MATH] and communicate the result to Bob, who will apply the corresponding correction [MATH].', '1603.08866-4-22-2': 'We define [MATH], where [MATH] denotes the transpose of the matrix [MATH], the symbol [MATH] denotes the 2-by-2 identity matrix, and [MATH] is the Bell state given above.', '1603.08866-4-22-3': 'The [MATH] are defined as follows: [EQUATION]', '1603.08866-4-22-4': 'It can readily be checked that this data forms an unitary error basis, and so by the results of Werner [CITATION] gives correct data for the execution of an ordinary quantum teleportation procedure for a single qubit, when the shared state is the Bell state [MATH].', '1603.08866-4-23-0': "If Bob's reference frame is correctly aligned with Alice's, then they are carrying out ordinary quantum teleportation, and the procedure will be successful.", '1603.08866-4-23-1': "However, if his reference direction is upside-down with respect to Alice's, then teleportation does not proceed successfully.", '1603.08866-4-23-2': "From Alice's perspective, Bob's correction is not in fact the unitary [MATH] corresponding to her measurement result, but rather the unitary [MATH]; a straightforward calculation then shows that Bob will receive a mixed state, and quantum information has been irrevocably lost.", '1603.08866-4-23-3': "From Bob's perspective, he correctly applied the unitary [MATH], but the teleportation failed because the measurement result [MATH] that Alice communicated to him did not correspond to the state she actually measured, which was [MATH].", '1603.08866-4-24-0': 'We now provide a resolution.', '1603.08866-4-24-1': 'In our new procedure, rather than communicating the two bits encoding the measurement result to Bob using their shared classical channel, Alice sends two physical objects to Bob: arrows, of the sort a medieval archer might use.', '1603.08866-4-24-2': 'She orients these arrows according to the measurement result that she obtained, using the following encoding, where her reference direction is written as [MATH]: [EQUATION]', '1603.08866-4-24-3': "One at a time, she physically sends these arrows through space to Bob's laboratory.", '1603.08866-4-25-0': 'Bob observes their local orientations and infers the measurement result 0, 1, 2 or 3 that Alice obtained.', '1603.08866-4-25-1': "Suppose Bob's laboratory is correctly aligned with Alice's; then he will correctly infer Alice's measurement result, and he will apply the corresponding unitary correction.", '1603.08866-4-25-2': 'In this case, the two parties have executed a traditional quantum teleportation protocol, albeit one where the two classical bits of information were transferred from Alice to Bob in an unusual way.', '1603.08866-4-26-0': "Now we suppose that Bob's laboratory is aligned upside-down with respect to Alice's.", '1603.08866-4-26-1': 'If Alice attempts to send the message 0, 1, 2 or 3, Bob will receive it as 1, 0, 3 or 2 respectively, since the arrows will appear to him with the opposite orientations.', '1603.08866-4-26-2': "Furthermore, just as before, when Bob applies a unitary [MATH], its action is seen in Alice's reference frame as [MATH].", '1603.08866-4-26-3': 'We now see the point of the entire construction: the unitary error basis is carefully chosen so that these effects cancel out.', '1603.08866-4-26-4': 'Indeed, the following equations can be easily verified: [EQUATION]', '1603.08866-4-26-5': "As a result, the quantum teleporation will conclude successfully, even though Alice's and Bob's reference directions were misaligned.", '1603.08866-4-27-0': 'In summary, by a careful choice of the unitary error basis, and by transferring the measurement result as unspeakable rather than speakable information, the quantum teleportation procedure can be carried out in a way which is robust against this restricted sort of reference frame error.', '1603.08866-4-27-1': 'Note in particular that only 2 bits of classical information were transferred from Alice to Bob, just as with the traditional teleportation procedure, and the Hilbert space of the entangled resource is of minimal dimension, so this procedure is tight in the sense of Werner [CITATION].', '1603.08866-4-27-2': "Also note that the unspeakable information Bob receives from Alice is uniformly random, since Alice's measurement results are; in particular, Bob receives no information during the protocol about the relative alignment between the two reference frames.", '1603.08866-4-27-3': "Finally, it is clear that the procedure would succeed even if Bob's reference frame were constantly changing between the two alignments, as long as the alignment stays constant between Bob's receipt of the arrows and his application of the unitary correction.", '1603.08866-4-28-0': '# Mathematical description of the proposal', '1603.08866-4-29-0': '## Traditional teleportation', '1603.08866-4-30-0': 'Teleportation is a well-understood procedure.', '1603.08866-4-30-1': 'It is traditionally formulated under the assumption that both parties have aligned reference frames [CITATION].', '1603.08866-4-30-2': "(Note that we only consider tight quantum teleportation in this paper, where the state spaces of the initial system and of the entangled systems all have the same dimension, which is equal to the number of classical bits transferred, and the procedure succeeds with probability 1; this corresponds to the informal restriction of 'no additional resources or prior communication'.)", '1603.08866-4-30-3': 'The traditional formulation is as follows.', '1603.08866-4-31-0': '[Teleportation without communication of unspeakable information]', '1603.08866-4-32-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-4-32-1': 'She performs a measurement with respect to an orthonormal basis of effects [MATH] on the bipartite system built from her initial system and her half of the entangled state.', '1603.08866-4-32-2': 'She sends the measurement result [MATH] to Bob through a generic classical channel.', '1603.08866-4-32-3': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-4-32-4': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-4-33-0': 'A complete description of correct data [MATH] was given by Werner [CITATION], as follows.', '1603.08866-4-34-0': 'For a Hilbert space [MATH], a unitary error basis is a basis of unitary operators [MATH], which are orthonormal under the Hilbert-Schmidt inner product: [EQUATION] [Werner]', '1603.08866-4-35-0': 'Up to equivalence, teleportation schemes for systems with Hilbert space [MATH] are in one to one correspondence with unitary error bases on [MATH].', '1603.08866-4-36-0': 'Under this correspondence, the shared entangled state [MATH] is the Bell state [MATH] for any orthonormal basis [MATH].', '1603.08866-4-36-1': 'Alice measures in the maximally-entangled basis [MATH], where [MATH] is defined as [MATH].', '1603.08866-4-36-2': "Bob's correction for the measurement outcome [MATH] is [MATH].", '1603.08866-4-37-0': '## Reference frame-independent quantum teleportation', '1603.08866-4-38-0': 'We now fully describe the problem we solve.', '1603.08866-4-39-0': '[Reference frame-independent quantum teleportation] Alice and Bob are spatially separated quantum information theorists capable of performing local operations and classical communication.', '1603.08866-4-39-1': 'Each party has one half of an entangled state which they created at some point in the past using a shared reference frame.', '1603.08866-4-39-2': 'Alice wants to use this entanglement to communicate a quantum state to Bob by teleportation.', '1603.08866-4-39-3': "However, their reference frames are now misaligned; in some observer's fixed reference frame, their frame alignments will be described by the action of unknown elements [MATH] of the group [MATH] of reference frame transformations.", '1603.08866-4-40-0': 'We will now show that the answer to (i) is almost always negative.', '1603.08866-4-40-1': 'First we note the following lemma.', '1603.08866-4-41-0': 'Under the conditions of Problem [REF], Procedure [REF] will work for all reference frame alignments if and only if the operations [MATH] and [MATH] are intertwiners for the group action.', '1603.08866-4-42-0': 'We express the operations with reference to the original shared reference frame.', '1603.08866-4-42-1': "Let Alice and Bob's frame shifts be described by group elements [MATH] and [MATH] respectively.", '1603.08866-4-42-2': 'Alice measures [MATH] relative to her reference frame; in the original frame the operation she has performed is [MATH].', '1603.08866-4-42-3': 'She then sends the result [MATH] to Bob, who performs the operation [MATH] relative to his frame; in the original frame the operation he has performed will be [MATH].', '1603.08866-4-42-4': 'In general, the channel will therefore only work for all reference frame configurations when, for all [MATH], [MATH] and [MATH], for some action of [MATH] on the set of measurement outcomes.', '1603.08866-4-42-5': 'Since [MATH] for the identity [MATH], this clearly implies that [MATH] for all [MATH].', '1603.08866-4-42-6': 'The result follows.', '1603.08866-4-43-0': 'We now demonstrate that Procedure [REF] works only for a trivial [MATH]-action, rendering it inadequate for reference frame-independent teleportation in any nontrivial case.', '1603.08866-4-44-0': 'Procedure [REF] will only work for all reference frame alignments when [MATH] acts by a global phase.', '1603.08866-4-45-0': 'By Theorem [REF] and Proposition [REF], Procedure [REF] will work only if all projections [MATH] and corrections [MATH] are intertwiners.', '1603.08866-4-45-1': 'By the definition of [MATH] in Theorem [REF], it is sufficient that all [MATH] be intertwiners.', '1603.08866-4-45-2': 'Let us assume that this is the case.', '1603.08866-4-45-3': 'Since the [MATH]-action is trivial on a basis of [MATH], it must be completely trivial on [MATH].', '1603.08866-4-45-4': 'Therefore we have [MATH].', '1603.08866-4-45-5': 'By straightforward character theory, there can only be one copy of [MATH] in the product of an irreducible representation with its dual.', '1603.08866-4-45-6': 'Breaking [MATH] up into simple factors, it follows by counting dimensions that they must all be identical and one dimensional.', '1603.08866-4-46-0': '## Our new scheme', '1603.08866-4-47-0': 'In answer to (ii), we will now present our new scheme for teleportation using unspeakable information transfer.', '1603.08866-4-47-1': '[Teleportation with communication of unspeakable information]', '1603.08866-4-48-0': 'Alice wants to teleport her state [MATH] to Bob; she has one half of a maximally-entangled bipartite state [MATH], and Bob the other.', '1603.08866-4-48-1': 'She forms the bipartite system given by her initial system together with her half of the entangled state, and sends it to Bob through a classical channel which is decoherent in the basis [MATH].', '1603.08866-4-48-2': 'Bob then performs a unitary operator [MATH] on his half of the entangled state.', '1603.08866-4-48-3': 'The data [MATH] is correct if Bob is guaranteed to receive the state [MATH] at the end of the procedure.', '1603.08866-4-49-0': 'The key aspect of Procedure [REF] is that misalignment of reference frames will not affect the way Bob perceives the data arriving through a generic classical channel, but it will affect his perception of the decohered bipartite system.', '1603.08866-4-49-1': 'In other words, the information Bob receives from Alice will depend in a nontrivial way on the alignment of his reference frame.', '1603.08866-4-50-0': 'The basic data of Procedure [REF] is the same as for Procedure [REF], and so Theorem [REF] still applies.', '1603.08866-4-50-1': 'However, not all unitary error bases give rise to successful teleporation schemes under this procedure.', '1603.08866-4-50-2': 'We now investigate which of them do.', '1603.08866-4-51-0': '[[MATH]-equivariant unitary error basis] For a Hilbert space [MATH] equipped with a unitary representation of [MATH], a unitary error basis is [MATH]-equivariant when the elements are permuted by the natural action [MATH] of [MATH] on [MATH].', '1603.08866-4-51-1': 'Explicitly, [MATH] for some permutation [MATH] of the set [MATH].', '1603.08866-4-52-0': 'Procedure [REF] will succeed for any reference frame misalignment [MATH] just when the unitary error basis [MATH] is [MATH]-equivariant.', '1603.08866-4-53-0': "We again work in Alice and Bob's original lab frame.", '1603.08866-4-53-1': 'Alice decoheres in the orthonormal basis [MATH].', '1603.08866-4-53-2': 'Bob then measures in the orthonormal basis [MATH], and, depending on his measurement outcome [MATH], performs the corresponding correction [MATH].', '1603.08866-4-54-0': "We first note that, putting Alice's decoherence and Bob's measurement together as one operation, we get a teleportation scheme under Definition [REF].", '1603.08866-4-54-1': "Therefore, by Theorem [REF], Alice's decoherence operation followed by Bob's measurement must be a measurement in some orthonormal basis of maximally-entangled states; clearly that must be the basis that Bob measures in.", '1603.08866-4-54-2': "Letting Bob's measurement channel be [MATH] and Alice's decohering channel be [MATH], it follows that [MATH]; this can clearly only be true if the projection basis for [MATH] is the same as the projection basis for [MATH].", '1603.08866-4-54-3': 'We therefore see that the basis [MATH] must be some reordering of the basis [MATH], for all [MATH].', '1603.08866-4-54-4': 'This is exactly [MATH]-equivariance of the UEB [MATH].', '1603.08866-4-55-0': 'We now demonstrate that this condition is sufficient to guarantee success for Procedure [REF].', '1603.08866-4-55-1': 'Suppose [MATH] is [MATH]-equivariant.', '1603.08866-4-55-2': "Then Alice's decohering operation is exactly the same as it would have been if her reference frame had not shifted at all.", '1603.08866-4-55-3': 'Bob measures and performs the correction; the correction therefore corresponds to the measurement and the result follows.', '1603.08866-4-56-0': 'In Procedure [REF], we have specified that Alice send the decohered bipartite system itself, since this is always theoretically possible.', '1603.08866-4-56-1': 'However, in practise it may be experimentally more practicable to use some other means of classically communicating unspeakable information; the important thing is that the classical data should itself carry the same [MATH]-action as the corresponding [MATH]-equivariant measurement basis.', '1603.08866-4-56-2': "An example was given in Section [REF], where Alice's measurement result was encoded in the spatial orientations of some physical objects.", '1603.08866-4-56-3': 'In order to demonstrate that this approach is applicable to other types of reference frame uncertainty, we provide two further examples of unspeakable encodings of classical information.', '1603.08866-4-57-0': '[(i)] Time.', '1603.08866-4-57-1': "Suppose the computational basis states of Alice and Bob's systems are nondegenerate energy eigenstates (for example, eigenstates of the photon number operator).", '1603.08866-4-57-2': 'Here they will need to share a time reference.', '1603.08866-4-57-3': 'Let the time translation operator [MATH] have periodicity [MATH] for some [MATH].', '1603.08866-4-57-4': 'Suppose that the group [MATH] of time translations has been discretised to some cyclic subgroup [MATH] of translations by [MATH].', '1603.08866-4-57-5': "Alice and Bob's reference frame configurations will correspond to their zeroes of time.", '1603.08866-4-57-6': 'Signals sent by Alice to to Bob which arrive, according to her reference frame, at time [MATH], arrive for Bob at a different time [MATH], depending on the difference between their reference frames.', '1603.08866-4-57-7': 'By encoding her measurement result in the time of arrival of signals, Alice may construct [MATH]-equivariant teleportation protocols.', '1603.08866-4-58-0': '[(ii)] Circular polarisation.', '1603.08866-4-58-1': 'Suppose Alice and Bob are working with photonic qubits whose computational basis states are left and right circular polarisation.', '1603.08866-4-58-2': 'In this case, the group of reference frame transformations will correspond to planar rotations of the axes perpendicular to the propagation direction.', '1603.08866-4-58-3': 'Suppose that the group [MATH] of planar rotations has been discretised to some cyclic subgroup [MATH] of rigid rotations by multiples of [MATH], and that Alice can classically communicate linearly polarised light to Bob.', '1603.08866-4-58-4': "By communicating frame configurations using beams of linearly polarised light, Alice may encode measurement results in the angle difference between Bob's frame and the commmunicated frame, allowing her to construct [MATH]-equivariant teleportation protocols.", '1603.08866-4-59-0': '# Classical structures in [MATH]', '1603.08866-4-60-0': 'Teleportation in the context of a finite group [MATH] can be described elegantly in the framework of categorical quantum mechanics [CITATION].', '1603.08866-4-60-1': 'One key strategy in this research programme is to understand features of quantum information in terms of the category [MATH] of finite-dimensional Hilbert spaces and linear maps, and then to generalize them by applying them in different categories.', '1603.08866-4-60-2': 'The concept of [MATH]-equivariant quantum teleportation arises by understanding the categorical structure of the traditional quantum teleportation procedure, and then applying it in [MATH], as we now explore.', '1603.08866-4-60-3': 'This technical section of the paper will make use of well-known ideas from categorical quantum mechanics, of which full details are available in the provided references.', '1603.08866-4-61-0': 'The following definition gives our abstract categorical description of quantum teleportation, in terms of classical structures in a symmetric monoidal category [CITATION].', '1603.08866-4-62-0': 'In a dagger-compact category, a quantum teleportation procedure on an object [MATH] with a right dual is a classical structure on the object [MATH], satisfying the following condition, where [MATH] is some scalar: [EQUATION]', '1603.08866-4-62-1': "This definition is motivated by the following theorem; recall Werner's Theorem [REF].", '1603.08866-4-63-0': 'Quantum teleportation procedures in [MATH] correspond precisely to unitary error bases.', '1603.08866-4-64-0': 'We now summarize the application of these ideas in a group representation category.', '1603.08866-4-65-0': 'For a group [MATH], the dagger-compact category [MATH] has objects given by unitary representations of [MATH], morphisms given by intertwiners, and a dagger-compact structure inherited from the underlying Hilbert spaces.', '1603.08866-4-66-0': 'Quantum teleportation procedures in [MATH] correspond precisely to [MATH]-equivariant unitary error bases.', '1603.08866-4-67-0': 'Finally, we observe that the constructions of unitary error bases in Theorem [REF] and Remark [REF] carry over straightforwardly to the [MATH]-equivariant setting because they are essentially categorical constructions; the Hadamard construction, for instance, is defined in terms of two special commutative dagger Frobenius algebras and an isomorphism between them.', '1603.08866-4-67-1': 'In Rep([MATH]), this reduces exactly to the intertwining Hadamard matrix and [MATH]-equivariant orthonormal basis of Theorem [REF].', '1603.08866-4-67-2': 'In this sense, these constructions are much more natural than, for instance, the construction of unitary error bases using projective group representations [CITATION]; indeed, it is difficult to see how the latter construction could be brought into the [MATH]-equivariant framework.', '1603.08866-4-68-0': '# Existence and construction of RFI teleportation protocols', '1603.08866-4-69-0': 'We have demonstrated that [MATH]-equivariant UEBs are exactly the structures we need to perform reference frame-independent teleportation protocols, but it is still unclear how to construct them for a given representation [MATH], if they exist at all.', '1603.08866-4-69-1': 'We cannot hope for a general classification of [MATH]-equivariant UEBs, since there is not even a classification in the case where the [MATH]-action is trivial, although many construction methods exist [CITATION].', '1603.08866-4-69-2': 'In this section we will demonstrate that [MATH]-equivariant unitary error bases need not exist on every representation, meaning that RFI teleportation is not always possible.', '1603.08866-4-69-3': 'We will then demonstrate that several UEB constructions carry over naturally to the [MATH]-equivariant setting, allowing us to construct RFI teleportation protcols for a wide variety of systems.', '1603.08866-4-70-0': 'We begin with a definition.', '1603.08866-4-71-0': 'A [MATH]-equivariant orthonormal basis for some representation [MATH] is an orthonormal basis of [MATH] whose elements are permuted by the action of [MATH].', '1603.08866-4-72-0': '[MATH]-equivariant unitary error bases are [MATH]-equivariant orthonormal bases of [MATH], all of whose elements are unitary maps.', '1603.08866-4-73-0': 'It will transpire that we can use [MATH]-equivariant orthonormal bases on [MATH] to construct [MATH]-equivariant UEBs for [MATH].', '1603.08866-4-73-1': 'Moreover, if we prove that there are no [MATH]-equivariant orthonormal bases on [MATH], it follows by Remark [REF] that there will be no [MATH]-equivariant UEBs for [MATH]; we will use this fact to demonstrate that RFI teleportation protocols need not always exist.', '1603.08866-4-73-2': 'Our first step is therefore a classification of [MATH]-equivariant orthonormal bases.', '1603.08866-4-74-0': '## A classification of [MATH]-equivariant orthonormal bases', '1603.08866-4-75-0': 'We begin with a simple lemma.', '1603.08866-4-75-1': 'Let [MATH]-Set be the category whose objects are sets carrying an action of [MATH], and whose morphisms are [MATH]-equivariant functions between them.', '1603.08866-4-75-2': 'Then there exists a functor [MATH]G[MATH]G[MATH], which, given a [MATH]-set, constructs the free Hilbert space on its elements, and extends the [MATH]-action and morphisms linearly.', '1603.08866-4-76-0': '[MATH]-equivariant orthonormal bases exist only on representations isomorphic to those in the image of [MATH].', '1603.08866-4-77-0': 'Immediate, since a [MATH]-equivariant orthonormal basis has an underlying Hilbert space isomorphic to the free Hilbert space on the elements of the chosen basis, which [MATH] acts on by permutations.', '1603.08866-4-78-0': 'We begin by presenting a simple classification of [MATH]-sets due to Burnside [CITATION].', '1603.08866-4-79-0': 'Given two [MATH]-sets [MATH] and [MATH], their disjoint union [MATH] is the disjoint union of [MATH] and [MATH] as sets with the natural induced action.', '1603.08866-4-80-0': 'Given a subgroup [MATH] of [MATH], the coset space [MATH] is the [MATH]-set whose elements are the cosets of [MATH] in [MATH], and whose [MATH]-action [MATH] is the natural action of [MATH] by left multiplication on those cosets.', '1603.08866-4-81-0': 'Any [MATH]-set is isomorphic to a disjoint union of coset spaces.', '1603.08866-4-81-1': 'Two coset spaces are isomorphic as [MATH]-sets if and only they correspond to conjugate subgroups.', '1603.08866-4-82-0': 'See [CITATION].', '1603.08866-4-83-0': 'In modern language, Lemma [REF] states that [MATH]-Set is a semisimple fusion category whose simple objects correspond to conjugacy classes of subgroups in [MATH].', '1603.08866-4-83-1': 'It is easy to see that the functor [MATH] is additive; the disjoint union of two [MATH]-sets will be sent under [MATH] to the direct sum of their corresponding representations.', '1603.08866-4-83-2': 'In order to classify all objects in the image of [MATH], therefore, it is sufficient to find the image of the coset spaces under [MATH].', '1603.08866-4-83-3': 'We will call those representations the basic permutation representations.', '1603.08866-4-84-0': 'In order to identify the basic permutation representations, we now state an obvious but critical lemma regarding the character of the permutation representation induced by [MATH] on a [MATH]-set.', '1603.08866-4-85-0': 'Given a [MATH]-set [MATH], let [MATH] be the character of [MATH].', '1603.08866-4-85-1': 'Then the following holds: [EQUATION]', '1603.08866-4-85-2': 'The character [MATH] is exactly the trace of the matrix representing [MATH]; the result follows trivially from the definition of [MATH].', '1603.08866-4-86-0': 'We may therefore identify the basic permutation representations by taking a representative of every conjugacy class of subgroups of [MATH], finding the number of fixed points of the action of each element of [MATH] on the corresponding coset spaces, then decomposing the resulting characters using the character table to find the corresponding representations.', '1603.08866-4-87-0': '## Existence of RFI teleportation protocols', '1603.08866-4-88-0': 'Using the results of Subsection [REF], we now exhibit a representation for which no [MATH]-equivariant UEBs exist, and on which quantum teleportation is therefore impossible.', '1603.08866-4-89-0': 'There is no RFI protocol to teleport the state of the 2-dimensional irreducible representation [MATH] of [MATH].', '1603.08866-4-90-0': 'Using the method outlined in Subsection [REF], we find that the characters of the basic permutation representations are as follows:', '1603.08866-4-91-0': 'The character of [MATH] is [MATH], which clearly cannot be composed as a sum of characters of basic permutation representations.', '1603.08866-4-91-1': 'By Remark [REF], the result follows.', '1603.08866-4-92-0': 'This argument does not extend to all irreducible representations.', '1603.08866-4-92-1': 'The endomorphism space of the 2-dimensional irreducible representation of [MATH], for instance, is a sum of basic permutation representations.', '1603.08866-4-93-0': '## Construction of RFI teleportation protocols', '1603.08866-4-94-0': 'Although RFI teleportation protocols need not always exist, they can often be constructed.', '1603.08866-4-94-1': 'We now demonstrate that, if we can find a [MATH]-equivariant orthonormal basis on [MATH], and a Hadamard matrix which commutes with all [MATH] in that basis, we can perform RFI teleportation on [MATH].', '1603.08866-4-95-0': 'Let [MATH] be a [MATH]-equivariant orthonormal basis on [MATH].', '1603.08866-4-95-1': 'In this basis all [MATH] will be permutation matrices.', '1603.08866-4-95-2': 'Let [MATH] be a Hadamard matrix that commutes with all [MATH] in this basis.', '1603.08866-4-95-3': 'Then the following family is a [MATH]-equivariant UEB: [EQUATION]', '1603.08866-4-95-4': 'It was already proved in [CITATION] that this is a UEB; we therefore need only show that it is [MATH]-equivariant.', '1603.08866-4-95-5': 'Since [MATH] we have that [MATH].', '1603.08866-4-95-6': 'We see easily that [MATH].', '1603.08866-4-95-7': 'Now note that the fact that [MATH] commutes with all elements of [MATH] means that permuting the columns of [MATH] is exactly the same as permuting the rows, since [MATH] for all [MATH].', '1603.08866-4-95-8': 'So [MATH].', '1603.08866-4-95-9': 'A similar argument works for [MATH].', '1603.08866-4-96-0': 'If the assumptions of Theorem [REF] are satisfied, it is possible to construct many more [MATH]-equivariant UEBs using quantum Latin squares (QLSs) [CITATION]; this construction will give [MATH]-equivariant UEBs provided the linear map defining the QLS is an intertwiner.', '1603.08866-4-97-0': 'We finish this section with a simple sufficient condition for the existence of tight RFI protocols on systems of dimension less than 5.', '1603.08866-4-97-1': 'Firstly we prove a lemma.', '1603.08866-4-98-0': 'Let [MATH] be a matrix of dimension [MATH] defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-4-98-1': 'Let [MATH] where [MATH] and [MATH].', '1603.08866-4-98-2': 'Then [MATH] is unitary precisely when the following conditions are satisfied:', '1603.08866-4-99-0': '0.3 [EQUATION] 0.33 [EQUATION] 0.36 [EQUATION] 0pt', '1603.08866-4-100-0': 'For unitarity it is sufficient that the rows form an orthonormal basis.', '1603.08866-4-100-1': 'It is clear from the symmetry of [MATH] that it is sufficient for one row vector to be normal, and one pair of row vectors to be orthogonal.', '1603.08866-4-100-2': 'This gives us two equations in [MATH] and [MATH]: [EQUATION]', '1603.08866-4-100-3': 'We will demonstrate that ([REF]) is necessary and sufficient for us to find [MATH] satisfying these equations.', '1603.08866-4-100-4': 'It is obvious that ([REF]) is satisfiable if and only if [MATH].', '1603.08866-4-100-5': 'Letting [MATH], Equation ([REF]) becomes [EQUATION].', '1603.08866-4-100-6': 'Since [MATH] and [MATH] can be freely adjusted to give [MATH] any value in that range, we see that the following is necessary and sufficient for ([REF]) to be soluble: [EQUATION]', '1603.08866-4-100-7': 'Use of the identity ([MATH]) and a short calculation demonstrates that this is equivalent to the lower bound in the inequality ([REF]).', '1603.08866-4-101-0': 'Suppose [MATH] admits a [MATH]-equivariant orthonormal basis, and is of dimension less than 5.', '1603.08866-4-101-1': 'Then there exists a RFI teleportation protocol for [MATH].', '1603.08866-4-102-0': 'We construct a [MATH]-equivariant UEB for [MATH].', '1603.08866-4-102-1': 'Expressed in the [MATH]-equivariant orthonormal basis, [MATH] will be some subgroup of the permutation matrices [MATH].', '1603.08866-4-102-2': 'To use Theorem [REF], we must find a Hadamard matrix commuting with [MATH].', '1603.08866-4-102-3': 'In the worst case, [MATH] will be the whole group [MATH] of permutation matrices.', '1603.08866-4-102-4': '(This situation is realised for the representation [MATH] of [MATH], where [MATH] is the fundamental [MATH]-dimensional representation of [MATH]).', '1603.08866-4-103-0': 'We will demonstrate that, when [MATH] is of dimension less than 5, we can find a Hadamard matrix which commutes with all the permutation matrices.', '1603.08866-4-103-1': 'First we eliminate the degenerate cases [MATH] and [MATH].', '1603.08866-4-103-2': 'Clearly for [MATH] we can perform RFI teleportation by Proposition [REF], For [MATH] the following family of Hadamard matrices commutes with [MATH], where [MATH] and [MATH]: [EQUATION]', '1603.08866-4-103-3': 'From now on we may therefore assume [MATH].', '1603.08866-4-104-0': 'It is easy to see that the centraliser [MATH] is the set of matrices defined by two complex parameters [MATH] and [MATH], where all entries on the diagonal are [MATH], and all other entries are [MATH].', '1603.08866-4-104-1': 'The conditions necessary for such a matrix to be unitary were given in Lemma [REF].', '1603.08866-4-104-2': 'Setting [MATH] in ([REF]), it follows that [MATH].', '1603.08866-4-104-3': 'This is compatible with ([REF]) only for [MATH].'}

1705.07904

{'1705.07904-1-0-0': 'We propose a new algorithm for training generative adversarial networks to jointly learn latent codes for both identities (e.g. individual humans) and observations (e.g. specific photographs).', '1705.07904-1-0-1': 'In practice, this means that by fixing the identity portion of latent codes, we can generate diverse images of the same subject, and by fixing the observation portion we can traverse the manifold of subjects while maintaining contingent aspects such as lighting and pose.', '1705.07904-1-0-2': 'Our algorithm features a pairwise training scheme in which each sample from the generator consists of two images with a common identity code.', '1705.07904-1-0-3': 'Corresponding samples from the real dataset consist of two distinct photographs of the same subject.', '1705.07904-1-0-4': 'In order to fool the discriminator, the generator must produce images that are both photorealistic, distinct, and appear to depict the same person.', '1705.07904-1-0-5': 'We augment both the DCGAN and BEGAN approaches with Siamese discriminators to accommodate pairwise training.', '1705.07904-1-0-6': "Experiments with human judges and an off-the-shelf face verification system demonstrate our algorithm's ability to generate convincing, identity-matched photographs.", '1705.07904-1-1-0': '# Introduction', '1705.07904-1-2-0': 'Generative adversarial networks (GANs) learn mappings from latent codes in some low-dimensional space [MATH] to points in the space of natural data [MATH] .', '1705.07904-1-2-1': 'They achieve this power through an adversarial training scheme pitting a generative model [MATH] against a discriminative model [MATH] in a minimax game.', '1705.07904-1-2-2': 'The generator learns to map low-dimensional vectors [MATH], sampled i.i.d. according to some prespecified distribution, to plausible counterfeits [MATH].', '1705.07904-1-2-3': 'The discriminator learns to differentiate between samples from the resulting generated distribution [MATH] over [MATH], and the real data distribution [MATH].', '1705.07904-1-2-4': "The generator's training objective is to fool the discriminator.", '1705.07904-1-3-0': 'Since their introduction, GANs have attracted widespread attention for their ability to produce high-fidelity images when trained on corpora of natural images.', '1705.07904-1-3-1': 'Several papers have since advanced these capabilities through architectural improvements and modifications to the training scheme .', '1705.07904-1-3-2': 'Another line of research addresses how to train GANs to produce class-conditional output.', '1705.07904-1-3-3': 'Conditional GANS (cGANs), proposed by [CITATION], require learning a distinct weight vector for each class.', '1705.07904-1-3-4': 'This is impractical when there are thousands or millions of classes and few instances of each.', '1705.07904-1-3-5': 'Moreover, in such cases we might want to learn a latent space corresponding to the classes themselves.', '1705.07904-1-4-0': "To make the discussion concrete, let's consider a dataset consisting of [MATH]M photographs of celebrity faces.", '1705.07904-1-4-1': 'This dataset contains [MATH] photographs (observations) each for 100,000 distinct people (identities).', '1705.07904-1-4-2': 'After training a traditional GAN on this dataset, we can synthesize face images at will.', '1705.07904-1-4-3': 'By traversing latent space, we might add glasses to a face, transition between a photograph of a woman and one of a man, or shift the pose from profile to center.', '1705.07904-1-4-4': 'But what if we want to fix the subject of a photograph and traverse the entire manifold of lighting, pose, expression, etc?', '1705.07904-1-4-5': 'Or instead fix these contingent aspects and vary the identity of the subject?', '1705.07904-1-4-6': 'The generic GAN framework offers no clear way to do this.', '1705.07904-1-5-0': 'We propose Semantically Decomposed GANs (SD-GANs), which coerce a specified portion of the latent space to correspond to a known source of variation.', '1705.07904-1-5-1': 'The technique decomposes [MATH] into one portion [MATH] corresponding to identity, and the remaining portion [MATH] corresponding to the other contingent aspects of observations.', '1705.07904-1-5-2': 'SD-GANs learn through a pairwise training scheme in which each sample from the generator consists of a pair of images with common [MATH] but differing [MATH].', '1705.07904-1-5-3': 'Each sample from the real dataset consists of two distinct images of the same person (i.e., with the same identity).', '1705.07904-1-5-4': 'In order to fool the discriminator, the generator must not only produce diverse and photorealistic images, but also images with the same identity when [MATH] is fixed.', '1705.07904-1-5-5': 'For SD-GANs, we modify the discriminator so that it can determine whether the pair of samples constitute a match.', '1705.07904-1-6-0': 'Our experiments with a dataset of celebrity faces demonstrate that SD-GANs can generate contrasting images of the same subject (Figure [REF]).', '1705.07904-1-6-1': 'The generator learns that certain properties are free to vary across observations but not identity.', '1705.07904-1-6-2': 'For example, SD-GANs learn that pose, facial expression, hair styles, black white vs. color, and lighting can all vary across different photographs of the same individual.', '1705.07904-1-6-3': 'On the other hand, the aspects that are more salient for facial verification remain consistent as we vary the observation code [MATH].', '1705.07904-1-6-4': 'We demonstrate that SD-GANs trained on faces generate stylistically contrasting, identity-matched image pairs that human annotators and a state-of-the-art face verification algorithm recognize as depicting the same subject.', '1705.07904-1-6-5': 'We also train SD-GANs on a dataset of product images, containing multiple photographs of each product from various perspectives (Figure [REF]).', '1705.07904-1-7-0': '# Generative adversarial networks', '1705.07904-1-8-0': 'GANs leverage the discriminative power of neural networks to learn generative models.', '1705.07904-1-8-1': 'The learning process consists of a minimax game between a generative model [MATH], parameterized by [MATH] and a discriminative model [MATH], parameterized by [MATH].', '1705.07904-1-8-2': 'In the original formulation, the discriminative model tries to maximize log likelihood, yielding: [EQUATION]', '1705.07904-1-8-3': 'Training proceeds as follows: For [MATH] iterations, sample one mini-batch from the real distribution [MATH] and one from the distribution of generated images [MATH], updating discriminator weights [MATH] to increase [MATH] by stochastic gradient ascent.', '1705.07904-1-8-4': 'Then sample a minibatch from [MATH], updating the [MATH] to decrease [MATH] by stochastic gradient descent.', '1705.07904-1-9-0': '[CITATION] show that GAN training corresponds to minimizing the Jensen-Shannon divergence between the generated distribution and the data-generating process.', '1705.07904-1-9-1': 'Following this work, several authors have explored minimizing other objectives.', '1705.07904-1-10-0': 'Notably, [CITATION] use the Wasserstein distance, which amounts to maximizing over a more restricted family of discriminator models, with parameters in some set [MATH] describing Lipschitz-constrained neural networks: [EQUATION]', '1705.07904-1-10-1': 'To enforce the Lipschitzness of the discriminator, they perform weight-clipping.', '1705.07904-1-10-2': 'They report more stable training than the original GAN and the useful property that discriminator loss tends to correlate with the perceptual quality of generated images.', '1705.07904-1-11-0': '[CITATION] propose energy-based GANs (EBGANs), in which the discriminator can be viewed as an energy function.', '1705.07904-1-11-1': 'Specifically, they devise a discriminator consisting of an autoencoder: [MATH].', '1705.07904-1-11-2': "In the minimax game, the discriminator's weights are updated to minimize the reconstruction error [MATH] for real data, while maximizing the error [MATH] for the generator.", '1705.07904-1-12-0': 'More recently, [CITATION] extend this work, introducing Boundary Equilibrium GANs (BEGANs), which optimize the Wasserstein distance between autoencoder loss distributions, yielding the formulation: [EQUATION]', '1705.07904-1-12-1': 'Additionally, they introduce a method for stabilizing training.', '1705.07904-1-12-2': 'Positing that training becomes unstable when the discriminator cannot distinguish between real and generated images, they introduce a new hyperparameter [MATH], updating the value function on each iteration to maintain a desired ratio between the two reconstruction errors: [MATH].', '1705.07904-1-12-3': 'The BEGAN model produces what appear to us, subjectively, to be the sharpest images of faces yet generated by a GAN.', '1705.07904-1-12-4': 'In this work, we adapt both the DCGAN and BEGAN algorithms to the SD-GAN training scheme.', '1705.07904-1-13-0': '# Semantically decomposed GANs', '1705.07904-1-14-0': '[t] Semantically Decomposed GAN Training [1] n in 1:NumberOfIterations m in 1:MinibatchSize Sample one identity vector [MATH].', '1705.07904-1-14-1': 'Sample two observation vectors [MATH].', '1705.07904-1-14-2': '[MATH], [MATH].', '1705.07904-1-14-3': 'Generate pair of images [MATH], adding them to the minibatch with label [MATH] (fake).', '1705.07904-1-14-4': 'm in 1:MinibatchSize Sample one identity [MATH] from the real data set.', '1705.07904-1-14-5': 'Sample two distinct images [MATH].', '1705.07904-1-14-6': 'Add the pair to the minibatch, assigning label [MATH] (real).', '1705.07904-1-14-7': 'Update discriminator weights by [MATH] using its stochastic gradient.', '1705.07904-1-14-8': 'Sample another minibatch of identity-matched latent vectors [MATH].', '1705.07904-1-14-9': 'Update generator weights by stochastic gradient descent [MATH].', '1705.07904-1-15-0': "To formulate our SD-GAN technique more fully, consider the data's identity as a random variable [MATH] in a discrete index set [MATH].", '1705.07904-1-15-1': 'We seek to learn a latent representation that conveniently decomposes the variation in the real data into two parts: due to [MATH], and due to the other factors of variation in the data, packaged as a random variable [MATH].', '1705.07904-1-15-2': 'Ideally, the decomposition of the variation in the data into [MATH] and [MATH] should correspond exactly to a decomposition of the latent space [MATH].', '1705.07904-1-15-3': 'This would permit convenient interpolation and other operations on the inferred subspaces [MATH] and [MATH].', '1705.07904-1-16-0': 'A conventional GAN samples [MATH] from their joint distribution.', '1705.07904-1-16-1': "Such a GAN's generative model samples directly from an unstructured prior over the latent space.", '1705.07904-1-16-2': 'It does not disentangle the variation in [MATH] and [MATH], for instance by modeling conditional distributions [MATH], but only models their average with respect to the prior on [MATH].', '1705.07904-1-17-0': 'Our SD-GAN method learns such a latent space decomposition, partitioning the coordinates of [MATH] into two parts representing the subspaces, so that any [MATH] can be written as the concatenation [MATH] of its identity representation [MATH] and its contingent aspect representation [MATH].', '1705.07904-1-17-1': 'SD-GANs achieve this through a pairwise training scheme in which each sample from the real data consists of [MATH], a pair of images with a common identity [MATH].', '1705.07904-1-17-2': 'Each sample from the generator consists of [MATH], a pair of images generated from a common identity vector [MATH] but i.i.d. observation vectors [MATH].', '1705.07904-1-17-3': 'We assign identity-matched pairs from [MATH] the label [MATH] and [MATH]-matched pairs from [MATH] the label [MATH].', '1705.07904-1-17-4': 'The discriminator can thus learn to reject pairs for either of two primary reasons: 1) not photorealistic or 2) not plausibly depicting the same subject.', '1705.07904-1-17-5': 'See Algorithm [REF] for pseudocode for SD-GAN training.', '1705.07904-1-18-0': '## Discriminator architecture', '1705.07904-1-19-0': 'With SD-GANs, we see no reason to alter the architecture of the generator.', '1705.07904-1-19-1': 'However, the discriminator must now act upon two images, producing a single output.', '1705.07904-1-19-2': 'Moreover, the effects of the two input images [MATH] on the output score are not independent.', '1705.07904-1-19-3': 'Two images might be otherwise photorealistic but deserve rejection because they clearly depict different identities.', '1705.07904-1-19-4': 'To this end, we devise two novel discriminator architectures to adapt DCGAN and BEGAN respectively.', '1705.07904-1-19-5': 'In both cases, we first separately encode each image using the same convolutional neural network [MATH].', '1705.07904-1-19-6': "We choose this Siamese setup as our problem is symmetrical in the images, and thus it's sensible to share weights between the encoders.", '1705.07904-1-20-0': 'To adapt the DCGAN (Figure [REF]), we stack the feature maps [MATH] and [MATH] along the channel axis, applying one additional strided convolution.', '1705.07904-1-20-1': 'This allows the network to further aggregate information from the two images before flattening and fully connecting to a [MATH] output neuron.', '1705.07904-1-20-2': 'For BEGAN, because the discriminator is an autoencoder, our architecture is more complicated.', '1705.07904-1-20-3': 'After encoding each image, we concatenate the representations [MATH] and apply one fully connected bottleneck layer [MATH] with linear activation.', '1705.07904-1-20-4': 'In alignment with BEGAN, the SD-BEGAN bottleneck has the same dimensionality as the tuple of latent codes ([MATH], [MATH], [MATH]) that generated the pair of images.', '1705.07904-1-20-5': 'Following the bottleneck, we apply a second FC layer [MATH], taking the first [MATH] components of its output to be the input to the first decoder and the second [MATH] components to be the input to the second decoder.', '1705.07904-1-20-6': 'The shared intermediate layer gives SD-BEGAN a mechanism to push apart matched and unmatched pairs.', '1705.07904-1-20-7': 'We specify our exact architectures in full detail in Appendix [REF].', '1705.07904-1-21-0': '# Experiments', '1705.07904-1-22-0': 'We experimentally validate SD-GANs using two datasets: 1) the MS-Celeb-1M dataset of celebrity face images and 2) a dataset of shoe images scraped from Amazon .', '1705.07904-1-22-1': 'Both datasets contain a large number of identities (faces and shoes, respectively) with multiple observations of each.', '1705.07904-1-22-2': 'Celebrity faces are a richer domain for our method as both identities and contingent factors are significant sources of variation.', '1705.07904-1-22-3': 'Thus, our primary experiments and evaluation focus on this data.', '1705.07904-1-22-4': "In contrast, Amazon's shoe images tend to vary only with camera perspective for a given product, making this data useful for sanity-checking our approach.", '1705.07904-1-23-0': '## Faces', '1705.07904-1-24-0': 'From the aligned face images in the MS-Celeb-1M dataset, we select 12,500 celebrities at random and [MATH] associated images of each, resizing them to [MATH]x[MATH] pixels.', '1705.07904-1-24-1': 'We split the celebrities into subsets of 10,000 (training), 1,250 (validation) and 1,250 (test).', '1705.07904-1-24-2': 'The dataset has a small number of duplicate images and some label noise (images matched to the wrong celebrity).', '1705.07904-1-24-3': 'To detect and remove duplicates, we hash the images.', '1705.07904-1-24-4': 'We do not explicitly rid the data of label noise, demonstrating the robustness of our algorithm.', '1705.07904-1-24-5': 'Because each training example consists of a pair, the training set contains a total of 280,000 examples ([MATH] 10,000).', '1705.07904-1-24-6': 'We scale the input values to [MATH], performing no additional preprocessing or data augmentation on these images.', '1705.07904-1-25-0': '## Shoes', '1705.07904-1-26-0': 'Product images are another promising application for our method.', '1705.07904-1-26-1': 'In this domain, we have access to multiple images of each product in different orientations.', '1705.07904-1-26-2': 'Generally, product photographs are captured against white backgrounds and primarily differ in orientation and distance.', '1705.07904-1-26-3': 'Accordingly, we expect that SD-GAN training will tie the observation latent space to capture these aspects.', '1705.07904-1-26-4': 'We choose to study shoes as a prototypical example of a category of product images.', '1705.07904-1-26-5': 'The Amazon dataset contains around 3,000 unique products with the category "Shoe" and multiple product images.', '1705.07904-1-26-6': 'We use the same [MATH] split and again hash the images to ensure that subsets are disjoint.', '1705.07904-1-26-7': 'There are an average of [MATH] photos of each product.', '1705.07904-1-27-0': '## Training details', '1705.07904-1-28-0': 'We train SD-DCGANs on both of our datasets for 500,000 iterations using batches of [MATH] identity-matched pairs.', '1705.07904-1-28-1': 'For MS-Celeb-1M, we compare results using the original GAN loss to those using the Wasserstein distance-based loss proposed by [CITATION] (SD-DCWGAN).', '1705.07904-1-28-2': 'To optimize SD-DCGAN, we use the Adam optimizer with hyperparameters [MATH] as recommended by [CITATION].', '1705.07904-1-28-3': 'To optimize SD-DCWGAN, we use RMS-prop with [MATH].', '1705.07904-1-28-4': 'We also consider a non-Siamese discriminator (SD-DCGAN-SC) that simply stacks the channels of the pair of real or fake images before encoding.', '1705.07904-1-29-0': 'As in , we sample latent vectors [MATH].', '1705.07904-1-29-1': 'For SD-GANs, we partition the latent codes according to [MATH] using values of [MATH].', '1705.07904-1-29-2': 'Our idea can be trivially applied with [MATH]-wise training (vs. pairwise).', '1705.07904-1-29-3': 'To explore the effects of using [MATH], we also implement an SD-DCGAN where we sample [MATH] instances each from [MATH] for some [MATH] and from [MATH] for some [MATH].', '1705.07904-1-29-4': 'For all experiments, unless otherwise stated, we use [MATH] and [MATH].', '1705.07904-1-30-0': 'We also train an SD-BEGAN on both of our datasets.', '1705.07904-1-30-1': 'For MS-Celeb-1M, we tried to train an SD-BEGAN with [MATH] but observed early mode collapse (Appendix [REF]).', '1705.07904-1-30-2': 'The increased complexity of the SD-BEGAN model significantly increases training time, requiring almost [MATH] hours to complete 500,000 iterations, thus limiting our ability to perform exhaustive hyperparameter validation.', '1705.07904-1-30-3': '[CITATION] sample latent vectors [MATH], however we use [MATH]-dimensional latent vectors for direct comparison to SD-DCGAN.', '1705.07904-1-30-4': 'We use the Adam optimizer with the default-hyperparameters from for our SD-BEGAN experiments.', '1705.07904-1-31-0': '# Evaluation', '1705.07904-1-32-0': 'The evaluation of generative models is a fraught topic.', '1705.07904-1-32-1': 'Quantitative measures of sample quality can be poorly correlated with each other.', '1705.07904-1-32-2': 'Accordingly, we design an evaluation to match conceivable uses of our algorithm.', '1705.07904-1-32-3': "Because we hope to produce samples which humans deem to depict the same person, we evaluate our generative model on a face verification task, using both a pretrained face verification model and crowd-sourced human judgments obtained through Amazon's Mechanical Turk platform.", '1705.07904-1-33-0': 'Recent advancements in face verification using deep convolutional neural networks have yielded accuracy rivaling humans.', '1705.07904-1-33-1': 'For our evaluation, we procure FaceNet, a publicly-available face verifier based on the popular Inception-ResNet CNN architecture .', '1705.07904-1-33-2': 'The FaceNet model was pretrained on the CASIA-WebFace dataset and achieves [MATH] accuracy on the benchmark LFW task .', '1705.07904-1-34-0': 'FaceNet ingests normalized, [MATH]x[MATH] color images and produces an embedding [MATH].', '1705.07904-1-34-1': 'FaceNet was trained to minimize the [MATH] distance between matched pairs of faces and to maximize the distance for mismatched pairs.', '1705.07904-1-34-2': 'Accordingly, the embedding space yields a function for measuring the similarity between two faces [MATH] and [MATH]: [MATH].', '1705.07904-1-34-3': 'Given two images, [MATH] and [MATH], we label them as a match if [MATH] where [MATH] is the threshold maximizing accuracy on a class-balanced set of pairs from MS-Celeb-1M validation data.', '1705.07904-1-34-4': 'We use the same threshold when evaluating both real and synthetic data with FaceNet.', '1705.07904-1-35-0': 'We compare the performance of FaceNet on pairs of images from the MS-Celeb-1M test set against generated samples from our trained SD-DCGAN and SD-BEGAN generative models.', '1705.07904-1-35-1': "To match FaceNet's training data, we preprocess all images by resizing from [MATH]x[MATH] to [MATH]x[MATH], normalizing each image individually.", '1705.07904-1-35-2': 'We prepare 10,000 pairs from MS-Celeb-1M, half identity-matched and half unmatched.', '1705.07904-1-35-3': 'From each generative model, we generate 5,000 pairs each with [MATH] and 5,000 pairs with [MATH].', '1705.07904-1-35-4': 'For each sample, we draw observation vectors [MATH] randomly.', '1705.07904-1-36-0': 'In Table [REF], we report the accuracy, true positive rate and false positive rate of FaceNet using threshold [MATH] for all datasets.', '1705.07904-1-36-1': 'We also report the AUROC and present the full ROC curve for all datasets in Figure [REF].', '1705.07904-1-36-2': 'Sample results from the best SD-DCGAN ([MATH], [MATH]) and SD-BEGAN ([MATH], [MATH]) model as determined by FaceNet are shown in Figures [REF] and [REF] respectively.', '1705.07904-1-37-0': 'In addition to validating that identity-matched SD-GAN samples are verified by FaceNet, we also demonstrate that humans are similarly convinced through experiments using Mechanical Turk.', '1705.07904-1-37-1': 'For these experiments, we use a balanced subset of 1,000 pairs from MS-Celeb-1M and the most promising SD-GANs from our FaceNet evaluation.', '1705.07904-1-37-2': 'We ask human annotators to determine if each pair depicts the "same person" or "different people".', '1705.07904-1-37-3': 'Random batches of ten pairs are evaluated by ensembles of three unique annotators and predictions are determined by majority vote.', '1705.07904-1-37-4': 'We also manually judge [MATH] pairs from MS-Celeb-1M to provide a benchmark for assessing the quality of the Mechanical Turk ensembles.', '1705.07904-1-37-5': 'Results are summarized in Table [REF].', '1705.07904-1-38-0': '# Discussion', '1705.07904-1-39-0': 'Our evaluation shows that FaceNet recognizes matched faces in the MS-Celeb-1M with [MATH] accuracy.', '1705.07904-1-39-1': 'We (the authors) achieve similar accuracy when manually annotating a small subset.', '1705.07904-1-39-2': 'These results indicate some amount of label noise in MS-Celeb-1M.', '1705.07904-1-39-3': 'Our most promising SD-DCGAN and SD-BEGAN models produce matched pairs of faces that the pretrained FaceNet model recognizes as matches with an accuracy similar to that achieved on real images.', '1705.07904-1-39-4': 'FaceNet verifies pairs of faces from SD-BEGAN with [MATH] accuracy, just shy of the [MATH] achieved on MS-Celeb-1M.', '1705.07904-1-39-5': 'At stricter thresholds ([MATH]), true positive rates are lower for SD-GAN pairs than for MS-Celeb-1M but interestingly, for less strict thresholds, the SD-BEGAN achieves a higher true positive rate than MS-Celeb-1M images (Figure [REF]).', '1705.07904-1-40-0': 'For all datasets, human annotators on Mechanical Turk answered "same person" less frequently than FaceNet at the accuracy-maximizing threshold [MATH].', '1705.07904-1-40-1': 'Even on real data, balanced so that [MATH] of pairs are identity-matched, human annotators report "same person" only [MATH] of the pairs.', '1705.07904-1-40-2': 'Annotators achieve [MATH] and [MATH] accuracy on SD-DCGAN and SD-BEGAN pairs respectively, compared to [MATH] accuracy on real data.', '1705.07904-1-40-3': 'Notably, annotators answered "same" for [MATH] of the SD-DCGAN examples, almost twice as often as for the MS-Celeb-1M and SD-BEGAN datasets.', '1705.07904-1-40-4': 'This provides some evidence that SD-DCGAN produces less identity diversity.', '1705.07904-1-41-0': 'Face samples from SD-BEGAN appear comparably crisp and globally coherent to those generated by a vanilla BEGAN.', '1705.07904-1-41-1': 'Given a SD-GAN trained generator, we can independently interpolate along the identity and observation manifolds (Figure [REF]).', '1705.07904-1-41-2': 'Here, the diagonal represents the entangled interpolation typically shown for ordinary GANs.', '1705.07904-1-41-3': 'In Figure [REF], we demonstrate that when we vary the observation vector [MATH], SD-GANs can change the color of clothing, add or remove sunglasses, or change pose.', '1705.07904-1-41-4': 'They can also perturb the lighting, color saturation, and contrast of an image, all while keeping the apparent identity fixed.', '1705.07904-1-41-5': 'We note, subjectively, that samples from SD-DCGAN tend to appear less photorealistic than those from SD-BEGAN.', '1705.07904-1-42-0': 'On the shoe dataset, we find that the SD-DCGAN model produces convincing results.', '1705.07904-1-42-1': 'As desired, manipulating [MATH] while keeping [MATH] fixed yields distinct shoes in consistent poses (Figure [REF]).', '1705.07904-1-42-2': 'The identity code [MATH] appears to capture the broad categories of shoes (sneakers, flip-flops, boots, etc.).', '1705.07904-1-42-3': 'We were surprised to find that both the original BEGAN and SD-BEGAN fail to produce diverse pictures of shoes (see Appendix [REF] for examples of SD-BEGAN).', '1705.07904-1-43-0': '# Related work', '1705.07904-1-44-0': '# Conclusions'}

{'1705.07904-2-0-0': 'We propose a new algorithm for training generative adversarial networks that jointly learns latent codes for both identities (e.g. individual humans) and observations (e.g. specific photographs).', '1705.07904-2-0-1': 'By fixing the identity portion of the latent codes, we can generate diverse images of the same subject, and by fixing the observation portion, we can traverse the manifold of subjects while maintaining contingent aspects such as lighting and pose.', '1705.07904-2-0-2': 'Our algorithm features a pairwise training scheme in which each sample from the generator consists of two images with a common identity code.', '1705.07904-2-0-3': 'Corresponding samples from the real dataset consist of two distinct photographs of the same subject.', '1705.07904-2-0-4': 'In order to fool the discriminator, the generator must produce pairs that are photorealistic, distinct, and appear to depict the same individual.', '1705.07904-2-0-5': 'We augment both the DCGAN and BEGAN approaches with Siamese discriminators to facilitate pairwise training.', '1705.07904-2-0-6': "Experiments with human judges and an off-the-shelf face verification system demonstrate our algorithm's ability to generate convincing, identity-matched photographs.", '1705.07904-2-1-0': '# Introduction', '1705.07904-2-2-0': 'In many domains, a suitable generative process might consist of several stages.', '1705.07904-2-2-1': 'To generate a photograph of a product, we might wish to first sample from the space of products, and then from the space of photographs of that product.', '1705.07904-2-2-2': 'Given such disentangled representations in a multistage generative process, an online retailer might diversify its catalog, depicting products in a wider variety of settings.', '1705.07904-2-2-3': 'A retailer could also flip the process, imagining new products in a fixed setting.', '1705.07904-2-2-4': 'Datasets for such domains often contain many labeled identities with fewer observations of each (e.g. a collection of face portraits with thousands of people and ten photos of each).', '1705.07904-2-2-5': 'While we may know the identity of the subject in each photograph, we may not know the contingent aspects of the observation (such as lighting, pose and background).', '1705.07904-2-2-6': 'This kind of data is ubiquitous; given a set of commonalities, we might want to incorporate this structure into our latent representations.', '1705.07904-2-3-0': 'Generative adversarial networks (GANs) learn mappings from latent codes [MATH] in some low-dimensional space [MATH] to points in the space of natural data [MATH] .', '1705.07904-2-3-1': 'They achieve this power through an adversarial training scheme pitting a generative model [MATH] against a discriminative model [MATH] in a minimax game.', '1705.07904-2-3-2': 'While GANs are popular, owing to their ability to generate high-fidelity images, they do not, in their original form, explicitly disentangle the latent factors according to known commonalities.', '1705.07904-2-4-0': 'In this paper, we propose Semantically Decomposed GANs (SD-GANs), which encourage a specified portion of the latent space to correspond to a known source of variation.', '1705.07904-2-4-1': 'The technique decomposes the latent code [MATH] into one portion [MATH] corresponding to identity, and the remaining portion [MATH] corresponding to the other contingent aspects of observations.', '1705.07904-2-4-2': 'SD-GANs learn through a pairwise training scheme in which each sample from the real dataset consists of two distinct images with a common identity.', '1705.07904-2-4-3': 'Each sample from the generator consists of a pair of images with common [MATH] but differing [MATH].', '1705.07904-2-4-4': 'In order to fool the discriminator, the generator must not only produce diverse and photorealistic images, but also images that depict the same identity when [MATH] is fixed.', '1705.07904-2-4-5': 'For SD-GANs, we modify the discriminator so that it can determine whether a pair of samples constitutes a match.', '1705.07904-2-5-0': 'As a case study, we experiment with a dataset of face photographs, demonstrating that SD-GANs can generate contrasting images of the same subject (Figure [REF]; interactive web demo in footnote on previous page).', '1705.07904-2-5-1': 'The generator learns that certain properties are free to vary across observations but not identity.', '1705.07904-2-5-2': 'For example, SD-GANs learn that pose, facial expression, hair styles, grayscale vs. color, and lighting can all vary across different photographs of the same individual.', '1705.07904-2-5-3': 'On the other hand, the aspects that are more salient for facial verification remain consistent as we vary the observation code [MATH].', '1705.07904-2-5-4': 'We also train SD-GANs on a dataset of product images, containing multiple photographs of each product from various perspectives (Figure [REF]).', '1705.07904-2-6-0': 'We demonstrate that SD-GANs trained on faces generate stylistically-contrasting, identity-matched image pairs that human annotators and a state-of-the-art face verification algorithm recognize as depicting the same subject.', '1705.07904-2-6-1': 'On measures of identity coherence and image diversity, SD-GANs perform comparably to a recent conditional GAN method ; SD-GANs can also imagine new identities, while conditional GANs are limited to generating existing identities from the training data.', '1705.07904-2-7-0': '# Semantically Decomposed Generative Adversarial Networks', '1705.07904-2-8-0': 'Before introducing our algorithm, we briefly review the prerequisite concepts.', '1705.07904-2-9-0': '## GAN preliminaries', '1705.07904-2-10-0': 'GANs leverage the discriminative power of neural networks to learn generative models.', '1705.07904-2-10-1': 'The generative model [MATH] ingests latent codes [MATH], sampled from some known prior [MATH], and produces [MATH], a sample of an implicit distribution [MATH].', '1705.07904-2-10-2': 'The learning process consists of a minimax game between [MATH], parameterized by [MATH], and a discriminative model [MATH], parameterized by [MATH].', '1705.07904-2-10-3': 'In the original formulation, the discriminative model tries to maximize log likelihood, yielding [EQUATION]', '1705.07904-2-10-4': 'Training proceeds as follows: For [MATH] iterations, sample one minibatch from the real distribution [MATH] and one from the distribution of generated images [MATH], updating discriminator weights [MATH] to increase [MATH] by stochastic gradient ascent.', '1705.07904-2-10-5': 'Then sample a minibatch from [MATH], updating [MATH] to decrease [MATH] by stochastic gradient descent.', '1705.07904-2-11-0': '[CITATION] propose energy-based GANs (EBGANs), in which the discriminator can be viewed as an energy function.', '1705.07904-2-11-1': 'Specifically, they devise a discriminator consisting of an autoencoder: [MATH].', '1705.07904-2-11-2': "In the minimax game, the discriminator's weights are updated to minimize the reconstruction error [MATH] for real data, while maximizing the error [MATH] for the generator.", '1705.07904-2-11-3': 'More recently, [CITATION] extend this work, introducing Boundary Equilibrium GANs (BEGANs), which optimize the Wasserstein distance (reminiscent of Wasserstein GANs ) between autoencoder loss distributions, yielding the formulation: [EQUATION]', '1705.07904-2-11-4': 'Additionally, they introduce a method for stabilizing training.', '1705.07904-2-11-5': 'Positing that training becomes unstable when the discriminator cannot distinguish between real and generated images, they introduce a new hyperparameter [MATH], updating the value function on each iteration to maintain a desired ratio between the two reconstruction errors: [MATH].', '1705.07904-2-11-6': 'The BEGAN model produces what appear to us, subjectively, to be the sharpest images of faces yet generated by a GAN.', '1705.07904-2-11-7': 'In this work, we adapt both the DCGAN and BEGAN algorithms to the SD-GAN training scheme.', '1705.07904-2-12-0': '## SD-GAN formulation', '1705.07904-2-13-0': '[t] Semantically Decomposed GAN Training [1] n in 1:NumberOfIterations m in 1:MinibatchSize Sample one identity vector [MATH].', '1705.07904-2-13-1': 'Sample two observation vectors [MATH].', '1705.07904-2-13-2': '[MATH], [MATH].', '1705.07904-2-13-3': 'Generate pair of images [MATH], adding them to the minibatch with label [MATH] (fake).', '1705.07904-2-13-4': 'm in 1:MinibatchSize Sample one identity [MATH] uniformly at random from the real data set.', '1705.07904-2-13-5': 'Sample two images of [MATH] without replacement [MATH].', '1705.07904-2-13-6': 'Add the pair to the minibatch, assigning label [MATH] (real).', '1705.07904-2-13-7': 'Update discriminator weights by [MATH] using its stochastic gradient.', '1705.07904-2-13-8': 'Sample another minibatch of identity-matched latent vectors [MATH].', '1705.07904-2-13-9': 'Update generator weights by stochastic gradient descent [MATH].', '1705.07904-2-14-0': "Consider the data's identity as a random variable [MATH] in a discrete index set [MATH].", '1705.07904-2-14-1': 'We seek to learn a latent representation that conveniently decomposes the variation in the real data into two parts: 1) due to [MATH], and 2) due to the other factors of variation in the data, packaged as a random variable [MATH].', '1705.07904-2-14-2': 'Ideally, the decomposition of the variation in the data into [MATH] and [MATH] should correspond exactly to a decomposition of the latent space [MATH].', '1705.07904-2-14-3': 'This would permit convenient interpolation and other operations on the inferred subspaces [MATH] and [MATH].', '1705.07904-2-15-0': 'A conventional GAN samples [MATH] from their joint distribution.', '1705.07904-2-15-1': "Such a GAN's generative model samples directly from an unstructured prior over the latent space.", '1705.07904-2-15-2': 'It does not disentangle the variation in [MATH] and [MATH], for instance by modeling conditional distributions [MATH], but only models their average with respect to the prior on [MATH].', '1705.07904-2-16-0': 'Our SD-GAN method learns such a latent space decomposition, partitioning the coordinates of [MATH] into two parts representing the subspaces, so that any [MATH] can be written as the concatenation [MATH] of its identity representation [MATH] and its contingent aspect representation [MATH].', '1705.07904-2-16-1': 'SD-GANs achieve this through a pairwise training scheme in which each sample from the real data consists of [MATH], a pair of images with a common identity [MATH].', '1705.07904-2-16-2': 'Each sample from the generator consists of [MATH], a pair of images generated from a common identity vector [MATH] but i.i.d. observation vectors [MATH].', '1705.07904-2-16-3': 'We assign identity-matched pairs from [MATH] the label [MATH] and [MATH]-matched pairs from [MATH] the label [MATH].', '1705.07904-2-16-4': 'The discriminator can thus learn to reject pairs for either of two primary reasons: 1) not photorealistic or 2) not plausibly depicting the same subject.', '1705.07904-2-16-5': 'See Algorithm [REF] for SD-GAN training pseudocode.', '1705.07904-2-17-0': '## SD-GAN discriminator architecture', '1705.07904-2-18-0': 'With SD-GANs, there is no need to alter the architecture of the generator.', '1705.07904-2-18-1': 'However, the discriminator must now act upon two images, producing a single output.', '1705.07904-2-18-2': 'Moreover, the effects of the two input images [MATH] on the output score are not independent.', '1705.07904-2-18-3': 'Two images might be otherwise photorealistic but deserve rejection because they clearly depict different identities.', '1705.07904-2-18-4': 'To this end, we devise two novel discriminator architectures to adapt DCGAN and BEGAN respectively.', '1705.07904-2-18-5': 'In both cases, we first separately encode each image using the same convolutional neural network [MATH] (Figure [REF]).', '1705.07904-2-18-6': "We choose this Siamese setup as our problem is symmetrical in the images, and thus it's sensible to share weights between the encoders.", '1705.07904-2-19-0': 'To adapt DCGAN, we stack the feature maps [MATH] and [MATH] along the channel axis, applying one additional strided convolution.', '1705.07904-2-19-1': 'This allows the network to further aggregate information from the two images before flattening and fully connecting to a sigmoid output.', '1705.07904-2-19-2': 'For BEGAN, because the discriminator is an autoencoder, our architecture is more complicated.', '1705.07904-2-19-3': 'After encoding each image, we concatenate the representations [MATH] and apply one fully connected bottleneck layer [MATH] with linear activation.', '1705.07904-2-19-4': 'In alignment with BEGAN, the SD-BEGAN bottleneck has the same dimensionality as the tuple of latent codes ([MATH], [MATH], [MATH]) that generated the pair of images.', '1705.07904-2-19-5': 'Following the bottleneck, we apply a second FC layer [MATH], taking the first [MATH] components of its output to be the input to the first decoder and the second [MATH] components to be the input to the second decoder.', '1705.07904-2-19-6': 'The shared intermediate layer gives SD-BEGAN a mechanism to push apart matched and unmatched pairs.', '1705.07904-2-19-7': 'We specify our exact architectures in full detail in Appendix [REF].', '1705.07904-2-20-0': '# Experiments', '1705.07904-2-21-0': 'We experimentally validate SD-GANs using two datasets: 1) the MS-Celeb-1M dataset of celebrity face images and 2) a dataset of shoe images collected from Amazon .', '1705.07904-2-21-1': 'Both datasets contain a large number of identities (people and shoes, respectively) with multiple observations of each.', '1705.07904-2-21-2': 'The "in-the-wild" nature of the celebrity face images offers a richer test bed for our method as both identities and contingent factors are significant sources of variation.', '1705.07904-2-21-3': "In contrast, Amazon's shoe images tend to vary only with camera perspective for a given product, making this data useful for sanity-checking our approach.", '1705.07904-2-22-0': 'Faces', '1705.07904-2-23-0': 'From the aligned face images in the MS-Celeb-1M dataset, we select 12,500 celebrities at random and [MATH] associated images of each, resizing them to [MATH]x[MATH] pixels.', '1705.07904-2-23-1': 'We split the celebrities into subsets of 10,000 (training), 1,250 (validation) and 1,250 (test).', '1705.07904-2-23-2': 'The dataset has a small number of duplicate images and some label noise (images matched to the wrong celebrity).', '1705.07904-2-23-3': 'We detect and remove duplicates by hashing the images, but we do not rid the data of label noise.', '1705.07904-2-23-4': 'We scale the pixel values to [MATH], performing no additional preprocessing or data augmentation.', '1705.07904-2-24-0': 'Shoes Synthesizing novel product images is another promising domain for our method.', '1705.07904-2-24-1': 'In our shoes dataset, product photographs are captured against white backgrounds and primarily differ in orientation and distance.', '1705.07904-2-24-2': 'Accordingly, we expect that SD-GAN training will allocate the observation latent space to capture these aspects.', '1705.07904-2-24-3': 'We choose to study shoes as a prototypical example of a category of product images.', '1705.07904-2-24-4': 'The Amazon dataset contains around 3,000 unique products with the category "Shoe" and multiple product images.', '1705.07904-2-24-5': 'We use the same [MATH] split and again hash the images to ensure that the splits are disjoint.', '1705.07904-2-24-6': 'There are [MATH] photos of each product on average.', '1705.07904-2-25-0': '## Training details', '1705.07904-2-26-0': 'We train SD-DCGANs on both of our datasets for 500,000 iterations using batches of [MATH] identity-matched pairs.', '1705.07904-2-26-1': 'To optimize SD-DCGAN, we use the Adam optimizer with hyperparameters [MATH] as recommended by [CITATION].', '1705.07904-2-26-2': 'We also consider a non-Siamese discriminator that simply stacks the channels of the pair of real or fake images before encoding (SD-DCGAN-SC).', '1705.07904-2-27-0': 'As in , we sample latent vectors [MATH].', '1705.07904-2-27-1': 'For SD-GANs, we partition the latent codes according to [MATH] using values of [MATH].', '1705.07904-2-27-2': 'Our algorithm can be trivially applied with [MATH]-wise training (vs. pairwise).', '1705.07904-2-27-3': 'To explore the effects of using [MATH], we also experiment with an SD-DCGAN where we sample [MATH] instances each from [MATH] for some [MATH] and from [MATH] for some [MATH].', '1705.07904-2-27-4': 'For all experiments, unless otherwise stated, we use [MATH] and [MATH].', '1705.07904-2-28-0': 'We also train an SD-BEGAN on both of our datasets.', '1705.07904-2-28-1': 'The increased complexity of the SD-BEGAN model significantly increases training time, limiting our ability to perform more-exhaustive hyperparameter validation (as we do for SD-DCGAN).', '1705.07904-2-28-2': 'We use the Adam optimizer with the default hyperparameters from for our SD-BEGAN experiments.', '1705.07904-2-28-3': 'While results from our SD-DCGAN [MATH] model are compelling, an experiment with a [MATH] variant of SD-BEGAN resulted in early mode collapse (Appendix [REF]); hence, we excluded SD-BEGAN [MATH] from our evaluation.', '1705.07904-2-29-0': 'We also compare to a DCGAN architecture trained using the auxiliary classifier GAN (AC-GAN) method .', '1705.07904-2-29-1': 'AC-GAN differs from SD-GAN in two key ways: 1) random identity codes [MATH] are replaced by a one-hot embedding over all the identities in the training set (matrix of size [MATH]x[MATH]); 2) the AC-GAN method enforces that generated photos depict the proper identity by tasking its discriminator with predicting the identity of the generated or real image.', '1705.07904-2-29-2': 'Unlike SD-GANs, the AC-DCGAN model cannot imagine new identities; when generating from AC-DCGAN (for our quantitative comparisons to SD-GANs), we must sample a random identity from those existing in the training data.', '1705.07904-2-30-0': '## Evaluation', '1705.07904-2-31-0': 'The evaluation of generative models is a fraught topic.', '1705.07904-2-31-1': 'Quantitative measures of sample quality can be poorly correlated with each other .', '1705.07904-2-31-2': 'Accordingly, we design an evaluation to match conceivable uses of our algorithm.', '1705.07904-2-31-3': "Because we hope to produce diverse samples that humans deem to depict the same person, we evaluate the identity coherence of SD-GANs and baselines using both a pretrained face verification model and crowd-sourced human judgments obtained through Amazon's Mechanical Turk platform.", '1705.07904-2-32-0': '### Quantitative', '1705.07904-2-33-0': 'Recent advancements in face verification using deep convolutional neural networks have yielded accuracy rivaling humans.', '1705.07904-2-33-1': 'For our evaluation, we procure FaceNet, a publicly-available face verifier based on the Inception-ResNet architecture .', '1705.07904-2-33-2': 'The FaceNet model was pretrained on the CASIA-WebFace dataset and achieves [MATH] accuracy on the LFW benchmark .', '1705.07904-2-34-0': 'FaceNet ingests normalized, [MATH]x[MATH] color images and produces an embedding [MATH].', '1705.07904-2-34-1': 'The training objective for FaceNet is to learn embeddings that minimize the [MATH] distance between matched pairs of faces and maximize the distance for mismatched pairs.', '1705.07904-2-34-2': 'Accordingly, the embedding space yields a function for measuring the similarity between two faces [MATH] and [MATH]: [MATH].', '1705.07904-2-34-3': 'Given two images, [MATH] and [MATH], we label them as a match if [MATH] where [MATH] is the accuracy-maximizing threshold on a class-balanced set of pairs from MS-Celeb-1M validation data.', '1705.07904-2-34-4': 'We use the same threshold for evaluating both real and synthetic data with FaceNet.', '1705.07904-2-35-0': 'We compare the performance of FaceNet on pairs of images from the MS-Celeb-1M test set against generated samples from our trained SD-GAN models and AC-DCGAN baseline.', '1705.07904-2-35-1': "To match FaceNet's training data, we preprocess all images by resizing from [MATH]x[MATH] to [MATH]x[MATH], normalizing each image individually.", '1705.07904-2-35-2': 'We prepare 10,000 pairs from MS-Celeb-1M, half identity-matched and half unmatched.', '1705.07904-2-35-3': 'From each generative model, we generate 5,000 pairs each with [MATH] and 5,000 pairs with [MATH].', '1705.07904-2-35-4': 'For each sample, we draw observation vectors [MATH] randomly.', '1705.07904-2-36-0': 'We also want to ensure that identity-matched images produced by the generative models are diverse.', '1705.07904-2-36-1': 'To this end, we propose an intra-identity sample diversity (ID-Div) metric.', '1705.07904-2-36-2': 'The multi-scale structural similarity (MS-SSIM) metric reports the similarity of two images on a scale from [MATH] (no resemblance) to [MATH] (identical images).', '1705.07904-2-36-3': 'We report [MATH] minus the mean MS-SSIM for all pairs of identity-matched images as ID-Div.', '1705.07904-2-36-4': 'To measure the overall sample diversity (All-Div), we also compute [MATH] minus the mean similarity of [MATH]k pairs with random identities.', '1705.07904-2-37-0': 'In Table [REF], we report the area under the receiver operating characteristic curve (AUC) and accuracy (at threshold [MATH]) of FaceNet on the real and generated data.', '1705.07904-2-37-1': 'We also report our proposed diversity statistics.', '1705.07904-2-37-2': 'FaceNet verifies pairs from the real data with [MATH] accuracy compared to [MATH] on pairs from our SD-BEGAN model.', '1705.07904-2-37-3': 'Though this is comparable to the accuracy achieved on pairs from the AC-DCGAN baseline, our model produces higher average sample diversity for a given identity.', '1705.07904-2-38-0': 'We also report the combined memory footprint of [MATH] and [MATH] for all methods in Table [REF].', '1705.07904-2-38-1': 'For conditional GAN approaches, the number of parameters grows linearly with the number of identities in the training data.', '1705.07904-2-38-2': 'Especially in the case of the AC-GAN, where the discriminator computes a softmax over all identities, linear scaling may be prohibitive.', '1705.07904-2-38-3': "While our [MATH]k-identity subset of MS-Celeb-1M requires a [MATH]MB AC-DCGAN model, an AC-DCGAN for all [MATH]M identities would be over [MATH]GB, with more than [MATH]% of the parameters devoted to the weights in the discriminator's softmax layer.", '1705.07904-2-38-4': 'In contrast, the complexity of SD-GAN is constant in the number of identities.', '1705.07904-2-39-0': '### Qualitative', '1705.07904-2-40-0': 'In addition to validating that identity-matched SD-GAN samples are verified by FaceNet, we also demonstrate that humans are similarly convinced through experiments using Mechanical Turk.', '1705.07904-2-40-1': 'For these experiments, we use balanced subsets of 1,000 pairs from MS-Celeb-1M and the most promising generative methods from our FaceNet evaluation.', '1705.07904-2-40-2': 'We ask human annotators to determine if each pair depicts the "same person" or "different people".', '1705.07904-2-40-3': 'Annotators are presented with batches of ten pairs at a time.', '1705.07904-2-40-4': 'Each pair is presented to three distinct annotators and predictions are determined by majority vote.', '1705.07904-2-40-5': 'Additionally, to provide a benchmark for assessing the quality of the Mechanical Turk ensembles, we (the authors) manually judged [MATH] pairs from MS-Celeb-1M.', '1705.07904-2-40-6': 'Results are in Table [REF].', '1705.07904-2-41-0': 'For all datasets, human annotators on Mechanical Turk answered "same person" less frequently than FaceNet when the latter uses the accuracy-maximizing threshold [MATH].', '1705.07904-2-41-1': 'Even on real data, balanced so that [MATH] of pairs are identity-matched, annotators report "same person" only [MATH] of the time (compared to the [MATH] of FaceNet).', '1705.07904-2-41-2': 'While annotators achieve higher accuracy on pairs from AC-DCGAN than pairs from SD-BEGAN, they also answer "same person" [MATH] more often for AC-DCGAN pairs than real data.', '1705.07904-2-41-3': 'In contrast, annotators answer "same person" at the same rate for SD-BEGAN pairs as real data.', '1705.07904-2-41-4': 'This may be attributable to the lower sample diversity produced by AC-DCGAN.', '1705.07904-2-41-5': 'Samples from SD-DCGAN and SD-BEGAN are shown in Figures [REF] and [REF] respectively.', '1705.07904-2-42-0': '# Related work', '1705.07904-2-43-0': '# Discussion', '1705.07904-2-44-0': 'Our evaluation demonstrates that SD-GANs can disentangle those factors of variation corresponding to identity from the rest.', '1705.07904-2-44-1': 'Moreover, with SD-GANs we can sample never-before-seen identities, a benefit not shared by conditional GANs.', '1705.07904-2-44-2': 'In Figure [REF], we demonstrate that by varying the observation vector [MATH], SD-GANs can change the color of clothing, add or remove sunglasses, or change facial pose.', '1705.07904-2-44-3': 'They can also perturb the lighting, color saturation, and contrast of an image, all while keeping the apparent identity fixed.', '1705.07904-2-44-4': 'We note, subjectively, that samples from SD-DCGAN tend to appear less photorealistic than those from SD-BEGAN.', '1705.07904-2-44-5': 'Given a generator trained with SD-GAN, we can independently interpolate along the identity and observation manifolds (Figure [REF]).', '1705.07904-2-44-6': 'Here, the diagonal represents the entangled interpolation typically shown for ordinary GANs.', '1705.07904-2-45-0': 'On the shoe dataset, we find that the SD-DCGAN model produces convincing results.', '1705.07904-2-45-1': 'As desired, manipulating [MATH] while keeping [MATH] fixed yields distinct shoes in consistent poses (Figure [REF]).', '1705.07904-2-45-2': 'The identity code [MATH] appears to capture the broad categories of shoes (sneakers, flip-flops, boots, etc.).', '1705.07904-2-45-3': 'Surprisingly, neither original BEGAN nor SD-BEGAN can produce diverse shoe images (Appendix [REF]).'}

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'1705.07904-3-14-4'], ['1705.07904-2-19-5', '1705.07904-3-14-5'], ['1705.07904-2-19-6', '1705.07904-3-14-6'], ['1705.07904-2-19-7', '1705.07904-3-14-7'], ['1705.07904-2-26-0', '1705.07904-3-21-0'], ['1705.07904-2-26-1', '1705.07904-3-21-1'], ['1705.07904-2-26-2', '1705.07904-3-21-2'], ['1705.07904-2-14-0', '1705.07904-3-9-0'], ['1705.07904-2-14-1', '1705.07904-3-9-1'], ['1705.07904-2-14-2', '1705.07904-3-9-2'], ['1705.07904-2-14-3', '1705.07904-3-9-3'], ['1705.07904-2-13-0', '1705.07904-3-8-0'], ['1705.07904-2-13-1', '1705.07904-3-8-1'], ['1705.07904-2-13-3', '1705.07904-3-8-3'], ['1705.07904-2-13-4', '1705.07904-3-8-4'], ['1705.07904-2-13-5', '1705.07904-3-8-5'], ['1705.07904-2-13-6', '1705.07904-3-8-6'], ['1705.07904-2-13-7', '1705.07904-3-8-7'], ['1705.07904-2-13-8', '1705.07904-3-8-8'], ['1705.07904-2-13-9', '1705.07904-3-8-9'], ['1705.07904-2-34-0', '1705.07904-3-29-0'], ['1705.07904-2-34-1', '1705.07904-3-29-1'], ['1705.07904-2-34-2', '1705.07904-3-29-2'], ['1705.07904-2-34-3', '1705.07904-3-29-3'], ['1705.07904-2-34-4', '1705.07904-3-29-4'], ['1705.07904-2-27-0', '1705.07904-3-22-0'], ['1705.07904-2-27-1', '1705.07904-3-22-1'], ['1705.07904-2-27-2', '1705.07904-3-22-2'], ['1705.07904-2-27-3', '1705.07904-3-22-3'], ['1705.07904-2-27-4', '1705.07904-3-22-4'], ['1705.07904-2-28-0', '1705.07904-3-23-0'], ['1705.07904-2-28-1', '1705.07904-3-23-1'], ['1705.07904-2-28-2', '1705.07904-3-23-2'], ['1705.07904-2-28-3', '1705.07904-3-23-3'], ['1705.07904-2-33-0', '1705.07904-3-28-0'], ['1705.07904-2-33-1', '1705.07904-3-28-1'], ['1705.07904-2-33-2', '1705.07904-3-28-2'], ['1705.07904-2-41-0', '1705.07904-3-36-0'], ['1705.07904-2-41-1', '1705.07904-3-36-1'], ['1705.07904-2-41-2', '1705.07904-3-36-2'], ['1705.07904-2-41-3', '1705.07904-3-36-3'], ['1705.07904-2-41-4', '1705.07904-3-36-4'], ['1705.07904-2-41-5', '1705.07904-3-36-5'], ['1705.07904-2-29-0', '1705.07904-3-24-0'], ['1705.07904-2-29-2', '1705.07904-3-24-2'], ['1705.07904-2-31-0', '1705.07904-3-26-0'], ['1705.07904-2-31-1', '1705.07904-3-26-1'], ['1705.07904-2-31-2', '1705.07904-3-26-2'], ['1705.07904-2-31-3', '1705.07904-3-26-3'], ['1705.07904-2-23-0', '1705.07904-3-18-0'], ['1705.07904-2-23-1', '1705.07904-3-18-1'], ['1705.07904-2-23-2', '1705.07904-3-18-2'], ['1705.07904-2-23-3', '1705.07904-3-18-3'], ['1705.07904-2-23-4', '1705.07904-3-18-4'], ['1705.07904-2-45-0', '1705.07904-3-40-0'], ['1705.07904-2-45-1', '1705.07904-3-40-1'], ['1705.07904-2-45-2', '1705.07904-3-40-2'], ['1705.07904-2-45-3', '1705.07904-3-40-3'], ['1705.07904-2-37-1', '1705.07904-3-32-1'], ['1705.07904-2-37-2', '1705.07904-3-32-2'], ['1705.07904-1-6-1', '1705.07904-2-5-1'], ['1705.07904-1-6-3', '1705.07904-2-5-3'], ['1705.07904-1-6-5', '1705.07904-2-5-4'], ['1705.07904-1-11-0', '1705.07904-2-11-0'], ['1705.07904-1-11-1', '1705.07904-2-11-1'], ['1705.07904-1-11-2', '1705.07904-2-11-2'], ['1705.07904-1-12-1', '1705.07904-2-11-4'], ['1705.07904-1-12-2', '1705.07904-2-11-5'], ['1705.07904-1-12-3', '1705.07904-2-11-6'], ['1705.07904-1-12-4', '1705.07904-2-11-7'], ['1705.07904-2-16-0', '1705.07904-3-11-0'], ['1705.07904-2-16-1', '1705.07904-3-11-1'], ['1705.07904-2-16-2', '1705.07904-3-11-2'], ['1705.07904-2-16-3', '1705.07904-3-11-3'], ['1705.07904-2-16-4', '1705.07904-3-11-4'], ['1705.07904-2-16-5', '1705.07904-3-11-5']]

[['1705.07904-1-20-0', '1705.07904-2-19-0'], ['1705.07904-1-20-1', '1705.07904-2-19-1'], ['1705.07904-1-28-4', '1705.07904-2-26-2'], ['1705.07904-1-33-1', '1705.07904-2-33-1'], ['1705.07904-1-33-2', '1705.07904-2-33-2'], ['1705.07904-1-40-0', '1705.07904-2-41-0'], ['1705.07904-1-40-1', '1705.07904-2-41-1'], ['1705.07904-1-29-2', '1705.07904-2-27-2'], ['1705.07904-1-29-3', '1705.07904-2-27-3'], ['1705.07904-1-22-0', '1705.07904-2-21-0'], ['1705.07904-1-22-1', '1705.07904-2-21-1'], ['1705.07904-1-35-0', '1705.07904-2-35-0'], ['1705.07904-1-34-1', '1705.07904-2-34-1'], ['1705.07904-1-34-3', '1705.07904-2-34-3'], ['1705.07904-1-34-4', '1705.07904-2-34-4'], ['1705.07904-1-2-0', '1705.07904-2-3-0'], ['1705.07904-1-5-0', '1705.07904-2-4-0'], ['1705.07904-1-5-1', '1705.07904-2-4-1'], ['1705.07904-1-5-4', '1705.07904-2-4-4'], ['1705.07904-1-5-5', '1705.07904-2-4-5'], ['1705.07904-1-41-1', '1705.07904-2-44-5'], ['1705.07904-1-41-3', '1705.07904-2-44-2'], ['1705.07904-1-14-4', '1705.07904-2-13-4'], ['1705.07904-1-15-1', '1705.07904-2-14-1'], ['1705.07904-1-8-1', '1705.07904-2-10-2'], ['1705.07904-1-8-2', '1705.07904-2-10-3'], ['1705.07904-1-8-3', '1705.07904-2-10-4'], ['1705.07904-1-8-4', '1705.07904-2-10-5'], ['1705.07904-1-30-4', '1705.07904-2-28-2'], ['1705.07904-1-32-1', '1705.07904-2-31-1'], ['1705.07904-1-19-5', '1705.07904-2-18-5'], ['1705.07904-1-0-0', '1705.07904-2-0-0'], ['1705.07904-1-0-1', '1705.07904-2-0-1'], ['1705.07904-1-0-4', '1705.07904-2-0-4'], ['1705.07904-1-0-5', '1705.07904-2-0-5'], ['1705.07904-1-26-2', '1705.07904-2-24-1'], ['1705.07904-1-26-3', '1705.07904-2-24-2'], ['1705.07904-1-26-6', '1705.07904-2-24-5'], ['1705.07904-1-37-1', '1705.07904-2-40-1'], ['1705.07904-1-37-4', '1705.07904-2-40-5'], ['1705.07904-1-37-5', '1705.07904-2-40-6'], ['1705.07904-2-44-2', '1705.07904-3-39-2'], ['1705.07904-2-29-1', '1705.07904-3-24-1'], ['1705.07904-2-37-0', '1705.07904-3-32-0'], ['1705.07904-1-6-2', '1705.07904-2-5-2'], ['1705.07904-1-6-4', '1705.07904-2-6-0'], ['1705.07904-1-12-0', '1705.07904-2-11-3']]

[]

[['1705.07904-1-40-2', '1705.07904-2-41-2'], ['1705.07904-1-22-2', '1705.07904-2-21-2'], ['1705.07904-1-17-5', '1705.07904-2-16-5'], ['1705.07904-1-5-2', '1705.07904-2-4-2'], ['1705.07904-1-5-2', '1705.07904-2-4-3'], ['1705.07904-1-5-3', '1705.07904-2-4-2'], ['1705.07904-1-5-3', '1705.07904-2-4-3'], ['1705.07904-1-14-5', '1705.07904-2-13-5'], ['1705.07904-1-15-0', '1705.07904-2-14-0'], ['1705.07904-1-30-1', '1705.07904-2-28-3'], ['1705.07904-1-30-2', '1705.07904-2-28-1'], ['1705.07904-1-32-3', '1705.07904-2-31-3'], ['1705.07904-1-19-0', '1705.07904-2-18-0'], ['1705.07904-1-26-0', '1705.07904-2-24-0'], ['1705.07904-1-26-7', '1705.07904-2-24-6'], ['1705.07904-1-24-6', '1705.07904-2-23-4'], ['1705.07904-2-37-3', '1705.07904-3-32-3'], ['1705.07904-1-6-0', '1705.07904-2-5-0']]

[]

['1705.07904-1-14-2', '1705.07904-2-8-0', '1705.07904-2-13-2', '1705.07904-2-22-0', '1705.07904-3-3-0', '1705.07904-3-8-2', '1705.07904-3-17-0']

{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}

https://arxiv.org/abs/1705.07904

{'1705.07904-3-0-0': 'We propose a new algorithm for training generative adversarial networks that jointly learns latent codes for both identities (e.g. individual humans) and observations (e.g. specific photographs).', '1705.07904-3-0-1': 'By fixing the identity portion of the latent codes, we can generate diverse images of the same subject, and by fixing the observation portion, we can traverse the manifold of subjects while maintaining contingent aspects such as lighting and pose.', '1705.07904-3-0-2': 'Our algorithm features a pairwise training scheme in which each sample from the generator consists of two images with a common identity code.', '1705.07904-3-0-3': 'Corresponding samples from the real dataset consist of two distinct photographs of the same subject.', '1705.07904-3-0-4': 'In order to fool the discriminator, the generator must produce pairs that are photorealistic, distinct, and appear to depict the same individual.', '1705.07904-3-0-5': 'We augment both the DCGAN and BEGAN approaches with Siamese discriminators to facilitate pairwise training.', '1705.07904-3-0-6': "Experiments with human judges and an off-the-shelf face verification system demonstrate our algorithm's ability to generate convincing, identity-matched photographs.", '1705.07904-3-1-0': '# Introduction', '1705.07904-3-2-0': '# Semantically Decomposed Generative Adversarial Networks', '1705.07904-3-3-0': 'Before introducing our algorithm, we briefly review the prerequisite concepts.', '1705.07904-3-4-0': '## GAN preliminaries', '1705.07904-3-5-0': 'GANs leverage the discriminative power of neural networks to learn generative models.', '1705.07904-3-5-1': 'The generative model [MATH] ingests latent codes [MATH], sampled from some known prior [MATH], and produces [MATH], a sample of an implicit distribution [MATH].', '1705.07904-3-5-2': 'The learning process consists of a minimax game between [MATH], parameterized by [MATH], and a discriminative model [MATH], parameterized by [MATH].', '1705.07904-3-5-3': 'In the original formulation, the discriminative model tries to maximize log likelihood, yielding [EQUATION]', '1705.07904-3-5-4': 'Training proceeds as follows: For [MATH] iterations, sample one minibatch from the real distribution [MATH] and one from the distribution of generated images [MATH], updating discriminator weights [MATH] to increase [MATH] by stochastic gradient ascent.', '1705.07904-3-5-5': 'Then sample a minibatch from [MATH], updating [MATH] to decrease [MATH] by stochastic gradient descent.', '1705.07904-3-6-0': '[CITATION] propose energy-based GANs (EBGANs), in which the discriminator can be viewed as an energy function.', '1705.07904-3-6-1': 'Specifically, they devise a discriminator consisting of an autoencoder: [MATH].', '1705.07904-3-6-2': "In the minimax game, the discriminator's weights are updated to minimize the reconstruction error [MATH] for real data, while maximizing the error [MATH] for the generator.", '1705.07904-3-6-3': 'More recently, [CITATION] extend this work, introducing Boundary Equilibrium GANs (BEGANs), which optimize the Wasserstein distance (reminiscent of Wasserstein GANs ) between autoencoder loss distributions, yielding the formulation: [EQUATION]', '1705.07904-3-6-4': 'Additionally, they introduce a method for stabilizing training.', '1705.07904-3-6-5': 'Positing that training becomes unstable when the discriminator cannot distinguish between real and generated images, they introduce a new hyperparameter [MATH], updating the value function on each iteration to maintain a desired ratio between the two reconstruction errors: [MATH].', '1705.07904-3-6-6': 'The BEGAN model produces what appear to us, subjectively, to be the sharpest images of faces yet generated by a GAN.', '1705.07904-3-6-7': 'In this work, we adapt both the DCGAN and BEGAN algorithms to the SD-GAN training scheme.', '1705.07904-3-7-0': '## SD-GAN formulation', '1705.07904-3-8-0': '[t] Semantically Decomposed GAN Training [1] n in 1:NumberOfIterations m in 1:MinibatchSize Sample one identity vector [MATH].', '1705.07904-3-8-1': 'Sample two observation vectors [MATH].', '1705.07904-3-8-2': '[MATH], [MATH].', '1705.07904-3-8-3': 'Generate pair of images [MATH], adding them to the minibatch with label [MATH] (fake).', '1705.07904-3-8-4': 'm in 1:MinibatchSize Sample one identity [MATH] uniformly at random from the real data set.', '1705.07904-3-8-5': 'Sample two images of [MATH] without replacement [MATH].', '1705.07904-3-8-6': 'Add the pair to the minibatch, assigning label [MATH] (real).', '1705.07904-3-8-7': 'Update discriminator weights by [MATH] using its stochastic gradient.', '1705.07904-3-8-8': 'Sample another minibatch of identity-matched latent vectors [MATH].', '1705.07904-3-8-9': 'Update generator weights by stochastic gradient descent [MATH].', '1705.07904-3-9-0': "Consider the data's identity as a random variable [MATH] in a discrete index set [MATH].", '1705.07904-3-9-1': 'We seek to learn a latent representation that conveniently decomposes the variation in the real data into two parts: 1) due to [MATH], and 2) due to the other factors of variation in the data, packaged as a random variable [MATH].', '1705.07904-3-9-2': 'Ideally, the decomposition of the variation in the data into [MATH] and [MATH] should correspond exactly to a decomposition of the latent space [MATH].', '1705.07904-3-9-3': 'This would permit convenient interpolation and other operations on the inferred subspaces [MATH] and [MATH].', '1705.07904-3-10-0': 'A conventional GAN samples [MATH] from their joint distribution.', '1705.07904-3-10-1': "Such a GAN's generative model samples directly from an unstructured prior over the latent space.", '1705.07904-3-10-2': 'It does not disentangle the variation in [MATH] and [MATH], for instance by modeling conditional distributions [MATH], but only models their average with respect to the prior on [MATH].', '1705.07904-3-11-0': 'Our SD-GAN method learns such a latent space decomposition, partitioning the coordinates of [MATH] into two parts representing the subspaces, so that any [MATH] can be written as the concatenation [MATH] of its identity representation [MATH] and its contingent aspect representation [MATH].', '1705.07904-3-11-1': 'SD-GANs achieve this through a pairwise training scheme in which each sample from the real data consists of [MATH], a pair of images with a common identity [MATH].', '1705.07904-3-11-2': 'Each sample from the generator consists of [MATH], a pair of images generated from a common identity vector [MATH] but i.i.d. observation vectors [MATH].', '1705.07904-3-11-3': 'We assign identity-matched pairs from [MATH] the label [MATH] and [MATH]-matched pairs from [MATH] the label [MATH].', '1705.07904-3-11-4': 'The discriminator can thus learn to reject pairs for either of two primary reasons: 1) not photorealistic or 2) not plausibly depicting the same subject.', '1705.07904-3-11-5': 'See Algorithm [REF] for SD-GAN training pseudocode.', '1705.07904-3-12-0': '## SD-GAN discriminator architecture', '1705.07904-3-13-0': 'With SD-GANs, there is no need to alter the architecture of the generator.', '1705.07904-3-13-1': 'However, the discriminator must now act upon two images, producing a single output.', '1705.07904-3-13-2': 'Moreover, the effects of the two input images [MATH] on the output score are not independent.', '1705.07904-3-13-3': 'Two images might be otherwise photorealistic but deserve rejection because they clearly depict different identities.', '1705.07904-3-13-4': 'To this end, we devise two novel discriminator architectures to adapt DCGAN and BEGAN respectively.', '1705.07904-3-13-5': 'In both cases, we first separately encode each image using the same convolutional neural network [MATH] (Figure [REF]).', '1705.07904-3-13-6': "We choose this Siamese setup as our problem is symmetrical in the images, and thus it's sensible to share weights between the encoders.", '1705.07904-3-14-0': 'To adapt DCGAN, we stack the feature maps [MATH] and [MATH] along the channel axis, applying one additional strided convolution.', '1705.07904-3-14-1': 'This allows the network to further aggregate information from the two images before flattening and fully connecting to a sigmoid output.', '1705.07904-3-14-2': 'For BEGAN, because the discriminator is an autoencoder, our architecture is more complicated.', '1705.07904-3-14-3': 'After encoding each image, we concatenate the representations [MATH] and apply one fully connected bottleneck layer [MATH] with linear activation.', '1705.07904-3-14-4': 'In alignment with BEGAN, the SD-BEGAN bottleneck has the same dimensionality as the tuple of latent codes ([MATH], [MATH], [MATH]) that generated the pair of images.', '1705.07904-3-14-5': 'Following the bottleneck, we apply a second FC layer [MATH], taking the first [MATH] components of its output to be the input to the first decoder and the second [MATH] components to be the input to the second decoder.', '1705.07904-3-14-6': 'The shared intermediate layer gives SD-BEGAN a mechanism to push apart matched and unmatched pairs.', '1705.07904-3-14-7': 'We specify our exact architectures in full detail in Appendix [REF].', '1705.07904-3-15-0': '# Experiments', '1705.07904-3-16-0': 'We experimentally validate SD-GANs using two datasets: 1) the MS-Celeb-1M dataset of celebrity face images and 2) a dataset of shoe images collected from Amazon .', '1705.07904-3-16-1': 'Both datasets contain a large number of identities (people and shoes, respectively) with multiple observations of each.', '1705.07904-3-16-2': 'The "in-the-wild" nature of the celebrity face images offers a richer test bed for our method as both identities and contingent factors are significant sources of variation.', '1705.07904-3-16-3': "In contrast, Amazon's shoe images tend to vary only with camera perspective for a given product, making this data useful for sanity-checking our approach.", '1705.07904-3-17-0': 'Faces', '1705.07904-3-18-0': 'From the aligned face images in the MS-Celeb-1M dataset, we select 12,500 celebrities at random and [MATH] associated images of each, resizing them to [MATH]x[MATH] pixels.', '1705.07904-3-18-1': 'We split the celebrities into subsets of 10,000 (training), 1,250 (validation) and 1,250 (test).', '1705.07904-3-18-2': 'The dataset has a small number of duplicate images and some label noise (images matched to the wrong celebrity).', '1705.07904-3-18-3': 'We detect and remove duplicates by hashing the images, but we do not rid the data of label noise.', '1705.07904-3-18-4': 'We scale the pixel values to [MATH], performing no additional preprocessing or data augmentation.', '1705.07904-3-19-0': 'Shoes Synthesizing novel product images is another promising domain for our method.', '1705.07904-3-19-1': 'In our shoes dataset, product photographs are captured against white backgrounds and primarily differ in orientation and distance.', '1705.07904-3-19-2': 'Accordingly, we expect that SD-GAN training will allocate the observation latent space to capture these aspects.', '1705.07904-3-19-3': 'We choose to study shoes as a prototypical example of a category of product images.', '1705.07904-3-19-4': 'The Amazon dataset contains around 3,000 unique products with the category "Shoe" and multiple product images.', '1705.07904-3-19-5': 'We use the same [MATH] split and again hash the images to ensure that the splits are disjoint.', '1705.07904-3-19-6': 'There are [MATH] photos of each product on average.', '1705.07904-3-20-0': '## Training details', '1705.07904-3-21-0': 'We train SD-DCGANs on both of our datasets for 500,000 iterations using batches of [MATH] identity-matched pairs.', '1705.07904-3-21-1': 'To optimize SD-DCGAN, we use the Adam optimizer with hyperparameters [MATH] as recommended by [CITATION].', '1705.07904-3-21-2': 'We also consider a non-Siamese discriminator that simply stacks the channels of the pair of real or fake images before encoding (SD-DCGAN-SC).', '1705.07904-3-22-0': 'As in , we sample latent vectors [MATH].', '1705.07904-3-22-1': 'For SD-GANs, we partition the latent codes according to [MATH] using values of [MATH].', '1705.07904-3-22-2': 'Our algorithm can be trivially applied with [MATH]-wise training (vs. pairwise).', '1705.07904-3-22-3': 'To explore the effects of using [MATH], we also experiment with an SD-DCGAN where we sample [MATH] instances each from [MATH] for some [MATH] and from [MATH] for some [MATH].', '1705.07904-3-22-4': 'For all experiments, unless otherwise stated, we use [MATH] and [MATH].', '1705.07904-3-23-0': 'We also train an SD-BEGAN on both of our datasets.', '1705.07904-3-23-1': 'The increased complexity of the SD-BEGAN model significantly increases training time, limiting our ability to perform more-exhaustive hyperparameter validation (as we do for SD-DCGAN).', '1705.07904-3-23-2': 'We use the Adam optimizer with the default hyperparameters from for our SD-BEGAN experiments.', '1705.07904-3-23-3': 'While results from our SD-DCGAN [MATH] model are compelling, an experiment with a [MATH] variant of SD-BEGAN resulted in early mode collapse (Appendix [REF]); hence, we excluded SD-BEGAN [MATH] from our evaluation.', '1705.07904-3-24-0': 'We also compare to a DCGAN architecture trained using the auxiliary classifier GAN (AC-GAN) method .', '1705.07904-3-24-1': 'AC-GAN differs from SD-GAN in two key ways: 1) random identity codes [MATH] are replaced by a one-hot embedding over all the identities in the training set (matrix of size [MATH]x[MATH]); 2) the AC-GAN method encourages that generated photos depict the proper identity by tasking its discriminator with predicting the identity of the generated or real image.', '1705.07904-3-24-2': 'Unlike SD-GANs, the AC-DCGAN model cannot imagine new identities; when generating from AC-DCGAN (for our quantitative comparisons to SD-GANs), we must sample a random identity from those existing in the training data.', '1705.07904-3-25-0': '## Evaluation', '1705.07904-3-26-0': 'The evaluation of generative models is a fraught topic.', '1705.07904-3-26-1': 'Quantitative measures of sample quality can be poorly correlated with each other .', '1705.07904-3-26-2': 'Accordingly, we design an evaluation to match conceivable uses of our algorithm.', '1705.07904-3-26-3': "Because we hope to produce diverse samples that humans deem to depict the same person, we evaluate the identity coherence of SD-GANs and baselines using both a pretrained face verification model and crowd-sourced human judgments obtained through Amazon's Mechanical Turk platform.", '1705.07904-3-27-0': '### Quantitative', '1705.07904-3-28-0': 'Recent advancements in face verification using deep convolutional neural networks have yielded accuracy rivaling humans.', '1705.07904-3-28-1': 'For our evaluation, we procure FaceNet, a publicly-available face verifier based on the Inception-ResNet architecture .', '1705.07904-3-28-2': 'The FaceNet model was pretrained on the CASIA-WebFace dataset and achieves [MATH] accuracy on the LFW benchmark .', '1705.07904-3-29-0': 'FaceNet ingests normalized, [MATH]x[MATH] color images and produces an embedding [MATH].', '1705.07904-3-29-1': 'The training objective for FaceNet is to learn embeddings that minimize the [MATH] distance between matched pairs of faces and maximize the distance for mismatched pairs.', '1705.07904-3-29-2': 'Accordingly, the embedding space yields a function for measuring the similarity between two faces [MATH] and [MATH]: [MATH].', '1705.07904-3-29-3': 'Given two images, [MATH] and [MATH], we label them as a match if [MATH] where [MATH] is the accuracy-maximizing threshold on a class-balanced set of pairs from MS-Celeb-1M validation data.', '1705.07904-3-29-4': 'We use the same threshold for evaluating both real and synthetic data with FaceNet.', '1705.07904-3-30-0': 'We compare the performance of FaceNet on pairs of images from the MS-Celeb-1M test set against generated samples from our trained SD-GAN models and AC-DCGAN baseline.', '1705.07904-3-30-1': "To match FaceNet's training data, we preprocess all images by resizing from [MATH]x[MATH] to [MATH]x[MATH], normalizing each image individually.", '1705.07904-3-30-2': 'We prepare 10,000 pairs from MS-Celeb-1M, half identity-matched and half unmatched.', '1705.07904-3-30-3': 'From each generative model, we generate 5,000 pairs each with [MATH] and 5,000 pairs with [MATH].', '1705.07904-3-30-4': 'For each sample, we draw observation vectors [MATH] randomly.', '1705.07904-3-31-0': 'We also want to ensure that identity-matched images produced by the generative models are diverse.', '1705.07904-3-31-1': 'To this end, we propose an intra-identity sample diversity (ID-Div) metric.', '1705.07904-3-31-2': 'The multi-scale structural similarity (MS-SSIM) metric reports the similarity of two images on a scale from [MATH] (no resemblance) to [MATH] (identical images).', '1705.07904-3-31-3': 'We report [MATH] minus the mean MS-SSIM for all pairs of identity-matched images as ID-Div.', '1705.07904-3-31-4': 'To measure the overall sample diversity (All-Div), we also compute [MATH] minus the mean similarity of [MATH]k pairs with random identities.', '1705.07904-3-32-0': 'In Table [REF], we report the area under the receiver operating characteristic curve (AUC), accuracy, and false accept rate (FAR) of FaceNet (at threshold [MATH]) on the real and generated data.', '1705.07904-3-32-1': 'We also report our proposed diversity statistics.', '1705.07904-3-32-2': 'FaceNet verifies pairs from the real data with [MATH] accuracy compared to [MATH] on pairs from our SD-BEGAN model.', '1705.07904-3-32-3': 'Though this is comparable to the accuracy achieved on pairs from the AC-DCGAN baseline, our model produces samples that are more diverse in pixel space (as measured by ID-Div and All-Div).', '1705.07904-3-32-4': 'FaceNet has a higher but comparable FAR for pairs from SD-GANs than those from AC-DCGAN; this indicates that SD-GANs may produce images that are less semantically diverse on average than AC-DCGAN.', '1705.07904-3-33-0': 'We also report the combined memory footprint of [MATH] and [MATH] for all methods in Table [REF].', '1705.07904-3-33-1': 'For conditional GAN approaches, the number of parameters grows linearly with the number of identities in the training data.', '1705.07904-3-33-2': 'Especially in the case of the AC-GAN, where the discriminator computes a softmax over all identities, linear scaling may be prohibitive.', '1705.07904-3-33-3': "While our [MATH]k-identity subset of MS-Celeb-1M requires a [MATH]MB AC-DCGAN model, an AC-DCGAN for all [MATH]M identities would be over [MATH]GB, with more than [MATH]% of the parameters devoted to the weights in the discriminator's softmax layer.", '1705.07904-3-33-4': 'In contrast, the complexity of SD-GAN is constant in the number of identities.', '1705.07904-3-34-0': '### Qualitative', '1705.07904-3-35-0': 'In addition to validating that identity-matched SD-GAN samples are verified by FaceNet, we also demonstrate that humans are similarly convinced through experiments using Mechanical Turk.', '1705.07904-3-35-1': 'For these experiments, we use balanced subsets of 1,000 pairs from MS-Celeb-1M and the most promising generative methods from our FaceNet evaluation.', '1705.07904-3-35-2': 'We ask human annotators to determine if each pair depicts the "same person" or "different people".', '1705.07904-3-35-3': 'Annotators are presented with batches of ten pairs at a time.', '1705.07904-3-35-4': 'Each pair is presented to three distinct annotators and predictions are determined by majority vote.', '1705.07904-3-35-5': 'Additionally, to provide a benchmark for assessing the quality of the Mechanical Turk ensembles, we (the authors) manually judged [MATH] pairs from MS-Celeb-1M.', '1705.07904-3-35-6': 'Results are in Table [REF].', '1705.07904-3-36-0': 'For all datasets, human annotators on Mechanical Turk answered "same person" less frequently than FaceNet when the latter uses the accuracy-maximizing threshold [MATH].', '1705.07904-3-36-1': 'Even on real data, balanced so that [MATH] of pairs are identity-matched, annotators report "same person" only [MATH] of the time (compared to the [MATH] of FaceNet).', '1705.07904-3-36-2': 'While annotators achieve higher accuracy on pairs from AC-DCGAN than pairs from SD-BEGAN, they also answer "same person" [MATH] more often for AC-DCGAN pairs than real data.', '1705.07904-3-36-3': 'In contrast, annotators answer "same person" at the same rate for SD-BEGAN pairs as real data.', '1705.07904-3-36-4': 'This may be attributable to the lower sample diversity produced by AC-DCGAN.', '1705.07904-3-36-5': 'Samples from SD-DCGAN and SD-BEGAN are shown in Figures [REF] and [REF] respectively.', '1705.07904-3-37-0': '# Related work', '1705.07904-3-38-0': '# Discussion', '1705.07904-3-39-0': 'Our evaluation demonstrates that SD-GANs can disentangle those factors of variation corresponding to identity from the rest.', '1705.07904-3-39-1': 'Moreover, with SD-GANs we can sample never-before-seen identities, a benefit not shared by conditional GANs.', '1705.07904-3-39-2': 'In Figure [REF], we demonstrate that by varying the observation vector [MATH], SD-GANs can change the color of clothing, add or remove sunnies, or change facial pose.', '1705.07904-3-39-3': 'They can also perturb the lighting, color saturation, and contrast of an image, all while keeping the apparent identity fixed.', '1705.07904-3-39-4': 'We note, subjectively, that samples from SD-DCGAN tend to appear less photorealistic than those from SD-BEGAN.', '1705.07904-3-39-5': 'Given a generator trained with SD-GAN, we can independently interpolate along the identity and observation manifolds (Figure [REF]).', '1705.07904-3-40-0': 'On the shoe dataset, we find that the SD-DCGAN model produces convincing results.', '1705.07904-3-40-1': 'As desired, manipulating [MATH] while keeping [MATH] fixed yields distinct shoes in consistent poses (Figure [REF]).', '1705.07904-3-40-2': 'The identity code [MATH] appears to capture the broad categories of shoes (sneakers, flip-flops, boots, etc.).', '1705.07904-3-40-3': 'Surprisingly, neither original BEGAN nor SD-BEGAN can produce diverse shoe images (Appendix [REF]).'}